% Lecture notes on $L^2$ estimates
% Ecole d'\'et\'e de Math\'ematiques "Analyse Complexe" 1996
% J.-P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France
        
%%% Macros

\font \tenbf                  = cmbx10
\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\leqslant="3\msa@36   \let\le\leqslant  \let\leq\leqslant
\mathchardef\geqslant="3\msa@3E   \let\ge\geqslant  \let\geq\geqslant
\mathchardef\smallsmile="3\msa@60
\mathchardef\compact="3\msa@62
\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\smallsetminus="2\msb@72   \let\ssm=\smallsetminus

\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=24.2true cm 
\hsize=15.3true cm
\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=20pt
\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\bbbr{{\Bbb R}}
\def\bbbn{{\Bbb N}}
\def\bbbh{{\Bbb H}}
\def\bbbk{{\Bbb K}}
\def\bbbc{{\Bbb C}}
\def\bbbd{{\Bbb D}}
\def\bbbz{{\Bbb Z}}
\def\bbbp{{\Bbb P}}
\def\bbbq{{\Bbb Q}}
\def\bbbone{\hbox{\bf 1}}

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

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

\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=18pt
  \lineskip=18pt
  \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}}

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

\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}

\newcount\numbersection \numbersection=0
\def\nextref#1{\advance\numbersection by 1
     \immediate\write1{\string\def \string\REF 
     \romannumeral\numbersection \string{%
     \noexpand#1 \string\dotfill \space \number\pageno \string}}}

 \def\titleb#1#2{%
     \nextref{#1 #2}
     \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\\{\hfil\break}
\def\ii{{\rm i}}
\def\bu{{\hbox{$\scriptstyle\bullet$}}}
\def\ld{,\ldots,}
\let\lra=\longrightarrow
\let\wh=\widehat
\let\wt=\widetilde
\let\ovl=\overline
\def\dbar{{\overline\partial}}
\def\ddbar{{\partial\overline\partial}}
\def\soul{{\raise-4.6pt\hbox{$-$}\kern-7pt}}
\def\ci{C^\infty}
\def\ampersand{\&}
\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\over#2{\mathrel{\mathop{\null#2}\limits^{#1}}}
\def\buildu#1\under#2{\mathrel{\mathop{\null#2}\limits_{#1}}}

\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\Dom{\mathop{\rm Dom}\nolimits}
\def\restr{\mathop{\rm restr}\nolimits}
\def\rank{\mathop{\rm rank}\nolimits}
\def\Sh{\mathop{\rm Sh}\nolimits}
\def\Psh{\mathop{\rm Psh}\nolimits}
\def\Conv{\mathop{\rm Conv}\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\grad{\mathop{\rm grad}\nolimits}
\def\diam{\mathop{\rm diam}\nolimits}
\def\Vect{\mathop{\rm Vect}\nolimits}
\def\cHom{\mathop{{\cal H}{\it om}}\nolimits}

\def\Ric{{\rm Ricci}}
\def\wtP{\smash{\wt P}}
\def\omcirc{({\ovl\Omega})^\circ}
\def\Gr{{\rm Gr}}
\def\Langle{\langle\!\langle}
\def\Rangle{\rangle\!\rangle}
\def\DR{{\rm DR}}
\def\FS{{\rm FS}}
\def\NS{{\rm NS}}
\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\mkk{&\kern-16pt}
\def\mk{&\kern-7pt}

\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\begstat#1 #2\endstat
{\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

\newdimen\refskip \refskip=1.6cm
\def\ref#1&#2&#3&#4&%
{\hangindent=\refskip\hangafter=1\noindent%
\rlap{\hbox{\bf #1}}\hbox to \refskip{\hfill}%
{\bf\ignorespaces #2}{\it #3}{\rm #4.}} 

\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}

\catcode`\@=11
\newcount\@tempcnta \newcount\@tempcntb 
\def\timeofday{{%
\@tempcnta=\time \divide\@tempcnta by 60 \@tempcntb=\@tempcnta
\multiply\@tempcntb by -60 \advance\@tempcntb by \time
\ifnum\@tempcntb > 9 \number\@tempcnta:\number\@tempcntb
  \else\number\@tempcnta:0\number\@tempcntb\fi}}
\catcode`\@=12

\catcode`\@=11
\newdimen\srdim \srdim=\hsize
\def\NOSECTREF#1{\noindent\hbox to \srdim{\null\dotfill ???(#1)}}
\def\SECTREF#1{
\mathchardef\@M=0
\noindent\hbox to \srdim{\csname REF\romannumeral#1\endcsname}
\mathchardef\@M=10000}
\newlinechar=`\^^J
\catcode`\@=12

\def\openauxfile{
  \immediate\openin1\jobname.aux
  \ifeof1
    \message{^^JCAUTION\string: you MUST run TeX a second time^^J}
    \let\sectref=\NOSECTREF \let\indref=\NOSECTREF
  \else
    \catcode`\@=11
    \input \jobname.aux
    \catcode`\@=12
    \message{^^JCAUTION\string: if the file has just been modified you may 
      have to run TeX twice^^J}
    \let\sectref=\SECTREF
  \fi
  \message{to get correct page numbers displayed in Contents or Index 
    Tables^^J}
  \immediate\openout1=\jobname.aux
  \let\END=\end \def\end{\immediate\closeout1\END}}

%%% Main File

\openauxfile

\titlea{}{On the Ohsawa-Takegoshi-Manivel\hfil\break $L^2$ extension theorem}
\leftrunningtitle{On the Ohsawa-Takegoshi-Manivel $L^2$ extension theorem}
\vskip10pt\noindent
{\tbfontt Jean-Pierre Demailly}
\vskip10pt\noindent
{\bf Universit\'e de Grenoble I\newline
Laboratoire de Math\'ematiques, UMR 5582 du CNRS\newline
Institut Fourier, BP74, 38402 Saint-Martin d'H\`eres, France}
\vskip10pt
\noindent
\line{\hfill \it en l'honneur de Monsieur Pierre Lelong, \`a l'occasion de son 
85\`eme anniversaire}
\vskip15pt
\begpet
\titled{R\'esum\'e}
Un des buts de ce travail est d'illustrer de diverses mani\`eres 
l'efficacit\'e des outils fondamentaux introduits par
Pierre Lelong dans l'\'etude de l'Analyse Complexe et de la
G\'eom\'etrie analytique ou alg\'ebrique.  Nous donnons d'abord une
pr\'esentation d\'etaill\'ee du th\'eor\`eme d'extension $L^2$ de
Ohsawa-Takegoshi, avec le m\^eme point de vue g\'eom\'etrique que celui
introduit par L.~Manivel. Ce faisant, nous simplifions la d\'emarche 
de Ohsawa-Takegoshi et Manivel, et mettons en \'evidence une difficult\'e
(non encore surmont\'ee) dans l'argument invoqu\'e par Manivel pour la 
r\'egularit\'e en bidegr\'e $(0,q)$, $q\,{\scriptstyle\ge}\,1$.  Nous donnons 
ensuite quelques applications frappantes du th\'eor\`eme d'extension, 
en particulier un th\'eor\`eme
d'approximation des fonctions plurisouharmoniques par des logarithmes
de fonctions holomorphes, pr\'eservant autant que possible les
singularit\'es et nombres de Lelong de la fonction plurisosuharmonique
donn\'ee. L'\'etude des singularit\'es de fonctions
plurisousharmoniques se poursuit par un th\'eor\`eme de type
Brian\c{c}on-Skoda nouveau pour les faisceaux d'id\'eaux
multiplicateurs de Nadel. En utilisant ce r\'esultat et des id\'ees de
R.~Lazarsfeld, nous donnons finalement une preuve nouvelle d'un
r\'esultat r\'ecent de T.~Fujita: la croissance du nombre des sections
des multiples d'un fibr\'e en droites gros sur une vari\'et\'e
projective est donn\'ee par la puissance d'intersection de plus haut 
degr\'e de la partie num\'eriquement effective dans la d\'ecomposition 
de Zariski du fibr\'e.

\titled{Abstract}
One of the goals of this work is to demonstrate in several
different ways the strength of the fundamental tools introduced by
Pierre Lelong for the study of Complex Analysis and Analytic or
Algebraic Geometry.  We first give a detailed presentation of the
Ohsawa-Takegoshi $L^2$ extension theorem, inspired by a geometric
viewpoint introduced by L.~Manivel in 1993.  Meanwhile, we simplify the
original approach of the above authors, and point out a difficulty 
(yet to be overcome) in the regularity argument invoked by Manivel
in bidegree $(0,q)$, $q\,{\scriptstyle\ge}\,1$. We then derive some striking
consequences of the $L^2$ extension theorem. In particular, we give an
approximation theorem of plurisubharmonic functions by logarithms of
holomorphic functions, preserving as much as possible the
singularities and Lelong numbers of the given function. The study of
plurisubharmonic singularities is pursued, leading to a new
Brian\c{c}on-Skoda type result concerning Nadel's multiplier ideal
sheaves. Using this result and some ideas of R.~Lazarsfeld, we finally
give a new proof of a recent result of T.~Fujita: the growth of the
number of sections of multiples of a big line bundle is given by the 
highest power of the first Chern class of the numerically effective part 
in the line bundle Zariski decomposition.
\vskip30pt 

\noindent
{\bf Contents}
\smallskip 
\sectref{0}
\sectref{1}
\sectref{2}
\sectref{3}
\sectref{4}
\sectref{5}
\sectref{6}
\sectref{7}
\sectref{8}
\sectref{9}
\sectref{10}
\endpet

\titleb{0.}{Introduction}
The Ohsawa-Takegoshi-Manivel $L^2$ extension theorem 
addresses the following basic problem.

\titled{Problem} {\it
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~$X$.}
\medskip

\noindent
The first satisfactory solution has been obtained by Ohsawa-Takegoshi
[OT87, Ohs88, Ohs94, Ohs95].  We follow here a more geometric approach
due to Manivel [Man93], which provides a more general extension
theorem in the framework of vector bundles and higher cohomology
groups. The first goal of this notes is to simplify further Manivel's
approach, as well as to point out a technical difficulty in Manivel's
proof. This difficulty occurs in the regularity argument for $(0,q)$
forms, when $q\ge 1\,$; it does not look to be very serious, so we
strongly hope that it will be overcome in a new future$\,$!

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]; see also Berndtsson [Ber96] for related calculations in the
special case of domains in $\bbbc^n$.

We then describe how the Oshawa-Takegoshi-Manivel extension theorem can be 
applied to solve several important problems of complex analysis or geometry.
The first of these is an approximation theorem for plurisubharmonic
functions. It is known since a long time that every plurisubharmonic
function can be written as a limit of logarithms of the modulus
of holomorphic functions, multiplied by suitable small positive numbers
(see Bremermann [Bre54] and Lelong [Lel72]). Here, we show that the 
approximation can be made with a uniform convergence of the Lelong numbers
of the holomorphic functions towards those of the given plurisubharmonic
function. This result contains as a special case Siu's theorem [Siu74] on 
the analyticity of Lelong number sublevel sets. A geometric (more or
less equivalent) form of the result is the existence of approximations 
of an arbitrary closed positive current of type $(1,1)$ of rational 
cohomology class by effective rational divisors. Somewhat surprisingly,
the proof of all the above only uses the $0$-dimensional case of the 
$L^2$ extension theorem$\,$! 

By combining some of the results provided by the proof of that
approximation theorem with Skoda's $L^2$ estimates for the division
of holomorphic functions, we obtain a Brian\c{c}on-Skoda type
theorem for Nadel's multiplier ideal sheaves. A weak form of it says
that $\cI(\ell\varphi)\subset\cI(\varphi)^{\ell-n}$ for every 
plurisubharmonic function on an open set of $\bbbc^n$. This result can 
in its turn be used to prove Fujita's ``asymptotic Zariski decomposition
result''. The result tells us that if we write a big $\bbbq$-divisor $D$ as
a sum $D=E+A$ with $E$ effective and $A$ ample, then the value of the
supremum of $A^n$ is determined by the growth of the number of sections,
and equal to $\limsup{n!\over k^n}h^0(X,kL)$ where $n=\dim X$.

I thank wholeheartedly Pierre Dolbeault, Andrei Iordan and Henri Skoda
for giving me the opportunity to present this work on the occasion of Pierre
Lelong's 85th birthday celebration in September 1997. 

\titleb{1.}{Notation and general setting}
Let $X$ be a complex $n$-dimensional manifold equipped with a hermitian
metric~$\omega$, viewed as a positive $(1,1)$-form
$$
\omega=\ii\sum_{1\le j,k\le n}\omega_{jk}(z)\,dz_j\wedge d\ovl z_k.
$$
The bundle of $(p,q)$-forms $\Lambda^{p,q}T^\star_X=
\Lambda^pT^\star_X\otimes\Lambda^q\ovl T^\star_X$ then inherits a natural
hermitian metric. If $(E,h)$ is a hermitian vector bundle, and 
$u,v:X\to\Lambda^{p,q}T^\star_X\otimes E$ are $(p,q)$-forms with values
in $E$ with measurable coefficients, we set
$$
\Vert u\Vert^2=\int_X|u|^2dV_\omega,\qquad
\Langle u,v\Rangle=\int_X\langle u,v\rangle dV_\omega
\leqno(1.1)
$$
where $|u|=|u|_{\omega,h}$ is the pointwise norm induced by $\omega$
and $h$ on $\Lambda^{p,q}T^\star_X\otimes E$, and $dV_\omega=
{1\over n!}\omega^n$ is the hermitian volume element. In this way, we
obtain a Hilbert space $L^2(X,\Lambda^{p,q}T^\star_X\otimes E)$ of sections,
containing the space $\cD(X,\Lambda^{p,q}T^\star_X\otimes E)$ of smooth
compactly supported sections as a dense subspace.

If we assume that the metric $h$ on $E$ is smooth, there is a unique 
smooth connection $D=D'+D''$ on $E$ (the so-called 
{\it Chern connection} of $(E,h)$) acting on forms with values in $E$,
such that:
\medskip\noindent
\bu\ $D'$ is of pure type $(1,0)$ and $D''$ is of pure type $(0,1)$;\\
\bu\ $D''$ coincides with the $\dbar$ operator;\\
\bu\ $D$ is compatible with $h$, that is, $D$ satisfies the Leibnitz rule
$$
d\{u,v\}=\{Du,v\}+(-1)^{\deg u}\{u,Dv\}
$$
where $\{~,~\}:\Lambda^{p,q}T^\star_X\otimes E\times
\Lambda^{r,s}T^\star_X\otimes E\to \Lambda^{p+s,q+r}T^\star_X$
is the sesquilinear product which combines the the wedge product 
$(u,v)\mapsto u\wedge\ovl v$ on scalar valued forms with the hermitian 
inner product on~$E$.

As usual one can view $D'$, $D''$ as closed and densely defined operators 
on the
Hilbert space \hbox{$L^2(X,\Lambda^{p,q}T^\star_X\otimes E)$}; the domain
of $D''$, for example, is the set of all $u\in L^2$ such that $D''u$
calculated in the sense of distributions satisfies $D''u\in L^2$. The 
formal adjoints $D^{\prime\star}$, $D^{\prime\prime\star}$ also have
closed extensions in the sense of distributions, which do not 
necessarily coincide with the Hilbert space adjoints in the sense 
of Von Neumann, since the latter ones may have strictly smaller domains. 
It is well known, however, that the domains coincide if the hermitian 
metric $\omega$ is (geodesically) complete. The complex {\it Laplace 
Beltrami operators} are defined by
$$
\Delta'=[D',D^{\prime\star}]=D'D^{\prime\star}+D^{\prime\star}D',\qquad
\Delta''=[D'',D^{\prime\prime\star}]=D''D^{\prime\prime\star}+
D^{\prime\prime\star}D''
\leqno(1.2)
$$
where $[A,B]=AB-(-)^{\deg A\,\deg B}BA$ is the graded commutator bracket of
operators. Other important operators of hermitian geometry are 
$Lu:=\omega\wedge u$ and its adjoint~$\Lambda$. Under the assumption that 
$\omega$ is {\it K\"ahler}, i.e.\ $d\omega=0$, we have the following 
basic commutation identities:
$$
\matrix{
&[D^{\prime\prime\star},L] = \ii D',\quad\hfill
&[D^{\prime\star},L] = -\ii D'',\cr
\noalign{\vskip-5pt}\cr
&\strut\kern4pt[\Lambda,D''] = -\ii D^{\prime\star},\hfill\quad
&[\Lambda,D'] = \ii D^{\prime\prime\star}.\cr}
\leqno(1.3)
$$
From there, one gets the fundamental {\it Bochner-Kodaira-Nakano identity}
$$
\Delta''=\Delta'+[\Lambda,\ii\Theta(E)],
\leqno(1.4)
$$
where $\Theta(E)=D^2=D'D''+D''D'\in C^\infty(X,\Lambda^{1,1}T^\star_X\otimes
\Hom(E,E))$ is the curvature tensor of $E$.

\titleb{2.}{Basic a priori inequality}

The standard $L^2$ estimates for solutions of $\dbar$ equations 
(Andreotti-Vesentini [AV65], H\"ormander [H\"or65, 66]) are based
on a direct application of the Bochner-Kodaira-Nakano identity
(1.4). In this setting, the curvature integrals are spread over $X$
and everything goes through in a rather straightforward manner. 
For the application to the $L^2$ extension theorem, however, one 
has to ``concentrate'' the effect of the curvature around the 
subvariety from which the extension is to be made. For this, a modified 
a priori inequality is required, involving ``bump functions'' in the
weight of the $L^2$ integrals. The following is an improved version, 
due to Ohsawa [Ohs95] of the original a priori inequality used by 
Ohsawa-Takegoshi [OT87, Ohs88].  Earlier similar estimates had been
used in a different context by Donnelly-Fefferman [DF83] and
Donnelly-Xavier [DX84].

\begstat{(2.1) Lemma {\rm ([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
$$ 
\eqalign{
\|(\eta^{{1\over2}}+\lambda^{{1\over2}})D^{\prime\prime\star}u\|^2&{}
+\|\eta^{{1\over2}}D''u\|^2+\|\lambda^{{1\over2}}D'u\|^2+
2\|\lambda^{-{1\over2}}d'\eta\wedge u\|^2\cr
 &\geq \Langle[\eta\,\ii\Theta(E)-\ii\,d'd''\eta-\ii\lambda^{-1}d'\eta\wedge 
d''\eta,\Lambda]u,u\Rangle.\cr}
$$
\endstat

\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
$$
\leqalignno{
D''\eta D^{\prime\prime\star}&{}+D^{\prime\prime\star}\eta  D'' -
D'\eta D^{\prime\star}  -D^{\prime\star}\eta  D' \cr
&{}=\eta [\ii\Theta(E),\Lambda]+(d''\eta)D^{\prime\prime\star}
-(d''\eta)^\star D''+(d'\eta)^\star D'-(d'\eta)D^{\prime\star}.
&(2.2)\cr}
$$
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~7.2. 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 (2.2), 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
$$
\Langle (D''\eta D^{\prime\prime\star})u,u\Rangle=\Langle\eta D^{\prime\prime\star}u,
D^{\prime\prime\star}u\Rangle=\|\eta^{{1\over2}}D^{\prime\prime\star}u\|^2,
$$
and likewise for the other similar terms. The above equalities
imply
$$
\eqalign{
&\|\eta^{{1\over 2}}D^{\prime\prime\star}u\|^2+\|\eta^{{1\over 2}}D''u\|^2
-\|\eta^{{1\over2}}D'u\|^2-\|\eta^{{1\over2}}D^{\prime\star}u\|^2={}\cr
&\qquad\Langle[\eta\,\ii\Theta(E)-\ii\,d'd''\eta,\Lambda]u,u\Rangle
+2\Re\,\Langle D^{\prime\prime\star}u,(d''\eta)^\star u\Rangle
+2\Re\,\Langle D'u,d'\eta\wedge u\Rangle.\cr}
$$
By neglecting the negative terms $-\|\eta^{{1\over2}}D'u\|^2
-\|\eta^{{1\over2}}D^{\prime\star}u\|^2$ and adding the squares
$$
\eqalign{
\|\lambda^{{1\over 2}}D^{\prime\prime\star}u\|^2+
2\Re\,\Langle D^{\prime\prime\star}u,(d''\eta)^\star u\Rangle+
\|\lambda^{-{1\over 2}}(d''\eta)^\star u\|^2
&\ge 0,\cr
\|\lambda^{{1\over 2}}D'u\|^2+
2\Re\,\Langle D'u,d'\eta\wedge u\Rangle+
\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2
&\ge0\cr}
$$
we get
$$
\eqalign{
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})
D^{\prime\prime\star}u\|^2&{}+\|\eta^{{1\over 2}}D''u\|^2
+\|\lambda^{{1\over 2}}D'u\|^2+\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2
+\|\lambda^{-{1\over 2}}(d''\eta)^\star u\|^2\cr
&\ge\Langle[\eta\,\ii\Theta(E)-\ii\,d'd''\eta,\Lambda]u,u\Rangle.\cr}
$$
Finally, we use the identities
$$
\eqalign{
(d'\eta)^\star(d'\eta)&{}-(d''\eta)(d''\eta)^\star
=\ii[d''\eta,\Lambda](d'\eta)+\ii(d''\eta)[d'\eta,\Lambda]
=[\ii d''\eta\wedge d'\eta,\Lambda],\cr
\|\lambda^{-{1\over 2}}d'\eta\wedge u\|^2&{}-
\|\lambda^{-{1\over2}}(d''\eta)^\star u\|^2
=-\Langle[\ii\lambda^{-1}d'\eta\wedge d''\eta,\Lambda]u,u\Rangle,\cr}
$$
The inequality asserted in Lemma~2.1 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
$$
\|(\eta^{{1\over2}}+\lambda^{{1\over2}})D^{\prime\prime\star}u\|^2{}
+\|\eta^{{1\over2}}D''u\|^2\ge
\Langle[\eta\,\ii\Theta(E)-\ii\,d'd''\eta
-\ii\lambda^{-1}\,d'\eta\wedge d''\eta,\Lambda]u,u\Rangle.\leqno(2.3)
$$

\titleb{3.}{$L^2$ existence theorem}

By essentially repeating the Hilbert space techniques already used by
Kohn\break [Ko63, 64] and H\"ormander [H\"or65, 66] in our context, we 
now derive from (2.3) the following existence theorem.

\begstat{(3.1) 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
$$
\int_X(\eta+\lambda)^{-1}
|f|^2\,dV_\omega\le 2\int_X\langle B^{-1}g,g\rangle\,dV_\omega.
$$
\endstat

\begproof{} Let $v\in L^2(X,\Lambda^{n,q}T^\star_X\otimes E)$ be an 
arbitrary element. Assume first that $\omega$ is complete, so that
$(\Ker D'')^\perp=\ovl{\Im D^{\prime\prime\star}}
\subset\Ker D^{\prime\prime\star}$. Then,
by using the decomposition \hbox{$v=v_1+v_2\in(\Ker D'')\oplus
(\Ker D'')^\perp$} and the fact that
$g\in\Ker D''$, we infer from Cauchy-Schwarz the inequality
$$
|\langle g,v\rangle|^2=|\langle g,v_1\rangle|^2\le
\int_X\langle B^{-1}g,g\rangle \,dV_\omega
\int_X\langle Bv_1,v_1\rangle \,dV_\omega.
$$
We have $v_2\in\Ker D^{\prime\prime\star}$, hence
$D^{\prime\prime\star}v=D^{\prime\prime\star}v_1$, and (2.3) implies
$$
\int_X\langle Bv_1,v_1\rangle \,dV_\omega\le
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v_1\|^2+
\|\eta^{{1\over 2}}D''v_1\|^2
=\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v\|^2
$$
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 
$$
\eqalign{
\|w\|^2&\le\int_X\langle B^{-1}g,g\rangle \,dV_\omega\qquad
\hbox{and}\cr
\Langle v,g\Rangle&=\Langle(\eta^{{1\over 2}}+\lambda^{{1\over2}})
D^{\prime\prime\star}v,w\Rangle\qquad
\forall g\in\Dom D''\cap\Dom D^{\prime\prime\star}.\cr}
$$
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~$\varepsilon$).
The technique is quite standard and entirely similar to the approach
described in [Dem82a], so we will not give any detail here.\qed
\endproof

\begstat{(3.2) 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
$$
\int_X\langle(B+\delta I)v_1,v_1\rangle \,dV_\omega\le
\|(\eta^{{1\over 2}}+\lambda^{{1\over 2}})D^{\prime\prime\star}v\|^2+
\delta\|v\|^2.
\eqno\qed
$$
\endstat

\titleb{4.}{The Ohsawa-Takegoshi-Manivel $L^2$ extension theorem}

We now derive the basic $L^2$ extension theorem, by using a variant of
the original ``weight bumping technique'' of Ohsawa-Takegoshi. At this point,
our approach is closer to Manivel's exposition [Man93].

\begstat{(4.1) 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 holomorphic section $f$ of $\Lambda^nT_X^\star\otimes L$ 
over $Y$, such that
$$
\int_Y|f|^2|\Lambda^r(ds)|^{-2}dV_\omega<+\infty,
$$ 
there  exists a holomorphic extension $F$ to $X$
such that $F_{\restriction Y}=f$ and
$$
\int_X{|F|^2\over|s|^{2r}(-\log|s|)^2}\,dV_{X,\omega}
\le C_r\int_Y{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}\,,
\leqno(\star)
$$
where $C_r$ is a numerical constant depending only on~$r$.
\endstat

We further state a conjecture for the extension $(0,q)$ forms, a variant 
of which was claimed as a theorem by L.~Manivel [Man93]. The proof given 
by Manivel is indeed essentially correct, apart from a minor regularity
argument which is incorrectly settled in [Man93]. Although we have 
been unable to fix the difficulty, we strongly hope that it will be
overcome in a near future.

\begstat{(4.2) Conjecture} If $f$ is a smooth $\dbar$-closed $(0,q)$-form 
over $Y$ satisfying the same $L^2$ condition as above, there exists a locally
square integrable extension $(0,q)$-form $F$ over $X$ which is an extension
of $f$, is smooth on $X\ssm\{s=\Lambda^r(ds)=0\}$, 
and satisfies $(\star)$.
\endstat

\begstat{(4.3) Remark} \rm
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 hypothesis
a) is the only one that really matters: if a) is satisfied for some 
choice of the function $\alpha\ge 1$, one can always achieve b) by 
multiplying the metric of $E$ with a sufficiently small weight
$e^{-\chi\circ\psi}$ (where $\psi$ is a psh exhaustion on~$X$ and $\chi$ 
a convex increasing function; property a) remains valid after we multiply
the metric of $L$ by $e^{-(r+\smash{\alpha_0^{-1}})\chi\circ\psi}$,
with $\alpha_0=\inf_{x\in X}\alpha(x)$. 
\endstat

We now split the proof of Theorem 4.1 in several steps, pushing forward
the general case of $(0,q)$-forms as long as we can (i.e.\ until the
check of regularity, where we unfortunately got stuck $\ldots$). By this
we mean that we consider a section $f$ of
$$
\Lambda^{0,q}T^*_Y\otimes (\Lambda^nT^*_X\otimes L)_{\restriction Y}
$$
with smooth coefficients on the regular part $Y_\reg\subset Y$, satisfying
a further ad hoc $L^2$ condition on~$Y$.

\begstat{(4.4) Construction of a smooth extension $\wt f_\infty$}
\rm 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
$$
\wt f_\infty\in C^\infty(X,\Lambda^{n,q}T^\star_X\otimes L)
=C^\infty(X,\Lambda^{0,q}T^\star_X\otimes\Lambda^nT^\star_X\otimes L)
$$ 
such that
{\parindent=0cm 
\smallskip
(a) $\wt f_\infty$ coincides with $f$ in restriction to $Y$,
\smallskip
(b) $|\wt f_\infty|=|f|$ at every point of~$Y$,
\smallskip
(c) $D''\wt f_\infty=0$ 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 certainly find a section $\smash{\wh f}\in
C^\infty(X,\Lambda^{n,q}T^\star_X\otimes L)$ 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~$\smash{\wh f}$ we have
$$
(D''\wh f)_{\restriction Y}=(D''\wh f_{\restriction Y})=D''f=0.
$$
It follows that we can divide $D''\wh f=\sum_{1\le \lambda\le r}
g_{j,\lambda}(z)\wedge d\ovl z_\lambda$ 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
$$
\wt f_\infty:=\wh f - \sum_j \theta_j(z)\sum_{1\le \lambda \le r}
\ovl z_\lambda g_{j,\lambda}(z)
$$
coincides with $\wh f$ on $Y$ and satisfies (c).
\endstat

\begstat{(4.5) Construction of weights, using a bumping technique} \rm
Since we do not know about $\wt f_\infty$ far away from~$Y$, we will consider
a truncation $\wt f_\varepsilon$ of $\wt f_\infty$ with support in a small
tubular neighborhood $|s|<\varepsilon$ of~$Y$, and solve the equation
$D''u_\varepsilon=D''\wt f_\varepsilon$ 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 apply Proposition 3.1
with a suitable choice of the functions $\eta=\eta_\varepsilon$ 
and $\lambda=\lambda_\varepsilon$, and an additional weight $|s|^{-2r}$ 
in the metric of $L$. The functions $\eta_\varepsilon$ and 
$\lambda_\varepsilon$ will present carefully constructed ``bumps'',
taking effect on the tubular neighborhood $|s|<\varepsilon$.

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
$$
\eqalign{
\ii\,d'\sigma_\varepsilon
&={\ii\{D's,s\}\over|s|^2+\varepsilon^2},\cr
\ii\,d'd''\sigma_\varepsilon
&={\ii\{D's,D's\}\over|s|^2+\varepsilon^2}-
{\ii\{D's,s\}\wedge\{s,D's\}\over(|s|^2+\varepsilon^2)^2}-
{\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2}\cr
&\ge{\varepsilon^2\over|s|^2}
{\ii\{D's,s\}\wedge\{s,D's\}\over(|s|^2+\varepsilon^2)^2}
-{\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2}\cr
&\ge{\varepsilon^2\over|s|^2}\ii d'\sigma_\varepsilon\wedge 
d''\sigma_\varepsilon-{\{\ii\Theta_Es,s\}\over|s|^2+\varepsilon^2},\cr}
$$
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 (the last equality $[\ii\,a\wedge
\ovl a,\Lambda]=(a)(a)^\star$ for $a=a^{1,0}$ is easily checked
and left as an exercise to the reader; recall that we denote
$(a)=a\wedge\bu$).
\endstat

\begstat{(4.6) Solving $\dbar$ with $L^2$ estimates, for suitably
truncated forms} \rm Let us fix a cut-off function $\theta:\bbbr\to[0,1]$ 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 $\wt f_\varepsilon=\theta(\varepsilon^{-2}|s|^2)\wt f_\infty$ and
its $D''$-derivative
$$
g_\varepsilon=D''\wt f_\varepsilon
=(1+\varepsilon^{-2}|s|^2)\theta'(\varepsilon^{-2}|s|^2)
d''\sigma_\varepsilon\wedge \wt f_\infty+
\theta(\varepsilon^{-2}|s|^2)D''\wt f_\infty
$$
[as is easily seen from the equality $1+\varepsilon^{-2}|s|^2=
\varepsilon^{-2}e^{\sigma_\varepsilon}\,$]; our later goal is to solve
the $\dbar$ equation $D''u_\varepsilon=g_\varepsilon=D''\wt f_\varepsilon$.
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
$$
g_\varepsilon^{(1)}:=
(1+\varepsilon^{-2}|s|^2)\theta'(\varepsilon^{-2}|s|^2)
\chi_0'(\sigma_\varepsilon)^{-1}d''\eta_\varepsilon\wedge \wt f_\infty.
$$
On $\Supp g_\varepsilon^{(1)} \subset\{|s|<\varepsilon\}$, since
$\chi_0'(\sigma_\varepsilon)\ge 1$, we thus find
$$
\langle B_\varepsilon^{-1}g_\varepsilon^{(1)},
g_\varepsilon^{(1)}\rangle\le 
(1+\varepsilon^{-2}|s|^2)^2\,\theta'(\varepsilon^{-2}|s|^2)^2|\wt f_\infty|^2.
$$
Instead of working on $X$ itself, we will work rather on the relatively
compact subset $X_c\ssm Y_c$, where $Y_c=Y\cap X_c=Y\cap\{\psi<c\}$. 
It is easy to check that $X_c\ssm Y_c$ is again complete K\"ahler 
(see e.g.\ [Dem82a]): a K\"ahler metric of the form
$$
\omega_c=C\omega+\ii d'd''\Big(\log{1\over c-\psi}+
\log|s|-\log(C-\log|s|)\Big),\qquad C\gg 0
$$
indeed satisfies $\omega_c\ge |d'\log(c-\psi)|^2+|d'\log(C-\log|s|)|^2$
and is therefore complete K\"ahler.
In this way, we avoid the singularity of the weight $|s|^{-2r}$ along~$Y$.
We find
$$
\int_{X_c\ssm Y_c}\langle B_\varepsilon^{-1}g_\varepsilon^{(1)},
g_\varepsilon^{(1)}\rangle\, |s|^{-2r}dV_\omega\le
\int_{X_c\ssm Y_c}|\wt f_\infty|^2
(1+\varepsilon^{-2}|s|^2)^2\theta'(\varepsilon^{-2}|s|^2)^2
|s|^{-2r}dV_\omega.
$$
Now, we let $\varepsilon\to 0$ and view $s$ as ``transverse local 
coordinates'' around $Y$.
As $\wt f_\infty$ coincides with $f$ on~$Y$, it is not hard to see 
from (4.4~b) 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
$$
c_r=\int_{z\in\bbbc^r,\,|z|\le 1}
(1+|z|^2)^2\theta'(|z|^2)^2{\ii^{r^2}\Lambda^r(dz)
\wedge\Lambda^r(d\ovl z)\over|z|^{2r}}<+\infty
$$
depending only on~$r$. The second term
$$
g_\varepsilon^{(2)}=\theta(\varepsilon^{-2}|s|^2)d''\wt f_\infty
$$
in $g_\varepsilon$ satisfies $\Supp(g_\varepsilon^{(2)})\subset\{|s|
<\varepsilon\}$ and $|g_\varepsilon^{(2)}|=O(|s|)$ (just look at the 
Taylor expansion of $d''\wt f_\infty$ 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~3.1
on $X_c\ssm Y_c$ with the additional weight factor~$|s|^{-2r}$. Otherwise, 
we use the modified estimate stated
in Remark~3.2 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 $D''u_{c,\varepsilon}=
g_\varepsilon=D''\wt f_\varepsilon$ 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 $\wt f_\varepsilon$ is uniformly bounded with support in $\{|s|<\varepsilon\}$,
we conclude from an obvious volume estimate that
$$
\int_{X_c}{|\wt f_\varepsilon|^2\over
(|s|^2+\varepsilon^2)^r(-\log(|s|^2+\varepsilon^2))^2}dV_{X,\omega}
\le {\Const \over (\log\varepsilon)^2}.
$$
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 $d''F_{c,\varepsilon}=0$ on $X_c\ssm Y_c$,
by construction. This equation actually extends from $X_c\ssm Y_c$ to 
$X_c$ because $F_{c,\varepsilon}$ is locally $L^2$ near $Y_c$. In fact, 
we have the following well-known lemma in $\dbar$-operator theory 
(see e.g.\ [Dem82a]).
\endstat

\begstat{(4.7) Lemma} Let $\Omega$ be an open subset of $\bbbc^n$ and
$Y$ a complex 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 $\dbar v=w$ on $\Omega\ssm Y$
$($in the sense of distribution theory$)$. Then $\dbar v=w$ on $\Omega$.
\endstat

\begstat{(4.8) Final check, and regularity of the solution} \rm
If $q=0$, then $u_{c,\varepsilon}$ must be smooth also by the
ellipticity of $\dbar$ in bidegree $(0,0)$. The
non integrability of the weight $|s|^{-2r}$ along $Y$ shows that
$u_{c,\varepsilon}$ vanishes on~$Y$, therefore
$$
F_{c,\varepsilon\restriction Y}=
\wt f_{\varepsilon\restriction Y}=\wt f_{\infty\restriction Y}=f.
$$
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 $\Sigma\subset\ovl Y\cap Z \subsetneq\ovl Y$, and 
then apply the extension theorem on the Stein manifold $X\ssm Z$, in 
combination with Lemma~11.10$\,$; ii) the whole procedure still works when
$\Sigma$ is nowhere dense in~$\ovl Y$ (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 $\wt f_\infty=\sum\theta_j\wt f_j$ and
observe that $|D''\wt f_\infty|^2|s|^{-2r}$ 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~$\ovl Y\,]$.
\endstat

\begstat{(4.9) Remarks} \rm Before discussing the difficulties to be
overcome to reach the required regularity result for bidegrees $(0,q)$,
$q\geq 1$, we make a few remarks.
\smallskip
\noindent{\rm a)} When $q=0$, the estimates provided by Theorem~4.1 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~\S$\,$1.
\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~4.1 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 4.1~a)). 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
$$
|\xi|^2_x={1\over c_r}
\int_{z\in E_x}(1+|z|^2)^2
\theta'(|z|^2)^2\,{\ii^{r^2}\xi\wedge\ovl\xi\over|z|^{2r}}
$$
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
$$
D''u_{c,\varepsilon,\delta}+\delta^{1/2}h_{c,\varepsilon,\delta}=g_\varepsilon
=D''\wt f_\varepsilon
$$
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 $(\Delta''+\delta\Id)h_{c,\varepsilon,\delta}=
(D''D^{\prime\prime\,*}+\delta\Id)h_{c,\varepsilon,\delta}=
\delta^{1/2}D''\wt f_\varepsilon$, and therefore $h_{c,\varepsilon,\delta}$ is 
smooth by the ellipticity of $\Delta''$.\qed
\endstat

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

\begstat{(4.10) Corollary}
Let $X$ be a weakly pseudoconvex K\"ahler manifold, $E$ a holomorphic
vector bundle of rank $r$ over~$X$, and $s$ 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 ample $($in the sense that the associated $\bbbq$-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~$E)$.
Then the restriction morphism
$$
H^0(X,\Lambda^nT^\star_X\otimes L)\to H^0\big(Y,(\Lambda^nT^\star_X
\otimes L)_{|Y}\big)
$$
is surjective. 
\endstat

Note that if conjecture 4.2 were true, we would also get the surjectivity
of the restriction morphism
$$
H^q(X,\Lambda^nT^\star_X\otimes L)\to H^q\big(Y,(\Lambda^nT^\star_X
\otimes L)_{|Y}\big),\qquad\forall q\ge 0,
$$
as asserted in [Man93]. However, this purely qualitative result is easy
to check directly [as we will see below, the real strength of Theorem 4.1
is in the quantitative $L^2$ estimate]. If $\codim Y=1$, the hypothesis 
says that $L\otimes E^\star=L\otimes\cO_X(-Y)$ is ample. We then conclude
from the exact sequence
$$
0\to\cO_X(-Y)\to\cO_X\to (i_Y)_\star\cO_Y\to0
$$ 
and the vanishing of $H^{q+1}(X,\Lambda^nT^\star_X\otimes L\otimes
\cO_X(-Y))$ by Kodaira's theorem. The case $r>1$ can be easily reduced 
to the case of codimension~$1$ by blowing up $X$ along~$Y$
(See also \S$\,$5 for more explanation on this strategy).

\begproof{} First assume for simplicity that $F$ is
Griffiths positive, i.e.\ that $E$ has a hermitian metric such that
$${1\over r} \ii\Theta(L)\otimes\Id_E-\ii\Theta(E)>_\Grif 0.$$ 
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~4.1 can thus be applied with $\alpha=r\delta/C$
and Corollary 4.10 follows. In the case when $L^{1/r}\otimes E^\star$ is 
just assumed to be ample, we can apply remark 4.9~b) and use the same 
arguments (with a Finsler metric on $E$ rather than a hermitian 
metric).\qed
\endproof

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

\begstat{(4.11) Corollary} Let $\Omega\subset\bbbc^n$ 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
$$
\int_\Omega{|F|^2\over|s|^{2r}(-\log|s|)^2}e^{-\varphi}dV_\Omega\le
C\int_Y{|f|^2\over|\Lambda^r(ds)|^2}e^{-\varphi}dV_Y.
$$
\endstat

\begproof{} We apply essentially the same idea as for the previous corollary,
in the special case when $L=\Omega\times\bbbc$ 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
4.1~a) is satisfied ($A$ just depends on the presupposed bound for the 
curvature tensor of~$E$).\qed
\endproof

\begstat{(4.12) 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 
$\Omega\times \bbbc^n$ 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
$$
\int_\Omega{|f(z)|^2\over|z-z_0|^{2(n-\varepsilon)}}e^{-\varphi(z)}dV_\Omega
\le{C_n\over\varepsilon^2(\diam\Omega)^{2(n-\varepsilon)}}|f_0|^2
e^{-\varphi(z_0)}.
$$
\endstat

\titleb{5.}{Regularity of the solution for bidegrees $(0,q)$, $q\ge 1$}

When $q\ge 1$, the arguments needed to get a smooth solution necessarily
involve much more delicate considerations. This is the part where the 
proof given by Manivel [Man93] appears to be incomplete. Actually, 
a natural idea is to consider the minimal $L^2$ solution $u_{c,\varepsilon}$
of the $D''$ equation considered in \S$\,$4, with respect to the weight
$(\log(|s|^2+\varepsilon^2))^{-2}|s|^{-2r}$. This minimal solution satisfies
$$
D''u_{c,\varepsilon}=g_\varepsilon=D''\wt f_\varepsilon,\qquad
D^{\prime\prime\star}((\log(|s|^2+\varepsilon^2))^{-2}
|s|^{-2r}u_{c,\varepsilon})=0
\leqno(5.1)
$$
on $X_c\ssm Y_c$, since $D^{\prime\prime\star}
((\log(|s|^2+\varepsilon^2))^{-2}|s|^{-2r}\bu)$ is the
adjoint of $D''$ for the $L^2$ norms involving the additional weight.
The main difficulty lies in the fact that the differential system (5.1) 
is singular along~$Y$. This forbids the use of a straightforward
elliptic regularity argument (as we did for the case $q=0$). We nevertheless
discuss a strategy which might possibly lead to $C^0$ or H\"older
regularity -- and one could then use conventional regularization 
techniques to obtain a smooth solution from there.
\medskip

\noindent{\bf Case of codimension $r=1$.} If $r=1$, the
subvariety $Y$ is a divisor; therefore, when we consider a $D''$~equation
with values in the line bundle $\Lambda^nT^\star_X\otimes L$,
a $L^2$ solution for the weight $|s|^{-2}$ can be interpreted as
a $L^2$ solution with values in the twisted line bundle
$\Lambda^nT^\star_X\otimes L\otimes\cO_X(-Y)$, equipped with a smooth
hermitian metric. Hence, if $r=1$, the minimal $L^2$ solution
$u_{c,\varepsilon}$ of the $D''$ equation considered earlier satisfies
the equations
$$
D''u_{c,\varepsilon}=g_\varepsilon=D''\wt f_\varepsilon,\qquad
D^{\prime\prime\star}
((\log(|s|^2+\varepsilon^2))^{-2}|s|^{-2}u_{c,\varepsilon})=0
$$
on $X_c\ssm Y_c$. These equations can be rewritten as
$$
D''(s^{-1}u_{c,\varepsilon})=s^{-1}D''\wt f_\varepsilon,\qquad
D^{\prime\prime\star}
((\log(|s|^2+\varepsilon^2))^{-2}s^{-1}u_{c,\varepsilon})=0
\leqno(5.2)
$$
where $s^{-1}u_{c,\varepsilon}$ is viewed as a $(0,q)$-form with values
in $\Lambda^nT^\star_X\otimes L\otimes\cO_X(-Y)$. 
By Lemma~4.7, the equalities (5.2) are valid on $X_c$ and not 
only on $X_c\ssm Y_c$, for $s^{-1}u_{c,\varepsilon}$ is locally $L^2$ and 
$s^{-1}D''\wt f_\varepsilon$ is locally bounded. From this, we infer that
$F_{c,\varepsilon}=\wt f_\varepsilon-u_{c,\varepsilon}$ satisfies
$$
\eqalign{
D''(s^{-1}F_{c,\varepsilon})
 &=D''(s^{-1}\wt f_\varepsilon)-s^{-1}D''\wt f_\varepsilon,\cr
D^{\prime\prime\star}
((\log(|s|^2+\varepsilon^2))^{-2}s^{-1}F_{c,\varepsilon})
 &=D^{\prime\prime\star}
   ((\log(|s|^2+\varepsilon^2))^{-2}s^{-1}\wt f_\varepsilon)\cr
 &=D^{\prime\prime\star}\big
  ((\log(|s|^2+\varepsilon^2))^{-2}
   \theta(\varepsilon^{-2}|s|^2)s^{-1}\wt f_\infty\big).
\cr}
$$
It is easy to show that $D''(s^{-1}\smash{\wt f})-s^{-1}D''\smash{\wt f}$
is independent of the choice of the smooth extension $\smash{\wt f}$ of $f$ 
(whether $\smash{\wt f}$ is $D''$-closed or not is irrelevant), and that 
it is equal to the current $D''(s^{-1})\wedge\smash{\wt f}$ with 
support in~$Y$. On the other hand, $s^{-1}\wt f_\infty$ is locally
integrable, hence $\theta(\varepsilon^{-2}|s|^2)s^{-1}\wt f_\infty$ converges
weakly to $0$ as $\varepsilon\to 0$. The uniform $L^2$ estimate on
$F_{c,\varepsilon}$ implies that there exists a weak limit
$F_{c,\varepsilon}\to F$ in $L^2_\loc((|s|\log|s|)^{-2})$.
From this we easily infer that 
$$
4(\log(|s|^2+\varepsilon^2))^{-2}
s^{-1}F_{c,\varepsilon}\to (\log|s|)^{-2}s^{-1}F
$$ 
in the weak topology of distributions, hence
$$
D''(s^{-1}F)=D''(s^{-1})\wedge\wt f,\qquad
D^{\prime\prime\star}((\log|s|)^{-2}s^{-1}F)=0
$$ 
in the limit. This is an elliptic differential system on $X\ssm Y$, 
therefore $F$ is smooth on~$X\ssm Y$. Unfortunately, the above
equations do not imply smoothness of the coefficients of $F$ near $Y$.
We hope that {\it they nevertheless imply H\"older continuity 
near~$Y$}, for any H\"older exponent~$\gamma<1$. 

In order to justify this, we select a smooth local extension 
$\smash{\wt f}$ such that $D''\smash{\wt f}=0$ and
$D's\ort\smash{\wt f}=0$ on $Y$ ($\ort$ denotes contraction by $(1,0)$-forms,
which is an operator of type $(0,-1)$). The form $\smash{\wt f}$
always exists: if the second condition is not satisfied, we can
replace $\smash{\wt f}$ with
$\smash{\wt f}-D''(\ovl s\,h)$, where $h$ is a suitable smooth
\hbox{$(n,q-1)$-form} on~$X\,$; the values taken by $\smash{\wt f}$
on~$Y$ are then uniquely defined. We then find
$$
D''(s^{-1}(F-\wt f))=0,\qquad
D^{\prime\prime\star}((\log|s|)^{-2}s^{-1}(F-\wt f))
=-D^{\prime\prime\star}((\log|s|)^{-2}s^{-1}\wt f).
$$
The main point with the choice of $\wt f$ is that no derivative of $s$
contributes in $D^{\prime\prime\star}((\log|s|)^{-2}s^{-1}\wt f)$, therefore
the singularity of this form along $Y$ is at most $(\log|s|)^{-2}s^{-1}\,$;
in particular it is in $L^2$ (and even a little bit better). We infer that 
$w:=(\log|s|)^{-2}s^{-1}(F-\wt f)$ satisfies
$$
D''((\log|s|)^2w)=0,\qquad
D^{\prime\prime\star}w=-D^{\prime\prime\star}((\log|s|)^{-2}s^{-1}\wt f).
\leqno(5.3)
$$
This is a smooth elliptic differential system on $X\ssm Y$, satisfied in the
sense of distributions on the whole of $X$, the section $w$ is known to be
$L^2$, and the principal terms in the differential system have mild 
singularities of the form $(\log|s|)^2$ at worse. Our hope is that 
one can prove from this that $w$ has
singularities of the form $O((\log|s|)^C)$. This would imply
$$
F-\wt f=O\big(|s|(\log|s|)^C\big),
$$
and thus $F$ would extend to a continuous form on $X$, whose restriction to
$Y$ is equal to~$f$. From this, it would not be very hard to regularize $F$ 
further (by local convolution procedures) to get a smooth solution.
\medskip

\noindent{\bf Case of arbitrary codimension $r>1$.}
When $r>1$, the above arguments can no longer be applied directly; 
one possibility to overcome the difficulty is to blow-up $Y$
so as to deal again with the case of a divisor. We may assume that
\hbox{$\Sigma=\emptyset$} (otherwise, we just replace $X_c$ with
$X_c\ssm\Sigma$, which is again complete K\"ahler). Instead of
working on $X_c\ssm Y_c$ as we did earlier, we work on the blow-up
$\wh X_c$ of $X_c$ along~$Y_c$. If $\mu:\wh X_c\to X_c$ is the blow-up
map, $\wh Y_c=\mu^{-1}(Y_c)$ the exceptional divisor and $\gamma$ a
positive constant, we equip $\wh X_c$ with the
smooth K\"ahler metric 
$$
\wh\omega_\gamma=\mu^\star\omega+\gamma\,
(\ii d'd''\log|s|^2+{\ii\over r}\Theta(L))\ge\mu^\star\omega.
$$ 
Then the minimal $L^2(\omega_\gamma)$ solution $u_{c,\varepsilon,\gamma}$
satisfies the equations
$$
D''u_{c,\varepsilon,\gamma}=\mu^\star g_\varepsilon
=D''(\mu^\star \wt f_\varepsilon),\qquad
D^{\prime\prime\star}_{\omega_\gamma}(|s|^{-2r}u_{c,\varepsilon,\gamma})=0
$$
on $\wh X_c\ssm\wh Y_c$, and $\wh F_{c,\varepsilon,\gamma}=
\mu^\star\wt f_\varepsilon-u_{c,\varepsilon,\gamma}$
satisfies the $L^2$ estimate
$$
\int_{\wh X_c}
{|\wh F_{c,\varepsilon,\gamma}|^2\over
(|\wh s|^2+\varepsilon^2)^r(-\log(|\wh s|^2+\varepsilon^2))^2}
dV_{\wh X_c,\omega_\gamma}
\le 8\,c_r\int_{Y_c}{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
+{\Const\over(\log\varepsilon)^2}
$$
where $\wh s=s\circ\mu$ [$\,$one can use the fact that for every $(n,q)$-form 
$u$, the integrands of $\int|u|^2_\omega dV_\omega$ and
$\int\langle B_\omega^{-1}u,u\rangle^2_\omega dV_\omega$ are 
decreasing functions of $\omega\,$; as $\omega_\gamma\ge\mu^\star\omega$,
we then infer that the right hand side always admits the given 
$\omega$-estimate as an upper bound; see e.g.\ [Dem82a] for details$\,$].
We can view $X_c$ as a submanifold of the projectivized bundle $P(E)$
of lines of~$E$, and $\cO_{\wh X_c}(-\wh Y_c)$ as the restriction to~$X_c$
of the tautological line bundle $\cO_{P(E)}(-1)$ on~$P(E)$. We thus view
$\wh s$ as a section of $\cO_{\wh Y_c}(-\wh Y_c)$ (actually, $\wh s$ is
a gene\-rator of that ideal sheaf). Since $|\wh s|^{-2r}|u_{c,\gamma,
\varepsilon}|^2$ is locally integrable by construction, we get
$$
D''({\wh s\,}^{-r}u_{c,\varepsilon,\gamma})=
{\wh s\,}^{-r}D''(\mu^\star \wt f_\varepsilon),\qquad
D^{\prime\prime\star}_{\omega_\gamma}({\wh s\,}^{-r}u_{c,\varepsilon,\gamma})=0
$$
on $\wh X_c$. Thanks to the equality
$$
\mu^\star(\Lambda^nT^\star_X)=\Lambda^nT^\star_{\wh X}\otimes
\cO_{\wh X}(-(r-1)\wh Y),
$$ 
we see that $\mu^\star\wt f_\varepsilon$ vanishes at order $r-1$ along $\wh Y$.
If we view our $(n,q)$-forms on $\wh X$ rather as $(0,q)$-forms with values
in $\mu^\star(\Lambda^nT^\star_X\otimes L)$, we may consider philosophically 
that we cancel out a factor ${\wh s\,}^{r-1}$ in the equations. The same
proof as in the case of codimension $1$ now shows that 
$\wh F_{c,\varepsilon,\gamma}$ is smooth on $\wh X_c\ssm\wh Y_c$ and has
H\"older continuous coefficients on $\wh X_c\,$; in particular, we have
a meaningful restriction equality
$$
\wh F_{c,\varepsilon,\gamma\restriction \wh Y_c}=\mu^\star f\qquad
\hbox{in $\mu^\star(\Lambda^nT^\star_X\otimes L)\otimes
\Lambda^{0,q}T^\star_{\wh X}$, over $\wh X_c$}.
$$
We now want to take the limit as $\varepsilon,\,\gamma$ tend to $0$ and
$c$ tends to $+\infty$. The trouble is that we lose control on
the regularity properties as $\gamma$ goes to zero ($\wh\omega_\gamma$ 
becomes a degenerate metric on $\wh X_c$ for $\gamma=0$). We can 
nevertheless let $\varepsilon$ go to $0$ and then $c$ to~$+\infty$.
In this way we find a section $\wh F_\gamma$
of $\mu^\star(\Lambda^nT^\star_X\otimes L)\otimes
\Lambda^{0,q}T^\star_{\wh X}$ on $\wh X$ such that
$$
\leqalignno{
&\wh F_{\gamma\restriction \wh Y}=\mu^\star f\qquad
\hbox{in $\mu^\star(\Lambda^nT^\star_X\otimes L)\otimes
\Lambda^{0,q}T^\star_{\wh X}$},&(5.4)\cr
&
\int_{\wh X}
{|\wh F_{\gamma}|^2\over|\wh s|^{2r}(-\log(|\wh s|^2))^2}
dV_{\wh X,\omega_\gamma}
\le 8\,c_r\int_Y{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}.
&(5.5)\cr}
$$
If what we have said earlier in codimension $1$ holds true, then
$\wh F_\gamma$ is continuous and we can smooth it further to get
a smooth solution on $\wh X$ satisfying essentially the same $L^2$ 
estimate. We still have to push forward the 
solution down to $X$ and obtain an $L^2$ estimate for it when $\gamma=0$
(and still without losing the regularity of the solution). For this, we 
observe that there is a commutative diagram
$$
\eqalign{
\hfill H^q(X,\Lambda^nT^\star_X\otimes L)&{}\buildo\mu^\star\over\lra
H^q\big(\wh X,\mu^\star(\Lambda^qT^\star_X\otimes L)\big)\hfill\cr
\hfill\restr\bigg\downarrow\kern30pt&
\kern60pt\bigg\downarrow\restr\cr
H^q(Y,(\Lambda^nT^\star_X\otimes L)_{|Y})&{}\buildo\mu^\star\over\lra
H^q\big(\wh Y,\mu^\star(\Lambda^nT^\star_X\otimes L)_{|\wh Y}\big).\cr}
\leqno(5.6)
$$
The horizontal arrows are isomorphisms, thanks to Leray's spectral sequence
and the fact that the higher direct sheaves $R^q\mu_\star(\cO_{\wh X})$
(resp.\ $R^q(\mu_{|\wh Y})_\star(\cO_{\wh Y})$) are zero on $X$
(resp.\ $Y$) for $q\ge 1$, by K\"unneth's formula and the well known 
cohomological properties of projective spaces. The left hand restriction arrow
is surjective by what we have just proved (any $(0,q)$ section becomes 
$L^2$ with a suitable rapidly decaying weight $e^{-\chi\circ\psi}$). Hence
the right hand vertical arrow is also surjective, and we infer that there
is a $\dbar$-closed form 
$$
F\in C^\infty\big(X,(\Lambda^nT^\star_X\otimes L)\otimes\Lambda^{0,q}
T^\star_X\big)
$$
such that $F_{|Y}=f$. [Note: a priori $F$ is obtained only as a cohomology
class, since every coboundary form $\dbar g$ on $Y$ extends to $X$, we even
conclude that the extension exists as a pointwise defined form]. This is
anyway enough to conclude the qualitative extension result stated after
Corollary~4.10, in the case of arbitrary degree $q$ and arbitrary
codimension~$r$.\qed

\titleb{6.}{Approximation of psh functions by logarithms of\ \newline
holomorphic functions}
\titlebrunning{6. Approximation of psh functions by logarithms of
holomorphic functions}
We prove here, as an application of the Ohsawa-Takegoshi extension theorem,
that every psh function on a pseudoconvex open set $\Omega\subset\bbbc^n$
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 [DK96]. 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 $\ord_{x_0}(f)=\sup\{k\in\bbbn\,;D^{\alpha}f(x_0)=0,~
\forall|\alpha|<k\}$.

\begstat{(6.1) Theorem} Let $\varphi$ be a plurisubharmonic function on a
bounded pseudoconvex open set~$\Omega\subset\bbbc^n$. 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 $z\in\Omega$.
\endstat

\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 (estimate 4.10) applied
to the $0$-dimensional subvariety $\{z\}\subset\Omega$ shows that for any
$a\in\bbbc$ 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 6.1 implies in a straightforward manner the deep result of
[Siu74] on the analyticity of the Lelong number sublevel sets.

\begstat{(6.2) 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~$X$.
\endstat

\begproof{} Since analyticity is a local property, it
is enough to consider the case of a psh function $\varphi$ on a 
pseudoconvex open set $\Omega\subset\bbbc^n$. The
inequalities obtained in 6.1~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 6.1 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
$L_{|U}\simeq U\times\bbbc$ and $\Vert\xi\Vert_h=|\xi|e^{-\varphi(x)}$,
$\xi\in L_x\simeq\bbbc$). 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 $\Omega\subset\bbbc^n$
with $H^2(\Omega,\bbbr)=0$, 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 $L^{\otimes k}$.

\begstat{(6.3) 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 $s\to+\infty$.
\endstat

\begstat{(6.4) Remark} {\rm The proof will actually show, with very slight
modifications, that Proposition~6.3 also holds when $X$ is a Stein manifold
and $L$ is an arbitrary holomorphic line bundle.}
\endstat

\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
$$
\nu(T,x)-{n\over m}\le\nu(T_m,x)\le\nu(T,x)
\leqno(6.4)
$$
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~6.1.
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
$$v_{m,\Omega}(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over 2s}
\log{C_2\over r^{2n}}.\leqno(6.5)$$
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 (6.5), 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 $\bbbc^n$.
By the Ohsawa-Takegoshi $L^2$ extension theorem (remark (4.11)), 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 $\theta:\bbbr\to[0,1]$ 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
$$
\nu(T,x)-{n\over m}\le{1\over m}\min\ord_x(f_j)\le\nu(T,x).\leqno(6.6)
$$

\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~6.7 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

\begstat{(6.7) Lemma} Let $\Omega$ be an open subset in $\bbbc^n$ 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
$$\nu(v,x)\le\nu\Big({1\over \ell }\log|h_\ell |\Big)\le\nu(v,x)+{1\over \ell },
~~~~\forall x\in X,~~\forall \ell \ge 1.$$
\endstat

\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{7.}{Multiplier ideal sheaves and the Brian\c{c}on-Skoda
theorem}
 In this section, we briefly recall the definition and main properties
of multiplier ideal sheaves. These have been originally introduced by
A.~Nadel [Nad89, 90] for the study of the existence of
K\"ahler-Einstein metrics.

\begstat{(7.1) 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 \hbox{$f\in\cO_{\Omega,x}$} such that 
$|f|^2e^{-2\varphi}$ is integrable with respect to the Lebesgue measure
in some local coordinates near~$x$.
\endstat

The zero variety $V(\cI(\varphi))$ is thus the set of points in
a neighborhood of which $e^{-2\varphi}$ is non integrable. Such points 
occur only if $\varphi$ has logarithmic poles, in virtue 
of the following basic Lemma due to Skoda [Sko72a].

\begstat{(7.2) Lemma {\rm (Skoda)}} 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~$\varphi$.\qed
\vskip0pt
\endstat

In fact, the ideal sheaf $\cI(\varphi)$ is always a coherent ideal sheaf,
and therefore its zero variety is an analytic set. This result is due to
Nadel ([Nad89], see also [Dem93b]).

\begstat{(7.3) Proposition {\rm(Nadel)}} For any psh function
$\varphi$ on~$\Omega\subset X$, the sheaf
$\cI(\varphi)$ is a coherent sheaf of ideals over $\Omega$.\qed
\endstat

\begproof{} As the main argument will be needed hereafter, we briefly 
reproduce the argument. As the result is local, we may assume that 
$\Omega$ is a bounded pseudoconvex open set in~$\bbbc^n$. Let $E$ be 
the set of all holomorphic functions $f$ 
on $\Omega$ such that $\int_\Omega|f|^2e^{-2\varphi}\,d\lambda<+\infty$.
By the strong noetherian property of coherent sheaves, the set $E$
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 $\dbar u=g:=\dbar(\theta f)$ by means of H\"ormander's
$L^2$ estimates applied with the strictly plurisubharmonic 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, $F\in E$ and 
$f_x-F_x=u_x\in\cI(\varphi)_x\cap\gm_{\Omega,x}^{s+1}$.
This proves our contention.\qed
\endproof

The importance of multiplier ideal sheaves stems from the following
basic vanishing theorem due to Nadel [Nad89] (see also [Dem93b]),
which is a direct consequence of the Andreotti-Vesentini-H\"ormander
$L^2$ estimates.  If $(L,h)$ is a pseudoeffective line
bundle, we denote $\cI(h)=\cI(\varphi)$ where $\varphi$ is the weight
function of~$h$ on any trivialization open set.

\begstat{(7.4) Nadel vanishing theorem} 
Let $(X,\omega)$ be a compact K\"ahler manifold, and let $L$ be a
holomorphic line bundle over $X$ equipped with a singular hermitian metric
$h$ such that $\ii\Theta_h(L)\ge\varepsilon
\omega$ for some continuous positive function $\varepsilon$ on~$X$. Then
$$
H^q\big(X,\cO(K_X\otimes L)\otimes\cI(h)\big)=0\qquad
\hbox{for all $q\ge 1$.}
\eqno\square
$$
\endstat

Our next goal is to understand somewhat better the behaviour of 
a multiplier ideal sheaf $\cI(k\varphi)$ as $k$ tends to $+\infty$.
The intuition is that the ideal grows more or less ``linearly''.
The following result provides a natural inclusion result for 
multiplier ideal sheaves, inspired by the classical Brian\c{c}on-Skoda 
theorem [BS74].

\begstat{(7.5) Theorem} Let $X$ be complex $n$-dimensional manifold
and let $\varphi$, $\psi$ be plurisubharmonic functions on $X$. Then
for any integer $k\ge n$ we have
$$
\cI(k\varphi+\psi)\subset \cI(\varphi)^{k-n}\cI(\psi).
$$
\endstat

\begproof{} Since the result is local, we can assume that $X=\Omega$ is
a bounded pseudoconvex open set. In that case, possibly after shrinking
$\Omega$ a little bit, the proof of (7.3) shows that $\cI(\varphi)$ is 
generated by a finite number of elements $g=(g_1,\ldots,g_N)\in\cO(\Omega)$
such that
$$
\int_\Omega|g_j|^2e^{-2\varphi}d\lambda<+\infty.
$$
We set as usual $|g|^2=|g_1|^2+\cdots+|g_N|^2$. It is a consequence of 
estimate (6.1a) (and thus of the Ohsawa-Takegoshi theorem)
that \hbox{$\varphi\le\log|g|+C$} for some constant $C>0$.
Now, let $f\in\cI(k\varphi+\psi)_{z_0}$ be a germ of holomorphic function 
defined on a neighborhood $V$ of $z_0\in\Omega$. If $V$ is small enough,
we have
$$
\int_V|f|^2|g|^{-2k}e^{-2\psi}d\lambda\le C'\int_V|f|^2e^{-2k\varphi-2\psi}
d\lambda<+\infty.
$$
By Skoda's division theorem (in its original form [Sko72b]; see also [Sko78]),
this implies that $f$ can be written as 
$$
f=g\cdot h=\sum_{1\le j\le N}g_jh_j
$$ 
with a $N$-tuple $h=(h_1,\ldots,h_N)$ of
holomorphic functions $h_j$ such that
$$
\int_V|h|^2|g|^{-2(k-1)}e^{-2\psi}d\lambda\le 
{k-n+1\over k-n}\int_V|f|^2|g|^{-2k}e^{-2\psi}
d\lambda,
$$
provided that $k>n$, i.e.\ $k-1\ge n$. By induction, we find a
multi-indexed collection 
$u_\ell=(u_{\ell,j_1j_2\ldots j_\ell})$ of holomorphic functions
on $V$ such that
$$
f=g^\ell\cdot u_\ell=\sum g_{j_1}g_{j_2}\cdots g_{j_\ell}u_{\ell,
j_1j_2\ldots j_\ell}
$$
and 
$$
\int_V\sum_\ell|u_\ell|^2|g|^{-2(k-\ell)}e^{-2\psi}d\lambda\le 
{k-n+1\over k-n+1-\ell}\int_V|f|^2|g|^{-2k}e^{-2\psi}
d\lambda
$$
whenever $k-\ell\ge n$. The last $L^2$ inequality shows that
$u_\ell\in\cI(\psi)_{z_0}$. The theorem follows by taking $\ell=k-n$.\qed
\endproof

\titleb{8.}{On Fujita's approximate Zariski decomposition of big line bundles}

Our goal here is to 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.
It turns out that Theorem 7.5 (of Brian\c{c}on-Skoda type) 
can be used to derive a simple direct proof, based on different
techniques (Theorem 8.5 below). The idea arose in the course of 
discussions 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.

\begstat{(8.1) Proposition} The line bundle $L$ is big if and only if there
a multiple $m_0L$ such that $m_0L=E+A$, where $E$ is an effective divisor
and $A$ an ample divisor.
\endstat

\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

\begstat{(8.2) 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.
\endstat

\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 $a\in\bbbn$ 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

The next lemma is due to Siu and was first observed in [Siu97] for the
proof of the invariance of plurigenera.

\begstat{(8.3) Lemma} 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 $G-(K_X+(n+1)A)$ is ample, e.g.\ $G=K_X+(n+2)A$.
\endstat

\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
$$
\int_B|u(z)|^2e^{-2\varphi(z)}|z-z_0|^{-2(n+1)}dV(z)<+\infty.
$$
Then Skoda's division theorem [Sko72b] implies $u(z)=\sum(z_j-z_{j,0})v_j(z)$
with
$$
\int_B|v_j(z)|^2e^{-2\varphi(z)}|z-z_0|^{-2(n+1)}dV(z)<+\infty,
$$
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 $|\sigma|^{-2(n+1)}$.
The weight of the above metric has singularity $(n+1)\log|z-z_0|^2$
at~$z_0$, and its curvature is
$$
\ii\Theta(G)+(n+1)\ii\ddbar\log|\sigma|^2\ge
\ii\Theta(G)-(n+1)\Theta(A).
\leqno(\star)
$$
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 7.4, the solution exists if the metric of 
$G+L-K_X$ has positive curvature. As $\Theta_h(L)\ge 0$ in the sense 
of currents, $(\star)$ shows that a sufficient condition is $G-K_X-(n+1)A>0$.
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
$\cO(G+L)\otimes\cI(h)$.\qed
\endproof

\noindent
We further need to invoke the existence of metrics with minimal singularities
(the reader will find more details about this topic in [Dem98] and [DPS99]).
This result can be seen as the analytic analogue of Zariski decomposition
for pseudo-effective divisors. Although algebraic Zariski decomposition
does not always exist and is a very hard problem to deal with, it turns out
that the existence of metrics with minimal singularities is a simple basic 
consequence of the existence of upper envelopes of plurisubharmonic functions,
as already mentioned in Lelong's early work on the subject.

\begstat{(8.4) Proposition} Let $L$ be a pseudoeffective line bundle~$X$.
There exists a singular hermitian metric $h_0$ satisfying
\smallskip
\item{\rm(i)} $\ii\Theta_{h_0}(L)\ge 0.$
\smallskip
\item{\rm(ii)} $h_0$ has minimal singularities among all metrics $h$ with
$\ii\Theta_h(L)\ge 0$, i.e.\ $h_0\le C h$, where $C$ is a constant, for
every such $h$.
\smallskip
\noindent
Moreover, for every nonnegative integer $k$, there is a natural isomorphism
$$
H^0(X,\cO(kL)\otimes\cI(h_0^k))\hookrightarrow H^0(X,\cO(kL)).
\leqno\hbox{\rm(iii)}
$$
\endstat

\begproof{} Let us fix once for all a smooth hermitian metric
$h_\infty$ on $L$ (with a curvature form
$\theta_\infty=\ii\Theta_{h_\infty}(L)$ of arbitrary signature). We
write any other singular hermitian metric as $h=e^{-\psi}h_\infty$
where $\psi\in L^1_\loc(X)$. Since $L$ is pseudoeffective, it is
possible to find $\psi$ such that
$\ii\Theta_h(L)=\ii\ddbar\psi+\theta_\infty\ge 0$, and we will always
assume that this is the case. In particular, $\psi$ is almost
plurisubharmonic and locally bounded from above. Also, we only deal
with $h$ up to equivalence of singularities, so we may adjust $\psi$
by a constant in such a way that $\sup_X\psi\le 0$. We define
$$
\psi_0(x)=\sup\big\{\psi(x)\,;\;\psi~\hbox{almost psh},
~\ii\ddbar\psi+\theta_\infty\ge 0,~\sup_X\psi\le 0\big\}.
$$ 
By the well known properties of plurisubharmonic functions (see 
Lelong [Lel68]), the upper semicon\-ti\-nuous regularization 
$\psi_0^\star$ is almost plurisubharmonic
and satisfies $\ii\ddbar\psi_0^\star+\theta_\infty\ge 0$, $\sup\psi_0^\star
\le 0$. We see that $\psi_0^\star$ concurs in the supremum defining $\psi_0$,
hence $\psi_0^\star=\psi_0$. The metric $h_0=e^{-\psi_0}\theta_\infty$ is
by definition a metric with minimal singularities which satisfies
$\ii\Theta_{h_0}(L)\ge 0$. If $\sigma\in H^0(X,kL)$, we define a 
corresponding singular hermitian metric on $L$ by
$$
\Vert\xi\Vert_h=\Big|{\xi^{\otimes k}\over\sigma(x)}\Big|^{1/k},
\qquad \xi\in L_x.
$$
Its curvature is $\Theta_h(L)={1\over k}[Z_\sigma]\ge 0$ where $Z_\sigma$ is
the zero divisor of $\sigma$, hence (up to a multiplicative constant) 
$h:=e^{-\psi}h_\infty$ also concurs in the definition of $\psi_0$. 
From this, we infer $\psi\le\psi_0+C$, hence
$$
\Vert\sigma\Vert_{h_0^{\otimes k}}\le e^{kC}\Vert\sigma\Vert_{h^{\otimes k}}
=e^{kC}.
$$
A fortiori $\sigma$ is a $L^2$ section with respect to the metric $h_0$,
i.e.\ 
$$
\sigma\in H^0(X,\cO(kL)\otimes\cI(h_0^{\otimes k})),
$$ 
therefore $H^0(X,\cO(kL)\otimes\cI(h_0^{\otimes k}))=
H^0(X,\cO(kL))$, as desired.\qed

\endproof

\noindent
We are now ready to prove Fujita's decomposition theorem.

\begstat{(8.5) 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 $\bbbq$-divisor
and $A$ an ample $\bbbq$-divisor, such that $A^n>v(L)-\varepsilon$.
\endstat

\begstat{(8.6) 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
$A^n\le v(L)$.}
\endstat

\begstat{(8.7) Remark} {\rm Once Theorem 8.4 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 $A$.}
\endstat

\begproof{of Theorem 8.5.} It is enough to prove the theorem with $A$
being a big and nef divisor. In fact, Proposition 8.1 then shows that we can
write $A=E'+A'$ where $E'$ is an effective $\bbbq$-divisor and $A'$ an
ample $\bbbq$-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 8.3. Lemma 8.2 implies that
$v(mL-G)>m^n(v(L)-\varepsilon)$ for $m$ large. By Proposition~8.4, 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 8.3 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
$$
\eqalign{
H^0(\wt X,\ell A)
  &=H^0\big(\wt X,\ell(m\,\mu^\star L-E)\big)\cr
  &\supset H^0\big(\wt X,\mu^\star\big(\cO(\ell mL)\otimes
       \cI(\varphi_m)^\ell\big)\big)\cr
  &\supset H^0\big(\wt X,\mu^\star\big(\cO(\ell mL)\otimes
       \cI((\ell+n)\varphi_m)\big)\big),\cr}
$$
thanks to the inclusion of sheaves
$\cI((\ell+n)\varphi_m)\subset \cI(\varphi_m)^\ell$ implied by the
Brian\c{c}on-Skoda theorem. 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
$$
\eqalign{
H^0(\wt X,\ell A)
  &\supset H^0\big(X,\cO(\ell mL)\otimes \cI((\ell+n)\varphi_m)\big)\cr
  &\supset H^0\big(X,\cO((\ell+n)(mL-G))\otimes \cI((\ell+n)\varphi_m)\big)\cr
  &= H^0\big(X,\cO((\ell+n)(mL-G))\big).\cr}
$$
The final equality is given by (8.4$\,$iii), whereas the second inclusion
holds true if $(\ell+n)G-nmL$ is effective. This is certainly the
case if $\ell\gg m$. By Lemma 8.2, we find
$$
h^0(\wt X,\ell A)\ge{(\ell+n)^n\over n!}m^n\big(v(L)-\varepsilon\big)
$$
for infinitely many $\ell$, therefore $v(A)=A^n\ge m^n(v(L)-\varepsilon)$.
Theorem 8.5 is proved, up to a minor change of notation
$E\mapsto{1\over m}E$, $A\mapsto{1\over m}A$.\qed
\endproof

\vskip28pt
\noindent{\tbfontt References} 
\nextref{References}
{\eightpoint\bigskip

\ref [AN54]& Akizuki, Y., Nakano, S.& Note on Kodaira-Spencer's
proof of Lefschetz theorems,& Proc.\ Jap.\ Acad.\ {\bf 30} (1954) 266--272&

\ref [AV65]& Andreotti, A., Vesentini, E.& Carleman estimates for the
Laplace-Beltrami equation in complex manifolds,& Publ.\ Math.\ I.H.E.S.\
{\bf 25} (1965) 81--130&

\ref [Ber96]& Berndtsson, B.& The extension theorem of Ohsawa-Takegoshi
and the theorem of Donnelly-Fefferman,& Ann.\ Inst.\ Fourier 
{\bf 14} (1996) 1087--1099&

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

\ref [Bom70]& Bombieri, E.& Algebraic values of meromorphic maps,& Invent.\ 
Math.\ {\bf 10} (1970) 267--287\ \ and\ \ {\it Addendum}, Invent.\ Math.\ 
{\bf 11} (1970), 163--166&    

\ref [Bre54]& Bremermann, H.& \"Uber die \"Aquivalenz der
pseudokonvexen Gebiete und der Holomorphiegebiete im Raum von $n$
komplexen Ver\"anderlichen,& Math.\ Ann.\ {\bf 128} (1954) 63--91&

\ref [BS74]& Brian\c{c}on, J., Skoda, H.& Sur la cl\^oture int\'egrale
d'un id\'eal de germes de fonctions holomorphes en un point de $\bbbc^n$,&
C.~R.\ Acad.\ Sci.\ s\'erie A {\bf 278} (1974) 949--951&

\ref [Dem82a]& Demailly, J.-P.& Estimations $L^2$ pour l'op\'erateur
$\ovl\partial$ d'un fibr\'e vectoriel holomorphe semi-positif au dessus
d'une vari\'et\'e k\"ahl\'erienne compl\`ete,& Ann.\ Sci.\ Ecole Norm.\
Sup.\ {\bf 15} (1982) 457--511&

\ref [Dem82b]& Demailly, J.-P.& Courants positifs extr\^emaux et conjecture
de Hodge,& Invent.\ Math.\ {\bf 69} (1982) 347--374&

\ref [Dem92]& Demailly, J.-P.& Regularization of closed
positive currents and intersection theory,& J.\ Alg.\ Geom.\ {\bf 1}
(1992) 361--409&

\ref [Dem93a]& Demailly, J.-P.& 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&

\ref [Dem93b]& Demailly, J.-P.& A numerical criterion for very ample line
bundles,& J.\ Differential Geom.\ {\bf 37} (1993) 323--374&

\ref [Dem96a]& Demailly, J.-P.& Effective bounds for very ample line 
bundles, & Invent.\ Math. {\bf 124} (1996) 243--261&

\ref [Dem96b]& Demailly, J.-P.& Complex analytic and differential geometry,&
preliminary draft, Institut Fourier, $\simeq 650\,$p&

\ref [DK96]& Demailly, J.-P., Koll\'ar, J.&
Semicontinuity of complex singularity exponents and K\"ahler-Einstein 
metrics on Fano orbifolds,& Manuscript, July 1996&

\ref [DPS99]& Demailly, J.-P., Peternell, T., Schneider, M.&
Pseudo-effective line bundles on projective varieties,& Manuscript in 
preparation, (1999)&

\ref [DF83]& Donnelly H., Fefferman C.& $L^2$-cohomology and
index theorem for the Bergman metric,& Ann.\ Math.\ {\bf 118} (1983) 
593--618&

\ref [DX84]& Donnelly H., Xavier F.& On the differential form spectrum 
of negatively curved Riemann manifolds,& Amer.\ J.\ Math.\ {\bf 106}
(1984) 169--185&

\ref [Fuj94]& Fujita, T.& Approximating Zariski decomposition of big
line bundles,& Kodai Math.~J.\ {\bf 17} (1994) 1--3&

\ref [Gri66]& Griffiths, P.A.& The extension problem in complex analysis~II;
embeddings with positive normal bundle,& Amer.\ J.~Math.\ {\bf 88} (1966) 
366--446&

\ref [Gri69]& Griffiths, P.A.& Hermitian differential geometry, Chern classes
and positive vector bundles,& Global Analysis, papers in honor of
K.~Kodaira, Princeton Univ.\ Press, Princeton (1969) 181--251&

\ref [Hir64]& Hironaka, H.& Resolution of singularities of an algebraic
variety over a field of characteristic zero,& Ann.\ of Math.\ {\bf 79}
(1964) 109--326&

\ref [H\"or65]& H\"ormander, L.& $L^2$ estimates and existence
theorems for the $\ovl\partial$ operator,& Acta Math.\ {\bf 113}
(1965) 89--152&

\ref [H\"or66]& H\"ormander, L.& An introduction to Complex
Analysis in several variables,& 1st edition, Elsevier Science Pub.,
New York, 1966, 3rd revised edition, North-Holland Math.\ library, Vol
7, Amsterdam (1990)&

\ref [Kod53a]& Kodaira, K.& On cohomology groups of compact analytic varieties
with coefficients in some analytic faisceaux,& Proc.\ Nat.\ Acad.\ Sci.\
U.S.A.\ {\bf 39} (1953) 868--872&

\ref [Kod53b]& Kodaira, K.& On a differential geometric method in the theory
of analytic stacks,& Proc.\ Nat.\ Acad.\ Sci.\ U.S.A.\ {\bf 39} (1953)&
1268--1273

\ref [Kohn63]& Kohn, J.J.& Harmonic integrals on strongly pseudo-convex
manifolds~I,& Ann.\ Math.\ (2), {\bf 78} (1963)& 206-213

\ref [Kohn64]& Kohn, J.J.& Harmonic integrals on strongly pseudo-convex
manifolds~II,& Ann.\ Math.\ {\bf 79} (1964)& 450-472

\ref [Kod54]& Kodaira, K.& On K\"ahler varieties of restricted type,& Ann.\ 
of Math.\ {\bf 60} (1954) 28--48&

\ref [Laz99]& Lazarsfeld, R.& Private communication,& Ann Arbor University,
April 1999&

\ref [Lel68]& Lelong, P.& Fonctions plurisousharmoniques et formes
diff\'erentielles positives,& Dunod, Paris, Gordon \ampersand Breach, 
New York (1968)&

\ref [Lel72]& Lelong, P.& 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&

\ref [Man93]& Manivel, L.& Un th\'eor\`eme de prolongement $L^2$ de
sections holomorphes d'un fibr\'e vectoriel,& Math.\ Zeitschrift {\bf 212}
(1993) 107--122&

\ref [Nak55]& Nakano, S.& On complex analytic vector bundles,& J.~Math.\ Soc.\
Japan {\bf 7} (1955) 1--12&

\ref [Nak73]& Nakano, S.& Vanishing theorems for weakly $1$-complete
manifolds,& Number Theory, Algebraic Geometry and Commutative
Algebra, in honor of Y.~Akizuki, Kinokuniya, Tokyo (1973) 169--179&

\ref [Nak74]& Nakano, S.& Vanishing theorems for weakly $1$-complete
manifolds~II,& Publ.\ R.I.M.S., Kyoto Univ.\ {\bf 10} (1974) 101--110&

\ref [Ohs88]& Ohsawa, T.& On the extension of $L^2$ holomorphic
functions, II,& Publ. RIMS, Kyoto Univ.\ {\bf 24} (1988) 265--275&

\ref [Ohs94]& Ohsawa, T.& 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&

\ref [Ohs95]& Ohsawa, T.& On the extension of $L^2$ holomorphic
functions, III$\,$: negligible weights,& Math.\ Zeitschrift {\bf 219}
(1995) 215--225&

\ref [OT87]& Ohsawa, T., Takegoshi, K.& On the extension of $L^2$
holomorphic functions,& Math.\ Zeitschrift {\bf 195} (1987) 197--204&

\ref [Siu74]& Siu, Y.T.& Analyticity of sets associated to Lelong numbers and
the extension of closed positive currents,& Invent.\ Math.\ {\bf 27}
(1974) 53--156&

\ref [Sko72a]& Skoda, H.& Sous-ensembles analytiques d'ordre fini ou infini
dans $\bbbc^n$,& Bull.\ Soc.\ Math.\ France, {\bf 100} (1972) 353--408&

\ref [Sko72b]& Skoda, H.& Application des techniques $L^2$ \`a l'\'etude des
id\'eaux d'une alg\`ebre de fonctions holomorphes avec poids,& Ann.\ Sci.\
Ecole Norm.\ Sup.\ {\bf 5} (1972) \hbox{545--579}&

\ref [Sko76]& Skoda, H.& Estimations $L^2$ pour l'op\'erateur $\ovl\partial$
et applications arithm\'etiques,& S\'eminaire P.\ Lelong (Analyse), ann\'ee
1975/76, Lecture Notes in Math., Vol.~538, Springer-Verlag, Berlin
(1977) 314--323&

\ref [Sko78]& Skoda, H.& Morphismes surjectifs de fibr\'es vectoriels
semi-positifs,& Ann.\ Sci.\ Ecole Norm.\ Sup.\ {\bf 11} (1978) 577--611&

}
\bigskip\noindent
Jean-Pierre Demailly\newline
Universit\'e de Grenoble I\newline
Institut Fourier, BP74\newline
38400 Saint-Martin d'H\`eres, France\newline
{\tt demailly@fourier.ujf-grenoble.fr}

\bigskip\noindent
(version of June 25, 1999, printed on \today, \timeofday)

\end

% Local Variables:
% TeX-command-default: "TeX"
% End:





, we find a $(n,q)$-form $\wh F$ with $L^2_\loc$ 
coefficients on $\wh X$ such that
$$
\int_{\wh X}
{|\wh F|^2\over
|\wh s|^{2r}(-\log|\wh s|)^2}
dV_{\wh X,\omega}
\le 256\,c_r\int_{Y_c}{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}
$$
and
$$
\eqalign{D''({\wh s\,}^{-r}\wh F)
&=D''({\wh s\,}^{-r}\mu^\star\wt f)-{\wh s\,}^{-r}D''(\mu^\star\wt f)
=D''({\wh s\,}^{-1})\wedge ({\wh s\,}^{-(r-1)}\mu^\star\wt f),\cr
D^{\prime\prime\star}_{\mu^\star\omega}({\wh s\,}^{-r}\wh F)&=0\cr}
$$
in the sense of distributions (everywhere on $\wh X$ and not only on
$\wh X\ssm\wh Y$); indeed, thanks to the equality
$$
\mu^\star(\Lambda^nT^\star_{X_c})=\Lambda^nT^\star_{\wh X_c}\otimes
\cO_{\wh X_c}(-(r-1)\wh Y_c),
$$ 
we see that $\mu^\star\wt f$ vanishes at order $r-1$ along $\wh Y$.
If we view $\wh F$ as a $(0,q)$-form with values in
$\mu^\star(\Lambda^nT^\star_X\otimes L)$ rather than as a $(n,q)$-form
on $\wh X$, we may consider philosophically that we cancel out a
factor ${\wh s\,}^{r-1}$ in the equations. This shows that we are
essentially in the same situation as in the earlier case $r=1$, except
that $D^{\prime\prime\star}_{\mu^\star\omega}$ is computed with
respect to a metric $\mu^\star\omega$ which is degenerate along~$\wh Y$.
A~finer analysis of the Green kernel of $\Delta''_{\mu^\star\omega}$
shows that $F:=\mu_\star F$ is smooth on $X\ssm Y$, that $F$ still has 
continuous coefficients near $Y$, and that $F_{\restriction Y}=f$. 
We then produce the desired solution by taking a small perturbation
of~$F$ as above. The details are rather tedious and will be left to
the reader.


