% \'Ecole d'\'et\'e de Grenoble, June-July 2007
% "Introductory course on H\"ormander's $L^2$ methods"
%
% Title:
% "Introductory course on H\"ormander's $L^2$ methods"
% J.P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France

\newcount \fullversion \fullversion=1
%%% Macros

\font \tenbf                  = cmb10
\font \tafontss               = cmbx5  scaled\magstep2
\font \tbfontss               = cmbx5  scaled\magstep1
\font \sixbf                  = cmbx6
\font \tafonts                = cmbx7  scaled\magstep2
\font \tbfonts                = cmbx7  scaled\magstep1
\font \ninebf                 = cmbx9
\font \bxf                    = cmbx10
\font \tbfontt                = cmbx10 scaled\magstep1
\font \tafontt                = cmbx10 scaled\magstep2
\font \tasys                  = cmex10 scaled\magstep1
\font \tasyt                  = cmex10 scaled\magstep2
\font \sixi                   = cmmi6
\font \ninei                  = cmmi9
\font \kleinhalbcurs          = cmmib10 scaled 833
\font \tamss                  = cmmib10
\font \tams                   = cmmib10 scaled\magstep1
\font \tamt                   = cmmib10 scaled\magstep2
\font \smallescriptscriptfont = cmr5
\font \smallescriptfont       = cmr5 at 7pt
\font \smalletextfont         = cmr5 at 10pt
\font \sixrm                  = cmr6
\font \ninerm                 = cmr9
\font \markfont               = cmr10 scaled\magstep1
\font \ninesl                 = cmsl9
\font \tbst                   = cmsy10 scaled\magstep1
\font \tbsss                  = cmsy5  scaled\magstep1
\font \tasss                  = cmsy5  scaled\magstep2
\font \sixsy                  = cmsy6
\font \tbss                   = cmsy7  scaled\magstep1
\font \tass                   = cmsy7  scaled\magstep2
\font \ninesy                 = cmsy9
\font \tast                   = cmsy10 scaled\magstep2
\font \nineit                 = cmti9
\font \ninett                 = cmtt9

\catcode`\@=11
\def\hexnumber@#1{\ifnum#1<10 \number#1\else
 \ifnum#1=10 A\else\ifnum#1=11 B\else\ifnum#1=12 C\else
 \ifnum#1=13 D\else\ifnum#1=14 E\else\ifnum#1=15 F\fi\fi\fi\fi\fi\fi\fi}
\def\mathhexbox@#1#2#3{\text{$\m@th\mathchar"#1#2#3$}}
\def\text{\relaxnext@\ifmmode\let\next\text@\else\let\next\text@@\fi\next}
\def\text@@#1{\leavevmode\hbox{#1}}
\font\tenmsa=msam10
\font\sevenmsa=msam7
\font\fivemsa=msam5
\font\tenmsb=msbm10
\font\sevenmsb=msbm7
\font\fivemsb=msbm5
\font\teneuf=eufm10
\font\seveneuf=eufm7
\font\fiveeuf=eufm5
\newfam\msafam
\newfam\msbfam
\newfam\euffam
\textfont\msafam=\tenmsa
\scriptfont\msafam=\sevenmsa
\scriptscriptfont\msafam=\fivemsa
\textfont\msbfam=\tenmsb
\scriptfont\msbfam=\sevenmsb
\scriptscriptfont\msbfam=\fivemsb
\textfont\euffam=\teneuf
\scriptfont\euffam=\seveneuf
\scriptscriptfont\euffam=\fiveeuf
\def\msa@{\hexnumber@\msafam}
\def\msb@{\hexnumber@\msbfam}
\mathchardef\boxtimes="2\msa@02
\mathchardef\square="0\msa@03
\mathchardef\twoheadrightarrow="3\msa@10
\mathchardef\twoheadleftarrow="3\msa@11
\mathchardef\upharpoonright="3\msa@16
\let\restriction\upharpoonright
\mathchardef\preccurlyeq="3\msa@34
\mathchardef\smallsmile="3\msa@60
\mathchardef\smallfrown="3\msa@61
\mathchardef\complement="0\msa@7B
\mathchardef\ltimes="2\msb@6E
\mathchardef\rtimes="2\msb@6F
\mathchardef\subsetneq="3\msb@28
\mathchardef\supsetneq="3\msb@29
\mathchardef\shortmid="3\msb@70
\mathchardef\shortparallel="3\msb@71
\mathchardef\ssm="2\msb@72
\def\Bbb{\relaxnext@\ifmmode\let\next\Bbb@\else
 \def\next{\errmessage{Use \string\Bbb\space only in math mode}}\fi\next}
\def\Bbb@#1{{\Bbb@@{#1}}}
\def\Bbb@@#1{\fam\msbfam#1}
\def\goth{\relaxnext@\ifmmode\let\next\goth@\else
 \def\next{\errmessage{Use \string\goth\space only in math mode}}\fi\next}
\def\goth@#1{{\goth@@{#1}}}
\def\goth@@#1{\fam\euffam#1}
\def\relaxnext@{\let\next\relax}
\catcode`\@=12 

\magnification=\magstep1
\vsize=25.2truecm 
\hsize=15.3truecm
\hoffset=3.3truemm
\parskip=2pt plus 2pt minus 0pt
\hfuzz=2pt
\tolerance=500
\abovedisplayskip=3 mm plus6pt minus 4pt
\belowdisplayskip=3 mm plus6pt minus 4pt
\abovedisplayshortskip=0mm plus6pt minus 2pt
\belowdisplayshortskip=2 mm plus4pt minus 4pt
\predisplaypenalty=0
\frenchspacing
\newdimen\oldparindent\oldparindent=15pt
\def\newline{\hfil\break}
\skewchar\ninei='177 \skewchar\sixi='177
\skewchar\ninesy='60 \skewchar\sixsy='60
\hyphenchar\ninett=-1

  \mathchardef\Gamma="0100
  \mathchardef\Delta="0101
  \mathchardef\Theta="0102
  \mathchardef\Lambda="0103
  \mathchardef\Xi="0104
  \mathchardef\Pi="0105
  \mathchardef\Sigma="0106
  \mathchardef\Upsilon="0107
  \mathchardef\Phi="0108
  \mathchardef\Psi="0109
  \mathchardef\Omega="010A

\def\bR{{\Bbb R}}
\def\bN{{\Bbb N}}
\def\bC{{\Bbb C}}
\def\bZ{{\Bbb Z}}
\def\bP{{\Bbb P}}
\def\bQ{{\Bbb Q}}
\def\bOne{\hbox{\bf 1}}

\def\begpet{\vskip6pt\bgroup\eightpoint}
\def\endpet{\vskip6pt\egroup}

\def\vec#1{\hbox{\textfont1=\bxf\textfont0=\bxf$#1$}}

\def\today{\ifcase\month\or
January\or February\or March\or April\or May\or June\or July\or August\or
September\or October\or November\or December\fi \space\number\day,
\number\year}

\def\eightpoint{\def\rm{\fam0\ninerm}%
\textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
 \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
 \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
 \def\it{\fam\itfam\nineit}%
 \textfont\itfam=\nineit
 \def\sl{\fam\slfam\ninesl}%
 \textfont\slfam=\ninesl
 \def\bf{\fam\bffam\ninebf}%
 \textfont\bffam=\ninebf \scriptfont\bffam=\sixbf
 \scriptscriptfont\bffam=\fivebf
 \normalbaselineskip=9pt
 \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}%
 \normalbaselines\rm
\def\vec##1{\setbox0=\hbox{$##1$}\hbox{\hbox
to0pt{\copy0\hss}\kern0.45pt\box0}}}%

\headline={\eightpoint\def\newline{ }\def\fonote#1{}\ifodd\pageno
\hfil\botmark\unskip\kern1.4true cm\llap{\folio}\else\leftheadline\fi}
\def\leftheadline{\rlap{\folio}\kern1.4true cm Chaptertitle\hfil}
\mark{Paragraphtitle}
\nopagenumbers
\output={\if N\header\headline={\hfil}\fi\plainoutput
\global\let\header=Y}

\let\lasttitle=N
\let\header=N
 \def\titlea#1#2{%
 {\baselineskip=20pt
  \lineskip=20pt
  \textfont0=\tafontt \scriptfont0=\tafonts \scriptscriptfont0=\tafontss
  \textfont1=\tamt \scriptfont1=\tams \scriptscriptfont1=\tamss
  \textfont2=\tast \scriptfont2=\tass \scriptscriptfont2=\tasss
  \textfont3=\tasyt \scriptfont3=\tasys \scriptscriptfont3=\tenex
  \tafontt
  \noindent#1#2\par}}

\def\leftrunningtitle#1{\gdef\leftheadline{\rlap{\folio}\kern1.4true cm
#1\hfil\ignorespaces}}

\def\titlebref#1{???}

\def\checkleftspace#1{\dimen0=\pagetotal
    \ifdim\dimen0<\pagegoal
       \dimen0=\ht0\advance\dimen0 by\dp0\advance\dimen0 by
       #1\normalbaselineskip
       \advance\dimen0 by\pagetotal
       \advance\dimen0 by-\pageshrink
       \ifdim\dimen0>\pagegoal
          \vfill\eject
       \fi
    \fi}

 \def\UNKNOWN#1{???(#1)}  
 \def\TITLEBREF#1{\csname REF\romannumeral#1\endcsname}
 \newlinechar=`\^^J

 \immediate\openin1\jobname.aux
 \ifeof1
   \message{^^JCAUTION\string: you MUST run TeX a second time^^J}
   \let\titlebpage=\UNKNOWN
 \else
   \input \jobname.aux
   \message{^^JCAUTION\string: if the file has just been modified you may 
    have to run TeX twice^^J}
   \let\titlebref=\TITLEBREF
 \fi
 \message{to get correct page numbers displayed in Table of Contents^^J}
 \immediate\openout1=\jobname.aux

 \newcount\counttitleb \counttitleb=-1
 \def\titleb#1#2{%
     \advance\counttitleb by 1
     \immediate\write1{\string\def \string\REF 
     \romannumeral\counttitleb \string{%
     \number\pageno \string} }
     \if N\lasttitle \removelastskip\else\vskip-29pt
     \fi
     \vskip25pt plus 4pt minus4pt
     \bgroup
 \textfont0=\tbfontt \scriptfont0=\tbfonts \scriptscriptfont0=\tbfontss
 \textfont1=\tams \scriptfont1=\tamss \scriptscriptfont1=\kleinhalbcurs
 \textfont2=\tbst \scriptfont2=\tbss \scriptscriptfont2=\tbsss
 \textfont3=\tasys \scriptfont3=\tenex \scriptscriptfont3=\tenex
     \baselineskip=16pt
     \lineskip=16pt
     \raggedright
     \pretolerance=10000
     \tbfontt
     \setbox0=\vbox{\vskip25pt
     \def\fonote##1{}%
     \noindent
     \ignorespaces#1\ #2
     \vskip15pt}%
     \checkleftspace 5
     \noindent\ignorespaces #1\ #2
     \vskip15true pt plus4pt minus4pt\egroup
     \setbox0=\hbox{\eightpoint\def\newline{ }\def\fonote##1{}%
     \kern1.4true cm#1\ #2}\ifdim\wd0>\hsize
     \message{Your TITLEB exceeds the headline, please use a
     short form with TITLEBRUNNING}\mark{Title of section
     suppressed due to excessive length}%
     \else{\def\newline{ }\def\fonote##1{}\mark{#1\ #2}}\fi
     \nobreak
     \global\let\lasttitle=B%
     \parindent=0pt
     \everypar={\global\parindent=\oldparindent
     \global\let\lasttitle=N\global\everypar={}}%
     \ignorespaces}

\def\titlebrunning#1{{\def\newline{ }\def\fonote##1{}\mark{#1}}%
\message{Excessive headline from TITLEB has been
corrected}\ignorespaces}

 \def\titlec#1#2{%
     \if N\lasttitle\else\vskip-3pt\vskip-\baselineskip
     \fi
     \vskip17pt plus 4pt minus 4pt
     \bgroup
     \bxf
     \raggedright
     \pretolerance=10000
     \setbox0=\vbox{\vskip 17pt
     \def\fonote##1{}%
     \noindent
     \ignorespaces#1\ #2
     \vskip10pt}%
     \checkleftspace 4
     \noindent
     \ignorespaces #1\ #2
     \vskip10pt plus4pt minus4pt\egroup
     \nobreak
     \global\let\lasttitle=C%
     \parindent=0pt
     \everypar={\global\parindent=\oldparindent
     \global\let\lasttitle=N\global\everypar={}}%
     \ignorespaces}

 \def \titled#1{
     \if N\lasttitle\else\vskip-3pt\vskip-\baselineskip
     \fi
     \vskip7true pt plus 4pt minus 4pt
     \bgroup
     \bf
     \noindent
     \ignorespaces #1\unskip.\ \egroup
     \ignorespaces}

\let\ts=\thinspace
\def\footnoterule{\kern-3pt\hrule width 2true cm\kern2.6pt}

\newcount\footcount \footcount=0
\def\advftncnt{\advance\footcount by1\global\footcount=\footcount}

\def\fonote#1{\advftncnt$^{\the\footcount}$\begingroup\eightpoint
       \def\textindent##1{\hangindent0.5\oldparindent\noindent\hbox
       to0.5\oldparindent{##1\hss}\ignorespaces}%
\vfootnote{$^{\the\footcount}$}{#1}\endgroup}

\def\item#1{\par\parindent=\oldparindent\noindent
\hangindent\parindent\hbox to\parindent{#1\hss}\ignorespaces}

\def\itemitem#1{\par\parindent=\oldparindent\noindent
\hangindent1.6\parindent\hangafter=0
\llap{#1\enspace}\ignorespaces}

\def\qed{\ifmmode\square\else{\unskip\nobreak\hfil
\penalty50\hskip1em\null\nobreak\hfil\hbox{$\square$}
\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi}

\def\ii{{\rm i}\,}
\def\bu{{\scriptstyle\bullet}}
\def\ld{,\ldots,}
\let\lra=\longrightarrow
\def\vlra{\mathrel{\smash-}\joinrel\mathrel{\smash-}\joinrel% 
\kern-2pt\longrightarrow}
\let\wh=\widehat
\let\wt=\widetilde
\let\ol=\overline
\def\ul#1{$\underline{\smash{\hbox{#1}}}$}
\def\dbar{{\overline\partial}}
\def\ddbar{{\partial\overline\partial}}
\def\soul{{\raise-4.6pt\hbox{$-$}\kern-7pt}}
\def\ci{C^\infty}
\def\compact{\mathrel{\subset\!\subset}}
\def\ort{\hbox{~\vrule width4.0pt height0.4pt depth0pt
                \vrule width0.4pt height6.0pt depth0pt~$\,$}}
\def\gge{\raise1.5pt\hbox{$\scriptscriptstyle\ge$}}
\def\dasharrow{\mathrel{\hbox{
  \kern1pt \vrule height2.45pt depth-2.15pt width2pt}
  \kern1pt{\vrule height2.45pt depth-2.15pt width2pt}
  \kern1pt{\vrule height2.45pt depth-2.15pt width2pt}
  \kern1pt{\vrule height2.45pt depth-2.15pt width1.7pt\kern-1.7pt}
  {\raise1.4pt\hbox{$\scriptscriptstyle\succ$}}}}
\def\lraww{\mathrel{\rlap{$\longrightarrow$}\kern-1pt\longrightarrow}}

\def\buildo#1^#2{\mathrel{\mathop{\null#1}\limits^{#2}}}
\def\buildu#1_#2{\mathrel{\mathop{\null#1}\limits_{#2}}}
\def\build#1^#2_#3{\mathrel{\mathop{\null#1}\limits^{#2}_{#3}}}

\def\Ker{\mathop{\rm Ker}\nolimits}
\def\Id{\mathop{\rm Id}\nolimits}
\def\Vol{\mathop{\rm Vol}\nolimits}
\def\rk{\mathop{\rm rk}\nolimits}
\def\Tr{\mathop{\rm Tr}\nolimits}
\def\deg{\mathop{\rm deg}\nolimits}
\def\Re{\mathop{\rm Re}\nolimits}
\def\Im{\mathop{\rm Im}\nolimits}
\def\mod{\mathop{\rm mod}\nolimits}
\def\codim{\mathop{\rm codim}\nolimits}
\def\Supp{\mathop{\rm Supp}\nolimits}
\def\Coker{\mathop{\rm Coker}\nolimits}
\def\Hom{\mathop{\rm Hom}\nolimits}
\def\End{\mathop{\rm End}\nolimits}
\def\Aut{\mathop{\rm Aut}\nolimits}
\def\Ext{\mathop{\rm Ext}\nolimits}
\def\Coker{\mathop{\rm Coker}\nolimits}
\def\Gl{\mathop{\rm Gl}\nolimits}
\def\Herm{\mathop{\rm Herm}\nolimits}
\def\rank{\mathop{\rm rank}\nolimits}
\def\sign{\mathop{\rm sign}\nolimits}
\def\Sh{\mathop{\rm Sh}\nolimits}
\def\Psh{\mathop{\rm Psh}\nolimits}
\def\Pic{\mathop{\rm Pic}\nolimits}
\def\Psh{\mathop{\rm Psh}\nolimits}
\def\div{\mathop{\rm div}\nolimits}
\def\Div{\mathop{\rm Div}\nolimits}
\def\pr{\mathop{\rm pr}\nolimits}
\def\ord{\mathop{\rm ord}\nolimits}
\def\Vect{\mathop{\rm Vect}\nolimits}
\def\Ll{\langle\!\langle}
\def\Rr{\rangle\!\rangle}
\def\Dom{\mathop{\rm Dom}\nolimits}
\def\diam{\mathop{\rm diam}\nolimits}

\def\DR{{\rm DR}}
\def\FS{{\rm FS}}
\def\NS{{\rm NS}}
\def\NE{\mathop{\rm NE}\nolimits}
\def\ME{\mathop{\rm ME}\nolimits}
\def\SME{\mathop{\rm SME}\nolimits}

\def\Const{{\rm Const}}
\def\Grif{{\rm Grif}}
\def\Nak{{\rm Nak}}
\def\red{{\rm red}}
\def\amp{{\rm amp}}
\def\nef{{\rm nef}}
\def\eff{{\rm eff}}
\def\psef{{\rm psef}}
\def\ac{{\rm ac}}
\def\loc{{\rm loc}}
\def\reg{{\rm reg}}
\def\sing{{\rm sing}}

\def\gm{{\goth m}}
\def\cHom{{\cal H}{\it om}}
\def\alg{{\rm alg}}
\def\nef{{\rm nef}}
\def\num{\nu}
\def\smallvee{{\scriptscriptstyle\vee}}

% figures inserted as PostScript files
\special{header=/home/demailly/psinputs/mathdraw/grlib.ps}
\long\def\InsertFig#1 #2 #3 #4\EndFig{\par
\hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{"
#3}}#4$}}
\long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise#2mm\hbox{#3}}}

\catcode`@=11    
\def\cmalign#1{\null\,\vcenter{\normalbaselines\m@th
    \ialign{\hfil$##$&&$##$\hfil\crcr
      \mathstrut\crcr\noalign{\kern-\baselineskip}
      #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,}
\catcode`@=12

\long\def\claim#1 #2\endclaim
{\removelastskip\vskip\baselineskip\noindent{\bf#1.}
{\it\ignorespaces#2}\vskip\baselineskip}

\def\nlni{%
   \par
   \if N\lasttitle
      \ifvmode \removelastskip \fi
      \bigskip
   \fi
   \noindent}
\def\newenvironment#1#2#3#4#5#6%
{\expandafter\gdef\csname beg#1\endcsname##1{#2%
{#3\csname #1\endcsname\if!##1!.\else\ \ignorespaces
##1\unskip\fi\kern0.4em}\bgroup
#4%
\ignorespaces
}%
\expandafter\gdef\csname #1\endcsname{#6}%
\expandafter\gdef\csname end#1\endcsname{\endgraf\egroup#5}%
}
\newenvironment{proof}{\nlni}{\it}{\rm}{\vskip\baselineskip}{Proof}

\def\cA{{\cal A}}
\def\cB{{\cal B}}
\def\cC{{\cal C}}
\def\cD{{\cal D}}
\def\cE{{\cal E}}
\def\cF{{\cal F}}
\def\cG{{\cal G}}
\def\cH{{\cal H}}
\def\cI{{\cal I}}
\def\cJ{{\cal J}}
\def\cK{{\cal K}}
\def\cL{{\cal L}}
\def\cM{{\cal M}}
\def\cN{{\cal N}}
\def\cO{{\cal O}}
\def\cP{{\cal P}}
\def\cQ{{\cal Q}}
\def\cR{{\cal R}}
\def\cS{{\cal S}}
\def\cT{{\cal T}}
\def\cU{{\cal U}}
\def\cV{{\cal V}}
\def\cW{{\cal W}}
\def\cX{{\cal X}}
\def\cY{{\cal Y}}
\def\cZ{{\cal Z}}

\def\ptstep{{\vrule height 0.4pt depth 0pt width 0.4pt}}
\def\verl{{\vrule width 0.4pt height 12pt depth 6pt}}
\def\table#1{\vbox{\baselineskip=0pt\lineskip=0pt
    \halign{$\displaystyle{}##\hfil{}$\hfil&&
    $\displaystyle{}##\hfil{}$\hfil\crcr#1\crcr}}}

\catcode`\@=11
\font\eightrm=cmr8
\font\eighti=cmmi8
\font\eightsy=cmsy8
\font\eightbf=cmbx8
\font\eighttt=cmtt8
\font\eightit=cmti8
\font\eightsl=cmsl8
\font\sixrm=cmr6
\font\sixi=cmmi6
\font\sixsy=cmsy6
\font\sixbf=cmbx6

\font\smallcap=cmcsc10
\font\bftwelve=cmbx10 at 12pt
\font\sytwelve=cmsy10 at 12pt

\def\eightpoint{%
  \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
  \def\rm{\fam\z@\eightrm}%
  \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei
  \def\oldstyle{\fam\@ne\eighti}%
  \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
  \textfont\itfam=\eightit
  \def\it{\fam\itfam\eightit}%
  \textfont\slfam=\eightsl
  \def\sl{\fam\slfam\eightsl}%
  \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf
  \scriptscriptfont\bffam=\fivebf
  \def\bf{\fam\bffam\eightbf}%
  \abovedisplayskip=9pt plus 2pt minus 6pt
  \abovedisplayshortskip=0pt plus 2pt
  \belowdisplayskip=9pt plus 2pt minus 6pt
  \belowdisplayshortskip=5pt plus 2pt minus 3pt
  \smallskipamount=2pt plus 1pt minus 1pt
  \medskipamount=4pt plus 2pt minus 1pt
  \bigskipamount=9pt plus 3pt minus 3pt
  \normalbaselineskip=9pt
  \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}%
  \let\bigf@ntpc=\eightrm \let\smallf@ntpc=\sixrm
  \normalbaselines\rm}

\def\tenpoint{%
  \textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm
  \def\rm{\fam\z@\tenrm}%
  \textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei
  \def\oldstyle{\fam\@ne\teni}%
  \textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy
  \textfont\itfam=\tenit
  \def\it{\fam\itfam\tenit}%
  \textfont\slfam=\tensl
  \def\sl{\fam\slfam\tensl}%
  \textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf
  \scriptscriptfont\bffam=\fivebf
  \def\bf{\fam\bffam\tenbf}%
  \textfont\ttfam=\tentt
  \def\tt{\fam\ttfam\tentt}%
  \abovedisplayskip=6pt plus 2pt minus 6pt
  \abovedisplayshortskip=0pt plus 3pt
  \belowdisplayskip=6pt plus 2pt minus 6pt
  \belowdisplayshortskip=7pt plus 3pt minus 4pt
  \smallskipamount=3pt plus 1pt minus 1pt
  \medskipamount=6pt plus 2pt minus 2pt
  \bigskipamount=12pt plus 4pt minus 4pt
  \normalbaselineskip=12pt
  \setbox\strutbox=\hbox{\vrule height8.5pt depth3.5pt width0pt}%
  \let\bigf@ntpc=\tenrm \let\smallf@ntpc=\sevenrm
  \normalbaselines\rm}
\catcode`\@=12
    
\def\pointdash{\discretionary{.}{}{.\kern.35em---\kern.7em}}
\newif\ifdigit
\def\digit{\digittrue}
\digit
\newdimen\laenge
\def\letter#1|{\setbox3=\hbox{#1}\laenge=\wd3\advance\laenge by 3mm
\digitfalse}

\def\article#1|#2|#3|#4|#5|#6|#7|%
    {{\ifdigit\leftskip=7mm\noindent
     \hangindent=2mm\hangafter=1
\llap{[#1]\hskip1.35em}{\smallcap #2}\pointdash {\it #3}, {\rm #4},
\nobreak{\bf #5} ({\rm #6}), \nobreak #7.\par\else\noindent
\advance\laenge by 4mm \hangindent=\laenge\advance\laenge by -4mm\hangafter=1
\rlap{[#1]}\hskip\laenge{\smallcap #2}\pointdash
{\it #3}, #4, {\bf #5} ({\rm #6}), #7.\par\fi}}

\def\bookref#1|#2|#3|#4|#5|%
    {{\ifdigit\leftskip=7mm\noindent
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\def\twolines{\newline\phantom{$~$}\hfill\newline}

\ifnum \fullversion=0
\titlea{}{Introductory course on H\"ormander's $L^2$ methods}
\leftrunningtitle{J.-P.\ Demailly,~~
Introductory course on H\"ormander's $L^2$ methods}
\fi

\ifnum \fullversion=1
\titlea{}{Applications of the theory of $L^2$ estimates\newline
and positive currents in algebraic geometry}
\leftrunningtitle{J.-P.\ Demailly,~~Complex analytic techniques 
in algebraic geometry}
\fi

\vskip20pt\noindent
{\tbfontt 
Jean-Pierre Demailly\hfil\break
\'Ecole d'\'et\'e de Math\'ematiques de Grenoble, 
June-July 2007}

\vskip25pt\noindent
These notes are derived in part from the lectures
``{\it Multiplier ideal sheaves and analytic methods in algebraic geometry}''
given at the ICTP School held in Trieste, Italy,
April 24 -- May 12, 2000
``{\it Vanishing theorems and effective results in Algebraic Geometry}'',
as well as from a more detailed online book on Algebraic\break Geometry 
available at the author's home page
\hfill\break
{\tt http://www-fourier.ujf-grenoble.fr/$\tilde{~}$demailly/ }

\titlec{}{\phantom{$~$}\newline\tbfontt Contents}
0. Introduction
   \dotfill \titlebref{0}\break
1. Preliminary Material
   \dotfill \titlebref{1}\break
2. Lelong Numbers and Intersection Theory 
   \dotfill \titlebref{2}\break
3. Holomorphic Vector Bundles, Connections and Curvature
   \dotfill \titlebref{3}\break
4. K\"ahler identities and Hodge Theory
   \dotfill \titlebref{4}\break
5. Bochner Technique and Vanishing Theorems
   \dotfill \titlebref{5}\break
6. $L^2$ Estimates and Existence Theorems
   \dotfill \titlebref{6}\break
7. Numerically Effective and Pseudoeffective Line Bundles
   \dotfill \titlebref{7}\break
\ifnum\fullversion=0
References
  \dotfill \titlebref{8}
\fi
\ifnum\fullversion=1
8. A Simple Algebraic Approach to Fujita's Conjecture
  \dotfill \titlebref{8}\break
9. Holomorphic Morse Inequalities
  \dotfill \titlebref{9}\break  
10. Effective Version of Matsusaka's Big Theorem
  \dotfill \titlebref{10}\break
11. Positivity Concepts for Vector Bundles
  \dotfill \titlebref{11}\break
12. Skoda's $L^2$ Estimates for Surjective Bundle Morphisms
  \dotfill \titlebref{12}\break
13. The Ohsawa-Takegoshi $L^2$ Extension Theorem
  \dotfill \titlebref{13}\break
14. Invariance of Plurigenera of Varieties of General Type
  \dotfill \titlebref{14}\break
15. Subadditivity of Multiplier Ideal Sheaves and Zariski Decomposition
  \dotfill \titlebref{15}\break
16. Hard Lefschetz Theorem with Multiplier Ideal Sheaves
  \dotfill \titlebref{16}\break
17. Nef and Pseudoeffective Cones in K\"ahler Geometry
   \dotfill \titlebref{17}\break
18. Numerical Characterization of the K\"ahler Cone
  \dotfill \titlebref{18}\break
19. Cones of Curves
  \dotfill \titlebref{19}\break  
20. Duality Results
  \dotfill \titlebref{20}\break
21. Approximation of psh functions by logarithms of holomorphic functions
  \dotfill \titlebref{21}\break
22. Zariski Decomposition and Movable Intersections
  \dotfill \titlebref{22}\break
23. The Orthogonality Estimate
  \dotfill \titlebref{23}\break
24. Proof of the Main Duality Theorem
  \dotfill \titlebref{24}\break
References
  \dotfill \titlebref{25}
\fi

\titleb{0.}{Introduction}
Transcendental methods of algebraic geometry have been extensively
studied since a long time, starting with the work of Abel, Jacobi
and Riemann in the nineteenth century. More recently, in the period
1940-1970, the work of Hodge, Hirzebruch, Kodaira, Atiyah revealed
deeper relations between complex analysis, topology, PDE theory
and algebraic geometry. In the last twenty years, gauge theory has proved
to be a very efficient tool for the study of many important questions:
moduli spaces, stable sheaves, non abelian Hodge theory, low
dimensional topology $\ldots$

Our main purpose here is to describe a few analytic tools which are useful
to study questions such as linear series and vanishing theorems for algebraic
vector bundles. One of the early successes of analytic methods in this context
is Kodaira's use of the Bochner technique in relation with the theory of
harmonic forms, during the decade 1950-60. The idea is to represent
cohomology classes by harmonic forms and to prove vanishing theorems by
means of suitable a priori curvature estimates. The prototype of such
results is the Akizuki-Kodaira-Nakano theorem (1954): if $X$ is a
nonsingular projective algebraic variety and $L$ is a holomorphic line
bundle on $X$ with positive curvature, then $H^q(X,\Omega^p_X\otimes L)=0$
for $p+q>\dim X$ (throughout the paper we set $\Omega^p_X=\Lambda^p
T^\star_X$ and $K_X=\Lambda^nT^\star_X$, $n=\dim X$, viewing these objects
either as holomorphic bundles or as locally free $\cO_X$-modules).
It is only much later that an algebraic proof of this result has been
proposed by Deligne-Illusie, via characteristic $p$ methods, in 1986.

A refinement of the Bochner technique used by Kodaira led, about ten years 
later, to fundamental $L^2$ estimates due to H\"ormander [H\"or65], concerning
solutions of the Cauchy-Riemann operator. Not only vanishing theorems
are proved, but more precise information of a quantitative nature is
obtained about solutions of $\dbar$-equations. The best way of
expressing these $L^2$ estimates is to use a geometric setting first
considered by Andreotti-Vesentini [AV65]. More explicitly, suppose
that we have a holomorphic line bundle $L$ equipped with a
hermitian metric of weight $e^{-2\varphi}$, where $\varphi$ is a
(locally defined) plurisubharmonic function; then explicit bounds on
the $L^2$ norm $\int_X|f|^2e^{-2\varphi}$ of solutions is obtained.
The result is still more useful if the plurisubharmonic weight
$\varphi$ is allowed to have singularities. Following Nadel [Nad89],
we define the {\it multiplier ideal sheaf} $\cI(\varphi)$ to be the
sheaf of germs of holomorphic functions $f$ such that
$|f|^2e^{-2\varphi}$ is locally summable.  Then $\cI(\varphi)$ is a
coherent algebraic sheaf over $X$ and $H^q(X,K_X\otimes
L\otimes\cI(\varphi))=0$ for all $q\ge 1$ if the curvature of $L$ is
positive (as a current). This important result can be seen as a
generalization of the Kawamata-Viehweg vanishing theorem ([Kaw82],
[Vie82]), which is one of the cornerstones of higher dimensional
algebraic geometry (especially of Mori's minimal model program).
In the dictionary between analytic geometry and algebraic geometry, the
ideal $\cI(\varphi)$ plays a very important role, since it directly converts
an analytic object into an algebraic one, and, simultaneously, takes care
of singularities in a very efficient way.

\ifnum\fullversion=1
Another analytic tool used to
deal with singularities is the theory of positive currents introduced by
Lelong [Lel57]. Currents can be seen as generalizations of algebraic cycles,
and many classical results of intersection theory still apply to currents.
The concept of Lelong number of a current is the analytic analogue of the
concept of multiplicity of a germ of algebraic variety. Intersections
of cycles correspond to wedge products of currents (whenever these products
are defined). 

Besides the Kodaira-Nakano vanishing theorem, one of the most basic
``effective result'' expected to hold in algebraic geometry is
expressed in the following conjecture of Fujita [Fuj87]: if $L$ is an
ample (i.e.\ positive) line bundle on a projective $n$-dimensional
algebraic variety $X$, then $K_X+(n+1)L$ is generated by sections and
$K_X+(n+2)L$ is very ample. In the last decade, a lot of effort has
been brought for the solution of this conjecture -- and it seems
indeed that a solution might finally emerge in the first years or the
third millenium -- hopefully during this Summer School!  The first
major results are the proof of the Fujita conjecture in the case of
surfaces by Reider [Rei88] (the case of curves is easy and has been
known since a very long time), and the numerical criterion for the
very ampleness of $2K_X+L$ given in [Dem93b], obtained by means of
analytic techniques and Monge-Amp\`ere equations with isolated
singularities. Alternative algebraic techniques were developed
slightly later by Koll\'ar [Kol92], Ein-Lazarsfeld [EL93], Fujita
[Fuj93], Siu [Siu95, 96], Kawamata [Kaw97] and Helmke [Hel97]. We
will explain here Siu's method because it is technically the simplest
method; one of the results obtained by this method is the following
effective result: $2K_X+mL$ is very ample for $m\ge 2+{3n+1\choose
n}$. The basic idea is to apply the Kawamata-Viehweg vanishing
theorem, and to combine this with the Riemann-Roch formula in order to
produce sections through a clever induction procedure on the dimension
of the base loci of the linear systems involved.

Although Siu's result is certainly not optimal, it is sufficient to
obtain a nice constructive proof of {\it Matsusaka's big
theorem} ([Siu93], [Dem96]). The result states that there is an effective
value $m_0$ depending only on the intersection numbers $L^n$ and $L^{n-1}\cdot
K_X$, such that $mL$ is very ample for $m\ge m_0$. The basic idea is to
combine results on the very ampleness of $2K_X+mL$ together with
the theory of holomorphic Morse inequalities ([Dem85b]). The Morse
inequalities are used to construct sections of $m'L-K_X$ for $m'$ large.
Again this step can be made algebraic (following suggestions by F.~Catanese
and R.~Lazarsfeld), but the analytic formulation apparently has a wider
range of applicability.

In the next sections, we pursue the study of $L^2$ estimates, in
relation with the Nullstellenstatz and with the extension
problem. Skoda [Sko72b, Sko78] showed that the division problem
$f=\sum g_jh_j$ can be solved holomorphically with very precise $L^2$
estimates, provided that the $L^2$ norm of $|f|\,|g|^{-p}$ is finite
for some sufficiently large exponent $p$ ($p>n=\dim X$ is enough).
Skoda's estimates have a nice interpretation in terms of local
algebra, and they lead to precise qualitative and quantitative
estimates in connection with the B\'ezout problem. Another very
important result is the $L^2$ extension theorem by Ohsawa-Takegoshi
[OT87, Ohs88], which has also been generalized later by Manivel
[Man93]. The main statement is that every $L^2$ section $f$ of a
suitably positive line bundle defined on a subavariety $Y\subset X$
can be extended to a $L^2$ section $\tilde f$ defined over the whole
of~$X$. The positivity condition can be understood in terms of the
canonical sheaf and normal bundle to the subvariety.  The extension
theorem turns out to have an incredible amount of important
consequences: among them, let us mention for instance Siu's theorem
[Siu74] on the analyticity of Lelong numbers, the basic approximation
theorem of closed positive $(1,1)$-currents by divisors, the
subadditivity property $\cI(\varphi+\psi)\subset\cI(\varphi)\cI(\psi)$
of multiplier ideals [DEL00], the restriction formula
$\cI(\varphi_{|Y})\subset\cI(\varphi)_{|Y}$, $\ldots$. A suitable
combination of these results can be used to reprove Fujita's result
[Fuj94] on approximate Zariski decomposition, as we show in section~15.

In the last section~16, we show how subadditivity can be used to derive 
an approximation theorem for (almost) plurisubharmonic functions: any
such function can be approximated by a sequence of (almost) plurisubharmonic 
functions which are smooth outside an analytic set, and which define
the same multiplier ideal sheaves. From this, we derive a generalized version
of the {\it hard Lefschetz theorem} for cohomology with values in a
pseudo-effective line bundle; namely, the Lefschetz map is surjective
when the cohomology groups are twisted by the relevant multiplier ideal 
sheaves.
\fi

These notes are essentially written with the idea of serving as an
analytic toolbox for algebraic geometers. Although efficient algebraic
techniques exist, our feeling is that the analytic techniques are very
flexible and offer a large variety of guidelines for more algebraic
questions (including applications to number theory which are not
discussed here). We made a special effort to use as little
prerequisites and to be as self-contained as possible; hence the
rather long preliminary sections dealing with basic facts of complex
differential geometry. I am indebted to L.\ Ein, J.\ Koll\'ar,
R. Lazarsfeld, Th.\ Peternell, M.\ Schneider and Y.T.\ Siu for many
discussions on these subjects over a period of time of two
decades or more. These discussions certainly had a great influence
on my research work and therefore on the contents of the present notes.
We refer to Rob Lazarsfeld's book [Laz04] for an extremely detailed 
algebraic exposition of the subject of positivity in algebraic
geometry.
\vfill\eject

\titleb{1.}{Preliminary Material}
\titlec{1.A.}{Dolbeault Cohomology and Sheaf Cohomology}
Let $X$ be a $\bC$-analytic manifold of dimension $n$. We denote by
$\Lambda^{p,q}T^\star_X$ the bundle of differential forms of bidegree
$(p,q)$ on $X$, i.e., differential forms which can be written as
$$u=\sum_{|I|=p,\,|J|=q}u_{I,J}dz_I\wedge d\ol z_J.$$
Here $(z_1\ld z_n)$ denote arbitrary local holomorphic coordinates,
\hbox{$I=(i_1\ld i_p)$}, \hbox{$J=(j_1\ld j_q)$} are multiindices
(increasing sequences of integers in the range $[1\ld n]$, of lengths
$|I|=p$, $|J|=q$), and
$$dz_I:=dz_{i_1}\wedge\ldots\wedge dz_{i_p},\qquad
d\ol z_J:=d\ol z_{j_1}\wedge\ldots\wedge d\ol z_{j_q}.$$
Let $\cE^{p,q}$ be the sheaf of germs of complex valued differential
$(p,q)$-forms with $C^\infty$ coefficients. Recall that the exterior
derivative $d$ splits as $d=d'+d''$ where
$$\eqalign{
d'u&=\sum_{|I|=p,\,|J|=q,1\le k\le n}{\partial u_{I,J}
\over\partial z_k}dz_k\wedge dz_I\wedge d\ol z_J,\cr
d''u&=\sum_{|I|=p,\,|J|=q,1\le k\le n}{\partial u_{I,J}
\over\partial\ol z_k}d\ol z_k\wedge dz_I\wedge d\ol z_J\cr}$$
are of type $(p+1,q)$, $(p,q+1)$ respectively. The well-known
Dolbeault-Grothendieck lemma asserts that any $d''$-closed
form of type $(p,q)$ with $q>0$ is locally $d''$-exact (this is the
analogue for $d''$ of the usual Poincar\'e lemma for~$d$, see e.g.\
[H\"or66]). In other words, the complex of sheaves
$(\cE^{p,\bu},d'')$ is exact in degree $q>0$; in degree $q=0$, $\Ker d''$
is the sheaf $\Omega^p_X$ of germs of holomorphic forms of degree $p$
on~$X$.

More generally, if $F$ is a holomorphic vector bundle
of rank $r$ over~$X$, there is a natural $d''$ operator acting
on the space $C^\infty(X,\Lambda^{p,q}T^\star_X\otimes F)$ of smooth
$(p,q)$-forms with values in~$F$; if $s=\sum_{1\le\lambda\le r}s_\lambda
e_\lambda$ is a $(p,q)$-form expressed in terms of a local holomorphic
frame of $F$, we simply define $d''s:=\sum d''s_\lambda\otimes e_\lambda$,
observing that the holomorphic transition matrices involved in changes
of holomorphic frames do not affect the computation of~$d''$.
It is then clear that the Dolbeault-Grothendieck lemma still
holds for $F$-valued forms. For every integer $p=0,1\ld n$, the
{\it Dolbeault Cohomology} groups $H^{p,q}(X,F)$ are defined to be the
cohomology groups of the complex of global $(p,q)$ forms (graded by $q$):
$$H^{p,q}(X,F)=H^q\big(C^\infty(X,\Lambda^{p,\bu}T^\star_X\otimes F)\big).
\leqno(1.1)$$
Now, let us recall the following fundamental result from sheaf theory
(De Rham-Weil isomorphism theorem): let $(\cL^{\bu},d)$ be a resolution
of a sheaf $\cA$ by acyclic sheaves, i.e.\ a complex of sheaves
$(\cL^\bu,\delta)$ such that there is an exact sequence of sheaves
$$0\lra\cA\buildo\lra^j \cL^0\buildo \lra^{\delta^0}\cL^1\lra\cdots
\lra\cL^q\buildo \lra^{\delta^q}\cL^{q+1}\lra\cdots~,$$
and $H^s(X,\cL^q)=0$ for all $q\ge 0$ and $s\ge 1$. Then there
is a functorial isomorphism 
$$H^q\big(\Gamma(X,\cL^\bu)\big)\lra H^q(X,\cA).\leqno(1.2)$$
We apply this to the following situation: let $\cE(F)^{p,q}$ be the
sheaf of germs of $C^\infty$ sections of $\Lambda^{p,q}T^\star_X\otimes
F$. Then $(\cE(F)^{p,\bu},d'')$ is a resolution of the locally free
$\cO_X$-module $\Omega^p_X\otimes\cO(F)$ (Dolbeault-Grothendieck
lemma), and the sheaves $\cE(F)^{p,q}$ are acyclic as modules over the
soft sheaf of rings~$\cC^\infty$. Hence by (1.2) we get

\claim{(1.3) Dolbeault Isomorphism Theorem {\rm (1953)}} For every
holomorphic vector bundle $F$ on~$X$, there is a canonical isomorphism
$$H^{p,q}(X,F)\simeq H^q(X,\Omega^p_X\otimes\cO(F)).\eqno\square$$
\endclaim

If $X$ is projective algebraic and $F$ is an algebraic vector bundle,
Serre's GAGA theorem [Ser56] shows that the algebraic sheaf cohomology
group $H^q(X,\Omega^p_X\otimes\cO(F))$ computed with algebraic sections
over Zariski open sets is actually isomorphic to the analytic
cohomology group. These results are the most basic tools to attack
algebraic problems via analytic methods. Another important tool
is the theory of plurisubharmonic functions and positive
currents originated by K.~Oka and P.~Lelong in the decades 1940-1960.

\titlec{1.B.}{Plurisubharmonic Functions}
Plurisubharmonic functions have been introduced independently
by Lelong and Oka in the study of holomorphic convexity. We refer to
[Lel67, 69] for more details.

\claim{(1.4) Definition} A function $u:\Omega\lra[-\infty,+\infty[$ defined
on an open subset $\Omega\subset\bC^n$ is said to be plurisubharmonic
$($psh for short$)$ if
\smallskip
\item{\rm a)} $u$ is upper semicontinuous~$;$
\smallskip
\item{\rm b)} for every complex line $L\subset\bC^n$, 
$u_{\restriction\Omega\cap L}$ is subharmonic on $\Omega\cap L$, that is,
for all $a\in\Omega$ and $\xi\in\bC^n$ with $|\xi|<d(a,\complement\Omega)$,
the function $u$ satisfies the mean value inequality
$$u(a)\le{1\over 2\pi}\int_0^{2\pi}u(a+e^{\ii\theta}\,\xi)\,d\theta.$$
\vskip0pt\noindent
The set of psh functions on $\Omega$ is denoted by
$\Psh(\Omega)$.
\endclaim

We list below the most basic properties of psh functions.
They all follow easily from the definition.

\claim{(1.5) Basic properties} \rm
\smallskip
\item{\rm a)} Every function $u\in\Psh(\Omega)$ is subharmonic, namely it
satisfies the mean value inequality on euclidean balls or spheres:
$$u(a)\le{1\over\pi^nr^{2n}/n!}\int_{B(a,r)}u(z)\,d\lambda(z)$$
for every $a\in\Omega$ and $r<d(a,\complement\Omega)$. Either $u\equiv-\infty$
or $u\in L^1_\loc$ on every connected component of~$\Omega$.
\smallskip
\item{\rm b)} For any decreasing sequence of psh
functions $u_k\in\Psh(\Omega)$, the limit $u=\lim u_k$ is psh
on $\Omega$.
\smallskip
\item{\rm c)} Let $u\in\Psh(\Omega)$ be such that $u\not\equiv-\infty$ 
on every connected component of~$\Omega$. If $(\rho_\varepsilon)$ is a 
family of smoothing kernels, then $u\star\rho_\varepsilon$ is $\ci$ and 
psh on
$$\Omega_\varepsilon=\big\{x\in\Omega\,;\,d(x,\complement\Omega)>\varepsilon
\big\},$$
the family $(u\star\rho_\varepsilon)$ is increasing in $\varepsilon$ and 
$\lim_{\varepsilon\to 0}u\star\rho_\varepsilon=u$.
\smallskip
\item{\rm d)} Let $u_1\ld u_p\in\Psh(\Omega)$ and
$\chi:\bR^p\lra\bR$ be a convex function such that $\chi(t_1\ld t_p)$
is increasing in each $t_j$. Then $\chi(u_1\ld u_p)$ is psh
on $\Omega$. In particular~ $u_1+\cdots+u_p$, $\max\{u_1\ld u_p\}$, 
$\log(e^{u_1}+\cdots+e^{u_p})$ are psh on~$\Omega$.\qed
\smallskip
\endclaim

\claim{(1.6) Lemma} A function $u\in C^2(\Omega,\bR)$ is psh on
$\Omega$ if and only if the hermitian form $Hu(a)(\xi)=\sum_{1\le j,k\le n}
\partial^2 u/\partial z_j\partial\ol z_k(a)\,\xi_j\ol \xi_k$
is semipositive at every point $a\in\Omega$.
\endclaim

\begproof{} This is an easy consequence of the following standard formula
$${1\over 2\pi}\int_0^{2\pi}u(a+e^{\ii\theta}\,\xi)\,d\theta - u(a)=
{2\over\pi}\int_0^1{dt\over t}\int_{|\zeta|<t}Hu(a+\zeta\xi)(\xi)\,
d\lambda(\zeta),$$
where $d\lambda$ is the Lebesgue measure on $\bC$.
Lemma~1.6 is a strong evidence that plurisubharmonicity is the natural
complex analogue of linear convexity.\qed
\endproof

For non smooth functions, a similar characterization of plurisubharmonicity
can be obtained by means of a regularization process.

\claim{(1.7) Theorem} If $u\in\Psh(\Omega)$, $u\not\equiv-\infty$ on every
connected component of $\Omega$, then for all $\xi\in\bC^n$
$$Hu(\xi)=\sum_{1\le j,k\le n}{\partial^2 u\over\partial z_j\partial\ol z_k}
\,\xi_j\ol\xi_k\in\cD'(\Omega)$$
is a positive measure. Conversely, if $v\in\cD'(\Omega)$ is such that 
$Hv(\xi)$ is a positive measure for every $\xi\in\bC^n$, there exists a 
unique function $u\in\Psh(\Omega)$ which is locally integrable on $\Omega$
and such that $v$ is the distribution associated to $u$.\qed
\endclaim

In order to get a better geometric insight of this notion, we assume more
generally that $u$ is a function on a complex $n$-dimensional manifold 
$X$. If $\Phi:X\to Y$ is a holomorphic mapping and if $v\in C^2(Y,\bR)$, we
have $\hbox{$d'd''(v\circ \Phi)$}=\Phi^\star d'd''v$, hence
$$H(v\circ \Phi)(a,\xi)=Hv\big(\Phi(a),\Phi'(a).\xi\big).$$
In particular $Hu$, viewed as a hermitian form on $T_X$, does not depend
on the choice of coordinates $(z_1\ld z_n)$. Therefore, the notion of
psh function makes sense on any complex manifold. More generally,
we have

\claim{(1.8) Proposition} If $\Phi:X\lra Y$ is a holomorphic map and
$v\in\Psh(Y)$, then $v\circ \Phi\in\Psh(X)$.\qed
\endclaim

\claim{(1.9) Example} \rm It is a standard fact that $\log|z|$ is psh
(i.e.\ subharmonic) on $\bC$. Thus $\log|f|\in\Psh(X)$ for every
holomorphic function $f\in H^0(X,\cO_X)$. More generally 
$$\log\big(|f_1|^{\alpha_1}+\cdots+|f_q|^{\alpha_q}\big)\in\Psh(X)$$
for every $f_j\in H^0(X,\cO_X)$ and $\alpha_j\ge 0$ (apply Property~1.5$\,$d
with $u_j=\alpha_j\,\log|f_j|$). We will be especially interested in the
singularities obtained at points of the zero variety $f_1=\ldots=f_q=0$,
when the $\alpha_j$ are rational numbers.\qed
\endclaim

\claim{(1.10) Definition} A psh function $u\in\Psh(X)$ will be said to
have analytic singularities if $u$ can be written locally as
$$u={\alpha\over 2}\log\big(|f_1|^2+\cdots+|f_N|^2\big)+v,$$
where $\alpha\in\bR_+$, $v$ is a locally bounded function and the $f_j$
are holomorphic functions. If $X$ is algebraic, we say that $u$ has
algebraic singularities if $u$ can be written as above on sufficiently
small Zariski open sets, with $\alpha\in\bQ_+$ and $f_j$ algebraic.
\endclaim

We then introduce the ideal $\cJ=\cJ(u/\alpha)$ of germs of holomorphic
functions $h$ such that $|h|\le Ce^{u/\alpha}$ for some constant~$C$, i.e.
$$|h|\le C\big(|f_1|+\cdots+|f_N|\big).$$
This is a globally defined ideal sheaf on $X$, locally equal to the
integral closure $\ol\cI$ of the ideal sheaf $\cI=(f_1\ld f_N)$, thus $\cJ$
is coherent on~$X$. If $(g_1\ld g_{N'})$ are local generators of $\cJ$,
we still have
$$u={\alpha\over 2}\log\big(|g_1|^2+\cdots+|g_{N'}|^2\big)+O(1).$$
If $X$ is projective algebraic and $u$ has analytic singularities with
$\alpha\in\bQ_+$, then $u$ automatically has algebraic singularities.
From an algebraic point of view, the singularities of $u$ are in 1:1
correspondence with the ``algebraic data'' $(\cJ,\alpha)$. Later
on, we will see another important method for associating an ideal sheaf
to a psh function.

\claim{(1.11) Exercise} \rm Show that the above definition of the integral
closure of an ideal $\cI$ is equivalent to the following more algebraic
definition: $\ol\cI$ consists of all germs $h$ satisfying
an integral equation
$$h^d+a_1h^{d-1}+\ldots+a_{d-1}h+a_d=0,\qquad a_k\in\cI^k.$$
{\it Hint}. One inclusion is clear. To prove the other inclusion, consider
the normalization of the blow-up of $X$ along the (non necessarily reduced)
zero variety $V(\cI)$.\qed
\endclaim

\titlec{1.C.}{Positive Currents}
The reader can consult [Fed69] for a more thorough treatment of
current theory. Let us first recall a few basic definitions. A {\it
current} of degree $q$ on an oriented differentiable manifold $M$ is
simply a differential $q$-form $\Theta$ with distribution coefficients.
The space of currents of degree $q$ over $M$ will be denoted by
$\cD^{\prime q}(M)$. Alternatively, a current of degree $q$ can be seen
as an element $\Theta$ in the dual space $\cD'_p(M):=\big(\cD^p(M)\big)'$ of
the space $\cD^p(M)$ of smooth differential forms of degree $p=\dim M-q$ with
compact support; the duality pairing is given by
$$\langle \Theta,\alpha\rangle=\int_M \Theta\wedge\alpha,~~~~
\alpha\in\cD^p(M).\leqno(1.12)$$
A basic example is the {\it current of integration} $[S]$ over a compact
oriented submanifold $S$ of $M\,$:
$$\langle[S],\alpha\rangle=\int_S\alpha,~~~~{\rm deg}\alpha=p=\dim_\bR S.
\leqno(1.13)$$
Then $[S]$ is a current with measure coefficients, and
Stokes' formula shows that $d[S]=(-1)^{q-1}[\partial S]$, in particular
$d[S]=0$ if $S$ has no boundary. Because of this example, the integer $p$
is said to be the dimension of~$\Theta$ when $\Theta\in\cD_p'(M)$. The
current $\Theta$ is said to be {\it closed} if $d\Theta=0$.

On a complex manifold $X$, we have similar notions of bidegree and
bidimension; as in the real case, we denote by
$$\cD^{\prime p,q}(X)=\cD'_{n-p,n-q}(X),\qquad n=\dim X,$$
the space of currents of bidegree $(p,q)$ and bidimension $(n-p,n-q)$ on~$X$.
According to [Lel57], a current $\Theta$
of bidimension $(p,p)$ is said to be {\it $($weakly$)$ positive} if
for every choice of smooth $(1,0)$-forms $\alpha_1\ld\alpha_p$ on~$X$
the distribution
$$\Theta\wedge\ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge
\ii\alpha_p\wedge\ol\alpha_p~~~~\hbox{\rm is a positive measure.}
\leqno(1.14)$$

\claim{(1.15) Exercise} \rm If $\Theta$ is positive, show that the
coefficients $\Theta_{I,J}$ of $\Theta$ are complex measures, and that,
up to constants, they are dominated by the trace measure
$$\sigma_\Theta=\Theta\wedge{1\over p!}\beta^p=2^{-p}\sum\Theta_{I,I},\qquad
\beta={\ii\over 2}d'd''|z|^2={\ii\over 2}\sum_{1\le j\le n}dz_j\wedge
d\ol z_j,$$
which is a positive measure.\newline
{\it Hint}. Observe that $\sum\Theta_{I,I}$ is invariant by unitary changes
of coordinates and that the $(p,p)$-forms
$\ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge\ii\alpha_p\wedge\ol\alpha_p$
generate $\Lambda^{p,p}T^\star_{\bC^n}$ as a $\bC$-vector space.\qed
\endclaim

A current $\Theta=\ii\sum_{1\le j,k\le n}\Theta_{jk}dz_j\wedge dz_k$
of bidegree $(1,1)$ is easily seen to be positive if and only if the
complex measure $\sum\lambda_j\ol\lambda_k \Theta_{jk}$ is a positive
measure for every $n$-tuple $(\lambda_1\ld\lambda_n)\in\bC^n$.

\claim{(1.16) Example} \rm If $u$ is a (not identically $-\infty$)
psh function on $X$, we can associate with $u$ a (closed) positive
current $\Theta=\ii\partial\ol\partial u$ of bidegree~$(1,1)$. Conversely,
every closed positive current of bidegree $(1,1)$ can be written under
this form on any open subset $\Omega\subset X$ such that
$H^2_{DR}(\Omega,\bR)=H^1(\Omega,\cO)=0$, e.g.\ on small coordinate
balls (exercise to the reader).\qed
\endclaim

It is not difficult to show that a product $\Theta_1\wedge\ldots\wedge
\Theta_q$ of positive currents of bidegree $(1,1)$ is positive whenever
the product is well defined (this is certainly the case if all
$\Theta_j$ but one at most are smooth; much finer conditions will be
discussed in Section~2).

We now discuss another very important example of closed positive current.
In fact, with every closed analytic set $A\subset X$ of pure dimension $p$
is associated a current of integration
$$\langle [A],\alpha\rangle=\int_{A_\reg}\alpha,~~~~\alpha\in\cD^{p,p}(X),
\leqno(1.17)$$
obtained by integrating over the regular points of~$A$. In order to show
that (1.17) is a correct definition of a current on~$X$, one must show that
$A_\reg$ has locally finite area in a neighborhood of $A_\sing$.
This result, due to [Lel57] is shown as follows. Suppose that $0$
is a singular point of $A$. By the local parametrization theorem for
analytic sets, there is a linear change of coordinates on $\bC^n$
such that all projections
$$\pi_I:(z_1\ld z_n)\mapsto (z_{i_1}\ld z_{i_p})$$
define a finite ramified covering of the intersection $A\cap\Delta$ with
a small polydisk $\Delta$ in $\bC^n$ onto a small
polydisk $\Delta_I$ in~$\bC^p$. Let $n_I$ be the sheet number. Then the
\hbox{$p$-dimensional} area of $A\cap\Delta$ is bounded above by the sum
of the areas of its projections counted with multiplicities, i.e.
$$\hbox{\rm Area}(A\cap\Delta)\le\sum n_I\hbox{\rm Vol}(\Delta_I).$$
The fact that $[A]$ is positive is also easy. In fact
$$\ii\alpha_1\wedge\ol\alpha_1\wedge\ldots\wedge
\ii\alpha_p\wedge\ol\alpha_p=|\det(\alpha_{jk})|^2\,
\ii w_1\wedge\ol w_1\wedge\ldots\wedge
\ii w_p\wedge\ol w_p$$
if $\alpha_j=\sum\alpha_{jk}dw_k$ in terms of local coordinates
$(w_1\ld w_p)$ on $A_\reg$. This shows that all such forms are $\ge 0$
in the canonical orientation defined by
$\ii w_1\wedge\ol w_1\wedge\ldots\wedge\ii w_p\wedge\ol w_p$.
More importantly, Lelong [Lel57] has shown that $[A]$ is $d$-closed in~$X$,
even at points of $A_\sing$. This last result can be seen today as a
consequence of the Skoda-El Mir extension theorem. For this we need the
following definition: a {\it complete pluripolar} set is a set $E$ such
that there is an open covering $(\Omega_j)$ of $X$ and psh functions $u_j$ on
$\Omega_j$ with $E\cap\Omega_j=u_j^{-1}(-\infty)$. Any (closed) analytic
set is of course complete pluripolar (take $u_j$ as in Example~1.9). 

\claim{(1.18) Theorem {\rm(Skoda [Sko82], El Mir [EM84], Sibony [Sib85])}}
Let $E$ be a closed complete pluripolar set in $X$,
and let $\Theta$ be a closed positive current on $X\ssm E$ such that the
coefficients $\Theta_{I,J}$ of $\Theta$ are measures with locally finite
mass near~$E$. Then the trivial extension $\wt \Theta$ obtained by extending
the measures $\Theta_{I,J}$ by $0$ on $E$ is still closed on~$X$.
\endclaim

Lelong's result $d[A]=0$ is obtained by applying the Skoda-El Mir
theorem to $\Theta=[A_\reg]$ on $X\ssm A_\sing$. 

\begproof{of Theorem 1.18.} The statement is local on $X$, so we may
work on a small open set $\Omega$ such that $E\cap\Omega=v^{-1}(-\infty)$,
$v\in\Psh(\Omega)$. Let $\chi:\bR\to\bR$ be a convex increasing function
such that $\chi(t)=0$ for $t\le -1$ and $\chi(0)=1$. By shrinking $\Omega$
and putting $v_k=\chi(k^{-1}v\star\rho_{\varepsilon_k})$ with
$\varepsilon_k\to 0$ fast, we get a sequence of functions
$v_k\in\Psh(\Omega)\cap\ci(\Omega)$ such that $0\le v_k\le 1$, $v_k=0$ in
a neighborhood of $E\cap\Omega$ and $\lim v_k(x)=1$ at every point of
$\Omega\ssm E$. Let $\theta\in\ci([0,1])$ be a
function such that $\theta=0$ on $[0,1/3]$, $\theta=1$ on $[2/3,1]$ and 
$0\le\theta\le 1$. Then $\theta\circ v_k=0$ near $E\cap\Omega$ and 
$\theta\circ v_k\to 1$ on $\Omega\ssm E$. Therefore 
$\tilde\Theta=\lim_{k\to+\infty}(\theta\circ v_k)\Theta$ and
$$d'\tilde\Theta=\lim_{k\to+\infty}\Theta\wedge d'(\theta\circ v_k)$$
in the weak topology of currents. It is therefore sufficient to verify that 
\hbox{$\Theta\wedge d'(\theta\circ v_k)$} converges weakly to $0$
(note that $d''\tilde \Theta$ is conjugate to $d'\tilde \Theta$, 
thus $d''\tilde \Theta$ will also vanish).

Assume first that $\Theta\in\cD^{\prime n-1,n-1}(X)$. Then
$\Theta\wedge d'(\theta\circ v_k)\in\cD^{\prime n,n-1}(\Omega)$, and we have
to show that
$$\langle \Theta\wedge d'(\theta\circ v_k),\ol\alpha\rangle=\langle\Theta,
\theta'(v_k)d'v_k\wedge\ol\alpha\rangle
\buildu\lra_{k\to+\infty} 0,~~~~\forall\alpha\in\cD^{1,0}(\Omega).$$
As $\gamma\mapsto\langle\Theta,\ii\gamma\wedge\ol\gamma\rangle$ is a
non-negative hermitian form on $\cD^{1,0}(\Omega)$, the Cauchy-Schwarz 
inequality yields
$$\big|\langle \Theta,\ii\beta\wedge\ol\gamma\rangle\big|^2\le
\langle \Theta,\ii\beta\wedge\ol\beta\rangle~
\langle \Theta,\ii\gamma\wedge\ol\gamma\rangle,~~~~
\forall\beta,\gamma\in\cD^{1,0}(\Omega).$$
Let $\psi\in\cD(\Omega)$, $0\le\psi\le 1$, be equal to $1$ in a neighborhood
of ${\rm Supp}\,\alpha$. We find
$$\big|\langle \Theta,\theta'(v_k)d'v_k\wedge\ol\alpha\rangle\big|^2\le
\langle \Theta,\psi \ii d'v_k\wedge d''v_k\rangle~
\langle \Theta,\theta'(v_k)^2 \ii\alpha\wedge\ol\alpha\rangle.$$
By hypothesis $\int_{\Omega\ssm E}\Theta\wedge \ii\alpha\wedge\ol\alpha<
+\infty$ and $\theta'(v_k)$ converges everywhere to $0$ on $\Omega$, thus
$\langle \Theta,\theta'(v_k)^2 \ii\alpha\wedge\ol\alpha\rangle$ converges
to $0$ by Lebesgue's dominated convergence theorem. On the other hand
$$\eqalign{
&\ii d'd''v_k^2=2v_k\,\ii d'd''v_k+2\ii d'v_k\wedge d''v_k\ge 2\ii d'v_k
\wedge d''v_k,\cr
&2\langle\Theta,\psi \ii d'v_k\wedge d''v_k\rangle\le\langle\Theta,\psi
\ii d'd''v_k^2\rangle.\cr}$$
As $\psi\in\cD(\Omega)$, $v_k=0$ near $E$ and $d\Theta=0$ on $\Omega\ssm E$,
an integration by parts yields
$$\langle \Theta,\psi \ii d'd''v_k^2\rangle=\langle \Theta,v_k^2
\ii d'd''\psi\rangle\le C\int_{\Omega\ssm E}\|\Theta\|<+\infty$$
where $C$ is a bound for the coefficients of $\ii d'd''\psi$. Thus 
$\langle \Theta,\psi \ii d'v_k\wedge d''v_k\rangle$ is bounded, 
and the proof is complete when $\Theta\in\cD^{\prime n-1,n-1}$.

In the general case $\Theta\in\cD^{\prime p,p}$, $p<n$, we simply apply the
result already proved to all positive currents $\Theta\wedge\gamma\in
\cD^{\prime n-1,n-1}$ where
$\gamma=\ii\gamma_1\wedge\ol\gamma_1\wedge\ldots\wedge\ii\gamma_{n-p-1,}
\wedge\ol\gamma_{n-p-1}$ runs over a basis of forms of
$\Lambda^{n-p-1,n-p-1}T^\star_\Omega$ with constant coefficients.
Then we get $d(\tilde \Theta\wedge\gamma)=d\tilde \Theta\wedge\gamma=0$ for
all such $\gamma$, hence $d\tilde \Theta=0$.\qed
\endproof

\claim{(1.19) Corollary} Let $\Theta$ be a closed positive current on
$X$ and let $E$ be a complete pluripolar set. Then $\bOne_E \Theta$
and $\bOne_{X\ssm E}\Theta$ are closed positive currents. In fact,
\hbox{$\wt\Theta=\bOne_{X\ssm E}\Theta$} is the trivial extension of
$\Theta_{\restriction X\ssm E}$ to~$X$, and
$\bOne_E\Theta=\Theta-\wt\Theta$.\qed
\endclaim

As mentioned above, any current $\Theta=\ii d'd''u$ associated with a psh
function $u$ is a closed positive $(1,1)$-current. In the special
case $u=\log|f|$ where $f\in H^0(X,\cO_X)$ is a non zero holomorphic
function, we have the important

\vbox{\claim{(1.20) Lelong-Poincar\'e equation}
Let $f\in H^0(X,\cO_X)$ be a non zero holomorphic function,
$Z_f=\sum m_jZ_j$, $m_j\in\bN$, the zero divisor of~$f$ and
$[Z_f]=\sum m_j[Z_j]$ the associated current of integration. Then
$${\ii\over\pi}\partial\ol\partial\log|f|=[Z_f].$$
\endclaim\removelastskip}

\begproof{(sketch).} It is clear that $\ii d'd''\log|f|=0$ in a neighborhood
of every point $x\notin\Supp(Z_f)=\bigcup Z_j$, so it is enough to check
the equation in a neighborhood of every point of $\Supp(Z_f)$. Let $A$ be
the set of singular points of $\Supp(Z_f)$, i.e.\ the union of the
pairwise intersections $Z_j\cap Z_k$ and of the singular loci
$Z_{j,\sing}$; we thus have $\dim A\le n-2$. In a neighborhood of any
point $x\in\Supp(Z_f)\ssm A$ there are local coordinates $(z_1\ld z_n)$
such that $f(z)=z_1^{m_j}$ where $m_j$ is the multiplicity of $f$ along
the component $Z_j$ which contains~$x$ and $z_1=0$ is an equation for
$Z_j$ near~$x$. Hence
$${\ii\over\pi}d'd''\log|f|=m_j{\ii\over\pi}d'd''\log|z_1|=m_j[Z_j]$$
in a neighborhood of $x$, as desired (the identity comes from the
standard formula ${\ii\over\pi}d'd''\log|z|={}$ Dirac measure $\delta_0$
in $\bC$). This shows that the equation holds on $X\ssm A$. Hence
the difference ${\ii\over\pi}d'd''\log|f|-[Z_f]$ is a closed current of
degree $2$ with measure coefficients, whose support is contained in~$A$.
By Exercise~1.21, this current must be $0$, for $A$ has too small
dimension to carry its support ($A$ is stratified by submanifolds of real
codimension $\ge 4$).\qed
\endproof

\claim{(1.21) Exercise} \rm Let $\Theta$ be a current of degree $q$
on a real manifold~$M$, such that both $\Theta$ and $d\Theta$ have
measure coefficients (``normal current''). Suppose that $\Supp \Theta$
is contained in a real submanifold $A$ with $\codim_\bR A>q$. Show
that $\Theta=0$.\newline
{\it Hint:} Let $m=\dim_\bR M$ and let $(x_1\ld x_m)$ be a coordinate
system in a neighborhood $\Omega$ of a point $a\in A$ such that
$A\cap\Omega=\{x_1=\ldots=x_k=0\}$, $k>q$. Observe that
$x_j\Theta=x_jd\Theta=0$ for $1\le j\le k$, thanks to the hypothesis on
supports and on the normality of~$\Theta$, hence $dx_j\wedge
\Theta=d(x_j\Theta)-x_jd\Theta=0$, $1\le j\le k$. Infer from this that
all coefficients in $\Theta=\sum_{|I|=q}\Theta_Idx_I$ vanish.\qed
\endclaim

We now recall a few basic facts of slicing theory (the reader will
profitably consult [Fed69] and [Siu74] for further developments).
Let \hbox{$\sigma:M\to M'$} be a submersion of smooth differentiable
manifolds and let $\Theta$ be a {\it locally flat} current on~$M$,
that is, a current which can be written locally as $\Theta=U+dV$
where $U$, $V$ have $L^1_\loc$ coefficients. It is a standard fact
(see Federer) that every current $\Theta$ such that both $\Theta$ and
$d\Theta$ have measure coefficients is locally flat; in particular,
closed positive currents are locally flat. Then, for almost every
$x'\in M'$, there is a well defined {\it slice} $\Theta_{x'}$,
which is the current on the fiber $\sigma^{-1}(x')$ defined by
$$\Theta_{x'}=U_{\restriction\sigma^{-1}(x')}+
  dV_{\restriction\sigma^{-1}(x')}.$$
The restrictions of $U$, $V$ to the fibers exist for almost all $x'$ by
the Fubini theorem. The slices $\Theta_{x'}$ are currents on the fibers with
the same degree as~$\Theta$ (thus of dimension $\dim \Theta-\dim
\hbox{(fibers)}$). Of course, every slice $\Theta_{x'}$ coincides with
the usual restriction of $\Theta$ to the fiber if $\Theta$ has smooth
coefficients. By using a regularization $\Theta_\varepsilon=\Theta\star
\rho_\varepsilon$, it is easy to show that the slices of a
closed positive current are again closed and positive: in fact
$U_{\varepsilon,x'}$ and $V_{\varepsilon,x'}$ converge to $U_{x'}$ and
$V_{x'}$ in $L^1_{\rm loc}(\sigma^{-1}(x'))$, thus
$\Theta_{\varepsilon,x'}$ converges weakly to $\Theta_{x'}$ for almost
every~$x'$. Now, the basic slicing formula is
$$\int_M \Theta\wedge\alpha\wedge\sigma^\star\beta=
\int_{x'\in M'}\Big(\int_{x''\in\sigma^{-1}(x')}\Theta_{x'}(x'')\wedge
\alpha_{\restriction\sigma^{-1}(x')}(x'')\Big)\beta(x')\leqno(1.22)$$
for every smooth form $\alpha$ on $M$ and $\beta$ on $M'$, such that
$\alpha$ has compact support and $\deg\alpha=\dim M-\dim M'-\deg \Theta$,
$\deg\beta=\dim M'$. This is an easy consequence of the usual Fubini
theorem applied to $U$ and $V$ in the decomposition $\Theta=U+dV$, if we
identify locally $\sigma$ with a projection map
$M=M'\times M''\to M'$, $x=(x',x'')\mapsto x'$, and use a partitition
of unity on the support of $\alpha$.

To conclude this section, we discuss De Rham and Dolbeault cohomology theory
in the context of currents. A basic observation is that the Poincar\'e and
Dolbeault-Grothendieck lemmas still hold for currents. Namely, if
$(\cD^{\prime q},d)$ and $(\cD'(F)^{p,q},d'')$ denote the complex of sheaves
of degree $q$ currents (resp.\ of $(p,q)$-currents with values in a
holomorphic vector bundle~$F$), we still have De Rham and Dolbeault sheaf
resolutions
$$0\to\bR\to\cD^{\prime \bu},\qquad 0\to\Omega^p_X\otimes\cO(F)\to
\cD'(F)^{p,\bu}.$$
Hence we get canonical isomorphisms
$$\leqalignno{
H^q_\DR(M,\bR)&=H^q\big((\Gamma(M,\cD^{\prime\bu}),d)\big),&(1.23)\cr
H^{p,q}(X,F)&=H^q\big((\Gamma(X,\cD'(F)^{p,\bu}),d'')\big).&\cr}$$
In other words, we can attach a cohomology class
$\{\Theta\}\in H^q_\DR(M,\bR)$ to any closed current $\Theta$ of degree~$q$,
resp.\ a cohomology class $\{\Theta\}\in H^{p,q}(X,F)$ to any $d''$-closed
current of bidegree~$(p,q)$. Replacing if necessary every current by a
smooth representative in the same cohomology class, we see that there is a
well defined cup product given by the wedge product of differential forms
$$\eqalign{
H^{q_1}(M,\bR)\times\ldots\times H^{q_m}(M,\bR)&\lra
H^{q_1+\ldots+q_m}(M,\bR),\cr
(\{\Theta_1\}\ld\{\Theta_1\})&\longmapsto\{\Theta_1\}\wedge\ldots\wedge
\{\Theta_m\}.\cr}$$
In particular, if $M$ is a compact oriented variety and
$q_1+\ldots+q_m=\dim M$, there is a well defined intersection number
$$\{\Theta_1\}\cdot\{\Theta_2\}\cdot\,\cdots\,\cdot
\{\Theta_m\}=\int_M\{\Theta_1\}\wedge\ldots\wedge\{\Theta_m\}.$$
However, as we will see in the next section, the pointwise product
\hbox{$\Theta_1\wedge\ldots\wedge\Theta_m$} need not exist in general.

\titleb{2.}{Lelong Numbers and Intersection Theory}
\titlec{2.A.}{Multiplication of Currents and Monge-Amp\`ere Operators}
Let $X$ be a $n$-dimensional complex manifold. We set
$$d^c={1\over2\ii\pi}(d'-d'').$$
It follows in particular that $d^c$ is a real operator, i.e. $\ol{d^cu}=
d^c\ol u$, and that $dd^c={\ii\over\pi}d'd''$. Although not quite standard,
the $1/2\ii\pi$ normalization is very convenient for many purposes, since
we may then forget the factor $\pi$ or $2\pi$ almost everywhere (e.g.\ in
the Lelong-Poincar\'e equation (1.20)).

Let $u$ be a psh function and let $\Theta$ be a closed
positive current on~$X$. Our desire is to define the wedge product
$dd^cu\wedge\Theta$ even when neither $u$ nor $\Theta$ are smooth.
In general, this product does not make sense because $dd^cu$ and $\Theta$
have measure coefficients and measures cannot be multiplied; see
Kiselman [Kis84] for interesting counterexamples. Even in the algebraic
setting considered here, multiplication of currents is not always possible:
suppose e.g.\ that $\Theta=[D]$ is the exceptional divisor of a blow-up
in a surface; then $D\cdot D=-1$ cannot be the cohomology class of a
closed positive current $[D]^2$. Assume however that
$u$ is a {\it locally bounded} psh function. Then the
current $u\Theta$ is well defined since $u$ is a locally bounded Borel
function and $\Theta$ has measure coefficients. According to
Bedford-Taylor [BT82] we define
$$dd^cu\wedge\Theta=dd^c(u\Theta)$$
where $dd^c(~~)$ is taken in the sense of distribution theory.

\claim{(2.1) Proposition} If $u$ is a locally bounded psh function, the
wedge product $dd^cu\wedge\Theta$ is again a closed positive current.
\endclaim

\begproof{} The result is local. Use a convolution $u_\nu=u\star\rho_{1/\nu}$
to get a decreasing sequence of smooth psh functions converging to~$u$.
Then write
$$dd^c(u\Theta)=\lim_{\nu\to+\infty}dd^c(u_\nu\Theta)=
dd^cu_\nu\wedge\Theta$$
as a weak limit of closed positive currents. Observe that $u_\nu\Theta$
converges weakly to $u\Theta$ by Lebesgue's monotone convergence theorem.\qed
\endproof

More generally, if $u_1\ld u_m$ are locally bounded
psh functions, we can define
$$dd^cu_1\wedge\ldots\wedge dd^cu_m\wedge\Theta=
dd^c\big(u_1dd^cu_2\wedge\ldots\wedge dd^cu_m\wedge\Theta\big)$$
by induction on~$m$. Chern, Levine and Nirenberg [CLN69] noticed
the following useful inequality. Define the {\it mass} of a current
$\Theta$ on a compact set $K$ to be
$$||\Theta||_K=\int_K\sum_{I,J}|\Theta_{I,J}|$$
whenever $K$ is contained in a coordinate patch and
$\Theta=\sum\Theta_{I,J}dz_I\wedge d\ol z_J$. Up to seminorm equivalence,
this does not depend on the choice of coordinates. If $K$ is not contained
in a coordinate patch, we use a partition of unity to define a suitable
seminorm $||\Theta||_K$. If $\Theta\ge 0$, Exercise~1.15 shows that the mass
is controlled by the trace measure, i.e.\
$||\Theta||_K\le C\int_K\Theta\wedge\beta^p$.

\claim{(2.2) Chern-Levine-Nirenberg inequality}
For all compact subsets $K,L$ of $X$ with $L\subset K^\circ$, there exists
a constant $C_{K,L}\ge0$ such that
$$||dd^cu_1\wedge\ldots\wedge dd^cu_m\wedge\Theta||_L\le C_{K,L}~
||u_1||_{L^\infty(K)}\ldots||u_m||_{L^\infty(K)}\,||\Theta||_K$$
\endclaim

\begproof{} By induction, it is sufficient to prove the result for
$m=1$ and $u_1=u$. There is a covering of $L$ by a family of open balls
$B'_j\compact B_j\subset K$ contained in coordinate patches of~$X$. Let
$(p,p)$ be the bidimension of $\Theta$, let $\beta={\ii\over 2}d'd''|z|^2$,
and let $\chi\in\cD(B_j)$ be equal to $1$ on $\smash{\ol B'_j}$. Then
$$||dd^cu\wedge\Theta||_{L\cap\ol B'_j}\le C\int_{\ol B'_j}dd^cu\wedge
\Theta\wedge\beta^{p-1}\le C\int_{B_j}\chi\,dd^c u\wedge\Theta\wedge
\beta^{p-1}.$$
As $\Theta$ and $\beta$ are closed, an integration by parts yields
$$||dd^cu\wedge\Theta||_{L\cap\ol B'_j}\le C\int_{B_j}u\,\Theta\wedge
dd^c\chi\wedge\beta^{p-1}\le C'||u||_{L^\infty(K)}||\Theta||_K$$
where $C'$ is equal to $C$ multiplied by a bound for the coefficients
of the smooth form $dd^c\chi\wedge\beta^{p-1}$.\qed
\endproof

Various examples (cf.\ [Kis84]) show however that products of
$(1,1)$-currents $dd^cu_j$ cannot be defined in a reasonable way for
arbitrary psh functions~$u_j$. However, functions $u_j$
with $-\infty$ poles can be admitted if the polar sets are sufficiently
small.

\claim{(2.3) Proposition} Let $u$ be a psh function
on $X$, and let $\Theta$ be a closed positive current
of bidimension $(p,p)$. Suppose that $u$ is locally bounded on $X\ssm A$,
where $A$ is an analytic subset of $X$ of dimension $<p$ at each point.
Then $dd^cu\wedge\Theta$ can be defined in such a way that 
$dd^cu\wedge\Theta=\lim_{\nu\to+\infty}dd^cu_\nu\wedge\Theta$
in the weak topology of currents, for any decreasing sequence
$(u_\nu)_{\nu\ge 0}$ of psh functions converging to~$u$.
\endclaim

\begproof{} When $u$ is locally bounded everywhere, we have 
$\,\lim u_\nu\,\Theta=u\,\Theta$ by the monotone convergence 
theorem and the result follows from the continuity of $dd^c$ with
respect to the weak topology.

First assume that $A$ is discrete. Since our results are local, we 
may suppose that $X$ is a ball $B(0,R)\subset\bC^n$ and that $A=\{0\}$. 
For every $s\le 0$, the function $u^{\gge s}=\max(u,s)$
is locally bounded on $X$, so the product $\Theta\wedge 
dd^cu^{\gge s}$ is well defined. For $|s|$ large, the function
$\smash{u^{\gge s}}$ differs from $u$ only in a small neighborhood 
of the origin, at which $u$ may have a $-\infty$ pole. Let $\gamma$
be a $(p-1,p-1)$-form with constant coefficients and set
$s(r)=\liminf_{|z|\to r-0}u(z)$. By Stokes' formula, we see that
the integral
$$I(s):=\int_{B(0,r)}dd^cu^{\gge s}\wedge\Theta\wedge\gamma\leqno(2.4)$$
does not depend on $s$ when $s<s(r)$, for the difference $I(s)-I(s')$
of two such integrals involves the $dd^c$ of a current
$(u^{\gge s}-u^{\gge s'})\wedge\Theta\wedge\gamma$ with compact support
in $B(0,r)$. Taking $\gamma=(dd^c|z|^2)^{p-1}$, we see that the current
$dd^cu\wedge\Theta$ has finite mass on $B(0,r)\ssm\{0\}$
and we can define $\langle\bOne_{\{0\}}(dd^cu\wedge\Theta),
\gamma\rangle$ to be the limit of the integrals $(2.4)$ as $r$ tends to
zero and $s<s(r)$. In this case, the weak convergence statement is easily
deduced from the locally bounded case discussed above.
 
In the case where $0<\dim A<p$, we use a slicing technique to reduce the 
situation to the discrete case. Set $q=p-1$. There are linear
coordinates $(z_1\ld z_n)$ centered at any point of $A$,
such that $0$ is an isolated point of $A\cap\big(\{0\}\times\bC^{n-q}\big)$.
Then there are small balls $B'=B(0,r')$ in $\bC^q$, $B''=B(0,r'')$ in
$\bC^{n-q}$ such that $A\cap(\smash{\ol B}'\times\partial B'')=
\emptyset$, and the projection map
$$\pi:\bC^n\to\bC^q,~~~~z=(z_1\ld z_n)\mapsto z'=(z_1\ld z_q)$$
defines a finite proper mapping $A\cap(B'\times B'')\to B'$.
These properties are preserved if we slightly change the direction of
projection. Take sufficiently many projections $\pi_m$ associated
to coordinate systems $(z^m_1\ld z^m_n)$, \hbox{$1\le m\le N$}, in such
a way that the family of $(q,q)$-forms
$$i\,dz^m_1\wedge d\ol z^m_1\wedge\ldots\wedge i\,dz^m_q\wedge d\ol z^m_q$$
defines a basis of the space of $(q,q)$-forms. Expressing any
compactly supported smooth $(q,q)$-form in such a basis, we see
that we need only define
$$\leqalignno{
&\int_{B'\times B''}dd^cu\wedge\Theta\wedge f(z',z'')\,
\ii dz_1\wedge d\ol z_1\wedge\ldots\wedge\ii dz_q\wedge d\ol z_q=&(2.5)\cr
&\int_{B'}\Big\{\int_{B''}f(z',\bu)\,
dd^cu(z',\bu)\wedge\Theta(z',\bu)\Big\}
\ii dz_1\wedge d\ol z_1\wedge\ldots\wedge\ii dz_q\wedge d\ol z_q\cr}$$
where $f$ is a test function with compact support in $B'\times B''$,
and $\Theta(z',\bu)$ denotes the slice of $\Theta$ on the fiber
$\{z'\}\times B''$ of the projection \hbox{$\pi:\bC^n\to\bC^q$}.
Each integral $\int_{B''}$ in the right hand side of (2.5) makes sense
since the slices $(\{z'\}\times B'')\cap A$ are discrete.
Moreover, the double integral $\int_{B'}\int_{B''}$ is convergent.
Indeed, observe that $u$ is bounded on any compact cylinder
$$K_{\delta,\varepsilon}=\smash{\ol B}\big((1-\delta)r'\big)\times
\Big(\smash{\ol B}(r'')\ssm\smash{\ol B}\big((1-\varepsilon)r''\big)\Big)$$
disjoint from $A$. Take $\varepsilon\ll\delta\ll 1$ so small that
$${\rm Supp}\,f\subset\smash{\ol B}\big((1-\delta)r'\big)\times
\smash{\ol B}\big((1-\varepsilon)r''\big).$$
For all $z'\in\smash{\ol B}((1-\delta)r')$, the proof of the
Chern-Levine-Nirenberg inequality (2.2) with a cut-off function $\chi(z'')$
equal to $1$ on $B((1-\varepsilon)r'')$ and with support in
$B((1-\varepsilon/2)r'')$ shows that
$$\eqalign{
&\int_{B((1-\varepsilon)r'')}dd^cu(z',\bu)\wedge\Theta(z',\bu)\cr
&\qquad\quad{}\le C_\varepsilon ||u||_{L^\infty(K_{\delta,\varepsilon})}
\int_{z''\in B((1-\varepsilon/2)r'')}
\Theta(z',z'')\wedge dd^c|z''|^2.\cr}$$
This implies that the double integral is convergent. Now replace $u$
everywhere by $u_\nu$ and observe that $\lim_{\nu\to+\infty}\int_{B''}$
is the expected integral for every $z'$ such that $\Theta(z',\bu)$ exists
(apply the discrete case already proven). Moreover, the
Chern-Levine-Nirenberg inequality yields uniform bounds for all
functions $u_\nu$, hence Lebesgue's dominated convergence theorem can
be applied to $\int_{B'}$. We conclude from this that the sequence of
integrals (2.5) converges when \hbox{$u_\nu\downarrow u$}, as
expected.\qed
\endproof

\claim{(2.6) Remark} \rm In the above proof, the fact that $A$ is an
analytic set does not play an essential role. The main point is just
that the slices $(\{z'\}\times B'')\cap A$ consist of isolated points
for generic choices of coordinates $(z',z'')$. In fact, the proof even
works if the slices are totally discontinuous, in particular if they are
of zero Hausdorff measure~$\cH_1$. It follows that Proposition~2.3 still
holds whenever $A$ is a closed set such that $\cH_{2p-1}(A)=0$.\qed
\endclaim

\titlec{2.B.}{Lelong Numbers}
The concept of Lelong number is an analytic analogue of the algebraic
notion of multiplicity. It is a very useful technique to extend
results of the intersection theory of algebraic cycles to currents.
Lelong numbers have been introduced for the first time by Lelong in [Lel57].
See also [Lel69], [Siu74], [Dem82a, 85a, 87] for further developments.

Let us first recall a few definitions.
Let $\Theta$ be a closed positive current of bidimension $(p,p)$
on a coordinate open set $\Omega\subset\bC^n$ of a complex manifold~$X$.
The Lelong number of $\Theta$ at a point $x\in\Omega$ is defined to be
the limit
$$\nu(\Theta,x)=\lim_{r\to 0{\scriptscriptstyle+}}\nu(\Theta,x,r),\qquad
\hbox{where}~~
\nu(\Theta,x,r)={\sigma_\Theta(B(x,r))\over\pi^p r^{2p}/p!}$$
measures the ratio of the area of $\Theta$ in the ball $B(x,r)$ to the 
area of the ball of radius $r$ in $\bC^p$. As $\sigma_\Theta=\Theta\wedge
{1\over p!}(\pi dd^c|z|^2)^p$ by 1.15, we also get
$$\nu(\Theta,x,r)={1\over r^{2p}}\int_{B(x,r)}\Theta(z)\wedge
(dd^c|z|^2)^p.\leqno(2.7)$$
The main results concerning Lelong numbers are summarized in the following
theorems, due respectively to Lelong, Thie and Siu.

\claim{(2.8) Theorem {\rm([Lel57])}}
\smallskip
\item{\rm a)}  For every positive current $\Theta$, the
ratio $\nu(\Theta,x,r)$ is a nonnegative increasing function of $r$, in
particular the limit $\nu(\Theta,x)$ as $r\to0{\scriptstyle+}$
always exists.
\smallskip
\item{\rm b)} If $\Theta=dd^cu$ is the bidegree $(1,1)$-current associated
with a psh function~$u$, then
$$\nu(\Theta,x)=\sup\big\{\gamma\ge 0\,;\,u(z)\le\gamma\log|z-x|+O(1)
~~\hbox{at $x$}\big\}.$$
In particular, if $u=\log|f|$ with $f\in H^0(X,\cO_X)$ and
$\Theta=dd^cu=[Z_f]$, we have
$$\nu([Z_f],x)=\ord_x(f)=\max\{m\in\bN\,;\,D^\alpha f(x)=0,\,|\alpha|<m\}.$$
\vskip0pt
\endclaim

\claim{(2.9) Theorem {\rm([Thi67])}} In the case where $\Theta$
is a current of integration $[A]$ over an analytic subvariety $A$, the
Lelong number $\nu([A],x)$ coincides with the multiplicity of $A$ at $x$
$($defined e.g.\ as the sheet number in the ramified covering obtained by
taking a generic linear projection of the germ $(A,x)$ onto a
$p$-dimensional linear subspace through $x$ in any coordinate
patch~$\Omega)$.
\endclaim

\claim{(2.10) Theorem {\rm([Siu74])}} Let $\Theta$ be a closed positive
current of bidimension $(p,p)$ on the complex manifold~$X$.
\smallskip
\item{\rm a)} The Lelong number $\nu(\Theta,x)$ is invariant by holomorphic
changes of local coordinates.
\smallskip
\item{\rm b)} For every $c>0$, the set
$E_c(\Theta)=\big\{x\in X\,;\,\nu(\Theta,x)\ge c\big\}$
is a closed analytic subset of~$X$ of dimension $\le p$.\vskip0pt
\endclaim

The most important result is 2.10~b), which is a deep application of
H\"ormander $L^2$ estimates (see Section~5). The earlier proofs of all
other results were rather intricate in spite of their rather simple nature.
We reproduce below a sketch of elementary arguments based on the use of
a more general and more flexible notion of Lelong number introduced in
[Dem87]. Let $\varphi$ be a continuous psh function
with an isolated $-\infty$ pole at $x$, e.g.\ a function of the form
$\varphi(z)=\log\sum_{1\le j\le N}|g_j(z)|^{\gamma_j}$, $\gamma_j>0$,
where $(g_1\ld g_N)$ is an ideal of germs of holomorphic functions in 
$\cO_x$ with $g^{-1}(0)=\{x\}$. The {\it generalized
Lelong number} $\nu(\Theta,\varphi)$ of $\Theta$ with respect to the
weight $\varphi$ is simply defined to be the mass of the measure
$\Theta\wedge(dd^c\varphi)^p$ carried by the point $x$
(the measure $\Theta\wedge(dd^c\varphi)^p$ is always well defined thanks
to Proposition~2.3). This number can also
be seen as the limit $\nu(\Theta,\varphi)=\lim_{t\to-\infty}
\nu(\Theta,\varphi,t)$, where
$$\nu(\Theta,\varphi,t)=\int_{\varphi(z)<t}\Theta\wedge
(dd^c\varphi)^p.\leqno(2.11)$$
The relation with our earlier definition of Lelong numbers (as well as part
a) of Theorem~2.8) comes from the identity
$$\nu(\Theta,x,r)=\nu(\Theta,\varphi,\log r),~~~~
\varphi(z)=\log|z-x|,\leqno(2.12)$$
in particular $\nu(\Theta,x)=\nu(\Theta,\log|\bu-x|)$. This equality is
in turn a consequence of the following general formula, applied to
$\chi(t)=e^{2t}$ and $t=\log r\,$:
$$\int_{\varphi(z)<t}\Theta\wedge(dd^c\chi\circ\varphi)^p=
\chi'(t-0)^p\int_{\varphi(z)<t}\Theta\wedge(dd^c\varphi)^p,
\leqno(2.13)$$
where $\chi$ is an arbitrary convex increasing function. To prove the
formula, we use a regularization and thus suppose that $\Theta$,
$\varphi$ and $\chi$ are smooth, and that $t$ is a non critical value of
$\varphi$. Then Stokes' formula shows that the integrals on the left and
right hand side of (2.13) are equal respectively to
$$\int_{\varphi(z)=t}\Theta\wedge(dd^c\chi\circ\varphi\big)^{p-1}
\wedge d^c(\chi\circ\varphi),~~~~
\int_{\varphi(z)=t}\Theta\wedge\big(dd^c\varphi\big)^{p-1}
\wedge d^c\varphi,$$
and the differential form of bidegree $(p-1,p)$ appearing in the integrand
of the first integral is equal to
$(\chi'\circ\varphi)^p\,(dd^c\varphi)^{p-1}\wedge d^c\varphi$.
The expected formula follows. Part b) of Theorem 2.8 is a consequence of
the Jensen-Lelong formula, whose proof is left as an exercise to the reader.

\claim{(2.14) Jensen-Lelong formula} Let $u$ be any psh function 
on~$X$. Then $u$ is integrable with respect to the measure
$\mu_r=(dd^c\varphi)^{n-1}\wedge d^c\varphi$ supported by the
pseudo-sphere $\{\varphi(z)=r\}$ and
$$\mu_r(u)=\int_{\{\varphi<r\}}u(dd^c\varphi)^n+
\int_{-\infty}^r\nu(dd^cu,\varphi,t)\,dt.\eqno\square$$
\endclaim

In our case, we set $\varphi(z)=\log|z-x|$. Then $(dd^c\varphi)^n=\delta_x$
and $\mu_r$ is just the unitary invariant mean value measure on the
sphere $S(x,e^r)$. For $r<r_0$, Formula 2.14 implies
$$\mu_r(u)-\mu_{r_0}(u)=\int_{r_0}^r\nu(dd^cu,x,t)\sim
(r-r_0)\nu(dd^cu,x)\qquad\hbox{as $r\to-\infty$}.$$
From this, using the Harnack inequality for subharmonic functions, we get
$$\liminf_{z\to x}{u(z)\over\log|z-x|}=
\lim_{r\to-\infty}{\mu_r(u)\over r}=\nu(dd^cu,x).$$
These equalities imply statement 2.8~b).

Next, we show that the Lelong numbers $\nu(T,\varphi)$ only
depend on the asymptotic behaviour of $\varphi$ near the polar set
$\varphi^{-1}(-\infty)$. In a precise way:

\claim{(2.15) Comparison theorem} Let $\Theta$ be a closed positive
current on~$X$, and let $\varphi,\psi:X\to[-\infty,+\infty[$
be continuous psh functions with isolated poles at some point
$x\in X$. Assume that
$$\ell:=\limsup_{z\to x}{\psi(z)\over\varphi(z)}<+\infty.$$
Then $\nu(\Theta,\psi)\le \ell^p\nu(\Theta,\varphi)$, and the equality holds
if $\ell=\lim\psi/\varphi$.
\endclaim

\begproof{} (2.12) shows that
$\nu(\Theta,\lambda\varphi)=\lambda^p\nu(\Theta,\varphi)$
for every positive constant $\lambda$. It is thus sufficient to verify the
inequality $\nu(\Theta,\psi)\le\nu(\Theta,\varphi)$ under the hypothesis 
$\limsup\psi/\varphi<1$. For any $c>0$, consider the psh function 
$$u_c=\max(\psi-c,\varphi).$$
Fix $r\ll 0$. For $c>0$ large enough, we have $u_c=\varphi$ on a
neighborhood of $\varphi^{-1}(r)$ and Stokes' formula gives
$$\nu(\Theta,\varphi,r)=\nu(\Theta,u_c,r)\ge\nu(\Theta,u_c).$$
On the other hand, the hypothesis $\limsup\psi/\varphi<1$ implies that
there exists $t_0<0$ such that $u_c=\psi-c$ on $\{u_c<t_0\}$. We thus get 
$$\nu(\Theta,u_c)=\nu(\Theta,\psi-c)=\nu(\Theta,\psi),$$
hence $\nu(\Theta,\psi)\le\nu(\Theta,\varphi)$. The equality case is obtained
by reversing the roles of $\varphi$ and $\psi$ and observing that
$\lim\varphi/\psi=1/l$.\qed
\endproof

Part a) of Theorem~2.10 follows immediately from 2.15 by considering
the weights $\varphi(z)=\log|\tau(z)-\tau(x)|$, $\psi(z)=\log
|\tau'(z)-\tau'(x)|$ associated to coordinates systems
$\tau(z)=(z_1\ld z_n)$, $\tau'(z)=(z'_1\ld z'_n)$ in a neighborhood
of~$x$. Another application is a direct simple proof of Thie's Theorem
2.9 when $\Theta=[A]$ is the current of integration over an
analytic set $A\subset X$ of pure dimension~$p$. For this, we have to
observe that Theorem 2.15 still holds provided that $x$ is an isolated
point in $\Supp(\Theta)\cap\varphi^{-1}(-\infty)$ and
$\Supp(\Theta)\cap\psi^{-1}(-\infty)$ (even though $x$ is not isolated in
$\varphi^{-1}(-\infty)$ or $\psi^{-1}(-\infty)$), under the weaker assumption
that $\limsup_{\Supp(\Theta)\ni z\to x}\psi(z)/\varphi(z)=\ell$. The reason
for this is that all integrals involve currents supported on $\Supp(\Theta)$.
Now, by a generic choice of local coordinates \hbox{$z'=(z_1\ld z_p)$}
and \hbox{$z''=(z_{p+1}\ld z_n)$} on $(X,x)$, the germ $(A,x)$ is contained
in a cone \hbox{$|z''|\le C|z'|$}. If
$B'\subset\bC^p$ is a ball of center $0$ and radius $r'$ small,
and \hbox{$B''\subset\bC^{n-p}$} is the ball of center $0$ and
radius~$r''=Cr'$, the projection
      $$\pr:A\cap(B'\times B'')\longrightarrow B'$$
is a ramified covering with finite sheet number $m$.
When $z\in A$ tends to $x=0$, the functions
$$\varphi(z)=\log|z|=\log(|z'|^2+|z''|^2)^{1/2},~~~~\psi(z)=\log|z'|.$$
satisfy $\lim_{z\to x}\psi(z)/\varphi(z)=1$. Hence Theorem~2.15 implies
      $$\nu([A],x)=\nu([A],\varphi)=\nu([A],\psi).$$
Now, Formula 2.13 with $\chi(t)=e^{2t}$ yields
$$\eqalign{\nu([A],\psi,\log t)
&=t^{-2p}\int_{\{\psi<\log t\}}[A]\wedge\Big({1\over 2}dd^ce^{2\psi}\Big)^p\cr
&=t^{-2p}\int_{A\cap\{|z'|<t\}}\Big({1\over 2}\pr^\star dd^c|z'|^2\Big)^p\cr
&=m\,t^{-2p}\int_{\bC^p\cap\{|z'|<t\}}\Big({1\over 2}dd^c|z'|^2\Big)^p=m,
\cr}$$
hence $\nu([A],\psi)=m$.  Here, we have used the fact that pr is an \'etale
covering with $m$ sheets over the complement of the ramification locus
$S\subset B'$, and the fact that $S$ is of zero Lebesgue measure in~$B'$.

\claim{(2.16) Proposition} Under the assumptions of Proposition 2.3,
we have
$$\nu(dd^cu\wedge\Theta,x)\ge\nu(u,x)\,\nu(\Theta,x)$$
at every point $x\in X$.
\endclaim

\begproof{} Assume that $X=B(0,r)$ and $x=0$. By definition
$$\nu(dd^c u\wedge\Theta,x)=
\lim_{r\to 0}\int_{|z|\le r}dd^cu\wedge\Theta\wedge(dd^c\log|z|)^{p-1}.$$
Set $\gamma=\nu(u,x)$ and
$$u_\nu(z)=\max\big(u(z),(\gamma-\varepsilon)\log|z|-\nu\big)$$
with $0<\varepsilon<\gamma$ (if $\gamma=0$, there is nothing to prove).
Then $u_\nu$ decreases to $u$ and
$$\int_{|z|\le r}\kern-7pt dd^c u\wedge\Theta\wedge(dd^c\log|z|)^{p-1}\ge
\limsup_{\nu\to+\infty}\int_{|z|\le r}\kern-7pt dd^cu_\nu\wedge
\Theta\wedge(dd^c\log|z|)^{p-1}$$
by the weak convergence of $dd^cu_\nu\wedge\Theta$; here 
$(dd^c\log|z|)^{p-1}$ is not
smooth on $\ol B(0,r)$, but the integrals remain unchanged if we replace
$\log|z|$ by $\chi(\log|z|/r)$ with a smooth convex function $\chi$ such
that $\chi(t)=t$ for $t\ge -1$ and $\chi(t)=0$ for $t\le -2$. Now, we have
$u(z)\le\gamma\log|z|+C$ near $0$, so $u_\nu(z)$ coincides with
$(\gamma-\varepsilon)\log|z|-\nu$ on a small ball $B(0,r_\nu)\subset
B(0,r)$ and we infer
$$\eqalign{
\int_{|z|\le r}dd^c u_\nu\wedge\Theta\wedge 
(dd^c\log|z|)^{p-1}&\ge(\gamma-\varepsilon)
\int_{|z|\le r_\nu}\Theta\wedge(dd^c\log|z|)^p\cr
&\ge(\gamma-\varepsilon)\nu(\Theta,x).\cr}$$
As $r\in{}]0,R[$ and $\varepsilon\in{}]0,\gamma[$ were arbitrary, the
desired inequality follows.\qed
\endproof

We will later need an important decomposition formula of [Siu74].
We start with the following lemma.

\claim{(2.17) Lemma} If $\Theta$ is a closed
positive current of bidimension $(p,p)$ and $Z$ is an irreducible
analytic set in $X$, we set
$$m_Z=\inf\{x\in Z\,;\,\nu(\Theta,x)\}.$$
\item{\rm a)} There is a countable family of proper analytic subsets
$(Z'_j)$ of $Z$ such that $\nu(\Theta,x)=m_Z$ for all
$x\in Z\ssm\bigcup Z'_j$. We say that $m_Z$ is the generic Lelong number
of $\Theta$ along $Z$.
\smallskip
\item{\rm b)} If $\dim Z=p$, then $\Theta\ge m_Z[Z]$ and
$\bOne_Z\Theta=m_Z[Z]$.\vskip0pt
\endclaim

\begproof{} a) By definition of $m_Z$ and $E_c(\Theta)$, we
have $\nu(\Theta,x)\ge m_Z$ for every $x\in Z$ and 
$$\nu(\Theta,x)=m_Z~~~~\hbox{\rm on}~~
Z\ssm\bigcup_{c\in\bQ,\,c>m_Z}Z\cap E_c(\Theta).$$
However, for $c>m_Z$, the intersection $Z\cap E_c(\Theta)$ is a proper
analytic subset of~$A$.

\noindent b) Left as an exercise to the reader. It is enough to prove
that $\Theta\ge m_Z[Z_\reg]$ at regular points of $Z$, so one may assume
that $Z$ is a $p$-dimensional linear subspace in~$\bC^n$. Show that
the measure $(\Theta-m_Z[Z])\wedge(dd^c|z|^2)^p$ has nonnegative mass
on every ball $|z-a|<r$ with center $a\in Z$. Conclude by using 
arbitrary affine changes of coordinates that $\Theta-m_Z[Z]\ge 0$.\qed
\endproof

\claim{(2.18) Decomposition formula {\rm([Siu74])}} Let $\Theta$
be a closed positive current of bidimension~$(p,p)$. Then
$\Theta$ can be written as a convergent series of
closed positive currents
$$\Theta=\sum_{k=1}^{+\infty}\lambda_k\,[Z_k]+R,$$
where $[Z_k]$ is a current of integration over an irreducible
analytic set of dimension~$p$, and $R$ is a residual current 
with the property that $\dim E_c(R)<p$ for every $c>0$.
This decomposition is locally and globally unique: the sets $Z_k$
are precisely the $p$-dimensional components occurring in the
sublevel sets $E_c(\Theta)$, and $\lambda_k=\min_{x\in Z_k}\nu(\Theta,x)$ is
the generic Lelong number of $\Theta$ along $Z_k$.
\endclaim

\begproof{of uniqueness.} If $\Theta$ has such a decomposition, the
$p$-dimensional components of $E_c(\Theta)$ are $(Z_j)_{\lambda_j\ge c}$,
for $\nu(\Theta,x)=\sum\lambda_j\nu([Z_j],x)+\nu(R,x)$ is non zero
only on $\bigcup Z_j\cup\bigcup E_c(R)$, and is equal to
$\lambda_j$ generically on $Z_j$ $\big($more precisely,
$\nu(\Theta,x)=\lambda_j$ at every regular
point of $Z_j$ which does not belong to any intersection $Z_j\cup Z_k$,
$k\ne j$ or to $\bigcup E_c(R)\big)$. In particular $Z_j$ and
$\lambda_j$ are unique.
\endproof

\begproof{of existence.} Let $(Z_j)_{j\ge 1}$ be the countable collection of
$p$-dimensional components occurring in one of the sets $E_c(\Theta)$,
$c\in\bQ_+^\star$, and let $\lambda_j>0$ be the generic Lelong
number of $\Theta$ along $Z_j$. Then Lemma~2.17 shows by induction on $N$
that $R_N=\Theta-\sum_{1\le j\le N}\lambda_j[Z_j]$ is positive. As $R_N$
is a decreasing sequence, there must be a
limit $R=\lim_{N\to+\infty}R_N$ in the weak topology. Thus we have
the asserted decomposition. By construction, $R$ has zero generic
Lelong number along $Z_j$, so $\dim E_c(R)<p$ for every $c>0$.\qed
\endproof

It is very important to note that some components of lower dimension
can actually occur in $E_c(R)$, but they cannot be subtracted because
$R$ has bidimension $(p,p)$. A typical case is the case of a
bidimension \hbox{$(n-1,n-1)$} current $\Theta=dd^cu$ with
$u=\log(|f_j|^{\gamma_1}+\ldots|f_N|^{\gamma_N})$ and
$f_j\in H^0(X,\cO_X)$. In general $\bigcup E_c(\Theta)=\bigcap f_j^{-1}(0)$
has dimension~$<n-1$. 

\claim{Corollary 2.19} Let $\Theta_j=dd^c u_j$, $1\le j\le p$,
be closed positive \hbox{$(1,1)$-currents} 
on a complex manifold $X$. Suppose that there are analytic sets 
$A_2\supset\ldots\supset A_p$ in $X$ with
$\codim A_j\ge j$ at every point such that each $u_j$, $j\ge 2$, 
is locally bounded on $X\ssm A_j$. Let $\{A_{p,k}\}_{k\ge 1}$ be the
irreducible components of $A_p$ of codimension $p$ exactly and let
$\nu_{j,k}=\min_{x\in A_{p,k}}\nu(\Theta_j,x)$ be the generic Lelong number
of $\Theta_j$ along $A_{p,k}$. Then $\Theta_1\wedge\ldots\wedge \Theta_p$
is well-defined and
$$\Theta_1\wedge\ldots\wedge \Theta_p\ge\sum_{k=1}^{+\infty}\nu_{1,k}\ldots
\nu_{p,k}\,[A_{p,k}].$$
\endclaim

\begproof{} By induction on $p$, Proposition~2.3 shows that 
$\Theta_1\wedge\ldots\wedge \Theta_p$ is well defined. Moreover, 
Proposition~2.16 implies 
$$\nu(\Theta_1\wedge\ldots\wedge \Theta_p,x)\ge
\nu(\Theta_1,x)\ldots\nu(\Theta_p,x)\ge\nu_{1,k}\ldots\nu_{p,k}$$
at every point $x\in A_{p,k}$. The desired inequality is then a consequence
of Siu's decomposition theorem.\qed
\endproof

\titleb{3.}{Hermitian Vector Bundles, Connections and Curvature}
The goal of this section is to recall the most basic definitions of
hemitian differential geometry related to the concepts of connection,
curvature and first Chern class of a line bundle.

Let $F$ be a complex vector bundle of rank $r$ over a smooth
differentiable manifold~$M$. A {\it connection} $D$ on $F$ is a
linear differential operator of order $1$
$$D:C^\infty(M,\Lambda^qT^\star_M\otimes F) \to C^\infty(M,
\Lambda^{q+1}T^\star_M\otimes F)$$
such that
$$D(f\wedge u) = df\wedge u + (-1)^{\deg\,f}f\wedge Du\leqno(3.1)$$
for all forms $f\in C^\infty(M,\Lambda^pT^\star_M)$, $u\in C^\infty(X,
\Lambda^qT^\star_M\otimes F)$. On an open set $\Omega \subset M$ where $F$
admits a trivialization $\theta:F_{|\Omega }\buildo\lra^\simeq \Omega
\times \bC^r$, a connection $D$ can be written
$$Du\simeq_\theta du+\Gamma \wedge u$$
where $\Gamma \in C^\infty(\Omega ,\Lambda^1T^\star_M\otimes\Hom
(\bC^r,\bC^r))$ is an arbitrary matrix of $1$-forms and $d$ acts
componentwise. It is then easy to check that
$$D^2u\simeq_\theta (d\Gamma +\Gamma \wedge \Gamma )\wedge u\quad
\hbox{on~}\Omega.$$
Since $D^2$ is a globally defined operator, there is a global $2$-form
$$
\Theta(D) \in C^\infty(M,\Lambda^2T^\star_M\otimes\Hom_\bC(F,F))
\leqno(3.2)
$$
such that $D^2u=\Theta(D)\wedge u$ for every form $u$ with values in~$F$.

Assume now that $F$ is endowed with a $C^\infty$ hermitian metric $h$ along
the fibers and that the isomorphism $F_{|\Omega }\simeq \Omega \times
\bC^r$ is given by a $C^\infty$ frame $(e_\lambda)$. We then have
a canonical sesquilinear pairing $\{\bu,\bu\}=\{\bu,\bu\}_h$
$$\leqalignno{
C^\infty(M,\Lambda^pT^\star_M\otimes F)\times
C^\infty(M,\Lambda^qT^\star_M\otimes F)
&\longrightarrow C^\infty(M,\Lambda^{p+q}T^\star_M\otimes\bC)&(3.3)\cr
(u,v) &\longmapsto \{u,v\}_h\cr}$$
given by
$$\{u,v\}_h=\sum_{\lambda ,\mu }u_\lambda \wedge \overline v_\mu \langle
e_\lambda,e_\mu \rangle_h,\qquad
u=\sum u_\lambda \otimes e_\lambda,\quad 
v=\sum v_\mu \otimes e_\mu.$$
We will frequently omit the subscript $h$ when no confusion can arise.
The connection $D$ is said to be {\it hermitian} (with respect to $h$) 
if it satisfies the additional property
$$d\{u,v\}=\{Du,v\}+(-1)^{{\rm deg~}u}\{u,Dv\}.$$
Assuming that $(e_\lambda )$ is orthonormal, one easily checks that $D$
is hermitian if and only if $\Gamma^\star=-\Gamma$. In this case
$\Theta(D)^\star=-\Theta(D)$ [$\,$observe that $(\Gamma\wedge\Gamma)^*=-
\Gamma^*\wedge\Gamma^*$ and more generally 
$(A\wedge B)^*=-B^*\wedge A^*$ for products of matrices of $1$-forms,
since reversing the order of the product of $1$-forms changes the sign$\,$].
Therefore the $2$-form $\ii\Theta(D)$ takes values in hermitian 
symmetric tensors $\Herm(F,F)$, i.e.
$$\ii\Theta(D) \in C^\infty(M,\Lambda^2 T^\star_M\otimes_\bR\Herm(F,F))$$
where $\Herm(F,F)\subset \Hom(F,F)$ is the real subspace of hermitian
endomorphisms.
\medskip

\claim{(3.4) Special case} \rm For a bundle $F$ of rank 1, the
connection form $\Gamma$ of a hermitian connection $D$ can be seen as
a $1$-form with purely imaginary coefficients (i.e.\ $\Gamma=\ii A$, 
$A$ real).
Then we have $\Theta(D)=d\Gamma=\ii dA$. In particular $\ii\Theta(F)$
is a closed 2-form. The {\it first Chern class} of $F$
is defined to be the cohomology class
$$c_1(F)_\bR=\Big\{{\ii\over 2\pi}\Theta(D)\Big\}\in H^2_\DR(M,\bR).$$
The cohomology class is actually independent of the connection, since
any other connection $D_1$ differs by a global 1-form, $D_1u=Du+B\wedge u$,
so that $\Theta(D_1)=\Theta(D)+dB$. It is well-known that $c_1(F)_\bR$
is the image in $H^2(M,\bR)$ of an integral class $c_1(F)\in
H^2(M,\bZ)\,$; by using the exponential exact sequence
$$0\to\bZ\to\cE\to\cE^\star\to 0,$$
$c_1(F)$ can be defined in \v Cech cohomology theory as the image
by the coboundary map $H^1(M,\cE^\star)\to H^2(M,\bZ)$ of the cocycle
$\{g_{jk}\}\in H^1(M,\cE^\star)$ defining $F\,$; see e.g.\ [GrH78] for
details.\qed
\endclaim

We now concentrate ourselves on the complex analytic case. If $M=X$ is a
complex manifold $X$, every connection $D$ on a complex $\ci$ vector
bundle $F$ can be splitted in a unique way as a
sum of a $(1,0)$ and of a $(0,1)$-connection, $D=D'+D''$. In a local
trivialization $\theta$ given by a $C^\infty$ frame, one can write 
$$\leqalignno{
D'u &\simeq_\theta d'u + \Gamma '\wedge u,&(3.5')\cr
D''u &\simeq_\theta d''u + \Gamma ''\wedge u,&(3.5'')\cr}$$
with $\Gamma =\Gamma '+\Gamma ''$. The connection is hermitian if and
only if $\Gamma '=-(\Gamma '')^\star$ in any orthonormal frame. Thus there
exists a unique hermitian connection $D$ corresponding to a prescribed
$(0,1)$ part $D''$. 

Assume now that the hermitian bundle $(F,h)$ itself has 
a {\it holomorphic} structure.
The unique hermitian connection $D_h$ for which $D''_h$ is the $d''$ operator
defined in \S~1 is called the {\it Chern connection} of $F$. In a local
holomorphic frame $(e_\lambda )$ of $E_{|\Omega }$, the metric $h$ is then 
given by a hermitian matrix $H=(h_{\lambda\mu})$, $h_{\lambda\mu}
=\langle e_\lambda,e_\mu \rangle$. We have
$$\{u,v\}=\sum_{\lambda,\mu}h_{\lambda\mu}u_\lambda \wedge
\ol v_\mu=u^\dagger\wedge H\ol v,$$
where $u^\dagger$ is the transposed matrix of $u$. Easy computations yield
$$\eqalign{d\{u,v\}
&=(du)^\dagger \wedge H\ol v+(-1)^{\deg u} u^\dagger
\wedge (dH\wedge \ol v + H\ol{dv})\cr
&=\big(du +\ol H{\,}^{-1}d'\ol H \wedge u\big)^\dagger \wedge H\ol v
+(-1)^{\deg u}u^\dagger\wedge(\ol{dv+ \ol H{\,}^{-1}d'\ol H\wedge v)}\cr}$$
using the fact that $dH = d'H+\ol{d'\ol H}$ and
$\ol H^\dagger=H$. Therefore the Chern connection $D_h$ coincides with the
hermitian connection defined by
$$
\left\{\eqalign{
D_hu&{}\simeq_\theta du+\ol H{\,}^{-1}d'\ol H \wedge u,\cr
D_h'&{}\simeq_\theta d'+\ol H{\,}^{-1} d'\ol H\wedge\bu=
\ol H{\,}^{-1} d'(\ol H\bu),~~~~D''_h = d''.\cr}\right.\leqno(3.6)
$$
It is clear from the above relations (3.6) that $D_h^{\prime 2}=
D_h^{\prime\prime 2}=0$.  Consequently $D_h^2$ is given by to
\hbox{$D_h^2=D_h'D_h''+D_h''D_h'$}, and the curvature tensor 
$\Theta(D_h)$ is of type~$(1,1)$. Since $d'd''+d''d'=0$, we get
$$
\eqalign{
(D_h'D_h''+D_h''D_h')u&\simeq_\theta\ol H{\,}^{-1}d'\ol H\wedge d''u 
+d''(\ol H{\,}^{-1}d'\ol H\wedge u)\cr
&=d''(\ol H{\,}^{-1}d'\ol H)\wedge u.\cr}
$$
By the above calculation $\Theta(D_h)$ is given by the  matrix of $(1,1)$-forms
$$
\Theta(D_h)\simeq_\theta d''(\ol H{\,}^{-1}d'\ol H)=\ol
H{\,}^{-1}d''d'\ol H -\ol H{\,}^{-1}d''\ol H\wedge \ol
H{\,}^{-1}d'\ol H
$$
Since $H=\ol H^\dagger$ is hermitian symmetric
and transposition reverses products, we find again in this setting 
that $\Theta(D_h)$ is hermitian skew symmetric
$$
\Theta(D_h)^* \simeq_\theta\ol H{\,}^{-1}
\ol{\Theta(F)}^\dagger \ol H=-\Theta(D_h).
$$

\claim{(3.7) Definition and proposition} The Chern curvature tensor of
$(F,h)$ is defined to be $\Theta_h(F):=\Theta(D_h)$. It is such that
$$
\ii\Theta_h(F) \in 
C^\infty\big(X,\Lambda^{1,1}_\bR T^\star_X\otimes_\bR \Herm(F,F)\big)\subset
C^\infty\big(X,\Lambda^{1,1} T^\star_X\otimes_\bC \Hom(F,F)\big).
$$
If $\theta: F_{\restriction\Omega}\to\Omega\times\bC^r$ is a holomorphic
trivialization and if $H$ is the hermitian matrix representing the
metric along the fibers of $F_{\restriction\Omega}$, then
$$\ii\Theta_h(F)\simeq_\theta
\ii d''(\ol H{\,}^{-1} d'\ol H)\quad\hbox{\rm on}~\Omega.\eqno\square$$
$[\,$We will frequently omit the subscript $h$ and write simply $D_h=D$,
$\Theta_h(F)=\Theta(F)$ when no confusion can arise$\,]$.
\endclaim

The next proposition shows that the Chern curvature tensor is the
obstruction to the existence of orthonormal holomorphic frames:
a holomorphic frame can be made ``almost orthonormal" only up to
curvature terms of order $2$ in a neighborhood of any point.

\claim{(3.8) Proposition} For every point $x_0\in X$ and every holomorphic
coordinate system $(z_j)_{1\le j\le n}$ at $x_0$, there exists a
holomorphic frame $(e_\lambda)_{1\le\lambda\le r}$ of $F$ in a neighborhood 
of $x_0$ such that
$$\langle e_\lambda(z),e_\mu(z)\rangle=\delta_{\lambda\mu}- 
\sum_{1\le j,k\le n}c_{jk\lambda\mu}\,z_j\ol z_k+O(|z|^3)$$ 
where $(c_{jk\lambda\mu})$ are the coefficients of the Chern curvature
tensor $\Theta(F)_{x_0}$, namely
$$\ii\Theta(F)=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}dz_j\wedge dz_k\otimes e_\lambda^\star\otimes e_\mu.$$
Such a frame $(e_\lambda)$ is called a normal
coordinate frame at~$x_0$.
\endclaim

\begproof{} Let $(h_\lambda)$ be a holomorphic frame of $F$. After replacing
$(h_\lambda)$ by suitable linear combinations with constant
coefficients, we may assume that $\big(h_\lambda(x_0)\big)$ is an
orthonormal basis of $F_{x_0}$.  Then the inner products 
$\langle h_\lambda,h_\mu\rangle$ have an expansion 
$$\langle h_\lambda(z),h_\mu(z)\rangle=\delta_{\lambda\mu}+
\sum_j(a_{j\lambda\mu}\,z_j+a'_{j\lambda\mu}\,\ol z_j)+O(|z|^2)$$
for some complex coefficients $a_{j\lambda\mu}$, $a'_{j\lambda\mu}$ such
that $a'_{j\lambda\mu}=\ol a_{j\mu\lambda}$.  Set first
$$g_\lambda(z)=h_\lambda(z)-\sum_{j,\mu}a_{j\lambda\mu}\,z_j\,h_\mu(z).$$
Then there are coefficients $a_{jk\lambda\mu}$, $a'_{jk\lambda\mu}$,
$a''_{jk\lambda\mu}$ such that
$$\eqalign{
\langle g_\lambda(z),g_\mu(z)\rangle&=\delta_{\lambda\mu}+O(|z|^2)\cr
&=\delta_{\lambda\mu}+\sum_{j,k}\big(a_{jk\lambda\mu}\,z_j\ol z_k+
a'_{jk\lambda\mu}\,z_jz_k+a''_{jk\lambda\mu}\ol z_j\ol z_k\big)+
O(|z|^3).}$$ 
The holomorphic frame $(e_\lambda)$ we are looking for is 
$$e_\lambda(z)=
g_\lambda(z)-\sum_{j,k,\mu}a'_{jk\lambda\mu}\,z_jz_k\,g_\mu(z).$$
Since $a''_{jk\lambda\mu}=\ol a'_{jk\mu\lambda}$, we easily find
$$\eqalign{
\langle e_\lambda(z),e_\mu(z)\rangle&=\delta_{\lambda\mu}+
\sum_{j,k}a_{jk\lambda\mu}\,z_j\ol z_k+O(|z|^3),\cr
d'\langle e_\lambda,e_\mu\rangle&=\{D'e_\lambda,e_\mu\}=
\sum_{j,k}a_{jk\lambda\mu}\,\ol z_k\,dz_j+O(|z|^2),\cr
\Theta(F)\cdot e_\lambda
&=D''(D'e_\lambda)=\sum_{j,k,\mu}a_{jk\lambda\mu}\,d\ol z_k\wedge dz_j
\otimes e_\mu+O(|z|),\cr}$$ 
therefore $c_{jk\lambda\mu}=-a_{jk\lambda\mu}$.\qed
\endproof

According to (3.8) and (3.9), one can identify the curvature tensor of
$F$ to a hermitian form
$$\wt\Theta(F)(\xi\otimes v)=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}\xi_j\ol\xi_kv_\lambda\ol v_\mu\leqno(3.9)$$
on $T_X\otimes F$. This leads in a natural way to positivity concepts,
following definitions introduced by Kodaira [Kod53], Nakano [Nak55] and
Griffiths [Gri69].

\claim{(3.10) Definition} The hermitian vector bundle $F$ is said to be
\smallskip
\item{\rm a)} positive in the sense of Nakano if
$\wt\Theta(F)(\tau)>0$ for all non zero tensors $\tau=\sum
\tau_{j\lambda}\partial/\partial z_j\otimes e_\lambda\in T_X\otimes F$.
\smallskip
\item{\rm b)} positive in the sense of Griffiths if
$\wt\Theta(F)(\xi\otimes v)>0$ for all non zero decomposable tensors
$\xi\otimes v\in T_X\otimes F\,;$
\smallskip\noindent
Corresponding semipositivity concepts are defined by relaxing the strict
inequalities.
\endclaim

\claim{(3.11) Special case of rank 1 bundles} \rm Assume that
$F$ is a line bundle. The hermitian matrix $H=(h_{11})$ associated
to a trivialization $\theta:F_{\restriction\Omega}\simeq\Omega\times\bC$
is simply a positive function which we find convenient to denote by
$e^{-2\varphi}$, \hbox{$\varphi\in C^\infty(\Omega,\bR)$}. In this case
the curvature form $\Theta_h(F)$ can be identified to the $(1,1)$-form
$2d'd''\varphi$, and
$$
{\ii\over 2\pi}\Theta_h(F)={\ii\over\pi}d'd''\varphi=dd^c\varphi
$$
is a real $(1,1)$-form. Hence $F$ is semipositive (in either the Nakano or
Griffiths sense) if and only if $\varphi$ is psh, resp.\ positive if
and only if $\varphi$ is {\it strictly psh}. In this setting, the
Lelong-Poincar\'e equation can be generalized as follows: let
$\sigma\in H^0(X,F)$ be a non zero holomorphic section. Then
$$
dd^c\log\|\sigma\|_h=[Z_\sigma]-{\ii\over 2\pi}\Theta_h(F).\leqno(3.12)
$$
Formula (3.12) is immediate if we write $\|\sigma\|_h=|\theta(\sigma)|
e^{-\varphi}$ and if we apply (1.20) to the holomorphic function
$f=\theta(\sigma)$. As we shall see later, it is very important for
the applications to consider also singular hermitian metrics.
\endclaim

\claim{(3.13) Definition} A singular $($hermitian$)$ metric $h$ on a line
bundle $F$ is a metric $h$ which is given in any trivialization
$\theta:F_{\restriction\Omega}
\buildo \lra^\simeq \Omega\times\bC$ by
$$\|\xi\|_h=|\theta(\xi)|\,e^{-\varphi(x)},~~~~x\in\Omega,~\xi\in F_x$$
where $\varphi$ is an arbitrary measurable function 
in $L^1_\loc(\Omega)$, called the
weight of the metric with respect to the trivialization $\theta$.
\endclaim

If $\theta':F_{\restriction\Omega'}\lra\Omega'\times\bC$ 
is another trivialization, $\varphi'$ the associated weight and
$g\in\cO^\star(\Omega\cap\Omega')$ the transition function, then
$\theta'(\xi)=g(x)\,\theta(\xi)$ for $\xi\in F_x$, and so
$\varphi'=\varphi+\log|g|$ on $\Omega\cap\Omega'$. The curvature form 
of $F$ is then given formally by the closed $(1,1)$-current
${\ii\over 2\pi}\Theta_h(F)=dd^c\varphi$ on $\Omega\,$; our assumption
$\varphi\in L^1_\loc(\Omega)$ guarantees that $\Theta_h(F)$ exists in
the sense of distribution theory. As in the smooth case, ${\ii\over 2\pi}
\Theta_h(F)$ is globally defined on $X$ and independent of the choice of
trivializations, and its De Rham cohomology class is the image of the first
Chern class $c_1(F)\in H^2(X,\bZ)$ in $H^2_{DR}(X,\bR)$.
Before going further, we discuss two basic examples. 

\claim{(3.14) Example} \rm Let $D=\sum\alpha_jD_j$ be a divisor with 
coefficients $\alpha_j\in\bZ$ and let $F=\cO(D)$ be the associated
invertible sheaf of meromorphic functions $u$ such that
${\rm div}(u)+D\ge 0\,$; the corresponding line bundle can be
equipped with the singular metric defined by $\|u\|=|u|$. If $g_j$ is a
generator of the ideal of $D_j$ on an open set $\Omega\subset X$ then 
$\theta(u)=u\prod\smash{g_j^{\alpha_j}}$ defines a trivialization of
$\cO(D)$ over $\Omega$, thus our singular metric is associated to the
weight $\varphi=\sum\alpha_j\log|g_j|$. By the Lelong-Poincar\'e equation,
we find
$${\ii\over 2\pi}\Theta\big(\cO(D)\big)=dd^c\varphi=[D],$$
where $[D]=\sum\alpha_j[D_j]$ denotes the current of integration
over~$D$.\qed
\endclaim

\claim{(3.15) Example} \rm Assume that $\sigma_1\ld \sigma_N$ are non zero 
holomorphic sections of $F$. Then we can define a natural (possibly
singular) hermitian metric $h^*$
on $F^\star$ by
$$\|\xi^\star\|^2_{h^*}=\sum_{1\le j\le n}\big|\xi^\star.\sigma_j(x)\big|^2
~~~~\hbox{\rm for}~~\xi^\star\in F^\star_x.$$
The dual metric $h$ on $F$ is given by
$$\|\xi\|^2_h={|\theta(\xi)|^2\over|\theta(\sigma_1(x))|^2+\ldots+
|\theta(\sigma_N(x))|^2}
\leqno(3.15\,{\rm a})
$$
with respect to any trivialization $\theta$. The associated weight function is
thus given by $\varphi(x)=\log\big(\sum_{1\le j\le N}|\theta(\sigma_j(x))|^2
\big){}^{1/2}$. In this case $\varphi$ is a psh function, thus
$\ii\Theta_h(F)$ is a closed positive current, given explicity by
$$
{\ii\over 2\pi}\Theta_h(F)=dd^c\varphi={\ii\over 2\pi}
\ddbar\log\Big(\sum_{1\le j\le N}|\theta(\sigma_j(x))|^2\Big).
\leqno(3.15\,{\rm b})
$$
Let us denote by $\Sigma$
the linear system defined by $\sigma_1\ld \sigma_N$ and by
$B_\Sigma=\bigcap\sigma_j^{-1}(0)$ its base locus. We have a meromorphic
map
$$\Phi_\Sigma:X\ssm B_\Sigma\to\bP^{N-1},\qquad
x\mapsto(\sigma_1(x):\sigma_2(x):\ldots:\sigma_N(x)).$$
Then ${\ii\over 2\pi}\Theta(F)$ is equal to the pull-back 
by $\Phi_\Sigma$ over $X\ssm B_\Sigma$ of the so called 
{\it Fubini-Study metric} on~$\bP^{N-1}$ 
(see also (4.15) in the next section)~:
$$
\omega_\FS={\ii\over 2\pi}
\ddbar\log(|z_1|^2+\ldots+ |z_N|^2)
\leqno\rlap{\kern\hsize\llap{\hbox{$\square$}}}(3.15\,{\rm c})
$$
\endclaim

\claim{(3.16) Ample and very ample line bundles} A holomorphic line
bundle $F$ over a compact complex manifold $X$ is said to be
\smallskip
\item{\rm a)} very ample if the map $\Phi_{|F|}:X\to\bP^{N-1}$ associated
to the complete linear system $|F|=P(H^0(X,F))$ is a regular embedding
$($by this we mean in particular that the base locus is empty, i.e.\
$B_{|F|}=\emptyset)$.
\smallskip
\item{\rm b)} ample if some multiple $mF$, $m>0$, is very ample.
\endclaim

Here we use an additive notation for $\Pic(X)=H^1(X,\cO^\star)$, hence
the symbol $mF$ denotes the line bundle $F^{\otimes m}$. By
Example~3.15, every ample line bundle $F$ has a smooth hermitian metric
with positive definite curvature form; indeed, if the linear system
$|mF|$ gives an embedding in projective space, then we get a smooth
hermitian metric on $F^{\otimes m}$, and the $m$-th root yields a
metric on $F$ such that ${\ii\over 2\pi}\Theta(F)={1\over m}
\Phi_{|mF|}^\star\omega_\FS$. Conversely, the Kodaira embedding theorem
[Kod54] tells us that every positive line bundle $F$ is ample (see
Exercise~6.14 for a straightforward analytic proof of the Kodaira
embedding theorem).

\titleb{4.}{K\"ahler identities and Hodge Theory}
We briefly explain here the most basic facts of $L^2$ Hodge theory. Assume for
the moment that $M$ is a differentiable manifold equipped with a Riemannian
metric $g=\sum g_{ij}dx_i\otimes dx_j$. Given a $q$-form $u$ on $M$ with
values in a hermitian vector bundle $(F,h)$, we consider the global
$L^2$ norm
$$
\|u\|^2 = \int_M |u(x)|^2dV_g(x)
\leqno(4.1)
$$
where $|u|=|u|_{g,h}$ is the pointwise hermitian norm on 
$\Lambda^q T^*_M\otimes F$ associated with $g,\,h$ and $dV_g$ is the
Riemannian volume form. Let $D$ be a hermitian connection on~$(F,h)$.
The Laplace-Beltrami operator associated with the connection $D$ is 
defined to be
$$
\Delta =DD^\star+D^\star D
\leqno(4.2)
$$
where
$$
D^\star:\ci(M,\Lambda^qT^\star_M\otimes F)\to
\ci(M,\Lambda^{q-1}T^\star_M\otimes F)
$$
is the (formal) adjoint of $D$ with respect to the $L^2$ inner product.
More generally, recall that if $P:\ci(M,E)\to\ci(M,F)$
is a differential operator and both $E$, $F$ are euclidean or hermitian
vector bundles,
there exists a unique differential operator
$$P^\star:\ci(M,F)\to\ci(M,E),$$
called the {\it formal adjoint} of $P$, such that for all sections
$u\in\ci(M,E)$ and $v\in\ci(M,F)$ there is an identity
$$
\Ll Pu,v\Rr=\Ll u,P^\star v\Rr,\qquad\hbox{\rm whenever
$\Supp u\cap\Supp v\compact M$}.
\leqno(4.3)
$$
In fact, let
$$
Pu(x)=\sum_{|\alpha|\le\delta}a_\alpha(x)D^\alpha u(x)
\leqno(4.4)
$$
be the expansion of $P$ with respect to trivializations of $E$, $F$ given
by orthonormal frames over some coordinate open set $\Omega\subset M$ and
let $dV_g(x)=\gamma(x)\,dx_1\ldots dx_n$ be the volume element. When
\hbox{$\Supp u\cap\Supp v\compact\Omega$}, an integration by parts
yields
$$\eqalign{
\Ll Pu,v\Rr&=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu}
a_{\alpha\lambda\mu}D^\alpha u_\mu(x)\ol v_\lambda(x)\,\gamma(x)\,
dx_1\ld dx_m\cr
&=\int_\Omega\sum_{|\alpha|\le\delta,\lambda,\mu}(-1)^{|\alpha|}
u_\mu(x)\ol{D^\alpha(\gamma(x)\,\ol a_{\alpha\lambda\mu}v_\lambda(x)}
\,dx_1\ld dx_m\cr
&=\int_\Omega\langle u,\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1}
D^\alpha\big(\gamma(x)\,{}^t\ol a_\alpha v(x)\big)\rangle\,dV(x).\cr}$$
Hence we see that $P^\star$ exists and is uniquely defined by
$$
P^\star v(x)=\sum_{|\alpha|\le\delta}(-1)^{|\alpha|}\gamma(x)^{-1}D^\alpha
\big(\gamma(x)\,{}^t\ol a_\alpha v(x)\big).
\leqno(4.5)$$
Now, assume that $M$ is {\it compact}. It can be checked that
$$\Delta:\ci(M,\Lambda^qT^\star_M\otimes F)\to
\ci(M,\Lambda^qT^\star_M\otimes F)$$
is a self-adjoint elliptic operator (for arbitrary $M$, $F$ and
arbitrary metrics on them). Standard results of
PDE theory show that there is an orthogonal decomposition
$$
\ci(M,\Lambda^qT^\star_M\otimes F)=\cH^q(M,F)\oplus\Im\Delta
\leqno(4.6)
$$
where $\cH^q(M,F)=\Ker\Delta$ is by definition the {\it space of harmonic 
forms of degree~$q$ in $F$} which, under these circumstances, must be 
a {\it finite dimensional space} by the ellipticity of~$P$ 
(This is the analogue of the theorem of finite
dimensional linear algebra which says that for a hermitian symmetric
operator, the kernel and the image are orthogonal and form a direct sum
decomposition of the space.) Notice also that by definition of
$\Delta=DD^*+D^*D$, we get
$$
\Ll \Delta u, u\Rr=\Vert Du\Vert^2+\Vert D^*u\Vert^2
\leqno(4.7)
$$
by means of an integration by parts. Therefore
the space $\cH^q(M,F)=\Ker\Delta$ consists of all forms 
$h$ such that $Dh=D^*h=0$ (This is not true if $M$ is non compact~:
for instance, on $M=\bC$, there are harmonic functions which are non 
constant$\,\ldots$).

\claim{(4.8) Basic $L^2$ decompositions}
Assume that the hermitian connection $D$ on $H$ is {\it integrable}, i.e.\ 
that $D^2=0$. Then we have orthogonal direct sums
$$
\leqalignno{
&\Im\Delta=\Im D\oplus\Im D^\star,
&(4.8\,{\rm a})\cr
&\ci(M,\Lambda^qT^\star_M\otimes F)=\cH^q(M,F)\oplus\Im D\oplus\Im D^\star,
&(4.8\,{\rm b})\cr
&\Ker D=\cH^q(M,F)\oplus\Im D.
&(4.8\,{\rm c})\cr}
$$
\endclaim

\begproof{} Indeed $\Ll Du,D^\star v\Rr=\Ll D^2u,v\Rr=0$ 
for all $u,v$, hence $\Im D\perp\Im D^*$. Likewise, we have 
$\Ll h,Du\Rr=\Ll D^*h,u\Rr=0$ and $\Ll h,D^*v\Rr=\Ll Dh,v\Rr=0$ 
for every harmonic form~$h$, hence
$\cH^q(M,F)\perp\Im D$ and $\cH^q(M,F)\perp\Im D^*$.
By the above and (4.6), it follows that
$$
\Im D\oplus\Im D^*\subset\cH^q(M,F)^\perp=\Im\Delta,
$$
and clearly we also have $\Im\Delta\subset \Im D+\Im D^*$,
since $\Delta=DD^*+D^*D$. Decompositions $(4.8\,{\rm a})$ and
$(4.8\,{\rm b})$ follow. Now, Let $w=h+Du+D^*v$ be an arbitrary element
in $\ci(M,\Lambda^qT^\star_M\otimes F)$ written according to decomposition
$(4.8\,{\rm b})$. We then find $Dw=DD^*v$, and since 
$\Ll Dw,v\Rr=\Ll DD^*v,v\Rr=\Vert D^*v\Vert^2$, we have $Dw=0$ if and only
if $D^*v=0$. This proves $(4.8\,{\rm c})$.\qed
\endproof

Notice that if $D^2=0$, one can define De Rham cohomology groups
$$
H^q_\DR(M,F)=\Ker D/\Im D
$$
which just generalize the usual De Rham cohomology groups associated with
the exterior derivative $d$ acting on the trivial bundle~$M\times\bC$.

\claim{(4.9) Corollary {\rm (Hodge Fundamental Theorem)}} 
If $M$ is a compact Riemannian manifold and $F$ a hermitian line bundle 
over $M$ with an integrable hermitian connection $D$,
there is an isomorphism
$$H^q_\DR(M,F)\simeq\cH^q(M,F)$$
from De Rham cohomology groups onto spaces of harmonic forms.
This applies in particular to the case of the trivial bundle
$F=M\times\bC$.\qed
\endclaim

A rather important consequence of the Hodge fundamental theorem is a proof
of the {\it Poincar\'e duality theorem}. Assume that the Riemannian manifold
$(M,g)$ is oriented. Then there
is a (conjugate linear) Hodge star operator
$$
\star:\Lambda^qT^\star_M\otimes\bC\to\Lambda^{m-q}T^\star_M\otimes\bC,
\qquad m=\dim_\bR M
$$
defined by $u\wedge\star v=\langle
u,v\rangle\,dV_g$ for any two complex valued $q$-forms $u$, $v$. With
respect to an orthonormal frame $(\xi_j)_{1\le j\le n}$ of $T_M$,
standard computations left to the reader show that
$$
u=\sum_{|J|=q}u_J\,\xi^*_J~~\longmapsto~~
*\,u=\sum_{|J|=q}\sign(J,\complement J)\,\ol u_J\,\xi^*_{\complement J},
$$
and as a consequence $**=\pm\Id$ and $D^*=\pm\,*\,D\,*$. It~follows
that $\star$ commutes with $\Delta$ and that $\star\,u$ is harmonic 
if and only if $u$ itself is harmonic. This implies 

\claim{(4.10) Poincar\'e duality theorem} If $M$ is a compact oriented 
differential manifold, the natural pairing
$$H^q_\DR(M,\bC)\times H^{m-q}_\DR(M,\bC),\qquad
(\{u\},\{v\})\mapsto\int_M u\wedge v$$
is a nondegenerate duality.
\endclaim

\begproof{} The statement follows from the following observations~:
\item{$\bu$} the dual of a
class $\{u\}$ represented by a harmonic form is $\{\star u\},$~~
\item{$\bu$} $(u,*\,u)$ is paired to $\int_Mu\wedge *\,u=\Vert u\Vert^2>0$
if~$u\ne 0$.\qed
\endproof

Let us now suppose that $X$ is a compact complex manifold
equipped with a hermitian metric $g=\sum\omega_{jk}dz_j\otimes d\ol
z_k$. Let $F$ be a holomorphic vector bundle on $X$ equipped with a
hermitian metric~$h$, and let $D=D'+D''$ be its Chern curvature form.
All that we said above for the Laplace-Beltrami operator $\Delta$
still applies to the complex Laplace operators
$$
\Delta'=D'D^{\prime\star}+D^{\prime\star}D',\qquad
\Delta''=D''D^{\prime\prime\star}+D^{\prime\prime\star}D'',
\leqno(4.11)
$$
with the great advantage that we always have 
$D^{\prime 2}=D^{\prime\prime 2}=0$ without making any special assumption
on the connection. Especially, if $X$ is a compact
complex manifold, there is an isomorphism
$$H^{p,q}(X,F)\simeq\cH^{p,q}(X,F)\leqno(4.12)$$
between the Dolbeault cohomology group $H^{p,q}(X,F)$ and the space
$\cH^{p,q}(X,F)$ of $\Delta''$-harmonic forms of bidegree $(p,q)$ with
values in~$F$. The isomorphism is derived as above from the $L^2$
direct sum decomposition
$$
\ci(M,\Lambda^{p,q}T^\star_M\otimes F)=\cH^{p,q}(M,F)\oplus
\Im D''\oplus\Im D^{\prime\prime\star},
$$
Now, there is a generalized (conjugate linear) Hodge star operator
such that 
$$
u\wedge\star v=\langle u,v\rangle dV_g,
$$
for any two $F$-valued $(p,q)$-forms $u$, $v$, when the wedge product
$u\wedge\star v$ is combined with the natural obvious
pairing $F\times F^\star\to\bC$. It is given by
$$
\eqalign{
&\star:\Lambda^{p,q}T^\star_X\otimes F\to
\Lambda^{n-p,n-q}T^\star_X\otimes F^\star,\qquad n=\dim_\bC X,\cr
&u=\sum_{I,J,\lambda}u_{I,J,\lambda}\,
\xi_I^*\wedge\ol\xi_J^*\otimes e_\lambda
~~\longmapsto
*\,u=\sum_{I,J,\lambda}\varepsilon_{I,J}\,
\ol u_{I,J,\lambda}\,
\xi_{\complement I}^*\wedge\ol\xi_{\complement J}^*\otimes e_\lambda^*\,,\cr
&\varepsilon_{I,J}:=(-1)^{q(n-p)}\sign(I,\complement I)\sign(J,\complement J),
\qquad \hbox{for $|I|=p$, $|J|=q$, $1\le\lambda\le r$,}\cr
}
$$
with respect to orthonormal frames $(\xi_j)_{1\le j\le n}$ of $(T_X,g)$
and $(e_\lambda)_{1\le\lambda\le r}$ of $(F,h)$.
This leads to the {\it Serre duality theorem} [Ser55]: 

\claim{(4.13) Serre duality theorem} If $X$ is a compact complex
manifold, the bilinear pairing
$$H^{p,q}(X,F)\times H^{n-p,n-q}(X,F^\star),\qquad
(\{u\},\{v\})\mapsto\int_X u\wedge v$$
is a nondegenerate duality.\qed
\endclaim

Combining this with the Dolbeault isomorphism (1.3),
we may restate the result in the form of the duality formula
$$H^q(X,\Omega^p_X\otimes\cO(F))^\star\simeq H^{n-q}(X,
\Omega^{n-p}_X\otimes\cO(F^\star)).\leqno(4.13')$$

We now proceed to explain the basic commutation relations satisfied by 
the operators of hermitian geometry. Great simplifications occur in the 
computations
if the hermitian metric on $X$ is supposed to be {\it K\"ahler}.

\claim{(4.14) Definition} A hermitian metric
$g=\sum_{j,k}\omega_{jk}dz_j\otimes d\ol z_k$ on $T_X$ is said to
be K\"ahler if the associated {\it fundamental $(1,1)$-form}
$$\omega=\ii\sum_{j,k}\omega_{jk}dz_j\wedge d\ol z_k$$
satisfies $d\omega=0$, i.e.\ is also a symplectic $2$-form.
\endclaim

\claim{(4.15) Examples}

\noindent
{\rm $\bu$ $\bC^n$ equipped with its canonical
metric $\omega=\ii\sum_{1\le j\le n}dz_j\wedge d\ol z_j$
(or any other hermitian metric
$\omega=\ii\sum_{1\le j,k\le n}\omega_{jk} dz_j\wedge d\ol z_k$ with
constant coefficients) is K\"ahler.}

\noindent
{\rm $\bu$ A {\it complex torus} is a quotient
$X=\bC^n/\Gamma$ by a lattice (closed discrete subgroup) $\Gamma$ of 
rank~$2n$. Then $X$ is a
compact complex manifold. Any positive definite hermitian form
$\omega=\ii\sum \omega_{jk}dz_j\wedge d\ol z_k$ with constant coefficients
defines a K\"ahler metric on~$X$.}

\noindent
{\rm $\bu$ The complex projective space 
$\bP^n$ is K\"ahler.
A natural K\"ahler metric $\omega_\FS$ on $\bP^n$ is the {\it Fubini-Study
metric} defined by
$$p^\star\omega_\FS={\ii\over 2\pi} d'd''\log\big(|\zeta_0|^2+|\zeta_1|^2
+\cdots+|\zeta_n|^2\big)$$
where $\zeta_0,\zeta_1\ld\zeta_n$ are coordinates of $\bC^{n+1}$ and
where \hbox{$p:\bC^{n+1}\setminus\{0\}\to\bP^n$} is the projection.
Let $z=(\zeta_1/\zeta_0\ld\zeta_n/\zeta_0)$ be non homogeneous coordinates
on $\bC^n\subset\bP^n$. A calculation shows that
$$\omega_\FS={\ii\over 2\pi}d'd''\log(1+|z|^2),
~~~~\int_{\bP^n}\omega_\FS^n=1.$$
It follows from this that $\{\omega_\FS\}=c_1(\cO(1))\in H^2(\bP^n,\bZ)$ is 
a generator of the cohomology algebra $H^\bu(\bP^n,\bZ)$.}

\noindent
{\rm $\bu$ Every (complex) submanifold $Y$ of a
K\"ahler manifold  $(X,\omega)$ is K\"ahler with metric
$\omega'=\omega_{\restriction Y}$.  Especially, all {\it projective manifolds},
i.e.\ all submanifolds $X\subset \bP^N$,  are K\"ahler with K\"ahler metric
$\omega=\omega_{\FS\restriction X}$. Since $\omega_\FS$ is
in $H^2(\bP,\bZ)$, the restriction $\omega$ is an 
integral class in $H^2(X,\bZ)$. Conversely, the Kodaira embedding
theorem [Kod54] states that every compact K\"ahler manifold $X$
possessing a K\"ahler metric $\omega$ with an integral cohomology
class $\{\omega\}\in H^2(X,\bZ)$ can be embedded in projective space
as a projective algebraic subvariety. We will be able to prove this in
section~6.}
\endclaim

\claim{(4.16) Lemma} Let $\omega$ be a $\ci$ positive definite $(1,1)$-form
on $X$. In order that $\omega$ be K\"ahler, it is necessary and sufficient
that to every point $x_0\in X$ corresponds a holomorphic coordinate system
$(z_1\ld z_n)$ centered at $x_0$ such that
$$\omega=\ii\sum_{1\le l,m\le n}\omega_{lm}\,dz_l\wedge d\ol z_m,~~~~
\omega_{lm}=\delta_{lm}+O(|z|^2).\leqno(4.16\,{\rm a})$$
If $\omega$ is K\"ahler, the coordinates $(z_j)_{1\le j\le n}$ can be
chosen such that
$$\omega_{lm}=\langle{\partial\over\partial z_l},{\partial\over\partial z_m}
\rangle=\delta_{lm}-\sum_{1\le j,k\le n}c_{jklm}\,z_j\ol z_k+O(|z|^3)
,\leqno(4.16\,{\rm b})$$
where $(c_{jklm})$ are the coefficients of the Chern curvature tensor
$$\Theta(T_X)_{x_0}=\sum_{j,k,l,m} c_{jklm}\,dz_j\wedge d\ol z_k\otimes
\Big({\partial\over\partial z_l}\Big)^\star
\otimes{\partial\over\partial z_m}\leqno(4.16\,{\rm c})$$
associated to $(T_X,\omega)$ at $x_0$. Such a system $(z_j)$
will be called a geodesic coordinate system at $x_0$.
\endclaim

\begproof{} It is clear that $(4.16\,{\rm a})$ implies $d_{x_0}\omega=0$, so the
condition is sufficient. Assume now that $\omega$ is K\"ahler and show
that $(4.16\,{\rm b})$ can be achieved. The calculations are somewhat
similar to those already made for the proof of~(3.8). One can
choose local coordinates $(x_1\ld x_n)$ such that $(dx_1\ld dx_n)$ is an
$\omega$-orthonormal basis of $T_{x_0}^\star X$. Therefore
$$\leqalignno{
\omega&=\ii\sum_{1\le l,m\le n}\wt\omega_{lm}\,dx_l\wedge d\ol x_m,
~~~~{\rm where}\cr
\qquad\wt\omega_{lm}&=\delta_{lm}+O(|x|)=\delta_{lm}+\sum_{1\le j\le n}
(a_{jlm}x_j+a'_{jlm}\ol x_j)+O(|x|^2).&(4.16\,{\rm d})\cr}$$
Since $\omega$ is real, we have $a'_{jlm}=\ol a_{jml}\,$; on the other
hand the K\"ahler condition $\partial\omega_{lm}/\partial x_j=
\partial\omega_{jm}/\partial x_l$ at $x_0$ implies $a_{jlm}=a_{ljm}$. Set now
$$z_m=x_m+{1\over 2}\sum_{j,l}a_{jlm}x_jx_l,~~~~1\le m\le n.$$
Then $(z_m)$ is a coordinate system at $x_0$, and
$$\eqalign{
dz_m&=dx_m+\sum_{j,l}a_{jlm}x_jdx_l,\cr
\ii\sum_mdz_m\wedge d\ol z_m&=\ii\sum_m dx_m\wedge d\ol x_m
+\ii\sum_{j,l,m}a_{jlm}x_j\,dx_l\wedge d\ol x_m\cr
&\phantom{=\ii\sum_m dx_m~\wedge d\ol x_m}
+\ii\sum_{j,l,m}\ol a_{jlm}\ol x_j\,dx_m\wedge d\ol x_l+O(|x|^2)\cr
&=\ii\sum_{l,m}\wt\omega_{lm}\,dx_l\wedge\ol dx_m+O(|x|^2)=
\omega+O(|z|^2).\cr}$$
Condition $(4.16\,{\rm a})$ is proved. Suppose the coordinates $(x_m)$
chosen from the beginning so that (4.9) holds with respect to $(x_m)$.
Then the Taylor expansion $(4.16\,{\rm d})$ can be refined into
$$\leqalignno{
\qquad\wt\omega_{lm}&=\delta_{lm}+O(|x|^2)&(4.16\,{\rm e})\cr
&=\delta_{lm}+\sum_{j,k}
\big(a_{jklm}x_j\ol x_k+a'_{jklm}x_jx_k+a''_{jklm}\ol x_j\ol x_k\big)+
O(|x|^3).}$$
These new coefficients satisfy the relations
$$a'_{jklm}=a'_{kjlm},~~~~a''_{jklm}=\ol a'_{jkml},~~~~\ol a_{jklm}=a_{kjml}.$$
The K\"ahler condition $\partial\omega_{lm}/\partial x_j=\partial
\omega_{jm}/\partial x_l$ at $x=0$ gives the equality
$a'_{jklm}=a'_{lkjm}\,;$ in particular $a'_{jklm}$ is invariant under all
permutations of~$j,k,l$. If we set
$$z_m=x_m+{1\over 3}\sum_{j,k,l}a'_{jklm}\,x_jx_kx_l,~~~~1\le m\le n,$$
then by $(4.16\,{\rm e})$ we find
$$\leqalignno{
\qquad dz_m&=dx_m+\sum_{j,k,l}a'_{jklm}\,x_jx_k\,dx_l,~~~~1\le m\le n,\cr
\omega&=\ii\sum_{1\le m\le n}dz_m\wedge d\ol z_m+\ii\sum_{j,k,l,m}a_{jklm}\,x_j
\ol x_k\,dx_l\wedge d\ol x_m+O(|x|^3),\cr
\omega&=\ii\sum_{1\le m\le n}dz_m\wedge d\ol z_m+\ii\sum_{j,k,l,m}a_{jklm}\,z_j
\ol z_k\,dz_l\wedge d\ol z_m+O(|z|^3).&(4.16\,{\rm f})\cr}$$
It is now easy to compute the Chern curvature tensor $\Theta(T_X)_{x_0}$ in terms of
the coefficients $a_{jklm}$. Indeed
$$\eqalign{
\langle{\partial\over\partial z_l},{\partial\over\partial z_m}\rangle&=
\delta_{lm}+\sum_{j,k}a_{jklm}\,z_j\ol z_k+O(|z|^3),\cr
d'\langle{\partial\over\partial z_l},{\partial\over\partial z_m}\rangle&=
\Big\{D'{\partial\over\partial z_l},{\partial\over\partial z_m}\Big\}=
\sum_{j,k}a_{jklm}\,\ol z_k\,dz_j+O(|z|^2),\cr
\Theta(T_X)\cdot{\partial\over\partial z_l}&=D''D'\Big({\partial\over\partial z_l}\Big)
=-\sum_{j,k,m}a_{jklm}\,dz_j\wedge d\ol z_k\otimes{\partial\over\partial z_m}
+O(|z|),\cr}$$
therefore $c_{jklm}=-a_{jklm}$ and the expansion $(4.16\,{\rm b})$ follows
from $(4.16\,{\rm f})$.\qed
\endproof

The next step is to compute the basic operators on $X=\bC^n$, in the case
of a trivial line bundle $F=X\times\bC$ (with the trivial constant metric!).
For any form 
$u\in C^\infty(\Omega,\Lambda^{p,q}T^\star_\Omega)$ we have
$$
\eqalign{
d'u&=\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
dz_k\wedge dz_I\wedge d\ol z_J=\sum_{1\le k\le n}dz_k\wedge
\Big({\partial u\over\partial z_k}\Big),\cr
d''u&=\sum_{I,J,k} {\partial u_{I,J}\over\partial\ol z_k}
d\ol z_k\wedge dz_I\wedge d\ol z_J=\sum_{1\le k\le n}d\ol z_k\wedge
\Big({\partial u\over\partial\ol z_k}\Big),\cr}$$
where $\partial u/\partial z_k$ and $\partial u/\partial\ol z_k$ are
the differentiations of $u$ in $z_k$, $\ol z_k$, taken componentwise on 
each coefficient~$u_{IJ}$. From this we easily get

\claim{(4.17) Lemma} On any open subset $\omega\subset\bC^n$ equipped
with the constant hermitian metric~$\omega=\ii\sum_{1\le j\le n}dz_j\wedge
d\ol z_j$, we have
$$
\eqalign{
d^{\prime\star}u&=-\sum_{I,J,k} {\partial u_{I,J}\over\partial\ol z_k}
{\partial\over\partial z_k}\ort (dz_I\wedge d\ol z_J),\cr
d^{\prime\prime\star}u&=-\sum_{I,J,k} {\partial u_{I,J}\over\partial z_k}
{\partial\over\partial\ol z_k}\ort (dz_I\wedge d\ol z_J),\cr}
$$
where $\ort$ denotes the interior product of a form by a vector field.
These formulas can be written more briefly as
$$
d^{\prime\star}u=-\sum_{1\le k\le n}
{\partial\over\partial z_k}\ort\Big({\partial u\over\partial\ol z_k}\Big),
\qquad
d^{\prime\prime\star}u=-\sum_{1\le k\le n}
{\partial\over\partial\ol z_k}\ort\Big({\partial u\over\partial z_k}\Big).
$$
\endclaim

\begproof{} The adjoint of $dz_j\wedge\bu$ is ${\partial\over\partial z_j}
\ort\bu~$. In the case of $d^{\prime\star}$, for instance, we get
$$
\eqalign{
\Ll d'u,v\Rr
&=\int_\Omega\langle\sum_{1\le k\le n}dz_k\wedge
\Big({\partial u\over\partial z_k}\Big),v\rangle\,dV
=\int_\Omega\sum_{1\le k\le n}\langle
{\partial u\over\partial z_k},{\partial\over\partial z_k}\ort v\rangle\,dV\cr
&=\int_\Omega \langle u,
-\sum_{1\le k\le n}{\partial\over\partial\ol z_k}\Big(
{\partial\over\partial z_k}\ort v\Big)\rangle\,dV
=\int_\Omega \langle u,
-\sum_{1\le k\le n}{\partial\over\partial z_k}\ort\Big(
{\partial v\over\partial\ol z_k}\Big)\rangle\,dV\cr}
$$
whenever $u$ (resp.\ $v$) is a $(p-1,q)$-form (resp.\ $(p,q)$-form), with
$\Supp u\cap\Supp v\compact\Omega$. The third equality is simply obtained
through an integration par parts, and amounts to observe that the formal 
adjoint of $\partial/\partial z_k$ is $-\partial/\partial\ol z_k$.\qed
\endproof

We now prove a useful lemma due to Akizuki and Nakano [AN54].
If $(X,\omega)$ is a K\"ahler manifold, we define the operators $L$ and 
$\Lambda$ by
$$
\leqalignno{
\kern45pt u\mapsto Lu=\omega\wedge u,\quad
L&:\ci(X,\Lambda^{p,q}T^*_X\otimes F)\longmapsto
\ci(X,\Lambda^{p+1,q+1}T^*_X\otimes F),&(4,18)\cr
\kern45pt\Lambda=L^*&:\ci(X,\Lambda^{p,q}T^*_X\otimes F)\longmapsto
\ci(X,\Lambda^{p-1,q-1}T^*_X\otimes F).&(4.18^*)\cr}
$$
Again, in the flat hermitian complex space $(\bC^n,\omega)$ we find~:

\claim{(4.19) Lemma} In $\bC^n$, we have
$[d^{\prime\prime\star},L]=\ii d'$.
\endclaim

\begproof{} Using Lemma~4.17, we get
$$
\eqalign{[d^{\prime\prime\star},L]u
&=d^{\prime\prime\star}(\omega\wedge u)-\omega\wedge d^{\prime\prime\star}u\cr
&=-\sum_k {\partial\over\partial\ol z_k}\ort\Big(
{\partial\over\partial z_k}(\omega\wedge u)\Big)+\omega\wedge\sum_k 
{\partial\over\partial\ol z_k}\ort\Big({\partial u\over\partial z_k}\Big).\cr}
$$
Since $\omega$ has constant coefficients, we have $\displaystyle{\partial\over
\partial z_k}(\omega\wedge u)=\omega\wedge{\partial u\over\partial z_k}$
and therefore
$$\eqalign{
[d^{\prime\prime\star},L]\,u&=-\sum_k\bigg({\partial\over\partial\ol z_k}\ort
\Big(\omega\wedge {\partial u\over\partial z_k}\Big)-\omega\wedge\Big(
{\partial\over\partial\ol z_k}\ort{\partial u\over\partial z_k}\Big)\bigg)\cr
&=-\sum_k\Big({\partial\over\partial\ol z_k}\ort\omega\Big)\wedge
{\partial u\over\partial z_k}\cr}$$
by the derivation property of $\ort$. Clearly
$$
{\partial\over\partial\ol z_k}\ort\omega=
{\partial\over\partial\ol z_k}\ort\ii\sum_{1\le j\le n}dz_j\wedge
d\ol z_j=-\ii dz_k,
$$
hence
$$
[d^{\prime\prime\star},L]\,u=\ii\sum_k dz_k\wedge
 {\partial u\over\partial z_k}=\ii d'u.
\eqno\square
$$
\endproof

The final step is to extend these results to an arbitrary K\"ahler manifold
$(X,\omega)$ and an arbitrary hermitian vector bundle $(F,h)$.
When we use a geodesic coordinate system $(z_1,\ldots,z_n)$ for
$\omega$ in combination with a normal coordinate frame for $(F,h)$, 
all norms expressed in a neighborhood of $x_0$ differ from constant
norms by $O(|z|^2)$ terms only.
It follows that all order~$1$ operators $D$, $D'$, $D''$ and their
adjoints $D^\star$, $D^{\prime\star}$, $D^{\prime\prime\star}$ admit at $x_0$
the same expansion as the analogous operators obtained when the hermitian
metrics on $X$ or $F$ are freezed at their values at $x_0$ (first order 
derivatives applied
to $O(|z|^2)$ terms yield~$0$). This can be used to derive the 
basic commutation relations of K\"ahler geometry from the simpler case of
constant metrics in an open subset of $\bC^n$. 

If $A,B$ are differential operators acting on the algebra
$C^\infty(X,\Lambda^{\bu,\bu}T^\star_X\otimes F)$, their graded
commutator (or graded Lie bracket) is defined by
$$
[A,B]= AB - (-1)^{ab} BA
\leqno(4.20)
$$
where $a,b$ are the degrees of $A$ and $B$ respectively. If $C$ is another
endomorphism of degree $c$, the following purely formal {\it Jacobi identity}
holds:
$$
(-1)^{ca} \big[A,[B,C]\big] + (-1)^{ab}\big[B,[C,A]\big]
+ (-1)^{bc} \big[C,[A,B]\big] = 0.
\leqno(4.21)
$$

\claim{(4.22) Basic commutation relations} Let $(X,\omega)$ be a K\"ahler
manifold and let $L$ be the operators defined by $Lu=\omega\wedge u$ and
$\Lambda=L^\star$. Then
$$\eqalign{
[D^{\prime\prime\star},L]&=\ii D',\cr
[\Lambda,D'']&=-\ii D^{\prime\star},\cr}
\qquad
\eqalign{
[D^{\prime\star},L]&=-\ii D'',\cr
[\Lambda,D']&=\ii D^{\prime\prime\star}.\cr}$$
\endclaim

\begproof{} The first formula follows from Lemma 4.19,
by looking at Taylor expansions of the
coefficients of the operators up to order~1.
The three other identities are obtained by taking conjugates or adjoints
in the first one.\qed
\endproof

\claim{(4.23) Bochner-Kodaira-Nakano identity} If $(X,\omega)$ is
K\"ahler, the complex Laplace operators $\Delta'$ and $\Delta''$ acting
on $F$-valued forms satisfy the identity
$$\Delta''=\Delta'+[\ii\Theta(F),\Lambda].$$
\endclaim

\begproof{} The last equality in (4.22) yields $D^{\prime\prime\star}
=-\ii[\Lambda,D']$, hence
$$\Delta''=[D'',\delta'']=-\ii[D'',\big[\Lambda,D']\big].$$
By the Jacobi identity (4.21) we get
$$\big[D'',[\Lambda,D']\big]=\big[\Lambda,[D',D'']]+\big[D',[D'',\Lambda]\big]
=[\Lambda,\Theta(F)]+\ii[D',D^{\prime\star}],$$
taking into account that $[D',D'']=D^2=\Theta(F)$. The formula follows.\qed
\endproof

\claim{(4.24) Corollary {\rm(Hodge decomposition)}} If $(X,\omega)$ is
a compact K\"ahler manifold, there is a canonical decomposition
$$
H^k_\DR(X,\bC)=\bigoplus_{p+q=k}H^{p,q}(X,\bC)
$$
of the De Rham cohomology groups in terms of the Dolbeault cohomology
groups.
\endclaim

\begproof{} If we apply the Bochner-Kodaira-Nakano identity to the
trivial bundle $F=X\times\bC$, we find $\Delta''=\Delta'$. Morever
$$
\Delta=[d'+d'',d^{\prime\star}+d^{\prime\prime\star}]=
\Delta'+\Delta''+[d',d^{\prime\prime\star}]+[d'',d^{\prime\star}].
$$
We claim that  $[d',d^{\prime\prime\star}]=[d'',d^{\prime\star}]=0$.
Indeed, we have 
$[d',d^{\prime\prime\star}]=-\ii\big[d',[\Lambda,d']\big]$ by (4.22), and 
the Jacobi identity (4.21) implies
$$
-\big[d',[\Lambda,d']\big]+\big[\Lambda,[d',d']\big]+
\big[d',[d',\Lambda]\big]=0,
$$
hence $-2\big[d',[\Lambda,d']\big]=0$ and $[d',d^{\prime\prime\star}]=0$. 
The second identity is similar. As a consequence
$$
\Delta'=\Delta''={1\over 2}\Delta.
$$
We infer that $\Delta$ preserves the bidegree of forms and operates
``separately'' on each term $C^\infty(X,\Lambda^{p,q}T^*_X)$. Hence,
on the level of harmonic forms we have
$$
\cH^k(X,\bC)=\bigoplus_{p+q=k}\cH^{p,q}(X,\bC).
$$
The decomposition theorem (4.24) now follows from the Hodge isomorphisms for 
De Rham and Dolbeault groups. The decomposition is canonical since $H^{p,q}(X)$
coincides with the set of classes in $H^k(X,\bC)$ which can be represented
by $d$-closed $(p,q)$-forms.\qed
\endproof

\claim{(4.25) Remark} {\rm The decomposition formula shows that the 
so called {\it Betti numbers} $b_k:=\dim H^k_\DR(X,\bC)$ satisfy the relation
$$
b_k=\sum_{p+q=k}h^{p,q}
$$
in terms of the {\it Hodge numbers} $h^{p,q}:=\dim H^{p,q}(X,\bC)$.
In addition to this, the complex conjugation $u\mapsto�\ol u$ takes
$(p,q)$-harmonic forms to $(q,p)$-harmonic forms, hence there is a
canonical conjugate linear isomorphism 
$$
H^{q,p}(X,\bC)\simeq \ol{H^{p,q}(X,\bC)}
$$
and $h^{q,p}=h^{p,q}$. In particular, if $X$
is compact K\"ahler, the Betti numbers $b_k$ of odd index are even.
The Serre duality theorem gives the further relation
$$
H^{n-p,n-q}(X,\bC)\simeq H^{p,q}(X,\bC)^*
$$
(actually, this is even true for an arbitrary compact complex
manifold~$X$). 

This observation implies the existence of compact complex manifolds $X$
which are non K\"ahler (hence non projective)~: for instance the {\it Hopf surface}
defined by $X=\bC^2\ssm\{0\}/\Gamma$, where $\Gamma\simeq \bZ$ is the discrete
group generated by a contraction $z\mapsto \lambda z$, $0<\lambda<1$,
is easily seen to be diffeomorphic to $S^1\times S^3$. Hence $b_1=1$
by the K\"unneth formula, and $X$ cannot be K\"ahler.}
\endclaim

\titleb{5.}{Bochner Technique and Vanishing Theorems}

Assume that $X$ is compact and that $u\in\ci(X,\Lambda^{p,q}T^\star
X\otimes F)$ is an arbitrary $(p,q)$-form. An integration by parts yields
$$\langle\Delta'u,u\rangle=\|D'u\|^2+\|D^{\prime\star}u\|^2\ge0$$
and similarly for $\Delta''$, hence we get the basic a priori inequality
$$\|D''u\|^2+\|D^{\prime\prime\star}u\|^2\ge\int_X
\langle[\ii\Theta(F),\Lambda]u,u\rangle dV_\omega.\leqno(5.1)$$
This inequality is known as the {\it Bochner-Kodaira-Nakano} inequality
(see [Boc48], [Kod53], [Nak55]). When $u$ is $\Delta''$-harmonic, we get
$$\int_X\big(\langle[\ii\Theta(F),\Lambda]u,u\rangle+
\langle T_\omega u,u\rangle\big)dV\le 0.$$
If the hermitian operator $[\ii\Theta(F),\Lambda]$ acting on
$\Lambda^{p,q}T^\star_X\otimes F$ is positive on each fiber, we infer
that $u$ must be zero, hence
$$H^{p,q}(X,F)=\cH^{p,q}(X,F)=0$$
by Hodge theory. The main point is thus to compute the curvature form
$\Theta(F)$ and find sufficient conditions under which the operator
$[\ii\Theta(F),\Lambda]$ is positive definite. Elementary (but somewhat
tedious) calculations yield the following formulae: if the curvature of
$F$ is written as in (3.9) and 
$u=\sum u_{J,K,\lambda}dz_I\wedge d\ol z_J\otimes e_\lambda$,
$|J|=p$, $|K|=q$, $1\le\lambda\le r$ is a $(p,q)$-form with values in~$F$,
then
$$\leqalignno{\langle[\ii\Theta(F),\Lambda]u,u\rangle=
&\phantom{{}+{}}\sum_{j,k,\lambda,\mu,J,S}{\,}
c_{jk\lambda\mu}\,u_{J,jS,\lambda}~\ol{u_{J,kS,\mu}}&(5.2)\cr
&{}+\sum_{j,k,\lambda,\mu,R,K}
c_{jk\lambda\mu}\,u_{kR,K,\lambda}~\ol{u_{jR,K,\mu}}\cr
&{}-{\,}\sum_{j,\lambda,\mu,J,K}\phantom{k}
c_{jj\lambda\mu}\,u_{J,K,\lambda}~\ol{u_{J,K,\mu}},\cr}$$
where the sum is extended to all indices $1\le j,k\le n$, $1\le\lambda,
\mu\le r$ and multiindices $|R|=p-1$, $|S|=q-1$ (here the notation
$u_{JK\lambda}$ is extended to non necessarily increasing multiindices by
making it alternate with respect to permutations). It is usually hard to
decide the sign of the curvature term (5.2), except in some special cases.

The easiest case is when $p=n$. Then all terms in the second summation of
(5.2) must have $j=k$ and $R=\{1\ld n\}\ssm\{j\}$, therefore the second
and third summations are equal. It follows that $[\ii\Theta(F),\Lambda]$ is
positive on $(n,q)$-forms under the assumption that $F$ is positive in
the sense of Nakano. In this case $X$ is automatically K\"ahler since
$$\omega=\Tr_F(\ii\Theta(F))=\ii\sum_{j,k,\lambda}c_{jk\lambda\lambda}dz_j
\wedge d\ol z_k=\ii\Theta(\det F)$$
is a K\"ahler metric.

\claim{(5.3) Nakano vanishing theorem {\rm ([Nak55])}} Let $X$ be a
compact complex manifold and let $F$ be a Nakano positive vector
bundle on~$X$. Then
$$H^{n,q}(X,F)=H^q(X,K_X\otimes F)=0\qquad\hbox{for every $q\ge 1$.}
\eqno\square$$
\endclaim

Another tractable case is the case where $F$ is a line bundle ($r=1$).
Indeed, at each point $x\in X$, we may then choose a coordinate
system which diagonalizes simultaneously the hermitians forms
$\omega(x)$ and $\ii\Theta(F)(x)$, in such a way that
$$\omega(x)=\ii\sum_{1\le j\le n}dz_j\wedge d\ol z_j,\qquad
\ii\Theta(F)(x)=\ii\sum_{1\le j\le n}\gamma_jdz_j\wedge d\ol z_j$$
with $\gamma_1\le\ldots\le\gamma_n$. The curvature eigenvalues
$\gamma_j=\gamma_j(x)$ are then uniquely defined and depend continuously
on~$x$. With our previous notation, we have $\gamma_j=c_{jj11}$ and all
other coefficients $c_{jk\lambda\mu}$ are zero. For any $(p,q)$-form
$u=\sum u_{JK}dz_J\wedge d\ol z_K\otimes e_1$, this gives
$$\leqalignno{\langle[\ii\Theta(F),\Lambda]u,u\rangle
&=\sum_{|J|=p,\,|K|=q}\Big(\sum_{j\in J}\gamma_j+\sum_{j\in K}\gamma_j-
\sum_{1\le j\le n}\gamma_j\Big)|u_{JK}|^2\cr
&\ge(\gamma_1+\ldots+\gamma_q-\gamma_{n-p+1}-\ldots-\gamma_n)|u|^2.
&(5.4)\cr}$$
Assume that $\ii\Theta(F)$ is positive. It is then natural to make the
special choice $\omega=\ii\Theta(F)$ for the K\"ahler metric. Then
$\gamma_j=1$ for $j=1,2\ld n$ and we obtain
$\langle[\ii\Theta(F),\Lambda]u,u\rangle=(p+q-n)|u|^2$.
As a consequence:

\claim{(5.5) Akizuki-Kodaira-Nakano vanishing theorem {\rm ([AN54])}}
If $F$ is a positive line bundle on a compact complex manifold~$X$, then
$$H^{p,q}(X,F)=H^q(X,\Omega^p_X\otimes F)=0\qquad\hbox{for}~~p+q\ge n+1.
\eqno\square$$
\endclaim

More generally, if $F$ is a Griffiths positive (or ample) vector bundle
of rank~$r\ge 1$, Le Potier [LP75] proved that $H^{p,q}(X,F)=0$ for
$p+q\ge n+r$. The proof is not a direct consequence of the Bochner
technique. A rather easy proof has been found by M.~Schneider [Sch74],
using the Leray spectral sequence associated to the projectivized bundle
projection $\bP(F)\to X$.

\claim{(5.7) Exercise} \rm It is important for various applications
to obtain vanishing theorems which are also valid in the case of semipositive
line bundles. The easiest case is the following result of Girbau [Gir76]:
let $(X,\omega)$ be compact K\"ahler; assume that $F$ is a line bundle and
that $\ii\Theta(F)\ge 0$ has at least $n-k$ positive eigenvalues at each
point, for some integer~$k\ge 0$; show that $H^{p,q}(X,F)=0$ for $p+q\ge
n+k+1$.\newline
{\it Hint:}\/ use the K\"ahler metric $\omega_\varepsilon=
\ii\Theta(F)+\varepsilon\omega$ with $\varepsilon>0$ small.
\smallskip
A stronger and more natural ``algebraic version'' of this result has been
obtained by Sommese [Som78]: define $F$ to be $k$-ample if some multiple
$mF$ is such that the canonical map
$$\Phi_{|mF|}:X\ssm B_{|mF|}\to\bP^{N-1}$$
has at most $k$-dimensional fibers and $\dim B_{|mF|}\le k$. If
$X$ is projective and $F$ is $k$-ample, show that $H^{p,q}(X,F)=0$ for
$p+q\ge n+k+1$.\newline
{\it Hint:}\/ prove the dual result $H^{p,q}(X,F^{-1})=0$ for $p+q\le n-k-1$
by induction on~$k$. First show that $F$ 0-ample $\Rightarrow$ $F$ positive;
then use hyperplane sections $Y\subset X$ to prove the induction step,
thanks to the exact sequences
$$\eqalignno{
&0\lra\Omega^p_X\otimes F^{-1}\otimes\cO(-Y)\lra\Omega^p_X\otimes F^{-1}
\lra\big(\Omega^p_X\otimes F^{-1}\big)_{\restriction Y}\lra 0,&\cr
&0\lra\Omega^{p-1}_Y\otimes F^{-1}_{\restriction Y}\otimes\cO(-Y)_{\restriction Y}
\lra\big(\Omega^p_X\otimes F^{-1}\big)_{\restriction Y}\lra
\Omega^p_Y\otimes F^{-1}_{\restriction Y}\lra 0.&\square\cr}$$
\endclaim

\titleb{6.}{$L^2$ Estimates and Existence Theorems}
The starting point is the following $L^2$ existence theorem, which is
essentially due to H\"ormander [H\"or65, 66], and Andreotti-Vesentini
[AV65]. We will only  outline the main ideas, referring e.g.\ to [Dem82b]
for a detailed exposition of the technical situation considered here.

\claim{(6.1) Theorem} Let $(X,\omega)$ be a K\"ahler manifold. Here $X$
is not necessarily compact, but we assume that the geodesic distance
$\delta_\omega$ is complete on~$X$. Let $F$ be a hermitian vector bundle of
rank $r$ over $X$, and assume that the curvature operator
$A=A^{p,q}_{F,\omega}=[\ii\Theta(F),\Lambda_\omega]$ is positive
definite everywhere on $\Lambda^{p,q}T^\star_X\otimes F$, $q\ge 1$.
Then for any form $g\in L^2(X,\Lambda^{p,q}T^\star_X\otimes F)$
satisfying $D''g=0$ and $\int_X\langle A^{-1}g,g\rangle
\,dV_\omega<+\infty$, there exists $f\in
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$ such that  $D''f=g$ and
$$\int_X|f|^2\,dV_\omega\le\int_X\langle A^{-1}g,g\rangle\,dV_\omega.$$
\endclaim

\begproof{} A well-known result of Riemannian geometry 
(Hopf-Rinow lemma) asserts that
the geodesic distance $\delta_\omega$ is complete if and only if all
closed geodesic balls $\ol B_\omega(x_0,r)$, $r>0$, are compact (none
can ``approach the boundary'', i.e.\ the boundary of $X$ is at infinity
from the point of view of distance).
This assumption implies (is in fact equivalent to) the
existence of cut-off functions $\psi_\nu$ with arbitrarily large compact
support such that $|d\psi_\nu|\le 2^{-\nu}$ (take $\psi_\nu$ to be a 
function of the rescaled distance $x\mapsto 2^{-\nu}\delta_\omega(x_0,x)$
(the distance function is an almost everywhere differentiable 1-Lipschitz 
function, so one might have to regularize it beforehand).

From this, it follows that very form $u\in
L^2(X,\Lambda^{p,q}T^\star_X\otimes F)$ such that $D''u\in L^2$ and
$D^{\prime\prime\star}u\in L^2$ in the sense of distribution theory is a
limit of a sequence of smooth forms $u_\nu$ with compact support, in such
a way that $u_\nu\to u$, $D''u_\nu\to D''u$ and $D^{\prime\prime\star}u_\nu
\to D^{\prime\prime\star}u$ in~$L^2$ (just take $u_\nu$ to be a
regularization of $\psi_\nu u$ and notice that the terms coming
from the differentiation of $\psi_\nu$ tend to $0$ in $L^2$~norm). 

As a consequence, the basic a priori
inequality (5.1) extends to arbitrary forms $u$ such that
$u,~D''u,D^{\prime\prime\star}u\in L^2$ (observe that the proof of (5.1) 
requires integration by parts; hence, if $X$ is non compact, the inequality
is a priori guaranteed only for compactly supported forms). Now, 
consider the Hilbert space orthogonal decomposition
$$L^2(X,\Lambda^{p,q}T^\star_X\otimes F)=\Ker
D''\oplus({\Ker D''})^\perp,$$
observing that $\Ker D''$ is weakly (hence strongly) closed. Let $v=v_1+v_2$
be the decomposition of a smooth form $v\in\cD^{p,q}(X,F)$ with compact
support according to this decomposition ($v_1$, $v_2$ do not have compact
support in general$\,$!).
Since $(\Ker D'')^\perp\subset\Ker D^{\prime\prime\star}$ by duality
and $g,v_1\in\Ker D''$ by hypothesis, we get
$D^{\prime\prime\star}v_2=0$ and
$$|\langle g,v\rangle|^2=|\langle g,v_1\rangle|^2\le
\int_X\langle A^{-1}g,g\rangle \,dV_\omega
\int_X\langle Av_1,v_1\rangle \,dV_\omega$$
thanks to the Cauchy-Schwarz inequality. The a priori inequality (5.1)
applied to $u=v_1$ yields
$$\int_X\langle Av_1,v_1\rangle \,dV_\omega\le
\|D''v_1\|^2+\|D^{\prime\prime\star}v_1\|^2=\|D^{\prime\prime\star}v_1\|^2=
\|D^{\prime\prime\star}v\|^2.$$
Combining both inequalities, we find
$$|\langle g,v\rangle|^2\le\Big(\int_X\langle A^{-1}g,g\rangle \,dV_\omega\Big)
\|D^{\prime\prime\star}v\|^2$$
for every smooth $(p,q)$-form $v$ with compact support. This shows that we
have a well defined linear form
$$w=D^{\prime\prime\star}v\longmapsto\langle v,g\rangle,\qquad
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)
\supset D^{\prime\prime\star}(\cD^{p,q}(F))\lra\bC$$
on the range of $D^{\prime\prime\star}$. This linear form
is continuous in $L^2$ norm and has norm $\le C$ with
$$C=\Big(\int_X\langle A^{-1}g,g\rangle \,dV_\omega\Big)^{1/2}.$$
By the Hahn-Banach theorem, there is an element
$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$ with $||f||\le C$, such that
$\langle v,g\rangle=\langle D^{\prime\prime\star}v,f\rangle$ for every $v$,
hence $D''f=g$ in the sense of distributions. The inequality $||f||\le C$
is equivalent to the last estimate in the theorem.\qed
\endproof

The above $L^2$ existence theorem can be applied in the fairly general
context of {\it weakly pseudoconvex} manifolds. By this, we mean a complex
manifold $X$ such that there exists a smooth psh exhaustion function
$\psi$ on $X$ ($\psi$ is said to be an exhaustion if for every $c>0$ the
sublevel set $X_c=\psi^{-1}(c)$ is relatively compact, i.e.\ $\psi(z)$
tends to $+\infty$ when $z$ is taken outside larger and larger compact
subsets of~$X$). In particular, every compact complex manifold $X$ is
weakly pseudoconvex (take $\psi=0$), as well as every Stein
manifold, e.g.\ affine algebraic submanifolds of $\bC^N$ (take
$\psi(z)=|z|^2$), open balls $X=B(z_0,r)$ $\big($take $\psi(z)=
1/(r-|z-z_0|^2)\big)$, convex open subsets, etc. Now, a basic observation
is that every weakly pseudoconvex K\"ahler manifold $(X,\omega)$ carries a
{\it complete} K\"ahler metric: let $\psi\ge 0$ be a psh exhaustion function
and set
$$\omega_\varepsilon=\omega+\varepsilon\,\ii d'd''\psi^2=
\omega+2\varepsilon(2\ii\psi d'd''\psi+\ii d'\psi\wedge d''\psi).$$
Then $|d\psi|_{\omega_\varepsilon}\le 1/\varepsilon$ and
$|\psi(x)-\psi(y)|\le\varepsilon^{-1}\delta_{\omega_\varepsilon}(x,y)$. 
It follows easily from this estimate that the geodesic balls are relatively
compact, hence $\delta_{\omega_\varepsilon}$ is complete
for every $\varepsilon>0$. Therefore, the $L^2$ existence theorem can be
applied to each K\"ahler metric $\omega_\varepsilon$, and by passing to
the limit it can even be applied to the non necessarily complete
metric~$\omega$. An important special case is the following

\claim{(6.2) Theorem} Let $(X,\omega)$ be a K\"ahler manifold,
$\dim X=n$. Assume that $X$ is weakly pseudoconvex. Let $F$ be a hermitian
line bundle and let
$$\gamma_1(x)\le\ldots\le\gamma_n(x)$$
be the curvature eigenvalues $($i.e.\ the eigenvalues of $\ii\Theta(F)$
with respect to the metric~$\omega)$ at every point. Assume that the
curvature is positive, i.e.\ $\gamma_1>0$ everywhere. Then for any form
$g\in L^2(X,\Lambda^{n,q}T^\star_X\otimes F)$ satisfying $D''g=0$ and
\hbox{$\int_X\langle(\gamma_1+\ldots+\gamma_q)^{-1}|g|^2\,dV_\omega<+\infty$},
there exists \hbox{$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$} such that
$D''f=g$ and
$$\int_X|f|^2\,dV_\omega\le\int_X
(\gamma_1+\ldots+\gamma_q)^{-1}|g|^2\,dV_\omega.$$
\endclaim

\begproof{} Indeed, for $p=n$, Formula~5.4 shows that
$$\langle Au,u\rangle\ge(\gamma_1+\ldots+\gamma_q)|u|^2,$$
hence $\langle A^{-1}u,u\rangle\ge(\gamma_1+\ldots+\gamma_q)^{-1}|u|^2$.\qed
\endproof

An important observation is that the above theorem still applies when the
hermitian metric on $F$ is a singular metric with positive curvature in
the sense of currents. In fact, by standard
regularization techniques (convolution of psh functions by smoothing
kernels), the metric can be made smooth and the solutions obtained by
(6.1) or (6.2) for the smooth metrics have limits satisfying the desired
estimates (see [Dem82b,~92]. Especially, we get the following

\claim{(6.3) Corollary} Let $(X,\omega)$ be a K\"ahler manifold,
$\dim X=n$. Assume that $X$ is weakly pseudoconvex. Let $F$ be a
holomorphic line bundle equipped with a singular metric whose local
weights are denoted $\varphi\in L^1_\loc$. Suppose that
$$\ii\Theta(F)=2\ii d'd''\varphi\ge\varepsilon\omega$$
for some $\varepsilon>0$. Then for any form $g\in L^2(X,\Lambda^{n,q}
T^\star_X\otimes F)$ satisfying $D''g=0$, there exists
$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$ such that  $D''f=g$ and
$$\int_X|f|^2e^{-2\varphi}\,dV_\omega\le{1\over q\varepsilon}\int_X
|g|^2e^{-2\varphi}\,dV_\omega.\eqno\square$$
\endclaim

Here we denoted somewhat incorrectly the metric by $|f|^2e^{-2\varphi}$,
as if the weight $\varphi$ was globally defined on~$X$ (of course, this
is so only if $F$ is globally trivial). We will use this notation anyway,
because it clearly describes the dependence of the $L^2$ norm on the psh
weights.

We now introduce the concept of {\it multiplier ideal sheaf},
following A.~Nadel [Nad89]. The main idea actually goes back to the
fundamental works of Bombieri [Bom70] and H.~Skoda [Sko72a].

\claim{(6.4) Definition} Let $\varphi$ be a
psh function on an open subset $\Omega\subset X\,$; to $\varphi$ is
associated the ideal subsheaf $\cI(\varphi)\subset\cO_\Omega$ of germs of
holomorphic functions $f\in\cO_{\Omega,x}$ such that $|f|^2e^{-2\varphi}$
is integrable with respect to the Lebesgue measure in some local
coordinates near~$x$.
\endclaim

The zero variety $V(\cI(\varphi))$ is thus the set of points in
a neighborhood of which $e^{-2\varphi}$ is non integrable. Of course,
such points occur only if $\varphi$ has logarithmic poles. This is made
precise as follows.

\claim{(6.5) Definition} A psh function $\varphi$ is said to have a
logarithmic pole of coefficient $\gamma$ at a point $x\in X$ if the Lelong
number
$$\nu(\varphi,x):=\liminf_{z\to x}{\varphi(z)\over\log|z-x|}$$
is non zero and if $\nu(\varphi,x)=\gamma$.
\endclaim

\claim{(6.6) Lemma {\rm (Skoda [Sko72a])}} Let $\varphi$ be a psh function
on an open set $\Omega$ and let $x\in\Omega$.
\smallskip
\item{\rm a)} If $\nu(\varphi,x)<1$, then $e^{-2\varphi}$ is integrable in
a neighborhood of~$x$, in particular $\cI(\varphi)_x=\cO_{\Omega,x}$.
\smallskip
\item{\rm b)} If $\nu(\varphi,x)\ge n+s$ for some integer $s\ge 0$, then
$e^{-2\varphi}\ge C|z-x|^{-2n-2s}$ in a neighborhood of $x$ and
$\cI(\varphi)_x\subset\gm_{\Omega,x}^{s+1}$, where $\gm_{\Omega,x}$ is the
maximal ideal of $\cO_{\Omega,x}$.
\smallskip
\item{\rm c)} The zero variety $V(\cI(\varphi))$ of $\cI(\varphi)$
satisfies 
$$E_n(\varphi)\subset V(\cI(\varphi))\subset E_1(\varphi)$$
where $E_c(\varphi)=\{x\in X\,;\,\nu(\varphi,x)\ge c\}$ is the $c$-sublevel
set of Lelong numbers of~$\varphi$.
\vskip0pt
\endclaim

\begproof{} a) Set $\Theta=dd^c\varphi$ and $\gamma=\nu(\Theta,x)=
\nu(\varphi,x)$. Let $\chi$ be a cut-off function with support in a small
ball $B(x,r)$, equal to $1$ in $B(x,r/2)$. As $(dd^c\log|z|)^n=\delta_0$,
we get
$$\eqalign{\varphi(z)
&=\int_{B(x,r)}\chi(\zeta)\varphi(\zeta)(dd^c\log|\zeta-z|)^n\cr
&=\int_{B(x,r)}dd^c(\chi(\zeta)\varphi(\zeta))\wedge
\log|\zeta-z|(dd^c\log|\zeta-z|)^{n-1}\cr}$$
for $z\in B(x,r/2)$. Expanding $dd^c(\chi\varphi)$ and observing that
$d\chi=dd^c\chi=0$ on $B(x,r/2)$, we find
$$\varphi(z)=\int_{B(x,r)}\chi(\zeta)\Theta(\zeta)\wedge
\log|\zeta-z|(dd^c\log|\zeta-z|)^{n-1}+\hbox{smooth terms}$$
on $B(x,r/2)$. Fix $r$ so small that
$$\int_{B(x,r)}\chi(\zeta)\Theta(\zeta)\wedge(dd^c\log|\zeta-x|)^{n-1}
\le\nu(\Theta,x,r)<1.$$
By continuity, there exists $\delta,\varepsilon>0$ such that
$$I(z):=\int_{B(x,r)}\chi(\zeta)\Theta(\zeta)\wedge(dd^c\log|\zeta-z|)^{n-1}
\le 1-\delta$$
for all $z\in B(x,\varepsilon)$. Applying Jensen's convexity inequality to
the probability measure
$$d\mu_z(\zeta)=I(z)^{-1}\chi(\zeta)\Theta(\zeta)\wedge
(dd^c\log|\zeta-z|)^{n-1},
$$
we find
$$\eqalign{
-\varphi(z)&=\int_{B(x,r)}I(z)\log|\zeta-z|^{-1}\,d\mu_z(\zeta)+O(1)\quad
\Longrightarrow\cr
e^{-2\varphi(z)}&\le C\int_{B(x,r)}|\zeta-z|^{-2I(z)}\,d\mu_z(\zeta).\cr}$$
As
$$d\mu_z(\zeta)\le C_1|\zeta-z|^{-(2n-2)}\Theta(\zeta)\wedge
(dd^c|\zeta|^2)^{n-1}=C_2|\zeta-z|^{-(2n-2)}d\sigma_\Theta(\zeta),$$
we get
$$e^{-2\varphi(z)}\le C_3\int_{B(x,r)}|\zeta-z|^{-2(1-\delta)-(2n-2)}
d\sigma_\Theta(\zeta),$$
and the Fubini theorem implies that $e^{-2\varphi(z)}$ is integrable on a
neighborhood of~$x$.
\smallskip

\noindent b) If $\nu(\varphi,x)=\gamma$, the convexity properties of
psh functions, namely, the convexity of $\log r\mapsto\sup_{|z-x|=r}
\varphi(z)$ implies that
$$\varphi(z)\le\gamma\log|z-x|/r_0+M,$$
where $M$ is the supremum on $B(x,r_0)$. Hence there exists a constant
$C>0$ such that $e^{-2\varphi(z)}\ge C|z-x|^{-2\gamma}$ in a neighborhood
of $x$. The desired result follows from the identity
$$\int_{B(0,r_0)}{\big|\sum a_\alpha z^\alpha\big|^2\over|z|^{2\gamma}}dV(z)=
{\rm Const}\int_0^{r_0}\Big(\sum |a_\alpha|^2r^{2|\alpha|}\Big)
r^{2n-1-2\gamma}\,dr,$$
which is an easy consequence of Parseval's formula. In fact, if $\gamma$
has integral part $[\gamma]=n+s$, the integral converges if and only if
$a_\alpha=0$ for $|\alpha|\le s$.
\smallskip
\noindent c) is just a simple formal consequence of a) and b).\qed
\endproof

\claim{(6.7) Proposition {\rm([Nad89])}} For any psh function
$\varphi$ on~$\Omega\subset X$, the sheaf
$\cI(\varphi)$ is a coherent sheaf of ideals over $\Omega$. Moreover,
if $\Omega$ is a bounded Stein open set, the sheaf $\cI(\varphi)$ is
generated by any Hilbert basis of the $L^2$ space $\cH^2(\Omega,\varphi)$
of holomorphic functions $f$ 
on $\Omega$ such that $\int_\Omega|f|^2e^{-2\varphi}\,d\lambda<+\infty$.
\endclaim

\begproof{} Since the result is local, we may assume that $\Omega$ is a
bounded pseudoconvex open set in $\bC^n$. 
By the strong noetherian property of coherent sheaves, the family of
sheaves generated by finite subsets of $\cH^2(\Omega,\varphi)$ has
a maximal element on each compact subset of $\Omega$, hence
$\cH^2(\Omega,\varphi)$ 
generates a coherent ideal sheaf $\cJ\subset\cO_\Omega$. It is clear that
$\cJ\subset\cI(\varphi)$; in order to prove the equality,
we need only check that $\cJ_x+\cI(\varphi)_x\cap
\gm^{s+1}_{\Omega,x}=\cI(\varphi)_x$ for every integer $s$, in
view of the Krull lemma. Let $f\in\cI(\varphi)_x$ be defined 
in a neighborhood $V$ of $x$ and let $\theta$ be a cut-off function
with support in $V$ such that $\theta=1$ in a neighborhood of $x$.
We solve the equation $d''u=g:=d''(\theta f)$ by means of H\"ormander's
$L^2$ estimates~6.3, where $F$ is the trivial line bundle
$\Omega\times\bC$ equipped with the strictly psh weight 
$$\wt\varphi(z)=\varphi(z)+(n+s)\log|z-x|+|z|^2.$$ 
We get a solution $u$ such that
$\int_\Omega |u|^2e^{-2\varphi}|z-x|^{-2(n+s)}d\lambda<\infty$, thus
$F=\theta f-u$ is holomorphic, $F\in \cH^2(\Omega,\varphi)$ and 
$f_x-F_x=u_x\in\cI(\varphi)_x\cap\gm_{\Omega,x}^{s+1}$.
This proves the coherence. Now, $\cJ$ is generated by any Hilbert basis
of $\cH^2(\Omega,\varphi)$, because it is well-known that the space of 
sections of any coherent sheaf is a Fr\'echet space, therefore
closed under local $L^2$ convergence.\qed
\endproof

The multiplier ideal sheaves satisfy the following basic
functoriality property with respect to direct images of sheaves
by modifications.

\claim{(6.8) Proposition} Let $\mu:X'\to X$ be a modification
of non singular complex manifolds $($i.e.\ a proper generically 1:1
holomorphic map$)$, and let $\varphi$ be a psh function on~$X$. Then 
$$\mu_\star\big(\cO(K_{X'})\otimes\cI(\varphi\circ\mu)\big)=
\cO(K_X)\otimes\cI(\varphi).$$
\endclaim

\begproof{} Let $n=\dim X=\dim X'$ and let $S\subset X$ be an analytic
set such that $\mu:X'\ssm S'\to X\ssm S$ is a biholomorphism. By
definition of multiplier ideal sheaves, $\cO(K_X)\otimes\cI(\varphi)$ is
just the sheaf of holomorphic $n$-forms $f$ on open sets $U\subset X$
such that
$\ii^{n^2}f\wedge\ol f\,e^{-2\varphi}\in L^1_\loc(U)$. Since
$\varphi$ is locally bounded from above, we may even consider forms $f$
which are a priori defined only on $U\ssm S$, because $f$ will be
in $L^2_\loc(U)$ and therefore will automatically extend through~$S$.
The change of variable formula yields
$$\int_U\ii^{n^2}f\wedge\ol f\,e^{-2\varphi}=
\int_{\mu^{-1}(U)}\ii^{n^2}\mu^\star f\wedge\ol{\mu^\star f}
\,e^{-2\varphi\circ\mu},$$
hence $f\in\Gamma(U,\cO(K_X)\otimes\cI(\varphi))$ iff
$\mu^\star f\in\Gamma(\mu^{-1}(U),\cO(K_{X'})\otimes\cI(\varphi\circ
\mu))$. Proposition~6.8 is proved.\qed
\endproof

\claim{(6.9) Remark} \rm If $\varphi$ has analytic singularities
(according to Definition~1.10), the computation of $\cI(\varphi)$
can be reduced to a purely algebraic problem.

The first observation is that $\cI(\varphi)$ can be computed easily if
$\varphi$ has the form $\varphi=\sum\alpha_j\log|g_j|$ where
$D_j=g^{-1}_j(0)$ are nonsingular irreducible divisors with normal
crossings. Then $\cI(\varphi)$ is the sheaf of functions $h$ on open
sets $U\subset X$ such that
$$\int_U|h|^2\prod|g_j|^{-2\alpha_j}dV<+\infty.$$
Since locally the $g_j$ can be taken to be coordinate functions from a
local coordinate system $(z_1\ld z_n)$, the condition is that
$h$ is divisible by $\prod g_j^{m_j}$ where $m_j-\alpha_j>-1$ for
each $j$, i.e.\ $m_j\ge\lfloor\alpha_j\rfloor$ (integer part). Hence
$$\cI(\varphi)=\cO(-\lfloor D\rfloor)=\cO(-\sum\lfloor\alpha_j\rfloor D_j)$$
where $\lfloor D\rfloor$ denotes the integral part of the $\bQ$-divisor
$D=\sum\alpha_jD_j$.

Now, consider the general case of analytic singularities and suppose
that $\varphi\sim{\alpha\over 2}\log\big(|f_1|^2+\cdots+|f_N|^2\big)$
near the poles. By the remarks after Definition~1.10, we may assume
that the $(f_j)$ are generators of the integrally closed ideal
sheaf $\cJ=\cJ(\varphi/\alpha)$, defined as the sheaf of holomorphic
functions $h$ such that $|h|\le C\exp(\varphi/\alpha)$. In this case,
the computation is made as follows (see also L.~Bonavero's work [Bon93],
where similar ideas are used in connection with ``singular'' holomorphic
Morse inequalities).

First, one computes a smooth modification $\mu:\wt X\to X$ of $X$ such
that $\mu^\star\cJ$ is an invertible sheaf $\cO(-D)$ associated with
a normal crossing divisor $D=\sum\lambda_jD_j$, where $(D_j)$ are the
components of the exceptional divisor of $\wt X$ (take the blow-up $X'$
of $X$ with respect to the ideal $\cJ$ so that the pull-back of $\cJ$
to $X'$ becomes an invertible sheaf $\cO(-D')$, then blow up again by
Hironaka [Hir64] to make $X'$ smooth and $D'$ have normal crossings).
Now, we have $K_{\wt X}=\mu^\star K_X+R$ where $R=\sum\rho_j
D_j$ is the zero divisor of the Jacobian function $J_\mu$ of the blow-up
map. By the direct image formula~6.8, we get
$$\cI(\varphi)=\mu_\star\big(\cO(K_{\wt X}-\mu^\star K_X)\otimes
\cI(\varphi\circ\mu)\big)=\mu_\star\big(\cO(R)\otimes
\cI(\varphi\circ\mu)\big).$$
Now, $(f_j\circ\mu)$ are generators of the ideal $\cO(-D)$, hence
$$\varphi\circ\mu\sim\alpha\sum\lambda_j\log|g_j|$$
where $g_j$ are local generators of $\cO(-D_j)$. We are thus reduced to
computing multiplier ideal sheaves in the case where the poles are
given by a $\bQ$-divisor with normal crossings $\sum\alpha\lambda_jD_j$.
We obtain $\cI(\varphi\circ\mu)=\cO(-\sum\lfloor\alpha\lambda_j\rfloor D_j)$,
hence
$$\cI(\varphi)=\mu_\star\cO_{\wt X}\big(\sum(\rho_j-\lfloor\alpha\lambda_j
\rfloor)D_j\big).\eqno\square$$
\endclaim

\claim{(6.10) Exercise} \rm Compute the multiplier ideal sheaf
$\cI(\varphi)$ associated with
$\varphi=\log(|z_1|^{\alpha_1}+\ldots+|z_p|^{\alpha_p})$
for arbitrary real numbers $\alpha_j>0$.\newline
{\it Hint:} using Parseval's formula and polar coordinates $z_j=r_j
e^{\ii\theta_j}$, show that the problem is equivalent to determining for
which $p$-tuples $(\beta_1\ld\beta_p)\in\bN^p$ the integral
$$\int_{[0,1]^p}{r_1^{2\beta_1}\ldots r_p^{2\beta_p}\,r_1dr_1\ldots r_pdr_p
\over r_1^{2\alpha_1}+\ldots+r_p^{2\alpha_p}}
=\int_{[0,1]^p}{t_1^{(\beta_1+1)/\alpha_1}\ldots t_p^{(\beta_p+1)/\alpha_p}
\over t_1+\ldots+t_p}\prod_{j=1}^p{1\over 2\alpha_j}{dt_j\over t_j}$$
is convergent. Conclude from this that $\cI(\varphi)$ is generated by
the monomials $z_1^{\beta_1}\ldots z_p^{\beta_p}$ such that
$\sum(\beta_p+1)/\alpha_p>1$. (This exercise shows that the analytic
definition of $\cI(\varphi)$ is sometimes also quite convenient for
computations).\qed
\endclaim

Let $F$ be a line bundle over $X$ with a singular metric $h$ of
curvature current $\Theta_h(F)$. If $\varphi$ is the weight
representing the metric in an open set $\Omega\subset X$, the ideal
sheaf $\cI(\varphi)$ is independent of the choice of the trivialization
and so it is the restriction to $\Omega$ of a global coherent sheaf
$\cI(h)$ on~$X$. We will sometimes still write $\cI(h)=\cI(\varphi)$ by
abuse of notation. In this context, we have the following fundamental
vanishing theorem, which is probably one of the most central results of
analytic and algebraic geometry (as we will see later, it contains the
Kawamata-Viehweg vanishing theorem as a special case).

\claim{(6.11) Nadel vanishing theorem {\rm([Nad89], [Dem93b])}} Let
$(X,\omega)$ be a K\"ahler weakly pseudoconvex manifold, and let $F$ be a
holomorphic line bundle over $X$ equipped with a singular hermitian metric
$h$ of weight~$\varphi$. Assume that $\ii\Theta_h(F)\ge\varepsilon\omega$
for some continuous positive function $\varepsilon$ on~$X$. Then
$$H^q\big(X,\cO(K_X+F)\otimes\cI(h)\big)=0\qquad
\hbox{for all $q\ge 1$.}$$
\endclaim

\begproof{} Let $\cL^q$ be the sheaf of germs of $(n,q)$-forms 
$u$ with values in $F$ and with measurable coefficients, such that
both $|u|^2e^{-2\varphi}$ and $|d''u|^2e^{-2\varphi}$ are locally
integrable. The $d''$ operator defines a complex of sheaves
$(\cL^\bu,d'')$ which is a resolution of the sheaf
$\cO(K_X+F)\otimes\cI(\varphi)$: indeed, the kernel of $d''$ in degree
$0$ consists of all germs of holomorphic $n$-forms with values in $F$
which satisfy the integrability condition; hence the coefficient
function lies in $\cI(\varphi)$; the exactness in degree $q\ge 1$
follows from Corollary~6.3 applied on arbitrary small balls.
Each sheaf $\cL^q$ is a ${\cal C}^\infty$-module, so $\cL^\bu$ 
is a resolution by acyclic sheaves. Let $\psi$ be a smooth psh
exhaustion function on~$X$. Let us apply Corollary~6.3 globally on~$X$,
with the original metric of $F$ multiplied by the factor
$e^{-\chi\circ\psi}$, where $\chi$ is a convex increasing function of
arbitrary fast growth at infinity. This factor can be used to ensure
the convergence of integrals at infinity. By Corollary~6.3, we conclude
that $H^q\big(\Gamma(X,\cL^\bu)\big)=0$ for $q\ge 1$. The theorem
follows.\qed
\endproof 

\claim{(6.12) Corollary} Let $(X,\omega)$, $F$ and $\varphi$ be as in
Theorem~6.11 and let $x_1\ld x_N$ be isolated points in the zero
variety~$V(\cI(\varphi))$. Then there is a surjective map
$$H^0(X,K_X+F)\lraww\bigoplus_{1\le j\le N}\cO(K_X+L)_{x_j}\otimes
\big(\cO_X/\cI(\varphi)\big)_{x_j}.$$
\endclaim

\begproof{} Consider the long exact sequence of cohomology associated to the
short exact sequence $0\to\cI(\varphi)\to\cO_X\to\cO_X/\cI(\varphi)\to 0$
twisted by $\cO(K_X+F)$, and apply Theorem~6.11 to obtain the vanishing of
the first $H^1$ group. The asserted surjectivity property follows.\qed
\endproof

\claim{(6.13) Corollary} Let $(X,\omega)$, $F$ and $\varphi$ be as in
Theorem~6.11 and suppose that the weight function $\varphi$ is such that
$\nu(\varphi,x)\ge n+s$ at some point $x\in X$ which is an isolated point
of $E_1(\varphi)$. Then $H^0(X,K_X+F)$ generates all $s$-jets at~$x$.
\endclaim

\begproof{} The assumption is that $\nu(\varphi,y)<1$ for $y$ near $x$,
$y\ne x$. By Skoda's lemma 6.6~b), we conclude that $e^{-2\varphi}$ is
integrable at all such points~$y$, hence $\cI(\varphi)_y=\cO_{X,y}$, whilst
$\cI(\varphi)_x\subset\gm_{X,x}^{s+1}$ by 6.6~a). Corollary 6.13 is thus
a special case of 6.12.\qed
\endproof

The philosophy of these results (which can be seen as generalizations
of the H\"ormander-Bombieri-Skoda theorem [Bom70], [Sko72a,~75]) is that
the problem of constructing holomorphic sections of $K_X+F$ can be
solved by constructing suitable hermitian metrics on $F$ such that the
weight $\varphi$ has isolated poles at given points~$x_j$. The following
result gives a somewhat general result in this direction.

\claim{(6.14) Theorem} Let $X$ be a compact complex manifold, $E$ a holomorphic
vector bundle and $(F,h_F)$ a hermitian line bundle with a smooth metric
$h$ such that 
$$
\omega=\ii\Theta_{h_F}(F)>0.
$$ 
Let $\varphi$ be a quasi-psh function on $X$, i.e.\ a function
$\varphi$ such that $\ii d'd''\varphi\ge -C\omega$ for some constant
$C>0$. Then 
\smallskip
\item{\rm a)} There exists an integer $m_0$ such that
$$
H^q(X,E\otimes F^{\otimes m}\otimes\cI(\varphi))=0
$$
for $q\ge 1$ and $m\ge m_0$. 
\smallskip
\item{\rm b)} The restriction map
$$
H^0(X,E\otimes F^{\otimes m}\otimes\cI(\varphi))\lra
H^0(X,E\otimes F^{\otimes m}\otimes\cO_X/\cI(\varphi))
$$
is surjective for $m\ge m_0$.
\smallskip
\item{\rm b)} The vector bundle  $E\otimes F^{\otimes m}$ generates its
sections $($or jets of any order $s)$ for $m\ge m_0(s)$ large enough.
\vskip 0pt
\endclaim

\begproof{} a) Put an arbitrary smooth hermitian metric $h_E$ en $E$ and
consider the singular hermitian metric $h_E\cdot h_F^m\cdot e^{-\varphi}$
on $E\otimes F^{\otimes m}$. The $L^2$ holomorphic sections of
$E\otimes F^{\otimes m}$ are exactly the sections of the sheaf 
$E\otimes F^{\otimes m}\otimes\cI(\varphi)$. On the other hand, the 
curvature of the metric is
$$
\ii\Theta_{h_E}(E)+(m\,\ii\Theta_{h_F}(F)+\ii d'd''\varphi)\otimes\Id_E
$$
and therefore the curvature is Nakano${}>0$ for $m$ large. This implies
the vanishing of $H^q(X,E\otimes F^{\otimes m}\otimes\cI(\varphi))$. 
\smallskip\noindent
b) The vanishing of $H^1$ itself implies the surjevtivity statement on the
$H^0$ groups by the same argument as in Corollary (6.12).
\smallskip\noindent
c) Clearly, one can construct a quasi-psh function $\varphi$ with
a single logarithmic pole at a point $x\in X$ by taking
$$
\varphi(z)=\theta(z)(n+s-1)\log\sum|z_j-x_j|^2
$$
in some local coordinates near $x$, where $\theta$ is a cut-off function
with support in the coordinate open set. Then $\cI(\varphi)=\gm_x^s$ and
we conclude by b).\qed
\endproof

\claim{(6.15) Remark} \rm Assume that $X$ is compact and that $F$ is
a positive line bundle on~$X$. Let $\{x_1\ld x_N\}$ be a finite set.
Show that there are constants $C_1,C_2\ge 0$ depending only on $X$ and $E$, $F$
such that $H^0(X,E\otimes F^{\otimes m})$ interpolates given jets of order 
$s_j$ at $x_j$ for $m\ge C_1\sum s_j+C_2$. To see this,  we take a quasi-psh 
weight
$$
\varphi(z)=\sum\theta_j(z)(n+s_j-1)\log|w^{(j)}(z)|
$$
with respect to coordinate systems $(w^{(j)}_k(z))_{1\le k\le n}$ centered 
at $x_j$. The cut-off functions can be taken of a fixed radius (bounded away
from~$0$) with respect to a finite collection of coordinate patches 
covering~$X$. It is then easy to see that $\ii d'd''\varphi\ge -C(\sum s_j+1)
\omega$.\qed
\endclaim

\claim{(6.16) Theorem {\rm(Kodaira [Kod54])}} Let $X$ be a compact complex
manifold. A line bundle $L$ on $X$ is ample if and only if $L$ is positive.
In particular, a manifold $X$ posessing a positive line bundle is
projective, and can be embedded in projective space via the canonical
map $\Phi_{|mL|}:X\to\bP^N$ for $N$ large.
\endclaim

\begproof{} If the line bundle $L$ is ample, then by definition the
canonical map $\Phi_{|mL|}:X\to\bP^N$ is an embedding for $m$ large,
and $mL=\smash{\Phi_{|mL|}^{-1}}\cO(1)$. This implies that $L$ can be
equipped with a metric of positive curvature, as we saw in (1.15).
Conversely, if $L$ possesses a metric with positive curvature, then
by Theorem 6.14$\,$c), there exists an integer $m_0$ such that for $m\ge m_0$
sections in $H^0(X,mL)$ separate any pair of points $\{x,y\}\subset X$
and generate $1$-jets of sections at every point $x\in X$. However,
as is easily seen, separation of points is equivalent to the injectivity
of the map $\Phi_{|mL|}$, and the generation of $1$-jets is equivalent
to the fact that $\Phi_{|mL|}$ is an immersion.\qed
\endproof

\claim{(6.17) Lemma} Let $X$ be a compact complex manifold~$X$ 
equipped with a K\"ahler metric 
$\omega=i\sum_{1\le j,k\le n}\omega_{jk}(z)dz_j\wedge d\ol z_k$
and let $Y\subset X$ be an analytic subset of~$X$. Then there exist
globally defined quasi-psh potentials $\psi$ and 
$(\psi_\varepsilon)_{\varepsilon\in{}]0,1]}$ on $X$,
satisfying the following properties.
{\oldparindent=6.6mm
\smallskip
\item {\rm (i)} The function $\psi$ is smooth on $X\ssm Y$, satisfies
$i\ddbar\psi\ge -A\omega$ for some $A>0$, and $\psi$
has logarithmic poles along~$Y$, i.e., locally near $Y$
$$
\psi(z)\sim \log\sum_k|g_k(z)|+O(1)
$$
where $(g_k)$ is a local system of generators of the ideal sheaf $\cI_Y$
of $Y$ in $X$.
\smallskip
\item {\rm(ii)} We have 
$\psi=\lim_{\varepsilon\to 0}\downarrow\psi_\varepsilon$ 
where the $\psi_\varepsilon$ are $C^\infty$ and possess a uniform 
Hessian estimate
$$
i\ddbar\psi_\varepsilon\ge -A\omega\qquad\hbox{on~$X$}.
$$}
\endclaim

\begproof{} By compactness of $X$, there is a covering of $X$ by open
coordinate balls~$B_j$, $1\le j\le N$, such that $\cI_Y$ is generated by 
finitely many holomorphic functions $(g_{j,k})_{1\le k\le m_j}$ on a 
neighborhood of $\ol B_j$. We take a partition of unity
$(\theta_j)$ subordinate to $(B_j)$ such that $\sum\theta_j^2=1$ on $X$,
and define
$$
\eqalign{
\psi(z)&={1\over 2}\log\sum_j\theta_j(z)^2\sum_k|g_{j,k}(z)|^2,\cr
\psi_\varepsilon(z)&={1\over 2}\log(e^{2\psi(z)}+\varepsilon^2)
={1\over 2}\log\Big(\sum_{j,k}\theta_j(z)^2|g_{j,k}(z)|^2+\varepsilon^2
\Big).\cr}
$$
Moreover, we consider the family of $(1,0)$-forms with support in $B_j$
such that
$$
\gamma_{j,k}=\theta_j\partial g_{j,k}+2g_{j,k}\partial\theta_j.
$$
Straightforward calculations yield
$$
\leqalignno{
\dbar\psi_\varepsilon&={1\over 2}
{\sum_{j,k}\theta_jg_{j,k}\ol{\gamma_{j,k}}\over e^{2\psi}+
\varepsilon^2},\cr
i\ddbar\psi_\varepsilon&={i\over 2}\left(
{\sum_{j,k}\gamma_{j,k}\wedge\ol{\gamma_{j,k}}\over e^{2\psi}+
\varepsilon^2}-
{\sum_{j,k}\theta_j\ol{g_{j,k}}\gamma_{j,k}
\wedge \sum_{j,k}\theta_jg_{j,k}\ol{\gamma_{j,k}}
\over (e^{2\psi}+\varepsilon^2)^2}\right),&(6.17\,{\rm a})\cr
&\qquad{}+i\;
{\sum_{j,k}|g_{j,k}|^2(\theta_j\ddbar\theta_j-\partial\theta_j\wedge
\dbar\theta_j)\over e^{2\psi}+\varepsilon^2}.\cr
}
$$
As $e^{2\psi}=\sum_{j,k}\theta_j^2|g_{j,k}|^2$, the first big sum in
$i\ddbar\psi_\varepsilon$ is nonnegative by the Cauchy-Schwarz inequality;
when viewed as a hermitian form, the value of this sum on a tangent 
vector $\xi\in T_X$ is simply
$$
{1\over 2}\left(
{\sum_{j,k}|\gamma_{j,k}(\xi)|^2\over e^{2\psi}+\varepsilon^2}-
{\big|\sum_{j,k}\theta_j\ol{g_{j,k}}\gamma_{j,k}(\xi)\big|^2\over
(e^{2\psi}+\varepsilon^2)^2}\right)\ge
{1\over 2}{\varepsilon^2\over(e^{2\psi}+\varepsilon^2)^2}
\sum_{j,k}|\gamma_{j,k}(\xi)|^2.\leqno(6.17\,{\rm b})
$$
Now, the second sum involving $\theta_j\ddbar\theta_j-\partial\theta_j\wedge
\dbar\theta_j$ in $(6.17\,{\rm a})$ is uniformly bounded below by a 
fixed negative 
hermitian form $-A\omega$, $A\gg 0$, and therefore
\hbox{$i\ddbar\psi_\varepsilon\ge -A\omega$}. Actually, for every pair
of indices $(j,j')$ we have a bound
$$
C^{-1}\le \sum_k|g_{j,k}(z)|^2/\sum_k|g_{j',k}(z)|^2\le C\qquad
\hbox{on $\ol B_j\cap\ol B_{j'}$},
$$
since the generators $(g_{j,k})$ can be expressed as holomorphic
linear combinations of the $(g_{j',k})$ by Cartan's theorem A (and
vice versa). It follows easily that all terms $|g_{j,k}|^2$ are uniformly
bounded by $e^{2\psi}+\varepsilon^2$. In particular, $\psi$ and
$\psi_\varepsilon$ are quasi-psh, and we see that 
(i) and (ii) hold true.\qed
\endproof

\claim{(6.18) Theorem {\rm(Chow [Chw49])}} Let $X\subset\bP^N$ be a 
$($closed$)$ complex analytic subset of $\bP^N$. Then $X$ is algebraic
and can be defined as the common zero set of a finite collection
of homogeneous polynomials $P_j(z_0,z_1,\ldots,z_N)=0$, $1\le j\le k$.
\endclaim

\begproof{} By Lemma 6.17 (i), there exists a quasi psh function $\psi$
with logarithmic poles along $X$. Then $\cI(2N\psi)\subset\cI_X$, since
$e^{-2N\psi}$ is certainly not integrable along~$Z$. For $x\in\bP^N\ssm X$, 
we consider a quasi-psh weight
$$
\varphi_x(z)=\psi(z)+\theta_x(z)(2N)\log|z-x|
$$
with a suitable cut-off function with support on a neighborhood of $x$
possessing holomorphic coordinates, in such a way that
$\cI(\varphi_x)_x=\gm_{\bP^N,x}$. We can arrange that $\ii\ddbar\varphi_x
\ge -C\omega$ uniformly for all $x$. Then we have vanishing of
$H^1(\bP^N,\cO(m)\otimes\cI(\varphi_x))$ for $m\ge m_0$ and therefore
we get a surjective map
$$
H^0(\bP^N,\cO(m))\to
H^0(\bP^N,\cO(m)\otimes\cO_{\bP^N}/\cI(\varphi_x)).
$$
As a consequence, we can find a homogeneous polynomial of degree $m$
which takes value $1$ at $x$ and vanishes on $X$ (as prescribed
by the ideal sheaf $\cI(\psi)$).\qed
\endproof

In the case of submanifolds, one can of course prove a slightly more
precise result, by demanding that the polynomials $P_j$ have non
vanishing differentials along~$X$.

\claim{(6.19) Theorem {\rm(Chow [Chw49])}} Let $X\subset\bP^N$ be a complex
analytic submanifold of dimension $n$. Then $X$ is projective algebraic 
and can be defined
by a collection of homogeneous polynomials $P_j(z_0,z_1,\ldots,z_N)=0$,
$1\le j\le k$, such that the system of differentials $(dP_j)$ has rank 
equal to $\codim X$ at every point of $X$.
\endclaim

\begproof{} Put $r=\codim X=N-n$. There exists a quasi-psh function 
$\varphi$ which has logarithmic poles along $X$. To see this, just take
an open overing of $\bP^N$ by open sets $U_j$ where $X$ is defined by
$w^{(j)}_1=\ldots=w^{(j)}_r=0$ and $(w^{(j)}_k)_{1\le k\le N}$ is
a suitable coordinate system on $U_j$. As in the proof of Lemma 6.17,
the function
$$
\varphi(z)=\log\Big(\sum\theta_j^2(z)\big(
|w^{(j)}_1|^2+\ldots+|w^{(j)}_r|^2\big)\Big)
$$
is quasi-psh if the functions $\theta_j$ are cut-off functions with support 
in $U_j$ such that $\sum\theta_j^2=1$. An easy calculation also shows that
$\cI((r+1)\varphi)=\cI_X^2$ where $\cI_X$ is the reduced ideal sheaf of $X$ 
in $\bP^N$. Hence for $m\ge m_0$ large enough, we have
$$
H^q(\bP^N,\cO(m)\otimes\cI_X^2)=0
$$
for $q\ge 1$. The long-exact sequence associated with
$$
0\to \cI_X^2\to \cI_X\to\cI_X/\cI_X^2\to 0
$$
twisted by $\cO(m)$ implies the surjectivity of
$$
H^0(\bP^N,\cO(m)\otimes\cI_X)\to H^0(X,\cO(m)\otimes\cI_X/\cI_X^2).
$$
However, $\cI_X/\cI_X^2$ can be identified with the conormal bundle
$N_X^*$ (where $N_X=T_{\bP^N|X}/T_X$), and we infer from this that
$H^0(X,\cO(m)\otimes\cI_X/\cI_X^2)$ is generated by its global sections 
for $m\ge m_1$ large
enough by Theorem 6.14$\,$c). This means that we can generate any
$1$-differential of $N_X^*$ at any point $x\in X$ as the differential
of a section 
$$
P\in H^0(\bP^N,\cO(m)\otimes\cI_X)\subset H^0(\bP^N,\cO(m)),
$$
i.e.\ a homogeneous polynomial of degree $m$ vanishing on $X$,
for $m\ge\max(m_0,m_1)$.\qed
\endproof

\claim{(6.20) Corollary {\rm(Kodaira [Kod54])}} Let $X$ be a compact complex 
manifold and $\dim_\bC X=n$. The following conditions are equivalent.
\smallskip
\item{\rm a)} $X$ is projective algebraic, i.e.\ $X$ can be embedded as an
algebraic submanifold of the complex projective space $\bP^N$ for $N$ large.
\smallskip
\item{\rm b)} $X$ carries a positive line bundle $L$.
\smallskip
\item{\rm c)} $X$ carries a Hodge metric,
i.e.\ a K\"ahler metric $\omega$ with rational cohomology class
$\{\omega\}\in H^2(X,\bQ)$.\smallskip
\endclaim

\begproof{} a) $\Leftrightarrow$ b). This follows from Theorems 6.16 and 6.19
combined.
\medskip
\noindent b) $\Rightarrow$ c). Take $\omega={\ii\over 2\pi}\Theta(L)\,;$
then $\{\omega\}$ is the image of $c_1(L)\in H^2(X,\bZ)$.
\medskip
\noindent c) $\Rightarrow$ b). We can multiply $\{\omega\}$ by a common
denominator of its coefficients and suppose that $\{\omega\}$ is in the image
of $H^2(X,\bZ)$. Then a classical result due to A.~Weil shows that
there exists a hermitian line bundle $(L,h)$ such that 
${\ii\over 2\pi}\Theta_h(L)=\omega$. This bundle $L$ is then positive.
\qed
\endproof

\claim{(6.21) Exercise {\rm (solution of the Levi problem)}} \rm Show
that the following two properties are equivalent.
\smallskip
\item{\rm a)} $X$ is strongly pseudoconvex, i.e.\ $X$ admits a strongly psh
exhaustion function.
\smallskip
\item{\rm b)} $X$ is Stein, i.e.\ the global holomorphic functions
$H^0(X,\cO_X)$ separate points and yield local coordinates at any point,
and $X$ is holomorphically convex $($this means that for any discrete
sequence $z_\nu$ there is a function $f\in H^0(X,\cO_X)$ such that
$|f(z_\nu)|\to\infty)$.\qed
\vskip0pt
\endclaim

\claim{(6.22) Remark} \rm As long as forms of bidegree $(n,q)$ are
considered, the $L^2$ estimates can be extended to complex spaces with
arbitrary singularities. In fact, if $X$ is a complex space and $\varphi$
is a psh weight function on~$X$, we may still define a sheaf $K_X(\varphi)$
on~$X$, such that the sections on an open set $U$ are the holomorphic
$n$-forms $f$ on the regular part $U\cap X_\reg$, satisfying the
integrability condition $\ii^{n^2}f\wedge\ol f\,e^{-2\varphi}\in
L^1_\loc(U)$. In this setting, the functoriality property 6.8 becomes
$$\mu_\star\big(K_{X'}(\varphi\circ\mu)\big)=K_X(\varphi)$$
for arbitrary complex spaces $X$, $X'$ such that $\mu:X'\to X$
is a modification. If $X$ is nonsingular we have $K_X(\varphi)=
\cO(K_X)\otimes\cI(\varphi)$, however, if $X$ is
singular, the symbols $K_X$ and $\cI(\varphi)$ must not be
dissociated. The statement of the Nadel vanishing theorem becomes
$H^q(X,\cO(F)\otimes K_X(\varphi))=0$ for $q\ge 1$, under the same
assumptions ($X$ K\"ahler and weakly pseudoconvex, curvature
$\ge\varepsilon\omega$). The proof can be obtained by restricting
everything to $X_\reg$. Although in general $X_\reg$ is not weakly
pseudoconvex (e.g.\ in case $\codim X_\sing\ge 2$), $X_\reg$ is
always K\"ahler complete (the complement of a proper analytic subset in
a K\"ahler weakly pseudoconvex space is complete K\"ahler, see
e.g.\ [Dem82a]). As a consequence, Nadel's vanishing theorem is
essentially insensitive to the presence of singularities.\qed
\endclaim

\titleb{7.}{Numerically Effective and Pseudoeffective Line Bundles}
Many problems of algebraic geometry (e.g.\ problems of classification of
algebraic surfaces or higher dimensional varieties) lead in a natural
way to the study of line bundles satisfying semipositivity conditions.
It turns out that semipositivity in the sense of curvature (at least,
as far as smooth metrics are considered) is not a very satisfactory
notion. More flexible notions perfectly suitable for algebraic purposes
are the notion of {\it numerical effectivity} and {\it pseudoeffectivity}.
The goal of this section
is to give the relevant algebraic definitions and to discuss their
differential geometric counterparts. We first suppose that $X$ is a
projective algebraic manifold, $\dim X=n$.

\claim{(7.1) Definition} A holomorphic line bundle $L$ over a projective
manifold $X$ is said to be numerically effective, nef for short, if
$L\cdot C=\int_C c_1(L)\ge 0$ for every curve $C\subset X$.
\endclaim

If $L$ is nef, it can be shown that $L^p\cdot Y=\int_Y c_1(L)^p\ge 0$
for any $p$-dimensional subvariety $Y\subset X$ (see e.g.\ [Har70]).
In relation to this, let us recall the Nakai-Moishezon ampleness
criterion: a line bundle $L$ is ample if and only if $L^p\cdot Y>0$ for
every $p$-dimensional subvariety~$Y$. From this, we easily infer

\claim{(7.2) Proposition} Let $L$ be a line bundle on a projective
algebraic manifold~$X$, on which an ample line bundle $A$ and a hermitian
metric $\omega$ are given. The following properties are equivalent:
\smallskip
\item{\rm a)} $L$ is nef$\,;$
\smallskip
\item{\rm b)} for any integer $k\ge 1$, the line bundle $kL+A$ is ample$\,;$
\smallskip
\item{\rm c)} for every $\varepsilon>0$, there is a smooth metric
$h_\varepsilon$ on $L$ such that $\ii\Theta_{h_\varepsilon}(L)\ge
-\varepsilon\omega$.
\vskip0pt
\endclaim

\begproof{} a) $\Rightarrow$ b). If $L$ is nef and $A$ is ample then
clearly $kL+A$ satisfies the Nakai-Moishezon criterion, hence
$kL+A$ is ample.
\smallskip
\noindent b) $\Rightarrow$ c). Condition c) is independent of the choice
of the hermitian metric, so we may select a metric $h_A$ on $A$ with
positive curvature and set $\omega=\ii\Theta(A)$. If $kL+A$ is ample,
this bundle has a metric $h_{kL+A}$ of positive curvature. Then the
metric $h_L=(h_{kL+A}\otimes h_A^{-1})^{1/k}$ has curvature
$$\ii\Theta(L)={1\over k}\big(\ii\Theta(kL+A)-\ii\Theta(A)\big)\ge
-{1\over k}\ii\Theta(A)\,;$$
in this way the negative part can be made smaller than $\varepsilon\,\omega$
by taking $k$ large enough.
\smallskip
\noindent c) $\Rightarrow$ a). Under hypothesis c), we get
$L\cdot C=\int_C{\ii\over2\pi}\Theta_{h_\varepsilon}(L)\ge-
{\varepsilon\over2\pi}\int_C\omega$ for every curve $C$ and every
$\varepsilon>0$, hence $L\cdot C\ge 0$ and $L$ is nef.\qed
\endproof

Let now $X$ be an arbitrary compact complex manifold. Since there need
not exist any curve in $X$, Property 7.2~c) is simply taken as a definition
of nefness ([DPS94]):

\claim{(7.3) Definition} A line bundle $L$ on a compact complex
manifold $X$ is said to be nef if for every $\varepsilon>0$, there is a
smooth hermitian metric $h_\varepsilon$ on $L$ such that
$\ii\Theta_{h_\varepsilon}(L)\ge -\varepsilon\omega$.
\endclaim

In general, it is not possible to extract a smooth limit $h_0$ such that
$\ii\Theta_{h_0}(L)\ge 0$. The following simple example is given in
[DPS94] (Example~1.7). Let $E$ be a non trivial extension
$0\to\cO\to E\to\cO\to 0$ over an elliptic curve $C$ and let $X=P(E)$
be the corresponding ruled surface over $C$. Then $L=\cO_{P(E)}(1)$ is
nef but does not admit any smooth metric of nonnegative curvature.
This example answers negatively a question raised by [Fuj83].

Let us now introduce the important concept of {\it Kodaira-Iitaka dimension}
of a line bundle.

\claim{(7.4) Definition} If $L$ is a line bundle, the Kodaira-Iitaka
dimension $\kappa(L)$ is the supremum of the rank of the canonical maps
$$\Phi_m:X\ssm B_m\lra P(V_m^\star),~~~~x\longmapsto H_x=\{\sigma\in
V_m\,;\,\sigma(x)=0\},~~~~m\ge 1$$
with $V_m=H^0(X,mL)$ and $B_m=\bigcap_{\sigma\in V_m}\sigma^{-1}(0)={}$base
locus of $V_m$. In case $V_m=\{0\}$ for all $m\ge 1$, we set $\kappa(L)=
-\infty$.\newline
A line bundle is said to be big if $\kappa(L)=\dim X$.
\endclaim

The following lemma is well-known (the proof is a rather elementary
consequence of the Schwarz lemma).

\claim{(7.5) Serre-Siegel lemma {\rm([Ser54], [Sie55])}} Let $L$ be any
line bundle on a compact complex manifold. Then we have
$$h^0(X,mL)\le O(m^{\kappa(L)})\qquad\hbox{for $m\ge 1$,}$$
and $\kappa(L)$ is the smallest constant for which this estimate holds.\qed
\endclaim

We now discuss the various concepts of positive cones in the space of
numerical classes of line bundles, and establish a simple dictionary
relating these concepts to corresponding concepts in the context of
differential geometry.

Let us recall that an integral cohomology class in
$H^2(X,\bZ)$ is the first Chern class of a holomorphic (or algebraic)
line bundle if and only if it lies in the
{\it Neron-Severi} group 
$$\NS(X)=\Ker\big(H^2(X,\bZ)\to H^2(X,\cO_X)\big)$$
(this fact is just an elementary consequence of the exponential exact
sequence $0\to\bZ\to\cO\to\cO^\star\to 0$). If $X$ is compact
K\"ahler, as we will suppose from now on in this section,
this is the same as saying that the class is of type $(1,1)$
with respect to Hodge decomposition.

Let $\NS_\bR(X)$ be the real vector space $\NS(X)\otimes\bR\subset
H^2(X,\bR)$. We define four convex cones
$$\eqalign{
&\cK_\NS^\amp(X)\subset \cE_\NS^\eff(X)\subset \NS_\bR(X),\cr
&\cK_\NS^\nef(X)\subset \cE_\NS^\psef(X)\subset\NS_\bR(X)\cr}$$
which are, respectively, the {\it convex cones} generated by Chern classes
$c_1(L)$ of ample and effective line bundles, resp.\ the {\it closure of
the convex cones} generated by numerically effective and pseudo-effective
line bundles; we say that $L$ is effective if $mL$ has a section for some
$m>0$, i.e.\ if $\cO(mL)\simeq\cO(D)$ for some effective divisor $D\,$; and
we say that $L$ pseudo-effective if $c_1(L)$ is the cohomology class of some
closed positive current~$T$, i.e.\ if $L$ can be equipped with a singular
hermitian metric $h$ with $T={\ii\over 2\pi}\Theta_h(L)\ge 0$ as a current.
For each of the ample, effective, nef and pseudo-effective
cones, the first Chern class $c_1(L)$ of a line bundle $L$ lies in the
cone if and only if $L$ has the corresponding property (for $\cE_\NS^\psef$
use the fact that the space of positive currents of mass $1$ is
weakly compact; the case of all other cones is obvious).

\claim{(7.6) Proposition} Let $(X,\omega)$ be a compact K\"ahler manifold.
The numerical cones satisfy the following properties.
\smallskip
\item{\rm a)} $\cK_\NS^\amp=(\cK_\NS^\amp)^\circ\subset(\cK_\NS^\nef)^\circ$,~~
$\cK_\NS^\nef\subset \cE_\NS^\psef$.
\smallskip
\item{\rm b)} If moreover $X$ is projective algebraic, we have
$\cK_\NS^\amp=(\cK_\NS^\nef)^\circ$ $($therefore $\ol{\cK_\NS^\amp}=\cK_\NS^\nef)$,
and $\ol{\cE_\NS^\eff}=\cE_\NS^\psef$.
\smallskip\noindent
If $L$ is a line bundle on $X$ and $h$ denotes a hermitian metric on $L$,
the following properties are equivalent:
\smallskip
\item{\rm c)} $c_1(L)\in \cK_\NS^\amp$
$\Leftrightarrow$ $\exists\varepsilon>0$, $\exists h$ smooth such that
$\ii\Theta_h(L)\ge\varepsilon\omega$.
\smallskip
\item{\rm d)} $c_1(L)\in \cK_\NS^\nef$
$\Leftrightarrow$ $\forall\varepsilon>0$, $\exists h$ smooth such that
$\ii\Theta_h(L)\ge-\varepsilon\omega$.
\smallskip
\item{\rm e)} $c_1(L)\in \cE_\NS^\psef$
$\Leftrightarrow$ $\exists h$ possibly singular such that
$\ii\Theta_h(L)\ge 0$.
\smallskip
\item{\rm f)} If moreover $X$ is projective algebraic, then\newline
$c_1(L)\in (\cE_\NS^\eff)^\circ$~$\Leftrightarrow\kappa(L)=\dim X$\newline
\phantom{$c_1(L)\in (\cE_\NS^\eff)^\circ$}~$\Leftrightarrow\exists\varepsilon>0$,
$\exists h$ possibly singular such that
$\ii\Theta_h(L)\ge\varepsilon\omega$.
\vskip0pt
\endclaim  

\begproof{} c) and d) are already known and e) is a definition.
\smallskip\noindent
a) The ample cone $\cK_\NS^\amp$ is always open by definition and contained
in $\cK_\NS^\nef$, so the first inclusion is obvious ($\cK_\NS^\amp$ is
of course empty if $X$ is not projective algebraic). Let us now prove
that $\cK_\NS^\nef\subset \cE_\NS^\psef$. Let $L$ be a line bundle with
$c_1(L)\in \cK_\NS^\nef$. Then for every $\varepsilon>0$, there is
a current $T_\varepsilon={\ii\over 2\pi}\Theta_{h_\varepsilon}(L)\ge
-\varepsilon\omega$. Then $T_\varepsilon+\varepsilon\omega$ is
a closed positive current and the family is uniformly bounded
in mass for $\varepsilon\in{}]0,1]$, since
$$\int_X(T_\varepsilon+\varepsilon\omega)\wedge\omega^{n-1}=
\int_X c_1(L)\wedge\omega^{n-1}+\varepsilon\int_X\omega^n.$$
By weak compactness, some subsequence converges to a weak limit
$T\ge 0$ and $T\in c_1(L)$ (the cohomology class $\{T\}$ of a current
is easily shown to depend continuously on $T$ with respect to the weak
topology; use e.g.\ Poincar\'e duality to check this).
\smallskip\noindent
b) If $X$ is projective, the equality $\cK_\NS^\amp=(\cK_\NS^\nef)^\circ$
is a simple consequence of 7.2~b) and of the fact that ampleness (or
positivity) is an open property. It remains to show that
$\cE_\NS^\psef\subset\ol{\cE_\NS^\eff}$. Let $L$ be a line bundle with
$c_1(L)\in \cE_\NS^\psef$ and let $h_L$ be a singular hermitian on $L$
such that $T={\ii\over 2\pi}\Theta(L)\ge 0$. Fix a point $x_0\in X$
such that the Lelong number of $T$ at $x_0$ is zero, and take a sufficiently
positive line bundle $A$ (replacing $A$ by a multiple if necessary), such
that $A-K_X$ has a singular metric $h_{A-K_X}$ of curvature
$\ge\varepsilon\omega$ and such that $h_{A-K_X}$ is smooth on
$X\ssm\{x_0\}$ and has an isolated logarithmic pole of Lelong number $\ge n$
at~$x_0$. Then apply Corollary 6.13 to $F=mL+A-K_X$ equipped with
the metric $h_L^{\otimes m}\otimes h_{A-K_X}$. Since the weight
$\varphi$ of this metric has a Lelong number $\ge n$ at $x_0$ and
a Lelong number equal to the Lelong number of $T={\ii\over2\pi}\Theta(L)$
at nearby points, $\limsup_{x\to x_0}\nu(T,x)=\nu(T,x_0)=0$,
Corollary 6.13 implies that $H^0(X,K_X+F)=H^0(X,mL+A)$ has a section
which does not vanish at~$x_0$. Hence there is an effective divisor $D_m$
such that $\cO(mL+A)=\cO(D_m)$ and $c_1(L)={1\over m}\{D_m\}-{1\over m}
c_1(A)=\lim{1\over m}\{D_m\}$ is in ${\ol\cE_\NS^\eff}$.\qed
\smallskip

\noindent
f) Fix a nonsingular ample divisor $A$. If $c_1(L)\in(\cE_\NS^\eff)^\circ$,
there
is an integer $m>0$ such that $c_1(L)-{1\over m}c_1(A)$ is still effective,
hence for $m,p$ large we have $mpL-pA=D+F$ with an effective divisor $D$
and a numerically trivial line bundle $F$. This implies
$\cO(kmpL)=\cO(kpA+kD+kF)\supset\cO(kpA+kF)$, hence $h^0(X,kmpL)\ge h^0(X,kpA+
kF)\sim (kp)^nA^n/n!$ by the Riemann-Roch formula. Therefore $\kappa(L)=n$.

If $\kappa(L)=n$, then $h^0(X,kL)\ge ck^n$ for $k\ge k_0$ and $c>0$. 
The exact cohomology sequence
$$0\lra H^0(X,kL-A)\lra H^0(X,kL)\lra H^0(A,kL_{\restriction A})$$
where $h^0(A,kL_{\restriction A})=O(k^{n-1})$ shows that
$kL-A$ has non zero sections for $k$ large. If $D$ is the divisor of
such a section, then $kL\simeq\cO(A+D)$. Select a smooth metric
on $A$ such that ${\ii\over2\pi}\Theta(A)\ge\varepsilon_0\omega$ for
some $\varepsilon_0>0$, and take the singular metric 
on $\cO(D)$ with weight function $\varphi_D=\sum\alpha_j\log|g_j|$
described in Example~3.14. Then the metric with weight $\varphi_L=
{1\over k}(\varphi_A+\varphi_D)$ on $L$ yields
$${\ii\over 2\pi}\Theta(L)={1\over k}
\Big({\ii\over2\pi}\Theta(A)+[D]\Big)\ge(\varepsilon_0/k)\,\omega,$$
as desired.

Finally, the curvature condition $\ii\Theta_h(L)\ge\varepsilon\omega$ in the
sense of currents yields by definition $c_1(L)\in(\cE_\NS^\psef)^\circ$.
Moreover, b) implies ($\cE_\NS^\psef)^\circ=(\cE_\NS^\eff)^\circ$.\qed
\endproof

Before going further, we need a lemma.

\claim{(7.7) Lemma} Let $X$ be a compact K\"ahler $n$-dimensional
manifold, let $L$ be a nef line bundle on~$X$, and let $E$ be an
arbitrary holomorphic vector bundle. Then
$h^q(X,\cO(E)\otimes\cO(kL))=o(k^n)$
as $k\to+\infty$, for every $q\ge1$. If $X$ is projective algebraic, the
following more precise bound holds:
$$h^q(X,\cO(E)\otimes\cO(kL))=O(k^{n-q}),\qquad\forall q\ge0.$$
\endclaim

\begproof{} The K\"ahler case will be proved in Section~12, as a
consequence of the holomorphic Morse inequalities. In the projective
algebraic case, we proceed by induction on $n=\dim X$. If $n=1$ the result
is clear, as well as if $q=0$. Now let $A$ be a nonsingular ample divisor
such that $E\otimes\cO(A-K_X)$ is Nakano positive. Then the Nakano
vanishing theorem applied to the vector bundle $F=E\otimes\cO(kL+A-K_X)$
shows that $H^q(X,\cO(E)\otimes\cO(kL+A))=0$ for all $q\ge 1$.
The exact sequence
$$0\to\cO(kL)\to\cO(kL+A)\to\cO(kL+A)_{\restriction A}\to0$$
twisted by $E$ implies
$$H^q(X,\cO(E)\otimes\cO(kL))\simeq H^{q-1}(A,\cO(E_{\restriction A}
\otimes\cO(kL+A)_{\restriction A}),$$
and we easily conclude by induction since $\dim A=n-1$. Observe that the
argument does not work any more if $X$ is not algebraic. It seems to be
unknown whether the $O(k^{n-q})$ bound still holds in that case.\qed
\endproof

\claim{(7.8) Corollary} If $L$ is nef, then $L$ is big $($i.e.\
$\kappa(L)=n)$ if and only if $L^n>0$. Moreover, if $L$ is nef and big,
then for every $\delta>0$, $L$ has a singular metric $h=e^{-2\varphi}$
such that $\,\max_{x\in X}\nu(\varphi,x)\le\delta$ and
$\ii\Theta_h(L)\ge\varepsilon\,\omega$ for some $\varepsilon>0$.
The metric $h$ can be chosen to be smooth on the complement of a fixed
divisor $D$, with logarithmic poles along $D$.
\endclaim

\begproof{} By Lemma~7.7 and the Riemann-Roch formula, we have
$h^0(X,kL)=\chi(X,kL)+o(k^n)=k^nL^n/n!+o(k^n)$, whence the first
statement. If $L$ is big, the proof made in (7.6~f) shows that
there is a singular metric $h_1$ on $L$ such that
$${\ii\over 2\pi}\Theta_{h_1}(L)={1\over k}
\Big({\ii\over2\pi}\Theta(A)+[D]\Big)$$
with a positive line bundle $A$ and an effective divisor $D$.
Now, for every $\varepsilon>0$, there is a smooth metric $h_\varepsilon$
on $L$ such that ${\ii\over2\pi}\Theta_{h_\varepsilon}(L)\ge-\varepsilon
\omega$, where $\omega={\ii\over2\pi}\Theta(A)$. The convex combination
of metrics $h'_\varepsilon=h_1^{k\varepsilon}h_\varepsilon^{1-k\varepsilon}$
is a singular metric with poles along $D$ which satisfies
$${\ii\over2\pi}\Theta_{h'_\varepsilon}(L)\ge \varepsilon(\omega+[D])-
(1-k\varepsilon)\varepsilon\omega\ge k\varepsilon^2\omega.$$
Its Lelong numbers are $\varepsilon\nu(D,x)$ and they can be made
smaller than $\delta$ by choosing $\varepsilon>0$ small.\qed
\endproof

We still need a few elementary facts about the numerical dimension of nef
line bundles.

\claim{(7.9) Definition} Let $L$ be a nef line bundle on a compact
K\"ahler manifold~$X$. One defines the numerical dimension of $L$ to be
$$\nu(L)=\max\big\{k=0\ld n\,;\,c_1(L)^k\ne0~\hbox{in}~
H^{2k}(X,\bR)\big\}.$$
\endclaim

By Corollary 7.8, we have $\kappa(L)=n$ if and only if $\nu(L)=n$.
In general, we merely have an inequality.

\claim{(7.10) Proposition} If $L$ is a nef line bundle on a compact
K\"ahler manifold, then $\kappa(L)\le\nu(L)$.
\endclaim

\begproof{} By induction on $n=\dim X$. If $\nu(L)=n$ or $\kappa(L)=n$
the result is true, so we may assume $r:=\kappa(L)\le n-1$ and $k:=\nu(L)\le
n-1$. Fix $m>0$ so that $\Phi=\Phi_{|mL|}$ has generic rank $r$.
Select a nonsingular ample divisor $A$ in $X$ such that the restriction of
$\Phi_{|mL|}$ to $A$ still has rank $r$ (for this, just take $A$
passing through a point $x\notin B_{|mL|}$ at which $\rank(d\Phi_x)=r<n$,
in such a way that the tangent linear map $d\Phi_{x\restriction T_{A,x}}$
still has rank~$r$). Then $\kappa(L_{\restriction A})\ge r=\kappa(L)$
(we just have an equality because there might exist sections in
$H^0(A,mL_{\restriction A})$ which do not extend to~$X$). On the other hand,
we claim that $\nu(L_{\restriction A})=k=\nu(L)$. The inequality
$\nu(L_{\restriction A})\ge\nu(L)$ is clear. Conversely, if we set
$\omega={\ii\over2\pi}\Theta(A)>0$, the cohomology class $c_1(L)^k$ can
be represented by a closed positive current of bidegree $(k,k)$
$$T=\lim_{\varepsilon\to0}\Big({\ii\over 2\pi}\Theta_{h_\varepsilon}(L)+
\varepsilon\omega\Big)^k$$
after passing to some subsequence (there is a uniform bound for the mass
thanks to the K\"ahler assumption, taking wedge products with
$\omega^{n-k}$). The current $T$ must be non zero since $c_1(L)^k\ne0$ by
definition of $k=\nu(L)$. Then $\{[A]\}=\{\omega\}$ as cohomology classes,
and
$$\int_A c_1(L_{\restriction A})^k\wedge\omega^{n-1-k}=
\int_X c_1(L)^k\wedge[A]\wedge\omega^{n-1-k}=
\int_X T\wedge\omega^{n-k}>0.$$
This implies $\nu(L_{\restriction A})\ge k$, as desired. The induction
hypothesis with $X$ replaced by $A$ yields
$$\kappa(L)\le\kappa(L_{\restriction A})\le
\nu(L_{\restriction A})\le\nu(L).\eqno\square$$
\endproof

\claim{(7.11) Remark} \rm It may happen that $\kappa(L)<\nu(L)$: take e.g.
$$L\to X=X_1\times X_2$$
equal to the total tensor product of an ample line bundle $L_1$ on a
projective manifold $X_1$ and of a unitary flat line bundle $L_2$ on an
elliptic curve $X_2$ given by a representation $\pi_1(X_2)\to U(1)$ such
that no multiple $kL_2$ with $k\ne0$ is trivial. Then
$H^0(X,kL)=H^0(X_1,kL_1)\otimes H^0(X_2,kL_2)=0$ for $k>0$, and thus
$\kappa(L)=-\infty$. However $c_1(L)=\pr_1^\star c_1(L_1)$ has
numerical dimension equal to $\dim X_1$. The same example shows
that the Kodaira dimension may increase by restriction to a
subvariety (if $Y=X_1\times\{{\rm point}\}$, then
$\kappa(L_{\restriction Y})=\dim Y$).\qed
\endclaim

We now derive an algebraic version of the Nadel vanishing theorem in the
context of nef line bundles. This algebraic vanishing theorem has been
obtained independently by Kawamata [Kaw82] and Viehweg [Vie82], who both
reduced it to the Kodaira-Nakano vanishing theorem by cyclic covering
constructions. Since then, a number of other proofs have been given,
one based on connections with logarithmic singularities [EV86], another
on Hodge theory for twisted coefficient systems [Kol85], a third one
on the Bochner technique [Dem89] (see also [EV92] for a general survey,
and [Eno93] for an extension to the compact K\"ahler case). Since the
result is best expressed in terms of multiplier ideal sheaves (avoiding
then any unnecessary desingularization in the statement), we feel that
the direct approach via Nadel's vanishing theorem is probably the most
natural one.

If $D=\sum\alpha_jD_j\ge 0$ is an effective $\bQ$-divisor, we define
the {\it multiplier ideal sheaf} $\cI(D)$ to be equal to $\cI(\varphi)$
where $\varphi=\sum\alpha_j|g_j|$ is the corresponding psh function
defined by generators $g_j$ of $\cO(-D_j)\,$; as we saw in Remark 6.9,
the computation of $\cI(D)$ can be made algebraically by using
desingularizations $\mu:\wt X\to X$ such that $\mu^\star D$ becomes a
divisor with normal crossings on~$\wt X$.

\claim{(7.12) Kawamata-Viehweg vanishing theorem} Let $X$ be a projective
algebraic manifold and let $F$ be a line bundle over $X$ such that some
positive multiple $mF$ can be written $mF=L+D$ where $L$ is a 
nef line bundle and $D$ an effective divisor. Then
$$H^q\big(X,\cO(K_X+F)\otimes\cI(m^{-1}D)\big)=0~~~~
{\it for}~~q>n-\nu(L).$$
\endclaim

\claim{(7.13) Special case} If $F$ is a nef line bundle, then
$$H^q\big(X,\cO(K_X+F)\big)=0~~~~{\it for}~~q>n-\nu(F).$$
\endclaim

\begproof{of Theorem 7.12.} First suppose that $\nu(L)=n$, i.e.\ that
$L$ is big. By the proof of 7.5~f), there is a singular hermitian
metric on $L$ such that the corresponding weight $\varphi_{L,0}$
has algebraic singularities and
$$\ii\Theta_0(L)=2\ii d'd''\varphi_L\ge\varepsilon_0\omega$$
for some $\varepsilon_0>0$. On the other hand, since $L$ is nef, there
are metrics given by weights $\varphi_{L,\varepsilon}$ such that
${\ii\over 2\pi}\Theta_\varepsilon(L)\ge\varepsilon\omega$ for
every $\varepsilon>0$, $\omega$ being a K\"ahler metric. Let
$\varphi_D=\sum\alpha_j\log|g_j|$ be the weight of the singular metric on
$\cO(D)$ described in Example~3.14. We define a singular metric on $F$ by
$$\varphi_F={1\over m}
\big((1-\delta)\varphi_{L,\varepsilon}+\delta\varphi_{L,0}+\varphi_D\big)$$
with $\varepsilon\ll\delta\ll 1$, $\delta$ rational. Then $\varphi_F$
has algebraic singularities, and by taking $\delta$ small enough we
find $\cI(\varphi_F)=\cI({1\over m}\varphi_D)=\cI({1\over m}D)$.
In fact, $\cI(\varphi_F)$ can be computed by taking integer parts
of $\bQ$-divisors (as explained in Remark~6.9), and adding
$\delta\varphi_{L,0}$ does not change the integer part of the rational
numbers involved when $\delta$ is small. Now
$$\eqalign{dd^c\varphi_F
&={1\over m}\big((1-\delta)dd^c\varphi_{L,\varepsilon}+
\delta dd^c\varphi_{L,0}+dd^c\varphi_D\big)\cr
&\ge{1\over m}\big(-(1-\delta)\varepsilon\omega+
\delta\varepsilon_0\omega+[D]\ge{\delta\varepsilon\over m}\omega,\cr}$$
if we choose $\varepsilon\le\delta\varepsilon_0$. Nadel's theorem thus
implies the desired vanishing result for all $q\ge 1$.

Now, if $\nu(L)<n$, we use hyperplane sections and argue by induction
on $n=\dim X$. Since the sheaf $\cO(K_X)\otimes\cI(m^{-1}D)$ behaves
functorially with respect to modifications (and since the $L^2$
cohomology complex is ``the same'' upstairs and downstairs),
we may assume after blowing-up that $D$ is a divisor with normal
crossings. By Remark~6.9, the multiplier ideal sheaf
$\cI(m^{-1}D)=\cO(-\lfloor m^{-1}D\rfloor)$ is locally free.
By Serre duality, the expected vanishing is equivalent to
$$H^q(X,\cO(-F)\otimes\cO(\lfloor m^{-1}D\rfloor))=0\qquad
\hbox{\rm for}~q<\nu(L).$$
Then select a nonsingular ample divisor $A$ such that $A$ meets all
components $D_j$ transversally. Select $A$ positive enough so that
$\cO(A+F-\lfloor m^{-1}D\rfloor)$ is ample. Then
$H^q(X,\cO(-A-F)\otimes\cO(\lfloor m^{-1}D\rfloor))=0$ for $q<n$
by Kodaira vanishing, and the exact sequence
$0\to\cO_X(-A)\to\cO_X\to(i_A)_\star\cO_A\to 0$ twisted by
$\cO(-F)\otimes\cO(\lfloor m^{-1}D\rfloor)$ yields an isomorphism
$$H^q(X,\cO(-F)\otimes\cO(\lfloor m^{-1}D\rfloor))\simeq
H^q(A,\cO(-F_{\restriction A})\otimes\cO(\lfloor m^{-1}
D_{\restriction A}\rfloor).$$
The proof of 6.8 showed that $\nu(L_{\restriction A})=\nu(L)$, hence
the induction hypothesis implies that the cohomology group on $A$ on
the right hand side is zero for $q<\nu(L)$.\qed
\endproof

\ifnum\fullversion=1
\titleb{8.}{A Simple Algebraic Approach to Fujita's Conjecture}
This section is devoted to a proof of various results related to the
Fujita conjecture. The main ideas occuring here are inspired by a
recent work of Y.T.~Siu [Siu96]. His method, which is algebraic in
nature and quite elementary, consists in a combination of the
Riemann-Roch formula together with Nadel's vanishing theorem (in fact,
only the algebraic case is needed, thus the original Kawamata-Viehweg
vanishing theorem would be sufficient). Slightly later, Angehrn and
Siu [AS95], [Siu95] introduced other closely related methods, producing
better bounds for the global generation question; since their method
is rather delicate, we can only refer the reader to the above references.
In the sequel, $X$ denotes a projective algebraic $n$-dimensional manifold.
The first observation is the following well-known consequence of the
Riemann-Roch formula.

\claim{(8.1) Special case of Riemann-Roch} Let $\cJ\subset\cO_X$ be a
coherent ideal sheaf on $X$ such that the subscheme $Y=V(\cJ)$ has
dimension $d$ $($with possibly some lower dimensional components$)$.
Let $[Y]=\sum\lambda_j[Y_j]$ be the effective algebraic cycle of dimension
$d$ associated to the $d$ dimensional components of~$Y$ $($taking
into account multiplicities $\lambda_j$ given by the ideal $\cJ)$. Then
for any line bundle $F$, the Euler characteristic
$$\chi(Y,\cO(F+mL)_{\restriction Y})=\chi(X,\cO(F+mL)\otimes\cO_X/\cJ)$$
is a polynomial $P(m)$ of degree $d$ and leading coefficient
$L^d\cdot[Y]/d!$
\endclaim

The second fact is an elementary lemma about numerical polynomials
(polynomials with rational coefficients, mapping $\bZ$ into $\bZ$).

\claim{(8.2) Lemma} Let $P(m)$ be a numerical polynomial of degree $d>0$
and leading coefficient $a_d/d!$, $a_d\in\bZ$, $a_d>0$. Suppose that
$P(m)\ge 0$ for $m\ge m_0$. Then
\smallskip
\item{\rm a)} For every integer $N\ge0$, there exists $m\in[m_0,m_0+Nd]$
such that~\hbox{$P(m)\ge N$}.
\smallskip
\item{\rm b)} For every $k\in\bN$, there exists $m\in[m_0,m_0+kd]$ such
that \hbox{$P(m)\ge a_dk^d/2^{d-1}$}.
\smallskip
\item{\rm c)} For every integer $N\ge 2d^2$, there exists
$m\in[m_0,m_0+N]$ such that \hbox{$P(m)\ge N$}.
\vskip0pt
\endclaim

\begproof{} a) Each of the $N$ equations $P(m)=0$, $P(m)=1$,
$\ldots$, $P(m)=N-1$ has at most $d$ roots, so there must be an integer
$m\in[m_0,m_0+dN]$ which is not a root of these.
\smallskip\noindent
b) By Newton's formula for iterated differences $\Delta P(m)=P(m+1)-P(m)$,
we get
$$\Delta^dP(m)=\sum_{1\le j\le d}(-1)^j{d\choose j}P(m+d-j)=a_d,
\qquad\forall m\in\bZ.$$
Hence if $j\in\big\{0,2,4\ld 2\lfloor d/2\rfloor\big\}\subset[0,d]$ is the
even integer achieving the maximum of $P(m_0+d-j)$ over this finite set,
we find
$$2^{d-1}P(m_0+d-j)=\left({d\choose 0}+{d\choose 2}+\ldots\right)P(m_0+d-j)
\ge a_d,$$
whence the existence of an integer $m\in[m_0,m_0+d]$ with $P(m)\ge
a_d/2^{d-1}$. The case $k=1$ is thus proved. In general, we apply the
above case to the polynomial $Q(m)=P(km-(k-1)m_0)$, which has leading
coefficient $a_dk^d/d!$
\smallskip\noindent
c) If $d=1$, part a) already yields the result. If $d=2$, a look at the
parabola shows that
$$\max_{m\in[m_0,m_0+N]}P(m)\ge\cases{
a_2N^2/8&if $N$ is even,\cr
a_2(N^2-1)/8&if $N$ is odd;\cr}$$
thus $\max_{m\in[m_0,m_0+N]}P(m)\ge N$ whenever $N\ge 8$. If $d\ge 3$,
we apply b) with $k$ equal to the smallest integer such
that $k^d/2^{d-1}\ge N$, i.e.\ $k=\lceil 2(N/2)^{1/d}\rceil$, where
$\lceil x\rceil\in\bZ$ denotes the round-up of $x\in\bR$.
\hbox{Then $kd\le (2(N/2)^{1/d}+1)d\le N$} whenever $N\ge 2d^2$, as a short
computation shows.\qed
\endproof

We now apply Nadel's vanishing theorem pretty much in the same way as
Siu [Siu96], but with substantial simplifications in the technique and
improvements in the bounds. Our method yields simultaneously a simple
proof of the following basic result.

\claim{(8.3) Theorem} If $L$ is an ample line bundle over a projective
$n$-fold~$X$, then the adjoint line bundle $K_X+(n+1)L$ is nef.
\endclaim

By using Mori theory and the base point free theorem ([Mor82],
[Kaw84]), one can even show that $K_X+(n+1)L$ is semiample, i.e., there
exists a positive integer $m$ such that $m(K_X+(n+1)L)$ is generated by
sections (see [Kaw85] and [Fuj87]). The proof rests on the observation
that $n+1$ is the maximal length of extremal rays of smooth projective
$n$-folds. Our proof of (8.3) is different and will be given simultaneously
with the proof of Th.~(8.4) below.

\claim{(8.4) Theorem} Let $L$ be an ample line bundle and let $G$ be
a nef line bundle on a projective $n$-fold~$X$. Then the following
properties hold.
\smallskip
\item{\rm a)} $2K_X+mL+G$ generates simultaneous jets of order
$s_1\ld s_p\in\bN$ at arbitrary points $x_1\ld x_p\in X$, i.e., there
is a surjective map
$$H^0(X,2K_X+mL+G)\lraww\bigoplus_{1\le j\le p}\cO(2K_X+mL+G)\otimes
\cO_{X,x_j}/\gm_{X,x_j}^{s_j+1},$$
provided that $\displaystyle m\ge 2+\sum_{1\le j\le p}{3n+2s_j-1\choose
n}$.\hfill\break In particular $2K_X+mL+G$ is very ample for
$\displaystyle m\ge 2+{3n+1\choose n}$.
\smallskip
\item{\rm b)} $2K_X+(n+1)L+G$ generates simultaneous jets of order
$s_1\ld s_p$ at arbitrary points $x_1\ld x_p\in X$ provided that
the intersection numbers $L^d\cdot Y$ of $L$ over all
$d$-dimensional algebraic subsets $Y$ of $X$ satisfy
$$L^d\cdot Y>{2^{d-1}\over\lfloor n/d\rfloor^d}\sum_{1\le j\le p}
{3n+2s_j-1\choose n}.$$
\endclaim

\begproof{} The proofs of (8.3) and (8.4$\,$a,$\,$b) go along the
same lines, so we deal with them simultaneously (in the case of
(8.3), we simply agree that $\{x_1\ld x_p\}=\emptyset$). The idea is
to find an integer (or rational number) $m_0$ and a singular hermitian
metric $h_0$ on $K_X+m_0L$ with strictly positive curvature current
$\ii\Theta_{h_0}\ge\varepsilon\omega$, such that $V(\cI(h_0))$ is
$0$-dimensional and the weight $\varphi_0$ of $h_0$ satis\-fies
$\nu(\varphi_0,x_j)\ge n+s_j$ for all~$j$. As $L$ and $G$ are nef,
$(m-m_0)L+G$ has for all $m\ge m_0$ a metric $h'$ whose
curvature $\ii\Theta_{h'}$ has arbitrary small negative part (see
[Dem90]), e.g., $\ii\Theta_{h'}\ge -{\varepsilon\over 2}\omega$. Then
$\ii\Theta_{h_0}+\ii\Theta_{h'}\ge{\varepsilon \over 2}\omega$ is again
positive definite. An application of Corollary (6.12) to
$F=K_X+mL+G=(K_X+m_0 L)+((m-m_0)L+G)$ equipped with the metric
$h_0\otimes h'$ implies the existence of the desired sections in
$K_X+F=2K_X+mL+G$ for $m\ge m_0$.

Let us fix an embedding $\Phi_{|\mu L|}:X\to\bP^N$, $\mu\gg 0$, given
by sections $\lambda_0\ld\lambda_N\in H^0(X,\mu L)$, and let $h_L$ be
the associated metric on $L$ of positive definite curvature form
$\omega={\ii\over 2\pi}\Theta(L)$. In order to obtain the desired
metric $h_0$ on $K_X+m_0 L$, we fix $a\in\bN^\star$ and use a double
induction process to construct singular metrics $(h_{k,\nu})_{\nu\ge
1}$ on $aK_X+b_kL$ for a non increasing sequence of positive integers
$b_1\ge b_2\ge\ldots\ge b_k\ge\ldots\,$. Such a sequence much be
stationary and $m_0$ will just be the stationary limit $m_0=\lim
b_k/a$. The metrics $h_{k,\nu}$ are taken to satisfy the following
properties:
\smallskip
\item{$\alpha\!)$} $h_{k,\nu}$ is an algebraic metric of the form
$$\|\xi\|_{h_{k,\nu}}^2={|\tau_k(\xi)|^2\over
\big(\sum_{1\le i\le\nu,\,0\le j\le N}\big|\tau_k^{(a+1)\mu}(\sigma_i^{a\mu}
\cdot\lambda_j^{(a+1)b_k-am_i})\big|^2\big)^{1/(a+1)\mu}},$$
defined by sections $\sigma_i\in H^0(X,(a+1)K_X+m_iL)$, $m_i<{a+1\over a}b_k$,
$1\le i\le\nu$, where $\xi\mapsto\tau_k(\xi)$ is an arbitrary local
trivialization of $aK_X+b_kL\,$; note that $\sigma_i^{a\mu}\cdot
\lambda_j^{(a+1)b_k-am_i}$ is a section of
$$a\mu((a+1)K_X+m_iL)+((a+1)b_k-am_i)\mu L=(a+1)\mu(aK_X+b_kL).$$
\item{$\beta\kern-1pt)$\kern-1pt} $\ord_{x_j}(\sigma_i)\ge(a+1)(n+s_j)$ for
all $i,j\,$;
\smallskip
\item{$\gamma)$} $\cI(h_{k,\nu+1})\supset\cI(h_{k,\nu})$~ and
~$\cI(h_{k,\nu+1})\ne\cI(h_{k,\nu})$ whenever the zero variety
$V(\cI(h_{k,\nu}))$ has positive dimension.
\medskip\noindent
The weight $\varphi_{k,\nu}={1\over 2(a+1)\mu}\log\sum\big|\tau_k^{(a+1)\mu}
(\sigma_i^{a\mu}\cdot\lambda_j^{(a+1)b_k-am_i})\big|^2$ of $h_{k,\nu}$ is
plurisubharmonic and the condition $m_i<{a+1\over a}b_k$ implies
$(a+1)b_k-am_i\ge 1$, thus the difference
$\varphi_{k,\nu}-{1\over2(a+1)\mu}\log\sum|\tau(\lambda_j)|^2$ is also
plurisubharmonic. Hence ${\ii\over 2\pi}\Theta_{h_{k,\nu}}(aK_X+b_kL)=
{\ii\over\pi}d'd''\varphi_{k,\nu}\ge{1\over(a+1)}\omega$. Moreover, condition
$\beta)$ clearly implies $\nu(\varphi_{k,\nu},x_j)\ge a(n+s_j)$. Finally,
condition $\gamma)$ combined with the strong Noetherian property of coherent
sheaves ensures that the sequence $(h_{k,\nu})_{\nu\ge 1}$ will finally
produce a zero dimensional subscheme $V(\cI(h_{k,\nu}))$. We agree that
the sequence $(h_{k,\nu})_{\nu\ge 1}$ stops at this point, and we denote
by $h_k=h_{k,\nu}$ the final metric, such that $\dim V(\cI(h_k))=0$.

For $k=1$, it is clear that the desired metrics $(h_{1,\nu})_{\nu\ge 1}$
exist if $b_1$ is taken large enough (so large, say, that
$(a+1)K_X+(b_1-1)L$ generates jets of order $(a+1)(n+\max s_j)$ at every
point; then the sections $\sigma_1\ld\sigma_\nu$ can be chosen with
$m_1=\ldots=m_\nu=b_1-1$). Suppose that the metrics $(h_{k,\nu})_{\nu\ge 1}$
and $h_k$ have been constructed and let us proceed with the construction
of $(h_{k+1,\nu})_{\nu\ge 1}$. We do this again by induction on~$\nu$,
assuming that $h_{k+1,\nu}$ is already constructed and that $\dim
V(\cI(h_{k+1,\nu}))>0$. We start in fact the induction with $\nu=0$, and
agree in this case that $\cI(h_{k+1,0})=0$ (this would correspond to an
infinite metric of weight identically equal to $-\infty$). By Nadel's
vanishing theorem applied to
$$F_m=aK_X+mL=(aK_X+b_kL)+(m-b_k)L$$
with the metric $h_k\otimes(h_L)^{\otimes m-b_k}$, we get
$$H^q(X,\cO((a+1)K_X+mL)\otimes\cI(h_k))=0\qquad
\hbox{\rm for $q\ge 1$, $m\ge b_k$.}$$
As $V(\cI(h_k))$ is $0$-dimensional, the sheaf $\cO_X/\cI(h_k)$ is a
skyscraper sheaf, and the exact sequence $0\to\cI(h_k)\to\cO_X\to
\cO_X/\cI(h_k)\to 0$ twisted with the invertible sheaf $\cO((a+1)K_X+mL)$
shows that
$$H^q(X,\cO((a+1)K_X+mL))=0\qquad \hbox{\rm for $q\ge 1$, $m\ge b_k$.}$$
Similarly, we find
$$H^q(X,\cO((a+1)K_X+mL)\otimes\cI(h_{k+1,\nu}))=0\qquad
\hbox{\rm for $q\ge 1$, $m\ge b_{k+1}$}$$
(also true for $\nu=0$, since $\cI(h_{k+1,0})=0$), and when
\hbox{$m\ge\max(b_k,b_{k+1})=b_k$}, the exact sequence $0\to\cI(h_{k+1,\nu})
\to\cO_X\to\cO_X/\cI(h_{k+1},\nu)\to 0$ implies
$$H^q(X,\cO((a+1)K_X+mL)\otimes\cO_X/\cI(h_{k+1,\nu}))=0\qquad
\hbox{\rm for $q\ge 1$, $m\ge b_k$.}$$
In particular, since the $H^1$ group vanishes, every section $u'$ of
$(a+1)K_X+mL$ on the subscheme $V(\cI(h_{k+1,\nu}))$ has an extension
$u$ to $X$. Fix a basis $u_1'\ld u_N'$ of the sections on
$V(\cI(h_{k+1,\nu}))$ and take arbitrary extensions $u_1\ld u_N$ to
$X$. Look at the linear map assigning the collection of jets of order
$(a+1)(n+s_j)-1$ at all points~$x_j$
$$u=\sum_{1\le j\le N}a_ju_j\longmapsto
\bigoplus J^{(a+1)(n+s_j)-1}_{x_j}(u).$$
Since the rank of the bundle of $s$-jets is ${n+s\choose n}$, the target
space has dimension
$$\delta=\sum_{1\le j\le p}{n+(a+1)(n+s_j)-1\choose n}.$$
In order to get a section $\sigma_{\nu+1}=u$ satisfying condition $\beta)$
with non trivial restriction $\sigma'_{\nu+1}$ to $V(\cI(h_{k+1,\nu}))$, we
need at least $N=\delta+1$ independent sections $u'_1\ld u'_N$. This
condition is achieved by applying Lemma~(8.2) to the numerical polynomial
$$\eqalign{P(m)
&=\chi(X,\cO((a+1)K_X+mL)\otimes\cO_X/\cI(h_{k+1,\nu}))\cr
&=h^0(X,\cO((a+1)K_X+mL)\otimes\cO_X/\cI(h_{k+1,\nu}))\ge 0,\qquad
m\ge b_k.\cr}$$
The polynomial $P$ has degree $d=\dim V(\cI(h_{k+1,\nu}))>0$.
We get the existence of an integer $m\in[b_k,b_k+\eta]$ such that
$N=P(m)\ge\delta+1$ with some explicit integer $\eta\in\bN$
(for instance $\eta=n(\delta+1)$ always works by (8.2$\,$a), but we will
also use other possibilities to find an optimal choice in each case).
Then we find a section $\sigma_{\nu+1}\in H^0(X,(a+1)K_X+mL)$
with non trivial restriction $\sigma'_{\nu+1}$ to $V(\cI(h_{k+1,\nu}))$,
vanishing at order $\ge(a+1)(n+s_j)$ at each point $x_j$. We just set
$m_{\nu+1}=m$, and the condition $m_{\nu+1}<{a+1\over a}b_{k+1}$ is
satisfied if $b_k+\eta<{a+1\over a}b_{k+1}$. This shows that we can take
inductively
$$b_{k+1}=\left\lfloor{a\over a+1}(b_k+\eta)\right\rfloor+1.$$
By definition, $h_{k+1,\nu+1}\le h_{k+1,\nu}$, hence $\cI(h_{k+1,\nu+1})
\supset\cI(h_{k+1,\nu})$. We necessarily have $\cI(h_{k+1,\nu+1})\ne
\cI(h_{k+1,\nu})$, for $\cI(h_{k+1,\nu+1})$ contains the ideal sheaf
associated with the zero divisor of $\sigma_{\nu+1}$, whilst
$\sigma_{\nu+1}$ does not vanish identically on~$V(\cI(h_{k+1,\nu}))$.
Now, an easy computation shows that the iteration of
$b_{k+1}=\lfloor{a\over a+1}(b_k+\eta)\rfloor+1$ stops at $b_k=a(\eta+1)+1$ for
any large initial value $b_1$. In this way, we obtain a metric $h_\infty$
of positive definite curvature on $aK_X+(a(\eta+1)+1)L$, with
$\dim V(\cI(h_\infty))=0$ and $\nu(\varphi_\infty,x_j)\ge a(n+s_j)$ at
each point~$x_j$.
\endproof

\begproof{of $(8.3)$.} In this case, the set $\{x_j\}$ is taken to be
empty, thus $\delta=0$. By (8.2$\,$a), the condition $P(m)\ge 1$ is
achieved for some $m\in[b_k,b_k+n]$ and we can take $\eta=n$. As $\mu L$ is
very ample, there exists on $\mu L$ a metric with an isolated logarithmic
pole of Lelong number $1$ at any given point~$x_0$ (e.g., the algebraic
metric defined with all sections of $\mu L$ vanishing at~$x_0$). Hence
$$F_a'=aK_X+(a(n+1)+1)L+n\mu L$$
has a metric $h_a'$ such that $V(\cI(h_a'))$ is zero dimensional and
contains $\{x_0\}$. By Corollary (6.12), we conclude that
$$K_X+F_a'=(a+1)K_X+(a(n+1)+1+n\mu)L$$
is generated by sections, in particular $K_X+{a(n+1)+1+n\mu\over a+1}L$
is nef. As $a$ tends to $+\infty$, we infer that $K_X+(n+1)L$ is nef.\qed
\endproof

\begproof{of $(8.4\,${\rm a).}} Here, the choice $a=1$ is sufficient for
our purposes. Then
$$\delta=\sum_{1\le j\le p}{3n+2s_j-1\choose n}.$$
If $\{x_j\}\ne\emptyset$, we have $\delta+1\ge{3n-1\choose n}+1\ge 2n^2$
for $n\ge 2$. Lemma (8.2$\,$c) shows that $P(m)\ge\delta+1$ for some
$m\in[b_k,b_k+\eta]$ with $\eta=\delta+1$. We can start in fact the
induction procedure $k\mapsto k+1$ with $b_1=\eta+1=\delta+2$, because the
only property needed for the induction step is the vanishing
property
$$H^0(X,2K_X+mL)=0\qquad\hbox{for $q\ge 1$, $m\ge b_1$},$$
which is true by the Kodaira vanishing theorem and the ampleness
of $K_X+b_1L$ (here we use Fujita's result (8.3), observing that $b_1>n+1$).
Then the recursion formula $b_{k+1}=\lfloor{1\over 2}(b_k+\eta)\rfloor+1$
yields $b_k=\eta+1=\delta+2$ for all $k$, and (8.4$\,$a) follows.\qed
\endproof

\begproof{of $(8.4\,${\rm b).}} Quite similar to (8.4$\,$a), except that we
take $\eta=n$, $a=1$ and $b_k=n+1$ for all~$k$. By Lemma (8.2$\,$b), we have
$P(m)\ge a_dk^d/2^{d-1}$ for some integer $m\in[m_0,m_0+kd]$,
where $a_d>0$ is the coefficient of highest degree in~$P$. By Lemma (8.2)
we have $a_d\ge\inf_{\dim Y=d}L^d\cdot Y$. We take $k=\lfloor n/d\rfloor$. The
condition $P(m)\ge\delta+1$ can thus be realized for some $m\in[m_0,m_0+kd]
\subset[m_0,m_0+n]$ as soon as
$$\inf_{\dim Y=d}L^d\cdot Y~\lfloor n/d\rfloor^d/2^{d-1}>\delta,$$
which is equivalent to the condition given in (8.4$\,$b).\qed
\endproof

\claim{(8.5) Corollary} Let $X$ be a smooth projective $n$-fold, let
$L$ be an ample line bundle and $G$ a nef line bundle over~$X$. Then
$m(K_X+(n+2)L)+G$ is very ample for $m\ge{3n+1\choose n}-2n$.
\endclaim

\begproof{} Apply Theorem (8.4$\,$a) with
$G'=a(K_X+(n+1)L)+G$, so that
$$2K_X+mL+G'=(a+2)(K_X+(n+2)L)+(m-2n-4-a)L+G,$$
and take $m=a+2n+4\ge 2+{3n+1\choose n}$.\qed
\endproof

The main drawback of the above technique is that multiples of $L$ at
least equal to $(n+1)L$ are required to avoid zeroes of the Hilbert
polynomial. In particular, it is not possible to obtain directly
a very ampleness criterion for $2K_X+L$ in the statement of (8.4$\,$b).
Nevertheless, using different ideas from Angehrn-Siu [AS95], [Siu96] has
obtained such a criterion. We derive here a slightly weaker version,
thanks to the following elementary Lemma.

\claim{(8.6) Lemma} Assume that for some integer $\mu\in\bN^\star$
the line bundle $\mu F$ gene\-rates simultaneously all jets of order
$\mu(n+s_j)+1$ at any point $x_j$ in a subset $\{x_1\ld x_p\}$ of~$X$.
Then $K_X+F$ generates simultaneously all jets of order $s_j$ at~$x_j$.
\endclaim

\begproof{} Take the algebraic metric on $F$ defined by a basis of
sections $\sigma_1\ld\sigma_N$ of $\mu F$ which vanish at order
$\mu(n+s_j)+1$ at all points~$x_j$. Since we are still free to choose
the homogeneous term of degree $\mu(n+s_j)+1$ in the Taylor expansion
at~$x_j$, we find that $x_1\ld x_p$ are isolated zeroes of
$\bigcap\sigma_j^{-1}(0)$. If $\varphi$ is the weight of the metric
of $F$ near~$x_j$, we thus have $\varphi(z)\sim (n+s_j+{1\over\mu})
\log|z-x_j|$ in suitable coordinates. We replace $\varphi$ in a
neighborhood of $x_j$ by
$$\varphi'(z)=\max\big(\varphi(z)\,,\,|z|^2-C+(n+s_j)\log|z-x_j|\big)$$
and leave $\varphi$ elsewhere unchanged (this is possible by taking
$C>0$ very large). Then $\varphi'(z)=|z|^2-C+(n+s_j)\log|z-x_j|$ near
$x_j$, in particular $\varphi'$ is strictly plurisubharmonic
near~$x_j$. In this way, we get a metric $h'$ on $F$ with semipositive
curvature everywhere on $X$, and with positive definite curvature on a
neighborhood of $\{x_1\ld x_p\}$. The conclusion then follows directly
from H\"ormander's $L^2$ estimates (6.1) and (6.2).\qed
\endproof

\claim{(8.7) Theorem} Let $X$ be a smooth projective $n$-fold, and let
$L$ be an ample line bundle over~$X$. Then $2K_X+L$ generates simultaneous
jets of order $s_1\ld s_p$ at arbitrary points $x_1\ld x_p\in X$ provided
that the intersection numbers $L^d\cdot Y$ of $L$ over all
\hbox{$d$-dimensional} algebraic subsets $Y$ of $X$ satisfy
$$L^d\cdot Y>{2^{d-1}\over\lfloor n/d\rfloor^d}
\sum_{1\le j\le p}{(n+1)(4n+2s_j+1)-2\choose n},\qquad 1\le d\le n.$$
\endclaim

\begproof{} By Lemma~(8.6) applied with $F=K_X+L$ and $\mu=n+1$, the
desired jet generation of $2K_X+L$ occurs if $(n+1)(K_X+L)$ generates jets
of order \hbox{$(n+1)(n+s_j)+1$} at~$x_j$. By Lemma~(8.6) again with
\hbox{$F=aK_X+(n+1)L$} and $\mu=1$, we see by backward induction on $a$
that we need the simultaneous generation of jets of order
\hbox{$(n+1)(n+s_j)+1+(n+1-a)(n+1)$} at~$x_j$. In particular, for
$2K_X+(n+1)L$ we need the generation of jets of order
\hbox{$(n+1)(2n+s_j-1)+1$}. Theorem (8.4$\,$b) yields the desired
condition.\qed
\endproof

We now list a few immediate consequences of Theorem~8.4, in
connection with some classical questions of algebraic geometry.

\claim{(8.8) Corollary} Let $X$ be a projective $n$-fold of
general type with $K_X$ ample. Then $mK_X$ is very ample for $m\ge
m_0={3n+1\choose n}+4$.
\endclaim

\claim{(8.9) Corollary} Let $X$ be a Fano $n$-fold, that is,
a $n$-fold such that $-K_X$ is ample. Then $-mK_X$ is very ample for
$m\ge m_0={3n+1\choose n}$.
\endclaim

\begproof{} Corollaries 8.8, 8.9 follow easily from Theorem 8.4$\,$a)
applied to \hbox{$L=\pm K_X$}. Hence we get pluricanonical embeddings
$\Phi:X\to\bP^N$ such that
$\Phi^\star\cO(1)=\pm m_0K_X$. The image $Y=\Phi(X)$ has
degree
$$\deg(Y)=\int_Y c_1\big(\cO(1)\big)^n=
\int_X c_1\big(\pm m_0K_X\big)^n=m_0^n|K_X^n|.$$
It can be easily reproved from this that there are only finitely
many deformation types of Fano $n$-folds, as well as of $n$-folds
of general type with $K_X$ ample, corresponding to a given
discriminant $|K_X^n|$ (from a theoretical viewpoint, this result is
a consequence of Matsusaka's big theorem [Mat72] and [KoM83], but
the bounds which can be obtained from it are probably extremely huge).
In the Fano case, a fundamental result obtained indepedently by
Koll\'ar-Miyaoka-Mori [KoMM92] and Campana [Cam92] shows that the
discriminant $K_X^n$ is in fact bounded by a constant $C_n$ depending
only on~$n$. Therefore, one can find an explicit bound $C'_n$ for the
degree of the embedding~$\Phi$, and it follows that there are only
finitely many families of Fano manifolds in each dimension.\qed
\endproof

In the case of surfaces, much more is known. We will content ourselves
with a brief account of recent results. If $X$ is a surface, the failure of
an adjoint bundle $K_X+L$ to be globally generated or very ample is
described in a very precise way by the following result of I.~Reider [Rei88].

\claim{(8.10) Reider's Theorem} Let $X$ be a smooth projective surface
and let $L$ be a nef line bundle on~$X$.
\smallskip
\item{\rm a)} Assume that $L^2\ge 5$ and let $x\in X$ be a given point.
Then $K_X+L$ has a section which does not vanish at~$x$, unless there is
an effective divisor $D\subset X$ passing through $x$ such that either
$$\eqalign{
&L\cdot D=0\quad\hbox{and}\quad D^2=-1\,;\qquad\hbox{or}\cr
&L\cdot D=1\quad\hbox{and}\quad D^2=0.\cr}$$
\smallskip
\item{\rm b)} Assume that $L^2\ge 10$. Then any two points $x,y\in X$
$($possibly infinitely near$)$ are separated by sections of $K_X+L$,
unless there is an effective divisor $D\subset X$ passing through $x$
and $y$ such that either
$$\eqalignno{
&L\cdot D=0\quad\hbox{and}\quad D^2=-1~\hbox{or}~{}-2\,;
\qquad\hbox{or}&\cr
&L\cdot D=1\quad\hbox{and}\quad D^2=0\phantom{-{}}~\hbox{or}~{}-1\,;
\qquad\hbox{or}&\cr
&L\cdot D=2\quad\hbox{and}\quad D^2=0.&\square\cr}$$
\vskip0pt
\endclaim

\claim{(8.11) Corollary} Let $L$ be an ample line bundle on a smooth
projective surface~$X$. Then $K_X+3L$ is globally generated and $K_X+4L$
is very ample. If $L^2\ge 2$ then $K_X+2L$ is globally generated and
$K_X+3L$ is very ample.\qed
\endclaim

The case of higher order jets can be treated similarly. The most general
result in this direction has been obtained by Beltrametti and Sommese
[BeS93].

\claim{(8.12) Theorem {\rm([BeS93])}} Let $X$ be a smooth projective surface
and let $L$ be a nef line bundle on~$X$. Let $p$ be a positive integer
such that $L^2>4p$. Then for every $0$-dimensional subscheme $Z\subset X$
of length $h^0(Z,\cO_Z)\le p$ the restriction
$$\rho_Z:H^0(X,\cO_X(K_X+L))\lra H^0(Z,\cO_Z(K_X+L))$$
is surjective, unless there is an effective divisor $D\subset X$
intersecting the support $|Z|$ such that
$$L\cdot D-p\le D^2<{1\over 2}L\cdot D.\eqno\square$$
\endclaim

\begproof{(Sketch).} The proof the above theorems rests in an essential way
on the construction of rank 2 vector bundles sitting in an exact sequence
$$0\to \cO_X\to E\to L\otimes\cI_Z\to 0.$$
Arguing by induction on the length of $Z$, we may assume that $Z$ is a
$0$-dimensional subscheme such that $\rho_Z$ is not surjective, but such
that $\rho_{Z'}$ is surjective for every proper subscheme $Z'\subset Z$.
The existence of $E$ is obtained by a classical construction of Serre
(unfortunately, this construction only works in dimension $2$).
The numerical condition on $L^2$ in the hypotheses ensures that
$c_1(E)^2-4\,c_2(E)>0$, hence $E$ is unstable in the sense of
Bogomolov. The existence of the effective divisor $D$ asserted in
8.10 or 8.12 follows. We refer to [Rei88], [BeS93] and [Laz97] for details.
The reader will find in [FdB95] a proof of the Bogomolov inequality
depending only on the Kawamata-Viehweg vanishing theorem.\qed
\endproof

\claim{(8.13) Exercise} \rm Prove the Fujita conjecture in the case of 
dimension $1$, according to the following steps.  
\smallskip 
\item{\rm a)}
By using Hodge theory, show that for every smooth function $f$ on a
compact K\"ahler manifold $(X,\omega)$, the equation $\Delta u=f$ has
a solution if and only if $\int_X f\,dV_\omega=0$.  
\smallskip
\item{\rm b)} Derive from (a), by using the local solvability of elliptic
operators, that one has a similar result when $f$ is a distribution.
\smallskip 
\item{\rm c)} If $X=C$ is a compact complex curve and $L$ a
positive line bundle, for every positive measure $\mu$ on $X$ such
that $\int_C\mu=\deg(L)=\int_C c_1(L)$, there exists a singular
hermitian metric $h$ on $L$ such that ${\ii\over 2\pi}\Theta_h(L)=\mu$
(with the obvious identification of measures and currents of bidegree
$(1,1)$).  
\smallskip 
\item{\rm d)} Given a finite collection of points $x_j\in C$ and
integers $s_j>0$, then $K_C+L$ generates jets of order $s_j$ at all
points $x_j$ as soon as $\deg(L)>\sum_j(s_j+1)$.  
\smallskip 
\item{\rm e)} If $L$ is positive on $C$, then $K_C+2L$ is globally
generated and $K_C+3L$ is very ample.\vskip0pt
\endclaim

\claim{(8.14) Exercise} \rm The goal of the exercise is to prove the
following weaker form of Theorems 8.10 and 8.12, by a simple direct method
based on Nadel's vanishing theorem:
\smallskip
\item{}{\it Let $L$ be a nef line bundle on a smooth projective surface~$X$.
Fix points $x_1\ld x_N$ and corresponding multiplicities $s_1\ld s_N$, and
set $p=\sum (2+s_j)^2$. Then $H^0(X,K_X+L)$ generates simultaneously
jets of order $s_j$ at all points $x_j$ provided that
$L^2>p$ and $L\cdot C>p$ for all curves $C$ passing through one of the
points~$x_j$.}
\smallskip
\item{\rm a)} Using the Riemann-Roch formula, show that the condition $L^2>p$
implies the existence of a section of a large multiple $mL$ vanishing
at order ${}>m(2+s_j)$ at each of the points.
\item{\rm b)} Construct a sequence of singular hermitian metrics on $L$
with positive definite curvature, such that the weights $\varphi_\nu$ have
algebraic singularities, $\nu(\varphi_\nu,x_j)\ge 2+s_j$ at each point,
and such that for some integer $m_1>0$ the multiplier ideal sheaves
satisfy $\cI(m_1\varphi_{\nu+1})\supsetneq\cI(m_1\varphi_\nu)$ if
$V(\cI(\varphi_\nu))$ is not $0$-dimensional near some $x_j$.
\smallskip\noindent
{\it Hint\/}: a) starts the procedure. Fix $m_0>0$ such that $m_0L-K_X$ is
ample. Use Nadel's vanishing theorem to show that
$$H^q(X,\cO((m+m_0)L)\otimes\cI(\lambda m\varphi_\nu))=0
\qquad\hbox{\rm for all $q\ge 1$, $m\ge 0$, $\lambda\in[0,1]$.}$$
Let $D_\nu$ be the effective $\bQ$-divisor describing the 1-dimensional
singularities of $\varphi_\nu$. Then $\cI(\lambda m\varphi_\nu)
\subset\cO(-\lfloor\lambda mD_\nu\rfloor)$ and the quotient has
$0$-dimensional support, hence
$$H^q(X,\cO((m+m_0)L)\otimes\cO(-\lfloor\lambda m D_\nu\rfloor))=0
\qquad\hbox{\rm for all $q\ge 1$, $m\ge 0$, $\lambda\in[0,1]$.}$$
By the Riemann-Roch formule again prove that
$$h^0(X,\cO((m+m_0)L)\otimes\cO/\cO(-\lfloor\lambda
mD_\nu\rfloor))={m^2\over 2}(2\lambda\, L\cdot D_\nu
-\lambda^2 D_\nu^2)+O(m).\leqno(\star)$$
As the left hand side of $(\star)$ is increasing with $\lambda$, one
must have $D_\nu^2\le L\cdot D_\nu$. If~$V(\cI(\varphi_\nu))$ is not
$0$-dimensional at $x_j$, then the coeffi\-cient of some component of
$D_\nu$ passing through $x_j$ is at least~$1$, hence
$$2\, L\cdot D_\nu -D_\nu^2\ge L\cdot D_\nu\ge p+1.$$
Show the existence of an integer $m_1>0$ independent of $\nu$ such that
$$h^0(X,\cO((m+m_0)L)\otimes\cO/\cO(-\lfloor mD_\nu\rfloor))>
\sum_{1\le j\le N}{(m+m_0)(2+s_j)+2\choose 2}$$
for $m\ge m_1$, and infer the existence of a suitable section of $(m_1+
m_0)L$ which is not in $H^0(X,\cO((m_1+m_0)L-\lfloor m_1D_\nu\rfloor))$.
Use this section to construct $\varphi_{\nu+1}$ such that
$\cI(m_1\varphi_{\nu+1})\supsetneq\cI(m_1\varphi_\nu)$.
\endclaim

\titleb{9.}{Holomorphic Morse Inequalities}
Let $X$ be a compact K\"ahler manifold, $E$ a holomorphic vector bundle of
rank $r$ and $L$ a line bundle over~$X$. If $L$ is equipped with a smooth 
metric of curvature form $\Theta(L)$, we define the $q$-index set of $L$
to be the open subset
$$X(q,L)=\left\{x\in X~;~\ii\Theta(L)_x \hbox{~has~}
{\displaystyle q\atop\displaystyle n-q}~
{\hbox{negative eigenvalues}\atop\hbox{positive eigenvalues}}\right\}
\leqno(9.1)$$
for $0\le q\le n$. Hence $X$ admits a partition $X=\Delta\cup
\bigcup_q X(q,L)$ where $\Delta=\{\hbox{$x\in X$}\,;\,\det(\Theta(L)_x)=0\}$
is the degeneracy set. We also introduce
$$X(\le q,L)=\bigcup_{0\le j\le q}X(j,L).\leqno(9.1')$$
It is shown in [Dem85b] that the cohomology groups
$H^q\big(X,E\otimes\cO(kL)\big)$ satisfy the following asymptotic
{\it weak Morse inequalities} as $k\to+\infty$
$$h^q\big(X,E\otimes\cO(kL)\big)\le
r{k^n\over n!}\int_{X(q,L)} (-1)^q\Big({\ii\over2\pi}\Theta(L)\Big)^n+o(k^n).
\leqno(9.2)$$
A sharper form is given by the {\it strong Morse inequalities}
$$\leqalignno{
\sum_{0\le j\le q}(-1)^{q-j}h^j\big(X,E&{}\otimes\cO(kL)\big)\cr
&\le r{k^n\over n!}\int_{X(\le q,L)}(-1)^q\Big({\ii\over2\pi}\Theta(L)\Big)^n
+o(k^n).&(9.2')\cr}$$
These inequalities are a useful complement to the Riemann-Roch formula
when information is needed about individual cohomology groups, and not
just about the Euler-Poincar\'e characteristic.

One difficulty in the application of these inequalities is that the
curvature integral is in general quite uneasy to compute, since it is
neither a topological nor an algebraic invariant. However, the Morse
inequalities can be reformulated in a more algebraic setting in which
only algebraic invariants are involved. We give here two such
reformulations.

\claim{(9.3) Theorem} Let $L=F-G$ be a holomorphic line bundle over a
compact K\"ahler manifold~$X$, where $F$ and $G$ are numerically effective
line bundles. Then for every $q=0,1,\ldots,n=\dim X$, there is an asymptotic
strong Morse inequality
$$\sum_{0\leq j\leq q}(-1)^{q-j}h^j(X,kL)\leq{k^n\over n!}
\sum_{0\leq j\leq q}(-1)^{q-j}{n\choose j}F^{n-j}\cdot G^j+o(k^n).$$
\endclaim

\begproof{} By adding $\varepsilon$ times a K\"ahler metric $\omega$ to the
curvature forms of $F$ and $G$, $\varepsilon>0$ one can write ${\ii\over
2\pi}\Theta(L)=\theta_\varepsilon(F)-\theta_\varepsilon(G)$ where
$\theta_\varepsilon(F)={\ii\over 2\pi}\Theta(F)+\varepsilon\omega$ and
$\theta_\varepsilon(G)={\ii\over 2\pi}\Theta(G)+\varepsilon\omega$ are
positive definite. Let $\lambda_1\geq\ldots\geq\lambda_n>0$ be the
eigenvalues of $\theta_\varepsilon(G)$ with respect to $\theta_\varepsilon
(F)$. Then the eigenvalues of ${\ii\over 2\pi}\Theta(L)$ with respect
to $\theta_\varepsilon(F)$ are the real numbers $1-\lambda_j$ and the
set $X(\le q,L)$ is the set $\{\lambda_{q+1}<1\}$ of points $x\in X$ such
that $\lambda_{q+1}(x)<1$. The strong Morse inequalities yield 
$$\sum_{0\leq j\leq q}(-1)^{q-j}h^j(X,kL)\leq{k^n\over n!}
\int_{\{\lambda_{q+1}<1\}}\!\!\!(-1)^q\!\!
\prod_{1\leq j\leq n}(1-\lambda_j)\theta_\varepsilon(F)^n+o(k^n).$$
On the other hand we have
$${n\choose j}\theta_\varepsilon(F)^{n-j}\wedge\theta_\varepsilon(G)^j=
\sigma_n^j(\lambda)\,\theta_\varepsilon(F)^n,$$
where $\sigma_n^j(\lambda)$ is the $j$-th elementary symmetric function in
$\lambda_1,\ldots,\lambda_n\,$, hence
$$\sum_{0\leq j\leq q}(-1)^{q-j}{n\choose j}F^{n-j}\cdot G^j=
\lim_{\varepsilon\to 0}\int_X\sum_{0\leq j\leq q}(-1)^{q-j}
\sigma_n^j(\lambda)\,\theta_\varepsilon(F)^n.$$
Thus, to prove the Lemma, we only have to check that
$$\sum_{0\leq j\leq n}(-1)^{q-j}\sigma_n^j(\lambda)-
\bOne_{\{\lambda_{q+1}<1\}}(-1)^q\prod_{1\leq j\leq n}(1-\lambda_j)\ge 0$$
for all $\lambda_1\geq\ldots\geq\lambda_n\geq 0$, where $\bOne_
{\{\ldots\}}$ denotes the characteristic function of a set.
This is easily done by induction on~$n$ (just split apart the parameter
$\lambda_n$ and write $\sigma_n^j(\lambda)=
\sigma_{n-1}^j(\lambda)+\sigma_{n-1}^{j-1}(\lambda)\,\lambda_n$).\qed
\endproof

In the case $q=1$, we get an especially interesting lower bound (this
bound has been observed and used by S.~Trapani [Tra95] in a similar
context).

\claim{(9.4) Consequence}
$h^0(X,kL)-h^1(X,kL)\geq{k^n\over n!}(F^n-nF^{n-1}\cdot G)-o(k^n)$.\newline
Therefore some multiple $kL$ has a section as soon as $F^n-nF^{n-1}\cdot
G>0$.
\endclaim

\claim{(9.5) Remark} \rm The weaker inequality
$$h^0(X,kL)\geq{k^n\over n!}(F^n-nF^{n-1}\cdot G)-o(k^n)$$
is easy to prove if $X$ is projective algebraic. Indeed, by adding a
small ample $\bQ$-divisor to $F$ and $G$, we may assume that
$F$, $G$ are ample. Let $m_0G$ be very ample and let $k'$ be the smallest
integer $\geq k/m_0$. Then $h^0(X,kL)\geq h^0(X,kF-k'm_0G)$. We select
$k'$ smooth members $G_j$, $1\leq j\leq k'$ in the linear system $|m_0G|$
and use the exact sequence
$$0\to H^0(X,kF-\sum G_j)\to H^0(X,kF)\to\bigoplus H^0(G_j,kF_{|G_j}).$$
Kodaira's vanishing theorem yields $H^q(X,kF)=0$ and $H^q(G_j,kF_{|G_j})=0$
for $q\geq 1$ and $k\geq k_0$. By the exact sequence combined with
Riemann-Roch, we get
$$\eqalign{
h^0(X,kL)&\geq h^0(X,kF-\sum G_j)\cr
&\geq{k^n\over n!}F^n-O(k^{n-1})-
\sum\Big({k^{n-1}\over(n-1)!}F^{n-1}\cdot G_j-O(k^{n-2})\Big)\cr
&\geq{k^n\over n!}\Big(F^n-n{k'm_0\over k}F^{n-1}\cdot G\Big)-O(k^{n-1})\cr
&\geq{k^n\over n!}\Big(F^n-n\,F^{n-1}\cdot G\Big)-O(k^{n-1}).\cr}$$
(This simple proof is due to F.~Catanese.)\qed
\endclaim

\claim{(9.6) Corollary} Suppose that $F$ and $G$ are nef and that $F$ is
big. Some multiple of $mF-G$ has a section as soon as
$$m>n\,{F^{n-1}\cdot G\over F^n}.$$
\endclaim

In the last condition, the factor $n$ is sharp: this is easily seen by taking
$X=\bP_1^n$ and $F=\cO(a,\ldots,a)$ and $G=\cO(b_1,\ldots,b_n)$ over
$\bP_1^n\,$; the condition of the Corollary is then
$m>\sum b_j/a$, whereas $k(mF-G)$ has a section if and only if
$m\ge\sup b_j/a$; this shows that we cannot replace $n$ by
$n(1-\varepsilon)$.\qed

\titleb{10.}{Effective Version of Matsusaka's Big Theorem}
An important problem of algebraic geometry is to find effective bounds
$m_0$ such that multiples $mL$ of an ample line bundle become very
ample for $m\ge m_0$. From a theoretical point of view, this problem
has been solved by Matsusaka [Mat72] and Koll\'ar-Matsusaka [KoM83].
Their result is that there is a bound $m_0=m_0(n,L^n,L^{n-1}\cdot K_X)$
depending only on the dimension and on the first two coefficients $L^n$
and $L^{n-1}\cdot K_X$ in the Hilbert polynomial of~$L$. Unfortunately,
the original proof does not tell much on the actual dependence of $m_0$
in terms of these coefficients. The goal of this section is to find
effective bounds for such an integer~$m_0$, along the lines of [Siu93].
However, one of the technical lemmas used in [Siu93] to deal with
dualizing sheaves can be sharpened. Using this sharpening of the lemma,
Siu's bound will be here substantially improved. We first start
with the simpler problem of obtaining merely a nontrivial section
in $mL$. The idea, more generally, is to obtain a criterion for the
ampleness of $mL-B$ when $B$ is nef. In this way, one is able to subtract
from $mL$ any multiple of $K_X$ which happens to get added by the
application of Nadel's vanishing theorem (for this, replace $B$ by $B$
plus a multiple of $K_X+(n+1)L$).

\claim{(10.1) Proposition} Let $L$ be an ample line bundle over a
projective $n$-fold $X$ and let $B$ be a nef line bundle over~$X$. Then
$K_X+mL-B$ has a nonzero section for some integer $m$ such that
$$m\le n\,{L^{n-1}\cdot B\over L^n}+n+1.$$
\endclaim  

\begproof{} Let $m_0$ be the smallest integer ${}>n\,{L^{n-1}\cdot B\over
L^n}$. Then $m_0L-B$ can be equipped with a singular hermitian metric
of positive definite curvature. Let $\varphi$ be the weight of this metric.
By Nadel's vanishing theorem, we have
$$H^q(X,\cO(K_X+mL-B)\otimes\cI(\varphi))=0\qquad\hbox{for $q\ge 1$,}$$
thus $P(m)=h^0(X,\cO(K_X+mL-B)\otimes\cI(\varphi))$ is a polynomial for
$m\ge m_0$. Since $P$ is a polynomial of degree $n$ and is not identically
zero, there must be an integer $m\in[m_0,m_0+n]$ which is not a root.
Hence there is a nontrivial section in
$$H^0(X,\cO(K_X+mL-B))\supset H^0(X,\cO(K_X+mL-B)\otimes\cI(\varphi))$$
for some $m\in[m_0,m_0+n]$, as desired.\qed
\endproof

\claim{(10.2) Corollary} If $L$ is ample and $B$ is nef, then $mL-B$ has
a nonzero section for some integer
$$m\le n\Big({L^{n-1}\cdot B+L^{n-1}\cdot K_X\over L^n}+n+1\Big).$$
\endclaim

\begproof{} By Fujita's result 9.3~a), $K_X+(n+1)L$ is nef. We can thus
replace $B$ by $B+K_X+(n+1)L$ in the result of Proposition~10.1.
Corollary 10.2 follows.\qed
\endproof

\claim{(10.3) Remark} \rm We do not know if the above Corollary is
sharp, but it is certainly not far from being so. Indeed, for $B=0$, the
initial constant $n$ cannot be replaced by anything smaller than
$n/2\,$: take $X$ to be a product of curves $C_j$ of large genus $g_j$ and
$B=0$; our bound for $L=\cO(a_1[p_1])\otimes\ldots\otimes\cO(a_n[p_n])$
to have $|mL|\neq\emptyset$ becomes $m\le\sum(2g_j-2)/a_j+n(n+1)$, which
fails to be sharp only by a factor $2$ when $a_1=\ldots=a_n=1$ and
$g_1\gg g_2\gg\ldots\gg g_n\to+\infty$. On the other hand, the additive
constant $n+1$ is already best possible when $B=0$ and $X=\bP^n$.\qed
\endclaim

So far, the method is not really sensitive to singularities (the Morse
inequalities are indeed still true in the singular case as is easily seen
by using desingularizations of the ambient variety). The same is true
with Nadel's vanishing theorem, provided that $K_X$ is replaced by the
$L^2$ dualizing sheaf $\omega_X$ (according to the notation introduced
in Remark 6.22, $\omega_X=K_X(0)$ is the sheaf of holomorphic $n$-forms
$u$ on $X_\reg$ such that $\smash{\ii^{n^2}}u\wedge\ol u$ is integrable
in a neighborhood of the singular set). Then Proposition~10.1 can be
generalized as
\medskip

\claim{(10.4) Proposition} Let $L$ be an ample line bundle over a
projective $n$-fold $X$ and let $B$ be a nef line bundle over~$X$. For every
$p$-dimensional $($reduced$)$ algebraic subvariety $Y$ of~$X$,
there is an integer
$$m\le p{L^{p-1}\cdot B\cdot Y\over L^p\cdot Y}+p+1$$
such that the sheaf $\omega_Y\otimes\cO_Y(mL-B)$ has a nonzero section.\qed
\endclaim

To proceed further, we need the following useful ``upper estimate''
about $L^2$ dualizing sheaves (this is one of the crucial steps in
Siu's approach; unfortunately, it has the effect of producing rather
large final bounds when the dimension increases).

\claim{(10.5) Proposition} Let $H$ be a very ample line bundle on a
projective algebraic mani\-fold~$X$, and let $Y\subset X$ be a
$p$-dimensional irreducible algebraic subvariety. If
\hbox{$\delta=H^p\cdot Y$} is the degree of $Y$ with respect to $H$,
the sheaf
$$\cHom\big(\omega_Y,\cO_Y((\delta-p-2)H)\big)$$
has a nontrivial section.
\endclaim

Observe that if $Y$ is a smooth hypersurface of degree $\delta$ in
$(X,H)=(\bP^{p+1},\cO(1))$, then $\omega_Y=\cO_Y(\delta-p-2)$ and the
estimate is optimal. On the other hand, if $Y$ is a smooth complete
intersection of multidegree $(\delta_1\ld\delta_r)$ in $\bP^{p+r}$,
then $\delta=\delta_1\ldots\delta_r$ whilst
$\omega_Y=\cO_Y(\delta_1+\ldots+\delta_r-p-r-1)\,$; in this case,
Proposition~(10.5) is thus very far from being sharp.

\begproof{} Let $X\subset\bP^N$ be the embedding given by $H$, so
that $H=\cO_X(1)$. There is a linear projection
$\bP^n\dasharrow\bP^{p+1}$ whose restriction $\pi:Y\to\bP^{p+1}$
to $Y$ is a finite and regular birational map of $Y$ onto an algebraic
hypersurface $Y'$ of degree $\delta$ in $\bP^{p+1}$.  Let $s\in
H^0(\bP^{p+1},\cO(\delta))$ be the polynomial of degree $\delta$
defining $Y'$. We claim that for any small Stein open set
$W\subset\bP^{p+1}$ and any $L^2$ holomorphic $p$-form $u$ on $Y'\cap
W$, there is a $L^2$ holomorphic $(p+1)$-form $\wt u$ on $W$ with
values in $\cO(\delta)$ such that $\wt u_{\restriction Y'\cap
W}=u\wedge ds$. In fact, this is precisely the conclusion of the
Ohsawa-Takegoshi extension theorem [OT87], [Ohs88] (see also [Man93]
for a more general version); one can also invoke more standard local
algebra arguments (see Hartshorne [Har77], Th.~III-7.11). As
$K_{\bP^{p+1}}=\cO(-p-2)$, the form $\wt u$ can be seen as a section
of $\cO(\delta-p-2)$ on~$W$, thus the sheaf morphism $u\mapsto u\wedge
ds$ extends into a global section of
$\cHom\big(\omega_{Y'},\cO_{Y'}(\delta-p-2)\big)$. The pull-back by
$\pi^\star$ yields a section of $\cHom\big(\pi^\star\omega_{Y'},
\cO_Y((\delta-p-2)H)\big)$. Since $\pi$ is finite and generically
$1:1$, it is easy to see that $\pi^\star\omega_{Y'}=\omega_Y$. The
Proposition follows.\qed
\endproof

By an appropriate induction process based on the above results, we can
now improve Siu's effective version of the Big Matsusaka Theorem
[Siu93]. Our version depends on a constant $\lambda_n$ such that
$m(K_X+(n+2)L)+G$ is very ample for $m\ge\lambda_n$ and every nef line
bundle~$G$. Corollary~(8.5) shows that $\lambda_n\le{3n+1\choose n}-2n$,
and a similar argument involving the recent results of Angehrn-Siu
[AS95] implies $\lambda_n\le n^3-n^2-n-1$ for $n\ge 2$.
Of course, it is expected that $\lambda_n=1$ in view of the Fujita
conjecture. 

\claim{(10.6) Effective version of the Big Matsusaka Theorem} Let $L$ and
$B$ be nef line bundles on a projective $n$-fold~$X$. Assume that $L$
is ample and let $H$ be the very ample line bundle $H=\lambda_n(K_X+(n+2)L)$.
Then $mL-B$ is very ample for
$$m\ge(2n)^{(3^{n-1}-1)/2}{(L^{n-1}\cdot(B+H))^{(3^{n-1}+1)/2}
(L^{n-1}\cdot H)^{3^{n-2}(n/2-3/4)-1/4}\over(L^n)^{3^{n-2}(n/2-1/4)+1/4}}.$$
In particular $mL$ is very ample for
$$m\ge C_n~(L^n)^{3^{n-2}}\left(n+2+{L^{n-1}\cdot K_X\over L^n}
\right)^{3^{n-2}(n/2+3/4)+1/4}$$
with $C_n=(2n)^{(3^{n-1}-1)/2}(\lambda_n)^{3^{n-2}(n/2+3/4)+1/4}$.
\endclaim

\begproof{} We use Proposition~(10.4) and Proposition~(10.5) to construct 
inductively a sequence of (non necessarily irreducible) algebraic subvarieties
$X=Y_n\supset Y_{n-1}\supset\ldots\supset Y_2\supset Y_1$ such that
$Y_p=\bigcup_jY_{p,j}$ is $p$-dimensional, and $Y_{p-1}$ is obtained
for each $p\ge 2$ as the union of zero sets of sections
$$\sigma_{p,j}\in H^0(Y_{p,j},\cO_{Y_{p,j}}(m_{p,j}L-B))$$
with suitable integers $m_{p,j}\ge 1$. We proceed by induction on
decreasing values of the dimension~$p$, and find inductively upper
bounds $m_p$ for the integers $m_{p,j}$. 

By Corollary (10.2), an integer $m_n$ for $m_nL-B$ to have a section
$\sigma_n$ can be found with
$$m_n\le n\,{L^{n-1}\cdot(B+K_X+(n+1)L)\over L^n}
\le n\,{L^{n-1}\cdot(B+H)\over L^n}.$$
Now suppose that the sections $\sigma_n$, $\ldots$, $\sigma_{p+1,j}$ have
been constructed. Then we get inductively a $p$-cycle $\wt
Y_p=\sum\mu_{p,j}Y_{p,j}$ defined by $\wt Y_p={}$ sum of zero divisors
of sections $\sigma_{p+1,j}$ in $\wt Y_{p+1,j}$, where the mutiplicity
$\mu_{p,j}$ on $Y_{p,j}\subset Y_{p+1,k}$ is obtained by multiplying
the corresponding multiplicity $\mu_{p+1,k}$ with the vanishing order
of $\sigma_{p+1,k}$ along $Y_{p,j}$. As cohomology classes, we find
$$\wt Y_p\equiv\sum(m_{p+1,k}L-B)\cdot (\mu_{p+1,k}Y_{p+1,k})\le
m_{p+1} L\cdot \wt Y_{p+1}.$$
Inductively, we thus have the numerical inequality
$$\wt Y_p\le m_{p+1}\ldots m_n L^{n-p}.$$
Now, for each component $Y_{p,j}$, Proposition~(10.4) shows that there exists
a section of $\omega_{Y_{p,j}}\otimes\cO_{Y_{p,j}}(m_{p,j}L-B)$ for some
integer
$$m_{p,j}\le p{L^{p-1}\cdot B\cdot Y_{p,j}\over L^p\cdot Y_{p,j}}+p+1
\le pm_{p+1}\ldots m_n\,L^{n-1}\cdot B+p+1.$$
Here, we have used the obvious lower bound $L^{p-1}\cdot Y_{p,j}\ge 1$
(this is of course a rather weak point in the argument). The degree of
$Y_{p,j}$ with respect to $H$ admits the upper bound
$$\delta_{p,j}:=H^p\cdot Y_{p,j}\le
m_{p+1}\ldots m_n H^p\cdot L^{n-p}.$$
We use the Hovanski-Teissier concavity inequality ([Hov79], [Tei79, Tei82])
$$(L^{n-p}\cdot H^p)^{1\over p}(L^n)^{1-{1\over p}}\leq L^{n-1}\cdot H$$
to express our bounds in terms of the intersection numbers $L^n$ and
$L^{n-1}\cdot H$ only. We then get
$$\delta_{p,j}\le m_{p+1}\ldots m_n{(L^{n-1}\cdot H)^p\over (L^n)^{p-1}}.$$
By Proposition~(10.5), there is a nontrivial section in
$$\cHom\big(\omega_{Y_{p,j}},\cO_{Y_{p,j}}((\delta_{p,j}-p-2)H)\big).$$
Combining this section with the section in $\omega_{Y_{p,j}}\otimes
\cO_{Y_{p,j}}(m_{p,j}L-B)$ already cons\-truc\-ted, we get a section of
$\cO_{Y_{p,j}}(m_{p,j}L-B+(\delta_{p,j}-p-2)H)$ on~$Y_{p,j}$. Since we do
not want $H$ to appear at this point, we replace $B$ with
$B+(\delta_{p,j}-p-2)H$ and thus get a section $\sigma_{p,j}$ of
$\cO_{Y_{p,j}}(m_{p,j}L-B)$ with some integer $m_{p,j}$ such that
$$\eqalign{m_{p,j}
&\le pm_{p+1}\ldots m_n\,L^{n-1}\cdot(B+(\delta_{p,j}-p-2)H)+p+1\cr
&\le p\,m_{p+1}\ldots m_n\,\delta_{p,j}\,L^{n-1}\cdot(B+H)\cr
&\le p\,(m_{p+1}\ldots m_n)^2{(L^{n-1}\cdot H)^p\over(L^n)^{p-1}}\,
L^{n-1}\cdot(B+H).\cr}$$
Therefore, by putting $M=n\,L^{n-1}\cdot(B+H)$, we get the recursion relation
$$m_p\le M\,{(L^{n-1}\cdot H)^p\over (L^n)^{p-1}}\,(m_{p+1}\ldots m_n)^2\qquad
\hbox{for $2\le p\le n-1$},$$
with initial value $m_n\le M/L^n$. If we let $(\ol m_p)$ be the sequence
obtained by the same recursion formula with equalities instead of
inequalities, we get $m_p\le\ol m_p$ with $\ol m_{n-1}=
M^3(L^{n-1}\cdot H)^{n-1}/(L^n)^n$ and
$$\ol m_p={L^n\over L^{n-1}\cdot H}\,\ol m_{p+1}^2\ol m_{p+1}$$
for $2\le p\le n-2$. We then find inductively
$$m_p\le\ol m_p=M^{3^{n-p}}{(L^{n-1}\cdot H)^{3^{n-p-1}(n-3/2)+1/2}\over
(L^n)^{3^{n-p-1}(n-1/2)+1/2}}.$$
We next show that $m_0L-B$ is nef for
$$m_0=\max\big(m_2\,,\,m_3\ld m_n\,,\,m_2\ldots m_n\,L^{n-1}\cdot B\big).$$
In fact, let $C\subset X$ be an arbitrary irreducible curve. Either
$C=Y_{1,j}$ for some $j$ or there exists an integer $p=2\ld n$ such that
$C$ is contained in $Y_p$ but not in~$Y_{p-1}$. If $C\subset
Y_{p,j}\ssm Y_{p-1}$, then $\sigma_{p,j}$ does not vanish identically
on~$C$. Hence $(m_{p,j}L-B)_{\restriction C}$ has nonnegative degree and
$$(m_0L-B)\cdot C\ge(m_{p,j}L-B)\cdot C\geq 0.$$
On the other hand, if $C=Y_{1,j}\,$, then
$$(m_0L-B)\cdot C\ge m_0-B\cdot\wt Y_1\ge m_0-m_2\ldots m_n\,L^{n-1}\cdot
B\ge 0.$$
By the definition of $\lambda_n$ (and by Corollary~(8.5) showing that such a
constant exists), $H+G$ is very ample for every nef line bundle~$G$, in
particular $H+m_0L-B$ is very ample. We thus replace again $B$ with $B+H$.
This has the effect of replacing $M$ with $M=n\,\big(L^{n-1}\cdot(B+2H)\big)$
and $m_0$ with
$$m_0=\max\big(m_n\,,\,m_{n-1}\ld m_2\,,\,m_2\ldots m_n\,L^{n-1}\cdot(B+H)
\big).$$
The last term is the largest one, and from the estimate on $\ol m_p\,$, we
get
$$\eqalignno{m_0
&\le M^{(3^{n-1}-1)/2}{(L^{n-1}\cdot H)^{(3^{n-2}-1)(n-3/2)/2+(n-2)/2}
(L^{n-1}\cdot(B+H))\over(L^n)^{(3^{n-2}-1)(n-1/2)/2+(n-2)/2+1}}\cr
&\le(2n)^{(3^{n-1}-1)/2}{(L^{n-1}\cdot(B+H))^{(3^{n-1}+1)/2}
(L^{n-1}\cdot H)^{3^{n-2}(n/2-3/4)-1/4}\over(L^n)^{3^{n-2}(n/2-1/4)+1/4}}
\cr}$$
\qed
\endproof

\claim{(10.7) Remark} \rm In the surface case $n=2$, one can take
$\lambda_n=1$ and our bound yields $mL$ very ample for
$$m\ge 4\,{(L\cdot(K_X+4L))^2\over L^2}.$$
If one looks more carefully at the proof, the initial constant $4$ can be
replaced by~$2$. In fact, it has been shown recently by Fern\'andez del
Busto that $mL$ is very ample for
$$m>{1\over 2}\left[{(L\cdot(K_X+4L)+1)^2\over L^2}+3\right],$$
and an example of G.~Xiao shows that this bound is essentially optimal
(see [FdB96]).
\endclaim

\titleb{11.}{Positivity concepts for vector bundles}
In the course of the proof of Skoda's $L^2$ estimates, we will have to
deal with dual bundles and exact sequences of hermitian vector bundles. 
The following fundamental differential geometric lemma will be needed.

\claim{(11.1) Lemma} Let $E$ be a hermitian holomorphic vector bundle
of rank $r$ on a complex $n$-dimensional manifold~$X$.
Then the Chern connections of $E$ and $E^\star$ are related by
$\Theta(E^\star)=-{}^t\Theta(E)$ where ${}^t$ denotes transposition.
In other words, the associated hermitian forms $\wt\Theta(E)$ and
$\wt\Theta(E^\star)$ are related by
$$
\eqalign{
\wt\Theta(E)(\tau,\tau)&=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}\tau_{j\lambda}\ol\tau_{k\mu},\qquad
\tau=\sum_{j,\lambda}\tau_{j,\lambda}{\partial\over\partial z_j}\otimes
e_\lambda,\cr
\wt\Theta(E^\star)(\tau,\tau)&=-\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\mu\lambda}\tau^\star_{j\lambda}\ol\tau_{k\mu}^\star,\qquad
\tau^\star=\sum_{j,\lambda}\tau^\star_{j,\lambda}
{\partial\over\partial z_j}\otimes e^\star_\lambda.\cr}
$$
In particular $E>_\Grif 0$ if and only if $E^\star<_\Grif 0$.
\endclaim

Notice that the corresponding duality statement for Nakano positivity
is wrong (because of the twist of indices, which is fortunately irrelevant
in the case of decomposable tensors).

\begproof{} The Chern connections of $E$ and $E^\star$ are related by the
Leibnitz rule
$$
d(\sigma\wedge s)=(D_{E^\star}\sigma)\wedge s+(-1)^{\deg\sigma}\sigma\wedge
D_E s
$$
whenever $s$, $\sigma$ are forms with values in $E$, $E^\star$ respectively,
and $\sigma\wedge s$ is computed using the pairing $E^\star\otimes E\to\bC$.
If we differentiate a second time, this yields the identity
$$
0=(D_{E^\star}^2\sigma)\wedge s+\sigma\wedge D_E^2s,
$$
which is equivalent to the formula $\Theta(E^\star)=-{}^t\Theta(E)$. All
other assertions follow.\qed
\endproof

\claim{(11.2) Lemma} Let
$$
0\lra S\buildo \lra^j E\buildo \lra^g Q\lra 0
$$
be an exact sequence of holomorphic vector bundles. Assume that $E$ is 
equipped with a smooth hermitian metric, and that $S$ and $Q$ are endowed 
with the metrics (restriction-metric and quotient-metric) induced by that 
of $E$. Then 
$$
j^\star\oplus g:E\to S\oplus Q,\qquad
j\oplus g^\star:S\oplus Q\to E
\leqno(11.3)
$$ 
are $C^\infty$ isomorphisms of bundles, which are inverse of each other. 
In the $\ci$-splitting $E\simeq S\oplus Q$, the Chern connection of $E$ 
admits a matrix decomposition
$$
D_E = \pmatrix{D_S & -\beta^\star\cr \beta & D_Q\cr}
\leqno(11.4)
$$
in terms of the Chern connections of $S$ and $Q$, where
$$
\beta\in\ci\big(X,\Lambda^{1,0}T^\star_X\otimes \Hom(S,Q)\big),\qquad
\beta^\star\in\ci\big(X,\Lambda^{0,1}T^\star_X\otimes \Hom(Q,S)\big).
$$
The form $\beta$ is called the second fundamental form associated with
the exact sequence. It is uniquely defined by each of the two formulas
$$
D'_{\Hom(S,E)}j=g^\star\circ\beta,\qquad 
j\circ\beta^\star=-D''_{\Hom(Q,E)}g^\star.
\leqno(11.5)
$$
We have $D'_{\Hom(S,Q)}\beta=0$, $D''_{\Hom(Q,S)}\beta^\star=0$, and the 
curvature form of $E$ splits as
$$
\Theta(E) = \pmatrix{
\Theta(S)-\beta^\star\wedge\beta & -D'_{\Hom(Q,S)}\beta^\star\cr 
D''_{\Hom(S,Q)}\beta & \Theta(Q)-\beta\wedge\beta^\star\cr},
\leqno(11.6)
$$
and the curvature forms of $S$ and $Q$ can be expressed as
$$
\Theta(S)=\Theta(E)_{\restriction S}+\beta^\star\wedge\beta,\qquad
\Theta(Q)=\Theta(E)_{\restriction Q}+\beta\wedge\beta^\star,
\leqno(11.7)
$$
where $\Theta(E)_{\restriction S}$, $\Theta(E)_{\restriction Q}$ stand for
$j^\star\circ\Theta(E)\circ j$ and $g\circ\Theta(E)\circ g^\star$.
\endclaim

\begproof{} Because of the uniqueness property of Chern connections, it
is easy to see that we have a Leibnitz formula
$$
D_F(f\wedge u)=(D_{\Hom(E,F)}f)\wedge u +(-1)^{\deg f}f\wedge D_E u
$$
whenever $u,\,f$ are forms with values in hermitian vector bundles 
$E$ and $\Hom(E,F)$ 
(where $\Hom(E,F)=E^\star\otimes F$ is equipped with the tensor product 
metric and $f\wedge u$ incorporates the evaluation mapping
$\Hom(E,F)\otimes E\to F$). In our case, given a form $u$ with values in $E$,
we write $u=ju_S+g^\star u_Q$ where $u_S=j^\star u$ and $u_Q=gu$ are the 
projections of $u$ on~$S$ and~$Q$. We then get
$$
\eqalign{
D_Eu&=D_E(ju_S+g^\star u_Q)\cr
&=(D_{\Hom(S,E)}j)\wedge u_S+ j\cdot D_Su_S+
(D_{\Hom(Q,E)}g^\star)\wedge u_Q+g^\star\cdot D_Qu_Q.\cr}
$$
Since $j$ is holomorphic as well as $j^\star\circ j=\Id_S$, we find
$D''_{Hom(S,E)}j=0$ and
$$
D''_{Hom(S,S)}\Id_S=0=D''_{\Hom(E,S)}j^\star\circ j.
$$
By taking the adjoint, we see that $j^\star\circ D'_{\Hom(S,E)}j=0$,
hence $D'_{\Hom(S,E)}j$ takes values in~$g^\star Q$ and we thus have a
unique form $\beta$ as in the Lemma such that
\hbox{$D'_{\Hom(S,E)}j=g^\star\circ\beta$}. Similarly, $g$ and 
$g\circ g^\star=\Id_Q$ are holomorphic, thus
$$
D''_{Hom(Q,Q)}\Id_Q=0=g\circ D''_{\Hom(Q,E)}g^\star
$$
and there is a form $\gamma\in
\ci\big(X,\Lambda^{0,1}T^\star_X\otimes \Hom(Q,S)\big)$ such that
\hbox{$D''_{\Hom(Q,E)}g^\star=j\circ\gamma$}. By adjunction, we get
$D'_{\Hom(E,Q)}g=\gamma^\star\circ j^\star$ and $D''_{Hom(E,Q)}g=0$ 
implies $D'_{\Hom(Q,E)}g^\star=0$. If we differentiate $g\circ j=0$ we
then get
$$
0=D'_{\Hom(E,Q)}g\circ j+g\circ D'_{\Hom(S,E)}j=
\gamma^\star\circ j^\star\circ j+g\circ g^\star\circ\beta=\gamma^\star+\beta,
$$
thus $\gamma=-\beta^\star$ and $D''_{\Hom(Q,E)}g^\star=-j\circ\beta^\star$.
Combining all this, we get
$$
\eqalign{
D_Eu&=g^\star\beta\wedge u_S+j\cdot D_Su_S-
j\beta^\star\wedge u_Q+g^\star\cdot D_Qu_Q\cr
&=j\big(D_Su_S-\beta^\star\wedge u_Q\big)
+g^\star\big(\beta\wedge u_S+D_Qu_Q\big),\cr}
$$
and the asserted matrix decomposition formula follows. By squaring the
matrix, we get
$$
D_E^2 = \pmatrix{
D_S^2-\beta^\star\wedge\beta & -D_S\circ\beta^\star-\beta^\star\circ D_Q\cr 
D_Q\circ\beta+\beta\circ D_S& D_Q^2-\beta\wedge\beta^\star\cr}.
$$
As $D_Q\circ\beta+\beta\circ D_S=D_{\Hom(S,Q)}\beta$ and
$D_S\circ\beta^\star+\beta^\star\circ D_Q=D_{\Hom(Q,S)}\beta^\star$
by the Leibnitz rule,
the curvature formulas follow (observe, since the Chern curvature form
is of type $(1,1)$, that we must have $D'_{\Hom(S,Q)}\beta=0$, 
$D''_{\Hom(Q,S)}\beta^\star=0$).\qed
\endproof

\claim{(11.8) Corollary} Let $0\to S \to E \to Q \to 0$ be an exact
sequence of hermitian vector bundles. Then
\medskip
\item{\rm a)} $E\ge_\Grif 0~~\Longrightarrow~~Q\ge_\Grif 0,$
\smallskip
\item{\rm b)} $E\le_\Grif 0~~\Longrightarrow~~S\le_\Grif 0,$
\smallskip
\item{\rm c)} $E\le_\Nak0~~\Longrightarrow~~S\le_\Nak 0,$
\smallskip
\noindent and analogous implications hold true for strict positivity.
\endclaim

\begproof{} If $\beta$ is written $\sum dz_j \otimes \beta_j$,~$\beta_j 
\in \Hom(S,Q)$, then formulas (11.7) yield
$$\eqalign{
\ii\Theta(S) &= \ii\Theta(E)_{\restriction S} - \sum dz_j \wedge d\ol z_k
\otimes \beta^\star_k \beta_j,\cr
\ii\Theta(Q) &= \ii\Theta(E)_{\restriction Q} + \sum dz_j \wedge d\ol z_k
\otimes \beta_j\beta^\star_k.\cr}$$
Since $\beta\cdot(\xi\otimes s)= \sum \xi_j\beta_j\cdot s$ and $\beta^\star
\cdot(\xi\otimes s)= \sum \ol \xi_k\beta^\star_k\cdot s$ we get
$$\wt\Theta(S)(\xi\otimes s,\xi'\otimes s') = \wt\Theta(E)(\xi\otimes s,\xi'\otimes s')
- \sum_{j,k} \xi_j \ol \xi'_k \langle \beta_j\cdot s,\beta_k\cdot s'\rangle,$$
$$\wt\Theta(S)(u,u) = \wt\Theta(E)(u,u) - |\beta\cdot u|^2,$$
$$\wt\Theta(Q)(\xi\otimes s,\xi'\otimes s') = \wt\Theta(E)(\xi\otimes s,\xi'\otimes s')
+ \sum_{j,k} \xi_j \ol \xi'_k \langle \beta^\star_k\cdot s,\beta^\star_j\cdot 
s'\rangle,
$$
$$
\wt\Theta(Q)(\xi\otimes s,\xi\otimes s) = \wt\Theta(E)(\xi\otimes s,\xi\otimes
s) = |\beta^\star\cdot(\xi\otimes s)|^2.\eqno{\square}
$$
\endproof

Next, we need positivity properties which somehow interpolate between 
Griffiths and Nakano positivity. This leads to the concept of $m$-tensor
positivity.

\claim{(11.9) Definition} Let $T$ and $E$ be complex vector spaces of 
dimensions $n,\,r$ respectively, and let $\Theta$ be a hermitian form on 
$T\otimes E$.
\smallskip
\item{\rm a)} A tensor $u\in T\otimes E$ is said to be of rank $m$ if $m$
is the smallest $\ge 0$ integer such that $u$ can be written
$$u = \sum_{j=1}^m \xi_j \otimes s_j,~~~~\xi_j\in T,~s_j\in E.$$
\smallskip
\item{\rm b)} $\Theta$ is said to be $m$-tensor positive $($resp. 
$m$-tensor semi-positive$)$ if
$\Theta(u,u)>0$ $($resp. $\Theta(u,u)\ge 0)$ for every tensor 
$u\in T\otimes E$ of rank $\le m$, $u\ne 0$. In this case, we write
$$
\Theta >_m 0~~~~(\hbox{\it resp.}~~\Theta \ge_m 0).
$$
\vskip-\belowdisplayskip
\endclaim

We say that a hermitian vector bundle $E$ is $m$-tensor positive 
if $\wt\Theta(E)>_m 0$. Griffiths positivity corresponds to $m=1$ and 
Nakano positivity to $m\ge \min(n,r)$. Recall from (5.2) that we have
$$
\langle[\ii\Theta(E),\Lambda]u,u\rangle=\sum_{|S|=q-1}\sum_{j,k,\lambda,\mu}
c_{jk\lambda\mu}\,u_{jS,\lambda}\ol u_{kS,\mu}
$$
for every $(n,q)$-form $u=\sum u_{K,\lambda}\,dz_1\wedge
\ldots\wedge dz_n\wedge d\ol z_K\otimes e_\lambda$ with values in~$E$.
Since $u_{jS,\lambda}=0$ for $j\in S$, the rank of the tensor
$(u_{jS,\lambda})_{j,\lambda}\in\bC^n\otimes\bC^r$ is in fact 
$\le\min\{n-q+1,r\}$. We obtain therefore:

\claim{(11.10) Lemma} Assume that $E\ge_m0$ $($resp.\ $E>_m0)$. Then the
hermitian operator $[\ii\Theta(E),\Lambda]$ is semipositive $($resp.\ positive 
definite$)$ on $\Lambda^{n,q}T^\star X\otimes E$ for $q\ge 1$ and 
$m\ge\min\{n-q+1,r\}.$
\endclaim

The Nakano vanishing theorem can then be improved as follows.

\claim{(11.11) Theorem} Let $X$ be a weakly pseudoconvex K\"ahler 
manifold of dimension $n$ and let $E$ a hermitian vector bundle of 
rank $r$ such that $\wt\Theta(E)>_m0$ over~$X$. Then
$$
H^{n,q}(X,E)=0~~~~\hbox{\rm for}~~q\ge 1~~\hbox{\rm and}~~
m\ge\min\{n-q+1,r\}.
$$
\endclaim

We next study some important relations which exist between the various
positivity concepts. Our starting point is the following result of
[DSk79].

\claim{(11.12) Theorem} For any hermitian vector bundle $E$, 
$$E >_\Grif 0~~\Longrightarrow~~E\otimes \det E >_\Nak 0.$$
\endclaim

To prove this result, we use the fact that
$$
\Theta(\det E)=\Tr_E \Theta(E)\leqno(11.13)
$$
where $\Tr_E:\Hom(E,E)\to\bC$ is the trace map, together with the identity
$$
\Theta(E\otimes \det E) = \Theta(E)+\Tr_E(\Theta(E)) \otimes \Id_E,
$$
which is itself a consequence of (11.13) and of the standard formula 
$$
\Theta(E\otimes F) = \Theta(E)\otimes\Id_F+\Id_E\otimes\Theta(F).
$$
In order to prove (11.13), for instance, we differentiate twice a wedge 
product, according to the formula
$$
D_{\Lambda^pE}(s_1\wedge\cdots\wedge s_p)
=\sum_{j=1}^p(-1)^{\deg s_1+\cdots+\deg
s_{j-1}} s_1\wedge\cdots\wedge s_{j-1}\wedge D_E s_j\wedge\cdots\wedge s_p.
$$
The corresponding hermitian forms on $T_X\otimes E$ are thus related by
$$
\wt\Theta(E\otimes \det E)= \wt\Theta(E)+\Tr_E\wt\Theta(E) \otimes h,
$$
where $h$ denotes the hermitian metric on $E$ and $\Tr_E\wt\Theta(E)$
is the hermitian form on $T_X$ defined by
$$
\Tr_E \wt\Theta(E)(\xi,\xi) = \sum_{1\le \lambda\le r} \wt\Theta(E)(\xi\otimes
e_\lambda,\xi\otimes e_\lambda),\qquad\xi\in T_X,
$$
for any orthonormal frame $(e_1\ld e_r)$ of $E$. Theorem 11.12 is now a
consequence of the following simple property of hermitian forms on a tensor
product of complex vector spaces.

\claim{(11.14) Proposition} Let $T,E$ be complex vector spaces of respective
dimensions $n,r,\,$ and $h$ a hermitian metric on $E$. Then for
every hermitian form $\Theta$ on $T\otimes E$
$$\Theta >_\Grif 0~~\Longrightarrow~~
\Theta + \Tr_E\Theta \otimes h >_\Nak 0.$$
\endclaim

We first need a lemma analogous to Fourier inversion formula for discrete
Fourier transforms.

\claim{(11.15) Lemma} Let $q$ be an integer $\ge 3$, and 
$x_\lambda,~y_\mu,~1\le \lambda,\mu \le r$, be complex numbers. 
Let $\sigma$ describe
the set $U^r_q$ of $r$-tuples of $q$-th roots of unity and put
$$x'_\sigma = \sum_{1\le \lambda\le r} x_\lambda \ol \sigma_\lambda,~~~~
y'_\sigma = \sum_{1\le \mu\le r} y_\mu \ol \sigma_\mu,~~~~ \sigma\in U^r_q.$$
Then for every pair $(\alpha,\beta),~1\le \alpha,\beta\le r$, the following 
identity holds:
$$q^{-r} \sum_{\sigma\in U^r_q} x'_\sigma \ol y'_\sigma \sigma_\alpha \ol 
\sigma_\beta=\cases{
x_\alpha \ol y_\beta\phantom{\sum_\sum}&if~~$\alpha\ne \beta,$\cr
\displaystyle\sum_{1\le \mu\le r} x_\mu \ol y_\mu&if~~$\alpha=\beta.$\cr}$$
\endclaim

\begproof{} The coefficient of $x_\lambda\ol y_\mu$ in the summation
$q^{-r} \sum_{\sigma\in U^r_q} x'_\sigma \ol y'_\sigma \sigma_\alpha \ol 
\sigma_\beta$ is given by
$$q^{-r} \sum_{\sigma\in U^r_q}\sigma_\alpha \ol \sigma_\beta\ol
\sigma_\lambda \sigma_\mu.$$
This coefficient equals 1 when the pairs $\{ \alpha,\mu\}$ and
$\{\beta,\lambda\}$ are equal (in which case $\sigma_\alpha
\ol \sigma_\beta \ol \sigma_\lambda \sigma_\mu = 1$ for any one of the $q^r$
elements of $U^r_q$). Hence, it is sufficient to prove that
$$\sum_{\sigma\in U^r_q} \sigma_\alpha \ol \sigma_\beta \ol \sigma_\lambda
\sigma_\mu = 0$$
when the pairs $\{ \alpha,\mu\}$ and $\{ \beta,\lambda\}$
are distinct.

If $\{ \alpha,\mu\} \ne \{\beta,\lambda\}$, then one of the
elements of one of the pairs does not belong to the other pair. As the four
indices $\alpha,\beta,\lambda,\mu$ play the same role, we may suppose for 
example that $\alpha\notin \{\beta,\lambda\}$. Let us apply to 
$\sigma$ the substitution $\sigma\mapsto \tau$, where $\tau$ is defined by
$$\tau_\alpha = e^{2\pi\ii/q}\sigma_\alpha,~\tau_\nu = 
\sigma_\nu \quad \hbox{\rm for}\quad \nu\ne \alpha.$$
We get
$$\sum_\sigma\sigma_\alpha\ol\sigma_\beta\ol\sigma_\lambda\sigma_\mu=\sum_\tau
=\cases{
\displaystyle e^{2\pi\ii/q}\sum_\sigma&if~~$\alpha\ne\mu,$\cr
\displaystyle e^{4\pi\ii/q}\sum_\sigma&if~~$\alpha=\mu,$\cr}$$
Since $q\ge 3$ by hypothesis, it follows that
$$\sum_\sigma\sigma_\alpha\ol\sigma_\beta\ol\sigma_\lambda\sigma_\mu=0.$$
\endproof

\begproof{of Proposition 11.14.} Let $(t_j)_{1\le j\le n}$ be a basis of
$T$, $(e_\lambda)_{1\le \lambda\le r}$ an orthonormal basis of $E$ and
$\xi=\sum_j\xi_jt_j\in T$, $u=\sum_{j,\lambda}u_{j\lambda}\,t_j\otimes 
e_\lambda \in T\otimes E$. The coefficients $c_{jk\lambda\mu}$ of $\Theta$
with respect to the basis $t_j\otimes e_\lambda$ satisfy the symmetry
relation $\ol c_{jk\lambda\mu}=c_{kj\mu\lambda}$, and we have the formulas 
$$\eqalign{\Theta(u,u)
&=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu}u_{j\lambda}\ol u_{k\mu},\cr
\Tr_E\Theta(\xi,\xi)&=\sum_{j,k,\lambda}c_{jk\lambda\lambda}\xi_j\ol\xi_k,\cr
(\Theta+\Tr_E\Theta\otimes h)(u,u)&=\sum_{j,k,\lambda,\mu}c_{jk\lambda\mu} 
u_{j\lambda} \ol u_{k\mu}+ c_{jk\lambda\lambda}u_{j\mu}\ol u_{k\mu}.\cr}$$
For every $\sigma\in U^r_q$ (cf.\ Lemma~11.15), put
$$\eqalign{
u'_{j\sigma}&=\sum_{1\le\lambda\le r}u_{j\lambda}\ol\sigma_\lambda\in\bC,\cr
\wh u_\sigma&=\sum_j u'_{j\sigma}t_j\in T\quad,\quad
\wh e_\sigma=\sum_\lambda\sigma_\lambda e_\lambda\in E.\cr}$$
Lemma 11.15 implies
$$\eqalign{
q^{-r} \sum_{\sigma\in U^r_q}\Theta(\wh u_\sigma\otimes\wh e_\sigma,
\wh u_\sigma\otimes\wh e_\sigma)&= q^{-r} \sum_{\sigma\in U^r_q} c_{jk\lambda\mu}
u'_{j\sigma} \ol u'_{k\sigma}\sigma_\lambda \ol \sigma_\mu\cr
&= \sum_{j,k,\lambda\ne \mu} c_{jk\lambda\mu} u_{j\lambda} \ol u_{k\mu} +
\sum_{j,k,\lambda,\mu} c_{jk\lambda\lambda} u_{j\mu} \ol u_{k\mu}.\cr}$$
The Griffiths positivity assumption shows that the left hand side is $\ge 0$,
hence
$$(\Theta + \Tr_E \Theta \otimes h)(u,u) \ge \sum_{j,k,\lambda} c_{jk\lambda\lambda} u_{j\lambda} \ol u_{k\lambda} \ge 0$$
with strict positivity if $\Theta >_\Grif 0$ and $u\ne 0$.\qed
\endproof

We now relate Griffiths positivity to $m$-tensor positivity. The most useful
result is the following

\claim{(11.16) Proposition} Let $T$ be a complex vector space and
$(E,h)$ a hermitian vector space of respective dimensions $n,r$ with
$r\ge 2$. Then for any hermitian form $\Theta$ on $T\otimes E$ and any
integer $m\ge 1$
$$\Theta>_\Grif0~~~\Longrightarrow~~~m\Tr_E\Theta\otimes h-\Theta>_m0.$$
\endclaim

\begproof{} Let us distinguish two cases.
\medskip
\noindent a)~ $m=1$.~
Let $u\in T \otimes E$ be a tensor of rank 1. Then $u$ can be written
$u = \xi_1 \otimes e_1$ with $\xi_1\in T,~\xi_1 \ne 0$, and
$e_1 \in E,~|e_1| = 1$. Complete $e_1$ into an orthonormal basis
$(e_1\ld e_r)$ of $E$. One gets immediately
$$\eqalign{
(\Tr_E \Theta \otimes h)(u,u) &= \Tr_E \Theta (\xi_1,\xi_1) = \sum_{1\le \lambda\le r} \Theta(\xi_1\otimes e_\lambda , \xi_1 \otimes e_\lambda)\cr
&> \Theta (\xi_1\otimes e_1,\xi_1\otimes e_1) = \Theta (u,u).\cr}$$
\medskip
\noindent b)~ $m\ge 2$.~
Every tensor $u\in T\otimes E$ of $\hbox{\rm rank~} \le m$ can be written
$$u=\sum_{1\le \lambda\le q} \xi_\lambda \otimes e_\lambda\quad,
\quad \xi_\lambda \in T,$$
with $q = \min(m,r)$ and $(e_\lambda)_{1\le \lambda\le r}$ an orthonormal basis
of $E$. Let $F$ be the vector subspace of $E$ generated by $(e_1\ld e_q)$
and $\Theta_F$ the restriction of $\Theta$ to $T\otimes F$. The first part
shows that
$$\Theta':= \Tr_F \Theta_F \otimes h - \Theta_F >_\Grif 0.$$
Proposition 11.14 applied to $\Theta'$ on $T\otimes F$ yields
$$\Theta'+\Tr_F \Theta'\otimes h=q\Tr_F\Theta_F\otimes h-\Theta_F>_q 0.$$
Since $u\in T\otimes F$ is of $\hbox{\rm rank~} \le q \le m$, we get
(for $u\ne 0$)
$$\eqalignno{
\Theta(u,u) = \Theta_F(u,u) &< q(\Tr_F \Theta_F \otimes h)(u,u)\cr
&=q\sum_{1\le j,\lambda\le q}\Theta(\xi_j\otimes e_\lambda,
\xi_j\otimes e_\lambda)\le m\Tr_E\Theta\otimes h(u,u).&\square\cr}$$
\endproof

Proposition 11.16 is of course also true in the semi-positive case. From these
facts, we deduce

\claim{(11.17) Theorem} Let $E$ be a Griffiths $($semi-$)$positive bundle of 
rank $r\ge 2$. Then for any integer $m\ge 1$
$$E^\star \otimes (\det E)^m >_m 0~~~~(\hbox{\it resp.}~~\ge_m 0).$$
\endclaim

\begproof{} We apply Proposition~11.16 to $\Theta = -\Theta (E^\star) =
{}^t\Theta(E)\ge_\Grif 0$ on $T_X\otimes E^\star$  and observe that
$$\Theta(\det E) = \Tr_E \Theta (E) = \Tr_{E^\star} \Theta.$$
\endproof

\claim{(11.18) Theorem} Let $0\to S \to E \to Q \to 0$ be an exact 
sequence of hermitian vector bundles. Then for any $m\ge 1$
$$E>_m0~~~\Longrightarrow~~~S\otimes (\det Q)^m >_m 0.$$
\endclaim

\begproof{} Formulas (11.7) imply
$$\ii\Theta(S) >_m \ii\beta^\star \wedge \beta\quad,\quad \ii\Theta(Q) >_m \ii\beta \wedge \beta^\star,$$
$$\ii\Theta(\det Q) = \Tr_Q(\ii\Theta(Q)) > \Tr_Q(\ii\beta \wedge \beta^\star).$$
If we write $\beta = \sum dz_j \otimes \beta_j$ as in the proof of
Corollary~11.8, then
$$\eqalign{ \Tr_Q(\ii\beta \wedge \beta^\star) &= \sum idz_j \wedge d\ol
z_k \Tr_Q(\beta_j\beta^\star_k)\cr &=\sum idz_j \wedge d\ol z_k
\Tr_S(\beta^\star_k\beta_j) = \Tr_S(-\ii\beta^\star \wedge \beta).\cr}$$
Furthermore, it has been already proved that $-\ii\beta^\star \wedge
\beta \ge_\Nak 0$. By Proposition~11.16 applied to the corresponding
hermitian form $\Theta$ on $T_X \otimes S$, we get 
$$
m\Tr_S(-\ii\beta^\star \wedge \beta) \otimes \Id_S + \ii\beta^\star\wedge
\beta \ge_m 0,
$$ 
and Theorem~11.18 follows.\qed
\endproof

\claim{(11.19) Corollary} Let $X$ be a weakly pseudoconvex K\"ahler 
$n$-dimensional manifold, $E$ a holomorphic vector bundle of rank 
$r\ge 2$ and $m\ge 1$ an integer. Then
\smallskip
\item{\rm a)} $E>_\Grif0 \Rightarrow H^{n,q}(X,E\otimes \det\,E)=0$
for $q\ge 1\,;$
\smallskip
\item{\rm b)} $E>_\Grif0\Rightarrow H^{n,q}\big
(X,E^\star\otimes(\det\,E)^m\big)=0$ for $q\ge 1$ and
\hbox{$m\ge\min\{n-q+1,r\}\,;$}
\smallskip
\item{\rm c)} Let~ $0\to S\to E\to Q\to 0$ be an exact sequence of
vector bundles and $m=\min\{n-q+1,\rk\,S\}$, $q\ge 1$. If $E>_m0$ and if
$L$ is a line bundle such that $L\otimes(\det Q)^{-m}\ge 0$, then
$$H^{n,q}(X,S\otimes L)=0.$$
\endclaim

\begproof{} Immediate consequence of Theorem 11.11, in combination with
11.12 for a), 11.17 for b) and 11.18 for c).\qed
\endproof

\titleb{12.}{Skoda's $L^2$ Estimates for Surjective Bundle Morphisms}
Let $(X,\omega)$ be a K\"ahler manifold, $\dim X=n$, and let $g:E\to Q$ a
holomorphic morphism of hermitian vector bundles over $X$. Assume
in the first instance that $g$ is {\it surjective}. We are interested in
conditions insuring that the induced morphisms $g:H^{n,k}(X,E)\lra 
H^{n,k}(X,Q)$ are also surjective (dealing with $(n,\bu)$ bidegrees is 
always easier, since we have to understand positivity conditions
for the curvature term). For that purpose, it is natural to consider the 
subbundle $S=\Ker g\subset E$ and the exact sequence
$$
0\lra S\buildo \lra^j E\buildo \lra^g Q\lra 0
\leqno(12.1)
$$
where $j:S\to E$ is the inclusion. In fact, we need a little more flexibility
to handle the curvature terms, so we take the tensor product of the exact 
sequence by a holomorphic line bundle $L$ (whose properties will be specified
later):
$$
0\lra S\otimes L\lra E\otimes L\buildo \lra^g Q\otimes L\lra0.
\leqno(12.2)
$$

\claim{(12.3) Theorem} Let $k$ be an integer such that $0\le k\le n$. Set
$r=\rk\,E$, $q={\rm rk Q}$, $s=\rk\,S=r-q$~ and
$$
m=\min\{n-k,s\}=\min\{n-k,r-q\}.
$$
Assume that $(X,\omega)$ possesses also a complete K\"ahler metric $\wh\omega$,
that $E\ge_m 0$, and that $L\lra X$ is a hermitian holomorphic line bundle such that
$$
\ii\Theta(L)-(m+\varepsilon)\ii\Theta(\det Q)\ge 0
$$
for some $\varepsilon>0$. Then for every $D''$-closed form $f$ of type 
$(n,k)$ with values in $Q\otimes L$ such that $\|f\|<+\infty$, there exists a 
$D''$-closed form $h$ of type $(n,k)$ with values in $E\otimes L$ such
that $f=g\cdot h$ and 
$$
\|h\|^2\le(1+m/\varepsilon)\,\|f\|^2.
$$
\endclaim

The idea of the proof is essentially due to [Sko78], who actually
proved the special case $k=0$. The general case appeared in [Dem82b].

\begproof{} Let $j:S\to E$ be the inclusion morphism, $g^\star:Q\to E$
and $j^\star:E\to S$ the adjoints of $g,j$, and
the matrix of $D_E$ with respect to the orthogonal splitting 
$E\simeq S\oplus Q$ (cf.\ Lemma~11.2). Then $g^\star f$ is a lifting of $f$
in $E\otimes L$. We will try to find $h$ under the form
$$
h=g^\star f+ju,~~~~u\in L^2(X,\Lambda^{n,k}T^\star_X\otimes S\otimes L).
$$
As the images of $S$ and $Q$ in $E$ are orthogonal, we have $|h|^2=|f|^2+|u|^2$
at every point of $X$. On the other hand $D''_{Q\otimes L}f=0$ by hypothesis 
and $D''g^\star=-j\circ\beta^\star$ by (11.5), hence
$$
D''_{E\otimes L}h=-j(\beta^\star\wedge f)+j\,D''_{S\otimes L}=
j(D''_{S\otimes L}-\beta^\star\wedge f).
$$
We are thus led to solve the equation
$$
D''_{S\otimes L}u=\beta^\star\wedge f,\leqno(12.4)
$$
and for that, we apply Theorem~6.1 to the $(n,k+1)$-form 
$\beta^\star\wedge f$. One now observes that the curvature of $S\otimes L$
can be expressed in terms of $\beta$. This remark will be used to prove:
\endproof

\claim{(12.5) Lemma} Let $A_k=[\ii\Theta(S\otimes L),\Lambda]$ be the 
curvature operator acting as an hermitian operator on the bundle of
\hbox{$(n,k+1)$-forms}. Then 
$$\langle A_k^{-1}(\beta^\star\wedge f),
(\beta^\star\wedge f)\rangle\le(m/\varepsilon)\,|f|^2.$$
\endclaim

If the Lemma is taken for granted, Theorem~9.4 yields a solution $u$ of~(12.4)
in $L^2(X,\Lambda^{n,q}T^\star_X\otimes S\otimes L)$ such that 
$\|u\|^2\le(m/\varepsilon)\,\|f\|^2$.
As \hbox{$\|h\|^2=\|f\|^2+\|u\|^2$}, the
proof of Theorem~12.3 is complete.\qed

\begproof{of Lemma~12.5.} Exactly as in the proof of Theorem~11.18,
the formulas (11.7) yield
$$
\ii\Theta(S)\ge_m \ii\beta^\star\wedge\beta,~~~~
\ii\Theta(\det Q)\ge\Tr_Q(\ii\beta\wedge
\beta^\star)=\Tr_S(-\ii\beta^\star\wedge\beta).
$$
Since $\ci(X,\Lambda^{1,1}T^\star_X\otimes \Herm\,S)
\ni\Theta:=-\ii\beta^\star\wedge\beta \ge_\Grif0$, Proposition~11.16 implies
$$
m\,\Tr_S(-\ii\beta^\star\wedge\beta)\otimes\Id_S+\ii\beta^\star\wedge
\beta\ge_m 0.
$$
From the hypothesis on the curvature of $L$ we get
$$
\eqalign{
\ii\Theta(S\otimes L)&\ge_m \ii\Theta(S)\otimes\Id_L+(m+\varepsilon)\,
\ii\Theta(\det Q)\otimes\Id_{S\otimes L}\cr
&\ge_m\big(\ii\beta^\star\wedge\beta+(m+\varepsilon)\,\Tr_S(-\ii\beta^\star
\wedge\beta)\otimes\Id_S\big)\otimes\Id_L\cr
&\ge_m(\varepsilon/m)\,(-\ii\beta^\star\wedge\beta)\otimes\Id_S\otimes
\Id_L.\cr}
$$
For any $v\in\Lambda^{n,k+1}T^\star_X\otimes S\otimes L$, Lemma~11.10 implies
$$
\langle A_kv,v\rangle\ge(\varepsilon/m)\,
\langle -\ii\beta^\star\wedge\beta\wedge\Lambda v,v\rangle,\leqno(12.6)
$$
because $\rk(S\otimes L)=s$ and $m=\min\{n-k,s\}$. Let $(dz_1\ld dz_n)$
be an orthonormal basis of $T^\star_X$ at a given point $x_0\in X$ and set 
$$
\beta=\sum_{1\le j\le n}dz_j\otimes\beta_j,~~~~\beta_j\in\Hom(S,Q).
$$
The adjoint of the operator $\beta^\star\wedge\bu=\sum d\ol z_j\wedge
\beta_j^\star\,\bu$ is the contraction operator $\beta\ort \bu$ defined by
$$
\beta\ort v=\sum{\partial\over\partial\ol z_j}\ort (\beta_j v)=
\sum -\ii dz_j\wedge\Lambda(\beta_j v)=-\ii\beta\wedge\Lambda v.
$$
Consequently, we get 
$\langle -\ii\beta^\star\wedge\beta\wedge\Lambda v,v\rangle
=|\beta\ort v|^2$ and (12.6) implies
$$
|\langle\beta^\star\wedge f,v\rangle|^2=|\langle f,\beta\ort v\rangle|^2\le
|f|^2\,|\beta\ort v|^2\le(m/\varepsilon)
\langle A_kv,v\rangle\,|f|^2.
$$
This is equivalent to the estimate asserted in the lemma.\qed
\endproof

If $X$ has a plurisubharmonic exhaustion function $\psi$, we can select
a convex increasing function $\chi\in\ci(\bR,\bR)$ and
multiply the metric of $L$ by the weight $\exp(-\chi\circ\psi)$
in order to make the $L^2$ norm of $f$ converge. Theorem 12.3 implies 
therefore:

\claim{(12.7) Corollary} Let $(X,\omega)$ be a weakly pseudoconvex K\"ahler 
manifold, \hbox{$g:E\to Q$} a surjective bundle morphism with
$r=\rk\,E$, $q=\rk\,Q$, and \hbox{$L\to X$} a hermitian holomorphic line 
bundle. We set \hbox{$m=\min\{n-k,r-q\}$} and assume that $E\ge_m 0$ and
$$
\ii\Theta(L)-(m+\varepsilon)\,\ii\Theta(\det Q)\ge 0
$$
for some $\varepsilon>0$. Then $g$ induces a surjective map
$$
H^{n,k}(X,E\otimes L)\lra H^{n,k}(X,Q\otimes L).
$$
\endclaim

The most remarkable feature of this result is that it does not require any 
strict positivity assumption on the curvature (for instance $E$ can be a flat
bundle). A careful examination of the proof shows that it amounts to verify 
that the image of the coboundary morphism
$$
-\beta^\star\wedge\bu~:~H^{n,k}(X,Q\otimes L)\lra H^{n,k+1}(X,S\otimes L)
$$
vanishes; however the cohomology group $H^{n,k+1}(X,S\otimes L)$
itself does not necessarily vanish, as it would do under a strict positivity
assumption.

We want now to get estimates also when $Q$ is endowed with a metric given
a priori, that can be distinct from the quotient metric of $E$ by $g$.
Then the map $g^\star(gg^\star)^{-1}~:~Q\lra E$ is the lifting of $Q$
orthogonal to $S=\Ker g$. The quotient metric $|\bu|'$ on $Q$ is
therefore defined in terms of the original metric $|\bu|$ by
$$
|v|^{\prime 2}=|g^\star(gg^\star)^{-1}v|^2=\langle(gg^\star)^{-1}v,v\rangle
=\det(gg^\star)^{-1}\,\langle\wt{gg^\star}v,v\rangle
$$
where $\wt{gg^\star}\in\End(Q)$ denotes the endomorphism of $Q$
whose matrix is the transposed comatrix of $gg^\star$. For every
$w\in\det Q$, we find
$$
|w|^{\prime2}=\det(gg^\star)^{-1}\,|w|^2.
$$
If $Q'$ denotes the bundle $Q$ with the quotient metric, we get
$$
\ii\Theta(\det Q')=\ii\Theta(\det Q)+\ii d'd''\log\det(gg^\star).
$$
In order that the hypotheses of Theorem~12.3 be satisfied, we are led to define
a new metric $|\bu|'$ on $L$ by $|u|^{\prime 2}=|u|^2\,\big(\det(gg^\star)
\big)^{-m-\varepsilon}$. Then
$$
\ii\Theta(L')=\ii\Theta(L)+(m+\varepsilon)\,\ii d'd''\log\det(gg^\star)\ge
(m+\varepsilon)\,\ii\Theta(\det Q').
$$
Theorem 12.3 applied to $(E,Q',L')$ can now be reformulated:

\claim{(12.8) Theorem} Let $X$ be a complete K\"ahler manifold equipped with
a K\"ahler metric $\omega$ on~$X$, let $E\to Q$ be
a surjective morphism of hermitian vector bundles and let
$L\to X$ be a hermitian holomorphic line bundle. Set $r=\rk\,E$,
$q=\rk\,Q$ and $m=\min\{n-k,r-q\}$, and assume that $E\ge_m 0$ and
$$
\ii\Theta(L)-(m+\varepsilon)\ii\Theta(\det Q)\ge 0
$$
for some $\varepsilon>0$. Then for every $D''$-closed form $f$ of type 
$(n,k)$ with values in $Q\otimes L$ such that 
$$
I=\int_X\langle\wt{gg^\star}f,f\rangle\,(\det gg^\star)^{-m-1-\varepsilon}\,
dV<+\infty,
$$
there exists a $D''$-closed form $h$ of type $(n,k)$ with values in 
$E\otimes L$ such that $f=g\cdot h$ and 
$$
\int_X|h|^2\,(\det gg^\star)^{-m-\varepsilon}\,dV
\le(1+m/\varepsilon)\,I.\eqno\square
$$
\endclaim

Our next goal is to extend Theorem~12.8 in the case when $g~:~E\lra Q$
is only {\it generically} surjective; this means that the analytic set
$$
Y=\{x\in X~;~g_x~:~E_x\lra Q_x~{\rm is~not~surjective~}\}
$$
defined by the equation $\Lambda^q g=0$ is nowhere dense in $X$.
Here $\Lambda^q g$ is a section of the bundle $\Hom(\Lambda^qE,\det Q)$.
The idea is to apply the above Theorem 12.8 to $X\ssm Y$. For this,
we have to know whether $X\ssm Y$ has a complete K\"ahler metric.

\claim{(12.9) Lemma} Let $(X,\omega)$ be a K\"ahler manifold, and
$Y=\sigma^{-1}(0)$ an analytic subset defined by a section of a
hermitian vector bundle $E\to X$.  If $X$ is weakly pseudoconvex and
exhausted by $X_c=\{x\in X~;~\psi(x)<c\}$, then $X_c\ssm Y$ has a
complete K\"ahler metric for all $c\in\bR$.  The same conclusion
holds for $X\ssm Y$ if $(X,\omega)$ is complete and if for some
constant $C\ge 0$ we have
$\Theta_E\le_\Grif C\,\omega\otimes\langle~,~\rangle_E$ on $X$.
\endclaim

\begproof{} Set $\tau=\log|\sigma|^2$. Then $d'\tau=\{D'\sigma,\sigma\}/
|\sigma|^2$ and $D''D'\sigma=D^2\sigma=\Theta(E)\sigma$, thus
$$\ii d'd''\tau=\ii{\{D'\sigma,D'\sigma\}\over|\sigma|^2}-
\ii{\{D'\sigma,\sigma\}\wedge\{\sigma,D'\sigma\}\over|\sigma|^4}-
{\{\ii\Theta(E)\sigma,\sigma\}\over|\sigma|^2}.$$
For every $\xi\in T_X$, we find therefore
$$\eqalign{
H\tau(\xi)&={|\sigma|^2\,|D'\sigma\cdot \xi|^2-|\langle D'\sigma\cdot \xi,
\sigma\rangle|^2\over|\sigma|^4}-{\wt\Theta(E)(\xi\otimes\sigma,\xi\otimes
\sigma)\over|\sigma|^2}\cr
&\ge -{\wt\Theta(E)(\xi\otimes \sigma,\xi\otimes\sigma)\over|\sigma|^2}\cr}$$
by the Cauchy-Schwarz inequality. If $C$ is a bound for the coefficients of
$\wt\Theta(E)$ on the compact
subset $\ol X_c$, we get $\ii d'd''\tau\ge -C\omega$ on
$X_c$. Let $\chi\in\ci(\bR,\bR)$ be a convex increasing function. We set
$$\wh\omega=\omega+\ii d'd''(\chi\circ\tau)=\omega+\ii\big(
\chi'\circ\tau~d'd''\tau+\chi''\circ\tau~d'\tau\wedge d''\tau\big).$$
We thus see that $\wh\omega$ is positive definite if
$\chi'\le 1/2C$, and by a computation similar to the one preceding
Theorem 6.2, we check that $\wh\omega$ is complete near $Y=\tau^{-1}(-\infty)$
as soon as
$$\int_{-\infty}^0\sqrt{\chi''(t)}\,dt=+\infty.$$
One can choose for example $\chi$ such that
$\chi(t)={1\over 5C}(t-\log|t|)$ for $t\le -1$. In order to obtain a
complete K\"ahler metric on $X_c\ssm Y$, we also need the metric 
to be complete near $\partial X_c$. If $\wh\omega$ is not, such a metric
can be defined by
$$\eqalign{
\wt\omega&=\wh\omega+\ii d'd''\log(c-\psi)^{-1}=\wh\omega+{\ii d'd''\psi\over
c-\psi}+{\ii d'\psi\wedge d''\psi\over(c-\psi)^2}\cr
&\ge \ii d'\log(c-\psi)^{-1}\wedge d''\log(c-\psi)^{-1}~;\cr}$$
$\wt\omega$ is complete on $X_c\ssm\Omega$ because $\log(c-\psi)^{-1}$
tends to $+\infty$ on $\partial X_c$.\qed
\endproof

We also need another elementary lemma dealing with the extension of
partial differential equalities across analytic sets.

\claim{(12.10) Lemma} Let $\Omega$ be an open subset of $\bC^n$ and
$Y$ an analytic subset of $\Omega$. Assume that $v$ is a $(p,q-1)$-form
with $L^2_\loc$ coefficients and $w$ a $(p,q)$-form with 
$L^1_\loc$ coefficients such that $d''v=w$ on $\Omega\ssm Y$
$($in the sense of distribution theory$)$. Then $d''v=w$ on $\Omega$.
\endclaim

\begproof{} An induction on the dimension of $Y$ shows that it is
sufficient to prove the result in a neighborhood of a regular point $a\in Y$.
By using a local analytic isomorphism, the proof is reduced to the case
where $Y$ is contained in the hyperplane $z_1=0$, with $a=0$. Let
$\lambda\in\ci(\bR,\bR)$ be a function such that $\lambda(t)=0$ for
$t\le{1\over 2}$ and $\lambda(t)=1$ for $t\ge 1$. We must show that
$$\int_\Omega w\wedge\alpha=(-1)^{p+q}\int_\Omega v\wedge d''\alpha
\leqno(12.11)$$
for all $\alpha\in\cD(\Omega,\Lambda^{n-p,n-q}T^\star_\Omega)$. 
Set $\lambda_\varepsilon(z)=\lambda(|z_1|/\varepsilon)$ and replace 
$\alpha$ in the integral by $\lambda_\varepsilon\alpha$. Then 
$\lambda_\varepsilon\alpha\in \cD(\Omega\ssm Y,\Lambda^{n-p,n-q}
T^\star_\Omega)$ and the hypotheses imply
$$
\int_\Omega w\wedge\lambda_\varepsilon\alpha=(-1)^{p+q}
\int_\Omega v\wedge d''(\lambda_\varepsilon\alpha)=(-1)^{p+q}
\int_\Omega v\wedge (d''\lambda_\varepsilon\wedge\alpha+\lambda_\varepsilon 
d''\alpha).
$$
As $w$ and $v$ have $L^1_\loc$ coefficients on $\Omega$, the integrals
of $w\wedge\lambda_\varepsilon\alpha$ and $v\wedge\lambda_\varepsilon 
d''\alpha$ converge respectively to the integrals of $w\wedge\alpha$
and $v\wedge d''\alpha$ as $\varepsilon$ tends to $0$. The remaining
term can be estimated by means of the Cauchy-Schwarz inequality:
$$\Big|\int_\Omega v\wedge d''\lambda_\varepsilon\wedge\alpha\Big|^2\le
\int_{|z_1|\le\varepsilon}|v\wedge\alpha|^2\,dV.~
\int_{\Supp\alpha}|d''\lambda_\varepsilon|^2\,dV\,;$$
as $v\in L^2_\loc(\Omega)$, the integral
$\int_{|z_1|\le\varepsilon}|v\wedge\alpha|^2\,dV$ converges to $0$ with 
$\varepsilon$, whereas
$$
\int_{\Supp\alpha}|d''\lambda_\varepsilon|^2\,dV\le
{C\over\varepsilon^2}\,{\rm Vol}\big(\Supp\,\alpha
\cap\{|z_1|\le\varepsilon\}\big)\le C'.
$$
Equality (12.11) follows when $\varepsilon$ tends to $0$.\qed
\endproof

\claim{(12.12) Theorem} The existence statement and the estimates of 
Theorem~$12.8$
remain true for a generically surjective morphism $g:E\to Q$,
provided that $X$ is weakly pseudoconvex.
\endclaim

\begproof{} Apply Theorem~12.8 to each relatively compact domain
$X_c\ssm Y$ (these domains are complete K\"ahler by Lemma~12.9).
From a sequence of solutions on $X_c\ssm Y$ we can extract a subsequence 
converging weakly on $X\ssm Y$ as c tends to $+\infty$. One gets a 
form $h$ satisfying the estimates, such that $D''h=0$ on $X\ssm Y$ and
$f=g\cdot h$. In~order to see that $D''h=0$ on $X$, it suffices 
to apply Lemma~12.10 and to observe that $h$ has $L^2_\loc$
coefficients on $X$ by our estimates.\qed
\endproof

A very special but interesting case is obtained for the trivial bundles
$E=\Omega\times\bC^r$, $Q=\Omega\times\bC$ over a pseudoconvex open set
\hbox{$\Omega\subset\bC^n$}. Then the morphism $g$ is given by a $r$-tuple
$(g_1\ld g_r)$ of holomorphic functions on $\Omega$. Let us take $k=0$
and $L=\Omega\times\bC$ with the metric given by a weight $e^{-\varphi}$.
If we observe that $\wt{gg^\star}=\Id$ when $\rk\,Q=1$, 
Theorem~12.8 applied on \hbox{$X=\Omega\ssm g^{-1}(0)$} and Lemmas~12.9,
12.10 give:

\claim{(12.13) Theorem {\rm(Skoda [Sko72b])}} Let $\Omega$ be a
complete K\"ahler open subset of $\bC^n$ and $\varphi$ a
plurisubharmonic function on $\Omega$.  Set $m=\min\{n,r-1\}$. Then
for every holomorphic function $f$ on $\Omega$ such that
$$
I=\int_{\Omega\ssm Z}|f|^2\,|g|^{-2(m+1+\varepsilon)}e^{-\varphi}\,dV
<+\infty,
$$
where $Z=g^{-1}(0)$, there exist holomorphic functions $(h_1\ld h_r)$
on $\Omega$ such that $f=\sum g_jh_j$ and
$$
\int_{\Omega\ssm Y}|h|^2\,|g|^{-2(m+\varepsilon)}e^{-\varphi}\,dV
\le(1+m/\varepsilon)I.\eqno{\square}
$$
\endclaim

We now show that Theorem~12.13 can be applied to get deep results
concerning ideals of the local ring $\cO_n=\bC\{z_1\ld z_n\}$ of
germs of holomorphic functions on $(\bC^n,0)$. Let $\cI=(g_1\ld
g_r)\ne(0)$ be an ideal of~$\cO_n$.

\claim{(12.14) Definition} Let $k\in\bR_+$.  We associate to $\cI$ the
following ideals:
\smallskip
\item{\rm a)} the ideal $\ol\cI^{(k)}$ of germs $u\in\cO_n$ such that 
$|u|\le C|g|^k$ for some constant $C\ge 0$, where 
$|g|^2=|g_1|^2+\cdots+|g_r|^2$.
\smallskip
\item{\rm b)} the ideal $\wh\cI^{(k)}$ of germs
$u\in\cO_n$ such that
$$
\int_\Omega|u|^2\,|g|^{-2(k+\varepsilon)}\,dV<+\infty
$$
on a small ball $\Omega$ centered at $0$, if $\varepsilon>0$ is small 
enough.\smallskip
\endclaim

\claim{(12.15) Proposition} For all $k,l\in\bR_+$ we have
\smallskip
\item{\rm a)} $\ol\cI^{(k)}\subset\wh\cI^{(k)}\,;$
\smallskip
\item{\rm b)} $\cI^k\subset\ol\cI^{(k)}$ if $k\in\bN\,;$
\smallskip
\item{\rm c)} $\ol\cI^{(k)}.\ol\cI^{(l)}\subset\ol\cI^{(k+l)}\,;$
\smallskip
\item{\rm d)} $\ol\cI^{(k)}.\wh\cI^{(l)}\subset\wh\cI^{(k+l)}.$
\endclaim

All properties are immediate from the definitions except a)
which is a consequence of the integrability of $|g|^{-\varepsilon}$
for $\varepsilon>0$ small (exercise to the reader!). Before stating 
the main result, we need a simple lemma.

\claim{(12.16) Lemma} If $\cI=(g_1\ld g_r)$ and $r>n$, we can find
elements $\wt g_1\ld\wt g_n\in\cI$ such that $C^{-1}|g|\le|\wt g|\le C|g|$
on a neighborhood of $0$. Each $\wt g_j$ can be taken to be a linear
combination
$$
\wt g_j=a_j.\,g=\sum_{1\le k\le r}a_{jk}g_k,~~~~a_j\in\bC^r\ssm\{0\}
$$
where the coefficients $([a_1]\ld[a_n])$ are chosen in the complement
of a proper ana\-lytic subset of $(\bP^{r-1})^n$.
\endclaim

It follows from the Lemma that the ideal $\cJ=(\wt g_1\ld\wt g_n)\subset\cI$
satisfies $\smash{\ol\cJ}^{(k)}=\smash{\ol\cI}^{(k)}$ and
$\smash{\wh\cJ}^{(k)}=\smash{\wh\cI}^{(k)}$ for all $k$.

\begproof{} Assume that $g\in\cO(\Omega)^r$. Consider the analytic subsets in
$\Omega\times(\bP^{r-1})^n$ defined by
$$
\eqalign{
A&=\big\{(z,[w_1]\ld[w_n])\,;\,w_j.\,g(z)=0\big\},\cr
A^\star&=\bigcup{\rm irreducible~components~of}~A~{\rm not~contained~in~}
g^{-1}(0)\times(\bP^{r-1})^n.\cr}
$$
For $z\notin g^{-1}(0)$ the fiber $A_z=\{([w_1]\ld[w_n])\,;\,w_j.\,g(z)=0\}
=A^\star_z$ is a product of $n$ hyperplanes in $\bP^{r-1}$, hence
$A\cap(\Omega\ssm g^{-1}(0))\times(\bP^{r-1})^n$ is a fiber bundle
with base $\Omega\ssm g^{-1}(0)$ and fiber $(\bP^{r-2})^n$. As
$A^\star$ is the closure of this set in $\Omega\times(\bP^{r-1})^n$,
we have
$$
\dim A^\star=n+n(r-2)=n(r-1)=\dim(\bP^{r-1})^n.
$$
It follows that the zero fiber
$$
A^\star_0=A^\star\cap\big(\{0\}\times(\bP^{r-1})^n\big)
$$
is a proper subset of $\{0\}\times(\bP^{r-1})^n$.
Choose $(a_1\ld a_n)\in(\bC^r\ssm\{0\})^n$ such that \hbox{
$(0,[a_1]\ld[a_n])$} is not in~$A^\star_0$. By an easy compactness argument
the set \hbox{$A^\star\cap\big(\ol
B(0,\varepsilon)\times(\bP^{r-1})^n\big)$} is disjoint from the
neighborhood \hbox{$B(0,\varepsilon)\times\prod[B(a_j,\varepsilon)]$} of
\hbox{$(0,[a_1]\ld[a_n])$} for $\varepsilon$ small enough. For $z\in
B(0,\varepsilon)$ we have \hbox{$|a_j.\,g(z)|\ge\varepsilon|g(z)|$} for
some~$j$, otherwise the inequa\-lity $|a_j.\,g(z)|<\varepsilon|g(z)|$ would
imply the existence of $h_j\in\bC^r$ with $|h_j|<\varepsilon$ and
$a_j.\,g(z)=h_j.\,g(z)$. Since $g(z)\ne 0$, we would have
$$
(z,[a_1-h_1]\ld[a_n-h_n])\in A^\star\cap\big(B(0,\varepsilon)\times
(\bP^{r-1})^n\big),
$$
a contradiction. We obtain therefore
$$
\varepsilon|g(z)|\le\max|a_j.\,g(z)|\le(\max|a_j|)\,|g(z)|~~~~{\rm on}~~
B(0,\varepsilon).\eqno{\square}
$$
\endproof

\claim{(12.17) Theorem {\rm(Brian\c{c}on-Skoda [BSk74])}}
Set $p=\min\{n-1,r-1\}$. Then
\smallskip
\item{\rm a)} $\wh\cI^{(k+1)}=\cI\,\wh\cI^{(k)}=\ol\cI\,\wh\cI^{(k)}$~~
for $k\ge p$.
\smallskip
\item{\rm b)} $\ol\cI^{(k+p)}\subset\wh\cI^{(k+p)}\subset\cI^k$~~
for all $k\in\bN$.
\endclaim

\begproof{} a) The inclusions $\cI\,\wh\cI^{(k)}\subset\ol\cI\,\wh\cI^{(k)}
\subset\wh\cI^{(k+1)}$ are obvious thanks to Proposition~12.15,
so we only have to prove that $\wh\cI^{(k+1)}\subset\cI\,\wh\cI^{(k)}$.
Assume first that $r\le n$. Let $f\in\wh\cI^{(k+1)}$ be such that
$$
\int_\Omega|f|^2\,|g|^{-2(k+1+\varepsilon)}\,dV<+\infty.
$$
For $k\ge p-1$, we can apply Theorem~12.13 with $m=r-1$ and
with the weight $\varphi=(k-m)\log|g|^2$. Hence $f$ can be written 
$f=\sum g_jh_j$ with
$$
\int_\Omega|h|^2\,|g|^{-2(k+\varepsilon)}\,dV<+\infty,
$$
thus $h_j\in\wh\cI^{(k)}$ and $f\in\cI\,\wh\cI^{(k)}$. When $r>n$,
Lemma~12.16 shows that there is an ideal $\cJ\subset\cI$ with $n$
generators such that $\smash{\wh\cJ}^{(k)}=\smash{\wh\cI}^{(k)}$. We find
$$
\wh\cI^{(k+1)}=\wh\cJ^{(k+1)}\subset\cJ\,\wh\cJ^{(k)}\subset
\cI\,\wh\cI^{(k)}~~~~{\rm for}~~k\ge n-1.
$$
\medskip
\noindent
b) Property a) implies inductively $\wh\cI^{(k+p)}=\cI^k\,\wh\cI^{(p)}$
for all $k\in\bN$. This gives in particular
$\wh\cI^{(k+p)}\subset\cI^k$.\qed
\endproof

\claim{(12.18) Corollary} \smallskip
\item{\rm a)} The ideal $\ol\cI$ is the integral
closure of $\cI$, i.e.\ by definition the set of germs $u\in\cO_n$ which
satisfy an equation
$$
u^d+a_1u^{d-1}+\cdots+a_d=0,~~~~a_s\in\cI^s,~~~~1\le s\le d.
$$
\smallskip
\item{\rm b)} Similarly, $\ol\cI^{(k)}$ is the set of germs $u\in\cO_n$ which
satisfy an equation
$$
u^d+a_1u^{d-1}+\cdots+a_d=0,~~~~a_s\in\cI^{\lceil ks\rceil},~~~~1\le s\le d,
$$
where $\lceil t\rceil$ denotes the smallest integer $\ge t$.\smallskip
\endclaim

As the ideal $\ol\cI^{(k)}$ is finitely generated, property b) shows
that there always exists a rational number $l\ge k$ such that 
$\smash{\ol\cI}^{(l)}=\smash{\ol\cI}^{(k)}$.

\begproof{} a) If $u\in\cO_n$ satisfies a polynomial equation with coefficients
$a_s\in\cI^s$, then clearly $|a_s|\le C_s\,|g|^s$
and the usual elementary bound 
$$
|\hbox{roots}|\le 2\max_{1\le s\le d}|a_s|^{1/s}
$$
for the roots of a monic polynomial implies $|u|\le C\,|g|$.

Conversely, assume that $u\in\ol\cI$. The ring $\cO_n$ is Noetherian, so
the ideal $\smash{\wh\cI}^{(p)}$ has a finite number of generators 
$v_1\ld v_N$. For every $j$ we have $uv_j\in\ol\cI\,\smash{\wh\cI}^{(p)}=
\cI\,\smash{\wh\cI}^{(p)}$, hence there exist
elements $b_{jk}\in\cI$ such that
$$
uv_j=\sum_{1\le k\le N}b_{jk}v_k.
$$
The matrix $(u\delta_{jk}-b_{jk})$ has the non zero vector $(v_j)$ in
its kernel, thus $u$ satisfies the equation $\det(u\delta_{jk}-b_{jk})=0$,
which is of the required type.
\medskip
\noindent b) Observe that $v_1\ld v_N$ satisfy simultaneously some
integrability condition $\int_\Omega|v_j|^{-2(p+\varepsilon)}<+\infty$,
thus $\smash{\wh\cI}^{(p)}=\smash{\wh\cI}^{(p+\eta)}$ for 
$\eta\in[0,\varepsilon[$. Let $u\in\smash{\ol\cI}^{(k)}$.
For every integer $m\in\bN$ we have
$$
u^mv_j\in\ol\cI^{(km)}\,\wh\cI^{(p+\eta)}\subset
\wh\cI^{(km+\eta+p)}.
$$
If $k\notin\bQ$, we can find $m$ such that $d(km+\varepsilon/2,\bZ)
<\varepsilon/2$,
thus $km+\eta\in\bN$ for some $\eta\in{}]0,\varepsilon[$. If
$k\in\bQ$, we take $m$ such that $km\in\bN$ and $\eta=0$. Then
$$
u^mv_j\in\wh\cI^{(N+p)}=\cI^N\,\wh\cI^{(p)}~~~~{\rm with}~~
N=km+\eta\in\bN,
$$
and the reasoning made in a) gives
$\det(u^m\delta_{jk}-b_{jk})=0$ for some \hbox{$b_{jk}\in\cI^N$}. This is an
equation of the type described in b), where the coefficients
$a_s$ vanish when $s$ is not a multiple of $m$ and
$a_{ms}\in\cI^{Ns}\subset\cI^{\lceil kms\rceil}$.\qed
\endproof

Let us mention that Brian\c{c}on and Skoda's result 12.17~b) is optimal for
$k=1$. Take for example $\cI=(g_1\ld g_r)$ with $g_j(z)=z_j^r$,
$1\le j\le r$, and $f(z)=z_1\ldots z_r$. Then $|f|\le C|g|$ and 12.17~b)
yields $f^r\in\cI\,;$ however, it is easy to verify that $f^{r-1}\notin\cI$.
The theorem also gives an answer to the following conjecture made 
by J.~Mather.

\claim{(12.19) Corollary} Let $f\in\cO_n$ and $\cI_f=(z_1\partial f/
\partial z_1\ld z_n\partial f/\partial z_n)$. Then
$f\in\ol\cI_f$, and for every integer $k\ge 0$, $f^{k+n-1}\in\cI^k_f$.
\endclaim

The Corollary is also optimal for $k=1\,:$ for example, one can verify that 
the function $f(z)=(z_1\ldots z_n)^3+z_1^{3n-1}+\cdots+z_n^{3n-1}$ is such 
that $f^{n-1}\notin\cI_f$.

\begproof{} Set $g_j(z)=z_j\,\partial f/\partial z_j$, $1\le j\le n$.
By 12.17~b), it suffices to show that $|f|\le C|g|$.
For every germ of analytic curve $\bC\ni t\longmapsto\gamma(t)$,
$\gamma\not\equiv 0$, the vanishing order of $f\circ\gamma(t)$ at $t=0$
is the same as that of
$$
t\,{d(f\circ\gamma)\over dt}=\sum_{1\le j\le n}t\,\gamma_j'(t)\,
{\partial f\over\partial z_j}\big(\gamma(t)\big).
$$
We thus obtain 
$$
|f\circ\gamma(t)|\le C_1\,|t|~\Big|{d(f\circ\gamma)\over dt}\Big|
\le C_2\,\sum_{1\le j\le n}|t\,\gamma_j'(t)|~\Big|
{\partial f\over\partial z_j}\big(\gamma(t)\big)\Big|
\le C_3\,|g\circ\gamma(t)|
$$
and conclude by the following elementary lemma.\qed
\endproof

\claim{(12.20) Curve selection lemma} Let $f,g_1\ld g_r\in\cO_n$ be
germs of holomorphic functions vanishing at $0$. Then we have $|f|\le
C|g|$ for some constant $C$ if and only if for every germ of analytic
curve $\gamma$ through $0$ there exists a constant $C_\gamma$ such
that \hbox{$|f\circ\gamma|\le C_\gamma|g\circ\gamma|$}.
\endclaim

\begproof{} If the inequality $|f|\le C|g|$ does not hold on any
neighborhood of $0$, the germ of analytic set $(A,0)\subset(\bC^{n+r},0)$ 
defined by
$$
g_j(z)=f(z)z_{n+j},~~~~1\le j\le r,
$$
contains a sequence of points $\big(z_\nu,g_j(z_\nu)/f(z_\nu)\big)$
converging to $0$ as $\nu$ tends to $+\infty$, with $f(z_\nu)\ne 0$.
Hence $(A,0)$ contains an irreducible component on which $f\not\equiv 0$
and there is a germ of curve \hbox{$\wt\gamma=(\gamma,\gamma_{n+j}):
(\bC,0)\to(\bC^{n+r},0)$} contained in $(A,0)$ such that $f\circ\gamma
\not\equiv 0$. We get \hbox{$g_j\circ\gamma=(f\circ\gamma)\gamma_{n+j}$},
hence \hbox{$|g\circ\gamma(t)|\le C|t|\,|f\circ\gamma(t)|$} and the
inequality \hbox{$|f\circ\gamma|\le C_\gamma|g\circ\gamma|$} does not
hold.\qed
\endproof

\titleb{13.}{The Ohsawa-Takegoshi $L^2$ Extension Theorem}
We address here the following extension problem: let $Y$ be a complex
analytic submanifold of a complex manifold~$X\,$; given a holomorphic 
function $f$ on $Y$ satisfying suitable $L^2$ conditions on $Y$, find
a holomorphic extension $F$ of $f$ to $X$, together with a good
$L^2$ estimate for $F$ on~$X$. The first satisfactory solution
has been obtained only rather recently by Ohsawa-Takegoshi [OT87, Ohs88]. 
We follow here a more geometric approach due to Manivel [Man93], which 
provides a generalized extension theorem in the general framework
of vector bundles. As in Ohsawa-Takegoshi's fundamental paper, the 
main idea is to use a modified Bochner-Kodaira-Nakano inequality. Such
inequalities were originally introduced in the work of 
Donnelly-Fefferman [DF83] and Donnelly-Xavier [DX84]. The main a priori 
inequality we are going to use is a simplified (and slightly extended)
version of the original Ohsawa-Takegoshi a priori inequality, as
proposed recently by Ohsawa [Ohs95].

\claim{(13.1) Lemma {\rm (Ohsawa [Ohs95])}} Let $E$ be a hermitian
vector bundle on a complex manifold $X$ equipped with a K\"ahler
metric~$\omega$. Let $\eta,\,\lambda>0$ be smooth functions
on~$X$. Then for every form $u\in\cD(X,\Lambda^{p,q} T^\star_X\otimes E)$
with compact support we have
$$ 
\eqalign{
\|(\eta^{{1\over2}}+\lambda^{{1\over2}})D^{\prime\prime\star}u\|^2&{}
+\|\eta^{{1\over2}}D''u\|^2+\|\lambda^{{1\over2}}D'u\|^2+
2\|\lambda^{-{1\over2}}d'\eta\wedge u\|^2\cr
 &\geq \Ll[\eta\,\ii\Theta(E)-\ii\,d'd''\eta-\ii\lambda^{-1}d'\eta\wedge 
d''\eta,\Lambda]u,u\Rr.\cr}
$$
\endclaim

\begproof{} Let us consider the ``twisted'' Laplace-Beltrami operators
$$
\eqalign{
D'\eta D^{\prime\star} +D^{\prime\star}\eta  D' 
& =  \eta [D',D^{\prime\star}]+[D',\eta]D^{\prime\star}+
[D^{\prime\star},\eta]D' \cr
& = \eta\Delta'+(d'\eta)D^{\prime\star}-(d'\eta)^*D',\cr
D''\eta D^{\prime\prime\star} +D^{\prime\prime\star}\eta  D'' 
& = \eta [D'',D^{\prime\prime\star}]+[D'',\eta]D^{\prime\prime\star}+
[D^{\prime\prime\star},\eta]D'' \cr
& = \eta\Delta''+(d''\eta)D^{\prime\prime\star}-(d''\eta)^*D'',\cr}
$$
where $\eta$, $(d'\eta)$, $(d''\eta)$ are abbreviated notations for the 
multiplication operators $\eta\bu$, $(d'\eta)\wedge\bu$, $(d''\eta)\wedge\bu$.
By subtracting the above equalities and taking into account the 
Bochner-Kodaira-Nakano identity $\Delta''-\Delta'=[\ii\Theta(E),\Lambda]$,
we get
$$
\leqalignno{
D''\eta D^{\prime\prime\star}&{}+D^{\prime\prime\star}\eta  D'' -
D'\eta D^{\prime\star}  -D^{\prime\star}\eta  D' \cr
&{}=\eta [\ii\Theta(E),\Lambda]+(d''\eta)D^{\prime\prime\star}
-(d''\eta)^\star D''+(d'\eta)^\star D'-(d'\eta)D^{\prime\star}.
&(13.2)\cr}
$$
Moreover, the Jacobi identity yields
$$
[D'',[d'\eta,\Lambda ]]-[d'\eta,[\Lambda,D'']]+[\Lambda,[D'',d'\eta]]=0,
$$
whilst $[\Lambda,D'']=-\ii D^{\prime\star}$ by the basic commutation 
relations~4.22. A straightforward computation shows that 
$[D'',d'\eta]=-(d'd''\eta)$ and $[d'\eta,\Lambda ]=\ii(d''\eta)^\star$.
Therefore we get
$$
\ii[D'',(d''\eta)^\star]+\ii[d'\eta,D^{\prime\star}]-[\Lambda,(d'd''\eta)]=0,
$$
that is,
$$
[\ii\,d'd''\eta,\Lambda]=
[D'',(d''\eta)^\star]+[D^{\prime\star},d'\eta]=
D''(d''\eta)^\star+(d''\eta)^\star D''+
D^{\prime\star}(d'\eta)+(d'\eta)D^{\prime\star}.
$$
After adding this to (13.2), we find
$$
\eqalign{
D''\eta D^{\prime\prime\star}&{}+D^{\prime\prime\star}\eta  D'' -
D'\eta D^{\prime\star}  -D^{\prime\star}\eta  D' 
+[\ii\,d'd''\eta,\Lambda]
\cr
&{}=\eta [\ii\Theta(E),\Lambda]+(d''\eta)D^{\prime\prime\star}+
D''(d''\eta)^\star+(d'\eta)^\star D'+D^{\prime\star}(d'\eta).
\cr}
$$
We apply this identity to a form $u\in\cD(X,\Lambda^{p,q}T^\star_X\otimes E)$
and take the inner bracket with~$u$. Then
$$
\Ll (D''\eta D^{\prime\prime\star})u,u\Rr=\Ll\eta D^{\prime\prime\star}u,
D^{\prime\prime\star}u\Rr=\|\eta^{{1\over2}}D^{\prime\prime\star}u\|^2,
$$
and likewise for the other similar terms. The above equalities
imply
$$
\eqalign{
&\|\eta^{{1\over 2}}D^{\prime\prime\star}u\|^2+\|\eta^{{1\over 2}}D''u\|^2
-\|\eta^{{1\over2}}D'u\|^2-\|\eta^{{1\over2}}D^{\prime\star}u\|^2={}\cr
&\qquad\Ll[\eta\,\ii\Theta(E)-\ii\,d'd''\eta,\Lambda]u,u\Rr
+2\Re\,\Ll D^{\prime\prime\star}u,(d''\eta)^\star u\Rr
+2\Re\,\Ll D'u,d'\eta\wedge u\Rr.\cr}
$$
By neglecting the negative terms $-\|\eta^{{1\over2}}D'u\|^2
-\|\eta^{{1\over2}}D^{\prime\star}u\|^2$ and adding the squares
$$
\eqalign{
\|\lambda^{{1\over 2}}D^{\prime\prime\star}u\|^2+
2\Re\,\Ll D^{\prime\prime\star}u,(d''\eta)^\star u\Rr+
\|\lambda^{-{1\over 2}}(d''\eta)^\star u\|^2
&\ge 0,\cr
\|\lambda^{{1\over 2}}D'u\|^2+
2\Re\,\Ll D'u,d'\eta\wedge u\Rr+
\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2
&\ge0\cr}
$$
we get
$$
\eqalign{
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})
D^{\prime\prime\star}u\|^2&{}+\|\eta^{{1\over 2}}D''u\|^2
+\|\lambda^{{1\over 2}}D'u\|^2+\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2
+\|\lambda^{-{1\over 2}}(d''\eta)^\star u\|^2\cr
&\ge\Ll[\eta\,\ii\Theta(E)-\ii\,d'd''\eta,\Lambda]u,u\Rr.\cr}
$$
Finally, we use the identities
$$
\eqalign{
(d'\eta)^\star(d'\eta)&{}-(d''\eta)(d''\eta)^\star
=\ii[d''\eta,\Lambda](d'\eta)+\ii(d''\eta)[d'\eta,\Lambda]
=[\ii d''\eta\wedge d'\eta,\Lambda],\cr
\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2&{}-
\|\lambda^{-{1\over2}}(d''\eta)^\star u\|^2
=-\Ll[\ii\lambda^{-1}d'\eta\wedge d''\eta,\Lambda]u,u\Rr,\cr}
$$
The inequality asserted in Lemma~13.1 follows by adding the second identity
to our last inequality.\qed
\endproof

In the special case of $(n,q)$-forms, the forms $D'u$ and $d'\eta\wedge u$
are of bidegree $(n+1,q)$, hence the estimate takes the simpler form
$$
\|(\eta^{{1\over2}}+\lambda^{{1\over2}})D^{\prime\prime\star}u\|^2{}
+\|\eta^{{1\over2}}D''u\|^2\ge
\Ll[\eta\,\ii\Theta(E)-\ii\,d'd''\eta
-\ii\lambda^{-1}\,d'\eta\wedge d''\eta,\Lambda]u,u\Rr.\leqno(13.3)
$$

\claim{(13.4) Proposition} Let $X$ be a complete K\"ahler manifold
equipped with a $($non ne\-ces\-sarily complete$)$ K\"ahler metric
$\omega$, and let $E$ be a hermitian vector bundle over~$X$.
Assume that there are smooth and bounded functions $\eta,\,\lambda>0$
on $X$ such that the $($hermitian$)$ curvature operator
$B=B^{n,q}_{E,\omega,\eta}=
[\eta\,\ii\Theta(E)-\ii\,d'd''\eta-\ii\lambda^{-1}d'\eta\wedge d''\eta,
\Lambda_\omega]$ is positive definite everywhere on 
$\Lambda^{n,q}T^\star_X\otimes E$, for some $q\ge 1$.
Then for every form $g\in L^2(X,\Lambda^{n,q}T^\star_X\otimes E)$
such that $D''g=0$ and $\int_X\langle B^{-1}g,g\rangle
\,dV_\omega<+\infty$, there exists $f\in
L^2(X,\Lambda^{n,q-1}T^\star_X\otimes E)$ such that  $D''f=g$ and
$$
\int_X(\eta+\lambda)^{-1}
|f|^2\,dV_\omega\le 2\int_X\langle B^{-1}g,g\rangle\,dV_\omega.
$$
\endclaim

\begproof{} The proof is almost identical to the proof of Theorem~6.1,
except that we use (13.3) instead of (5.1). Assume first that $\omega$ is
complete. With the same notation as in 8.4, we get for every 
$v=v_1+v_2\in(\Ker D'')\oplus(\Ker D'')^\perp$ the inequalities
$$
|\langle g,v\rangle|^2=|\langle g,v_1\rangle|^2\le
\int_X\langle B^{-1}g,g\rangle \,dV_\omega
\int_X\langle Bv_1,v_1\rangle \,dV_\omega,
$$
and
$$
\int_X\langle Bv_1,v_1\rangle \,dV_\omega\le
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v_1\|^2+
\|\eta^{{1\over 2}}D''v_1\|^2
=\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v\|^2
$$
provided that $v\in\Dom D^{\prime\prime\star}$. Combining both, we find
$$
|\langle g,v\rangle|^2
\le \Big(\int_X\langle B^{-1}g,g\rangle \,dV_\omega\Big)
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v\|^2.
$$
This shows the existence of an element $w\in L^2(X,\Lambda^{n,q}T^\star_X
\otimes E)$ such that 
$$
\eqalign{
\|w\|^2&\le\int_X\langle B^{-1}g,g\rangle \,dV_\omega\qquad
\hbox{and}\cr
\Ll v,g\Rr&=\Ll(\eta^{{1\over 2}}+\lambda^{{1\over2}})
D^{\prime\prime\star}v,w\Rr\qquad
\forall g\in\Dom D''\cap\Dom D^{\prime\prime\star}.\cr}
$$
As $(\eta^{1/2}+\lambda^{{1\over2}})^2\le 2(\eta+\lambda)$, it follows 
that $f=(\eta^{1/2}+\lambda^{{1\over2}})w$ satisfies
$D''f=g$ as well as the desired $L^2$ estimate. If $\omega$ is not
complete, we set $\omega_\varepsilon=\omega+\varepsilon\wh\omega$ with
some complete K\"ahler metric~$\wh\omega$. The final conclusion is
then obtained by passing to the limit and using a monotonicity argument
(the integrals are monotonic with respect to~$\varepsilon$).\qed
\endproof

\claim{(13.5) Remark} \rm We will also need a variant of the
$L^2$-estimate, so as to obtain approximate solutions with weaker
requirements on the data$\,$: given $\delta>0$ and $g\in
L^2(X,\Lambda^{n,q}T^\star_X\otimes E)$ such that $D''g=0$ and
$\int_X\langle (B+\delta I)^{-1}g,g\rangle\,dV_\omega<+\infty$, there
exists an approximate solution 
\hbox{$f\in L^2(X,\Lambda^{n,q-1}T^\star_X \otimes E)$} and a correcting
term $h\in L^2(X,\Lambda^{n,q}T^\star_X \otimes E)$ such that
$D''f+\delta^{1/2}h=g$ and
$$
\int_X(\eta+\lambda)^{-1}|f|^2\,dV_\omega+
\int_X|h|^2\,dV_\omega
\le 2\int_X\langle (B+\delta I)^{-1}g,g\rangle\,dV_\omega.
$$
The proof is almost unchanged, we rely instead on the estimates
$$
|\langle g,v_1\rangle|^2\le
\int_X\langle(B+\delta I)^{-1}g,g\rangle \,dV_\omega
\int_X\langle(B+\delta I)v_1,v_1\rangle \,dV_\omega,
$$
and
$$
\int_X\langle(B+\delta I)v_1,v_1\rangle \,dV_\omega\le
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v\|^2+
\delta\|v\|^2.
\eqno\qed
$$
\endclaim

\claim{(13.6) Theorem}
Let $X$ be a weakly pseudoconvex $n$-dimensional complex manifold
equipped with a K\"ahler metric $\omega$, let $L$ $($resp.\ $E)$ be
a hermitian holomorphic line bundle $($resp.\ a hermitian holomorphic 
vector bundle of rank $r$ over~$X)$, and $s$ a global holomorphic 
section of~$E$. Assume that $s$ is generically transverse to the zero
section, and let
$$
Y=\big\{x\in X\,;\;s(x)=0, \Lambda^r ds(x)\not= 0\big\},\qquad
p=\dim Y=n-r.
$$
Moreover, assume that the $(1,1)$-form 
$\ii\Theta(L)+r\,\ii\,d'd''\log|s|^2$ is semipositive and that there is
a continuous function $\alpha\ge 1$ such that the following two inequalities
hold everywhere on $X:$
\smallskip
\item{\rm a)}
$\displaystyle\ii\Theta(L)+r\,\ii\,d'd''\log|s|^2\ge\alpha^{-1}
{\{\ii\Theta(E)s,s\}\over|s|^2}\,,$
\smallskip
\item{\rm b)} $|s|\le e^{-\alpha}$.
\smallskip\noindent
Then for every smooth $D''$-closed \hbox{$(0,q)$-form $f$} over $Y$ with 
values in the line bundle $\Lambda^nT_X^\star\otimes L$ 
$($restricted to~$Y)$, such that
$\int_Y|f|^2|\Lambda^r(ds)|^{-2}dV_\omega<+\infty$, 
there  exists a $D''$-closed $(0,q)$-form $F$ over $X$ with values 
in $\Lambda^nT^\star_X\otimes L$, such that $F$ is smooth over 
$X\ssm\{s=\Lambda^r(ds)=0\}$, satisfies $F_{\restriction Y}=f$ and
$$
\int_X{|F|^2\over|s|^{2r}(-\log|s|)^2}\,dV_{X,\omega}
\le C_r\int_Y{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}\,,
$$
where $C_r$ is a numerical constant depending only on~$r$.
\endclaim

Observe that the differential $ds$ (which is intrinsically defined only at 
points where $s$ vanishes) induces a vector bundle isomorphism
$ds:T_X/T_Y\to E$ along $Y$, hence a non vanishing section 
$\Lambda^r(ds)$, taking values in 
$$
\Lambda^r(T_X/T_Y)^\star\otimes \det E\subset
\Lambda^rT_X^\star\otimes \det E.
$$
The norm $|\Lambda^r(ds)|$ is computed here with respect to the metrics
on $\Lambda^rT_X^\star$ and $\det E$ induced by the K\"ahler metric
$\omega$ and by the given metric on~$E$. Also notice that if
hypothesis a) is satisfied for some $\alpha$, one can always achieve
b) by multiplying the metric of $E$ with a sufficiently small weight
$e^{-\chi\circ\psi}$ (with $\psi$ a psh exhaustion on~$X$ and $\chi$ a
convex increasing function; property a) remains valid after we multiply
the metric of $L$ by $e^{-(r+\smash{\alpha_0^{-1}})\chi\circ\psi}$,
where $\alpha_0=\inf_{x\in X}\alpha(x)$.

\begproof{} Let us first assume that the singularity set $\Sigma=
\{s=0\}\cap\{\Lambda^r(ds)=0\}$ is empty, so that $Y$ is closed and
nonsingular. We claim that there exists a smooth section
$$
F_\infty\in C^\infty(X,\Lambda^{n,q}T^\star_X\otimes L)
=C^\infty(X,\Lambda^{0,q}T^\star_X\otimes\Lambda^nT^\star_X\otimes L)
$$ 
such that
{\parindent=0cm 
\smallskip
(a) $F_\infty$ coincides with $f$ in restriction to $Y$,
\smallskip
(b) $|F_\infty|=|f|$ at every point of~$Y$,
\smallskip
(c) $D''F_\infty=0$ at every point of~$Y$.
\smallskip}

\noindent
For this, consider coordinates patches $U_j\subset X$ biholomorphic 
to polydiscs such that 
$$
U_j\cap Y=\{z\in U_j\,;\;z_1=\ldots=z_r=0\}
$$
in the corresponding
coordinates. We can find a section $\smash{\wt f}$ in
$C^\infty(X,\Lambda^{n,q}T^\star_X\otimes L)$ which achieves a) and b),
since the restriction map $(\Lambda^{0,q}T_X^*)_{\restriction Y}\to
\Lambda^{0,q}T_Y^*$ can be viewed as an orthogonal projection onto a 
$C^\infty$-subbundle of~$(\Lambda^{0,q}T_X^*)_{\restriction Y}$. 
It is enough to extend this subbundle from \hbox{$U_j\cap Y$} to $U_j$ 
(e.g.\ by extending each component of a frame), and then to extend 
$f$ globally via local smooth extensions and a partition of unity.
For any such extension~$\smash{\wt f}$ we have
$$
(D''\wt f)_{\restriction Y}=(D''\wt f_{\restriction Y})=D''f=0.
$$
It follows that we can divide $D''\wt f=\sum_{1\le \lambda\le r}
g_{j,\lambda}(z)\wedge d\ol z_\lambda$ on $U_j\cap Y$, with suitable
smooth $(0,q)$-forms $g_{j,\lambda}$ which we also extend arbitrarily 
from $U_j\cap Y$ to~$U_j$. Then
$$
F_\infty:=\wt f - \sum_j \theta_j(z)\sum_{1\le \lambda \le r}
\ol z_\lambda g_{j,\lambda}(z)
$$
coincides with $\wt f$ on $Y$ and satisfies (c).
Since we do not know about $F_\infty$ except in an infinitesimal
neighborhood of~$Y$, we will consider a
truncation $F_\varepsilon$ of $F_\infty$ with support in a small
tubular neighborhood $|s|<\varepsilon$ of~$Y$, and solve the equation
$D''u_\varepsilon=D''F_\varepsilon$ with the constraint that
$u_\varepsilon$ should be $0$ on~$Y$. As $\codim Y=r$, this will be
the case if we can guarantee that $|u_\varepsilon|^2|s|^{-2r}$ is
locally integrable near~$Y$. For this, we will apply Proposition 13.4
with a suitable choice of the functions $\eta$ and $\lambda$, and an
additional weight $|s|^{-2r}$ in the metric of $L$.

Let us consider the smooth strictly convex function $\chi_0:{}]-\infty,0]
\to{}]-\infty,0]$ defined by $\chi_0(t)=t-\log(1-t)$ for $t\le 0$, which
is such that \hbox{$\chi_0(t)\le t$}, \hbox{$1\le\chi_0'\le2$} and 
\hbox{$\chi_0''(t)=1/(1-t)^2$}. 
We set
$$
\sigma_\varepsilon=\log(|s|^2+\varepsilon^2),\qquad
\eta_\varepsilon=\varepsilon-\chi_0(\sigma_\varepsilon).
$$
As $|s|\le e^{-\alpha}\le e^{-1}$, we have $\sigma_\varepsilon\le 0$
for $\varepsilon$ small, and
$$
\eta_\varepsilon\ge\varepsilon-\sigma_\varepsilon
\ge\varepsilon-\log(e^{-2\alpha}+\varepsilon^2).
$$
Given a relatively compact subset $X_c=\{\psi<c\}\compact X$, we thus
have $\eta_\varepsilon\ge 2\alpha$ for $\varepsilon<\varepsilon(c)$ small
enough. Simple calculations yield
$$
\eqalign{
\ii\,d'\sigma_\varepsilon
&={\ii\{D's,s\}\over|s|^2+\varepsilon^2},\cr
\ii\,d'd''\sigma_\varepsilon
&={\ii\{D's,D's\}\over|s|^2+\varepsilon^2}-
{\ii\{D's,s\}\wedge\{s,D's\}\over(|s|^2+\varepsilon^2)^2}-
{\{\ii\Theta(E)s,s\}\over|s|^2+\varepsilon^2}\cr
&\ge{\varepsilon^2\over|s|^2}
{\ii\{D's,s\}\wedge\{s,D's\}\over(|s|^2+\varepsilon^2)^2}
-{\{\ii\Theta(E)s,s\}\over|s|^2+\varepsilon^2}\cr
&\ge{\varepsilon^2\over|s|^2}\ii d'\sigma_\varepsilon\wedge 
d''\sigma_\varepsilon-{\{\ii\Theta(E)s,s\}\over|s|^2+\varepsilon^2},\cr}
$$
thanks to Lagrange's inequality $\ii\{D's,s\}\wedge\{s,D's\}\le|s|^2
\ii\{D's,D's\}$. On the other hand, we have $d'\eta_\varepsilon=
-\chi_0'(\sigma_\varepsilon)d\sigma_\varepsilon$ with 
$1\le\chi_0'(\sigma_\varepsilon)\le 2$, hence
$$
\eqalign{
-\ii d'd''\eta_\varepsilon&=\chi_0'(\sigma_\varepsilon)\ii 
d'd''\sigma_\varepsilon+\chi_0''(\sigma_\varepsilon)\ii d'\sigma_\varepsilon
\wedge d''\sigma_\varepsilon\cr
&\ge \Big({1\over \chi_0'(\sigma_\varepsilon)}{\varepsilon^2\over |s|^2}+
{\chi_0''(\sigma_\varepsilon)\over\chi_0'(\sigma_\varepsilon)^2}\Big)
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon
-\chi_0'(\sigma_\varepsilon){\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2}.\cr}
$$
We consider the original metric of $L$ multiplied by the weight $|s|^{-2r}$. 
In this way, we get a curvature form
$$
\ii\Theta_L+r\,\ii d'd''\log|s|^2
\ge{1\over 2}\chi_0'(\sigma_\varepsilon)
\alpha^{-1}{\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2}
$$
by hypothesis a), thanks to the semipositivity of the
left hand side and the fact that ${1\over 2}\chi_0'(\sigma_\varepsilon){1\over
|s|^2+\varepsilon^2}\le {1\over |s|^2}$. As $\eta_\varepsilon\ge 2\alpha$ 
on $X_c$ for $\varepsilon$ small, we infer
$$
\eta_\varepsilon(\ii\Theta_L+\ii d'd''\log|s|^2)
-\ii d'd''\eta_\varepsilon-
{\chi_0''(\sigma_\varepsilon)\over\chi_0'(\sigma_\varepsilon)^2}
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon\ge
{\varepsilon^2\over \chi_0'(\sigma_\varepsilon)|s|^2}
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon
$$
on~$X_c$. Hence, if $\lambda_\varepsilon=
\chi_0'(\sigma_\varepsilon)^2/\chi_0''(\sigma_\varepsilon)$,
we obtain
$$
\eqalign{
B_\varepsilon&:=\big[\eta_\varepsilon(\ii\Theta_L+\ii d'd''\log|s|^2)
-\ii d'd''\eta_\varepsilon-\lambda_\varepsilon^{-1}
\ii d'\eta_\varepsilon\wedge d''\eta_\varepsilon\;,\;\Lambda\big]\cr
&\phantom{:}\ge\Big[{\varepsilon^2\over \chi_0'(\sigma_\varepsilon)|s|^2}\ii 
d'\eta_\varepsilon\wedge d''\eta_\varepsilon\,,\,\Lambda\Big]
={\varepsilon^2\over \chi_0'(\sigma_\varepsilon)|s|^2}
(d''\eta_\varepsilon)(d''\eta_\varepsilon)^\star\cr}
$$
as an operator on $(n,q)$-forms (see the proof of Lemma~13.1).

Let $\theta:\bR\to[0,1]$ be a smooth cut-off function such 
that $\theta(t)=1$ on $]-\infty,1/2]$, $\Supp\theta\subset{}]-\infty,1[$
and $|\theta'|\le 3$. For $\varepsilon>0$ small, we consider the 
$(n,q)$-form $F_\varepsilon=\theta(\varepsilon^{-2}|s|^2)F_\infty$ and
its $D''$-derivative
$$
g_\varepsilon=D''F_\varepsilon
=(1+\varepsilon^{-2}|s|^2)\theta'(\varepsilon^{-2}|s|^2)
d''\sigma_\varepsilon\wedge F_\infty+
\theta(\varepsilon^{-2}|s|^2)D''F_\infty
$$
[as is easily seen from the equality $1+\varepsilon^{-2}|s|^2=
\varepsilon^{-2}e^{\sigma_\varepsilon}\,$].
We observe that $g_\varepsilon$ has its support contained in the tubular
neighborhood $|s|<\varepsilon\,$; moreover, as $\varepsilon\to 0$, the 
second term in the right hand side converges uniformly to $0$ on every 
compact set; it will therefore produce no contribution in the
limit. On the other hand, the first term has the same order of magnitude
as $d''\sigma_\varepsilon$ and $d''\eta_\varepsilon$, and can be controlled
in terms of~$B_\varepsilon$. In fact, for any $(n,q)$-form $u$ and any
$(n,q+1)$-form $v$ we have
$$
\eqalign{
|\langle d''\eta_\varepsilon\wedge u,v\rangle|^2
&=|\langle u,(d''\eta_\varepsilon)^\star v\rangle|^2
\le|u|^2|(d''\eta_\varepsilon)^\star v|^2
=|u|^2\langle (d''\eta_\varepsilon)(d''\eta_\varepsilon)^\star v,v
\rangle\cr
&\le {\chi_0'(\sigma_\varepsilon)|s|^2\over\varepsilon^2}
|u|^2\langle B_\varepsilon v,v\rangle.\cr}
$$
This implies
$$
\langle B_\varepsilon^{-1}(d''\eta_\varepsilon\wedge u),
(d''\eta_\varepsilon\wedge u)\rangle\le{\chi_0'(\sigma_\varepsilon)
|s|^2\over\varepsilon^2}|u|^2.
$$
The main term in $g_\varepsilon$ can be written
$$
g_\varepsilon^{(1)}:=
(1+\varepsilon^{-2}|s|^2)\theta'(\varepsilon^{-2}|s|^2)
\chi_0'(\sigma_\varepsilon)^{-1}d''\eta_\varepsilon\wedge F_\infty.
$$
On $\Supp g_\varepsilon^{(1)}\subset\{|s|<\varepsilon\}$, since
$\chi_0'(\sigma_\varepsilon)\ge 1$, we thus find
$$
\langle B_\varepsilon^{-1}g_\varepsilon^{(1)},
g_\varepsilon^{(1)}\rangle\le (1+\varepsilon^{-2}|s|^2)^2
\,\theta'(\varepsilon^{-2}|s|^2)^2|F_\infty|^2.
$$
Instead of working on $X$ itself, we will work rather on the relatively
compact subset $X_c\ssm Y_c$, where $Y_c=Y\cap X_c=Y\cap\{\psi<c\}$. We know
that $X_c\ssm Y_c$ is again complete K\"ahler by Lemma 12.9.
In this way, we avoid the singularity of the weight $|s|^{-2r}$ along~$Y$.
We find
$$
\int_{X_c\ssm Y_c}\langle B_\varepsilon^{-1}g_\varepsilon^{(1)},
g_\varepsilon^{(1)}\rangle\, |s|^{-2r}dV_\omega\le
\int_{X_c\ssm Y_c}|F_\infty|^2
(1+\varepsilon^{-2}|s|^2)^2\theta'(\varepsilon^{-2}|s|^2)^2
|s|^{-2r}dV_\omega.
$$
Now, we let $\varepsilon\to 0$ and view $s$ as ``transverse local 
coordinates'' around $Y$. As $F_\infty$ coincides with $f$ on~$Y$, 
it is not hard to see that the 
right hand side converges to $c_r\int_{Y_c}|f|^2|\Lambda^r(ds)|^{-2}
dV_{Y,\omega}$ where $c_r$ is the ``universal'' constant
$$
c_r=\int_{z\in\bC^r,\,|z|\le 1}
(1+|z|^2)^2\theta'(|z|^2)^2{\ii^{r^2}\Lambda^r(dz)
\wedge\Lambda^r(d\ol z)\over|z|^{2r}}<+\infty
$$
depending only on~$r$. The second term
$$
g_\varepsilon^{(2)}=\theta(\varepsilon^{-2}|s|^2)d''F_\infty
$$
in $g_\varepsilon$ satisfies $\Supp(g_\varepsilon^{(2)})\subset\{|s|
<\varepsilon\}$ and $|g_\varepsilon^{(2)}|=O(|s|)$ (just look at the 
Taylor expansion of $d''F_\infty$ near $Y$). From this we easily conclude that
$$
\int_{X_c\ssm Y_c}\langle B_\varepsilon^{-1}g_\varepsilon^{(2)},
g_\varepsilon^{(2)}\rangle\, |s|^{-2r}dV_{X,\omega}=O(\varepsilon^2),
$$
provided that $B_\varepsilon$ remains locally uniformly bounded below near~$Y$
(this is the case for instance if we have strict inequalities in the
curvature assumption a)). If this holds true, we apply Proposition~8.4
on $X_c\ssm Y_c$ with the additional weight factor~$|s|^{-2r}$. Otherwise, 
we use the modified estimate stated
in Remark~8.5 in order to solve the approximate equation $D''u+
\delta^{1/2}h=g_\varepsilon$ with $\delta>0$ small. This yields sections
$u=u_{c,\varepsilon,\delta}$, $h=h_{c,\varepsilon,\delta}$ such that
$$
\eqalign{
\int_{X_c\ssm Y_c}(\eta_\varepsilon+\lambda_\varepsilon)^{-1}
|u_{c,\varepsilon,\delta}|^2|s|^{-2r}\,dV_\omega
&{}+
\int_{X_c\ssm Y_c}|h_{c,\varepsilon,\delta}|^2|s|^{-2r}\,dV_\omega\cr
&\le 2\int_{X_c\ssm Y_c}\langle (B_\varepsilon+\delta I)^{-1}g_\varepsilon,
g_\varepsilon\rangle|s|^{-2r}\,dV_\omega,\cr}
$$ 
and the right hand side is under control in all cases. The extra error
term $\delta^{1/2}h$ can be removed at the end by letting $\delta$ tend
to~$0$.  Since there is essentially no additional difficulty involved
in this process, we will assume for simplicity of exposition that we
do have the required lower bound for~$B_\varepsilon$ and the estimates
of $\smash{g_\varepsilon^{(1)}}$ and $\smash{g_\varepsilon^{(2)}}$ as
above. For $\delta=0$, the above estimate provides a solution
$u_{c,\varepsilon}$ of the equation $D''u_{c,\varepsilon}=
g_\varepsilon=D''F_\varepsilon$ on $X_c\ssm Y_c$, such that
$$
\eqalign{
\int_{X_c\ssm Y_c}(\eta_\varepsilon+\lambda_\varepsilon)^{-1}
|u_{c,\varepsilon}|^2|s|^{-2r}dV_{X,\omega}
&\le 2\int_{X_c\ssm Y_c}\langle B_\varepsilon^{-1}g_\varepsilon,
g_\varepsilon\rangle\, |s|^{-2r}dV_{X,\omega}\cr
&\le 2\,c_r\int_{Y_c}{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
+O(\varepsilon).\cr}
$$
Here we have
$$
\eqalign{
\sigma_\varepsilon&=\log(|s|^2+\varepsilon^2)\le
\log(e^{-2\alpha}+\varepsilon^2)\le -2\alpha+O(\varepsilon^2)\le
-2+O(\varepsilon^2),\cr
\eta_\varepsilon&=\varepsilon-\chi_0(\sigma_\varepsilon)\le
(1+O(\varepsilon))\sigma_\varepsilon^2,\cr
\lambda_\varepsilon&={\chi_0'(\sigma_\varepsilon)^2\over
\chi_0''(\sigma_\varepsilon)}=(1-\sigma_\varepsilon)^2+
(1-\sigma_\varepsilon)\le(3+O(\varepsilon))\sigma_\varepsilon^2,\cr
\eta_\varepsilon+\lambda_\varepsilon&\le
(4+O(\varepsilon))\sigma_\varepsilon^2\le(4+O(\varepsilon))
\big(-\log(|s|^2+\varepsilon^2)\big)^2.\cr}
$$
As $F_\varepsilon$ is uniformly bounded with support in $\{|s|<\varepsilon\}$,
we conclude from an obvious volume estimate that
$$
\int_{X_c}{|F_\varepsilon|^2\over
(|s|^2+\varepsilon^2)^r(-\log(|s|^2+\varepsilon^2))^2}dV_{X,\omega}
\le {\Const \over (\log\varepsilon)^2}.
$$
Therefore, thanks to the usual inequality
$|t+u|^2\le (1+k)|t|^2+(1+k^{-1})|u|^2$ applied to the sum
$F_{c,\varepsilon}=\wt f_\varepsilon-u_{c,\varepsilon}$
with $k=|\log\varepsilon|$, we obtain
from our previous estimates
$$
\int_{X_c\ssm Y_c}
{|F_{c,\varepsilon}|^2\over
(|s|^2+\varepsilon^2)^r(-\log(|s|^2+\varepsilon^2))^2}dV_{X,\omega}
\le 8\,c_r\int_{Y_c}{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
+O(|\log\varepsilon|^{-1}).
$$
In addition to this, we have $d''F_{c,\varepsilon}=0$ by construction,
and this equation extends from $X_c\ssm Y_c$ to $X_c$ by Lemma~12.10.

If $q=0$, then $u_{c,\varepsilon}$ must also be smooth, and the
non integrability of the weight $|s|^{-2r}$ along $Y$ shows that
$u_{c,\varepsilon}$ vanishes on~$Y$, therefore
$$
F_{c,\varepsilon\restriction Y}=
F_{\varepsilon\restriction Y}=F_{\infty\restriction Y}=f.
$$
The theorem and its final estimate are thus obtained by extracting
weak limits, first as $\varepsilon\to 0$, and then as $c\to+\infty$. The
initial assumption that $\Sigma=\hbox{$\{s=\Lambda^r(ds)=0\}$}$ is empty
can be easily removed in two steps: i) the result is true if $X$ is 
Stein, since we can always find a complex hypersurface $Z$ in $X$ such 
that $\Sigma\subset\ol Y\cap Z \subsetneq\ol Y$, and 
then apply the extension theorem on the Stein manifold $X\ssm Z$, in 
combination with Lemma~12.10$\,$; ii) the whole procedure still works when
$\Sigma$ is nowhere dense in~$\ol Y$ (and possibly nonempty). Indeed 
local $L^2$ extensions $\wt f_j$ still exist by step~i) applied on small 
coordinate balls~$U_j\,$; we then set $F_\infty=\sum\theta_j\wt f_j$ and
observe that $|D''F_\infty|^2|s|^{-2r}$ is locally integrable, thanks to the
estimate $\int_{\smash{U_j}}|\wt f_j|^2|s|^{-2r}(\log|s|)^{-2}dV<+\infty$ 
and the fact that $|\sum d''\theta_j\wedge\wt f_j|=O(|s|^\delta)$ for 
suitable $\delta>0$ [as follows from Hilbert's Nullstensatz applied 
to $\wt f_j-\wt f_k$ at singular points of~$\ol Y\,]$.

When $q\ge 1$, the arguments needed to get a smooth solution involve
more delicate considerations, and we will skip the details, which are
extremely technical and not very enlightening.
\endproof

\claim{(13.7) Remarks} \rm
\smallskip
\noindent{\rm a)} When $q=0$, the estimates provided by Theorem~13.6 are
independent of the K\"ahler metric~$\omega$. In fact, if $f$ and $F$
are holomorphic sections of $\Lambda^nT^\star_X\otimes L$ over $Y$ 
(resp.~$X$), viewed as $(n,0)$-forms with values in~$L$, we can ``divide''
$f$ by $\Lambda^r(ds)\in\Lambda^r(TX/TY)^\star\otimes\det E$ to get a 
section $f/\Lambda^r(ds)$ of $\Lambda^pT^\star_Y\otimes L\otimes(\det E)^{-1}$
over~$Y$. We then find
$$
\eqalign{
|F|^2dV_{X,\omega}&=\ii^{n^2}\{F,F\},\cr
{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
&=\ii^{p^2}\{f/\Lambda^r(ds),f/\Lambda^r(ds)\},\cr}
$$
where $\{\bu,\bu\}$ is the canonical bilinear pairing described in~(3.3).
\medskip
\noindent{\rm b)} The hermitian structure on $E$ is not really used
in~depth. In fact, one only needs $E$ to be equipped with a {\it Finsler
metric}, that is, a smooth complex homogeneous function of degree~$2$
on~$E$ [or equivalently, a smooth hermitian metric on the tautological
bundle $\cO_{P(E)}(-1)$ of lines of~$E$ over the projectivized 
bundle~$P(E)$]. The section $s$ of $E$ induces a section $[s]$ of
$P(E)$ over $X\ssm s^{-1}(0)$ and a corresponding section $\wt s$ of
the pull-back line bundle $[s]^\star\cO_{P(E)}(-1)$.  A trivial check
shows that Theorem~13.6 as well as its proof extend to the case of a
Finsler metric on~$E$, if we replace everywhere $\{\ii\Theta(E)s,s\}$
by $\{\ii\Theta([s]^\star\cO_{P(E)}(-1))\wt s,\wt s\,\}$ (especially in
hypothesis 13.6~b)). A minor issue is that $|\Lambda^r(ds)|$ is 
(a priori) no longer defined, since no obvious hermitian norm exists
on~$\det E$. A posteriori, we have the following ad hoc definition of 
a metric on $(\det E)^\star$ which makes the $L^2$~estimates work as 
before: for $x\in X$ and $\xi\in\Lambda^rE^\star_x$, we set
$$
|\xi|^2_x={1\over c_r}
\int_{z\in E_x}
(1+|z|^2)^2\theta'(|z|^2)^2\,{\ii^{r^2}\xi\wedge\ol\xi\over|z|^{2r}}
$$
where $|z|$ is the Finsler norm on~$E_x$ [the constant $c_r$ is there to 
make the result agree with the hermitian case; it is not hard to see that
this metric does not depend on the choice of~$\theta\,$].
\medskip
\noindent{\rm c)} Even when $q=0$, the regularity of $u_{c,\varepsilon,\delta}$
requires some explanations, in case $\delta>0$. In fact, the equation
$$
D''u_{c,\varepsilon,\delta}+\delta^{1/2}h_{c,\varepsilon,\delta}=g_\varepsilon
=D''F_\varepsilon
$$
does not immediately imply smoothness of $u_{c,\varepsilon,\delta}$
(since $h_{c,\varepsilon,\delta}$ need not be smooth in gene\-ral). However,
if we take the pair $(u_{c,\varepsilon,\delta},h_{c,\varepsilon,\delta})$ to 
be the minimal $L^2$ solution orthogonal to 
the kernel of $D''\oplus\delta^{1/2}\Id$, then it must be in the 
closure of the image of
the adjoint ope\-rator $D^{\prime\prime\,*}\oplus\delta^{1/2}\Id$, i.e.\ it must 
satisfy the  additional condition 
\hbox{$D^{\prime\prime\,*}h_{c,\varepsilon,\delta}=\delta^{1/2}u_{c,\varepsilon,\delta}$},
whence $(\Delta''+\delta\Id)h_{c,\varepsilon,\delta}=
(D''D^{\prime\prime\,*}+\delta\Id)h_{c,\varepsilon,\delta}=
\delta^{1/2}D''F_\varepsilon$, and therefore $h_{c,\varepsilon,\delta}$ is 
smooth by the ellipticity of $\Delta''$.\qed
\endclaim

We now present a few interesting corollaries. The first one is a
surjectivity theorem for restriction morphisms in Dolbeault cohomology.

\claim{(13.8) Corollary}
Let $X$ be a projective algebraic manifold and $E$ a holomorphic
vector bundle of rank $r$ over~$X$, $s$ a holomorphic section of $E$
which is everywhere transverse to the zero section, $Y=s^{-1}(0)$, and
let $L$ be a holomorphic line bundle such that \hbox{$F=L^{1/r}\otimes
E^\star$} is Griffiths positive $($we just mean formally that
\hbox{${1\over r} \ii\Theta(L)\otimes\Id_E-\ii\Theta(E)>_\Grif
0)$}. Then the restriction morphism
$$
H^{0,q}(X,\Lambda^nT^\star_X\otimes L)\to H^{0,q}(Y,\Lambda^nT^\star_X
\otimes L)
$$
is surjective for every $q\ge 0$.
\endclaim

\begproof{} A short computation gives
$$
\eqalign{
\ii\,d'&d''\log|s|^2=\ii\,d'\Big({\{s,D's\}\over|s|^2}\Big)\cr
&=\ii\Big({\{D's,D's\}\over|s|^2}-{\{D's,s\}\wedge \{s,D's\}\over |s|^4}+
{\{s,\Theta(E)s\}\over|s|^2}\Big)
\ge -{\{\ii\Theta(E)s,s\}\over|s|^2}\cr}
$$
thanks to Lagrange's inequality and the fact that $\Theta(E)$ is
antisymmetric. Hence, if $\delta$ is a small positive constant
such that
$$
-\ii\Theta(E)+{1\over r}\ii\Theta(L)\otimes\Id_E\ge_\Grif\delta\,
\omega\otimes\Id_E>0,
$$
we find
$$
\ii\Theta(L)+r\,\ii\,d'd''\log|s|^2\ge r\delta\,\omega.
$$
The compactness of $X$ implies $\ii\Theta(E)\le C\omega\otimes\Id_E$
for some $C>0$. Theorem~13.6 can thus be applied with $\alpha=r\delta/C$
and Corollary 13.8 follows. By remark 13.7~b), the above surjectivity
property even holds if $L^{1/r}\otimes E^\star$ is just assumed to be
ample (in the sense that the associated line bundle $\pi^\star L^{1/r}
\otimes\cO_{P(E)}(1)$ is positive on the projectivized bundle
$\pi:P(E)\to X$ of lines of~$E$).\qed
\endproof

Another interesting corollary is the following special case, 
dealing with bounded pseudoconvex domains $\Omega\compact\bC^n$. Even
this simple version retains highly interesting information on the behavior
of holomorphic and plurisubharmonic functions.

\claim{(13.9) Corollary} Let $\Omega\subset\bC^n$ be a bounded
pseudoconvex domain, and let $Y\subset X$ be a nonsingular complex
submanifold defined by a section $s$ of some hermitian vector
bundle $E$ with {\rm bounded} curvature tensor on~$\Omega$. Assume
that $s$ is everywhere transverse to the zero section and that
$|s|\le e^{-1}$ on~$\Omega$. Then there is a constant $C>0$ $($depending 
only on $E)$, with the following property:
for every psh function $\varphi$ on $\Omega$, every holomorphic 
function $f$ on $Y$ with $\int_Y|f|^2|\Lambda^r(ds)|^{-2}e^{-\varphi}dV_Y
<+\infty$, there exists an extension $F$ of $f$ to $\Omega$ such that
$$
\int_\Omega{|F|^2\over|s|^{2r}(-\log|s|)^2}e^{-\varphi}dV_\Omega\le
C\int_Y{|f|^2\over|\Lambda^r(ds)|^2}e^{-\varphi}dV_Y.
$$
\endclaim

\begproof{} We apply essentially the same idea as for the previous corollary,
in the special case when $L=\Omega\times\bC$ is the trivial bundle
equipped with a weight function $e^{-\varphi-A|z|^2}$. The choice of a
sufficiently large constant $A>0$ guarantees that the curvature assumption
13.6~a) is satisfied ($A$ just depends on the presupposed bound for the 
curvature tensor of~$E$).\qed
\endproof

\claim{(13.10) Remark} \rm The special case when $Y=\{z_0\}$ is a point is 
especially interesting. In that case, we just take $s(z)=(e\diam\Omega)^{-1}
(z-z_0)$, viewed as a section of the rank $r=n$ trivial vector bundle 
$\Omega\times \bC^n$ with $|s|\le e^{-1}$. We take $\alpha=1$ 
and replace $|s|^{2n}(-\log|s|)^2$ in the denominator by 
$|s|^{2(n-\varepsilon)}$, using the inequality
$$
-\log|s|={1\over\varepsilon}\log|s|^{-\varepsilon}\le {1\over\varepsilon}
|s|^{-\varepsilon},\qquad\forall \varepsilon>0.
$$ 
For any given value $f_0$, we then find a holomorphic function $f$ such 
that $f(z_0)=f_0$ and
$$
\int_\Omega{|f(z)|^2\over|z-z_0|^{2(n-\varepsilon)}}e^{-\varphi(z)}dV_\Omega
\le{C_n\over\varepsilon^2(\diam\Omega)^{2(n-\varepsilon)}}|f_0|^2
e^{-\varphi(z_0)}.
$$
\endclaim

We prove here, as an application of the Ohsawa-Takegoshi extension theorem,
that every psh function on a pseudoconvex open set $\Omega\subset\bC^n$
can be approximated very accurately by functions of the form
$c\log|f|$, where $c>0$ and $f$ is a holomorphic function. The main idea
is taken from [Dem92]. For other applications to algebraic geometry,
see [Dem93b] and Demailly-Koll\'ar [DK99]. Recall that 
the Lelong number of a function $\varphi\in\Psh(\Omega)$ at a point $x_0$ 
is defined to be
$$
\nu(\varphi,x_0)=\liminf_{z\to x_0}{\log\varphi(z)\over\log|z-x_0|}=
\lim_{r\to 0_+}{\sup_{B(x_0,r)}\varphi\over \log r}.
$$
In particular, if $\varphi=\log|f|$ with $f\in\cO(\Omega)$, then 
$\nu(\varphi,x_0)$ is equal to
the vanishing order $\ord_{x_0}(f)=\sup\{k\in\bN\,;D^{\alpha}f(x_0)=0,~
\forall|\alpha|<k\}$.

\claim{(13.11) Theorem} Let $\varphi$ be a plurisubharmonic function on a
bounded pseudoconvex open set~$\Omega\subset\bC^n$. For every $m>0$, let
$\cH_\Omega(m\varphi)$ be the Hilbert space of holomorphic functions $f$
on $\Omega$ such that $\int_\Omega|f|^2e^{-2m\varphi}d\lambda<+\infty$ and
let $\varphi_m={1\over 2m}\log\sum|\sigma_\ell|^2$ where $(\sigma_\ell)$
is an orthonormal basis of~$\cH_\Omega(m\varphi)$. Then there are
constants $C_1,C_2>0$ independent of $m$ such that
\item{\rm a)}$\displaystyle\varphi(z)-{C_1\over m}\le
\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over m}
\log{C_2\over r^n}$\hfill\break
for every $z\in\Omega$ and $r<d(z,\partial\Omega)$. In particular,
$\varphi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$ and
\item{\rm b)}$\displaystyle\nu(\varphi,z)-{n\over m}\le\nu(\varphi_m,z)\le
\nu(\varphi,z)$~ for every $z\in\Omega$.
\endclaim

\begproof{} Note that $\sum|\sigma_\ell(z)|^2$ is the square of the norm
of the evaluation linear form $f\mapsto f(z)$ on $\cH_\Omega(m\varphi)$. As
$\varphi$ is locally bounded above, the $L^2$ topology is actually stronger
than the topology of uniform convergence on compact subsets of~$\Omega$.
It follows that the series $\sum|\sigma_\ell|^2$ converges uniformly on
$\Omega$ and that its sum is real analytic. Moreover we have
$$\varphi_m(z)=\sup_{f\in B(1)}{1\over m}\log|f(z)|$$
where $B(1)$ is the unit ball of $\cH_\Omega(m\varphi)$. For
$r<d(z,\partial\Omega)$, the mean value inequality applied to the psh
function $|f|^2$ implies
$$\eqalign{
|f(z)|^2&\le{1\over\pi^nr^{2n}/n!}\int_{|\zeta-z|<r}
|f(\zeta)|^2d\lambda(\zeta)\cr
&\le{1\over\pi^nr^{2n}/n!}\exp\Big(2m\sup_{|\zeta-z|<r}\varphi(\zeta)\Big)
\int_\Omega|f|^2e^{-2m\varphi}d\lambda.\cr}$$
If we take the supremum over all $f\in B(1)$ we get
$$
\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over 2m}
\log{1\over\pi^nr^{2n}/n!}
$$
and the second inequality in a) is proved. Conversely, the
Ohsawa-Takegoshi extension theorem (Corollary 13.9) applied
to the $0$-dimensional subvariety $\{z\}\subset\Omega$ shows that for any
$a\in\bC$ there is a holomorphic function $f$ on $\Omega$ such that
$f(z)=a$ and
$$\int_\Omega|f|^2e^{-2m\varphi}d\lambda\le C_3|a|^2e^{-2m\varphi(z)},$$
where $C_3$ only depends on $n$ and~${\rm diam}\,\Omega$. We fix $a$
such that the right hand side is~$1$. This gives the other
inequality
$$\varphi_m(z)\ge{1\over m}\log|a|=\varphi(z)-{\log C_3\over 2m}.$$
The above inequality implies $\nu(\varphi_m,z)\le\nu(\varphi,z)$. In the
opposite direction, we find
$$\sup_{|x-z|<r}\varphi_m(x)\le\sup_{|\zeta-z|<2r}\varphi(\zeta)+
{1\over m}\log{C_2\over r^n}.$$
Divide by $\log r$ and take the limit as $r$ tends to~$0$. The quotient
by $\log r$ of the supremum of a psh function over $B(x,r)$ tends to the
Lelong number at~$x$. Thus we obtain
$$
\nu(\varphi_m,x)\ge\nu(\varphi,x)-{n\over m}.\eqno\square
$$
\endproof

Theorem 13.11 implies in a straightforward manner the deep result of
[Siu74] on the analyticity of the Lelong number sublevel sets.

\claim{(13.12) Corollary} Let $\varphi$ be a plurisubharmonic function
on a complex manifold~$X$. Then, for every $c>0$, the Lelong number
sublevel set
$$
E_c(\varphi)=\big\{z\in X\,;\;\nu(\varphi,z)\ge c\big\}
$$
is an analytic subset of~$X$.
\endclaim

\begproof{} Since analyticity is a local property, it
is enough to consider the case of a psh function $\varphi$ on a 
pseudoconvex open set $\Omega\subset\bC^n$. The
inequalities obtained in 13.11~b) imply that 
$$
E_c(\varphi)=\bigcap_{m\ge m_0}E_{c-n/m}(\varphi_m).
$$
Now, it is clear that $E_c(\varphi_m)$ is the analytic set defined by the
equations $\sigma_\ell^{(\alpha)}(z)=0$ for all multi-indices $\alpha$
such that~$|\alpha|<mc$. Thus $E_c(\varphi)$ is analytic as a (countable) 
intersection of analytic sets.\qed
\endproof

We now translate Theorem 13.11 into a more geometric setting. Let $X$ be
a projective manifold and $L$ a line bundle over~$X$. A singular hermitian
metric $h$ on $L$ is a metric such that the weight function $\varphi$ of
$h$ is $L^1_\loc$ in any local trivialization (such that
$L_{|U}\simeq U\times\bC$ and $\Vert\xi\Vert_h=|\xi|e^{-\varphi(x)}$,
$\xi\in L_x\simeq\bC$). The curvature of $L$ can then be computed in
the sense of distributions
$$
T={\ii\over 2\pi}\Theta_h(L)={\ii\over \pi}\ddbar\varphi,
$$
and $L$ is said to be pseudoeffective if $L$ admits a singular hermitian
metric $h$ such that the curvature current $T={\ii\over 2\pi}\Theta_h(L)$
is semipositive [The weight functions $\varphi$ of $L$ are thus
plurisubharmonic]. 
Our goal is to approximate $T$ in the weak topology by divisors which
have roughly the same Lelong numbers as~$T$. The existence of weak
approximations by divisors has already been proved in [Lel72] for
currents defined on a pseudoconvex open set $\Omega\subset\bC^n$
with $H^2(\Omega,\bR)=0$, and in [Dem92, 93b] in the situation considered
here (cf.\ also [Dem82b], although the argument given there is somewhat
incorrect). We take the opportunity to present here a slightly simpler
derivation. In what follows, we use an additive notation for $\Pic(X)$,
i.e.\ $kL$ is meant for the line bundle $L^{\otimes k}$.

\claim{(13.13) Proposition} For any $T={\ii\over 2\pi}\Theta_h(L)\ge 0$
and any ample line bundle $F$, there is a sequence of non zero sections
$h_s\in H^0(X,p_sF+q_sL)$ with $p_s,q_s>0$, $\lim q_s=+\infty$
and $\lim p_s/q_s=0$, such that the divisors $D_s=\smash{1\over
q_s}\div(h_s)$ satisfy $T=\lim D_s$ in the weak topology and
$\,\sup_{x\in X}|\nu(D_s,x)-\nu(T,x)|\to 0$ as $s\to+\infty$.
\endclaim

\claim{(13.14) Remark} {\rm The proof will actually show, with very slight
modifications, that Proposition~13.13 also holds when $X$ is a Stein manifold
and $L$ is an arbitrary holomorphic line bundle.}
\endclaim

\begproof{} We first use H\"ormander's $L^2$ estimates to construct 
a suitable family of holomorphic sections and combine this with some ideas
of [Lel72] in a second step. Select a smooth metric with positive curvature
on~$F$, choose $\omega={\ii\over 2\pi}\Theta(F)>0$ as a K\"ahler metric 
on $X$ and fix some large integer $k$ (how large $k$ must be will be 
specified later). For all $m\ge 1$ we define
$$w_m(z)=\sup_{1\le j\le N}{1\over m}\log\Vert f_j(z)\Vert ,$$
where $(f_1\ld f_N)$ is an orthonormal basis of the space of sections of
\hbox{$\cO(kF+mL)$} with finite global $L^2$ norm 
$\int_X\Vert f\Vert ^2dV_\omega$. Let
$e_F$ and $e_L$ be non vanishing holomorphic sections of $F$ and $L$ on a
trivializing open set $\Omega$, and let
$e^{-\psi}=\Vert e_F\Vert $, $e^{-\varphi}=\Vert e_L\Vert $ be the corresponding weights.
If $f$ is a section of $\cO(kF+mL)$ and if we still denote by $f$ the
associated complex valued function on $\Omega$ with respect to the
holomorphic frame $e_F^k\otimes e_L^m$, we have
$\Vert f(z)\Vert =|f(z)|e^{-k\psi(z)-m\varphi(z)}\,$; here $\varphi$ is
plurisubharmonic, $\psi$ is smooth and
strictly plurisubharmonic, and $T={i\over\pi}\partial\dbar\varphi$,~
$\omega={i\over\pi}\partial\dbar\psi$. In $\Omega$, we can write
$$w_m(z)=\sup_{1\le j\le N}{1\over m}\log|f_j(z)|-\varphi(z)-
{k\over m}\psi(z).$$
In particular $T_m:={i\over\pi}\partial\dbar w_m+T+{k\over m}\omega$ is a 
closed positive current belonging to the cohomology class $c_1(L)+
{k\over m}c_1(F)$.\smallskip

\noindent
{\bf Step 1}. We claim that $T_m$ converges to $T$ as $m$ tends to $+\infty$
and that $T_m$ satisfies the inequalities
$$
\nu(T,x)-{n\over m}\le\nu(T_m,x)\le\nu(T,x)
\leqno(13.15)
$$
at every point $x\in X$. Note that $T_m$ is defined on $\Omega$ by
$T_m={i\over\pi}\partial\dbar v_{m,\Omega}$ with 
$$v_{m,\Omega}(z)=\sup_{1\le j\le N}{1\over m}\log|f_j(z)|,~~~~
\int_\Omega|f_j|^2e^{-2k\psi-2m\varphi}dV_\omega\le 1.$$
We proceed in the same way as for the proof of Theorem~13.11.
We suppose here that $\Omega$ is a coordinate open set with analytic 
coordinates $(z_1\ld z_n)$. Take $z\in\Omega'\compact\Omega$ and
$r\le r_0={1\over 2}d(\Omega',\partial\Omega)$. By the $L^2$ estimate and the
mean value inequality for subharmonic functions, we obtain
$$|f_j(z)|^2\le{C_1\over r^{2n}}\int_{|\zeta-z|<r}
|f_j(\zeta)|^2d\lambda(\zeta)\le
{C_2\over r^{2n}}\sup_{|\zeta-z|<r}e^{2m\varphi(\zeta)}$$
with constants $C_1$, $C_2$ independent of $m$ and $r$ (the smooth function 
$\psi$ is bounded on any compact subset of $\Omega$). Hence we infer
$$v_{m,\Omega}(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over 2s}
\log{C_2\over r^{2n}}.\leqno(13.16)$$
If we choose for example $r=1/m$ and use the upper semi-continuity of
$\varphi$, we infer $\limsup_{s\to+\infty}v_{m,\Omega}\le\varphi$.
Moreover, if $\gamma=\nu(\varphi,x)=\nu(T,x)$, then
$\varphi(\zeta)\le\gamma\log|\zeta-x|+O(1)$ near $x$. By taking $r=|z-x|$ 
in (13.16), we find
$$
\eqalign{
&v_{m,\Omega}(z)\le\sup_{|\zeta-x|<2r}\varphi(\zeta)-{n\over m}\log r+O(1)
\le\Big(\gamma-{n\over m}\Big)\log|z-x|+O(1),\cr
&\nu(T_m,x)=\nu(v_{m,\Omega},x)\ge\Big(\gamma-{n\over m}\Big)_+\ge
\nu(T,x)-{n\over m}.\cr}
$$
In the opposite direction, the inequalities require deeper arguments since 
we actually have to construct sections in $H^0(X,kF+mL)$. Assume that
$\Omega$ is chosen isomorphic to a bounded pseudoconvex open set in $\bC^n$.
By the Ohsawa-Takegoshi $L^2$ extension theorem (Corollary 13.9), for every 
point $x\in\Omega$, there is a holomorphic function $g$ on $\Omega$ such that
$g(x)=e^{m\varphi(x)}$ and 
$$\int_\Omega|g(z)|^2e^{-2m\varphi(z)}d\lambda(z)\le C_3,$$ 
where $C_3$ depends only on $n$ and $\diam(\Omega)$. For $x\in\Omega'$, set
$$\sigma(z)=\theta\big(|z-x|/r\big)\,g(z)\,e_F(z)^k\otimes e_L(z)^m,~~~~
r=\min\big(1,2^{-1}d(\Omega',\partial\Omega)\big),$$
where $\theta:\bR\to[0,1]$ is a cut-off function such that
$\theta(t)=1$ for $t<1/2$ and $\theta(t)=0$ for $t\ge 1$. We solve the
global equation $\dbar u=v$ on $X$ with $v=\dbar\sigma$, after multiplication
of the metric of $kF+mL$ with the weight
$$e^{-2n\rho_x(z)},~~~~\rho_x(z)=\theta\big(|z-x|/r\big)\,\log|z-x|\le 0.$$
The $(0,1)$-form $v$ can be considered as a $(n,1)$-form with values in
the line bundle $\cO(-K_X+kF+mL)$ and the resulting curvature form of this
bundle is
$$
{\rm Ricci}(\omega)+k\omega+mT+n{i\over\pi}\partial\dbar\rho_x.
$$
Here the first two summands are smooth, $i\partial\dbar\rho_x$ is 
smooth on $X\ssm\{x\}$ and $\ge 0$ on $B(x,r/2)$, and $T$ is a positive
current. Hence by choosing $k$ large enough, we can suppose that this 
curvature form is $\ge\omega$, uniformly for $x\in\Omega'$. By
H\"ormander's standard $L^2$ estimates [AV65, H\"or65, 66], we get a
solution $u$ on $X$ such that
$$
\int_X\Vert u\Vert ^2e^{-2n\rho_x}dV_\omega\le C_4
\int_{r/2<|z-x|<r}|g|^2e^{-2k\psi-2m\varphi-2n\rho_x}dV_\omega\le C_5\,;
$$
to get the estimate, we observe that $v$ has support in the corona
$r/2<|z-x|<r$ and that $\rho_x$ is bounded there. Thanks to the logarithmic
pole of $\rho_x$, we infer that $u(x)=0$. Moreover
$$
\int_\Omega\Vert \sigma\Vert ^2dV_\omega\le
\int_{\Omega'+B(0,r/2)}|g|^2e^{-2k\psi-2m\varphi}dV_\omega\le C_6,
$$
hence $f=\sigma-u\in H^0(X,kF+mL)$ satisfies $\int_X\Vert f\Vert ^2
dV_\omega\le C_7$ and
$$
\Vert f(x)\Vert =\Vert \sigma(x)\Vert =
\Vert g(x)\Vert \,\Vert e_F(x)\Vert ^m\Vert e_L(x)\Vert ^m=
\Vert e_F(x)\Vert ^k=e^{-k\psi(x)}.
$$
In our orthonormal basis $(f_j)$, we can write $f=\sum\lambda_jf_j$
with $\sum|\lambda_j|^2\le C_7$. Therefore
$$\eqalign{
&e^{-k\psi(x)}=\Vert f(x)\Vert \le\sum|\lambda_j|\,\sup\Vert f_j(x)\Vert \le
\sqrt{C_7N}\,e^{mw_m(x)},\cr
&w_m(x)\ge{1\over m}\log(C_7N)^{-1/2}\Vert f(x)\Vert \ge
-{1\over m}\Big(\log(C_7N)^{1/2}+k\,\psi(x)\Big)\cr}$$
where $N=\dim H^0(X,kF+mL)=O(m^n)$. By adding $\varphi+{k\over m}\psi$, we
get $v_{m,\Omega}\ge\varphi-C_8m^{-1}\log m$. Thus $\lim_{m\to+\infty}
v_{m,\Omega}=\varphi$ everywhere, $T_m={i\over\pi}\partial\dbar v_{m,\Omega}$
converges weakly to $T={i\over\pi}\partial\dbar\varphi$, and
$$
\nu(T_m,x)=\nu(v_{m,\Omega},x)\le\nu(\varphi,x)=\nu(T,x).
$$
Note that $\nu(v_{m,\Omega},x)={1\over m}\min\ord_x(f_j)$ where
$\ord_x(f_j)$ is the vanishing order of $f_j$ at $x$, so our initial 
lower bound for $\nu(T_m,x)$ combined with the last inequality gives
$$
\nu(T,x)-{n\over m}\le{1\over m}\min\ord_x(f_j)\le\nu(T,x).\leqno(13.17)
$$

\smallskip

\noindent
{\bf Step 2:} {\it Construction of the divisors $D_s$}.

Select sections $(g_1\ld g_N)\in H^0(X,k_0F)$ with $k_0$ so large that
$k_0F$ is very ample, and set 
$$
h_{\ell,m}=f_1^\ell g_1+\ldots+f_N^\ell g_N\in H^0\big(X,(k_0+\ell k)F
+\ell mL\big).
$$
For almost every $N$-tuple $(g_1\ld g_N)$, Lemma~13.18 below and the weak
continuity of $\partial\dbar$ show that
$$
\Delta_{\ell,m}={1\over \ell m}{i\over\pi}\partial\dbar\log|h_{\ell ,m}|=
{1\over \ell m}\,{\rm div}(h_{\ell ,m})
$$
converges weakly to $T_m={i\over\pi}\partial\dbar v_{m,\Omega}$ as $\ell$
tends to $+\infty$, and that
$$
\nu(T_m,x)\le\nu\Big({1\over \ell m}\Delta_{\ell ,m},x\Big)\le\nu(T,x)+
{1\over \ell m}.
$$
This, together with the first step, implies the proposition for some
subsequence $D_s=\Delta_{\ell(s),s}$, $\ell(s)\gg s\gg 1$. We even obtain 
the more explicit inequality
$$
\nu(T,x)-{n\over m}\le\nu\Big({1\over \ell m}\Delta_{\ell,m},x\Big)\le
\nu(T,x)+{1\over\ell m}.\qquad\qed
$$
\endproof

\claim{(13.18) Lemma} Let $\Omega$ be an open subset in $\bC^n$ and let
$f_1\ld f_N\in H^0(\Omega,\cO_\Omega)$ be non zero functions. 
Let $G\subset H^0(\Omega,\cO_\Omega)$ be a finite dimensional subspace
whose elements generate all $1$-jets at any point of $\Omega$. 
Finally, set $v=\sup\log|f_j|$ and
$$h_\ell =f_1^\ell g_1+\ldots+f_N^\ell g_N,~~~~g_j\in G\ssm\{0\}.$$
Then for all $(g_1\ld g_N)$ in $(G\ssm\{0\})^N$ except a set of measure~$0$,
the sequence ${1\over \ell }\log|h_\ell |$ converges to $v$ in 
$L^1_{\rm loc}(\Omega)$ and
$$\nu(v,x)\le\nu\Big({1\over \ell }\log|h_\ell |\Big)\le\nu(v,x)+
{1\over \ell },~~~~\forall x\in X,~~\forall \ell \ge 1.$$
\endclaim

\begproof{} The sequence ${1\over \ell }\log|h_\ell |$ is locally uniformly
bounded above and we have
$$\lim_{\ell \to+\infty}{1\over \ell }\log\big|h_\ell (z)\big|=v(z)$$
at every point $z$ where all absolute values $|f_j(z)|$ are distinct
and all $g_j(z)$ are nonzero. This is a set of full measure in $\Omega$
because the sets $\{|f_j|^2=|f_l|^2,\,j\ne l\}$ and $\{g_j=0\}$ are real
analytic and thus of zero measure (without loss of generality, we may
assume that $\Omega$ is connected and that the $f_j$'s are not pairwise
proportional). The well-known uniform integrability properties of 
plurisubharmonic functions then show that ${1\over \ell }\log|h_\ell |$ converges
to $v$ in $L^1_{\rm loc}(\Omega)$. It is easy to see that $\nu(v,x)$ is the
minimum of the vanishing orders ${\rm ord}_x(f_j)$, hence
$$\nu(\log|h_\ell |,x)={\rm ord}_x(h_\ell )\ge \ell \,\nu(v,x).$$
In the opposite direction, consider the set ${\cal E}_\ell $ of all $(N+1)$-tuples
$$(x,g_1\ld g_N)\in\Omega\times G^N$$
for which $\nu(\log|h_\ell |,x)\ge \ell \,\nu(v,x)+2$. Then ${\cal E}_\ell $ is a
constructible set in $\Omega\times G^N$: it has a locally finite
stratification by analytic sets, since
$${\cal E}_\ell =\bigcup_{s\ge 0}\Big(\bigcup_{j,\,|\alpha|=s}
\big\{x\,;\,D^\alpha f_j(x)\ne 0\big\}\times G^N\Big)\cap
\bigcap_{|\beta|\le \ell s+1}\big\{(x,(g_j))\,;\,D^\beta h_\ell (x)=0\big\}.$$
The fiber ${\cal E}_\ell \cap(\{x\}\times G^N)$ over a point $x\in\Omega$
where $\nu(v,x)=\min{\rm ord}_x(f_j)=s$ is the vector space of $N$-tuples
$(g_j)\in G^N$ satisfying the equations $D^\beta
\big(\sum f_j^\ell g_j(x)\big)=0$, $|\beta|\le \ell s+1$. However, if
${\rm ord}_x(f_j)=s$, the linear map
$$(0\ld 0,g_j,0\ld 0)\longmapsto
\big(D^\beta(f_j^\ell g_j(x))\big)_{|\beta|\le \ell s+1}$$
has rank $n+1$, because it factorizes into an injective map
$J^1_xg_j\mapsto J^{\ell s+1}_x(f_j^\ell g_j)$. It follows that the fiber
${\cal E}_\ell \cap(\{x\}\times G^N)$ has codimension at least $n+1$.
Therefore
$$\dim {\cal E}_\ell \le\dim(\Omega\times G^N)-(n+1)=\dim G^N-1$$
and the projection of ${\cal E}_\ell $ on $G^N$ has measure zero by Sard's
theorem. By definition of ${\cal E}_\ell $, any choice of $(g_1\ld g_N)\in
G^N\ssm\bigcup_{\ell \ge 1} {\rm pr}({\cal E}_\ell )$ produces functions $h_\ell $
such that  $\nu(\log|h_\ell |,x)\le \ell \,\nu(v,x)+1$ on~$\Omega$.\qed
\endproof

\titleb{14.}{Invariance of Plurigenera of Varieties of General Type}
The goal of this section is to give a proof of the following 
fundamental result on the invariance of plurigenera, proved
by Y.T.~Siu [Siu98]. A generalized version has been obtained shortly
afterwards by Y.~Kawamata [Kaw99], using only algebraic tools.

\claim{(14.0) Theorem (Siu)} Let $X\to S$ be a smooth projective family of
varieties of general type on a connected base~$S$.
Then the plurigenus $p_m(X_t)=h^0(X_t,mK_{X_t})$ of fibers is
independent of~$t$ for all $m\ge 0$.
\endclaim

Given a family $\gamma:X\to S$ of projective varieties, the proof of
the invariance of the plurigenera of fibers $X_t=\gamma^{-1}(t)$ is
easily reduced to the case when the base is the unit disk
$\Delta\subset\bC$ (in general, one can always connect two arbitrary
points of $S$ by a chain of small analytic disks, and one can take the
pull-back of the family in restriction to each of those disks). We can
therefore suppose that $\gamma:X\to\Delta$.  In that case, we have a
canonical isomorphism $K_{X_t}\simeq K_{X|X_t}$ on each fiber, given
by $u\mapsto dt\wedge u$ [in this way, we will allow ourselves to
identify $K_{X_t}$ and $K_{X|X_t}$ in the sequel]. We know from
Grauert's direct image theorem [Gra60] that the direct image sheaves
$\gamma_\star\cO(mK_X)$ are coherent, and moreover, the plurigenera
$p_m(X_t)=h^0(X_t,(mK_X)_{|X_t})$ are upper semi-continuous functions
of~$t$. The jumps occur precisely if a section on some fiber $X_{t_0}$
does not extend to nearby fibers. Proving the invariance of
plurigenera is thus equivalent to proving that a section of
$mK_{X|X_{t_0}}$ on a fiber $X_{t_0}$ can be extended to a
neighborhood of $X_{t_0}$ in~$X$.  We can assume without loss of
generality that~$t_0=0$.  The strategy of Siu's proof consists more or
less in the use of suitable singular hermitian metrics on $K_X$ and
$K_{X_0}$ (with ``minimal singularities''), combined with $L^2$
extension theorems with respect to these metrics.

\titlec{14.1.}{Metrics with minimal singularities}
One of the main ideas in the proof of the invariance of plurigenera -- 
although it is not completely explicit in [Siu98] -- rests on the fact
that the singularities of hermitian metrics on a line bundle $L$ 
can reflect very accurately the base loci of the sequence
of linear systems $|mL|$, precisely when metrics with minimal 
singularities are used. We follow here the approach of~[DPS00].

\claim{(14.1.1) Definition} Let $L$ be a pseudo-effective line bundle
on a compact complex manifold~$X$. Consider two hermitian metrics 
$h_1$, $h_2$ on $L$ with curvature $i\Theta_{h_j}(L)\ge 0$
in the sense of currents.
\smallskip
\item{\rm(i)} We will write $h_1\preccurlyeq h_2$, and say that $h_1$ 
is less singular than $h_2$, if there exists a constant $C>0$ such that
$h_1\le C h_2$.
\smallskip
\item{\rm(ii)} We will write $h_1\sim h_2$, and say that $h_1$, $h_2$ are
equivalent with respect to singularities, if there exists a constant
$C>0$ such that $C^{-1}h_2\le h_1\le C h_2$.
\vskip0pt
\endclaim

\noindent
Of course $h_1\preccurlyeq h_2$ if and only if the associated weights
in suitable trivializations locally satisfy $\varphi_2\le \varphi_1+C$.
This implies in particular $\nu(\varphi_1,x)\le\nu(\varphi_2,x)$ at each
point. The above definition is motivated by the following observation.

\claim{(14.1.2) Theorem} For every pseudo-effective line bundle $L$ over
a compact complex manifold $X$, there exists up to equivalence of
singularities a unique class of hermitian metrics $h$ with minimal
singularities such that $i\Theta_h(L)\ge 0$.
\endclaim

\begproof{} The proof is almost trivial. We fix once for all a smooth
metric $h_\infty$ (whose curvature is of random sign and signature),
and we write singular metrics of $L$ under the form $h=h_\infty e^{-2\psi}$. 
The condition $i\Theta_h(L)\ge 0$ is equivalent to 
${i\over\pi}\ddbar\psi\ge -u$ where $u=i\Theta_{h_\infty}(L)$. This
condition implies that $\psi$ is plurisubharmonic up to the addition
of the weight $\varphi_\infty$ of $h_\infty$, and therefore locally
bounded from above.  Since we are concerned with metrics only up to
equivalence of singularities, it is always possible to adjust
$\psi$ by a constant in such a way that $\sup_X\psi=0$. We now set
$$
h_{\min}=h_\infty e^{-2\psi_{\min}},\qquad \psi_{\min}(x)=\sup_\psi\psi(x)
$$
where the supremum is extended to all functions $\psi$ such that
$\sup_X\psi=0$ and\break ${i\over\pi}\ddbar\psi\ge -u$. By standard
results on plurisubharmonic functions (see Lelong [Lel69]), 
$\psi_{\min}$ still satisfies ${i\over\pi}\ddbar\psi_{\min}\ge -u$
(i.e.\ the weight $\varphi_\infty+\psi_{\min}$ de $h_{\min}$ is
plurisubharmonic), and $h_{\min}$ is obviously the metric with
minimal singularities that we were looking for.\qed
\endproof

Now, given a section $\sigma\in H^0(X,mL)$, the expression $h(\xi)=
|\xi^m/\sigma(x)|^{2/m}$ defines a singular metric on $L$, 
which therefore necessarily has at least as much singularity as
$h_{\min}$ as,
i.e.\ ${1\over m}\log|\sigma|^2\le \varphi_{\min}+C$ locally. In particular,
$|\sigma|^2e^{-m\varphi_{\min}}$ is locally bounded, hence
$\sigma\in H^0(X,mL\otimes\cI(h_{\min}^{\otimes m}))$. For all 
$m>0$ we therefore have an isomorphism
$$
H^0(X,mL\otimes\cI(h_{\min}^{\otimes m}))\buildo\vlra^\simeq H^0(X,mL).
$$
By the well-known properties of Lelong numbers (see Skoda [Sko72a]), the
union of all zero varieties of the ideals $\cI(h_{\min}^{\otimes m})$
is equal to the Lelong sublevel set
$$
E_+(h_{\min})=\big\{x\in X\,;~\nu(\varphi_{\min},x)>0\big\}.
\leqno(14.1.3)
$$
We will call this set the {\it virtual base locus} of $L$. It is
always contained in the ``algebraic'' base locus
$$B_{\Vert L\Vert}=\bigcap_{m>0}B_{|mL|},\qquad
B_{|mL|}=\bigcap_{\sigma\in H^0(X,mL)}\sigma^{-1}(0),
$$ 
but there may be a strict inclusion. This is the case for instance if 
$L\in\Pic^0(X)$ is such that all positive multiples $mL$ have no 
nonzero sections; in that case $E_+(h_{\min})=\emptyset$ but
$\bigcap_{m>0}B_{|mL|}=X$. Another general situation where
$E_+(h_{\min})$ and $B_{\Vert L\Vert}$ can differ is given by the 
following result.

\claim{(14.1.4) Proposition} Let $L$ be a big nef line bundle. Then
$h_{\min}$ has zero Lelong numbers everywhere, i.e.\
$E_+(h_{\min})=\emptyset$.
\endclaim

\begproof{} Recall that $L$ is big if its Kodaira-Iitaka dimension 
$\kappa(L)$ is equal to $n=\dim X$. In that case, it is well known that
one can write $m_0L=A+E$ with $A$ ample and $E$ effective, for $m_0$
sufficiently large. Then $mL=((m-m_0)L+A) +E$ is the sum of an
ample divisor $A_m=(m-m_0)L+A$ plus a (fixed) effective divisor,
so that there is a hermitian metric $h_m$ on $L$ for which
$i\Theta_{h_m}(L)={1\over m}i\Theta(A_m)+{1\over m}[E]$,
with a suitable smooth positive form $i\Theta(A_m)$. This shows that
the Lelong numbers of the weight of $h_m$ are $O(1/m)$, hence in
the limit those of $h_{\min}$ are zero.
\endproof

If $h$ is a singular hermitian metric such that $i\Theta_h(L)\ge 0$ and
$$
H^0(X,mL\otimes\cI(h^{\otimes m}))\simeq H^0(X,mL)\qquad
\hbox{for all $m\ge 0$},
\leqno(14.1.5)
$$
we say that $h$ is an {\it analytic Zariski decomposition} of~$L$. 
We have just seen that such a decomposition always exists and
that $h=h_{\min}$ is a solution. The concept of analytic Zariski decomposition
is motivated by its algebraic counterpart (the existence of which generally
fails)$\,$: one says that $L$ admits an {\it algebraic Zariski decomposition}
if there exists an integer $m_0$ such that $m_0L\simeq\cO(E+D)$ where 
$E$ is an effective divisor and $D$ a nef divisor, in such a way that
$H^0(X,kD)\simeq H^0(X,km_0L)$ for all $k\ge 0$. If $\cO(*D)$ is generated
by sections, there is a smooth metric with semipositive curvature on 
$\cO(D)$, and this metric induces a singular hermitian metric
$h$ on $L$ of curvature current ${1\over m_0}(i\Theta(\cO(D))+ [E])$.
Its poles are defined by the effective $\bQ$-divisor ${1\over m_0}E$. 
For this metric, we of course have $\cI(h^{\otimes km_0})=\cO(-kE)$, hence
(14.1.5) holds true at least when $m$ is a multiple of $m_0$.

\titlec{14.2.}{A uniform global generation property}
The ``uniform global generation property'' shows in some sense that the
curvature of the tensor product sheaf $L\otimes\cI(h)$ is uniformly
bounded below, for any singular hermitian metric 
$h$ with nonnegative curvature on~$L$.

\claim{(14.2.1) Proposition} There exists an ample line bundle $G$ on $X$
such that for every pseudoeffective line bundle $(L,h)$, the sheaf
$\cO(G+L)\otimes\cI(h)$ is generated by its global sections. In fact, $G$
can be chosen as follows: pick any very ample line bundle $A$, and take
$G$ such that $G-(K_X+nA)$ is ample, e.g.\ $G=K_X+(n+1)A$.
\endclaim

\begproof{} Let $\varphi$ be the weight of the metric $h$ on a small
neighborhood of a point~$z_0\in X$. Assume that we have a local section 
$u$ of $\cO(G+L)\otimes\cI(h)$ on a coordinate open ball $B=B(z_0,\delta)$, 
such that
$$
\int_B|u(z)|^2e^{-2\varphi(z)}|z-z_0|^{-2(n+\varepsilon)}dV(z)<+\infty.
$$
Then Skoda's division theorem [Sko72b] implies $u(z)=\sum(z_j-z_{j,0})v_j(z)$
with
$$
\int_B|v_j(z)|^2e^{-2\varphi(z)}|z-z_0|^{-2(n-1+\varepsilon)}dV(z)<+\infty,
$$
in particular $u_{z_0}\in\cO(G+L)\otimes\cI(h)\otimes\gm_{X,z_0}$. Select
a very ample line bundle $A$ on~$X$. We take a basis
$\sigma=(\sigma_j)$ of sections of $H^0(X,G\otimes\gm_{X,z_0})$ and
multiply the metric $h$ of $G$ by the factor $|\sigma|^{-2(n+\varepsilon)}$.
The weight of the above metric has singularity $(n+\varepsilon)\log|z-z_0|^2$
at~$z_0$, and its curvature is
$$
i\Theta(G)+(n+\varepsilon)i\ddbar\log|\sigma|^2\ge
i\Theta(G)-(n+\varepsilon)\Theta(A).
\leqno(14.2.2)
$$
Now, let $f$ be a local section in $H^0(B,\cO(G+L)\otimes\cI(h))$ on 
$B=B(z_0,\delta)$, $\delta$ small. We solve the global $\dbar$ equation 
$$
\dbar u =\dbar(\theta f)\qquad\hbox{on $X$}
$$ 
with a cut-off function $\theta$ supported near $z_0$ and with the 
weight associated with our above choice of metric on $G+L$. 
Thanks to Nadel's Theorem 6.11, the solution exists if the metric of 
$G+L-K_X$ has positive curvature. As $i\Theta_h(L)\ge 0$ in the sense 
of currents, $(14.2.2)$ shows that a sufficient condition is $G-K_X-nA>0$
(provided that $\varepsilon$ is small enough).
We then find a smooth solution $u$ such that $u_{z_0}\in 
\cO(G+L)\otimes\cI(h)\otimes\gm_{X,z_0}$, hence
$$
F:=\theta f -u \in H^0(X,\cO(G+L)\otimes\cI(h))
$$ 
is a global section differing from $f$ by a germ in 
$\cO(G+L)\otimes\cI(h)\otimes\gm_{X,z_0}$. Nakayama's lemma implies 
that $H^0(X,\cO(G+L)\otimes\cI(h))$ generates the stalks of
$\cO(G+L)\otimes\cI(h)$.
\endproof

\titlec{14.3.}{A special case of the Ohsawa-Takegoshi-Manivel
$L^2$ extension theorem}
We will need the special case below of the Ohsawa-Takegoshi $L^2$ 
extension theorem. (Notice that, in this way, the proof of the theorem 
on invariance of plurigenera requires $3$ essentially different types of
$L^2$ existence theorems~!).

\claim{(14.3.1) Theorem} Let $\gamma:X\rightarrow\Delta$ be a projective
family of projective manifolds parametrized by the open unit disk
$\Delta\subset\bC$. Let $X_0=\gamma^{-1}(0)$, \hbox{$n=\dim_\bC X_0$}, 
and let $L\to X$ be a line bundle equipped with a hermitian metric
written locally as $e^{-\chi}$, such that
$i\partial\bar\partial\chi\geq \omega$ in the sense of
currents, with a suitable positive $(1,1)$-form $\omega$ on~$X$.
Let $0<r<1$ and $\Delta_r=\{\,t\in\Delta\,;~|t|<r\,\}$. Then there
exists a positive constant $A_r$ such that the following 
statement holds. For every holomorphic $n$-form $f$ on $X_0$ 
with values in $L$ such that
$$
\int_{X_0}|f|^2e^{-\chi}dV<\infty,
$$
there exists a holomorphic $(n+1)$-form $\tilde f$ on $\gamma^{-1}(\Delta_r)$
with values in $L$, such that $\tilde f|_{X_0}=f\wedge\gamma^*(dt)$ 
on $X_0$ and
$$
\int_X|\tilde f|^2e^{-\chi}dV\leq A_r
\int_{X_0}|f|^2e^{-\chi}dV.
$$
\endclaim

\claim{(14.3.2) Remark} {\rm
It should be noticed that no hermitian metric on the tangent bundle of $X_0$ 
or $X$ is required in order to define the integral of the square of the
norm of holomorphic forms $f$ and $\tilde f$ on $X_0$ and $X\,$;
in fact, it suffices to integrate the volume forms $i^{n^2}
f\wedge\ol f$ and $i^{(n+1)^2} \tilde f\wedge\ol{\tilde f}$.}
\endclaim

\titlec{14.4.}{Construction of a hermitian metric with positive 
curvature on~$K_{X_0}$\kern-2pt}
From now on, we suppose that $X\to\Delta$ is a family of projective
manifolds of general type (i.e.\ that all fibers $X_t$ are of general 
type). A technical point to be settled first is that the hermitian
metric on $K_X$ must be chosen in such a way that its
curvature current dominates a smooth positive definite $(1,1)$-form, 
so that the Skoda division theorem and the Ohsawa-Takegoshi-Manivel
$L^2$ extension theorem can be applied. This point is easily settled
by means of a variation of Kodaira's technique which consists in
expressing a sufficiently large multiple of a big line bundle as
the sum of an effective divisor and of an ample line bundle.

\claim{(14.4.1) Lemma} There exists a positive integer $a$ such that
$aK_X=D+F$, where $D$ is an effective divisor on $X$ which does not
contain $X_0$, and where $F$ is a positive line bundle on~$X$.
\endclaim

\begproof{} Let $F$ be a positive line bundle on $X$, and let $r_a$
be the generic rank of $H^0(X_t,aK_{X_t}-F)$, which is achieved
for $t\in \Delta\ssm S_a$ in the complement of a suitable locally
finite subset $S_a\subset\Delta$. Fix $t_1\in\Delta\ssm\bigcup S_a$. 
Since $X_{t_1}$ is of general type, we know that
$h^0(X_{t_1},aK_{X_{t_1}})\geq c\,a^n$, hence
$h^0(X_{t_1},aK_{X_{t_1}}-F)\geq c'a^n$ for suitable $c,c'>0$ and
$a$ large enough. By the choice of $t_1$, every non zero section
of $H^0(X_{t_1},aK_{X_{t_1}}-F)$ can be extended into a section $s$ of
$H^0(X,aK_X-F)$, hence $aK_X=D+F$ where $D$ is the zero divisor of $s$. 
If necessary, we can eliminate any unwanted component $X_0$ in the
decomposition of $D$ by dividing $s$ with a suitable power of~$t$.\qed
\medskip

The next step is to construct hermitian metrics on $K_{X_0}$ and $K_X$, 
respectively, and to compare their multiplier ideal sheaves.
By the results of section 14.1, there exists a metric with minimal
singularities on $X_0$ (unique up to equivalence of singularities). 
We will denote by $\varphi_0$ the weight of this metric.
Similarly, there exists a metric with minimal singularities 
on every relatively compact neighborhood of $X_0$ in $X$, 
and by shrinking the base $\Delta$, we can suppose that this
metric exists on the whole space $X$. We will denote by $\varphi$ 
the {\it restriction to $X_0$} of the weight of this metric. By 
definition, $\varphi$ is at least as singular as $\varphi_0$ on $X_0$, 
hence by adding a constant, we may eventually assume that
$\varphi\le\varphi_0$.

On the other hand, Lemma 14.4.1 shows that we can choose an integer
$a\geq 2$ such that $aK_X=D+F$, where $D$ is an effective divisor on
$X$ which does not contain $X_0$ as a component, and $F$ is an ample 
line bundle on~$X$. After $a$ has been replaced with a sufficiently 
large multiple, we can make the following additional assumptions:
\smallskip

\noindent 
(14.4.2) {\it $F-K_X>K_X+nA$ for some very ample line bundle $A$ on $X$.}
\smallskip

\noindent 
(14.4.3) {\it there is a basis of sections of $H^0(X,F)_{|X_0}$
providing an embedding of $X_0$ onto a subvariety of projective
space.}

Let $s_D$ be the canonical section of the line bundle $\cO(D)$,
so that the divisor of $s_D$ is $D$. Let $u_1,\ldots,u_N\in H^0(X,F)$ 
be sections such that
$$
u_{1|X_0},\ldots,u_{N|X_0}
$$ 
form a basis of $H^0(X,F)_{|X_0}$. Since 
$s_Du_j\in H^0(X,aK_X)$ ($1\leq j\leq N$), we get a hermitian metric
$$
e^{-\psi}=\left({1\over|s_D|^2\sum|u_j|^2}\right)^{1\over a} 
$$ 
on the line bundle $K_X$. Moreover
$$
\psi={1\over a}\left(\log|s_D|^2+\chi\right)
$$ 
where $\chi=\log(\sum|u_j|^2)$ defines a smooth hermitian metric with
positive curvature on~$F$. Therefore, $i\ddbar\psi$ is a positive definite
current. Furthermore, the singularities of $\psi$ on $X$ are at least as
large as those of the weight $\varphi$ which is by definition the weight
with minimal singularities on~$X$. By adjusting again the weights with
constants, we can state:
\endproof

\claim{(14.4.4) Lemma} The hermitian metric $e^{-\psi}$ has positive 
definite curvature and\break \hbox{$\psi\leq\varphi\leq\varphi_0$} on $X_0$.
\endclaim

The crucial argument is a comparison result for the multiplier ideal sheaves
on $X_0$ defined by $\ell\varphi_0$ and $\ell\varphi$, respectively, 
when $\ell$ is large. In the sequel, the notation
$\cI((\ell+a-\varepsilon)\varphi+\varepsilon\psi)$
will stand for an ideal sheaf on $X_0$ (and not for an ideal sheaf on
$X$, even when the weight is possibly defined on the whole space~$X$).

\claim{(14.4.5) Proposition} Select $0<\varepsilon<1$ so small that
$e^{-\varepsilon\psi}$ is locally integrable on $X$ 
$($possibly after shrinking the base $\Delta)$ and
$e^{-\varepsilon\psi}_{|X_0}$ is locally integrable on~$X_0$. Then
$$
\cI((\ell-\varepsilon)\varphi_0+(a+\varepsilon)\psi)
\subset \cI((\ell-1+a-\varepsilon)\varphi+\varepsilon\psi)
$$
for every integer $\ell\geq 1$.
\endclaim

\begproof{} We argue by induction on $\ell$. For $\ell=1$,
Lemma 14.4.4 implies
$$(1-\varepsilon)\varphi_0+(a+\varepsilon)\psi\le 
(1-\varepsilon)\varphi_0+a\varphi+\varepsilon\psi\le
C+(a-\varepsilon)\varphi+\varepsilon\psi$$
since $\varphi_0$ is locally bounded above by a constant $C$.
Thus we get
$$
\cI((1-\varepsilon)\varphi_0+(a+\varepsilon)\psi)
\subset \cI((a-\varepsilon)\varphi+\varepsilon\psi),
$$
as desired.

Now, suppose that the induction step $\ell$ has been settled.  
Take an arbitrary germ of function $f$ in the ideal sheaf
$\cI((\ell+1-\varepsilon)\varphi_0 +(a+\varepsilon)\psi)$, defined on
a small neighborhood $U$ of a point $P\in X_0$. Fix a local non
vanishing holomorphic section $e$ of $(\ell+a)K_{X_0}$ on $U$. Then $s=fe$ 
is a section of
$$
\cO((\ell+a)K_{X_0})\otimes\cI((\ell+1-\varepsilon)\varphi_0
+(a+\varepsilon)\psi)
$$ 
on $U$. Observe that, given an arbitrary plurisubharmonic function~$\xi$, 
we have
$$\cI(\xi+\log|s_D|^2)=\cI(\xi)\otimes \cO(-D).$$
Writing $aK_{X_0}=(D+F)_{|X_0}$ thanks to Lemma 14.4.1 and
$a\psi=\log|s_D|^2+\chi$ by definition of $\psi$, we can
reinterpret $s$ as a section of
$$
\eqalign{
\cO(\ell K_{X_0}+(D+F)_{|X_0})&{}\otimes
\cI((\ell+1-\varepsilon)\varphi_0+\varepsilon\psi+\log|s_D|^2+\chi)\cr
&=\cO((\ell+2)K_{X_0}+E_{|X_0})\otimes
\cI((\ell+1-\varepsilon)\varphi_0+\varepsilon\psi),
\cr}
$$
where $E=F-2K_X$ (since $\chi$ is smooth, adding $\chi$ does not change 
the multiplier ideal sheaf). Let us observe that 
$(\ell+1-\varepsilon)\varphi_0+\varepsilon\psi$ defines a 
hermitian metric with positive curvature on $(\ell+1)K_{X_0}$. By
the above hypothesis (14.4.2) and Proposition~14.2.1 applied with $Y=X_0$ 
and $L=(\ell+1)K_{X_0}$, we conclude that
$$
\cO((\ell+2)K_{X_0}+E_{|X_0})\otimes
\cI((\ell+1-\varepsilon)\varphi_0+\varepsilon\psi)
$$
is generated by global sections on~$X_0$. We can therefore, without loss
of generality, restrict ourselves to the case of germs $f$ such that $fe$
coincides on $U$ with a global section
$$
s\in H^0\left(X_0,\cO((\ell+2)K_{X_0}+E_{|X_0})\otimes
\cI((\ell+1-\varepsilon)\varphi_0+\varepsilon\psi)
\right).$$
By reversing the order of calculations, we find
$$
\eqalign{
H^0\left(X_0,\cO((\ell+2)K_{X_0}\right.&{}+E_{|X_0})\otimes
\left.\cI((\ell+1-\varepsilon)\varphi_0+\varepsilon\psi)\right)\cr
&=H^0\left(X_0,\cO((\ell+a)K_{X_0})\otimes
\cI((\ell+1-\varepsilon)\varphi_0+(a+\varepsilon)\psi)\right)\cr
&\subset H^0\left(X_0,\cO((\ell+a)K_{X_0})\otimes
\cI((\ell-\varepsilon)\varphi_0+(a+\varepsilon)\psi)\right)\cr
&\subset H^0\left(X_0,\cO((\ell+a)K_{X_0})\otimes
\cI((\ell+a-1-\varepsilon)\varphi+\varepsilon\psi)\right),\cr}
$$
[the first inclusion is obtained by neglecting the term $\varphi_0$
in the weight, and the second one is a consequence of the induction
hypothesis for step $\ell$]. Now, the weight
$$
(\ell+a-1-\varepsilon)\varphi+\varepsilon\psi
$$
defines a hermitian metric with positive definite curvature on 
$L=(\ell+a-1)K_X$, and Theorem 14.3.1 implies that $s$ can be extended 
into a global section $\tilde s\in H^0(X,(\ell+a)K_X)$ (maybe after
shrinking~$\Delta$). The definition of $\varphi$ implies
$|\tilde s|^2\leq C\,e^{(\ell+a)\varphi}$, hence
$$
|\tilde s|^2e^{-(\ell+a)\varphi-\varepsilon\psi}\leq C\,e^{-\varepsilon\psi}
$$
is integrable on $X$. from there, we conclude that
$$
fe=s=\tilde s_{|X_0}\in\cO((\ell+a)K_{X_0})\otimes
\cI((\ell+a)\varphi+\varepsilon\psi),
$$
hence $f\in \cI((\ell+a)\varphi+\varepsilon\psi)$. The step $\ell+1$
of the induction is proved.
\endproof

\titlec{14.5.}{Proof of the invariance of plurigenera.} 
Fix an integer $m>0$ and an arbitrary section $s\in H^0(X_0,mK_{X_0})$.
By definition of $\varphi_0$, we have
$|s|^2\le Ce^{m\varphi_0}$ on $X_0$. If $s_D$ is the canonical section 
of $\cO(-D)$ of divisor $D$, we conclude that $s^\ell s_D$ 
is locally $L^2$ with respect to the weight
$e^{-\ell m\varphi_0-(a+\varepsilon)\psi}$, for $a\psi$ has the same
singularities as $\log|s_D|^2$ and $e^{-\varepsilon\psi}$ is supposed
to be locally integrable on~$X_0$. The functions $s^\ell s_D$ are
locally in the ideal sheaf
$\cI((\ell m-\varepsilon)\varphi_0+(a+\varepsilon)\psi)$. By
Proposition 14.4.5, they belong to 
$\cI((\ell m-1-\varepsilon)\varphi+\varepsilon\psi)$, i.e.\
$$
\int_U|s^\ell s_D|^2e^{-(\ell m-1-\varepsilon)\varphi-\varepsilon\psi}
<+\infty
$$
on each sufficiently small open set $U$ such that $K_{X_0}|U$ and
$\cO(-D)|U$ are trivial. In an equivalent way, we can write
$$
\int_U|s|^{2\ell}e^{-(\ell m-1-\varepsilon)\varphi+(a-\varepsilon)\psi}
<+\infty.
$$
Take $\ell$ large enough so that $a/(\ell-1)\leq\varepsilon$. Then
$e^{\varphi-{a\over\ell-1}\psi}\le Ce^{-\varepsilon\psi}$ is integrable
on $U$. By H\"older inequality with conjugate exponents $\ell$, 
$\ell'=\ell/(\ell-1)$, we find
$$
+\infty>
\left(\int_U|s|^{2\ell}e^{-(\ell m-1-\varepsilon)\varphi+(a-\varepsilon)\psi}
\right)^{1/\ell}
\left(\int_U e^{\varphi-{a\over \ell-1}\psi}\right)^{(\ell-1)/\ell}
\phantom{-------}
$$
$$
~~~~~{}\geq\int_U |s|^2e^{-(m-{1\over\ell}-{\varepsilon\over\ell})\varphi+
({a\over\ell}-{\varepsilon\over\ell})\psi}
e^{(1-{1\over\ell})\varphi-{a\over\ell}\psi}
=\int_U |s|^2e^{-(m-1-\delta)\varphi-\delta\psi}
$$
with $\delta=\varepsilon/\ell$. We can consider $s$ as a
section of $K_{X_0}+L_{|X_0}$ with \hbox{$L=(m-1)K_X$}. The weight
$(m-1-\delta)\varphi- \delta\psi$ defines a hermitian metric
on $L$ with positive definite curvature, and $s$ is globally $L^2$
with respect to this metric.  By the Ohsawa-Takegoshi-Manivel extension
theorem, we can extend $s$ into a section 
$\tilde s\in H^0(X,K_X+L)=H^0(X,mK_X)$, possibly after
shrinking $\Delta$ again. This achieves the proof of the main theorem.\qed

\titleb{15.}{Subadditivity of Multiplier Ideals and Fujita's\hfil\break 
Approximate Zariski Decompostion}
\titlebrunning{15. Subadditivity of Multiplier Ideals and Fujita's
Approximate Zariski Decompostion}

We first notice the following basic restriction formula, which is just
a rephrasing of the Ohsawa-Takegoshi extension theorem.

\claim{(15.1) Restriction formula} Let $\varphi$ be a
plurisubharmonic function on a complex mani\-fold $X$, and let $Y\subset
X$ be a submanifold. Then
$$\cI(\varphi_{|Y})\subset\cI(\varphi)_{|Y}.$$
\endclaim

Thus, in some sense, the singularities of $\varphi$ can only get worse
if we restrict to a submanifold (if the restriction of $\varphi$ to some
connected component of $Y$ is identically $-\infty$, we agree that the 
corresponding multiplier ideal sheaf is zero). The proof is straightforward
and just amounts to extending locally a germ of function $f$ on $Y$ near 
a point $y_0\in Y$ to a function $\tilde f$ on a small Stein neighborhood 
of $y_0$ in $X$, which is possible by the Ohsawa-Takegoshi extension theorem. 
As a direct consequence, we get:

\claim{(15.2) Subadditivity Theorem}
\item{\rm(i)} Let $X_1$, $X_2$  be complex manifolds,
$\pi_i:X_1\times X_2\to X_i$, $i=1,2$ the projections, and
let $\varphi_i$ be a plurisubharmonic function on~$X_i$.
Then
$$
\cI(\varphi_1\circ \pi_1 + \varphi_2\circ \pi_2) = \pi_1^\star
\cI(\varphi_1) \cdot  \pi_2^\star\cI(\varphi_2).
$$
\item{\rm(ii)} Let $X$ be a complex manifold and let
$\varphi$, $\psi$ be plurisubharmonic functions on $X$.
Then
$$
\cI(\varphi + \psi) \subset \cI(\varphi) \cdot \cI(\psi)
$$
\endclaim

\begproof{} (i) Let us fix two relatively
compact Stein open subsets $U_1\subset X_1$, 
$U_2\subset X_2$. Then $\cH^2(U_1\times U_2,
\varphi_1\circ \pi_1 + \varphi_2\circ \pi_2, \pi_1^\star
dV_1\otimes \pi_2^\star dV_2)$ is the Hilbert tensor
product of $\cH^2(U_1,\varphi_1,dV_1)$ and
$\cH^2(U_2,\varphi_2,dV_2)$, and admits $(f'_k\boxtimes
f''_l)$ as a Hilbert  basis, where $(f'_k)$ and
$(f''_l)$ are respective Hilbert bases. Since
$\cI(\varphi_1\circ \pi_1 + \varphi_2\circ \pi_2)_{|U_1\times
U_2}$ is generated  as an $\cO_{U_1\times U_2}$ module
by the $(f'_k\boxtimes f''_l)$ (Proposition 5.7), we  conclude that (i)
holds true.
\medskip

\noindent
(ii) We apply (i) to $X_1=X_2=X$ and the restriction
formula to $Y={}$diagonal of $X\times X$. Then
$$
\eqalign{
\cI(\varphi+\psi)
&=\cI\big((\varphi\circ \pi_1 + \psi\circ \pi_2)_{|Y}\big)
\subset \cI\big(\varphi\circ \pi_1 + \psi\circ \pi_2\big)_{|Y}\cr
&=\Big(\pi_1^\star\cI(\varphi)\otimes \pi_2^\star\cI(\psi)\Big)_{|Y}
=\cI(\varphi)\cdot\cI(\psi).\cr}
$$
\endproof

\claim{(15.3) Proposition} Let $f:X\to Y$ be an arbitrary holomorphic 
map, and let $\varphi$ be a plurisubharmonic function on~$Y$. Then
$\cI(\varphi\circ f)\subset f^\star\cI(\varphi)$.
\endclaim

\begproof{} Let 
$$
\Gamma_f=\{(x,f(x)\,;\,x\in X\}\subset X\times Y
$$ 
be the graph of $f$, and let $\pi_X:X\times Y\to X$, $\pi_Y:X\times
Y\to Y$ be the natural projections. Then we can view $\varphi\circ f$
as the restriction of $\varphi\circ\pi_Y$ to $\Gamma_f$, as $\pi_X$ is
a biholomorphism from $\Gamma_f$ to $X$. Hence the restriction formula
implies
$$
\cI(\varphi\circ f)=\cI\big((\varphi\circ\pi_Y)_{|\Gamma_f}\big)
\subset\cI(\varphi\circ\pi_Y)_{|\Gamma_f}
=\big(\pi_Y^\star\cI(\varphi)\big)_{|\Gamma_f}=f^\star\cI(\varphi).
\eqno\square
$$
\endproof

As an application of subadditivity, we now reprove a result of Fujita
[Fuj93], relating the growth of sections of multiples of a line bundle
to the Chern numbers of its ``largest nef part''. Fujita's original
proof is by contradiction, using the Hodge index theorem and
intersection inequalities. The present method arose in
the course of joint work with R.~Lazarsfeld~[Laz99].

Let $X$ be a projective $n$-dimensional algebraic variety and $L$
a line bundle over~$X$. We define the {\it volume} of $L$ to be
$$
v(L)=\limsup_{k\to+\infty}{n!\over k^n}h^0(X,kL)\in[0,+\infty[.
$$
The line bundle is said to be {\it big} if $v(L)>0$. If $L$ is ample, 
we have $h^q(X,kL)=0$ for $q\ge 1$ and $k\gg 1$ by the Kodaira-Serre
vanishing theorem, hence 
$$
h^0(X,kL)\sim\chi(X,kL)\sim{L^n\over n!}k^n
$$ 
by the Riemann-Roch formula.  Thus $v(L)=L^n\,({}=c_1(L)^n)$ if $L$ is
ample. This is still true if $L$ is nef (numerically effective), i.e.\
if $L\cdot C\ge 0$ for every effective curve~$C$. In fact, one can show 
that $h^q(X,kL)=O(k^{n-q})$ in that case. The following
well-known proposition characterizes big line bundles.

\claim{(15.4) Proposition} The line bundle $L$ is big if and only if there
is a multiple $m_0L$ such that $m_0L=E+A$, where $E$ is an effective divisor
and $A$ an ample divisor.
\endclaim

\begproof{} If the condition is satisfied, the decomposition 
$km_0L=kE+kA$ gives
rise to an injection $H^0(X,kA)\hookrightarrow H^0(X,km_0L)$, thus
$v(L)\ge m_0^{-n}v(A)>0$. Conversely, assume that $L$ is big, and take 
$A$ to be a very ample nonsingular divisor in $X$. The exact sequence
$$
0\lra \cO_X(kL-A)\lra \cO_X(kL)\lra \cO_A(kL_{|A})\lra 0
$$
gives rise to a cohomology exact sequence
$$
0\to H^0(X,kL-A) \lra H^0(X,kL) \lra H^0(A,kL_{|A}), 
$$
and $h^0(A,kL_{|A})=O(k^{n-1})$ since $\dim A=n-1$. Now, the assumption that
$L$ is big implies that $h^0(X,kL)> ck^n$ for infinitely many $k$, hence
$H^0(X,m_0L-A)\ne 0$ for some large integer $m_0$. If $E$ is the divisor 
of a section in $H^0(X,m_0L-A)$, we find $m_0L-A=E$, as required.\qed
\endproof

\claim{(15.5) Lemma} Let $G$ be an arbitrary line bundle. For every
$\varepsilon>0$, there exists a positive integer $m$ and a sequence
$\ell_\nu\uparrow+\infty$ such that
$$
h^0\big(X,\ell_\nu(mL-G)\big)\ge {\ell_\nu^mm^n\over n!}
\big(v(L)-\varepsilon\big),
$$
in other words, $v(mL-G)\ge m^n(v(L)-\varepsilon)$ for $m$ large enough.
\endclaim

\begproof{} Clearly, $v(mL-G)\ge v(mL-(G+E))$ for every effective divisor
$E$. We can take $E$ so large that $G+E$ is very ample, and we are thus 
reduced to the case where $G$ is very ample by replacing $G$ with $G+E$.
By definition of $v(L)$, there exists a sequence $k_\nu\uparrow+\infty$
such that
$$
h^0(X,k_\nu L)\ge {k_\nu^n\over n!}\Big(v(L)-{\varepsilon\over 2}\Big).
$$
We take $m\gg 1$ (to be precisely chosen later), and $\ell_\nu=
\big[{k_\nu\over m}\big]$, so that $k_\nu=\ell_\nu m+r_\nu$,
$0\le r_\nu<m$. Then
$$
\ell_\nu(mL-G)=k_\nu L-(r_\nu L+\ell_\nu G).
$$
Fix a constant $a\in\bN$ such that $aG-L$ is an effective divisor. Then
$r_\nu L\le maG$ (with respect to the cone of effective divisors), hence
$$
h^0\big(X,\ell_\nu(mL-G)\big)\ge
h^0\big(X,k_\nu L-(\ell_\nu+am)G\big).
$$
We select a smooth divisor $D$ in the very ample linear system
$|G|$. By looking at global sections associated with the exact
sequences of sheaves
$$
0\to\cO(-(j+1)D)\otimes\cO(k_\nu L)\to\cO(-jD)\otimes\cO(k_\nu L)
\to\cO_D(k_\nu L-jD)\to 0,
$$
$0\le j<s$, we infer inductively that
$$
\eqalign{
h^0(X,k_\nu L-sD)
&\ge h^0(X,k_\nu L)-\sum_{0\le j< s}h^0\big(D,\cO_D(k_\nu L-jD)\big)\cr
&\ge h^0(X,k_\nu L)- s\,h^0\big(D,k_\nu L_{|D})\cr
&\ge {k_\nu^n\over n!}\Big(v(L)-{\varepsilon\over 2}\Big) -s\,Ck_\nu^{n-1}
\cr}
$$
where $C$ depends only on $L$ and~$G$. Hence, by putting $s=\ell_\nu+am$,
we get
$$
\eqalign{
h^0\big(X,\ell_\nu(mL-G)\big) 
&\ge {k_\nu^n\over n!}\Big(v(L)-{\varepsilon\over 2}\Big) 
-C(\ell_\nu+am)k_\nu^{n-1}\cr
&\ge{\ell_\nu^nm^n\over n!}\Big(v(L)-{\varepsilon\over 2}\Big)
-C(\ell_\nu+am)(\ell_\nu+1)^{n-1}m^{n-1}\cr}
$$
and the desired conclusion follows by taking $\ell_\nu\gg m\gg 1$.\qed
\endproof

We are now ready to prove Fujita's decomposition theorem, as reproved
in [DEL00].

\claim{(15.6) Theorem (Fujita)} Let $L$ be a big line bundle. Then for 
every $\varepsilon>0$, there exists a modification $\mu:\wt X\to X$ and
a decomposition $\mu^\star L=E+A$, where $E$ is an effective $\bQ$-divisor
and $A$ an ample $\bQ$-divisor, such that $A^n>v(L)-\varepsilon$.
\endclaim

\claim{(15.7) Remark} {\rm Of course, if $\mu^\star L= E+A$ with $E$ 
effective and $A$ nef, we get an injection
$$
H^0(\wt X,kA)\hookrightarrow H^0(\wt X, kE+kA)=H^0(\wt X, k\mu^\star L)=
H^0(X,kL)
$$
for every integer $k$ which is a multiple of the denominator of $E$, hence
$A^n\le v(L)$.}
\endclaim

\claim{(15.8) Remark} {\rm Once Theorem 15.6 is proved, the same kind
of argument easily shows that
$$
v(L)=\lim_{k\to +\infty}{n!\over k^n}h^0(X,kL),
$$
because the formula is true for every ample line bundle $A$.}
\endclaim

\begproof{of Theorem 15.6.} It is enough to prove the theorem with $A$
being a big and nef divisor. In fact, Proposition 15.4 then shows that we can
write $A=E'+A'$ where $E'$ is an effective $\bQ$-divisor and $A'$ an
ample $\bQ$-divisor, hence
$$
E+A=E+\varepsilon E'+(1-\varepsilon)A+\varepsilon A'
$$
where $A''=(1-\varepsilon)A+\varepsilon A'$ is ample and the intersection
number $A^{\prime\prime n}$ approaches $A^n$ as closely as we want.
Let $G$ be as in Lemma 14.2.1 (uniform global generation). Lemma 15.5 
implies that
$v(mL-G)>m^n(v(L)-\varepsilon)$ for $m$ large. By Theorem~14.1.2, there
exists a hermitian metric $h_m$ of weight $\varphi_m$ on $mL-G$ such that
$$
H^0\big(X,\ell(mL-G)\big)=H^0\big(X,\ell(mL-G)\otimes\cI(\ell\varphi_m)\big)
$$
for every $\ell\ge 0$. We take a smooth modification $\mu:\wt X\to X$ such
that 
$$
\mu^\star\cI(\varphi_m)=\cO_{\wt X}(-E)
$$
is an invertible ideal sheaf in $\cO_{\wt X}$. This is possible by
taking the blow-up of $X$ with respect to the ideal $\cI(\varphi_m)$
and by resolving singularities (Hironaka [Hir64]). Lemma 14.2.1 applied 
to $L'=mL-G$
implies that $\cO(mL)\otimes\cI(\varphi_m)$ is generated by its
global sections, hence its pull-back $\cO(m\,\mu^\star L-E)$ is 
also generated. This implies
$$
m\,\mu^\star L=E+A
$$ 
where $E$ is an effective divisor and $A$ is a nef (semi-ample)
divisor in~$\wt X$. We find
$$
\eqalign{
H^0(\wt X,\ell A)
  &=H^0\big(\wt X,\ell(m\,\mu^\star L-E)\big)\cr
  &\supset H^0\big(\wt X,\mu^\star\big(\cO(\ell mL)\otimes
       \cI(\varphi_m)^\ell\big)\big)\cr
  &\supset H^0\big(\wt X,\mu^\star\big(\cO(\ell mL)\otimes
       \cI(\ell\varphi_m)\big)\big),\cr}
$$
thanks to the subadditivity property of multiplier ideals.
Moreover, the direct image 
$\mu_\star\mu^\star\cI(\ell\varphi_m)$ coincides with the integral
closure of $\cI(\ell\varphi_m)$, hence with $\cI(\ell\varphi_m)$,
because a multiplier ideal sheaf is always integrally closed.
From this we infer
$$
\eqalign{
H^0(\wt X,\ell A)
  &\supset H^0\big(X,\cO(\ell mL)\otimes \cI(\ell\varphi_m)\big)\cr
  &\supset H^0\big(X,\cO(\ell(mL-G))\otimes \cI(\ell\varphi_m)\big)\cr
  &= H^0\big(X,\cO(\ell(mL-G))\big).\cr}
$$
By Lemma 15.5, we find
$$
h^0(\wt X,\ell A)\ge{\ell^n\over n!}m^n\big(v(L)-\varepsilon\big)
$$
for infinitely many $\ell$, therefore $v(A)=A^n\ge m^n(v(L)-\varepsilon)$.
Theorem 15.6 is proved, up to a minor change of notation
$E\mapsto{1\over m}E$, $A\mapsto{1\over m}A$.\qed
\endproof

We conclude by using Fujita's theorem to establish a geometric
interpretation of the volume $v(L)$. Suppose as above that $X$ is a
smooth projective variety of dimension $n$, and that $L$ is a big line
bundle on $X$. Given a large integer $k \gg 0$, denote by $B_k
\subset X$ the base-locus of the linear system $| kL |$. The
{\it moving self-intersection number} $(kL)^{[n]}$ of
$|kL|$ is defined by choosing $n$ general divisors $D_1, \dots , D_n
\in |kL|$ and putting
$$( kL)^{[n]} = \# \Big( D_1 \cap \ldots
\cap D_n \cap ( X - B_k)  \Big).
$$
In other words, we simply count the number of intersection points
away from the base locus of $n$ general divisors in the linear system
$|kL|$. This notion arises for example in Matsusaka's proof of his
``big theorem". We show that the volume $v(L)$ of $L$ 
measures the rate of growth with respect to $k$ of these moving 
self-intersection numbers:

\claim{(15.9) Proposition} One has
$$v(L) = \limsup_{k \to \infty} { (kL)^{[n]} \over k^n}.$$
\endclaim

\begproof{} We start by interpreting 
$(kL)^{[n]}$ geometrically. Let $\mu_k: X_k \lra X$
be a modification of $|kL|$ such that $\mu_k^\star |kL| = |V_k| +  F_k $, 
where
$$ P_k := \mu_k^\star (kL) - F_k$$  
is generated by sections, and
$H^0(X,\cO_X(kL)) = V_k = H^0(X_k,\cO_{X_k}(P_k))$, so that $B_k =
\mu_k(F_k)$. Then evidently
$(kL)^{[n]}$ counts the number of intersection points
of $n$ general divisors in $P_k$, and consequently
$$(kL)^{[n]} = (P_k)^n .$$  
Since then $P_k$ is big (and nef) for $k\gg 0$, 
we have  $v(P_k)=(P_k)^n$. Also, $v(kL) \ge v(P_k)$ 
since $P_k$ embeds in $\mu_k^\star(k L)$.  Hence
$$
v(kL) \ge  ( kL)^{[n]}\qquad \forall k \gg 0.
$$
On the other hand, an easy argument in the spirit of Lemma (15.5)
shows that $v(kL) = k^n \cdot v(L)$
(cf.\ also [ELN96], Lemma 3.4), and so we conclude that
$$
v(L) \ge { (kL)^{[n]} \over k^n}. \leqno(15.10)
$$
for every $k \gg 0$. 

For the reverse inequality we use Fujita's theorem. Fix
$\varepsilon > 0$, and consider the  decomposition $\mu^\star
L = A + E$ on $\mu : \wt X \lra X$ constructed in
Fujita's theorem. Let $k$ be any positive integer
such that $kA$ is integral and globally generated. By taking a
common resolution we can assume that $X_k$ dominates
$\wt X$, and hence we can write 
$$
\mu_k^\star  kL \sim  A_k + E_k
$$
with $A_k$ globally generated and 
$$
(A_k)^n\ge k^n \cdot ( v(L) - \varepsilon ).
$$  
But then $A_k$ embeds in $P_k$ and both $\cO(A_k)$ and $\cO(P_k)$
are globally generated, consequently 
$$
(A_k)^n \le (P_k)^n =  ( kL)^{[n]}.
$$
Therefore
$$
{(kL)^{[n]}\over k^n} \ge  v(L) - \varepsilon. \leqno(15.11)
$$
But (15.11) holds for any sufficiently large and divisible $k$, and in 
view of (15.10) the Proposition follows.\qed
\endproof

\titleb{16.}{Hard Lefschetz Theorem with Multiplier Ideal Sheaves}
 
\titlec{16.1.}{Main statement}

The goal of this section is to prove the following surjectivity theorem,
which can be seen as an extension of the hard Lefschetz theorem. We
closely follow the exposition of [DPS00].

\claim{(16.1.1) Theorem} Let $(L,h)$ be a pseudo-effective line bundle on a
compact K\"ahler manifold $(X,\omega)$, of dimension $n$, let 
$\Theta_h(L)\ge 0$ be its curvature current and $\cI(h)$ the 
associated multiplier ideal sheaf. Then, the wedge multiplication 
operator $\omega^q\wedge\bu$ induces a surjective morphism
$$
\Phi^q_{\omega,h}:
H^0(X,\Omega_X^{n-q}\otimes L\otimes\cI(h))\lra
H^q(X,\Omega_X^n\otimes L\otimes\cI(h)).
$$
\endclaim

\noindent
The special case when $L$ is nef is due to Takegoshi [Tak97]. An even
more special case is when $L$ is semi-positive, i.e.\ possesses a
smooth metric with semi-positive curvature.
In that case the multiple ideal sheaf $\cI(h)$ coincides with $\cO_X$
and we get the following consequence already observed by Mourougane
[Mou99].

\claim{(16.1.2) Corollary} Let $(L,h)$ be a semi-positive line bundle
on a compact K\"ahler mani\-fold $(X,\omega)$ of dimension~$n$. Then, the wedge
multiplication operator $\omega^q\wedge\bu$ induces a surjective morphism
$$
\Phi^q_\omega:H^0(X,\Omega_X^{n-q}\otimes L)\lra
H^q(X,\Omega_X^n\otimes L).
$$
\endclaim

The proof of Theorem (16.1.1) is based on the Bochner formula, combined
with a use of harmonic forms with values in the hermitian line bundle
$(L,h)$. The method can be applied only after $h$ has been made smooth
at least in the complement of an analytic set. However, we have to
accept singularities even in the regularized metrics because only a
very small incompressible loss of positivity is acceptable in the
Bochner estimate (by the results of [Dem92], singularities can only be
removed at the expense of a fixed loss of positivity). Also, we need
the multiplier ideal sheaves to be preserved by the smoothing process.
This is possible thanks to a suitable ``equisingular'' regularization
process.

\titlec{16.2.}{Equisingular approximations of quasi plurisubharmonic
functions}

A quasi-plurisubharmonic (quasi-psh) function is by definition a
function $\varphi$ which is locally equal to the sum of a psh function
and of a smooth function, or equivalently, a locally integrable function
$\varphi$ such that $i\ddbar\varphi$ is locally bounded below by
$-C\omega$ where $\omega$ is a hermitian metric and $C$ a constant.
We say that $\varphi$ has logarithmic poles if $\varphi$ is locally
bounded outside an analytic set $A$ and has singularities of the form
$$\varphi(z)=c\log\sum_k|g_k|^2+O(1)$$
with $c>0$ and $g_k$ holomorphic, on a neighborhood of every point
of~$A$. Our goal is to show the following

\claim{(16.2.1) Theorem} Let $T=\alpha+i\ddbar\varphi$ be a closed
$(1,1)$-current on a compact hermitian manifold $(X,\omega)$, where
$\alpha$ is a smooth closed $(1,1)$-form and $\varphi$ a quasi-psh
function. Let $\gamma$ be a continuous real $(1,1)$-form such that
$T\ge\gamma$. Then one can write
\hbox{$\varphi=\lim_{\nu\to+\infty}\varphi_\nu$} where
\smallskip
\item{\rm a)} $\varphi_\nu$ is smooth in the complement $X\ssm Z_\nu$
of an analytic set $Z_\nu\subset X\,;$
\smallskip
\item{\rm b)} $(\varphi_\nu)$ is a decreasing sequence, and $Z_\nu\subset
Z_{\nu+1}$ for all~$\nu\,;$
\smallskip
\item{\rm c)} $\int_X(e^{-2\varphi}-e^{-2\varphi_\nu})dV_\omega$
is finite for every $\nu$ and converges to $0$ as $\nu\to+\infty\,;$
\smallskip
\item{\rm d)} $\cI(\varphi_\nu)=\cI(\varphi)$ for all $\nu$
$($``equisingularity''$)\,;$
\smallskip
\item{\rm e)} $T_\nu=\alpha+i\ddbar\varphi_\nu$ satisfies $T_\nu\ge
\gamma-\varepsilon_\nu\omega$, where $\lim_{\nu\to+\infty}\varepsilon_\nu=0$.
\vskip0pt
\endclaim

\claim{(16.2.2) Remark} {\rm It would be interesting to know whether the
$\varphi_\nu$ can be taken to have logarithmic poles along $Z_\nu$.
Unfortunately, the proof given below destroys this property in the
last step. Getting it to hold true seems to be more or less
equivalent to proving the semi-continuity property
$$\lim_{\varepsilon\to 0_+}
\cI((1+\varepsilon)\varphi)=\cI(\varphi).$$
Actually, this can be checked in dimensions $1$ and $2$, but is unknown in
higher dimensions (and probably quite hard to establish).}
\endclaim


\proof\ {\it of Theorem 16.2.1}. Clearly, by replacing $T$ with
$T-\alpha$ and $\gamma$ with $\gamma-\alpha$, we may assume that
$\alpha=0$ and $T=i\ddbar\varphi\ge \gamma$. We divide the proof in
four steps.
\medskip

\noindent
{\it Step 1. Approximation by quasi-psh functions with logarithmic
poles.}

\noindent
By [Dem92], there is a decreasing sequence $(\psi_\nu)$ of quasi-psh
functions with logarithmic poles such that $\varphi=\lim\psi_\nu$ and
$i\ddbar\psi_\nu\ge\gamma- \varepsilon_\nu\omega$. We need a little bit
more information on those functions, hence we first recall the main
techniques used for the construction of~$(\psi_\nu)$. For $\varepsilon>0$
given, fix a covering of $X$ by open balls $B_j=\{|z^{(j)}|<r_j\}$ with
coordinates $z^{(j)}=(z^{(j)}_1,\ldots, z^{(j)}_n)$, such that
$$
0\le\gamma+c_j\,i\ddbar|z^{(j)}|^2\le \varepsilon\omega
\qquad\hbox{on}\quad B_j,
\leqno(16.2.3)
$$
for some real number~$c_j$. This is possible by selecting coordinates
in which $\gamma$ is diagonalized at the center of the ball, and by
taking the radii $r_j>0$ small enough (thanks to the fact that $\gamma$
is continuous). We may assume that these coordinates come from a finite
sample of coordinates patches covering $X$, on which we perform suitable
linear coordinate changes (by invertible matrices lying in some
compact subset of the complex linear group). By taking additional balls,
we may also assume that $X=\bigcup B''_j$ where
$$
B''_j\compact B'_j\compact B_j
$$
are concentric balls $B'_j=\{|z^{(j)}|<r'_j=r_j/2\}$,
$B''_j=\{|z^{(j)}|<r''_j=r_j/4\}$. We define
$$
\psi_{\varepsilon,\nu,j}=
{1\over 2\nu}\log\sum_{k\in\bN}|f_{\nu,j,k}|^2-c_j|z^{(j)}|^2
\qquad\hbox{on}\quad B_j,
\leqno(16.2.4)
$$
where $(f_{\nu,j,k})_{k\in\bN}$ is an orthonormal basis of the Hilbert
space ${\cal H}_{\nu,j}$ of holomorphic functions on $B_j$ with finite
$L^2$ norm
$$
\Vert u\Vert^2 = \int_{B_j} |u|^2 e^{-2\nu(\varphi+c_j|z^{(j)}|^2)}
d\lambda(z^{(j)}).
$$
(The dependence of $\psi_{\varepsilon,\nu,j}$ on $\varepsilon$ is through
the choice of the open covering~$(B_j)$). Observe that the choice of
$c_j$ in (16.2.3) guarantees that $\varphi+c_j|z^{(j)}|^2$ is
plurisubharmonic on $B_j$, and notice also that
$$
\sum_{k\in\bN}|f_{\nu,j,k}(z)|^2=\sup_{f\in\cH_{\nu,j},\,\Vert f\Vert\le 1}
|f(z)|^2
\leqno(16.2.5)
$$
is the square of the norm of the continuous linear form $\cH_{\nu,j}\to\bC$,
$f\mapsto f(z)$. We claim that there exist
constants $C_i$, $i=1,2,\ldots$ depending only on $X$ and $\gamma$
(thus independent of $\varepsilon$ and $\nu$), such that the following
uniform estimates hold:
$$
\leqalignno{\qquad\qquad
&i\ddbar\psi_{\varepsilon,\nu,j}\ge -c_j\,i\ddbar|z^{(j)}|^2\ge\gamma-
\varepsilon\omega\qquad\hbox{on}~~B'_j~~(B'_j\compact B_j),
&(16.2.6)\cr
&\varphi(z)\le\psi_{\varepsilon,\nu,j}(z)\le\sup_{|\zeta-z|\le r}\varphi(\zeta)
+{n\over\nu}\log{C_1\over r}+C_2r^2
\qquad\hbox{$\forall z\in B'_j$,~ $r<r_j-r'_j$,}
&(16.2.7)\cr
&|\psi_{\varepsilon,\nu,j}-\psi_{\varepsilon,\nu,k}|\le {C_3\over\nu}
+C_4\varepsilon\big(\min(r_j,r_k)\big)^2\qquad
\hbox{on}~~B'_j\cap B'_k.
&(16.2.8)\cr}
$$
Actually, the Hessian estimate (16.2.6) is obvious from (16.2.3) and (16.2.4).
As in the proof of ([Dem92], Prop.~3.1), (16.2.7) results from the
Ohsawa-Takegoshi $L^2$ extension theorem (left hand inequality)
and from the mean value inequality (right hand inequa\-lity).
Finally, as in ([Dem92], Lemma~3.6 and Lemma~4.6), (16.2.8) is a
consequence of H\"ormander's $L^2$ estimates. We briefly sketch the idea.
Assume that the balls $B_j$ are small enough, so that the
coordinates $z^{(j)}$ are still defined on a neighborhood of all
balls $\overline B_k$ which intersect~$B_j$ (these coordinates can be
taken to be linear transforms of coordinates belonging to a fixed
finite set of coordinate patches covering~$X$, selected once for all).
Fix a point
$z_0\in B'_j\cap B'_k$. By (16.2.4) and (16.2.5), we have
$$
\psi_{\varepsilon,\nu,j}(z_0)={1\over \nu}\log|f(z_0)|-c_j|z^{(j)}|^2
$$
for some holomorphic function $f$ on $B_j$ with $\Vert f\Vert=1$. We
consider the weight function
$$
\Phi(z)=2\nu(\varphi(z)+c_k|z^{(k)}|^2)+2n\log|z^{(k)}-z^{(k)}_0|,
$$
on both $B_j$ and $B_k$. The trouble is that a priori we have to deal with
different weights, hence a comparison of weights is needed. By the Taylor
formula applied at $z_0$, we get
$$
\Big|c_k|z^{(k)}-z^{(k)}_0|^2 - c_j|z^{(j)}-z^{(j)}_0|^2\Big|\le
C\varepsilon\big(\min(r_j,r_k)\big)^2\qquad\hbox{on $B_j\cap B_k$}
$$
[the only nonzero term of degree 2 has type $(1,1)$ and its Hessian satisfies
$$
-\varepsilon\omega\le i\ddbar(c_k|z^{(k)}|^2-c_j|z^{(j)}|^2)\le
\varepsilon\omega
$$
by (16.2.3); we may suppose $r_j\ll\varepsilon$ so that the terms of order
$3$ and more are negligible]. By writing
$|z^{(j)}|^2=|z^{(j)}-z^{(j)}_0|^2+|z^{(j)}_0|^2+
2\Re\langle z^{(j)}-z^{(j)}_0,z^{(j)}_0\rangle$, we obtain
$$\eqalign{
c_k|z^{(k)}|^2-c_j|z^{(j)}|^2
&=2c_k\Re\langle z^{(k)}-z^{(k)}_0,z^{(k)}_0\rangle
-2c_j\Re\langle z^{(j)}-z^{(j)}_0,z^{(j)}_0\rangle\cr
&\qquad{}+c_k|z^{(k)}_0|^2-c_j|z^{(j)}_0|^2
\pm C\varepsilon(\min(r_j,r_k))^2.\cr}
$$
We use a cut-off function $\theta$ equal to $1$ in a neighborhood of
$z_0$ and with support in $B_j\cap B_k$; as $z_0\in B'_j\cap B'_k$, the
function $\theta$ can be taken to have its derivatives uniformly bounded
when $z_0$ varies. We solve the equation
$\dbar u=\dbar(\theta f e^{\nu g})$ on $B_k$, where $g$ is the holomorphic
function
$$
g(z)=c_k\langle z^{(k)}-z^{(k)}_0,z^{(k)}_0\rangle
-c_j\langle z^{(j)}-z^{(j)}_0,z^{(j)}_0\rangle.
$$
Thanks to H\"ormander's $L^2$ estimates [H\"or66], the $L^2$ solution
for the weight $\Phi$ yields a holomorphic function
$f'=\theta fe^{\nu g}-u$ on $B_k$ such that $f'(z_0)=f(z_0)$ and
$$
\eqalign{
&\int_{B_k}|f'|^2e^{-2\nu(\varphi+c_k|z^{(k)}|^2)}
d\lambda(z^{(k)})
\le C'\int_{B_j\cap B_k} |f|^2 |e^{\nu g}|^2e^{-2\nu(\varphi+c_k|z^{(k)}|^2)}
d\lambda(z^{(k)})\le{}\cr
&
C'\exp\!\Big(\!2\nu\big(c_k|z^{(k)}_0|^2-c_j|z^{(j)}_0|^2
+C\varepsilon(\min(r_j,r_k))^2\big)\!\Big)\!
\int_{B_j}\kern-1pt|f|^2 e^{-2\nu(\varphi+c_j|z^{(j)}|^2)}
d\lambda(z^{(j)}).\cr}
$$
Let us take the supremum of ${1\over\nu}\log|f(z_0)|={1\over\nu}\log|f'(z_0)|$
over all $f$ with $\Vert f\Vert\le 1$. By the definition of
$\psi_{\varepsilon,\nu,k}$ ((16.2.4) and (16.2.5)) and the bound on
$\Vert f'\Vert$, we find
$$
\psi_{\varepsilon,\nu,k}(z_0)\le \psi_{\nu,j}(z_0)+{\log C'\over 2\nu}
+C\varepsilon(\min(r_j,r_k))^2,
$$
whence (16.2.8) by symmetry. Assume that $\nu$ is so large that
$C_3/\nu<C_4\varepsilon(\inf_j r_j)^2$.
We ``glue'' all functions $\psi_{\varepsilon,\nu,j}$
into a function $\psi_{\varepsilon,\nu}$ globally defined on~$X$,
and for this we set
$$
\psi_{\varepsilon,\nu}(z)=\sup_{j,~B'_j\ni z}
\Big(\psi_{\varepsilon,\nu,j}(z)+
12\,C_4\varepsilon(r^{\prime 2}_j-|z^{(j)}|^2)\Big)
\qquad\hbox{on}~~X.
$$
Every point of $X$ belongs to some ball $B''_k$, and for such a point
we get
$$
12\,C_4\varepsilon(r^{\prime 2}_k-|z^{(k)}|^2)\ge
12\,C_4\varepsilon(r^{\prime 2}_k-r^{\prime\prime2}_k)>2C_4r_k^2>
{C_3\over\nu}+C_4\varepsilon(\min(r_j,r_k))^2.
$$
This, together with (16.2.8), implies that in $\psi_{\varepsilon,\nu}(z)$
the supremum is never reached for indices $j$ such that
$z\in \partial B'_j$, hence $\psi_{\varepsilon,\nu}$ is well defined
and continuous, and by standard properties of upper envelopes of
(quasi)-plurisubharmonic functions we get
$$
i\ddbar\psi_{\varepsilon,\nu}\ge \gamma-C_5\varepsilon\omega
\leqno(16.2.9)
$$
for $\nu\ge\nu_0(\varepsilon)$ large enough. By inequality (16.2.7)
applied with $r=e^{-\sqrt{\nu}}$, we see that
$\lim_{\nu\to+\infty}\psi_{\varepsilon,\nu}(z) = \varphi(z)$.
At this point, the difficulty is to show that
$\psi_{\varepsilon,\nu}$ is decreasing with $\nu$ -- this may not be 
formally true, but we will see at Step 3 that this is essentially true.
Another difficulty is that we must simultaneously let $\varepsilon$
go to $0$, forcing us to change the covering as we want the error
to get smaller and smaller in (16.2.9).
\medskip

\noindent{\it Step 2. A comparison of integrals.}

\noindent We claim that
$$
I:=\int_X\big(e^{-2\varphi}-e^{-2\max(\varphi,{\ell\over\ell-1}
\psi_{\nu,\varepsilon})+a}\big)dV_\omega<+\infty
\leqno(16.2.10)
$$
for every $\ell\in{}]1,\nu]$ and $a\in\bR$. In fact
$$
\eqalign{
I&\le\int_{\{\varphi<{\ell\over\ell-1}\psi_{\varepsilon,\nu}+a\}}
e^{-2\varphi}dV_\omega
=\int_{\{\varphi<{\ell\over\ell-1}\psi_{\varepsilon,\nu}\}+a}
e^{2(\ell-1)\varphi-2\ell\varphi}dV_\omega\cr
&\le e^{2(\ell-1)a}\int_Xe^{2\ell(\psi_{\varepsilon,\nu}-\varphi)}dV_\omega
\le C\Big(\int_Xe^{2\nu(\psi_{\varepsilon,\nu}-\varphi)}
dV_\omega\Big)^{\ell\over\nu}\cr}
$$
by H\"older's inequality. In order to show that these integrals are finite,
it is enough, by the definition and properties of the functions
$\psi_{\varepsilon,\nu}$ and $\psi_{\varepsilon,\nu,j}$, to prove that
$$
\int_{B'_j}e^{2\nu\psi_{\varepsilon,\nu,j}-2\nu\varphi}d\lambda=
\int_{B'_j}\Big(\sum_{k=0}^{+\infty}|f_{\nu,j,k}|^2\Big)
e^{-2\nu\varphi}d\lambda<+\infty.
$$
By the strong Noetherian property of coherent ideal sheaves
(see e.g.\ [GR84]), we know that the sequence of ideal sheaves
generated by the holomorphic functions
$(f_{\nu,j,k}(z)\overline{f_{\nu,j,k}(\overline w)})_{k\le k_0}$ on
$B_j\times B_j$ is locally statio\-nary as $k_0$ increases, hence
independant of $k_0$ on $B'_j\times B'_j\compact B_j\times B_j$ for
$k_0$ large enough. As the sum of the series $\sum_k f_{\nu,j,k}(z)\overline
{f_{\nu,j,k}(\overline w)}$ is bounded by
$$\Big(\sum_k |f_{\nu,j,k}(z)|^2 \sum_k|f_{\nu,j,k}(\overline w)|^2
\Big)^{1/2}$$
and thus uniformly covergent on every compact subset of $B_j\times B_j$,
and as the space of sections of a coherent ideal sheaf is closed
under the topology of uniform convergence on compact subsets, we infer
from the Noetherian property that the holomorphic function
$\sum_{k=0}^{+\infty}f_{\nu,j,k}(z)\overline{f_{\nu,j,k}(\overline w)}$
is a section of the coherent ideal sheaf generated by
$(f_{\nu,j,k}(z)\overline{f_{\nu,j,k}(\overline w)})_{k\le k_0}$ over
$B'_j\times B'_j$, for $k_0$ large enough. Hence, by restricting
to the conjugate diagonal $w=\overline z$, we get
$$
\sum_{k=0}^{+\infty}|f_{\nu,j,k}(z)|^2\le C
\sum_{k=0}^{k_0}|f_{\nu,j,k}(z)|^2\qquad\hbox{on}~~B'_j.
$$
This implies
$$
\int_{B'_j}\Big(\sum_{k=0}^{+\infty}|f_{\nu,j,k}|^2\Big)e^{-2\varphi}d\lambda
\le C
\int_{B'_j}\Big(\sum_{k=0}^{k_0}|f_{\nu,j,k}|^2\Big)e^{-2\varphi}d\lambda
= C(k_0+1).
$$
Property (16.2.10) is proved.\medskip

\noindent{\it Step 3. Subadditivity of the approximating
sequence $\psi_{\varepsilon,\nu}$.}

\noindent
We want to compare $\psi_{\varepsilon,\nu_1+\nu_2}$ and
$\psi_{\varepsilon,\nu_1}$, $\psi_{\varepsilon,\nu_2}$
for every pair of indices~$\nu_1$, $\nu_2$, first when the functions are
associated with the same covering $X=\bigcup B_j$. Consider a function
$f\in\cH_{\nu_1+\nu_2,j}$ with
$$
\int_{B_j}|f(z)|^2e^{-2(\nu_1+\nu_2)\varphi_j(z)}d\lambda(z)\le 1,\qquad
\varphi_j(z)=\varphi(z)+c_j|z^{(j)}|^2.
$$
We may view $f$ as a function $\hat f(z,z)$ defined on the diagonal
$\Delta$ of $B_j\times B_j$. Consider the Hilbert space of holomorphic
functions $u$ on $B_j\times B_j$ such that
$$
\int_{B_j\times B_j}|u(z,w)|^2
e^{-2\nu_1\varphi_j(z)-2\nu_2\varphi_j(w)}d\lambda(z)d\lambda(w)<+\infty.
$$
By the Ohsawa-Takegoshi $L^2$ extension theorem [OT87], there exists a
function $\wt f(z,w)$ on $B_j\times B_j$ such that $\wt f(z,z)=f(z)$ and
$$
\eqalign{
\int_{B_j\times B_j}|\wt f(z,w)|^2
& e^{-2\nu_1\varphi_j(z)-2\nu_2\varphi_j(w)}d\lambda(z)d\lambda(w)\cr
&\qquad{}
\le C_7\int_{B_j}|f(z)|^2e^{-2(\nu_1+\nu_2)\varphi_j(z)}d\lambda(z)=C_7,\cr}
$$
where the constant $C_7$ only depends on the dimension $n$
(it is actually independent of the radius $r_j$ if say $0<r_j\le 1$).
As the Hilbert
space under consideration on $B_j\times B_j$ is the completed tensor
product $\cH_{\nu_1,j}\mathop{\wh\otimes}\cH_{\nu_2,j}$, we infer that
$$
\wt f(z,w)=\sum_{k_1,k_2}c_{k_1,k_2}f_{\nu_1,j,k_1}(z)
f_{\nu_2,j,k_2}(w)
$$
with $\sum_{k_1,k_2}|c_{k_1,k_2}|^2\le C_7$. By restricting to the
diagonal, we obtain
$$
|f(z)|^2=|\wt f(z,z)|^2\le\sum_{k_1,k_2}|c_{k_1,k_2}|^2
\sum_{k_1}|f_{\nu_1,j,k_1}(z)|^2\sum_{k_2}|f_{\nu_2,j,k_2}(z)|^2.
$$
From (16.2.3) and (16.2.4), we get
$$
\psi_{\varepsilon,\nu_1+\nu_2,j}\le {\log C_7\over\nu_1+\nu_2}+
{\nu_1\over\nu_1+\nu_2}\psi_{\varepsilon,\nu_1,j}+{\nu_2\over\nu_1+\nu_2}
\psi_{\varepsilon,\nu_2,j},
$$
in particular
$$
\psi_{\varepsilon,2^\nu,j}\le \psi_{\varepsilon,2^{\nu-1},j}+{C_8\over 2^\nu},
$$
and we see that $\psi_{\varepsilon,2^\nu}+C_8 2^{-\nu}$ is a decreasing
sequence. By Step~2 and Lebesgue's monotone convergence theorem,
we infer that for every $\varepsilon,\delta>0$ and $a\le a_0\ll 0$ fixed,
the integral
$$
I_{\varepsilon,\delta,\nu}=\int_X
\Big(e^{-2\varphi}-e^{-2\max(\varphi,(1+\delta)(\psi_{2^\nu,\varepsilon}+a
))}\Big)dV_\omega
$$
converges to $0$ as $\nu$ tends to~$+\infty$ (take
$\ell={1\over\delta}+1$ and $2^\nu>\ell$ and $a_0$ such that
$\delta\sup_X\varphi+a_0\le 0$; we do not have monotonicity strictly speaking
but need only replace $a$ by $a+C_8 2^{-\nu}$ to get it, thereby slightly
enlarging the integral).
\medskip

\noindent
{\it Step 4. Selection of a suitable upper envelope.}

\noindent For the simplicity of notation, we assume here that
$\sup_X\varphi=0$ (possibly after subtracting a constant), hence we can take
$a_0=0$ in the above. We may even further assume that all our
functions $\psi_{\varepsilon,\nu}$ are nonpositive. By Step~3,
for each $\delta=\varepsilon=2^{-k}$, we can select an index
$\nu=p(k)$ such that
$$
I_{2^{-k},2^{-k},p(k)}=
\int_X\Big(e^{-2\varphi}-e^{-2\max(\varphi,(1+2^{-k})\psi_{2^{-k},2^{p(k)}})}
\Big)dV_\omega\le 2^{-k}
\leqno(16.2.11)
$$
By construction, we have an estimate $i\ddbar\psi_{2^{-k},2^{p(k)}}\ge
\gamma-C_52^{-k}\omega$, and the functions $\psi_{2^{-k},2^{p(k)}}$ are
quasi-psh with logarithmic poles. Our estimates (especially (16.2.7))
imply that $\lim_{k\to+\infty}\psi_{2^{-k},2^{p(k)}}(z)=\varphi(z)$
as soon as $2^{-p(k)}\log\big(1/\inf_j r_j(k)\big)\to 0$ (notice that
the $r_j$'s now depend on $\varepsilon=2^{-k}$). We set
$$
\varphi_\nu(z)=\sup_{k\ge\nu}(1+2^{-k})\psi_{2^{-k},2^{p(k)}}(z).
\leqno(16.2.12)
$$
By construction $(\varphi_\nu)$ is a decreasing sequence and satisfies the
estimates
$$\varphi_\nu\ge
\max\big(\varphi,(1+2^{-\nu})\psi_{2^{-\nu},2^{p(\nu)}}\big),
\qquad
i\ddbar\varphi_\nu\ge \gamma-C_52^{-\nu}\omega.$$
Inequality (16.2.11) implies that
$$
\int_X(e^{-2\varphi}-e^{-2\varphi_\nu})dV_\omega\le
\sum_{k=\nu}^{+\infty}2^{-k}=2^{1-\nu}.
$$
Finally, if $Z_\nu$ is the set of poles of $\psi_{2^{-\nu},2^{p(\nu)}}$, then
$Z_\nu\subset Z_{\nu+1}$ and $\varphi_\nu$ is continuous on~$X\ssm Z_\nu$.
The reason is that in a neighborhood of every point
\hbox{$z_0\in X\ssm Z_\nu$},
the term \hbox{$(1+2^{-k})\psi_{2^{-k},2^{p(k)}}$} contributes to
$\varphi_\nu$ only when it is larger than
$(1+2^{-\nu})\psi_{2^{-\nu},2^{p(\nu)}}$.
Hence, by the almost-monotonicity, the relevant terms of the sup
in (16.2.12) are squeezed
between $(1+2^{-\nu})\psi_{2^{-\nu},2^{p(\nu)}}$ and
$(1+2^{-k})(\psi_{2^{-\nu},2^{p(\nu)}}+C_82^{-\nu})$, and therefore
there is uniform convergence in a neighborhood of~$z_0$.
Finally, condition c) implies that
$$
\int_U|f|^2(e^{-2\varphi}-e^{-2\varphi_\nu})dV_\omega<+\infty
$$
for every germ of holomorphic function $f\in\cO(U)$ at a point~$x\in X$.
Therefore both integrals $\int_U|f|^2e^{-2\varphi}dV_\omega$ and
$\int_U|f|^2e^{-2\varphi_\nu}dV_\omega$ are simultaneously convergent
or divergent, i.e.\ $\cI(\varphi)=\cI(\varphi_\nu)$. Theorem 16.2.1 is
proved, except that $\varphi_\nu$ is possibly just continuous instead of
being smooth. This can be arranged by Richberg's regularization theorem
[Ri68], at the expense of an arbitrary small loss in the Hessian form.\qed

\claim{(16.2.13) Remark} {\rm By a very slight variation of the proof, we
can strengthen condition c) and obtain that for every $t>0$
$$
\int_X(e^{-2t\varphi}-e^{-2t\varphi_\nu})dV_\omega
$$
is finite for $\nu$ large enough and converges to $0$ as $\nu\to+\infty$.
This implies that the sequence of multiplier ideals
$\cI(t\varphi_\nu)$ is a stationary decreasing
sequence, with \hbox{$\cI(t\varphi_\nu)=\cI(t\varphi)$} for $\nu$ large.}
\endclaim

\titlec{16.3.}{A Bochner type inequality}

Let $(L,h)$ be a smooth hermitian line bundle on a (non necessarily
compact) K\"ahler manifold $(Y,\omega)$.  We denote by
$|~~|=|~~|_{\omega,h}$ the pointwise hermitian norm on
\hbox{$\Lambda^{p,q}T^\star_Y\otimes L$} associated with $\omega$ and $h$,
and by $\Vert~~\Vert=\Vert~~\Vert_{\omega,h}$ the global $L^2$ norm
$$
\Vert u\Vert^2 = \int_Y |u|^2 dV_\omega \qquad
\hbox{where}\quad  dV_\omega={\omega^n\over n!}
$$
We consider the $\dbar$ operator acting on $(p,q)$-forms with values in
$L$, its adjoint $\dbar_h^\star$ with respect to $h$ and the complex
Laplace-Beltrami operator $\Delta''_h=\dbar\dbar_h^\star+\dbar_h^\star\dbar$.
Let $v$ be a smooth $(n-q,0)$-form with compact support
in~$Y$. Then $u=\omega^q\wedge v$ satisfies
$$
\Vert\dbar u\Vert^2+\Vert\dbar^\star_h u\Vert^2=
\Vert\dbar v\Vert^2+\int_Y\sum_{I,J}\Big(\sum_{j\in J}\lambda_j\Big)|u_{IJ}|^2
\leqno(16.3.1)
$$
where $\lambda_1\le\ldots\le\lambda_n$ are the curvature
eigenvalues of $\Theta_h(L)$ expressed in an orthonormal frame
$(\partial/\partial z_1,\ldots,\partial/\partial z_n)$ (at some fixed
point $x_0\in Y$), in such a way that
$$
\omega_{x_0}=i\sum_{1\le j\le n}dz_j\wedge d\overline z_j,\qquad
\Theta_h(L)_{x_0}=i\ddbar\varphi_{x_0}=
i\sum_{1\le j\le n} \lambda_jdz_j\wedge d\overline z_j.
$$
The proof of (16.3.1) proceeds by checking that
$$
(\dbar^\star_\varphi\,\dbar+\dbar\,\dbar^\star_\varphi)(v\wedge\omega^q)-
(\dbar^\star_\varphi\,\dbar v)\wedge\omega^q=
q\,i\ddbar\varphi\wedge\omega^{q-1}\wedge v,
\leqno(16.3.2)
$$
taking the inner product with $u=\omega^q\wedge v$ and integrating by
parts in the left hand side. In order to check (16.3.2), we use the
identity $\dbar^\star_\varphi=e^\varphi\dbar^\star(e^{-\varphi}\bu)
=\dbar^\star+\nabla^{0,1}\varphi\ort\bu$~. Let us work in
a local trivialization of $L$ such that $\varphi(x_0)=0$ and
$\nabla\varphi(x_0)=0$. At~$x_0$ we then find
$$
\eqalign{
(\dbar^\star_\varphi\,\dbar+\dbar\,\dbar^\star_\varphi)(\omega^q\wedge v)&{}-
\omega^q\wedge(\dbar^\star_\varphi\,\dbar v)={}\cr
&\big[(\dbar^\star\,\dbar+\dbar\,\dbar^\star)(\omega^q\wedge v)-
\omega^q\wedge(\dbar^\star\,\dbar v)\big]+\dbar(\nabla^{0,1}\varphi\ort(
\omega^q\wedge v)).\cr}
$$
However, the term $[\,\ldots\,]$ corresponds to the case of a trivial vector
bundle and it is well known in that case that $[\Delta'',
\omega^q\wedge\bu]=0$, hence $[\,\ldots\,]=0$. On the other hand
$$\nabla^{0,1}\varphi\ort(\omega^q\wedge v)=
q(\nabla^{0,1}\varphi\ort\omega)\wedge \omega^{q-1}\wedge v=
-q\,i\partial\varphi\wedge \omega^{q-1}\wedge v,$$
and so
$$
(\dbar^\star_\varphi\,\dbar+\dbar\,\dbar^\star_\varphi)(\omega^q\wedge v)-
\omega^q\wedge(\dbar^\star_\varphi\,\dbar v)=q\,i\ddbar\varphi
\wedge \omega^{q-1}\wedge v.$$
Our formula is thus proved when $v$ is smooth and compactly
supported. In general, we have:

\claim{(16.3.3) Proposition} Let $(Y,\omega)$ be a {\rm complete} K\"ahler
manifold and $(L,h)$ a smooth hermitian line bundle such that the
curvature possesses a uniform lower bound\break $\Theta_h(L)\ge -C\omega$.
For every measurable $(n-q,0)$-form $v$ with $L^2$ coefficients and
values in $L$ such that $u=\omega^q\wedge v$ has differentials
$\dbar u$, $\dbar^\star u$ also in $L^2$, we have
$$
\Vert\dbar u\Vert^2+\Vert\dbar^\star_h u\Vert^2=
\Vert\dbar v\Vert^2+\int_Y\sum_{I,J}\Big(\sum_{j\in J}\lambda_j\Big)|u_{IJ}|^2
$$
$($here, all differentials are computed in the sense of distributions$)$.
\endclaim

\begproof{} Since $(Y,\omega)$ is assumed to be complete,
there exists a sequence of smooth forms $v_\nu$ with compact support in $Y$
(obtained by truncating $v$ and taking the convolution with a regularizing
kernel) such that $v_\nu\to v$ in $L^2$ and such that
$u_\nu=\omega^q\wedge v_\nu$ satisfies $u_\nu\to u$, $\dbar u_\nu\to\dbar u$,
$\dbar^\star u_\nu\to\dbar^\star u$ in~$L^2$. By the curvature assumption,
the final integral in the right hand side of (16.3.1) must be under control
(i.e.\ the integrand becomes nonnegative if we add a term $C\Vert u\Vert^2$
on both sides, $C\gg 0$). We thus get the equality by passing to the limit
and using Lebesgue's monotone convergence theorem.\qed
\endproof

\titlec{16.4.}{Proof of Theorem (16.1.1)}

To fix the ideas, we first indicate the proof in the much simpler case
when $(L,h)$ is hermitian semipositive, and then treat the general
case.\medskip

\noindent
{\bf (16.4.1) Special case.} $(L,h)$ is (smooth) hermitian semipositive.

Let $\{\beta\}\in H^q(X,\Omega^n_X\otimes L)$ be an arbitrary cohomology class.
By standard $L^2$ Hodge theory, $\{\beta\}$ can be represented by a smooth
harmonic $(0,q)$-form $\beta$ with values in $\Omega^n_X\otimes L$. We can
also view $\beta$ as a $(n,q)$-form with values in $L$. The pointwise
Lefschetz isomorphism produces a unique $(n-q,0)$-form $\alpha$ such
that $\beta=\omega^q\wedge\alpha$. Proposition 16.3.3 then yields
$$
\Vert\dbar\alpha\Vert^2+\int_Y\sum_{I,J}\Big(\sum_{j\in J}\lambda_j\Big)
|\alpha_{IJ}|^2=\Vert\dbar\beta\Vert^2+\Vert\dbar^\star_h \beta\Vert^2=0,
$$
and the curvature eigenvalues $\lambda_j$ are nonnegative by our assumption.
Hence $\dbar\alpha=0$ and $\{\alpha\}\in H^0(X,\Omega^{n-q}_X\otimes L)$
is mapped to $\{\beta\}$ by $\Phi^q_{\omega,h}=\omega^q\wedge\bu~$.
\medskip

\noindent
{\bf (16.4.2) General case.}

There are several difficulties. The first difficulty is that the
metric $h$ is no longer smooth and we cannot directly represent
cohomology classes by harmonic forms. We circumvent this problem by
smoothing the metric on an (analytic) Zariski open subset and by 
avoiding the remaining poles on the complement. However, some careful 
estimates have to be made in order to take the error terms into account.

Fix $\varepsilon=\varepsilon_\nu$ and let $h_\varepsilon=h_{\varepsilon_\nu}$
be an approximation of~$h$, such that $h_\varepsilon$ is smooth on
$X\ssm Z_\varepsilon$ ($Z_\varepsilon$ being an analytic subset of $X$),
$\Theta_{h_\varepsilon}(L)\ge -\varepsilon\omega$,
$h_\varepsilon\le h$ and $\cI(h_\varepsilon)=\cI(h)$. This is possible
by Theorem~16.2.1. Now, we can find a family
$$
\omega_{\varepsilon,\delta}=\omega+\delta(i\ddbar \psi_\varepsilon+\omega),
\qquad \delta>0
$$
of {\it complete K\"ahler} metrics on $X\ssm Z_\varepsilon$, where
$\psi_\varepsilon$ is a quasi-psh function on $X$ with
$\psi_\varepsilon=-\infty$ on $Z_\varepsilon$,
$\psi_\varepsilon$ on $X\ssm Z_\varepsilon$ and $i\ddbar
\psi_\varepsilon+\omega\ge 0$ (see e.g.\ [Dem82], Th\'eor\`eme 1.5).
By construction, $\omega_{\varepsilon,\delta}\ge\omega$ and
$\lim_{\delta\to 0}\omega_{\varepsilon,\delta}=\omega$.
We look at the $L^2$ Dolbeault complex $L^\bu_{\varepsilon,\delta}$
of $(n,\bu)$-forms on $X\ssm Z_\varepsilon$, where the $L^2$ norms are
induced by $\omega_{\varepsilon,\delta}$ on differential forms and by
$h_\varepsilon$ on elements in~$L$. Specifically
$$
L^q_{\varepsilon,\delta}=\Big\{u{:}X\ssm Z_\varepsilon{\to}\Lambda^{n,q}
T^\star_X\otimes L;\int_{X\ssm Z_\varepsilon}\kern-15pt
(|u|^2_{\Lambda^{n,q}\omega_{\varepsilon,\delta}
\otimes h_\varepsilon}+|\dbar u|^2_{\Lambda^{n,q+1}\omega_{\varepsilon,\delta}
\otimes h_\varepsilon})dV_{\omega_{\varepsilon,\delta}}<\infty\Big\}.
$$
Let $\cL^q_{\varepsilon,\delta}$ be the corresponding sheaf of germs
of locally $L^2$ sections on $X$ (the local $L^2$ condition
should hold on $X$, not only on $X\ssm Z_\varepsilon\,$!). Then,
for all $\varepsilon>0$ and $\delta\ge 0$,
$(\cL^q_{\varepsilon,\delta},\dbar)$ is a resolution of the sheaf
$\Omega^n_X\otimes L\otimes\cI(h_\varepsilon)=
\Omega^n_X\otimes L\otimes\cI(h)$. This is
because $L^2$ estimates hold locally on small Stein open sets, and the
$L^2$ condition on $X\ssm Z_\varepsilon$ forces holomorphic sections
to extend across~$Z_\varepsilon$ ([Dem82b], Lemme 6.9).

Let $\{\beta\}\in H^q(X,\Omega^n_X\otimes L\otimes\cI(h))$ be a
cohomology class represented by a smooth form with values in
$\Omega^n_X\otimes L\otimes\cI(h)$ (one can use a \v Cech cocycle
and convert it to an element in the $C^\infty$ Dolbeault complex by
means of a partition of unity, thanks to the usual De Rham-Weil
isomorphism). Then
$$
\Vert\beta\Vert_{\varepsilon,\delta}^2\le \Vert\beta\Vert^2=
\int_X|\beta|^2_{\Lambda^{n,q}\omega\otimes h}dV_\omega<+\infty.
$$
The reason is that $|\beta|^2_{\Lambda^{n,q}\omega\otimes h}dV_\omega$
decreases as $\omega$ increases. This is just an easy calculation,
shown by comparing two metrics $\omega$, $\omega'$ which are
expressed in diagonal form in suitable coordinates; the norm
$|\beta|^2_{\Lambda^{n,q}\omega\otimes h}$ turns out to decrease
faster than the volume $dV_\omega$ increases; see e.g.\ [Dem82b], Lemme 3.2;
a special case is $q=0$, then $|\beta|^2_{\Lambda^{n,q}\omega\otimes h}
dV_\omega=i^{n^2}\beta\wedge\overline\beta$ with the identification
$L\otimes\overline L\simeq\bC$ given by the metric $h$, hence the
integrand is even independent of $\omega$ in that case. 

By the proof of the De Rham-Weil isomorphism, the map
$\alpha\mapsto\{\alpha\}$ from the cocycle space
$Z^q(\cL^\bu_{\varepsilon,\delta})$ equipped with its $L^2$ topology,
into $H^q(X,\Omega^n_X\otimes L\otimes\cI(h))$ equipped with its
finite vector space topology, is continuous.  Also, Banach's open
mapping theorem implies that the coboundary space
$B^q(\cL^\bu_{\varepsilon,\delta})$ is closed in
$Z^q(\cL^\bu_{\varepsilon,\delta})$. This is true for all $\delta\ge
0$ (the limit case $\delta=0$ yields the strongest $L^2$ topology in
bidegree $(n,q)$). Now, $\beta$ is a $\dbar$-closed form in the Hilbert space
defined by $\omega_{\varepsilon,\delta}$ on $X\ssm Z_\varepsilon$, so there
is a $\omega_{\varepsilon,\delta}$-harmonic form
$u_{\varepsilon,\delta}$ in the same cohomology class
as $\beta$, such that
$$
\Vert u_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}\le
\Vert\beta\Vert_{\varepsilon,\delta}.
$$
\claim{(16.4.3) Remark} {\rm
The existence of a harmonic representative holds true only for $\delta>0$, 
because we need to have a complete K\"ahler metric on $X\ssm Z_\varepsilon$.
The trick of employing $\omega_{\varepsilon,\delta}$ instead of a fixed
metric $\omega$, however, is not needed when $Z_\varepsilon$ is (or can 
be taken to be) empty. This is the case if $(L,h)$ is such that 
$\cI(h)=\cO_X$ and $L$ is nef. Indeed, in that case, from the very 
definition of nefness, it is easy to prove that we can take the 
$\varphi_\nu$'s to be everywhere smooth in Theorem~16.2.1. However,
we will see in \S~16.5 that multiplier ideal sheaves are needed even
in case $L$ is nef, when $\cI(h)\ne\cO_X$.}
\endclaim

Let $v_{\varepsilon,\delta}$ be the unique $(n-q,0)$-form such that
$u_{\varepsilon,\delta}=v_{\varepsilon,\delta}\wedge
\omega_{\varepsilon,\delta}^q$ ($v_{\varepsilon,\delta}$ exists by the
pointwise Lefschetz isomorphism). Then
$$
\Vert v_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}=
\Vert u_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}\le
\Vert\beta\Vert_{\varepsilon,\delta}\le\Vert\beta\Vert.
$$
As $\sum_{j\in J}\lambda_j\ge -q\varepsilon$ by the assumption on
$\Theta_{h_\varepsilon}(L)$, the Bochner formula yields
$$
\Vert\dbar v_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}^2\le
q\varepsilon\Vert u_{\varepsilon,\delta}\Vert_{\varepsilon,\delta}^2
\le q\varepsilon\Vert\beta\Vert^2.
$$
These uniform bounds imply that there are subsequences $u_{\varepsilon,
\delta_\nu}$ and $v_{\varepsilon,\delta_\nu}$ with $\delta_\nu\to 0$,
possessing weak-$L^2$ limits $u_\varepsilon=
\lim_{\nu\to+\infty}u_{\varepsilon,\delta_\nu}$
and $v_\varepsilon=\lim_{\nu\to+\infty}v_{\varepsilon,\delta_\nu}$.
The limit $u_\varepsilon=
\lim_{\nu\to+\infty}u_{\varepsilon,\delta_\nu}$ is with respect to
$L^2(\omega)=L^2(\omega_{\varepsilon,0})$. To check this, notice that
in bidegree $(n-q,0)$, the space $L^2(\omega)$ has the weakest topology
of all spaces $L^2(\omega_{\varepsilon,\delta})$; indeed, an easy calculation
as in ([Dem82b], Lemme 3.2) yields
$$
|f|^2_{\Lambda^{n-q,0}\omega\otimes h}dV_\omega\le
|f|^2_{\Lambda^{n-q,0}\omega_{\varepsilon,\delta}\otimes h}
dV_{\omega_{\varepsilon,\delta}}\qquad
\hbox{if $f$ is of type $(n-q,0)$}.
$$
On the other hand, the limit
$v_\varepsilon=\lim_{\nu\to+\infty}v_{\varepsilon,\delta_\nu}$
takes place in all spaces $L^2(\omega_{\varepsilon,\delta})$, $\delta>0$,
since the topology gets stronger and stronger as $\delta\downarrow 0$
[$\,$possibly not in $L^2(\omega)$, though, because in bidegree $(n,q)$
the topology of $L^2(\omega)$ might be strictly stronger than that 
of all spaces $L^2(\omega_{\varepsilon,\delta})\,$].
The above estimates yield
$$
\eqalign{
&\Vert v_\varepsilon\Vert^2_{\varepsilon,0}=
\int_X|v_\varepsilon|^2_{\Lambda^{n-q,0}\omega\otimes h_\varepsilon}
dV_\omega\le\Vert\beta\Vert^2,\cr
&\Vert \dbar v_\varepsilon\Vert^2_{\varepsilon,0}\le q\varepsilon
\Vert\beta\Vert^2_{\varepsilon,0},\cr
\noalign{\vskip3pt}
&u_\varepsilon=\omega^q\wedge v_\varepsilon\equiv\beta
\qquad\hbox{in}~~H^q(X,\Omega^n_X\otimes L\otimes\cI(h_\varepsilon)).\cr}
$$
Again, by arguing in a given Hilbert space $L^2(h_{\varepsilon_0})$,
we find $L^2$ convergent subsequences $u_\varepsilon\to u$,
$v_\varepsilon\to v$ as $\varepsilon\to 0$, and in this way get
$\dbar v=0$ and
$$
\eqalignno{
&\Vert v\Vert^2\le \Vert \beta\Vert^2,\cr
&u=\omega^q\wedge v\equiv \beta
\qquad\hbox{in}~~H^q(X,\Omega^n_X\otimes L\otimes\cI(h)).\cr}
$$
Theorem 16.1.1 is proved. Notice that the equisingularity property
$\cI(h_\varepsilon)=\cI(h)$ is crucial in the above proof,
otherwise we could not infer that $u\equiv \beta$ from the fact
that $u_\varepsilon\equiv \beta$. This is true only because
all cohomology classes $\{u_\varepsilon\}$ lie in the same fixed 
cohomology group $H^q(X,\Omega^n_X\otimes L\otimes\cI(h))$, whose
topology is induced by the topology of $L^2(\omega)$ on $\dbar$-closed
forms (e.g.\ through the De Rham-Weil isomorphism).\qed

\titlec{16.5.}{A counterexample}

In view of Corollary 16.1.2, one might wonder whether the morphism
$\Phi^q_\omega$ would not still be surjective when $L$ is a nef
vector bundle. We will show that this is unfortunately not so,
even in the case of algebraic surfaces.

Let $B$ be an elliptic curve and let $V$ be the rank $2$ vector bundle 
over $B$ which is defined as the (unique) non split
extension
$$0 \to \cO_B \to V \to \cO_B \to 0.$$
In particular, the bundle $V$ is numerically flat, i.e.\ $c_1(V)=0$,
$c_2(V)=0$. We consider the ruled surface $X=\bP(V)$. On that surface
there is a unique section $C=\bP(\cO_B) \subset X$ with $C^2=0$ and
$$\cO_X (C)= \cO_{\bP (V)} (1)$$
is a nef line bundle. It is easy to see that
$$h^0 (X,\cO_{\bP (V)} (m))= h^0 (B,S^m V)=1$$
for all $m \in \bN$ (otherwise we would have $mC=aC+M$ where
$aC$ is the fixed part of the linear system $|mC|$ and $M\ne 0$ the
moving part, thus $M^2\ge 0$ and $C\cdot M>0$, contradiction).
We claim that
$$h^0 (X,\Omega_X^1 (kC))=2$$
for all $k \ge 2.$ This follows by tensoring the exact sequence
$$
0 \to \Omega^1_{X \vert C} \to \Omega_X^1 \to \pi^* \Omega^1_{C}
\simeq \cO_{C} \to 0
$$
by $\cO_X (kC)$ and observing that
$$\Omega^1_{X \vert C}= K_X=\cO_X (-2C).$$
From this, we get
$$
0 \to H^0(X,\cO_X((k-2)C)) \to H^0(X,\Omega_X^1\cO(kC))
\to H^0(X,\cO_X(kC))
$$
where $h^0 (X,\cO_X ((k-2)C))=h^0 (X,\cO_X (kC))=1$ for
all $k\ge 2$. Moreover, the last arrow is surjective because we
can multiply a section of $H^0(X,\cO_X(kC))$ by a nonzero section
in $H^0(X,\pi^*\Omega_B^1)$ to get a preimage. Our claim follows.
We now consider the diagram
$$
\matrix{
H^0 (X,\Omega_X^1 (2C)) & \buildo\vlra^{\textstyle \wedge \omega} &
H^1 (X,K_X (2C)) \cr
\noalign{\vskip4pt}
\simeq \Big\downarrow & & \Big\downarrow \varphi \cr
H^0 (X,\Omega_X^1 (3C)) &
\build\vlra^{\textstyle\wedge \omega}_{\textstyle\psi} & H^1(X,K_X (3C)).\cr
}
$$
Since $K_X (2C) \simeq \cO_X$ and $K_X (3C) \simeq \cO_X(C),$ the
cohomology sequence of
$$
0 \to K_X (2C) \to K_X (3C) \to K_X (3C) \vert C \simeq \cO_{C}
\to 0
$$
immediately implies $\varphi=0$ (notice that $h^1 (X,K_X (2C))=h^1 (X,K_X
(3C))=1,$ since $h^1 (B,\cO_B)=h^1 (B,V)=1)$, and
$h^2(X,K_X(2C))=h^2(B,\cO_B)=0$). Therefore the diagram
implies $\psi=0$, and we get:

\claim{(16.5.1) Proposition} $L=\cO_{\bP(V)}(3)$ is a counterample to
$(16.1.2)$ in the nef case.\qed
\endclaim

\noindent
By Corollary (16.1.2), we infer that $\cO_X(3)$ cannot be hermitian
semi-positive and we thus again obtain -- by a quite different method --
the result of [DPS94], example~1.7.

\claim{(16.5.2) Corollary} Let $B$ be an elliptic curve, $V$ the vector
bundle given by the unique non-split extension
$$0 \to \cO_B \to V\to \cO_B \to 0.$$
Let $X=\bP(V)$. Then $L=\cO_X(1)$ is nef but not hermitian
semi-positive $($nor does any multiple, e.g.\ the
anticanonical line bundle $-K_X=\cO_X(-2)$ is nef but not
semi-positive$)$.
\endclaim


\titleb{17.}{Nef and Pseudo-Effective Cones in K\"ahler Geometry}

We now introduce important concepts of positivity for
cohomology classes of type~$(1,1)$ in the general K\"ahler context.

\claim{(17.1) Definition} Let $X$ be a compact K\"ahler manifold.
\smallskip
\item{\rm(i)} The K\"ahler cone is the set 
$\cK\subset H^{1,1}(X,\bR)$ of cohomology classes $\{\omega\}$
of K\"ahler forms. This is an \ul{open convex cone}.
\smallskip
\item{\rm(ii)}The pseudo-effective cone is the set
$\cE\subset H^{1,1}(X,\bR)$ of cohomology classes $\{T\}$ of closed positive 
currents of type $(1,1)$. This is a \ul{closed convex cone}.\vskip0pt
\endclaim

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\LabelTeX -19 29 $\cE$\ELTX
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\LabelTeX 30 20 $\cE=\hbox{pseudo-effective cone in $H^{1,1}(X,\bR)$}$\ELTX
\EndFig
\vskip8mm

The openness of $\cK$ is clear by definition, and the closedness of $\cE$
follows from the fact that bounded sets of currents are weakly
compact (as follows from the similar weak compactness property 
for bounded sets of positive measures). It is then clear that 
$\overline\cK\subset\cE$.

In spite of the fact that cohomology groups can be defined either in terms
of forms or currents, it turns out that the cones $\overline\cK$ and
$\cE$ are in general different. To see this, it is enough to observe
that a K\"ahler class $\{\alpha\}$ satisfies $\int_Y\alpha^p>0$ for
every $p$-dimensional analytic set. On the other hand, if $X$ is the surface
obtained by blowing-up $\bP^2$ in one point, then the exceptional
divisopr $E\simeq\bP^1$ has a cohomology class $\{\alpha\}$ such that 
$\int_E\alpha= E^2=-1$, hence $\{\alpha\}\notin\overline\cK$, although
$\{\alpha\}=\{[E]\}\in\cE$.

In case $X$ is projective, it is interesting to consider also the
algebraic analogues of our ``transcendental cones'' $\cK$ and $\cE$,
which consist of suitable integral divisor classes. Since the cohomology
classes of such divisors live in $H^2(X,\bZ)$, we are led to introduce
the Neron-Severi lattice and the associated Neron-Severi space
$$
\eqalign{
\NS(X)&:= H^{1,1}(X,\bR)\cap \big(H^2(X,\bZ)/\{\hbox{torsion}\}\big),\cr
\NS_\bR(X)&:=\NS(X)\otimes_\bZ\bR.\cr
}
$$
All classes of real divisors $D=\sum c_jD_j$, $c_j\in\bR$, lie by definition 
in $\NS_\bR(X)$.
Notice that the integral lattice $H^2(X,\bZ)/\{\hbox{torsion}\}$ need not 
hit at all the subspace\break 
$H^{1,1}(X,\bR)\subset H^2(X,\bR)$ in the Hodge decomposition,
hence in general the Picard number 
$$\rho(X)=\rank_\bZ \NS(X)=\dim_\bR \NS_\bR(X)$$
satisfies
$\rho(X)\le h^{1,1}=\dim_\bR H^{1,1}(X,\bR)$,
but the equality can be strict (actually, it is well known that a generic 
complex torus $X=\bC^n/\Lambda$ satisfies $\rho(X)=0$ and $h^{1,1}=n^2$). 
In order to deal with the case of algebraic varieties we introduce
$$\cK_{\NS} = \cK\cap \NS_\bR(X),\qquad \cE_{\NS}=\cE\cap \NS_{\bR}(X).$$ 

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\LabelTeX -3 46 $\cK_{\NS}$\ELTX
\LabelTeX -19 27 $\cE_{\NS}$\ELTX
\LabelTeX -38.5 55 $\NS_\bR(X)$\ELTX
\EndFig
\vskip8mm

A very important fact is that the ``Neron-Severi part'' of any of
the open or closed transcendental cones $\cK$, $\cE$, $\ol\cK$, $\cE^\circ$
is algebraic, i.e.\ can be characterized in simple algebraic terms.

\claim{(17.2) Theorem} Let $X$ be a projective manifold. Then
{\oldparindent=6.6mm
\smallskip
\item{\rm (i)} $\cE_{\NS}$ is the closure of the cone generated by classes
of \ul{effective} divisors, i.e.\ divisors $D=\sum c_jD_j$, $c_j\in\bR_+$.
\smallskip
\item{\rm (ii)} $\cK_{\NS}$ is the open cone generated by classes of
\ul{ample} $($or very ample$)$ divisors $A$ $($Recall that a divisor
$A$ is said to be very ample if the linear system $H^0(X,\cO(A))$
provides an embedding of $X$ in projective space$)$.
\smallskip
\item{\rm (iii)} The interior $\cE_{\NS}^\circ$ is the cone generated
by classes of \ul{big} divisors, namely divisors $D$ such that
$h^0(X,\cO(kD))\ge c\,k^{\dim X}$ for $k$ large.
\smallskip
\item{\rm (iv)} The closed cone $\ol\cK_{\NS}$ consists of the closure
of the cone generated by \ul{nef} divisors $D$ (or nef line bundles $L$), 
namely effective integral divisors $D$ such that $D\cdot C\ge 0$ for
every curve $C$.\vskip0pt
}
\endclaim

\noindent
{\it Sketch of proof.} These results were already observed (maybe in a slightly
different terminology) in [Dem90]. If we denote by $\cK_\alg$ the open cone
generated by ample divisors, resp.\ by $\cE_\alg$ the closure of the cone
generated by effective divisors, it is obvious that
$$
\cK_\alg\subset \cK_{\NS},\qquad \cE_\alg\subset\cE_{\NS}.
$$
As was to be expected, the interesting part lies in the converse
inclusions. The inclusion $K_{\NS}\subset\cK_\alg$ is more or 
less equivalent
to the Kodaira embedding theorem~: if a rational class $\{\alpha\}$ is
in $\cK$, then some multiple of $\{\alpha\}$ is the first Chern class
of a hermitian line bundle $L$ whose curvature form is K\"ahler. Therefore $L$
is ample and $\{\alpha\}\in \cK_\alg\,$; property (ii) follows.

Similarly, if we take a
rational class $\{\alpha\}\in\cE_{\NS}^\circ$, then it is still in
$\cE$ by subtracting a small multiple $\varepsilon\omega$ of a
K\"ahler class, hence some multiple of $\{\alpha\}$ is the first Chern class
of a hermitian line bundle $(L,h)$ with curvature form
$$
T=\Theta_h(L):=-{i\over 2\pi}i\ddbar\log h \ge \varepsilon\omega.
$$
Let us apply Theorem 6.14 to a metric of the form $h^ke^{-m\psi}$ on 
$L^{\otimes k}$, where $\psi$ has logarithmic poles at given points $x_j$ 
in $X$. It is then easily shown that $L^{\otimes k}$ admits sections
which have given $m$-jets at the point $x_j$, provided that $k\ge Cm$,
$C\gg 1$, and the $x_j$ are chosen outside the Lelong sublevel sets
of $\log h$. From this we get $h^0(X,L^{\otimes k})\ge m^n/n!\ge ck^n$, 
hence the linear system $kL$ can be represented by a big divisor. 
This implies (iii) and also that $\cE_{\NS}^\circ\subset\cE_\alg$. Therefore 
$\cE_{\NS}\subset\cE_\alg$ by passing to the closure$\,$; (i) follows.
The statement (iv) about nef divisors follows
e.g.\ from [Kle66], [Har70], since every nef divisor is a limit 
of a sequence of ample rational divisors.
\qed

As a natural extrapolation of the algebraic situation, we say that 
$\overline{\cK}$ is the cone of {\it nef} $(1,1)$-cohomology classes 
(even though these classes are not necessarily integral). Property 
17.2~(i) also explains the terminology used for the pseudo-effective cone.
\bigskip

\titleb{18.}{Numerical Characterization of the K\"ahler Cone}

We describe here the main results obtained in [DP03]. The upshot is
that the K\"ahler cone depends only on the intersection product
of the cohomology ring, the Hodge structure and the homology classes
of analytic cycles. More precisely, we have$\,$:

\claim{(18.1) Theorem} Let $X$ be a compact K\"ahler manifold. Let $\cP$ 
be the set of real $(1,1)$ cohomology classes $\{\alpha\}$
which are \ul{numerically} \ul{positive} on analytic cycles, i.e.\ such 
that $\int_Y\alpha^p>0$ for every irreducible analytic set $Y$ in $X$, 
$p=\dim Y$. Then the K\"ahler cone $\cK$ of $X$ is one of the 
connected components of $\cP$.
\endclaim

\claim{(18.2) Special case} If $X$ is projective algebraic, then $\cK=\cP$.
\endclaim

These results (which are new even in the projective case) can be seen 
as a generalization of the well-known Nakai-Moishezon criterion. Recall
that the Nakai-Moishezon criterion provides a necessary and sufficient 
criterion for a line bundle
to be ample: {\it a line bundle $L\rightarrow X$ on a projective
algebraic manifold $X$ is ample if and only if 
$$L^p\cdot Y = \int_Yc_1(L)^p> 0,$$ 
for every algebraic subset $Y\subset X$, $p=\dim Y$.} 

It turns out that the numerical conditions $\int_Y\alpha^p> 0$ also 
characterize arbitrary transcendental K\"ahler classes when  $X$ is 
projective$\,$: this is precisely the meaning of the special case 18.2.

\claim{(18.3) Example} {\rm The following example shows that the cone $\cP$
need not be connected (and also that the components of $\cP$ need not 
be convex, either). Let us consider for instance a complex torus 
$X=\bC^n/\Lambda$. It is well-known that a generic torus $X$ does not
possess any analytic subset except finite subsets and $X$ itself. 
In that case, the numerical positivity is expressed by the single
condition $\int_X\alpha^n>0$. However, on a torus, $(1,1)$-classes are 
in one-to-one correspondence with constant hermitian forms $\alpha$ on
$\bC^n$. Thus, for $X$ generic, $\cP$ is the set of hermitian forms on 
$\bC^n$ such that $\det(\alpha)>0$, and Theorem 18.1 just expresses the 
elementary result of linear algebra saying that the set $\cK$ of 
positive definite forms is one of the connected components of the open set
$\cP=\{\det(\alpha)>0\}$ of hermitian forms of positive determinant
(the other components, of course, are the sets of forms of signature 
$(p,q)$, $p+q=n$, $q$ even. They are not convex when $p>0$ and $q>0$).}
\endclaim

\noindent
{\it Sketch of proof of Theorems 18.1 and 18.2.} By definition a 
\ul{\it K\"ahler current} is a closed positive current $T$ of type $(1,1)$
such that $T\ge\varepsilon\omega$ for some smooth
K\"ahler metric $\omega$ and $\varepsilon>0$ small enough.
The crucial steps of the proof of Theorem~18.1 are contained in the 
following statements. 

\claim{(18.4) Proposition {\rm (Paun [Pau98a, 98b])}} Let $X$ be a compact 
complex manifold $($or more generally a compact complex space$)$. Then
\smallskip
\item{\rm (i)} The cohomology class of a closed positive 
$(1,1)$-current $\{T\}$ is nef if and only if the restriction 
$\{T\}_{|Z}$ is nef for every irreducible component $Z$ in any of
the Lelong sublevel sets~$E_c(T)$.

\smallskip
\item{\rm (ii)} The cohomology class of a K\"ahler current $\{T\}$ is 
a K\"ahler class $($i.e.\ the class of a \ul{\rm smooth} K\"ahler form$)$ 
if and only if the restriction $\{T\}_{|Z}$ is a K\"ahler class for 
every irreducible component $Z$ in any of the Lelong sublevel sets~$E_c(T)$. 
\vskip0pt
\endclaim

The proof of Proposition 18.4 is not extremely hard if we take for granted the
fact that K\"ahler currents can be approximated by K\"ahler currents
with logarithmic poles, a fact which was first proved in [Dem92] (see also 
section 9 below). Thus in (ii), we may assume that $T=
\alpha+\ii \ddbar\varphi$ is a current with analytic singularities,
where $\varphi$ is a quasi-psh function with logarithmic poles on some
analytic set $Z$, and $\varphi$ smooth on $X\ssm Z$. Now, we proceed by 
an induction on dimension (to do this, we have to consider analytic spaces
rather than with complex manifolds, but it turns out that this makes no
difference for the proof). Hence, by the induction hypothesis, there exists
a smooth potential $\psi$ on $Z$ such that 
$\alpha_{|Z}+\ii \ddbar\psi>0$ along $Z$. It is well known that
one can then find a potential $\wt\psi$ on $X$ such that
$\alpha+\ii \ddbar\wt\psi>0$ in a neighborhood $V$ of $Z$ (but possibly
non positive elsewhere). Essentially, it is enough to take an arbitrary
extension of $\psi$ to $X$ and to add a large multiple of the square of 
the distance to $Z$, at
least near smooth points; otherwise, we stratify $Z$ by its successive
singularity loci, and proceed again by induction on the dimension of these
loci. Finally, we use a
a standard gluing procedure~: the current $T=\alpha+
\ii \max_\varepsilon(\varphi,\wt\psi-C)$, $C\gg 1$, will be equal
to $\alpha+\ii \ddbar\varphi>0$ on $X\ssm V$, and to a smooth
K\"ahler form on $V$.\qed

The next (and more substantial step) consists of the following result
which is reminiscent of the Grauert-Riemenschneider conjecture
([Siu84], [Dem85]).

\claim{(18.5) Theorem {\rm ([DP03])}} Let $X$ be a compact K\"ahler manifold
and let $\{\alpha\}$ be a nef class $($i.e.\ $\{\alpha\}\in\ol\cK)$.
Assume that $\int_X\alpha^n>0$. Then $\{\alpha\}$ contains a K\"ahler
current~$T$, in other words $\{\alpha\}\in\cE^\circ$.
\endclaim

\noindent
{\it Step 1}. The basic argument is to prove that for every irreducible 
analytic set $Y\subset X$ of
codimension $p$, the class $\{\alpha\}^p$ contains a closed positive
$(p,p)$-current $\Theta$ such that $\Theta\ge\delta[Y]$ for some $\delta>0$.
For this, we use two ingredients.  The first is a ``local'' calculation 
which completes Lemma 6.17 by analyzing the behavior of
Monge-Amp�re masses concentrated near an analytic set. The second is 
based on the exact solution of Monge-Amp�re equations by means of 
the Calabi-Yau theorem [Yau78].

\claim{(18.6) Lemma} Let $X$ be a compact complex manifold~$X$ 
equipped with a K\"ahler metric 
$\omega=i\sum_{1\le j,k\le n}\omega_{jk}(z)dz_j\wedge d\ol z_k$
and let $Y\subset X$ be an analytic subset of~$X$. Then there exist
globally defined quasi-plurisubharmonic potentials $\psi$ and 
$(\psi_\varepsilon)_{\varepsilon\in{}]0,1]}$ on $X$,
satisfying the following properties.
{\oldparindent=6.6mm
\smallskip
\item {\rm (i)} The function $\psi$ is smooth on $X\ssm Y$, satisfies
$i\ddbar\psi\ge -A\omega$ for some $A>0$, and $\psi$
has logarithmic poles along~$Y$, i.e., locally near $Y$
$$
\psi(z)\sim \log\sum_k|g_k(z)|+O(1)
$$
where $(g_k)$ is a local system of generators of the ideal sheaf $\cI_Y$
of $Y$ in $X$.
\smallskip
\item {\rm(ii)} We have 
$\psi=\lim_{\varepsilon\to 0}\downarrow\psi_\varepsilon$ 
where the $\psi_\varepsilon$ are $C^\infty$ and possess a uniform 
Hessian estimate
$$
i\ddbar\psi_\varepsilon\ge -A\omega\qquad\hbox{on~$X$}.
$$
\smallskip
\item {\rm(iii)} Consider the family of hermitian metrics
$$
\omega_\varepsilon := \omega + {1\over 2A} i\ddbar\psi_\varepsilon
\ge {1\over 2}\omega.
$$
For any point $x_0\in Y$ and any neighborhood $U$ of $x_0$, 
the volume element of $\omega_\varepsilon$ has
a uniform lower bound 
$$
\int_{U\cap V_\varepsilon}\omega_\varepsilon^n\ge\delta(U)>0,
$$
where $V_\varepsilon=\{z\in X\,;\;\psi(z)<\log\varepsilon\}$
is the ``tubular neighborhood'' of radius $\varepsilon$ around $Y$.
\smallskip
\item {\rm(iv)} For every integer $p\ge 0$, the family of positive 
currents $\omega_\varepsilon^p$ is bounded in mass. Moreover,
if $Y$ contains an irreducible component $Y'$ of codimension $p$,
there is a uniform lower bound
$$
\int_{U\cap V_\varepsilon}\omega_\varepsilon^p\wedge\omega^{n-p}\ge
\delta_p(U)>0
$$
in any neighborhood $U$ of a regular point $x_0\in Y'$. In particular, 
any weak limit $\Theta$ of $\omega_\varepsilon^p$ as $\varepsilon$
tends to $0$ satisfies $\Theta\ge\delta'[Y']$ for some
$\delta'>0$.
\vskip0pt
}
\endclaim

\begproof{} Properties (i) and (ii) were already proved in Lemma
6.17. We continue our calculations with the same functions $\psi$ and
$\psi_\varepsilon$.By construction, the real $(1,1)$-form
$\omega_\varepsilon:=\omega+{1\over 2A} i\ddbar\psi_\varepsilon$
satisfies $\omega_\varepsilon\ge {1\over 2}\omega$, hence it is
K\"ahler and its eigenvalues with respect to $\omega$ are at least
equal to~$1/2$.

Assume now that we are in a neighborhood $U$ of a regular point
$x_0\in Y$ where $Y$ has codimension~$p$.  Then
\hbox{$\gamma_{j,k}=\theta_j\partial g_{j,k}$} at $x_0$, hence the
rank of the system of $(1,0)$-forms $(\gamma_{j,k})_{k\ge 1}$ is at
least equal to $p$ in a neighborhood of~$x_0$. Fix a holomorphic
local coordinate system $(z_1,\ldots,z_n)$ such that 
\hbox{$Y=\{z_1=\ldots=z_p=0\}$} near $x_0$, and let $S\subset T_X$ 
be the holomorphic subbundle generated by $\partial/\partial z_1,\ldots,
\partial/\partial z_p$. This choice ensures that the rank of the system
of $(1,0)$-forms $(\gamma_{j,k|S})$ is everywhere equal to~$p$.
By (1,3) and the minimax principle applied to the $p$-dimensional subspace
$S_z\subset T_{X,z}$, we see that the $p$-largest eigenvalues of 
$\omega_\varepsilon$ are bounded below by 
$c\varepsilon^2/(e^{2\psi}+\varepsilon^2)^2$.
\smallskip
However, we can even restrict the form defined in (3.6) to the 
$(p-1)$-dimen\-sional subspace $S\cap\Ker\tau$ where 
$\tau(\xi):=\sum_{j,k}\theta_j\ol{g_{j,k}}\gamma_{j,k}(\xi)$,
to see that the $(p-1)$-largest eigenvalues of $\omega_\varepsilon$
are bounded below by $c/(e^{2\psi}+\varepsilon^2)$, $c>0$. The
$p$-th eigenvalue is then bounded by 
$c\varepsilon^2/(e^{2\psi}+\varepsilon^2)^2$ and the
remaining $(n-p)$-ones by~$1/2$. From this we infer
$$
\eqalign{
\omega_\varepsilon^n
&\ge c\,{\varepsilon^2\over(e^{2\psi}+\varepsilon^2)^{p+1}}\omega^n
\qquad\hbox{near $x_0$},\cr
\omega_\varepsilon^p
&\ge c\,{\varepsilon^2\over(e^{2\psi}+\varepsilon^2)^{p+1}}\Big
(i\sum_{1\le\ell\le p}\gamma_{j,k_\ell}\wedge
\ol{\gamma_{j,k_\ell}}\Big)^p
\cr}
$$
where $(\gamma_{j,k_\ell})_{1\le \ell\le p}$ is a suitable $p$-tuple
extracted from the $(\gamma_{j,k})$, such that\break 
$\bigcap_\ell\Ker\gamma_{j,k_\ell}$ is a smooth complex
(but not necessarily holomorphic) subbundle of codimension $p$ of $T_X$;
by the definition of the forms $\gamma_{j,k}$, this
subbundle must coincide with $T_Y$ along $Y$. From this, properties
(iii) and (iv) follow easily; actually, up to constants,
we have $e^{2\psi}+\varepsilon^2\sim|z_1|^2+\ldots+|z_p|^2+\varepsilon^2$
and
$$
i\sum_{1\le\ell\le p}\gamma_{j,k_\ell}\wedge
\ol{\gamma_{j,k_\ell}}\ge c\,i\ddbar(|z_1|^2+\ldots+|z_p|^2)-
O(\varepsilon)i\ddbar|z|^2\qquad\hbox{on $U\cap V_\varepsilon$},
$$
hence, by a straightforward calculation,
$$
\omega_\varepsilon^p\wedge\omega^{n-p}\ge
c\big(i\ddbar\log(|z_1|^2+\ldots+|z_p|^2+\varepsilon^2)\big)^p\wedge
\big(i\ddbar(|z_{p+1}|^2+\ldots+|z_n|^2)\big)^{n-p}
$$
on $U\cap V_\varepsilon$; notice also that
$\omega_\varepsilon^n\ge 2^{-(n-p)}\omega_\varepsilon^p\wedge\omega^{n-p}$,
so any lower bound for the volume of 
$\omega_\varepsilon^p\wedge\omega^{n-p}$ will also produce a bound
for the volume of $\omega_\varepsilon^n$. As it is well known, the $(p,p)$-form
$$
\Big({i\over 2\pi}\ddbar\log(|z_1|^2+\ldots+|z_p|^2+\varepsilon^2)\Big)^p
\qquad\hbox{on $\bC^n$}
$$
can be viewed as the pull-back to $\bC^n=\bC^p\times\bC^{n-p}$ of 
the Fubini-Study volume form of the complex $p$-dimensional
projective space of dimension $p$ containing $\bC^p$ as an affine
Zariski open set, rescaled by the dilation ratio~$\varepsilon$. Hence
it converges weakly to the current of integration on the $p$-codimensional
subspace $z_1=\ldots=z_p=0$. Moreover the volume contained in any
compact tubular cylinder
$$
\{|z'|\le C\varepsilon\}\times K''\subset \bC^p\times\bC^{n-p}
$$
depends only on $C$ and $K$ (as one sees after rescaling by $\varepsilon$).
The fact that $\omega_\varepsilon^p$ is uniformly bounded in mass can
be seen easily from the fact that
$$
\int_X \omega_\varepsilon^p\wedge\omega^{n-p}=\int_X\omega^n,
$$
as $\omega$ and $\omega_\varepsilon$ are in the same K\"ahler class.
Let $\Theta$ be any weak limit of $\omega_\varepsilon^p$. By what we
have just seen, $\Theta$ carries non zero mass on every 
$p$-codimensional component $Y'$ of $Y$, for instance near every
regular point. However, standard results of the theory of currents
(support theorem and Skoda's extension result) imply that
${\bf 1}_{Y'}\Theta$ is a closed positive current and that
${\bf 1}_{Y'}\Theta=\lambda[Y']$ is a nonnegative multiple
of the current of integration on $Y'$. The fact that the mass
of $\Theta$ on $Y'$ is positive yields~$\lambda>0$. 
Lemma~18.6 is proved.\qed

\claim{(18.7) Lemma {\rm([Yau78])}} Let $(X,\omega)$ 
be a compact K\"ahler manifold and $n=\dim X$. Then for any smooth
volume form $f>0$ such that $\int_X f = \int_X\omega^n$,
there exist a K\"ahler metric $\widetilde \omega = \omega+i\ddbar\varphi$
in the same K\"ahler class as $\omega$, such that 
$\widetilde \omega^n=f$.\qed
\endclaim

\noindent
We exploit this by observing that $\alpha+\varepsilon\omega$ is a
K\"ahler class, and by solving the Monge-Amp\`ere equation
$$
(\alpha+\varepsilon\omega+i\ddbar\varphi_\varepsilon)^n=C_\varepsilon
\omega_\varepsilon^n
\leqno(18.7\hbox{a})
$$
where $(\omega_\varepsilon)$ is the family of K\"ahler metrics on $X$
produced by Lemma 18.6~(iii), such that their volume is concentrated
in an $\varepsilon$-tubular neighborhood of $Y$.
$$
C_\varepsilon = {\int_X\alpha_\varepsilon^n\over\int_X\omega_\varepsilon^n}
={\int_X(\alpha+\varepsilon\omega)^n\over\int_X\omega^n}\ge
C_0={\int_X\alpha^n\over\int_X\omega^n}>0.
$$
Let us denote by 
$$
\lambda_1(z)\le\ldots\le\lambda_n(z)
$$
the eigenvalues of $\alpha_\varepsilon(z)$ with respect to
$\omega_\varepsilon(z)$, at every point $z\in X$ (these functions
are continuous with respect to $z$, and of course depend also 
on~$\varepsilon$). The equation (18.7a) is equivalent to the fact that
$$
\lambda_1(z)\ldots\lambda_n(z)=C_\varepsilon\leqno(18.7\hbox{b})
$$
is constant, and the most important observation for us is that the
constant $C_\varepsilon$ is bounded away from $0$, thanks to
our assumption $\int_X\alpha^n>0$.

Fix a regular point $x_0\in Y$ and a small neighborhood $U$ (meeting only
the irreducible component of $x_0$ in $Y$). By Lemma 18.6, we have a
uniform lower bound
$$
\int_{U\cap V_\varepsilon}\omega_\varepsilon^p\wedge\omega^{n-p}
\ge\delta_p(U)>0.\leqno(18.7\hbox{c})
$$
Now, by looking at the $p$ smallest (resp.\ $(n-p)$ largest) eigenvalues
$\lambda_j$ of $\alpha_\varepsilon$ with respect to $\omega_\varepsilon$, 
we find
$$
\leqalignno{
\alpha_\varepsilon^p&\ge \lambda_1\ldots\lambda_p\,\omega_\varepsilon^p,
&(18.7\hbox{d})\cr
\alpha_\varepsilon^{n-p}\wedge\omega_\varepsilon^p
&\ge {1\over n!}\lambda_{p+1}\ldots\lambda_n\,
\omega_\varepsilon^n,&(18.7\hbox{e})\cr
}
$$
The last inequality $(18.7\hbox{e})$ implies
$$
\int_X\lambda_{p+1}\ldots\lambda_n\,\omega_\varepsilon^n\le
n!\int_X\alpha_\varepsilon^{n-p}\wedge\omega_\varepsilon^p=
n!\int_X(\alpha+\varepsilon\omega)^{n-p}\wedge\omega^p\le M
$$
for some constant $M>0$ (we assume $\varepsilon\le 1$, say). 
In particular, for every $\delta>0$, the subset 
$E_\delta\subset X$ of points $z$ such that 
$\lambda_{p+1}(z)\ldots\lambda_n(z)>M/\delta$ satisfies
$\int_{E_\delta}\omega_\varepsilon^n\le\delta$, hence
$$
\int_{E_\delta}\omega_\varepsilon^p\wedge\omega^{n-p}\le 2^{n-p}
\int_{E_\delta}\omega_\varepsilon^n\le 2^{n-p}\delta.\leqno(18.7\hbox{f})
$$
The combination of (18.7c) and (18.7f) yields
$$
\int_{(U\cap V_\varepsilon)\ssm E_\delta}\omega_\varepsilon^p\wedge\omega^{n-p}
\ge \delta_p(U)-2^{n-p}\delta.
$$
On the other hand (18.7b) and (18.7d) imply
$$
\alpha_\varepsilon^p\ge{C_\varepsilon\over\lambda_{p+1}\ldots\lambda_n}
\omega_\varepsilon^p
\ge{C_\varepsilon\over M/\delta}\omega_\varepsilon^p\qquad
\hbox{on $(U\cap V_\varepsilon)\ssm E_\delta$}.
$$
From this we infer
$$
\int_{U\cap V_\varepsilon}\alpha_\varepsilon^p\wedge\omega^{n-p}\ge
{C_\varepsilon\over M/\delta}
\int_{(U\cap V_\varepsilon)\ssm E_\delta}
\omega_\varepsilon^p\wedge\omega^{n-p}\ge
{C_\varepsilon\over M/\delta}(\delta_p(U)-2^{n-p}\delta)>0
\leqno(18.7\hbox{g})
$$
provided that $\delta$ is taken small enough, e.g.\ 
$\delta=2^{-(n-p+1)}\delta_p(U)$. The family of $(p,p)$-forms
$\alpha_\varepsilon^p$ is uniformly bounded in mass since
$$
\int_X\alpha_\varepsilon^p\wedge\omega^{n-p}
=\int_X(\alpha+\varepsilon\omega)^p\wedge\omega^{n-p}\le\hbox{Const}.
$$
Inequality (18.7g) implies that any weak limit $\Theta$ of 
$(\alpha_\varepsilon^p)$
carries a positive mass on~$U\cap Y$. By Skoda's extension theorem [Sko82], 
${\bf 1}_Y\Theta$ is a closed positive current with support in $Y$, 
hence ${\bf 1}_Y\Theta=\sum c_j[Y_j]$ is a combination of the 
various components $Y_j$ of $Y$ with coefficients $c_j>0$. 
Our construction shows that $\Theta$ belongs to the cohomology class 
$\{\alpha\}^p$. Step 1 of Theorem 18.5 is proved.
\medskip

\noindent
{\it Step 2.} The second and final step consists in using
a ``diagonal trick'': for this, we apply Step 1 to 
$$
\wt X=X\times X,\qquad \wt Y=\hbox{diagonal}\,\Delta\subset \wt X,\qquad
\wt\alpha=\pr_1^*\alpha+\pr_2^*\alpha.
$$
It is then clear that $\wt\alpha$ is nef on $\wt X$ and that
$$
\int_{\wt X}(\wt\alpha)^{2n}=
{2n\choose n}\Big(\int_X\alpha^n\Big)^2>0.
$$
It follows by Step 1 that the
class $\smash{\{\wt\alpha\}}^n$ contains a K\"ahler current $\Theta$ of 
bidegree $(n,n)$ such that
$\Theta\ge\delta[\Delta]$ for some $\delta>0$. Therefore 
the push-forward
$$
T:=(\pr_1)_*(\Theta\wedge\pr_2^*\omega)
$$
is a positive $(1,1)$-current such that
$$
T\ge\delta(\pr_1)_*([\Delta]\wedge\pr_2^*\omega)=\delta\omega.
$$
It follows that $T$ is a K\"ahler current. On the other hand, $T$
is numerically equivalent to $(\pr_1)_*(\wt\alpha^n\wedge\pr_2^*\omega)$, 
which is the form given in coordinates by
$$
x\mapsto\int_{y\in X}\big(\alpha(x)+\alpha(y)\big)^n\wedge \omega(y)=
C\alpha(x)
$$
where $C=n\int_X\alpha(y)^{n-1}\wedge\omega(y)$. Hence $T\equiv C\alpha$,
which implies that $\{\alpha\}$ contains a K\"ahler current. 
Theorem 18.5 is proved.\qed
\medskip

\noindent
{\it End of Proof of Theorems 18.1 and 18.2}. Clearly the open cone $\cK$ 
is contained in $\cP$, hence in order to show that $\cK$ is one of the
connected components of $\cP$, we need only show that $\cK$ is \ul{closed}
in $\cP$, i.e.\ that $\ol\cK\cap\cP\subset\cK$. Pick a class
$\{\alpha\}\in\ol\cK\cap\cP$. In particular $\{\alpha\}$ is
nef and satisfies $\int_X\alpha^n>0$. By Theorem~18.5 we conclude that
$\{\alpha\}$ contains a K\"ahler current $T$. However, an induction on
dimension using the assumption $\int_Y\alpha^p$ for all analytic
subsets $Y$ (we also use resolution of singularities for $Y$ at this step)
shows that the restriction $\{\alpha\}_{|Y}$ is the class of a K\"ahler
current on~$Y$. We conclude that $\{\alpha\}$ is a K\"ahler class by
18.4~(ii), therefore $\{\alpha\}\in\cK$, as desired.\qed

The projective case 18.2 is a consequence of the following variant of
Theorem 18.1.

\claim{(18.8) Corollary} Let $X$ be a compact
K\"ahler manifold. A $(1,1)$ cohomology class $\{\alpha\}$ on $X$
is K\"ahler if and only if there exists a 
K\"ahler metric $\omega$ on $X$ such that $\int_Y\alpha^k\wedge\omega^{p-k}>0$
for all irreducible analytic sets $Y$ and all $k=1,2,\ldots,p=\dim Y$.
\endclaim

\begproof{} The assumption clearly implies that
$$
\int_Y(\alpha+t\omega)^p>0
$$
for all $t\in\bR_+$, hence the half-line $\alpha+(\bR_+)\omega$ is 
entirely contained in the cone $\cP$ of numerically positive classes.
Since $\alpha+t_0\omega$ is K\"ahler for $t_0$ large, we conclude that
the half-line in entirely contained in the connected component $\cK$,
and therefore $\alpha\in\cK$.\qed
\endproof

In the projective case, we can take $\omega=c_1(H)$ for a given very ample
divisor $H$, and the condition
 $\int_Y\alpha^k\wedge\omega^{p-k}>0$ is equivalent to
$$
\int_{Y\cap H_1\cap\ldots\cap H_{p-k}}\alpha^k>0
$$
for a suitable complete intersection $Y\cap H_1\cap\ldots\cap H_{p-k}$,
$H_j\in|H|$. This shows that algebraic cycles are sufficient to test
the K\"ahler property, and the special case 18.2 follows. On the other 
hand, we can pass to the limit in 18.8 by replacing $\alpha$ 
by $\alpha+\varepsilon\omega$,
and in this way we get also a characterization of nef classes.

\claim{(18.9) Corollary} Let $X$ be a compact
K\"ahler manifold. A $(1,1)$ cohomology class $\{\alpha\}$ on $X$
is nef if and only if there exists a 
K\"ahler metric $\omega$ on $X$ such that $\int_Y\alpha^k\wedge\omega^{p-k}
\ge 0$ for all irreducible analytic sets $Y$ and all $k=1,2,\ldots,p=\dim Y$.
\endclaim

By a formal convexity argument, one can derive from 18.8 or 18.9 the following
interesting consequence about the dual of the cone $\cK$. We will not 
give the proof here, because it is just a simple tricky argument which 
does not require any new analysis.

\claim{(18.10) Theorem} Let $X$ be a compact K\"ahler manifold. 
A $(1,1)$ cohomology class $\{\alpha\}$ on $X$ is nef if and only 
for every irreducible analytic set $Y$ in $X$, $p=\dim X$
and every K\"ahler metric $\omega$ on $X$ we have
$\int_Y\alpha\wedge\omega^{p-1}\ge 0$. In other words, the dual of the
nef cone $\smash{\ol\cK}$ is the closed convex cone 
in $H^{n-1,n-1}_\bR(X)$ generated by cohomology 
classes of currents of the form $[Y]\wedge\omega^{p-1}$ in 
$H^{n-1,n-1}(X,\bR)$, where $Y$ runs over the 
collection of irreducible analytic subsets of $X$ and $\{\omega\}$ over
the set of K\"ahler classes of~$X$.
\endclaim

Our main Theorem 18.1 has an important application to the deformation theory of
compact K\"ahler manifolds.

\claim{(18.11) Theorem} Let $\pi:\cX\to S$ be a deformation of compact K\"ahler
manifolds over an irreducible base~$S$. Then there exists a countable union 
$S'=\bigcup S_\nu$ of analytic subsets $S_\nu\subsetneq S$, such that 
the K\"ahler cones $\cK_t\subset H^{1,1}(X_t,\bC)$ of the fibers
$X_t=\pi^{-1}(t)$ are invariant over $S\ssm S'$ under 
parallel transport with respect to the $(1,1)$-projection $\nabla^{1,1}$ of
the Gauss-Manin connection $\nabla$ in the decomposition of
$$
\nabla=\pmatrix{
\nabla^{2,0} & * & 0\cr 
* & \nabla^{1,1} & * \cr
0 & * & \nabla^{0,2}\cr}
$$
on the Hodge bundle $H^2=H^{2,0}\oplus H^{1,1}\oplus H^{0,2}$.
\endclaim

We moreover conjecture that for an arbitrary deformation $\cX\to S$ of 
compact complex manifolds, the K\"ahler property is
open with respect to the countable Zariski topology on the base $S$ of
the deformation.
\smallskip

Let us recall the general fact that all fibers $X_t$ of a deformation over a 
connected base $S$ are diffeomorphic, since $\cX\to S$ is a locally trivial
differentiable bundle. This implies that the cohomology bundle
$$
S\ni t\mapsto H^k(X_t,\bC)
$$
is locally constant over the base $S$. The corresponding (flat) connection
of this bundle is called the Gauss-Manin connection, and will be denoted here
by $\nabla$. As is well known, the Hodge filtration
$$
F^p(H^k(X_t,\bC))=\bigoplus_{r+s=k,r\ge p}H^{r,s}(X_t,\bC)
$$
defines a {\sl holomorphic} subbundle of $H^k(X_t,\bC)$ (with respect to its
locally constant structure). On the other hand, the Dolbeault groups are 
given by
$$
H^{p,q}(X_t,\bC) = F^p(H^k(X_t,\bC))\cap\overline{F^{k-p}(H^k(X_t,\bC))},\qquad
k=p+q,
$$
and they form {\sl real analytic} subbundles of $H^k(X_t,\bC)$. We are 
interested especially in the decomposition
$$
H^2(X_t,\bC)=H^{2,0}(X_t,\bC)\oplus H^{1,1}(X_t,\bC)\oplus H^{0,2}(X_t,\bC)
$$
and the induced decomposition of the Gauss-Manin connection acting on $H^2$
$$
\nabla=\pmatrix{\nabla^{2,0}&*&*\cr *&\nabla^{1,1}&*\cr *&*&\nabla^{0,2}\cr}.
$$
Here the stars indicate suitable bundle morphisms -- actually with the 
lower left and upper right stars being zero by Griffiths' transversality 
property, but we do not really care here. The notation $\nabla^{p,q}$ stands 
for the induced (real analytic, not necessarily flat) connection on the
subbundle $t\mapsto H^{p,q}(X_t,\bC)$. 

\noindent
{\it Sketch of Proof of Theorem 18.11.} The result is local on the base,
hence we may assume that $S$ is contractible. Then the family is
differentiably trivial, the Hodge bundle $t\mapsto H^2(X_t,\bC)$ is
the trivial bundle and $t\mapsto H^2(X_t,\bZ)$ is a trivial lattice.
We use the existence of a relative cycle space $C^p(\cX/S)\subset C^p(\cX)$
which consists of all cycles contained in the fibres of $\pi:X\to S$.
It is equipped with a canonical holomorphic projection 
$$
\pi_p:C^p(\cX/S)\to S.
$$
We then define 
the $S_\nu$'s to be the images in $S$ of those connected components of 
$C^p(\cX/S)$ which do not project onto $S$. By the fact that the projection 
is proper on each component, we infer that $S_\nu$ is an analytic subset 
of $S$. The definition of the $S_\nu$'s imply that the cohomology classes
induced by the analytic cycles $\{[Z]\}$, $Z\subset X_t$, remain exactly 
the same for all $t\in S\ssm S'$. This result implies in its turn 
that the conditions defining the numerically positive cones $\cP_t$
remain the same, except for the fact that the spaces $H^{1,1}_\bR(X_t)\subset
H^2(X_t,\bR)$ vary along with the Hodge decomposition. At this point,
a standard calculation implies that the $\cP_t$ are invariant by parallel
transport under $\nabla^{1,1}$. This is done as follows.

Since $S$ is irreducible and $S'$ is a countable union of analytic sets, 
it follows that $S\ssm S'$ is arcwise connected by piecewise smooth analytic 
arcs. Let 
$$
\gamma:[0,1]\to S\ssm S',\qquad u\mapsto t=\gamma(u)
$$ 
be such a smooth arc,
and let $\alpha(u)\in H^{1,1}(X_{\gamma(u)},\bR)$ be a family of real 
$(1,1)$-cohomology classes which are constant by parallel transport
under $\nabla^{1,1}$. This is equivalent to assuming that 
$$
\nabla(\alpha(u))\in H^{2,0}(X_{\gamma(u)},\bC)\oplus 
H^{0,2}(X_{\gamma(u)},\bC)
$$
for all $u$. Suppose that $\alpha(0)$ is a numerically positive class
in~$X_{\gamma(0)}$. We then have
$$
\alpha(0)^p\cdot \{[Z]\}=\int_Z \alpha(0)^p>0
$$
for all $p$-dimensional analytic cycles $Z$ in $X_{\gamma(0)}$. Let us 
denote by 
$$
\zeta_Z(t)\in H^{2q}(X_t,\bZ),\qquad q=\dim X_t -p,
$$ 
the family of cohomology classes equal to $\{[Z]\}$ at $t=\gamma(0)$,
such that $\nabla\zeta_Z(t)=0$ (i.e.\ constant with respect to the Gauss-Manin
connection).  By the above discussion, $\zeta_Z(t)$ is of type
$(q,q)$ for all $t\in S$, and when $Z\subset X_{\gamma(0)}$ varies,
$\zeta_Z(t)$ generates all classes of analytic cycles in $X_t$ if
$t\in S\ssm S'$.  Since $\zeta_Z$ is $\nabla$-parallel and
$\nabla\alpha(u)$ has no component of type $(1,1)$, we find
$$
{d\over du}(\alpha(u)^p\cdot\zeta_Z(\gamma(u)) =
p \alpha(u)^{p-1}\cdot \nabla\alpha(u)\cdot \zeta_Z(\gamma(u))= 0.
$$
We infer from this that $\alpha(u)$ is a numerically positive class
for all $u\in[0,1]$. This argument shows that the set $\cP_t$ of 
numerically positive classes in $H^{1,1}(X_t,\bR)$ is invariant
by parallel transport under $\nabla^{1,1}$ over $S\ssm S'$.

By a standard result of Kodaira-Spencer [KS60] relying on elliptic PDE
theory, every K\"ahler class in $X_{t_0}$ can be deformed to a nearby 
K\"ahler class in nearby fibres $X_t$. This implies that the connected 
component of $\cP_t$ which corresponds to the K\"ahler cone $\cK_t$ must 
remain the same. The theorem is proved.\qed
\medskip

As a by-product of our techniques, especially the regularization
theorem for currents, we also get the following result
for which we refer to [DP03].

\claim{(18.12) Theorem} A compact complex manifold carries a K\"ahler
current if and only if it is bimeromorphic to a K\"ahler manifold
$($or equivalently, dominated by a K\"ahler manifold$)$.
\endclaim

This class of manifolds is called the {\it Fujiki class $\cC$}. If we 
compare this result with the solution of the Grauert-Riemenschneider
conjecture, it is tempting to make the following conjecture which would
somehow encompass both results.

\claim{(18.13) Conjecture} Let $X$ be a compact complex manifold of
dimension~$n$. Assume that $X$ possesses a nef cohomology class $\{\alpha\}$
of type $(1,1)$ such that $\int_X\alpha^n>0$. Then $X$ is in the
Fujiki class~$\cC$. $[$Also, $\{\alpha\}$ would contain a K\"ahler 
current, as it follows from Theorem 18.5 if Conjecture 18.13 is proved$\,]$.
\endclaim

We want to mention here that most of the above results were already known in 
the cases of complex surfaces (i.e.\ in dimension $2$),
thanks to the work of N.~Buchdahl [Buc99, 00] and A.~Lamari [Lam99a, 99b].

Shortly after the original [DP03] manuscript appeared 
in April 2001, Daniel Huybrechts [Huy01] informed us Theorem 18.1 can be 
used to calculate the K\"ahler cone of
a very general hyperk\"ahler manifold: the K\"ahler cone is then equal
to a suitable connected component of the positive cone defined by the
Beauville-Bogomolov quadratic form. In the case of an  arbitrary
hyperk\"ahler manifold, S.Boucksom [Bou02] later showed that a $(1,1)$ class
$\{\alpha\}$ is K\"ahler if and only if it lies in the positive part
of the Beauville-Bogomolov quadratic cone and moreover $\int_C\alpha>0$
for all \ul{\it rational curves} $C\subset X$ (see also [Huy99]).

\titleb{19.}{Cones of Curves}

In a dual way, we consider in $H^{n-1,n-1}_\bR(X)$ the cone 
$\cN$ gene\-rated by classes of positive currents $T$ of type
\hbox{$(n-1,n-1)$} (i.e., of bidimension $(1,1)$). In the projective
case, we also consider the intersection

By extension, we will say that $\overline{\cK}$ is the cone of {\it nef}
$(1,1)$-cohomology classes (even though they are not necessarily integral).
We now turn ourselves to cones in cohomo\-logy of bidegree $(n-1,n-1)$.
\bigskip

\claim{(19.1) Definition} Let $X$ be a compact K\"ahler manifold.
{\oldparindent=6.6mm
\smallskip
\item {\rm (i)} We define $\cN$ to be the $($closed$)$ convex cone in 
$H^{n-1,n-1}_\bR(X)$ gene\-rated by classes of positive currents $T$ of type
\hbox{$(n-1,n-1)$} $($i.e., of bidimension $(1,1))$.

\item{\rm (ii)} We define the cone $\cM\subset H^{n-1,n-1}_\bR(X)$ of
\ul{movable classes} to be the closure of the convex cone generated by
classes of currents of the form
$$\mu_\star(\wt\omega_1\wedge\ldots\wedge\wt\omega_{n-1})$$
where $\mu:\wt X\to X$ is an arbitrary modification $($one could just
restrict oneself to compositions of blow-ups with smooth centers$)$, and 
the $\wt\omega_j$ are K\"ahler forms on~$\wt X$. Clearly $\cM\subset\cN$.

\item {\rm (iii)} Correspondingly, we introduce the intersections
$$ 
\cN_{\NS}=\cN\cap N_1(X),\qquad \cM_{\NS}=\cM\cap N_1(X),\qquad 
$$
in the space of integral bidimension $(1,1)$-classes
$$
N_1(X):=(H^{n-1,n-1}_\bR(X)\cap H^{2n-2}(X,\bZ)/{\rm tors})\otimes_\bZ\bR.
$$
\item{\rm (iv)} If $X$ is projective, we define $\NE(X)$ to be the
convex cone generated by all effective curves. Clearly  
$\overline{\NE(X)}\subset\cN_{\NS}$.

\item{\rm (v)} If $X$ is projective, we say that $C$ is a \ul{strongly 
movable curve} if 
$$
C=\mu_\star(\wt A_1\cap\ldots\cap \wt A_{n-1})
$$ 
for suitable very ample divisors $\wt A_j$ on $\wt X$, where 
$\mu:\wt X\to X$ is a modification. We let ${\rm SME}(X)$ 
to be the convex cone generated by all strongly movable $($effective$)$ 
curves. Clearly  $\overline{{\rm SME}(X)}\subset\cM_{\NS}$.
\smallskip
\item{\rm (vi)} We say that $C$ is a \ul{movable curve} if $C=C_{t_0}$
is a member of an analytic family $(C_t)_{t\in S}$ such that 
$\bigcup_{t\in S}C_t=X$
and, as such, is a reduced irreducible $1$-cycle. We let ${\rm ME}(X)$ 
to be the convex cone generated by all movable $($effective$)$ curves.
\vskip0pt
}
\endclaim

The upshot of this definition lies in the following easy observation.

\claim{(19.2) Proposition} Let $X$ be a compact K\"ahler manifold. 
Consider the Poincar\'e duality pairing
$$
H^{1,1}(X,\bR)\times H^{n-1,n-1}_\bR(X)\lra \bR,\qquad
(\alpha,\beta)\longmapsto\int_X \alpha\wedge\beta.
$$
Then the duality pairing takes nonnegative values
\smallskip
\item{\rm (i)} for all pairs $(\alpha,\beta)\in\ol\cK\times\cN$;
\item{\rm (ii)} for all pairs $(\alpha,\beta)\in\cE\times\cM$.
\item{\rm (iii)} for all pairs $(\alpha,\beta)$ where $\alpha\in\cE$ and 
$\beta=[C_t]\in{\rm ME}(X)$ is the class of a movable curve.
\vskip0pt
\endclaim

\begproof{} (i) is obvious. In order to prove (ii), we may assume that
$\beta=\mu_\star(\wt\omega_1\wedge\ldots\wedge\wt\omega_{n-1})$ for
some modification $\mu:\wt X\to X$, where $\alpha=\{T\}$ is the class
of a positive $(1,1)$-current on $X$ and $\wt\omega_j$ are K\"ahler
forms on~$\wt X$. Then
$$
\int_X\alpha\wedge\beta=\int_X T\wedge
\mu_\star(\wt\omega_1\wedge\ldots\wedge\wt\omega_{n-1})=
\int_X \mu^* T\wedge
\wt\omega_1\wedge\ldots\wedge\wt\omega_{n-1}\ge 0.
$$
Here, we have used the fact that a closed positive $(1,1)$-current $T$
always has a pull-back $\mu^\star T$, which follows from the fact that
if $T=i\ddbar\varphi$ locally for some plurisubharmonic function in~$X$, 
we can set 
$\mu^\star T=i\ddbar(\varphi\circ\mu)$. For (iii), we suppose
$\alpha=\{T\}$ and $\beta=\{[C_t]\}$. Then we take an open covering
$(U_j)$ on $X$ such that $T=i\ddbar\varphi_j$ with suitable plurisubharmonic 
functions $\varphi_j$ on $U_j$. If we select a smooth partition of unity 
$\sum\theta_j=1$ subordinate to $(U_j)$, we then get
$$
\int_X\alpha\wedge\beta=\int_{C_t} T_{|C_t}=\sum_j\int_{C_t\cap U_j}
\theta_j i\ddbar\varphi_{j|C_t}\ge 0.
$$
For this to make sense, it should be noticed that $T_{|C_t}$ is a well 
defined closed positive $(1,1)$-current (i.e.\ measure) on $C_t$ for almost 
every $t\in S$, in the sense of Lebesgue measure. This is true only because
$(C_t)$ covers $X$, thus $\varphi_{j|C_t}$ is not identically
$-\infty$ for almost every $t\in S$. The equality in the last formula
is then shown by a regularization argument for $T$, writing $T=\lim T_k$
with $T_k=\alpha+i\ddbar\psi_k$ and a decreasing sequence of smooth almost 
plurisubharmonic potentials $\psi_k\downarrow\psi$ such that the Levi forms 
have a uniform lower bound $i\ddbar\psi_k\ge -C\omega$ (such a
sequence exists by [Dem92]). Then, writing $\alpha=i\ddbar v_j$ for some
smooth potential $v_j$ on $U_j$, we have $T=i\ddbar\varphi_j$ on $U_j$
with $\varphi_j=v_j+\psi$, and this is the decreasing limit of the smooth
approximations $\varphi_{j,k}=v_j+\psi_k$ on $U_j$. Hence 
$T_{k|C_t}\to T_{|C_t}$ for the weak topology of measures on $C_t$.\qed
\endproof

If $\cC$ is a convex cone in a finite dimensional vector space $E$,
we denote by $\cC^\smallvee$ the dual cone, i.e.\ the set of linear
forms $u\in E^\star$ which take nonnegative values on all elements of $\cC$. 
By the Hahn-Banach theorem, we always have $\cC^{\smallvee\smallvee}=
\overline{\cC}$. 

Proposition 19.2 leads to the natural question whether the 
cones $(\cK,\cN)$ and $(\cE,\cM)$ are dual under Poincar\'e duality. 
This question is addressed in the next section. Before doing so,
we observe that the algebraic and transcendental cones
of $(n-1,n-1)$ cohomology classes are related by the following 
equalities.

\claim{(19.3) Theorem} Let $X$ be a projective manifold. Then
\smallskip
\item{\rm(i)} $\overline{\NE(X)} =\cN_{\NS}$.
\smallskip
\item{\rm(ii)} $\overline{{\rm SME}(X)}=\overline{{\rm ME}(X)} =\cM_{\NS}$.
\endclaim

\begproof{} (i) It is a standard result of algebraic geometry (see e.g.\
[Har70]), that the cone of effective cone $\NE(X)$ is dual to the cone
$\overline{\cK_{\NS}}$ of nef divisors, hence 
$$\cN_{\NS}\supset\overline{\NE(X)}=\cK^\vee.$$ 
On the other hand, (19.3)~(i) implies that $\cN_{\NS}\subset\cK^\vee$, so 
we must have equality and (i) follows.

Similarly, (ii) requires a duality statement which will be established only
in the next sections, so we postpone the proof.\qed
\endproof

\titleb{20.}{Main Duality Results}

It is very well-known that the cone $\overline{\cK_{\NS}}$ of nef divisors
is dual to the cone $\cN_{\NS}$ of effective curves if $X$ is projective.
The transcendental case can be stated as follows.

\claim{(20.1) Theorem {\rm(Demailly-Paun, 2001)}} If $X$ is K\"ahler,
the cones $\overline{\cK}\subset H^{1,1}(X,\bR)$ and 
$\cN\subset H^{n-1,n-1}_\bR(X)$ are dual by Poincar\'e duality, and
$\cN$ is the closed convex cone generated by classes $[Y]\wedge\omega^{p-1}$
where $Y\subset X$ ranges over $p$-dimensional analytic subsets,
$p=1,2,\ldots, n$, and $\omega$ ranges over K\"ahler forms.
\endclaim

\begproof{} Indeed, Prop.~19.4 shows that the dual cone $\cK^\vee$ contains
$\cN$ which itself contains the cone $\cN'$ of all classes of the form
$\{[Y]\wedge\omega^{p-1}\}$. The main result of [DP03] conversely shows
that the dual of $(\cN')^\vee$ is equal to $\overline\cK$, so we must
have
$$\cK^\vee = \overline{\cN'} = \cN.\eqno\square$$
\endproof

The other important duality result is the following characterization of
pseudo-effective classes, proved in [BDPP03] (the ``only if'' part 
already follows from 19.4~(iii)).

\claim{(20.2) Theorem} If $X$ is projective, then a class $\alpha\in\NS_\bR(X)$
is pseudo-effective if $($and only if$\,)$ it is in the dual cone of the cone
$\SME(X)$ of strongly movable curves.
\endclaim

In other words, a line bundle $L$ is pseudo-effective if (and
only if) $L\cdot C\ge 0$ for all {\it movable curves}, i.e., $L\cdot C\ge 0$
for every very generic curve $C$ (not contained in a countable union of 
algebraic subvarieties). In fact, by definition of $\SME(X)$, it is enough 
to consider only those curves $C$ which are images of generic complete 
intersection of very ample divisors on some variety $\wt X$, under
a modification $\mu:\wt X\to X$.

By a standard blowing-up argument, it also follows that a line bundle
$L$ on a normal Moishezon variety is pseudo-effective if and only if
$L \cdot C \geq 0$ for every movable curve~$C$. 
\smallskip

The K\"ahler analogue should be~:

\claim{(20.3) Conjecture} For an arbitrary compact 
K\"ahler manifold $X$, the cones $\cE$ and $\cM$ are dual.
\endclaim


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\vskip8mm

The relation between the various cones of movable curves and currents in 
(19.5) is now a rather direct consequence of Theorem 20.2. In fact, using 
ideas hinted in [DPS96], we can say a little bit more. Given 
an irreducible curve $C \subset X$, we consider its normal ``bundle''  
$N_C = \Hom(\cI/\cI^2,\cO_C)$, where $\cI$ is the ideal sheaf of $C$. 
If $C$ is a general member of a covering family $(C_t)$, then $N_C$ 
is nef. Now [DPS96] says that the dual cone of the pseudo-effective cone
of $X$ contains the closed cone spanned by curves with nef normal 
bundle, which in turn contains the cone of movable curves. In this
way we get~:

\claim{(20.4) Theorem} Let $X$ be a projective manifold. Then the following 
cones coincide.
{\oldparindent=6.6mm
\smallskip
\item {\rm (i)} the cone $\cM_{\NS}=\cM\cap N_1(X)\,;$
\smallskip
\item {\rm (ii)} the closed cone $\overline{\SME(X)}$ of strongly movable 
curves$\,;$
\smallskip
\item {\rm (iii)} the closed cone $\overline{\ME(X)}$ of movable curves$\,;$
\smallskip
\item {\rm (iv)} the closed cone $\overline{\ME_{\nef}(X)}$ of curves with nef 
normal bundle.\vskip0pt
}
\endclaim

\begproof{} We have already seen that 
$$\SME(X)\subset\ME(X)\subset\ME_{\nef}(X)\subset(\cE_{\NS})^\smallvee$$
and 
$$
\SME(X)\subset\ME(X)\subset\cM_{\NS}\subset(\cE_{\NS})^\smallvee
$$ 
by 19.4~(iii). Now Theorem 20.2 implies $(\cM_{\NS})^\smallvee =
\overline{\SME(X)}$, and 20.4 follows.\qed
\endproof

\claim{(20.5) Corollary} Let $X$ be a projective manifold and $L$ a line bundle 
on $X$.
\smallskip
\item {\rm (i)} $L$ is pseudo-effective if and only if $L \cdot C \geq 0$ 
for all curves $C$ with nef
normal sheaf $N_C$.
\smallskip
\item {\rm (ii)} If $L$ is big, then $L \cdot C > 0$ for all curves $C$ 
with nef normal sheaf $N_C$.\vskip0pt
\endclaim

20.5~(i) strenghtens results from [PSS99]. It is however not yet clear whether 
$\cM_{\NS}=\cM\cap N_1(X)$ is equal to the closed cone of curves with 
{\it ample} normal bundle (although we certainly expect this to be true).

The most important special case of Theorem 20.2 is

\claim{(20.6) Theorem} If $X$ is a projective manifold and is not uniruled,
then $K_X$ is pseudo-effective, i.e.\ $K_X\in\cE_{\NS}$.
\endclaim

\begproof{} If $K_X\notin\cE_{\NS}$, Theorem 19.2 shows that
there is a moving curve $C_t$ such that $K_X\cdot C_t<0$. The
``bend-and-break'' lemma then implies that there is family $\Gamma_t$ of 
\ul{rational curves} with $K_X\cdot\Gamma_t<0$, so $X$ is uniruled.\qed
\endproof

A stronger result is expected to be true, namely~:

\claim{(20.7) Conjecture {\rm (special case of the ``abundance conjecture'')}}
If $K_X$ is pseudo-effective, then $\kappa(X)\ge 0$.
\endclaim

\titleb{21.}{Approximation of psh functions by logarithms of holomor\-phic
functions}

The fundamental tool is the Ohsawa-Takegoshi extension theorem
in the following form ([OT87], see also [Dem00]).

\claim{(21.1) Theorem} Let $\Omega\subset\bC^n$ be a bounded
pseudoconvex domain, and let $Y\subset X$ be a nonsingular complex
submanifold defined by a section $s$ of some hermitian vector
bundle $E$ with {\rm bounded} curvature tensor on~$\Omega$. Assume
that $s$ is everywhere transverse to the zero section and that
$|s|\le e^{-1}$ on~$\Omega$. Then there is a constant $C>0$ $($depending 
only on $E)$, with the following property:
for every psh function $\varphi$ on $\Omega$, every holomorphic 
function $f$ on $Y$ with $\int_Y|f|^2|\Lambda^r(ds)|^{-2}e^{-\varphi}dV_Y
<+\infty$, there exists an extension $F$ of $f$ to $\Omega$ such that
$$
\int_\Omega{|F|^2\over|s|^{2r}(-\log|s|)^2}e^{-\varphi}dV_\Omega\le
C\int_Y{|f|^2\over|\Lambda^r(ds)|^2}e^{-\varphi}dV_Y.
$$
\endclaim

\noindent
Here we simply take $Y$ to be a point $\{z_0\}$. In this case, the
theorem says that we can find $F\in\cO(\Omega)$ with a prescribed
value $F(z_0)$, such that
$$
\int_\Omega|F|^2e^{-\varphi}d\lambda\le C|F(z_0)|^2.
$$
We now show that every psh function on a pseudoconvex open set 
$\Omega\subset\bC^n$ can be approximated very accurately by psh functions 
with analytic singularities. The main idea is taken from [Dem92]. 

\claim{(21.2) Theorem} Let $\varphi$ be a plurisubharmonic function on a
bounded pseudoconvex open set~$\Omega\subset\bC^n$. For every $m>0$, let
$\cH_\Omega(m\varphi)$ be the Hilbert space of holomorphic functions $f$
on $\Omega$ such that $\int_\Omega|f|^2e^{-2m\varphi}d\lambda<+\infty$ and
let $\varphi_m={1\over 2m}\log\sum|\sigma_\ell|^2$ where $(\sigma_\ell)$
is an orthonormal basis of~$\cH_\Omega(m\varphi)$. Then there are
constants $C_1,C_2>0$ independent of $m$ such that
\item{\rm a)}$\displaystyle\varphi(z)-{C_1\over m}\le
\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over m}
\log{C_2\over r^n}$\hfill\break
for every $z\in\Omega$ and $r<d(z,\partial\Omega)$. In particular,
$\varphi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$ and
\item{\rm b)}$\displaystyle\nu(\varphi,z)-{n\over m}\le\nu(\varphi_m,z)\le
\nu(\varphi,z)$~ for every $z\in\Omega$.
\endclaim

\begproof{} Note that $\sum|\sigma_\ell(z)|^2$ is the square of the norm
of the evaluation linear form $f\mapsto f(z)$ on $\cH_\Omega(m\varphi)$. As
$\varphi$ is locally bounded above, the $L^2$ topology is actually stronger
than the topology of uniform convergence on compact subsets of~$\Omega$.
It follows that the series $\sum|\sigma_\ell|^2$ converges uniformly on
$\Omega$ and that its sum is real analytic. Moreover we have
$$\varphi_m(z)=\sup_{f\in B(1)}{1\over m}\log|f(z)|$$
where $B(1)$ is the unit ball of $\cH_\Omega(m\varphi)$. For
$r<d(z,\partial\Omega)$, the mean value inequality applied to the psh
function $|f|^2$ implies
$$\eqalign{
|f(z)|^2&\le{1\over\pi^nr^{2n}/n!}\int_{|\zeta-z|<r}
|f(\zeta)|^2d\lambda(\zeta)\cr
&\le{1\over\pi^nr^{2n}/n!}\exp\Big(2m\sup_{|\zeta-z|<r}\varphi(\zeta)\Big)
\int_\Omega|f|^2e^{-2m\varphi}d\lambda.\cr}$$
If we take the supremum over all $f\in B(1)$ we get
$$
\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over 2m}
\log{1\over\pi^nr^{2n}/n!}
$$
and the second inequality in a) is proved. Conversely, the
Ohsawa-Takegoshi extension theorem applied
to the $0$-dimensional subvariety $\{z\}\subset\Omega$ shows that for any
$a\in\bC$ there is a holomorphic function $f$ on $\Omega$ such that
$f(z)=a$ and
$$\int_\Omega|f|^2e^{-2m\varphi}d\lambda\le C_3|a|^2e^{-2m\varphi(z)},$$
where $C_3$ only depends on $n$ and~${\rm diam}\,\Omega$. We fix $a$
such that the right hand side is~$1$. This gives the other
inequality
$$\varphi_m(z)\ge{1\over m}\log|a|=\varphi(z)-{\log C_3\over 2m}.$$
The above inequality implies $\nu(\varphi_m,z)\le\nu(\varphi,z)$. In the
opposite direction, we find
$$\sup_{|x-z|<r}\varphi_m(x)\le\sup_{|\zeta-z|<2r}\varphi(\zeta)+
{1\over m}\log{C_2\over r^n}.$$
Divide by $\log r$ and take the limit as $r$ tends to~$0$. The quotient
by $\log r$ of the supremum of a psh function over $B(x,r)$ tends to the
Lelong number at~$x$. Thus we obtain
$$
\nu(\varphi_m,x)\ge\nu(\varphi,x)-{n\over m}.\eqno\square
$$
\endproof

Theorem 21.2 implies in a straighforward manner a deep result of
[Siu74] on the analyticity of the Lelong number sublevel sets.

\claim{(21.3) Corollary} Let $\varphi$ be a plurisubharmonic function
on a complex manifold~$X$. Then, for every $c>0$, the Lelong number
sublevel set
$$
E_c(\varphi)=\big\{z\in X\,;\;\nu(\varphi,z)\ge c\big\}
$$
is an analytic subset of~$X$.
\endclaim

\begproof{} Since analyticity is a local property, it
is enough to consider the case of a psh function $\varphi$ on a 
pseudoconvex open set $\Omega\subset\bC^n$. The
inequalities obtained in 21.2~b) imply that 
$$
E_c(\varphi)=\bigcap_{m\ge m_0}E_{c-n/m}(\varphi_m).
$$
Now, it is clear that $E_c(\varphi_m)$ is the analytic set defined by the
equations $\sigma_\ell^{(\alpha)}(z)=0$ for all multi-indices $\alpha$
such that~$|\alpha|<mc$. Thus $E_c(\varphi)$ is analytic as a (countable) 
intersection of analytic sets.\qed
\endproof

\claim{(21.4) Regularization theorem for currents} 
Let $X$ be a compact complex manifold equipped with a hermitian metric
$\omega$. Let $T=\alpha+i\ddbar\psi$ be a closed $(1,1)$-current on~$X$,
where $\alpha$ is smooth and $\psi$ is a quasi-plurisubharmonic function. 
Assume that $T\ge\gamma$ for some real $(1,1)$-form $\gamma$ on $X$ with 
real coefficients. Then there exists 
a sequence $T_k=\alpha+i\ddbar\psi_k$ of closed $(1,1)$-currents such that
{\oldparindent=6.6mm
\smallskip
\item{\rm (i)} $\psi_k$ $($and thus $T_k)$ is smooth on the complement
$X\ssm Z_k$ of an analytic set $Z_k$, and the $Z_k$'s form an increasing
sequence
$$
Z_0\subset Z_1\subset\ldots\subset Z_k\subset\ldots \subset X.
$$
\smallskip
\item{\rm (ii)} There is a uniform estimate $T_k\ge \gamma-\delta_k\omega$
with $\lim\downarrow\delta_k=0$ as $k$ tends to $+\infty$.
\smallskip
\item{\rm (iii)} The sequence $(\psi_k)$ is non increasing, and we 
have $\lim\downarrow\psi_k=\psi$. As a consequence, $T_k$ converges
weakly to $T$ as  $k$ tends to $+\infty$.
\smallskip
\item{\rm (iv)} Near $Z_k$, the potential $\psi_k$ has logarithmic poles,
namely, for every $x_0\in Z_k$, there is a neighborhood $U$ of $x_0$
such that $\psi_k(z)=\lambda_k\log\sum_\ell|g_{k,\ell}|^2+O(1)$ for 
suitable holomorphic functions $(g_{k,\ell})$ on $U$ and $\lambda_k>0$.
Moreover, there is a $($global$)$ proper modification $\mu_k:\wt X_k\to X$ 
of $X$, obtained as a sequence of blow-ups with smooth centers, such that 
$\psi_k\circ\mu_k$ can be written locally on $\wt X_k$ as
$$
\psi_k\circ\mu_k(w)=\lambda_k\big(\sum n_\ell\log|\wt g_\ell|^2+f(w)\big)
$$
where $(\wt g_\ell=0)$ are local generators of suitable $($global$)$
divisors $D_\ell$ on $\wt X_k$ such that $\sum D_\ell$ has normal crossings,
$n_\ell$ are positive integers, and the $f$'s are smooth functions 
on~$\wt X_k$.
\vskip0pt
}
\endclaim

\noindent
\begproof{\it Sketch of proof.} We briefly indicate the main ideas, since the
proof can only be reconstructed by patching together arguments which
appeared in different places (although the core the proof is entirely
in [Dem92]). After replacing $T$ with $T-\alpha$, we can assume that 
$\alpha=0$ and
$T=i\ddbar\psi\ge\gamma$. Given a small $\varepsilon>0$, we select a 
covering of $X$ by open balls $B_j$ together with holomorphic coordinates 
$(z^{(j)})$ and real numbers $\beta_j$ such that
$$
0\le \gamma-\beta_j\,i\ddbar|z^{(j)}|^2\le \varepsilon\,i\ddbar|z^{(j)}|^2
\qquad\hbox{on $B_j$}
$$
(this can be achieved just by continuity of $\gamma$, after 
diagonalizing $\gamma$ at the center of the balls). We now take a partition
of unity $(\theta_j)$ subordinate to $(B_j)$ such that $\sum\theta_j^2=1$,
and define
$$
\psi_k(z)={1\over 2k}\log\sum_j\theta_j^2
e^{2k\beta_j|z^{(j)}|^2}\sum_{\ell\in\bN}|g_{j,k,\ell}|^2
$$
where $(g_{j,k,\ell})$ is a Hilbert basis of the Hilbert space of holomorphic
functions $f$ on $B_j$ such that
$$
\int_{B_j}|f|^2e^{-2k(\psi-\beta_j|z^{(j)}|^2)}<+\infty.
$$
Notice that by the Hessian estimate $i\ddbar\psi\ge\gamma\ge\beta_j\,
i\ddbar|z^{(j)}|^2$, the weight involved in the $L^2$ norm is
plurisubharmonic. It then follows from the proof of Proposition 3.7
in [Dem92] that all properties (i)--(iv) hold true, except possibly the
fact that the sequence $\psi_k$ can be chosen to be non increasing, and
the existence of the modification in (iv). However, the multiplier ideal 
sheaves of the weights \hbox{$k(\psi-\beta_j|z^{(j)}|^2)$} are generated by
the $(g_{j,k,\ell})_\ell$ on $B_j$, and these sheaves glue together into
a global coherent multiplier ideal sheaf $\cI(k\psi)$ on $X$
(see [DEL99]); the modification $\mu_k$ is then obtained by blowing-up 
the ideal sheaf $\cI(k\psi)$ so that $\mu_k^*\cI(k\psi)$ is an invertible
ideal sheaf associated with a normal crossing divisor (Hironaka [Hir64]).
The fact that $\psi_k$ can be chosen to be non increasing follows from 
a quantitative version of the ``subadditivity of multiplier ideal sheaves'' 
which is proved in Step 3 of section 16.2 (originally appeared as 
Theorem 2.2.1 in [DPS01], see also ([DEL99]).\qed
\endproof

\titleb{22.}{Zariski Decomposition and Movable Intersections}

Let $X$ be compact K\"ahler and let $\alpha\in\cE^\circ$ be in the
{\it interior} of the pseudo--effective cone. In analogy with the
algebraic context such a class $\alpha$ is called ``big'', and it can
then be represented by a {\it K\"ahler current} $T$, i.e.\ a closed
positive $(1,1)$-current $T$ such that $T\ge \delta\omega$ for some
smooth hermitian metric $\omega$ and a constant $\delta\ll 1$.

\claim{(22.1) Theorem {\rm (Demailly [Dem92], [Bou02, 3.1.24]}} If $T$
is a K\"ahler current, then one can write $T=\lim T_m$ for a sequence of 
K\"ahler currents $T_m$ which have
logarithmic poles with coefficients in ${1\over m}\bZ$, i.e.\ there
are modifications $\mu_m:X_m\to X$ such that
$$\mu_m^\star T_m=[E_m]+\beta_m$$
where $E_m$ is an effective $\bQ$-divisor on $X_m$ with coefficients
in ${1\over m}\bZ$ $($the ``fixed part''$)$ and $\beta_m$ is a closed 
semi-positive form $($the ``movable part''$)$.
\endclaim

\begproof{} This is a direct consequence of the results of section~21.
Locally we can write $T=i\ddbar\varphi$ for some strictly 
plurisubharmonic potential $\varphi$. By the Bergman kernel trick and the
Ohsawa-Takegoshi $L^2$ extension theorem, we get local approximations
$$
\varphi=\lim\varphi_m,\qquad \varphi_m(z)={1\over 2m}\log\sum_\ell
|g_{\ell,m}(z)|^2
$$
where $(g_{\ell,m})$ is a Hilbert basis of the set of holomorphic
functions which are $L^2$ with respect to the weight $e^{-2m\varphi}$.
This Hilbert basis is also a family of local generators of the globally defined
multiplier ideal sheaf $\cI(mT)=\cI(m\varphi)$. Then $\mu_m:X_m\to X$
is obtained by blowing-up this ideal sheaf, so that 
$$\mu_m^\star\cI(mT)=\cO(-mE_m).$$
We should notice that by approximating $T-{1\over m}\omega$ instead of $T$,
we can replace $\beta_m$ by $\beta_m+{1\over m}\mu^\star\omega$ which is
a big class on $X_m\,$; by playing with the multiplicities of the components
of the exceptional divisor, we could even achieve that $\beta_m$ is a 
K\"ahler class on $X_m$, but this will not be needed here.\qed
\endproof

The more familiar algebraic analogue would be to take \hbox{$\alpha=c_1(L)$}
with a big line bundle $L$ and to blow-up the base locus of $|mL|$,
$m\gg 1$, to get a $\bQ$-divisor decomposition
$$
\mu_m^\star L\sim E_m+D_m,\qquad E_m~~\hbox{effective},~~D_m~~\hbox{free}.
$$
Such a blow-up is usually referred to as a ``log resolution'' of the
linear system $|mL|$, and we say that $E_m+D_m$ is an approximate
Zariski decomposition of $L$. We will also use this terminology for K\"ahler
currents with logarithmic poles.
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\vskip4mm

\claim{(22.2) Definition} We define the \ul{volume}, or \ul{movable
self-intersection} of a big class $\alpha\in\cE^\circ$ to be
$$
\Vol(\alpha)=\sup_{T\in \alpha}\int_{\wt X}\beta^n>0
$$
where the supremum is taken over all K\"ahler currents $T\in \alpha$
with logarithmic poles, and $\mu^\star T=[E]+\beta$ with
respect to some modification $\mu:\wt X\to X$.
\endclaim

By Fujita [Fuj94] and Demailly-Ein-Lazarsfeld [DEL00], if $L$ 
is a big line bundle, we have
$$
\Vol(c_1(L))=\lim_{m\to+\infty}D_m^n=\lim_{m\to+\infty}{n!\over m^n}h^0(X,mL),
$$
and in these terms, we get the following statement.

\claim{(22.3) Proposition} Let $L$ be a big line bundle on the
projective manifold $X$.  Let $\epsilon > 0$. Then there exists a
modification $\mu: X_{\epsilon} \to X$ and a decomposition $\mu^*(L) =
E + \beta $ with $E$ an effective $\bQ$-divisor and $\beta$ a big and
nef $\bQ$-divisor such that
$$\Vol(L) -\varepsilon\le \Vol(\beta) \le \Vol(L).$$
\endclaim

It is very useful to observe that the supremum in Definition 22.2 is actually
achieved by a collection of currents whose singularities satisfy a filtering 
property.
Namely, if $T_1=\alpha+i\ddbar\varphi_1$ and $T_2=\alpha+i\ddbar\varphi_2$ 
are two K\"ahler currents with logarithmic poles in the class of $\alpha$, 
then 
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi=\max(\varphi_1,\varphi_2)\leqno(22.4)$$ 
is again a K\"ahler current with weaker 
singularities than $T_1$ and $T_2$. One could define as well 
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi={1\over 2m}\log(e^{2m\varphi_1}+e^{2m\varphi_2}),\leqno(22.4')$$
where $m={\rm lcm}(m_1,m_2)$ is the lowest common multiple of the denominators
occuring in $T_1$, $T_2$. Now, take a simultaneous log-resolution
$\mu_m:X_m\to X$ for which the singularities of $T_1$ and $T_2$ are resolved 
as $\bQ$-divisors $E_1$
and $E_2$. Then clearly the associated divisor in the decomposition 
$\mu_m^\star T=[E]+\beta$ is given by $E=\min(E_1,E_2)$. By doing so, 
the volume $\int_{X_m}\beta^n$ gets increased, as we shall see in the
proof of Theorem 22.5 below.

\claim{(22.5) Theorem {\rm (Boucksom [Bou02])}} Let $X$ be a compact K\"ahler
manifold. We denote here by $H^{k,k}_{\ge 0}(X)$ the cone of cohomology classes
of type $(k,k)$ which have non-negative intersection with all closed
semi-positive smooth forms of bidegree $(n-k,n-k)$.
\smallskip
\item{\rm(i)} For each integer $k=1,2,\ldots,n$, there exists a canonical
``movable intersection product''
$$
\cE\times\cdots\times\cE\to H^{k,k}_{\ge 0}(X), \quad
(\alpha_1,\ldots,\alpha_k)\mapsto \langle\alpha_1\cdot\alpha_2\cdots
\alpha_{k-1}\cdot \alpha_k\rangle
$$
such that $\Vol(\alpha)=\langle\alpha^n\rangle$ whenever $\alpha$ is
a big class.
\smallskip
\item{\rm(ii)} The product is increasing, homogeneous of 
degree $1$ and superadditive in each argument, i.e.\
$$
\langle\alpha_1\cdots(\alpha'_j+\alpha''_j)\cdots \alpha_k\rangle\ge
\langle\alpha_1\cdots\alpha'_j\cdots \alpha_k\rangle+
\langle\alpha_1\cdots\alpha''_j\cdots \alpha_k\rangle.
$$
It coincides with the ordinary intersection
product when the $\alpha_j\in\overline{\cK}$ are nef classes.
\smallskip
\item{\rm(iii)} 
The movable intersection product satisfies the Teissier-Hovanskii inequalities
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_n\rangle\ge
(\langle\alpha_1^n\rangle)^{1/n}\ldots(\langle\alpha_n^n\rangle)^{1/n}\qquad
\hbox{$($with $\langle\alpha_j^n\rangle=\Vol(\alpha_j)\,)$}.
$$
\smallskip
\item{\rm(iv)} For $k=1$, the above ``product'' reduces to a
$($non linear$)$ projection operator 
$$
\cE\to\cE_1,\qquad \alpha\to\langle\alpha\rangle
$$
onto a certain convex subcone $\cE_1$ of $\cE$ such that
$\overline{\cK}\subset\cE_1\subset\cE$. Moreover, there is
a ``divisorial Zariski decomposition''
$$
\alpha=\{N(\alpha)\}+\langle\alpha\rangle
$$
where $N(\alpha)$ is a uniquely defined effective divisor which is
called the ``negative divisorial part'' of $\alpha$. The map
$\alpha\mapsto N(\alpha)$ is homogeneous and subadditive, and 
$N(\alpha)=0$ if and only if $\alpha\in\cE_1$.
\smallskip
\item{\rm(v)} The components of $N(\alpha)$ always consist of
divisors whose cohomology classes are linearly independent,
especially $N(\alpha)$ has at most $\rho=\rank_\bZ\NS(X)$ components.
\vskip0pt
\endclaim

\begproof{} We essentially repeat the arguments developped in [Bou02], with
some simplifications arising from the fact that $X$ is supposed to
be K\"ahler from the start.
\smallskip

\noindent
(i) First assume that all classes $\alpha_j$ are big, i.e.\ 
$\alpha_j\in\cE^\circ$. Fix a smooth closed $(n-k,n-k)$ {\it semi-positive}
form $u$ on $X$. We select K\"ahler currents $T_j\in\alpha_j$ with
logarithmic poles, and a simultaneous log-resolution
$\mu:\wt X\to X$ such that 
$$
\mu^\star T_j=[E_j]+\beta_j.
$$
We consider the direct image current
$\mu_\star(\beta_1\wedge\ldots\wedge\beta_k)$ (which is a closed
positive current of bidegree $(k,k)$ on $X$) and the corresponding
integrals
$$
\int_{\wt X}\beta_1\wedge\ldots\wedge\beta_k\wedge\mu^\star u\ge 0.
$$
If we change the representative $T_j$ with another current $T_j'$,
we may always take a simultaneous log-resolution such that
$\mu^\star T_j'=[E_j']+\beta_j'$, and by using $(22.4')$ we can always
assume that $E'_j\le E_j$. Then $D_j=E_j-E'_j$ is an effective
divisor and we find $[E_j]+\beta_j\equiv [E'_j]+\beta'_j$, hence
$\beta'_j\equiv\beta_j+[D_j]$. A substitution in the integral implies
$$
\eqalign{
\int_{\wt X}\beta_1'\wedge\beta_2
&{}\wedge\ldots\wedge\beta_k\wedge\mu^\star u\cr
&=\int_{\wt X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u+
\int_{\wt X}[D_1]\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u\cr
&\ge \int_{\wt X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge
\mu^\star u.\cr}
$$
Similarly, we can replace successively all forms $\beta_j$ by the $\beta'_j$,
and by doing so, we find
$$
\int_{\wt X}\beta'_1\wedge\beta'_2\wedge\ldots\wedge\beta'_k\wedge\mu^\star u
\ge\int_{\wt X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u.
$$
We claim that the closed positive currents $\mu_\star(\beta_1\wedge\ldots
\wedge\beta_k)$ are uniformly bounded in mass. In fact, if $\omega$ is a 
K\"ahler metric in $X$, there exists a constant $C_j\ge 0$
such that $C_j\{\omega\}-\alpha_j$ is a K\"ahler class. Hence
$C_j\omega-T_j\equiv\gamma_j$ for some K\"ahler form $\gamma_j$ on $X$.
By pulling back with $\mu$, we find
$C_j\mu^\star \omega-([E_j]+\beta_j)\equiv\mu^\star \gamma_j$, hence 
$$\beta_j\equiv C_j\mu^\star \omega-([E_j]+\mu^\star \gamma_j).$$
By performing again a substitution in the integrals, we find
$$
\int_{\wt X}\beta_1\wedge\ldots\wedge\beta_k\wedge\mu^\star u
\le C_1\ldots C_k\int_{\wt X}\mu^\star\omega^k\wedge\mu^\star u
= C_1\ldots C_k\int_X\omega^k\wedge u
$$
and this is true especially for $u=\omega^{n-k}$. We can now arrange
that for each of the integrals associated with a countable dense family 
of forms $u$,  the supremum is achieved by a sequence of currents 
$(\mu_m)_\star(\beta_{1,m}\wedge\ldots\wedge\beta_{k,m})$ obtained
as direct images by a suitable sequence of modifications $\mu_m:
\smash{\wt X_m}\to X$.
By extracting a subsequence, we can achieve that this sequence is weakly 
convergent and we set
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle =
\mathop{\lim\uparrow}\limits_{m\to+\infty}
\{(\mu_m)_\star(\beta_{1,m}\wedge\beta_{2,m}\wedge\ldots\wedge\beta_{k,m})\}
$$
(the monotonicity is not in terms of the currents themselves, but in terms
of the integrals obtained when we evaluate against a smooth closed
semi-positive form~$u$). By evaluating against a basis of positive classes
$\{u\}\in H^{n-k,n-k}(X)$, we infer by Poincar\'e duality that the class of
$\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle$ is uniquely defined
(although, in general, the representing current is not unique). 
\smallskip

\noindent
(ii) It is indeed clear from the definition that the movable intersection 
product is homogeneous, increasing and superadditive in each argument, at least
when the $\alpha_j$'s are in $\cE^\circ$. However, we can extend the product 
to the closed cone $\cE$ by monotonicity, by setting
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle =
\mathop{\lim\downarrow}\limits_{\delta\downarrow 0}
\langle(\alpha_1+\delta\omega)\cdot(\alpha_2+\delta\omega)\cdots 
(\alpha_k+\delta\omega)\rangle
$$
for arbitrary classes $\alpha_j\in\cE$ (again, monotonicity occurs
only where we evaluate against closed semi-positive forms~$u$).
By weak compactness, the movable intersection product can always be 
represented by a closed positive current of bidegree $(k,k)$.
\smallskip

\noindent
(iii) The Teissier-Hovanskii inequalities are a direct consequence of the
fact that they hold true for nef classes, so we just have to apply them
to the classes $\beta_{j,m}$ on $\smash{\wt X_m}$ and pass to the limit.
\smallskip

\noindent
(iv) When $k=1$ and $\alpha\in\cE^0$, we have 
$$
\alpha=\lim_{m\to+\infty}\{(\mu_m)_\star T_m\}=
\lim_{m\to+\infty}(\mu_m)_\star[E_m]+\{(\mu_m)_\star\beta_m\}
$$
and $\langle\alpha\rangle=\lim_{m\to+\infty}\{(\mu_m)_\star\beta_m\}$ by 
definition. However, the images $F_m=(\mu_m)_\star F_m$ are effective
$\bQ$-divisors in $X$, and the filtering property implies that $F_m$ is a
decreasing sequence. It must therefore converge to a (uniquely defined)
limit $F=\lim F_m := N(\alpha)$ which is an effective $\bR$-divisor, 
and we get the asserted decomposition in the limit. 

Since $N(\alpha)=\alpha-\langle\alpha\rangle$ we easily see that
$N(\alpha)$ is subadditive and that $N(\alpha)=0$ if $\alpha$ is the
class of a smooth semi-positive form. When $\alpha$ is no longer a big 
class, we define
$$
\langle\alpha\rangle=\lim_{\delta\downarrow 0}\downarrow\langle\alpha+
\delta\omega\rangle,\qquad
N(\alpha)=\lim_{\delta\downarrow 0}\uparrow N(\alpha+\delta\omega)
$$
(the subadditivity of $N$ implies 
$N(\alpha+(\delta+\varepsilon)\omega)\le N(\alpha+\delta\omega)$).
The divisorial Zariski decomposition follows except maybe for the fact 
that $N(\alpha)$ might be a convergent countable sum of divisors. 
However, this will be ruled out when (v) is proved. As 
$N({\scriptscriptstyle\bullet})$ is subadditive
and homogeneous, the set $\cE_1=\{\alpha\in\cE\;;\;N(\alpha)=0\}$ is a 
closed convex conne, and we find that $\alpha\mapsto\langle\alpha\rangle$
is a projection of $\cE$ onto $\cE_1$ (according to [Bou02], $\cE_1$
consists of those pseudo-effective classes which are ``nef in 
codimension~$1$'').
\smallskip

\noindent
(v) Let $\alpha\in\cE^\circ$, and assume that $N(\alpha)$ contains
linearly dependent components $F_j$. Then already all currents
$T\in \alpha$ should be such that $\mu^\star T=[E]+\beta$ where 
$F=\mu_\star E$ contains those linearly dependent components. 
Write $F=\sum\lambda_jF_j$, $\lambda_j>0$ and assume that 
$$\sum_{j\in J}c_jF_j\equiv 0$$
for a certain non trivial linear combination. Then some of the
coefficients $c_j$ must be negative (and some other positive).
Then $E$ is numerically equivalent to
$$
E'\equiv E+t\mu^\star\Big(\sum\lambda_jF_j\Big),
$$
and by choosing $t>0$ appropriate, we obtain an effective divisor $E'$ 
which has a zero coefficient on one of the components $\mu^\star F_{j_0}$.
By replacing $E$ with $\min(E,E')$ via $(22.4')$, we eliminate the component
$\mu^\star F_{j_0}$. This is a contradiction since $N(\alpha)$
was supposed to contain $F_{j_0}$.\qed
\endproof

\claim{(22.6) Definition} For a class $\alpha\in H^{1,1}(X,\bR)$, 
we define the \ul{numerical dimension} $\num(\alpha)$ to be 
$\num(\alpha)=-\infty$ if $\alpha$ is not pseudo-effective, and
$$
\num(\alpha)=\max\{p\in\bN\;;\;\langle\alpha^p\rangle\ne 0\},\qquad
\num(\alpha)\in\{0,1,\ldots,n\}
$$
if $\alpha$ is pseudo-effective.
\endclaim

By the results of [DP03], a class is big ($\alpha\in\cE^\circ$) if and only
if $\num(\alpha)=n$. Classes of numerical dimension $0$ can be
described much more precisely, again following Boucksom [Bou02].

\claim{(22.7) Theorem} Let $X$ be a compact
K\"ahler manifold. Then the subset $\cD_0$ of irreducible divisors 
$D$ in $X$ such that $\num(D)=0$ is countable, and these divisors 
are rigid as well as their multiples. If $\alpha\in\cE$ is a 
pseudo-effective class of numerical dimension~$0$, then $\alpha$ 
is numerically equivalent to 
an effective $\bR$-divisor $D=\sum_{j\in J}\lambda_j D_j$,
for some finite subset $(D_j)_{j\in J}\subset\cD_0$ such that the 
cohomology classes $\{D_j\}$ are linearly independent and
some \hbox{$\lambda_j>0$}. If such a linear combination is of
numerical dimension~$0$, then so is any other linear combination of the
same divisors.
\endclaim

\begproof{} It is immediate from the definition that a pseudo-effective
class is of numerical dimension $0$ if and only if $\langle\alpha\rangle=0$,
in other words if $\alpha=N(\alpha)$. Thus $\alpha\equiv\sum\lambda_jD_j$
as described in 22.7, and since $\lambda_j\langle D_j\rangle\le 
\langle\alpha\rangle$, the divisors $D_j$ must themselves have numerical 
dimension $0$. There is at most one such divisor $D$ in any given cohomology
class in $NS(X)\cap\cE\subset H^2(X,\bZ)$, otherwise two such divisors 
$D\equiv D'$ would yield a blow-up $\mu:\wt X\to X$ resolving the 
intersection, and by taking
$\min(\mu^\star D,\mu^\star D')$ via $(22.4')$, we would find 
$\mu^\star D\equiv E+\beta$, $\beta\ne 0$, so that
$\{D\}$ would not be of numerical dimension $0$. This implies 
that there are at most countably many divisors of numerical 
dimension~$0$, and that these divisors are rigid as well as their
multiples.\qed
\endproof

The above general concept of numerical dimension leads to a very natural 
formulation of the abundance conjecture for non-minimal (K\"ahler) varieties.

\claim{(22.8) Generalized abundance conjecture} For an arbitrary compact 
K\"ahler manifold~$X$, the Kodaira dimension should be equal to the
numerical dimension~:
$$\kappa(X)=\num(X):=\num(c_1(K_X)).$$
\endclaim

This appears to be a fairly strong statement. In fact, it is not difficult
to show that the generalized abundance conjecture would contain the
$C_{n,m}$ conjectures.

\claim{(22.9) Remark} {\rm Using the Iitaka fibration, it is immediate to 
see that $\kappa(X)\leq\num(X)$.}
\endclaim

\claim{(22.10) Remark} {\rm It is known that abundance holds in case
\hbox{$\num(X)=-\infty$} (if $K_X$ is not pseudo-effective, no
multiple of $K_X$ can have sections), or in case $\num(X)=n$. The
latter follows from the solution of the Grauert-Riemenschneider
conjecture in the form proven in [Dem85] (see also [DP03]).

In the remaining cases, the most tractable situation is probably the
case when $\num(X)=0$.
In fact Theorem 22.7 then gives $K_X\equiv\sum\lambda_j D_j$ for some
effective divisor with numerically independent components, $\num(D_j)=0$. 
It follows that the $\lambda_j$ are rational and therefore
$$
K_X\sim \sum\lambda_j D_j+F\qquad
\hbox{where $\lambda_j\in\bQ^+$, $\num(D_j)=0$ and $F\in\Pic^0(X)$.}
\leqno(*)
$$
Especially, if we assume additionally that
$q(X)=h^{0,1}(X)$ is zero, then $mK_X$ is linearly equivalent
to an integral divisor for some multiple $m$, and it follows 
immediately that $\kappa(X)=0$. The case of a general projective
(or compact K\"ahler) manifold with \hbox{$\num(X)=0$} and positive 
irregularity $q(X)>0$ would be interesting to understand.}
\endclaim

\titleb{23.}{The Orthogonality Estimate}

The goal of this section is to show that, in an appropriate sense,
approximate Zariski decompositions are almost orthogonal.

\claim{(23.1) Theorem} Let $X$ be a projective manifold, and
let $\alpha=\{T\}\in\cE^\circ_{\NS}$ be a big class represented by
a K\"ahler current~$T$. Consider
an approximate Zariski decomposition
$$\mu_m^\star T_m = [E_m]+[D_m]$$
Then
$$
(D_m^{n-1}\cdot E_m)^2\le 20\,(C\omega)^n\big(\Vol(\alpha)-D_m^n\big)
$$
where $\omega=c_1(H)$ is a K\"ahler form and $C\ge 0$ is a constant such that
$\pm\alpha$ is dominated by $C\omega$ $($i.e., $C\omega\pm\alpha$ is nef$\,)$.
\endclaim

\begproof{} For every $t\in[0,1]$, we have
$$
\Vol(\alpha)=\Vol(E_m+D_m)\ge \Vol(tE_m+D_m).
$$
Now, by our choice of $C$, we can write $E_m$ as a difference of two
nef divisors
$$E_m=\mu^\star\alpha-D_m=\mu_m^\star(\alpha+C\omega)-(D_m+C\mu_m^\star
\omega).$$
\endproof

\claim{(23.2) Lemma} For all nef $\bR$-divisors $A$, $B$ we have
$$\Vol(A-B)\ge A^n - nA^{n-1}\cdot B$$
as soon as the right hand side is positive.
\endclaim

\begproof{} In case $A$ and $B$ are integral (Cartier) divisors, this is a 
consequence of the holomorphic Morse inequalities, [Dem01, 8.4]; one can
also argue by an elementary estimate of 
to $H^0(X, mA-B_1-\ldots-B_m)$ via the Riemann-Roch formula
(assuming $A$ and $B$ very ample, $B_1,\ldots,B_m\in|B|$ generic). If $A$ and 
$B$ are $\bQ$-Cartier, we conclude by the homogeneity of the volume. The 
general case of $\bR$-divisors follows by approximation using the
upper semi-continuity of the volume [Bou02, 3.1.26].\qed 
\endproof

\claim{(23.3) Remark} {\rm We hope that Lemma 23.2 also holds true on an
arbitrary K\"ahler mani\-fold for arbitrary nef (non neces\-sarily
integral) classes. This would follow from a generalization of
holomorphic Morse inequalities to non integral classes. However 
the proof of such a result seems technically much more involved than
in the case of integral classes.}
\endclaim
 
\claim{(23.4) Lemma} Let $\beta_1,\ldots,\beta_n$ and 
$\beta'_1,\ldots,\beta'_n$ be nef classes on a compact K\"ahler manifold
$\wt X$ such that each difference $\beta'_j-\beta_j$ is pseudo-effective. Then
the $n$-th intersection products satisfy
$$
\beta_1\cdots\beta_n \le \beta'_1\cdots\beta'_n.
$$ 
\endclaim

\begproof{} We can proceed step by step and replace just one $\beta_j$ by
$\beta'j\equiv\beta_j+T_j$ where $T_j$ is a closed positive
$(1,1)$-current and the other classes $\beta'_k=\beta_k$, $k\ne j$ are
limits of K\"ahler forms. The inequality is then obvious.\qed
\endproof

\noindent
{\it End of proof of Theorem 23.1.}
In order to exploit the lower bound of the volume, we write
$$
tE_m+D_m=A-B,\qquad
A=D_m+t\mu_m^\star(\alpha+C\omega),\quad B=t(D_m+C\mu_m^\star\omega).
$$
By our choice of the constant $C$, both $A$ and $B$ are nef. Lemma 23.2 and 
the binomial formula imply
$$\eqalign{
\Vol(tE_m+&D_m)\ge A^n - nA^{n-1}\cdot B\cr
&=D_m^n+nt\,D_m^{n-1}\cdot\mu_m^\star(\alpha+C\omega)+\sum_{k=2}^n
t^k{n\choose k}D_m^{n-k}\cdot\mu_m^\star(\alpha+C\omega)^k\cr
&\phantom{{}=D_m^n}-nt\,D_m^{n-1}\cdot(D_m+C\mu_m^\star\omega)\cr
&\phantom{{}=D_m^n}-nt^2\sum_{k=1}^{n-1}t^{k-1}{n-1\choose k}
D_m^{n-1-k}\cdot\mu_m^\star(\alpha+C\omega)^k\cdot(D_m+C\mu_m^\star\omega).
\kern-4pt\cr}
$$
Now, we use the obvious inequalities
$$
D_m\le \mu_m^\star(C\omega),\qquad
\mu_m^\star(\alpha+C\omega)\le 2\mu_m^\star(C\omega),\qquad
D_m+C\mu_m^\star\omega\le 2\mu_m^\star(C\omega)
$$
in which all members are nef (and where the inequality${}\le$ means 
that the difference of classes is pseudo-effective). We use Lemma~23.4
to bound the last summation in the estimate of the volume, and in this
way we get
$$
\Vol(tE_m+D_m)\ge D_m^n+nt D_m^{n-1}\cdot E_m-nt^2\sum_{k=1}^{n-1}2^{k+1}
t^{k-1}{n-1\choose k}(C\omega)^n.
$$
We will always take $t$ smaller than $1/10n$ so that the last summation is 
bounded by $4(n-1)(1+1/5n)^{n-2}<4ne^{1/5}<5n$. This implies
$$
\Vol(tE_m+D_m)\ge D_m^n+nt\,D_m^{n-1}\cdot E_m-5n^2t^2(C\omega)^n.
$$
Now, the choice $t={1\over 10n}(D_m^{n-1}\cdot E_m)((C\omega)^n)^{-1}$ 
gives by substituting
$$
\eqalign{
{1\over 20}{(D_m^{n-1}\cdot E_m)^2\over (C\omega)^n}
&\le \Vol(E_m+D_m)-D_m^n\le \Vol(\alpha)-D_m^n\cr}
$$
(and we have indeed $t\le {1\over 10n}$ by Lemma 23.4), whence Theorem 23.1. 
Of course, the constant 20 is certainly not optimal.\qed

\claim{(23.5) Corollary} If $\alpha\in\cE_{\NS}$, then the divisorial
Zariski decomposition $\alpha=N(\alpha)+\langle\alpha\rangle$
is such that
$$
\langle\alpha^{n-1}\rangle\cdot N(\alpha)=0.
$$
\endclaim

\begproof{} By replacing $\alpha$ by $\alpha+\delta c_1(H)$, one sees that
it is sufficient to consider the case where $\alpha$ is big. Then the 
orthogonality estimate implies
$$
(\mu_m)_\star(D_m^{n-1})\cdot (\mu_m)_\star E_m
= D_m^{n-1}\cdot(\mu_m)^\star (\mu_m)_\star E_m
\le D_m^{n-1}\cdot E_m \le C (\Vol(\alpha)-D_m^n)^{1/2}.
$$
Since $\langle\alpha^{n-1}\rangle = \lim (\mu_m)_\star(D_m^{n-1})$,
$N(\alpha)=\lim (\mu_m)_\star E_m$ and $\lim D_m^n=\Vol(\alpha)$,
we get the desired conclusion in the limit.\qed
\endproof

\titleb{24.}{Proof of the Main Duality Theorem}

The proof is reproduced from [BDPP03]. We want to show that 
$\cE_{\NS}$ and $\SME(X)$ are dual (Theorem 20.2). By 19.4~(iii) we have 
in any case
$$\cE_{\NS}\subset (\SME(X))^\smallvee.$$ 
If the inclusion is strict, there is an element
$\alpha\in\partial\cE_{\NS}$ on the boundary of $\cE_{\NS}$ which is in
the interior of $\SME(X)^\smallvee$.


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\vskip8mm

Let $\omega=c_1(H)$ be an ample class. Since $\alpha\in\partial\cE_{\NS}$, 
the class $\alpha+\delta\omega$ is big for every $\delta>0$, and since
$\alpha\in((\SME(X))^{\smallvee})^\circ$ we still have 
$\alpha-\varepsilon\omega\in(\SME(X))^{\smallvee}$ for
$\varepsilon>0$ small. Therefore
$$
\alpha\cdot \Gamma\ge\varepsilon\omega\cdot \Gamma\leqno(24.1)
$$
for every movable curve $\Gamma$. We are going to contradict (24.1).
Since $\alpha+\delta\omega$ is big, we have an approximate Zariski
decomposition
$$
\mu_\delta^\star(\alpha+\delta\omega)=E_\delta+D_\delta.
$$
We pick $\Gamma=(\mu_\delta)_\star(D_\delta^{n-1})$. By the Hovanskii-Teissier
concavity inequality
$$
\omega\cdot \Gamma\ge (\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}.
$$
On the other hand
$$
\eqalign{
\alpha\cdot \Gamma&=\alpha\cdot(\mu_\delta)_\star(D_\delta^{n-1})\cr
&=\mu_\delta^\star\alpha\cdot D_\delta^{n-1}\le
\mu_\delta^\star(\alpha+\delta\omega)\cdot D_\delta^{n-1}\cr
&=(E_\delta+D_\delta)\cdot D_\delta^{n-1}
=D_\delta^n+D_\delta^{n-1}\cdot E_\delta.\cr
}
$$
By the orthogonality estimate, we find
$$\eqalign{
{\alpha\cdot \Gamma\over
\omega\cdot \Gamma}
&\le {D_\delta^n + \big(20(C\omega)^n
(\Vol(\alpha+\delta\omega)-D_\delta^n)\big)^{1/2}\over
(\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}}\cr
&\le C'(D_\delta^n)^{1/n}+C''{(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}\over
(D_\delta^n)^{(n-1)/n}}.\cr}
$$
However, since $\alpha\in\partial\cE_{\NS}$, the class $\alpha$ cannot be big 
so
$$\lim_{\delta \to 0}D_\delta^n=\Vol(\alpha)=0.$$ 
We can also take $D_\delta$ to approximate $\Vol(\alpha+\delta\omega)$ 
in such a way that
$(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}$ tends to $0$ much faster than
$D_\delta^n$. Notice that \hbox{$D_\delta^n\ge\delta^n\omega^n$}, so in fact
it is enough to take
$$
\Vol(\alpha+\delta\omega)-D_\delta^n\le \delta^{2n}.
$$
This is the desired contradiction by (24.1).\qed
\bigskip

\claim{(24.2) Remark} {\rm If holomorphic Morse inequalities were known also
in the K\"ahler case, we would infer by the same proof that
``$\alpha$ not pseudo-effective'' implies the existence of a blow-up
$\mu:\wt X\to X$ and a K\"ahler metric 
$\wt\omega$ on $\smash{\wt X}$ such that 
$\alpha\cdot\mu_\star(\wt\omega)^{n-1} < 0$. In the special case
when $\alpha=K_X$ is not pseudo-effective, we would expect the 
K\"ahler manifold $X$ to be covered by rational curves.
The main trouble is that characteristic $p$ techniques are no longer
available. On the other hand it is tempting to approach the
question via techniques of symplectic geometry~: } \endclaim

\claim{(24.3) Question} Let $(M,\omega)$ be a compact real symplectic manifold.
Fix an almost\break complex structure $J$ compatible with $\omega$, and
for this structure, assume that\break \hbox{$c_1(M)\cdot\omega^{n-1}>0$}. 
Does it follow that $M$ is covered by rational $J$-pseudoholomorphic curves~?
\endclaim
\vskip20pt


\titleb{13.}{References}
{
\eightpoint
\letter KoMM87| 

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\article Boc48|S.\ Bochner|Curvature and Betti numbers (I) and (II)|Ann.\
of Math.|49|1948|379--390$\,$; {\bf 50} (1949), 77--93|

\article Bom70|E.\ Bombieri|Algebraic values of meromorphic maps|Invent.\ 
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\miscell Bon93|L.\ Bonavero|In\'egalit\'es de Morse holomorphes
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\miscell Bou02|S.\ Boucksom|C\^ones positifs des vari\'et\'es
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\miscell BDPP03|S.\ Boucksom, J.-P.\ Demailly, M.\ Paun, Th.\ Peternell|The 
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\article Buc99|N.\ Buchdahl|On compact K\"ahler surfaces|Ann.\ 
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\article Buc00|N.\ Buchdahl|A Nakai-Moishezon criterion for
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\article Chw49|W.L.\ Chow|On compact complex analytic varieties|Amer.\ 
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\article Dem82a|J.-P.\ Demailly|Sur les nombres de Lelong associ\'es \`a
l'image directe d'un courant positif ferm\'e|Ann.\ Inst.\ Fourier 
(Grenoble)|32|1982|37--66|
  
\article Dem82b|J.-P.\ Demailly|Estimations $L^2$ pour l'op\'erateur $\dbar$
d'un fibr\'e vectoriel holomorphe semi-positif au dessus d'une vari\'et\'e
k\"ahl\'e\-rienne compl\`ete|Ann.\ Sci.\ Ec.\ Norm.\ Sup.|15|1982|457--511|

\article Dem85a|J.-P.\ Demailly|Mesures de Monge-Amp\`ere et caract\'erisation 
g\'eo\-m\'e\-trique des vari\'et\'es alg\'ebriques affines|M\'em.\ Soc.\ 
Math.\ France (N.S.)|19|1985|1--124|

\article Dem85b|J.-P.\ Demailly|Champs magn\'etiques et in\'egalit\'es de
Morse pour la $d''$-coho\-mo\-logie|Ann.\ Inst.\ Fourier
(Grenoble)|35|1985|189--229|

\article Dem87|J.-P.\ Demailly|Nombres de Lelong g\'en\'eralis\'es, 
th\'eor\`emes d'int\'e\-gralit\'e et 
d'ana\-lyticit\'e|Acta Math.|159|1987|153--169|

\article Dem89|J.-P.\ Demailly|Transcendental proof of a generalized 
Ka\-wa\-mata-Viehweg vanishing theorem|C.\ R.\ Acad.\ Sci.\ Paris S\'er.\ 
I Math.|309|1989|123--126~~~and:\hfill\break
Proceedings of the Conference ``Geometrical and algebraical aspects
in several complex variables" held at Cetraro, Univ.\ della Calabria,
June 1989|

\miscell Dem90|J.-P.\ Demailly|Singular hermitian metrics on positive
line bundles|Proc.\ Conf.\ Complex algebraic varieties (Bayreuth,
April~2--6, 1990), edited by K.~Hulek, T.~Peternell, M.~Schneider,
F.~Schreyer, Lecture Notes in Math., Vol.~1507, Springer-Verlag, Berlin,
1992|

\article Dem92|J.-P.\ Demailly|Regularization of closed positive currents
and Intersection Theory|J.\ Alg.\ Geom.|1|1992|361--409|

\miscell Dem93a|J.-P.\ Demailly|Monge-Amp\`ere operators, Lelong numbers
and intersection theory|Complex Analysis and Geometry, Univ.\ Series in
Math., edited by V.~Ancona and A.~Silva, Plenum Press, New-York,
1993, 115--193|

\article Dem93b|J.-P.\ Demailly|A numerical criterion for very ample line
bundles|J.\ Differential Geom.|37|1993|323--374|

\miscell Dem94|J.-P.\ Demailly|$L^2$ vanishing theorems for positive
line bundles and adjunction theory|Lecture Notes of the CIME Session
``Transcendental methods in Algebraic Geometry'', Cetraro, Italy, July
1994, Ed.\ F.~Catanese, C.~Ciliberto, Lecture Notes in Math.,
Vol.~1646, 1--97|

\article Dem96|J.-P.\ Demailly|Effective bounds for very ample line
  bundles|Invent.\ Math.|124|1996|243--261|

\miscell Dem98|J.-P.\ Demailly|M\'ethodes $L^2$ et r\'esultats
  effectifs en g\'eom\'etrie alg\'ebrique|S\'em.\ Bourbaki, Novembre
  1998, expos\'e n${}^\circ$852|

\miscell Dem00|J.-P.\ Demailly|On the Ohsawa-Takegoshi-Manivel $L^2$ extension
 theorem|Proceedings of the Conference in honour of the 85th birthday
 of Pierre Lelong, Paris, September 1997, \'ed.\ P.~Dolbeault,
 Progress in Mathematics, Birkh\"auser, Vol.~{\bf 188} (2000) 47-82|

\miscell Dem01|J.-P.\ Demailly|Multiplier ideal sheaves and 
  analytic methods in algebraic geometry|Lecture Notes, School on 
  ``Vanishing theorems and effective results in Algebraic Geometry, 
  ICTP Trieste, April 2000, Publications of ICTP, 2001|

\miscell DEL00|J.-P.\ Demailly, L.\ Ein {\rm and} R.\ Lazarsfeld|A
  subadditivity property of multiplier ideals|Michigan Math.\ J.,
  special volume in honor of William Fulton, {\bf 48} (2000), 137-156|


\article DK01|J.-P.\ Demailly, J.\ Koll\'ar|Semicontinuity of complex 
  singularity exponents and K\"ahler-Einstein metrics on Fano 
  orbifolds|Ann.\ Ec.\ Norm.\ Sup|34|2001|525--556|

\article DPS94|J.-P.\ Demailly, Th.\ Peternell, M.\ Schneider|Compact
complex manifolds with numerically effective tangent bundles|J.\
Algebraic Geometry|3|1994|295--345|

\article DPS96|J.-P.\ Demailly, Th.\ Peternell, M.\ Schneider|Holomorphic
  line bundles with partially vanishing cohomology|Israel 
  Math.\ Conf.\ Proc.\ vol.|9|1996|165--198|

\article DPS01|J.-P.\ Demailly, Th.\ Peternell, M.\ 
Schneider|Pseudo-effec\-tive line bundles on compact K\"ahler 
manifolds|International Journal of Math|6|2001|689--741|

\miscell DP03|J.-P.\ Demailly, M.\ Paun|Numerical characterization
  of the K\"ahler cone of a compact K\"ahler manifold|manuscript
  Universit\'e Joseph fourier, Grenoble et Universit\'e Louis Pasteur,
  Strasbourg, septembre 2001, math.AG/0105176, accepted for
  publication in Annals of Math|

\miscell DSk79|J.-P.\ Demailly, H.\ Skoda|Relations entre les notions de
positivit\'e de P.A.~Griffiths et de S.~Nakano|S\'eminaire P.~Lelong-H.~Skoda
(Analyse), ann\'ee 1978/79, Lecture notes in Math.\ no~{\bf 822},
Springer-Verlag, Berlin (1980) 304--309|

\article DX84|H.\ Donnelly, F.\ Xavier|On the differential form spectrum 
of negatively curved Riemann manifolds|Amer.\ J.\ Math.|106|1984|169--185|

\miscell Ein94|L.\ Ein|Note on higher dimensional adjoint linear
systems|preprint 1994, personal communication to the
author|

\article EKL95|L.\ Ein, O.\ K\"uchle {\rm and} R.\ Lazarsfeld|Local
positivity of ample line bundles|J.\ Differ.\ Geom.|42|1995|193--219|

\miscell EL92|L.\ Ein and R.\ Lazarsfeld|Seshadri constants on smooth
surfaces|Journ\'ees de G\'eom\'etrie Alg\'ebrique d'Orsay, July
1992, Ast\'e\-risque {\bf 282}, 1993, 177--186|

\article EL93|L.\ Ein {\rm and} R.\ Lazarsfeld|Global generation of
pluricanonical and adjoint linear series on smooth projective
threefolds|Jour.\ of Am.\ Math.\ Soc.|6|1993|875--903|

\article ELM95|L.\ Ein, R.\ Lazarsfeld {\rm and} V.\ Masek|Global 
generation of linear series on terminal threefolds|Intern.\ J.\ 
Math.|6|1995|1--18|

\miscell ELN96|L.\ Ein, R.\ Lazarsfeld {\rm and} M.\ Nakamaye|Zero 
estimates, intersection theory, and a theorem
of Demailly|in Higher Dimensional Complex
Varieties (Trento, 1994), de Gruyter (1996), 183--207|

\article EM84|H.\ El Mir|Sur le prolongement des courants positifs
ferm\'es|Acta Math.|153|1984|1--45|

\miscell Eno93|I.\ Enoki|Kawamata-Viehweg vanishing theorem for compact
K\"ahler manifolds|Einstein metrics and Yang-Mills connections
(ed.\ T.~Mabu\-chi, S.~Mukai), Marcel Dekker, 1993, 59--68|

\article EV86|H.\ Esnault {\rm and} E.\ Viehweg|Logarithmic De Rham
complexes and vanishing theorems|Invent.\ Math.|86|1986|161--194|

\miscell EV92|H.\ Esnault {\rm and} E.\ Viehweg|Lectures on vanishing
theorems|DMV Seminar, Band {\bf 20}, Birkh\"auser Verlag, 1992|

\bookref Fed69|H.\ Federer|Geometric measure theory|Springer Verlag,
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\miscell FdB94|G.\ Fern\'andez del Busto|UCLA Thesis|1994|

\article FdB95|G.\ Fern\'andez del Busto|Bogomolov instability and
Kawamata-Viehweg vanishing|J.\ Algebr.\ Geom.|4|1995|693--700|

\article FdB96|G.\ Fernandez del Busto|A Matsusaka-type theorem on
surfaces|J.\ Algebr.\ Geom.|5|1996|513--520|

\article Fuj83|T.\ Fujita|Semipositive line bundles|J.\ Fac.\ Sci.\ Univ.\
of Tokyo|30|1983|353--378|

\miscell Fuj87|T.\ Fujita|On polarized manifolds whose adjoint bundles
are not semipositive|Algebraic Geometry, Sendai, 1985,
Adv.\ Stud.\ in Pure Math., Vol.~10, North Holland, T.~Oda (ed.),
1987, 167--178|

\miscell Fuj88|T.\ Fujita|Problem list|Conference held at the 
Taniguchi Foundation, Katata, Japan, August 1988|

\miscell Fuj93|T.\ Fujita|Remarks on Ein-Lazarsfeld criterion of
spannedness of adjoint bundles of polarized threefolds|Preprint
1993|

\article Fuj94|T.\ Fujita|Approximating Zariski decomposition of big line
bundles|Kodai Math.\ J.|17|1994|1--3|

\article Gir76|J.\ Girbau|Sur le th\'eor\`eme de Le Potier d'annulation de
la cohomologie|C.~R.\ Acad.\ Sci.\ Paris, s\'erie A|283|1976|355--358|

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die Modulr\"aume komplexer Strukturen|Publ.\ Math.\ I.H.E.S.|5|1960|233--292|

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geometry|Wiley, New York|1978|

\miscell Gri69|P.A.\ Griffiths|Hermitian differential geometry, Chern
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K.~Kodaira, Princeton Univ.\ Press, Princeton, 1969, 181--251|

\miscell Har70|R.\ Hartshorne|Ample subvarieties of algebraic 
varieties|\rm Lecture Notes in Math., Vol.~156, Springer-Verlag,
Berlin (1970)|

\article Hel97|S.\ Helmke|On Fujita's conjecture|Duke Math.\ 
J.|88|1997|201--231|

\miscell Hel98|S.\ Helmke|Global generation of adjoint linear 
systems|Communication to the Oberwolfach Conference ``Komplexe
Analysis'', September 1998|

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variety over a field of characteristic zero|Ann.\ of Math.|79|1964|109--326|

\article H\"or65|L.\ H\"ormander|$L^2$ estimates and existence theorems for
the $\ol\partial$ operator|Acta Math.|113|1965|89--152|

\bookref H\"or66|L.\ H\"ormander|An introduction to Complex Analysis in
several variables|1966, 3rd edition, North-Holland Math.\ 
Libr., vol.7, Amsterdam, London|1990|

\article Huy99|D.\ Huybrechts|Compact Hyperk\"ahler Manifolds:
  Basic Results|Invent.\break Math.|135|1999|63--113|

\miscell Huy01|D.\ Huybrechts|The Projectivity Criterion for
  Hyperk\"ahler manifolds as a Consequence of the Demailly-Paun
  Theorem|personal communication, manuscript K\"oln University, May
  2001, 3~p, to appear in Invent.\ Math|

\article Kaw82|Y.\ Kawamata|A generalization of Kodaira-Ramanujam's
va\-nishing theorem|Math.\ Ann.|261|1982|43--46|

\article Kaw84|Y.\ Kawamata|The cone of curves of algebraic varieties|Ann.\
of Math.|119|1984|603--633|

\article Kaw85|Y.\ Kawamata|Pluricanonical systems of minimal
algebraic varieties|Invent.\ Math.|79|1985|567--588|

\article Kaw97|Y.\ Kawamata|On Fujita's freeness conjecture 
for $3$-folds and $4$-folds|Math.\ Ann.|308|1997|491--505|

\miscell Kaw98|Y.\ Kawamata|On the extension problem of
pluricanonical forms|Preprint Univ.\ of Tokyo at Komaba,
September 1998|

\article Kaw99|Y.\ Kawamata|Deformation of canonical 
singularities|J.~Amer.\ Math.\ Soc.|12|1999|85--92|

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plu\-ri\-sub\-harmonic functions|Invent.\ Math.|49|1978|137--148|

\article Kis79|C.O.\ Kiselman|Densit\'e des fonctions
plurisousharmoniques|Bull.\ Soc.\ Math.\ France|107|1979|295--304|

\article Kis84|C.O.\ Kiselman|Sur la d\'efinition de l'op\'erateur de
Monge-Amp\`ere complexe|Analyse Complexe, Proceedings of the Jour\-n\'ees
Fermat (SMF), Toulouse, 1983, Lecture Notes in Math., Vol.\ 1094,
Springer-Verlag|\rm Berlin|1984|139--150|

\article Kle66|S.\ Kleiman|Toward a numerical theory of
  ampleness|Ann.\ Math.|84|1966|293--344|

\miscell KMM87|Y.\ Kawamata, K.\ Matsuda {\rm and} K.\ Matsuki|Introduction
to the minimal problem|Algebraic Geometry, Sendai, 1985,
Adv.\ Stud.\ in Pure Math., Vol.~10, T.~Oda (ed.), North Holland, Amsterdam,
1987, 283--360|

\article KobO73|S.\ Kobayashi {\rm and} T.\ Ochiai|Characterisations of
complex projective spaces and hyperquadrics|J.\ Math.\ Kyoto
Univ.|13|1973|31--47|

\article Kod53|K.\ Kodaira|On a differential geometric method in the theory
of analytic stacks|Proc.\ Nat.\ Acad.\ Sci.\ USA|39|1953|1268--1273|

\article Kod54|K.\ Kodaira|On K\"ahler varieties of restricted type|Ann.\
of Math.|60|1954|28--48|

\miscell Kol85|J.\ Koll\'ar|Vanishing theorems for cohomology groups|
Algebraic Geometry, Bowdoin 1985, Proc.\ Symp.\ Pure Math.,
Vol.~46, 1987, 233--243|

\article Kol92|J.\ Koll\'ar|Effective basepoint freeness|Math.\
Ann.|296|1993|595--605|

\article KoM83|J.\ Koll\'ar {\rm and} T.\ Matsusaka|Riemann-Roch type
inequalities|Amer.\ J.\ of Math.|105|1983|229--252|

\article KoMM92|J.\ Koll\'ar, Y.\ Miyaoka {\rm and} S.\ Mori|Rational
connectedness and boundedness of Fano manifolds|J.\ Differential
Geom.|36|1992|765--779|

\article KS60|K.\ Kodaira, D.C.\ Spencer|On deformations of
  complex analytic structures. III. Stability theorems for complex
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\article Lam99a|A.\ Lamari|Courants k\"ahl\'eriens et surfaces
  compactes|Ann.\ Inst.\ Fourier|49|1999|263--285|
  
\article Lam99b|A.\ Lamari|Le cone K\"ahl\'erien d'une surface|J.\ 
  Math.\ Pures Appl.|78|1999|249--263|

\miscell Laz97|R.\ Lazarsfeld, {\rm with the assistance of} G.\ Fern\'andez
del Busto|Lec\-tu\-res on linear series|Complex Algebraic Geometry
(Park City, UT, 1993), Amer.\ Math.\ Soc., Providence, RI (1997), 
161--219|

\miscell Laz99|R.\ Lazarsfeld|Private communication|University of Ann Arbor,
1999|

\article Lel57|P.\ Lelong|Int\'egration sur un ensemble analytique 
complexe|Bull.\ Soc.\ Math.\ France|85|1957|239--262|

\miscell Lel67|P.\ Lelong|Fonctionnelles analytiques et fonctions enti\`eres
($n$ variables)|S\'em.\ de Math.\ Sup\'erieures, 6e session, \'et\'e 1967,
Presses Univ.\ Montr\'eal 1968|

\bookref Lel69|P.\ Lelong|Plurisubharmonic functions and positive differential
forms|Gordon and Breach, New-York, and Dunod, Paris|1969|

\miscell Lel72|P.\ Lelong|El\'ements extr\^emaux sur le c\^one des
courants positifs ferm\'es|S\'eminaire P. Lelong (Analyse), ann\'ee
1971/72, Lecture Notes in Math., Vol.~332, Springer-Verlag, Berlin,
(1972) 112--131|

\article LP75|J.\ Le Potier|Annulation de la cohomologie \`a valeurs dans un
  fibr\'e vectoriel holomorphe de rang quelconque|Math.\ Ann.|218|1975|35--53|

\article Man93|L.\ Manivel|Un th\'eor\`eme de prolongement $L^2$ de
sections holomorphes d'un fibr\'e vectoriel|Math.\
Zeitschrift|212|1993|107--122|

\article Mat72|T.\ Matsusaka|Polarized varieties with a given Hilbert
polynomial|Amer.\ J.\ of Math.|94|1972|1027--1077|

\article Mor82|S.\ Mori|Threefolds whose canonical bundles are not
numerically effective|Ann.\ of Math.|116|1982|133--176|

\article Mou99|Ch.\ Mourougane|Th\'eor\`emes d'annulation g\'en\'eriques pour
les fibr\'es vectoriels semi-n\'egatifs|Bull.\ Soc.\ Math.\ 
Fr.|127|1999|115--133|

\article Nad89|A.M.\ Nadel|Multiplier ideal sheaves and 
  K\"ahler-Einstein metrics of positive scalar curvature|Proc.\ Nat.\
  Acad.\ Sci.\ U.S.A.|86|1989|7299--7300~~~and~~
  Annals of Math., {\bf 132} (1990), 549--596|

\article Nak55|S.\ Nakano|On complex analytic vector bundles|J.~Math.\
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\article Ohs88|T.\ Ohsawa|On the extension of $L^2$ holomorphic
functions, II|Publ.\ RIMS, Kyoto Univ.|24|1988|265--275|

\miscell Ohs94|T.\ Ohsawa|On the extension of $L^2$ holomorphic functions,
IV$\,$: A new density concept|Mabuchi, T.\ (ed.) et al.,
Geometry and analysis on complex manifolds. Festschrift for Professor
S.~Kobayashi's 60th birthday. Singapore: World Scientific,
(1994) 157--170|

\article Ohs95|T.\ Ohsawa|On the extension of $L^2$ holomorphic
functions, III$\,$: negligible weights|Math.\ Zeitschrift|219|1995|215--225|

\article OT87|T.\ Ohsawa, K.\ Takegoshi|On the extension of $L^2$
  holomorphic functions|Math.\ Zeitschrift|195|1987|197--204|

\article Pau98a|M.\ Paun|Sur l'effectivit\'e num\'erique des
  images inverses de fibr\'es en droites|Math.\ Ann.|310|1998|411--421|
  
\miscell Pau98b|M.\ Paun|Fibr\'e en droites num\'eriquement
  effectifs et vari\'et\'es K\"ahl\'eriennes compactes \`a courbure de
  Ricci nef|Th\`ese, Universit\'e de Grenoble 1 (1998), 80$\,$p|
  
\miscell Pau00|M.\ Paun|Semipositive $(1,1)$-cohomology classes on
  projective manifolds|Pre\-print de l'I.R.M.A.\ Strasbourg|

\article PSS99|Th.\ Peternell, M.\ Schneider, A.J.\ Sommese|Kodaira 
  dimension of subvarieties|Intl.\ J.\ Math.|10|1999|1065--1079|

\article Rei88|I.\ Reider|Vector bundles of rank 2 and linear systems on
algebraic surfaces|Ann.\ of Math.|127|1988|309--316|

\article Ri68|R.\ Richberg|Stetige streng pseudokonvexe Funktionen|Math.\ 
Ann.|175|1968|257--286|

\miscell Sak88|F.\ Sakai|Reider-Serrano's method on normal surfaces|Proc.\
Intern.\ Conf.\ on Algebraic Geometry (L'Aquila, June 1988),
Lecture Notes in Math., Vol.~1417, Springer-Verlag, Berlin, 1990, 301--319|

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f\"ur positive holomorphe Vektorraumb\"undel|Manuscripta Math.|11|1974|95--101|

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dans le cas o\`u $X/G$ est compact|S\'em.\ Cartan (1953-54) 2--1 \`a 2--9|

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Math.|29|1955|9--26|

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analytique|Ann.\ Inst.\ Fourier (Grenoble)|6|1956|1--42|

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complex manifolds|Progress in Math.\ no~{\bf 56}, Birkh\"auser|1985|

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en analyse complexe|Duke Math.\ J.|52|1985|157--197|

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Mannigfaltig\-keiten~|Nachrichten der Akademie der Wissenschaften in
G\"ottingen, Math.-Phys.\ Klasse|4|1955|71--77|

\article Siu74|Y.T.\ Siu|Analyticity of sets associated to Lelong numbers
and the extension of closed positive currents|Invent.\ Math.|27|1974|53--156|

\article Siu84|Y.T.\ Siu|A vanishing theorem for semipositive line
bundles over non-K\"ahler manifolds|J.\ Diff.\ Geom.|19|1984|431--452|


\article Siu93|Y.T.\ Siu|An effective Matsusaka big theorem|Ann.\ Inst.\
Fourier|43|1993|1387--1405|

\miscell Siu95|Y.T.\ Siu|Very ampleness criterion of double
adjoint of ample line bundles|Princeton Conf.\ in honor of Robert C.~Gunning 
and Joseph J.~Kohn, Princeton University, NJ, USA, Mar.~16-20, 1992;
Ann.\ Math.\ Stud.\ {\bf 137} (1995), 291--318|

\article Siu96|Y.T.\ Siu|Effective Very Ampleness|Invent.\ 
Math.|124|1996|563--571|

\article Siu98|Y.T.\ Siu|Invariance of Plurigenera|Invent.\ 
Math.|134|1998|661--673|

\article Sko72a|H.\ Skoda|Sous-ensembles analytiques d'ordre fini ou infini
dans $\bC^n$|Bull.\ Soc.\ Math.\ France|100|1972|353--408|

\article Sko72b|H.\ Skoda|Applications des techniques $L^2$ \`a la th\'eorie
des id\'eaux d'une alg\`ebre de fonctions holomorphes avec poids|Ann.\
Scient.\ Ec.\ Norm.\ Sup.\ 4e S\'erie|5|1972|545--579|

\article Sko75|H.\ Skoda|Estimations $L^2$ pour l'op\'erateur $\ol\partial$
  et applications arith\-m\'etiques|S\'emi\-naire P.\ Lelong (Analyse), ann\'ee
  1975/76, Lecture Notes in Math., Vol.~538, 
  Springer-Verlag|\rm Berlin|1977|314--323|

\article Sko82|H.\ Skoda|Prolongement des courants positifs ferm\'es de masse
  finie|Invent.\ Math.|66|1982|361--376|

\article Som78|A.J.\ Sommese|Submanifolds of abelian varieties|Math.\
Ann.|233|1978|229--256|

\article Sug88|K-I.\ Sugiyama|A geometry of K\"ahler cones|
Math.\ Ann.|281|1988|135--144|

\article Tak97|K.\ Takegoshi|On cohomology groups of nef line
bundles tensorized with multiplier ideal sheaves on compact
K\"ahler manifolds|Osaka J.\ Math.|34|1997|783--802|

\article Tei79|B.\ Teissier|Du th\'eor\`eme de l'index de Hodge aux
in\'egalit\'es isop\'eri\-m\'etriques|C.\ R.\ Acad.\ Sc.\ Paris,
s\'er.\ A|288|29\ {\rm Janvier}\ 1979|287--289|

\miscell Tei82|B.\ Teissier|Bonnesen-type inequalities in algebraic
geometry|Sem.\ on Diff.\ Geom.\ edited by S.T.\ Yau, 1982,
Princeton Univ.\ Press, 85--105|

\article Thi67|P.\ Thie|The Lelong number of a point of a complex analytic 
set|Math.\ Annalen|172|1967|269--312|

\article Tra95|S.\ Trapani|Numerical criteria for the positivity of the
difference of ample divisors|Math.\ Zeitschrift|219|1995|387--401|

\article Vie82|E.\ Viehweg|Vanishing theorems|J.\ Reine Angew.\ 
Math.|335|1982|1--8|

\article Yau78|S.-T.\ Yau|On the Ricci curvature of a complex
  K\"ahler manifold and the complex Monge-Amp\`ere equation|Comm.\ 
  Pure Appl.\ Math.|31|1978|339--411|

}

\vskip30pt
\noindent(June 18, 2007, printed on \today)
\vskip10pt
\noindent
Jean-Pierre Demailly\hfill\break
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, UMR 5582 du CNRS, BP$\,$74,\hfill\break
F-38402 Saint-Martin d'H\`eres, France

\noindent{\it e-mail address:}\/ demailly@fourier.ujf-grenoble.fr
\end
\fi

\ifnum\fullversion=0
\titleb{}{\kern-3.7pt References}
{
\eightpoint
\letter KoMM87| 

\article AN54|Y.\ Akizuki {\rm and} S.\ Nakano|Note on Kodaira-Spencer's
proof of Lefschetz theorems|Proc.\ Jap.\ Acad.|30|1954|266--272|

\article AV65|A.\ Andreotti {\rm and} E.\ Vesentini|Carleman estimates
for~the Laplace-Beltrami equation in complex manifolds|Publ.\ Math.\ 
I.H.E.S.|25|1965|81--130|

\article Boc48|S.\ Bochner|Curvature and Betti numbers (I) and (II)|Ann.\
of Math.|49|1948|379--390$\,$; {\bf 50} (1949), 77--93|

\article Bom70|E.\ Bombieri|Algebraic values of meromorphic maps|Invent.\ 
Math.|10|1970|267--287 and {\it Addendum}, Invent.\ Math.\ {\bf 11} 
(1970), 163--166|    

\article Bom73|E.\ Bombieri|Canonical models of surfaces of general 
type|Publ.\ Math.\ IHES|42|1973|171--219|

\miscell Bon93|L.\ Bonavero|In\'egalit\'es de Morse holomorphes
singuli\`eres|Pr\'epu\-blication Universit\'e de Grenoble~I, Institut
Fourier, n${}^\circ\,259$, 1993|

\article BT82|E.\ Bedford {\rm and} B.A.\ Taylor|A new capacity for 
plurisubharmonic functions|Acta Math.|149|1982|1--41|

\article Chw49|W.L.\ Chow|On compact complex analytic varieties|Amer.\ 
J.~Math.|71|1949|893-914|

\bookref CLN69|S.S.\ Chern, H.I.\ Levine {\rm and} L.\ Nirenberg|Intrinsic
norms on a complex manifold|Global Analysis (papers in honor of K.Kodaira), 
p.~119--139, Univ.\ of Tokyo Press, Tokyo|1969|

\article Dem82a|J.-P.\ Demailly|Sur les nombres de Lelong associ\'es \`a
l'image directe d'un courant positif ferm\'e|Ann.\ Inst.\ Fourier 
(Grenoble)|32|1982|37--66|

\article Dem82b|J.-P.\ Demailly|Estimations $L^2$ pour l'op\'erateur $\dbar$
d'un fibr\'e vectoriel holomorphe semi-positif au dessus d'une vari\'et\'e
k\"ahl\'e\-rienne compl\`ete|Ann.\ Sci.\ Ec.\ Norm.\ Sup.|15|1982|457--511|

\article Dem85a|J.-P.\ Demailly|Mesures de Monge-Amp\`ere et caract\'erisation 
g\'eo\-m\'e\-trique des vari\'et\'es alg\'ebriques affines|M\'em.\ Soc.\ 
Math.\ France (N.S.)|19|1985|1--124|

\article Dem85b|J.-P.\ Demailly|Champs magn\'etiques et in\'egalit\'es de
Morse pour la $d''$-coho\-mo\-logie|Ann.\ Inst.\ Fourier
(Grenoble)|35|1985|189--229|

\article Dem87|J.-P.\ Demailly|Nombres de Lelong g\'en\'eralis\'es, 
th\'eor\`emes d'int\'e\-gralit\'e et 
d'analyticit\'e|Acta Math.|159|1987|153--169|

\article Dem89|J.-P.\ Demailly|Transcendental proof of a generalized 
Ka\-wa\-mata-Viehweg vanishing theorem|C.\ R.\ Acad.\ Sci.\ Paris S\'er.\ 
I Math.|309|1989|123--126~~~and:\hfill\break
Proceedings of the Conference ``Geometrical and algebraical aspects
in several complex variables" held at Cetraro, Univ.\ della Calabria,
June 1989|

\miscell Dem90|J.-P.\ Demailly|Singular hermitian metrics on positive
line bundles|Proc.\ Conf.\ Complex algebraic varieties (Bayreuth,
April~2--6, 1990), edited by K.~Hulek, T.~Peternell, M.~Schneider,
F.~Schreyer, Lecture Notes in Math., Vol.~1507, Springer-Verlag, Berlin,
1992|

\article Dem92|J.-P.\ Demailly|Regularization of closed positive currents
and Intersection Theory|J.\ Alg.\ Geom.|1|1992|361--409|

\miscell Dem93a|J.-P.\ Demailly|Monge-Amp\`ere operators, Lelong numbers
and intersection theory|Complex Analysis and Geometry, Univ.\ Series in
Math., edited by V.~Ancona and A.~Silva, Plenum Press, New-York,
1993, 115--193|

\article Dem93b|J.-P.\ Demailly|A numerical criterion for very ample line
bundles|J.\ Differential Geom.|37|1993|323--374|

\article DPS94|J.-P.\ Demailly, Th.\ Peternell, M.\ Schneider|Compact
complex manifolds with numerically effective tangent bundles|J.\
Algebraic Geometry|3|1994|295--345|

\article EM84|H.\ El Mir|Sur le prolongement des courants positifs
ferm\'es|Acta Math.|153|1984|1--45|

\miscell Eno93|I.\ Enoki|Kawamata-Viehweg vanishing theorem for compact
K\"ahler manifolds|Einstein metrics and Yang-Mills connections
(ed.\ T.~Mabu\-chi, S.~Mukai), Marcel Dekker, 1993, 59--68|

\article EV86|H.\ Esnault {\rm and} E.\ Viehweg|Logarithmic De Rham
complexes and vanishing theorems|Invent.\ Math.|86|1986|161--194|

\miscell EV92|H.\ Esnault {\rm and} E.\ Viehweg|Lectures on vanishing
theorems|DMV Seminar, Band {\bf 20}, Birkh\"auser Verlag, 1992|

\bookref Fed69|H.\ Federer|Geometric measure theory|Springer Verlag,
Vol. 153, Berlin, Heidelberg, New-York|1969|

\article Fuj83|T.\ Fujita|Semipositive line bundles|J.\ Fac.\ Sci.\ Univ.\
of Tokyo|30|1983|353--378|

\article Gir76|J.\ Girbau|Sur le th\'eor\`eme de Le Potier d'annulation de
la cohomologie|C.~R.\ Acad.\ Sci.\ Paris, s\'erie A|283|1976|355--358|

\article Gra60|H.\ Grauert|Ein Theorem der analytischen Garbentheorie und
die Modulr\"aume komplexer Strukturen|Publ.\ Math.\ I.H.E.S.|5|1960|233--292|

\miscell Gri69|P.A.\ Griffiths|Hermitian differential geometry, Chern
clas\-ses and positive vector bundles|Global Analysis, papers in honor of
K.~Kodaira, Princeton Univ.\ Press, Princeton, 1969, 181--251|

\bookref GrH78|P.A.\ Griffiths, J.\ Harris|Principles of algebraic
geometry|Wiley, New York|1978|

\miscell Har70|R.\ Hartshorne|Ample subvarieties of algebraic 
varieties|\rm Lecture Notes in Math., Vol.~156, Springer-Verlag,
Berlin (1970)|

\article Hir64|H.\ Hironaka|Resolution of singularities of an algebraic 
variety over a field of characteristic zero|Ann.\ of Math.|79|1964|109--326|

\article H\"or65|L.\ H\"ormander|$L^2$ estimates and existence theorems for
the $\ol\partial$ operator|Acta Math.|113|1965|89--152|

\bookref H\"or66|L.\ H\"ormander|An introduction to Complex Analysis in
several variables|1966, 3rd edition, North-Holland Math.\ 
Libr., vol.7, Amsterdam, London|1990|

\article Kaw82|Y.\ Kawamata|A generalization of Kodaira-Ramanujam's
va\-nishing theorem|Math.\ Ann.|261|1982|43--46|

\article Kis84|C.O.\ Kiselman|Sur la d\'efinition de l'op\'erateur de
Monge-Amp\`ere complexe|Analyse Complexe, Proceedings of the Jour\-n\'ees
Fermat (SMF), Toulouse, 1983, Lecture Notes in Math., Vol.\ 1094,
Springer-Verlag|\rm Berlin|1984|139--150|

\article Kod53|K.\ Kodaira|On a differential geometric method in the theory
of analytic stacks|Proc.\ Nat.\ Acad.\ Sci.\ USA|39|1953|1268--1273|

\article Kod54|K.\ Kodaira|On K\"ahler varieties of restricted type|Ann.\
of Math.|60|1954|28--48|

\miscell Kol85|J.\ Koll\'ar|Vanishing theorems for cohomology groups|
Algebraic Geometry, Bowdoin 1985, Proc.\ Symp.\ Pure Math.,
Vol.~46, 1987, 233--243|

\miscell Laz04|R.\ Lazarsfeld|Positivity in Algebraic Geometry I, II|Ergebnisse
der Mathematik und ihrer Grenzgebiete, Springer, Vol.\ 48 and 49, Berlin,
Heidelberg, New-York, 2004|

\article Lel57|P.\ Lelong|Int\'egration sur un ensemble analytique 
complexe|Bull.\ Soc.\ Math.\ France|85|1957|239--262|

\miscell Lel67|P.\ Lelong|Fonctionnelles analytiques et fonctions enti\`eres
($n$ variables)|S\'em.\ de Math.\ Sup\'erieures, 6e session, \'et\'e 1967,
Presses Univ.\ Montr\'eal 1968|

\bookref Lel69|P.\ Lelong|Plurisubharmonic functions and positive differential
forms|Gordon and Breach, New-York, and Dunod, Paris|1969|

\article LP75|J.\ Le Potier|Annulation de la cohomologie \`a valeurs dans un
fibr\'e vectoriel holomorphe de rang quelconque|Math.\ Ann.|218|1975|35--53|

\article Nad89|A.M.\ Nadel|Multiplier ideal sheaves and 
K\"ahler-Einstein metrics of positive scalar curvature|Proc.\ Nat.\
Acad.\ Sci.\ U.S.A.|86|1989|7299--7300~~~and~~
Annals of Math., {\bf 132} (1990), 549--596|

\article Nak55|S.\ Nakano|On complex analytic vector bundles|J.~Math.\
Soc.\ Japan|7|1955|1--12|

\article Sch74|M.\ Schneider|Ein einfacher Beweis des Verschwindungs\-satzes
f\"ur positive holomorphe Vektorraumb\"undel|Manuscripta Math.|11|1974|95--101|

\miscell Ser54|J.-P.\ Serre|Fonctions automorphes: quelques majorations
dans le cas o\`u $X/G$ est compact|S\'em.\ Cartan (1953-54) 2--1 \`a 2--9|

\article Ser55|J.-P.\ Serre|Un th\'eor\`eme de dualit\'e|Comment.\
Math.|29|1955|9--26|

\article Ser56|J.-P.\ Serre|G\'eom\'etrie alg\'ebrique et g\'eom\'etrie
analytique|Ann.\ Inst.\ Fourier (Grenoble)|6|1956|1--42|

\article Sie55|C.L.\ Siegel|Meromorphic Funktionen auf kompakten
Mannigfaltig\-keiten~|Nachrichten der Akademie der Wissenschaften in
G\"ottingen, Math.-Phys.\ Klasse|4|1955|71--77|

\article Siu74|Y.T.\ Siu|Analyticity of sets associated to Lelong numbers
and the extension of closed positive currents|Invent.\ Math.|27|1974|53--156|

\article Sko72a|H.\ Skoda|Sous-ensembles analytiques d'ordre fini ou infini
dans $\bC^n$|Bull.\ Soc.\ Math.\ France|100|1972|353--408|

\article Sko72b|H.\ Skoda|Applications des techniques $L^2$ \`a la th\'eorie
des id\'eaux d'une alg\`ebre de fonctions holomorphes avec poids|Ann.\
Scient.\ Ec.\ Norm.\ Sup.\ 4e S\'erie|5|1972|545--579|

\article Sko75|H.\ Skoda|Estimations $L^2$ pour l'op\'erateur $\ol\partial$
et applications arith\-m\'etiques|S\'eminaire P.\ Lelong (Analyse), ann\'ee
1975/76, Lecture Notes in Math., Vol.~538, 
Springer-Verlag|\rm Berlin|1977|314--323|

\article Sko82|H.\ Skoda|Prolongement des courants positifs ferm\'es de
masse finie|Invent.\ Math.|66|1982|361--376|

\article Som78|A.J.\ Sommese|Submanifolds of abelian varieties|Math.\
Ann.|233|1978|229--256|

\article Thi67|P.\ Thie|The Lelong number of a point of a complex analytic 
set|Math.\ Annalen|172|1967|269--312|

\article Vie82|E.\ Viehweg|Vanishing theorems|J.\ Reine Angew.\ 
Math.|335|1982|1--8|

}

\vskip20pt
\noindent(June 17, 2007, typeset on \today)
\vskip10pt
\noindent
Jean-Pierre Demailly\hfill\break
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, UMR 5582 du CNRS, BP$\,$74,\hfill\break
F-38402 Saint-Martin d'H\`eres, France

\noindent{\it e-mail address:}\/ demailly@fourier.ujf-grenoble.fr
\fi

\ifnum\fullversion=1
\titleb{}{\kern-3.7pt References}
{
\eightpoint
\letter KoMM87| 

\article AN54|Y.\ Akizuki {\rm and} S.\ Nakano|Note on Kodaira-Spencer's
proof of Lefschetz theorems|Proc.\ Jap.\ Acad.|30|1954|266--272|

\article AS95|U.\ Angehrn {\rm and} Y.T.\ Siu|Effective freeness and point
separation for adjoint bundles|Invent.\ Math.|122|1995|291-308|
 
\article Aub78|T.\ Aubin|Equations du type Monge-Amp\`ere sur les vari\'et\'es
k\"ahl\'e\-riennes compactes|C.R.\ Acad.\ Sci.\ Paris Ser.\ A {\bf
283}$\,$(1976), 119--121$\,$; Bull.\ Sci.\ Math|102|1978|63--95|

\article AV65|A.\ Andreotti {\rm and} E.\ Vesentini|Carleman estimates
for~the Laplace-Beltrami equation in complex manifolds|Publ.\ Math.\ 
I.H.E.S.|25|1965|81--130|

\article BBS89|M.\ Beltrametti, A.\ Biancofiore {\rm and} A.J.\
Sommese|Pro\-jective $n$-folds of log-general type, I|Trans.\ Amer.\
Math.\ Soc.|314|1989|825--849|

\miscell BDPP03|S.\ Boucksom, J.-P.\ Demailly, M.\ Paun, Th.\ Peternell|The 
  pseudo-effective cone of a compact K\"ahler manifold and
  varieties of negative Kodaira dimension|manuscript May 2004,
  math.AG/0405285|

\miscell BeS93|M.\ Beltrametti {\rm and} A.J.\ Sommese|On $k$-jet
ampleness|Complex Analysis and Geometry, Univ.\ Series in
Math., edited by V.~Ancona and A.~Silva, Plenum Press, New-York, 1993,
355--376|

\article BFS89|M.\ Beltrametti, P.\ Francia {\rm and} A.J.\ Sommese|On
Reider's method and higher order embeddings|Duke Math.\ J.|58|1989|425--439|

\article Boc48|S.\ Bochner|Curvature and Betti numbers (I) and (II)|Ann.\
of Math.|49|1948|379--390$\,$; {\bf 50} (1949), 77--93|

\article Bom70|E.\ Bombieri|Algebraic values of meromorphic maps|Invent.\ 
Math.|10|1970|267--287 and {\it Addendum}, Invent.\ Math.\ {\bf 11} 
(1970), 163--166|    

\miscell Bon93|L.\ Bonavero|In\'egalit\'es de Morse holomorphes
singuli\`eres|Pr\'epu\-blication Universit\'e de Grenoble~I, Institut
Fourier, n${}^\circ\,259$, 1993|

\miscell Bou02|S.\ Boucksom|C\^ones positifs des vari\'et\'es
  complexes compactes|Thesis, Grenoble 2002|

\article BSk74|J.\ Brian\c con {\rm et} H.\ Skoda|Sur la cl\^oture
int\'egrale d'un id\'eal de germes de fonctions holomorphes en un point
de $\bC^n$|C.\ R.\ Acad.\ Sc.\ Paris, s\'er.\ A|278|1974|949--951|

\article BT76|E.\ Bedford {\rm and} B.A.\ Taylor|The Dirichlet problem for the 
complex Monge-Amp\`ere equation|Invent.\ Math.|37|1976|1--44|

\article BT82|E.\ Bedford {\rm and} B.A.\ Taylor|A new capacity for 
plurisubharmonic functions|Acta Math.|149|1982|1--41|
  
\article Buc99|N.\ Buchdahl|On compact K\"ahler surfaces|Ann.\ 
  Inst.\ Fourier|49|1999|287--302|
  
\article Buc00|N.\ Buchdahl|A Nakai-Moishezon criterion for
  non-K\"ahler surfaces|Ann.\ Inst.\ Fourier|50|2000|1533--1538|

\article Cam92|F.\ Campana|Connexit\'e rationnelle des vari\'et\'es de
Fano|Ann.\ Sci.\ Ecole Norm.\ Sup.|25|1992|539--545|

\article Chw49|W.L.\ Chow|On compact complex analytic varieties|Amer.\ 
  J.~Math.|71|1949|893-914|
  
\article Dem82a|J.-P.\ Demailly|Sur les nombres de Lelong associ\'es \`a
l'image directe d'un courant positif ferm\'e|Ann.\ Inst.\ Fourier 
(Grenoble)|32|1982|37--66|

\article Dem82b|J.-P.\ Demailly|Estimations $L^2$ pour l'op\'erateur $\dbar$
d'un fibr\'e vectoriel holomorphe semi-positif au dessus d'une vari\'et\'e
k\"ahl\'e\-rienne compl\`ete|Ann.\ Sci.\ Ec.\ Norm.\ Sup.|15|1982|457--511|

\article Dem85a|J.-P.\ Demailly|Mesures de Monge-Amp\`ere et caract\'erisation 
g\'eo\-m\'e\-trique des vari\'et\'es alg\'ebriques affines|M\'em.\ Soc.\ 
Math.\ France (N.S.)|19|1985|1--124|

\article Dem85b|J.-P.\ Demailly|Champs magn\'etiques et in\'egalit\'es de
Morse pour la $d''$-coho\-mo\-logie|Ann.\ Inst.\ Fourier
(Grenoble)|35|1985|189--229|

\article Dem87|J.-P.\ Demailly|Nombres de Lelong g\'en\'eralis\'es, 
th\'eor\`emes d'int\'e\-gralit\'e et 
d'ana\-lyticit\'e|Acta Math.|159|1987|153--169|

\article Dem89|J.-P.\ Demailly|Transcendental proof of a generalized 
Ka\-wa\-mata-Viehweg vanishing theorem|C.\ R.\ Acad.\ Sci.\ Paris S\'er.\ 
I Math.|309|1989|123--126~~~and:\hfill\break
Proceedings of the Conference ``Geometrical and algebraical aspects
in several complex variables" held at Cetraro, Univ.\ della Calabria,
June 1989|

\miscell Dem90|J.-P.\ Demailly|Singular hermitian metrics on positive
line bundles|Proc.\ Conf.\ Complex algebraic varieties (Bayreuth,
April~2--6, 1990), edited by K.~Hulek, T.~Peternell, M.~Schneider,
F.~Schreyer, Lecture Notes in Math., Vol.~1507, Springer-Verlag, Berlin,
1992|

\article Dem92|J.-P.\ Demailly|Regularization of closed positive currents
and Intersection Theory|J.\ Alg.\ Geom.|1|1992|361--409|

\miscell Dem93a|J.-P.\ Demailly|Monge-Amp\`ere operators, Lelong numbers
and intersection theory|Complex Analysis and Geometry, Univ.\ Series in
Math., edited by V.~Ancona and A.~Silva, Plenum Press, New-York,
1993, 115--193|

\article Dem93b|J.-P.\ Demailly|A numerical criterion for very ample line
bundles|J.\ Differential Geom.|37|1993|323--374|

\miscell Dem94|J.-P.\ Demailly|$L^2$ vanishing theorems for positive
line bundles and adjunction theory|Lecture Notes of the CIME Session
``Transcendental methods in Algebraic Geometry'', Cetraro, Italy, July
1994, Ed.\ F.~Catanese, C.~Ciliberto, Lecture Notes in Math.,
Vol.~1646, 1--97|

\article Dem96|J.-P.\ Demailly|Effective bounds for very ample line
  bundles|Invent.\ Math.|124|1996|243--261|

\miscell Dem98|J.-P.\ Demailly|M\'ethodes $L^2$ et r\'esultats
  effectifs en g\'eom\'etrie alg\'ebrique|S\'em.\ Bourbaki, Novembre
  1998, expos\'e n${}^\circ$852|

\miscell Dem00|J.-P.\ Demailly|On the Ohsawa-Takegoshi-Manivel $L^2$ extension
 theorem|Proceedings of the Conference in honour of the 85th birthday
 of Pierre Lelong, Paris, September 1997, \'ed.\ P.~Dolbeault,
 Progress in Mathematics, Birkh\"auser, Vol.~{\bf 188} (2000) 47-82|

\miscell Dem01|J.-P.\ Demailly|Multiplier ideal sheaves and 
  analytic methods in algebraic geometry|Lecture Notes, School on 
  ``Vanishing theorems and effective results in Algebraic Geometry, 
  ICTP Trieste, April 2000, Publications of ICTP, 2001|

\miscell DEL00|J.-P.\ Demailly, L.\ Ein {\rm and} R.\ Lazarsfeld|A
  subadditivity property of multiplier ideals|Michigan Math.\ J.,
  special volume in honor of William Fulton, {\bf 48} (2000), 137-156|


\article DK01|J.-P.\ Demailly, J.\ Koll\'ar|Semicontinuity of complex 
  singularity exponents and K\"ahler-Einstein metrics on Fano 
  orbifolds|Ann.\ Ec.\ Norm.\ Sup|34|2001|525--556|

\article DPS94|J.-P.\ Demailly, Th.\ Peternell, M.\ Schneider|Compact
complex manifolds with numerically effective tangent bundles|J.\
Algebraic Geometry|3|1994|295--345|

\article DPS96|J.-P.\ Demailly, Th.\ Peternell, M.\ Schneider|Holomorphic
  line bundles with partially vanishing cohomology|Israel 
  Math.\ Conf.\ Proc.\ vol.|9|1996|165--198|

\article DPS01|J.-P.\ Demailly, Th.\ Peternell, M.\ 
Schneider|Pseudo-effec\-tive line bundles on compact K\"ahler 
manifolds|International Journal of Math|6|2001|689--741|

\miscell DP03|J.-P.\ Demailly, M.\ Paun|Numerical characterization
  of the K\"ahler cone of a compact K\"ahler manifold|manuscript
  Universit\'e Joseph fourier, Grenoble et Universit\'e Louis Pasteur,
  Strasbourg, septembre 2001, math.AG/0105176, accepted for
  publication in Annals of Math|

\miscell Gri69|P.A.\ Griffiths|Hermitian differential geometry, Chern
clas\-ses and positive vector bundles|Global Analysis, papers in honor of
K.~Kodaira, Princeton Univ.\ Press, Princeton, 1969, 181--251|

\bookref GrH78|P.A.\ Griffiths, J.\ Harris|Principles of algebraic
geometry|Wiley, New York|1978|

\miscell Har70|R.\ Hartshorne|Ample subvarieties of algebraic 
varieties|\rm Lecture Notes in Math., Vol.~156, Springer-Verlag,
Berlin (1970)|

\article Hir64|H.\ Hironaka|Resolution of singularities of an algebraic 
variety over a field of characteristic zero|Ann.\ of Math.|79|1964|109--326|

\article H\"or65|L.\ H\"ormander|$L^2$ estimates and existence theorems for
the $\ol\partial$ operator|Acta Math.|113|1965|89--152|

\bookref H\"or66|L.\ H\"ormander|An introduction to Complex Analysis in
several variables|1966, 3rd edition, North-Holland Math.\ 
Libr., vol.7, Amsterdam, London|1990|

\article Huy99|D.\ Huybrechts|Compact Hyperk\"ahler Manifolds:
  Basic Results|Invent.\break Math.|135|1999|63--113|

\miscell Huy01|D.\ Huybrechts|The Projectivity Criterion for
  Hyperk\"ahler manifolds as a Consequence of the Demailly-Paun
  Theorem|personal communication, manuscript K\"oln University, May
  2001, 3~p, to appear in Invent.\ Math|
  
\article Kle66|S.\ Kleiman|Toward a numerical theory of
  ampleness|Ann.\ Math.|84|1966|293--344|

\article Kod53|K.\ Kodaira|On a differential geometric method in the theory
of analytic stacks|Proc.\ Nat.\ Acad.\ Sci.\ USA|39|1953|1268--1273|

\article Kod54|K.\ Kodaira|On K\"ahler varieties of restricted type|Ann.\
of Math.|60|1954|28--48|

\article KS60|K.\ Kodaira, D.C.\ Spencer|On deformations of
  complex analytic structures. III. Stability theorems for complex
  structures|Ann.\ of Math.|71|1960|43--76|

\article Lam99a|A.\ Lamari|Courants k\"ahl\'eriens et surfaces
  compactes|Ann.\ Inst.\ Fourier|49|1999|263--285|
  
\article Lam99b|A.\ Lamari|Le cone K\"ahl\'erien d'une surface|J.\ 
  Math.\ Pures Appl.|78|1999|249--263|

\article LP75|J.\ Le Potier|Annulation de la cohomologie \`a valeurs dans un
  fibr\'e vectoriel holomorphe de rang quelconque|Math.\ Ann.|218|1975|35--53|

\article Nad89|A.M.\ Nadel|Multiplier ideal sheaves and 
  K\"ahler-Einstein metrics of positive scalar curvature|Proc.\ Nat.\
  Acad.\ Sci.\ U.S.A.|86|1989|7299--7300~~~and~~
  Annals of Math., {\bf 132} (1990), 549--596|

\article Nak55|S.\ Nakano|On complex analytic vector bundles|J.~Math.\
  Soc.\ Japan|7|1955|1--12|

\article OT87|T.\ Ohsawa, K.\ Takegoshi|On the extension of $L^2$
  holomorphic functions|Math.\ Zeitschrift|195|1987|197--204|

\article Pau98a|M.\ Paun|Sur l'effectivit\'e num\'erique des
  images inverses de fibr\'es en droites|Math.\ Ann.|310|1998|411--421|
  
\miscell Pau98b|M.\ Paun|Fibr\'e en droites num\'eriquement
  effectifs et vari\'et\'es K\"ahl\'eriennes compactes \`a courbure de
  Ricci nef|Th\`ese, Universit\'e de Grenoble 1 (1998), 80$\,$p|
  
\miscell Pau00|M.\ Paun|Semipositive $(1,1)$-cohomology classes on
  projective manifolds|Preprint de l'I.R.M.A.\ Strasbourg|

\article PSS99|Th.\ Peternell, M.\ Schneider, A.J.\ Sommese|Kodaira 
  dimension of subvarieties|Intl.\ J.\ Math.|10|1999|1065--1079|
    
\article Sch74|M.\ Schneider|Ein einfacher Beweis des Verschwindungs\-satzes
f\"ur positive holomorphe Vektorraumb\"undel|Manuscripta Math.|11|1974|95--101|

\article Ser55|J.-P.\ Serre|Un th\'eor\`eme de dualit\'e|Comment.\
Math.|29|1955|9--26|

\article Sko72a|H.\ Skoda|Sous-ensembles analytiques d'ordre fini ou infini
dans $\bC^n$|Bull.\ Soc.\ Math.\ France|100|1972|353--408|

\article Sko72b|H.\ Skoda|Applications des techniques $L^2$ \`a la th\'eorie
des id\'eaux d'une alg\`ebre de fonctions holomorphes avec poids|Ann.\
Scient.\ Ec.\ Norm.\ Sup.\ 4e S\'erie|5|1972|545--579|

\article Sko75|H.\ Skoda|Estimations $L^2$ pour l'op\'erateur $\ol\partial$
  et applications arith\-m\'etiques|S\'emi\-naire P.\ Lelong (Analyse), ann\'ee
  1975/76, Lecture Notes in Math., Vol.~538, 
  Springer-Verlag|\rm Berlin|1977|314--323|

\article Sko82|H.\ Skoda|Prolongement des courants positifs ferm\'es de masse
  finie|Invent.\ Math.|66|1982|361--376|

\article Siu74|Y.T.\ Siu|Analyticity of sets associated to Lelong numbers
  and the extension of closed positive currents|Invent.\ Math.|27|1974|53--156|

\article Yau78|S.-T.\ Yau|On the Ricci curvature of a complex
  K\"ahler manifold and the complex Monge-Amp\`ere equation|Comm.\ 
  Pure Appl.\ Math.|31|1978|339--411|

}

\vskip30pt
\noindent(June 17, 2007, typeset on \today)
\vskip10pt
\noindent
Jean-Pierre Demailly\hfill\break
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, UMR 5582 du CNRS, BP$\,$74,\hfill\break
F-38402 Saint-Martin d'H\`eres, France

\noindent{\it e-mail address:}\/ demailly@fourier.ujf-grenoble.fr
\fi

\end
