%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Holomorphic Morse inequalities and entire curves in projective varieties
%
% Jean-Pierre Demailly (Universit\'e de Grenoble I, Institut Fourier)
%
% July 2011
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% Personal macros
\def\bigger{1200} \def\smaller{1000} \let\textsize=\bigger

\def\swapleft#1#2{\hbox{\rlap{#2}\kern\parindent\kern-#1}}
\newcount\doextras \doextras=0

\ifx\textsize\smaller
\magnification=\textsize
\vsize=19.5cm \hsize=12.5cm
\parskip=3pt plus 2pt minus 1pt
\parindent=6.24mm
\let\sbreak=\break
\def\bbreak{\ }
\else
\magnification=\textsize
\vsize=20.4cm \hsize=13.5cm
\parskip=4pt plus 2pt minus 1pt
\parindent=5.2mm
\def\sbreak{\ }
\let\bbreak=\break
\fi

\font\twentybf = cmbx10 at 20.736pt
\font\fourteenbf = cmbx10 at 14.4pt
\font\twelvebf = cmbx10 at 12pt

\font\fourteenbsy=cmbsy10 at 14.4pt
\font\twelvebsy=cmbsy10 at 12pt
\font\tenbsy=cmbsy10
\font\eightbsy=cmbsy8
\font\sevenbsy=cmbsy7
\font\sixbsy=cmbsy6
\font\fivebsy=cmbsy5

%\font\tensans = cmss10
%\font\eightsans = cmss8
%\font\sevensans = cmss7
%\font\sixsans = cmss6
%\font\fivesans = cmss5
%\newfam\sansfam
%\textfont\sansfam=\tensans
%\scriptfont\sansfam=\sevensans
%\scriptscriptfont\sansfam=\fivesans
%\def\sans{\fam\sansfam\tensans}

\font\tenCal=caljpd10 at 10pt
\font\eightCal=caljpd10 at 8pt
\font\sevenCal=caljpd10 at 7pt
\font\sixCal=caljpd10 at 6pt
\font\fiveCal=caljpd10 at 5pt

%\font\tenCal=mrsfs10 at 10pt
%\font\eightCal=mrsfs10 at 8pt
%\font\sevenCal=mrsfs10 at 7pt
%\font\sixCal=mrsfs10 at 6pt
%\font\fiveCal=mrsfs10 at 5pt

\newfam\Calfam
  \textfont\Calfam=\tenCal
  \scriptfont\Calfam=\sevenCal
  \scriptscriptfont\Calfam=\fiveCal
\def\Cal{\fam\Calfam\tenCal}

%% AMS stuff

\font\tenmsa = msam10
\font\sevenmsa = msam7
\font\fivemsa = msam5
\newfam\msafam
\textfont\msafam=\tenmsa
\scriptfont\msafam=\sevenmsa
\scriptscriptfont\msafam=\fivemsa

\font\tenmsb = msbm10
\font\sevenmsb = msbm7
\font\fivemsb = msbm5
\newfam\msbfam
\textfont\msbfam=\tenmsb
\scriptfont\msbfam=\sevenmsb
\scriptscriptfont\msbfam=\fivemsb

\font\teneuf = eufm10
\font\seveneuf = eufm7
\font\fiveeuf = eufm5
\newfam\euffam
\textfont\euffam=\teneuf
\scriptfont\euffam=\seveneuf
\scriptscriptfont\euffam=\fiveeuf

\catcode`\@=11
\def\frak#1{{\fam\euffam#1}}
\def\Bbb#1{{\fam\msbfam#1}}
\def\hexnumber@#1{\ifcase#1 0\or1\or2\or3\or4\or5\or6\or7\or8\or9\or
 A\or B\or C\or D\or E\or F\fi}
\edef\msa@{\hexnumber@\msafam}
\edef\msb@{\hexnumber@\msbfam}
\mathchardef\boxdot="2\msa@00
\mathchardef\boxplus="2\msa@01
\mathchardef\boxtimes="2\msa@02
\mathchardef\square="0\msa@03
\mathchardef\blacksquare="0\msa@04
\mathchardef\centerdot="2\msa@05
\mathchardef\lozenge="0\msa@06
\mathchardef\blacklozenge="0\msa@07
\mathchardef\circlearrowright="3\msa@08
\mathchardef\circlearrowleft="3\msa@09
\mathchardef\rightleftharpoons="3\msa@0A
\mathchardef\leftrightharpoons="3\msa@0B
\mathchardef\boxminus="2\msa@0C
\mathchardef\Vdash="3\msa@0D
\mathchardef\Vvdash="3\msa@0E
\mathchardef\vDash="3\msa@0F
\mathchardef\twoheadrightarrow="3\msa@10
\mathchardef\twoheadleftarrow="3\msa@11
\mathchardef\leftleftarrows="3\msa@12
\mathchardef\rightrightarrows="3\msa@13
\mathchardef\upuparrows="3\msa@14
\mathchardef\downdownarrows="3\msa@15
\mathchardef\upharpoonright="3\msa@16
\let\restriction\upharpoonright
\mathchardef\downharpoonright="3\msa@17
\mathchardef\upharpoonleft="3\msa@18
\mathchardef\downharpoonleft="3\msa@19
\mathchardef\rightarrowtail="3\msa@1A
\mathchardef\leftarrowtail="3\msa@1B
\mathchardef\leftrightarrows="3\msa@1C
\mathchardef\rightleftarrows="3\msa@1D
\mathchardef\Lsh="3\msa@1E
\mathchardef\Rsh="3\msa@1F
\mathchardef\rightsquigarrow="3\msa@20
\mathchardef\leftrightsquigarrow="3\msa@21
\mathchardef\looparrowleft="3\msa@22
\mathchardef\looparrowright="3\msa@23
\mathchardef\circeq="3\msa@24
\mathchardef\succsim="3\msa@25
\mathchardef\gtrsim="3\msa@26
\mathchardef\gtrapprox="3\msa@27
\mathchardef\multimap="3\msa@28
\mathchardef\therefore="3\msa@29
\mathchardef\because="3\msa@2A
\mathchardef\doteqdot="3\msa@2B
\let\Doteq\doteqdot
\mathchardef\triangleq="3\msa@2C
\mathchardef\precsim="3\msa@2D
\mathchardef\lesssim="3\msa@2E
\mathchardef\lessapprox="3\msa@2F
\mathchardef\eqslantless="3\msa@30
\mathchardef\eqslantgtr="3\msa@31
\mathchardef\curlyeqprec="3\msa@32
\mathchardef\curlyeqsucc="3\msa@33
\mathchardef\preccurlyeq="3\msa@34
\mathchardef\leqq="3\msa@35
\mathchardef\leqslant="3\msa@36 \let\le=\leqslant
\mathchardef\lessgtr="3\msa@37
\mathchardef\backprime="0\msa@38
\mathchardef\risingdotseq="3\msa@3A
\mathchardef\fallingdotseq="3\msa@3B
\mathchardef\succcurlyeq="3\msa@3C
\mathchardef\geqq="3\msa@3D
\mathchardef\geqslant="3\msa@3E  \let\ge=\geqslant
\mathchardef\gtrless="3\msa@3F
\mathchardef\sqsubset="3\msa@40
\mathchardef\sqsupset="3\msa@41
\mathchardef\vartriangleright="3\msa@42
\mathchardef\vartriangleleft ="3\msa@43
\mathchardef\trianglerighteq="3\msa@44
\mathchardef\trianglelefteq="3\msa@45
\mathchardef\bigstar="0\msa@46
\mathchardef\between="3\msa@47
\mathchardef\blacktriangledown="0\msa@48
\mathchardef\blacktriangleright="3\msa@49
\mathchardef\blacktriangleleft="3\msa@4A
\mathchardef\vartriangle="0\msa@4D
\mathchardef\blacktriangle="0\msa@4E
\mathchardef\triangledown="0\msa@4F
\mathchardef\eqcirc="3\msa@50
\mathchardef\lesseqgtr="3\msa@51
\mathchardef\gtreqless="3\msa@52
\mathchardef\lesseqqgtr="3\msa@53
\mathchardef\gtreqqless="3\msa@54
\mathchardef\Rrightarrow="3\msa@56
\mathchardef\Lleftarrow="3\msa@57
\mathchardef\veebar="2\msa@59
\mathchardef\barwedge="2\msa@5A
\mathchardef\doublebarwedge="2\msa@5B
\mathchardef\angle="0\msa@5C
\mathchardef\measuredangle="0\msa@5D
\mathchardef\sphericalangle="0\msa@5E
\mathchardef\varpropto="3\msa@5F
\mathchardef\smallsmile="3\msa@60
\mathchardef\smallfrown="3\msa@61
\mathchardef\Subset="3\msa@62
\mathchardef\Supset="3\msa@63
\mathchardef\Cup="2\msa@64
\let\doublecup\Cup
\mathchardef\Cap="2\msa@65
\let\doublecap\Cap
\mathchardef\curlywedge="2\msa@66
\mathchardef\curlyvee="2\msa@67
\mathchardef\leftthreetimes="2\msa@68
\mathchardef\rightthreetimes="2\msa@69
\mathchardef\subseteqq="3\msa@6A
\mathchardef\supseteqq="3\msa@6B
\mathchardef\bumpeq="3\msa@6C
\mathchardef\Bumpeq="3\msa@6D
\mathchardef\lll="3\msa@6E
\let\llless\lll
\mathchardef\ggg="3\msa@6F
\let\gggtr\ggg
\mathchardef\circledS="0\msa@73
\mathchardef\pitchfork="3\msa@74
\mathchardef\dotplus="2\msa@75
\mathchardef\backsim="3\msa@76
\mathchardef\backsimeq="3\msa@77
\mathchardef\complement="0\msa@7B
\mathchardef\intercal="2\msa@7C
\mathchardef\circledcirc="2\msa@7D
\mathchardef\circledast="2\msa@7E
\mathchardef\circleddash="2\msa@7F
\def\ulcorner{\delimiter"4\msa@70\msa@70 }
\def\urcorner{\delimiter"5\msa@71\msa@71 }
\def\llcorner{\delimiter"4\msa@78\msa@78 }
\def\lrcorner{\delimiter"5\msa@79\msa@79 }
\def\yen{{\mathhexbox@\msa@55 }}
\def\checkmark{{\mathhexbox@\msa@58 }}
\def\circledR{{\mathhexbox@\msa@72 }}
\def\maltese{{\mathhexbox@\msa@7A }}
\mathchardef\lvertneqq="3\msb@00
\mathchardef\gvertneqq="3\msb@01
\mathchardef\nleq="3\msb@02
\mathchardef\ngeq="3\msb@03
\mathchardef\nless="3\msb@04
\mathchardef\ngtr="3\msb@05
\mathchardef\nprec="3\msb@06
\mathchardef\nsucc="3\msb@07
\mathchardef\lneqq="3\msb@08
\mathchardef\gneqq="3\msb@09
\mathchardef\nleqslant="3\msb@0A
\mathchardef\ngeqslant="3\msb@0B
\mathchardef\lneq="3\msb@0C
\mathchardef\gneq="3\msb@0D
\mathchardef\npreceq="3\msb@0E
\mathchardef\nsucceq="3\msb@0F
\mathchardef\precnsim="3\msb@10
\mathchardef\succnsim="3\msb@11
\mathchardef\lnsim="3\msb@12
\mathchardef\gnsim="3\msb@13
\mathchardef\nleqq="3\msb@14
\mathchardef\ngeqq="3\msb@15
\mathchardef\precneqq="3\msb@16
\mathchardef\succneqq="3\msb@17
\mathchardef\precnapprox="3\msb@18
\mathchardef\succnapprox="3\msb@19
\mathchardef\lnapprox="3\msb@1A
\mathchardef\gnapprox="3\msb@1B
\mathchardef\nsim="3\msb@1C
\mathchardef\napprox="3\msb@1D
\mathchardef\ncong="3\msb@1D
\def\napprox{\not\approx}
\mathchardef\varsubsetneq="3\msb@20
\mathchardef\varsupsetneq="3\msb@21
\mathchardef\nsubseteqq="3\msb@22
\mathchardef\nsupseteqq="3\msb@23
\mathchardef\subsetneqq="3\msb@24
\mathchardef\supsetneqq="3\msb@25
\mathchardef\varsubsetneqq="3\msb@26
\mathchardef\varsupsetneqq="3\msb@27
\mathchardef\subsetneq="3\msb@28
\mathchardef\supsetneq="3\msb@29
\mathchardef\nsubseteq="3\msb@2A
\mathchardef\nsupseteq="3\msb@2B
\mathchardef\nparallel="3\msb@2C
\mathchardef\nmid="3\msb@2D
\mathchardef\nshortmid="3\msb@2E
\mathchardef\nshortparallel="3\msb@2F
\mathchardef\nvdash="3\msb@30
\mathchardef\nVdash="3\msb@31
\mathchardef\nvDash="3\msb@32
\mathchardef\nVDash="3\msb@33
\mathchardef\ntrianglerighteq="3\msb@34
\mathchardef\ntrianglelefteq="3\msb@35
\mathchardef\ntriangleleft="3\msb@36
\mathchardef\ntriangleright="3\msb@37
\mathchardef\nleftarrow="3\msb@38
\mathchardef\nrightarrow="3\msb@39
\mathchardef\nLeftarrow="3\msb@3A
\mathchardef\nRightarrow="3\msb@3B
\mathchardef\nLeftrightarrow="3\msb@3C
\mathchardef\nleftrightarrow="3\msb@3D
\mathchardef\divideontimes="2\msb@3E
\mathchardef\varnothing="0\msb@3F
\mathchardef\nexists="0\msb@40
\mathchardef\mho="0\msb@66
\mathchardef\eth="0\msb@67
\mathchardef\eqsim="3\msb@68
\mathchardef\beth="0\msb@69
\mathchardef\gimel="0\msb@6A
\mathchardef\daleth="0\msb@6B
\mathchardef\lessdot="3\msb@6C
\mathchardef\gtrdot="3\msb@6D
\mathchardef\ltimes="2\msb@6E
\mathchardef\rtimes="2\msb@6F
\mathchardef\shortmid="3\msb@70
\mathchardef\shortparallel="3\msb@71
\mathchardef\smallsetminus="2\msb@72
\mathchardef\thicksim="3\msb@73
\mathchardef\thickapprox="3\msb@74
\mathchardef\approxeq="3\msb@75
\mathchardef\succapprox="3\msb@76
\mathchardef\precapprox="3\msb@77
\mathchardef\curvearrowleft="3\msb@78
\mathchardef\curvearrowright="3\msb@79
\mathchardef\digamma="0\msb@7A
\mathchardef\varkappa="0\msb@7B
\mathchardef\hslash="0\msb@7D
\mathchardef\hbar="0\msb@7E
\mathchardef\backepsilon="3\msb@7F
\catcode`\@=12

%% Private stuff

%%\def\ii{{\rm i}}
\def\ii{i}
\def\bu{{\scriptstyle\bullet}}
\let\lra=\longrightarrow
\def\vlra{\mathrel{\smash-}\joinrel\mathrel{\smash-}\joinrel%
\kern-2pt\longrightarrow}
\def\lraww{\mathrel{\rlap{$\longrightarrow$}\kern-1pt\longrightarrow}}
\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{$\scriptscriptstyle\succ$}}}}
\let\wh=\widehat
\let\wt=\widetilde
\let\ol=\overline
\let\ssm=\smallsetminus
\def\soul{{\raise-4.6pt\hbox{$-$}\kern-7pt}}
\def\compact{\subset\!\subset}
\def\ort{\hbox{~\vrule width4.0pt height0.4pt depth0pt
                \vrule width0.4pt height6.0pt depth0pt~$\,$}}

\def\lhra{\lhook\joinrel\longrightarrow}
\def\srelbar{\vrule width0.6ex height0.65ex depth-0.55ex}
\def\merto{\mathrel{\srelbar\kern1.3pt\srelbar\kern1.3pt\srelbar
    \kern1.3pt\srelbar\kern-1ex\raise0.3ex\hbox{${\scriptscriptstyle>}$}}}
\def\larex#1 {\hbox to #1 pt{\rightarrowfill}}
\def\mapstex#1 {\hbox to #1 pt{$\mapstochar$\rightarrowfill}}

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

\def\emph#1{{\it #1}}

\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

\font\fourteenmib = cmmib10 at 14.4pt
\font\twelvemib = cmmib10 at 12pt
\font\tenmib = cmmib10
\font\sevenmib = cmmib7

\font\eightrm = cmr8
\font\eightbf = cmbx8
\font\eighti = cmmi8
\font\eightit = cmti8
\font\eightsy = cmsy8
\font\eighttt = cmtt8
\font\eightex = cmex8
\font\eightmib = cmmib8
\font\eightmsa = msam8
\font\eightmsb = msbm8
\font\eighteuf = eufm8

\font\sixrm = cmr6
\font\sixbf = cmbx6
\font\sixi = cmmi6
\font\sixit = cmti6
\font\sixsy = cmsy6
\font\sixtt = cmtt6
\font\sixex = cmex6
\font\sixmib = cmmib6
\font\sixmsa = msam6
\font\sixmsb = msbm6
\font\sixeuf = eufm6

\font\fiverm = cmr5
\font\fivebf = cmbx5
\font\fivei = cmmi5
\font\fiveit = cmti5
\font\fivesy = cmsy5
\font\fivett = cmtt5
\font\fiveex = cmex5
\font\fivemib = cmmib5
\font\fivemsa = msam5
\font\fivemsb = msbm5

\def\eightpoint{\def\rm{\fam0\eightrm}% redefine \rm
   % set up smaller fonts
   \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
   \textfont1=\eighti  \scriptfont1=\sixi  \scriptscriptfont1=\fivei
   \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
   \textfont3=\eightex \scriptfont3=\eightex\scriptscriptfont3=\eightex
   % redefine all other font-family commands
   \def\it{\fam\itfam\eightit}%
   \textfont\itfam=\eightit
   \def\bf{\fam\bffam\eightbf}%
   \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf
   \scriptscriptfont\bffam=\fivebf
   \def\tt{\fam\ttfam\eighttt}%
   \textfont\ttfam=\eighttt
   % redefine AmS fonts if they exist
   \textfont\msafam=\eightmsa \scriptfont\msafam=\sixmsa
   \textfont\msbfam=\eightmsb \scriptfont\msbfam=\sixmsb
   \textfont\euffam=\eighteuf \scriptfont\euffam=\sixeuf
   \textfont\Calfam=\eightCal
   \scriptfont\Calfam=\sixCal
   % new \baselineskip and \strut
   \normalbaselineskip=9.6pt
   % establish those values
   \normalbaselines\rm}

\def\eightpointbf{%
 \textfont0=\eightbf   \scriptfont0=\sixbf   \scriptscriptfont0=\fivebf
 \textfont1=\eightmib  \scriptfont1=\sixmib  \scriptscriptfont1=\fivemib
 \textfont2=\eightbsy  \scriptfont2=\sixbsy  \scriptscriptfont2=\fivebsy
 \tenbf
 \baselineskip=14.4pt}

\def\tenpoint{%
 \textfont0=\tenrm   \scriptfont0=\sevenrm   \scriptscriptfont0=\fiverm
 \textfont1=\teni  \scriptfont1=\seveni  \scriptscriptfont1=\fivei
 \textfont2=\tensy  \scriptfont2=\sevensy  \scriptscriptfont2=\fivesy
 \tenbf
 \baselineskip=14.4pt}

\newskip\eightpointsurround
\eightpointsurround=6pt plus 3pt minus 2pt
\def\begineightpoint{\vskip\eightpointsurround \bgroup\eightpoint}
\def\endeightpoint{\vskip\eightpointsurround \egroup}

\def\fourteenpointbf{%
 \textfont0=\fourteenbf   \scriptfont0=\twelvebf   \scriptscriptfont0=\sevenbf
 \textfont1=\fourteenmib  \scriptfont1=\twelvemib \scriptscriptfont1=\sevenmib
 \textfont2=\fourteenbsy  \scriptfont2=\twelvebsy  \scriptscriptfont2=\sevenbsy
 \fourteenbf
 \baselineskip=14.4pt}

\def\twelvepointbf{%
 \textfont0=\twelvebf   \scriptfont0=\eightbf   \scriptscriptfont0=\sixbf
 \textfont1=\twelvemib  \scriptfont1=\eightmib \scriptscriptfont1=\sixmib
 \textfont2=\twelvebsy  \scriptfont2=\eightbsy  \scriptscriptfont2=\sixbsy
 \twelvebf
 \baselineskip=14.4pt}

\def\tenpointbf{%
 \textfont0=\tenbf   \scriptfont0=\sevenbf   \scriptscriptfont0=\fivebf
 \textfont1=\tenmib  \scriptfont1=\sevenmib  \scriptscriptfont1=\fivemib
 \textfont2=\tenbsy  \scriptfont2=\sevenbsy  \scriptscriptfont2=\fivebsy
 \tenbf
 \baselineskip=14.4pt}

\def\proof#1{\removelastskip\vskip4pt\noindent{\it #1.}}
\def\dummy{\phantom{$\hat{\hat{\hat{\hat f}}}$}}

\def\endproof{\vskip5pt plus 1pt minus 2pt}

\def\qed{\relax\ifmmode
            \hbox{$\square$}%
         \else
            {\unskip\nobreak\hfil
            \penalty50\hskip1em\null\nobreak\hfil\hbox{$\square$}%
            \parfillskip=0pt\finalhyphendemerits=0\endgraf}%
         \fi}
\def\bigsquare{{\kern-0.3ex\hbox{
\vrule height 1.7ex  width 0.093ex  depth 0ex\kern-0.093ex
\vrule height 1.8ex  width 1.7ex  depth -1.707ex\kern-0.093ex
\vrule height 1.7ex  width 0.093ex  depth 0ex\kern-1.65ex
\vrule height 0.093ex  width 1.6ex  depth 0ex}\kern0.3ex}}

\long\def\claim#1 #2\endclaim
{\removelastskip\vskip7pt plus 1pt minus 1pt\noindent{\bf#1.}
{\it\ignorespaces#2}\vskip\baselineskip}

\newdimen\plainitemindent\plainitemindent=18pt

\def\\{\hfil\break}
\def\Fbreak{\hfill\break}

%% Frequently used symbols
\def\bC{{\Bbb C}}
\def\bG{{\Bbb G}}
\def\bH{{\Bbb H}}
\def\bK{{\Bbb K}}
\def\bN{{\Bbb N}}
\def\bP{{\Bbb P}}
\def\bQ{{\Bbb Q}}
\def\bR{{\Bbb R}}
\def\bZ{{\Bbb Z}}
\def\bOne{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
  {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}

\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}}

\special{header=/home/demailly/psinputs/mathdraw/grlib.ps}
\newdimen\figmm
\figmm=0.694444mm
\long\def\InsertFig#1 #2 #3 #4\EndFig{\par
\hbox{\hskip #1\figmm$\vbox to #2\figmm{\vfil\special{"
0.694444 dup scale
#3}}#4$}}
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\twolines
\title{Applications of Pluripotential}\vskip-1cm
\title{Theory to Algebraic Geometry}
\titlerunning{Applications of Pluripotential theory to Algebraic Geometry}
%\title{Holomorphic Morse inequalities and}\vskip-1cm
%\title{entire curves on projective varieties}
%\titlerunning{Holomorphic Morse inequalities and entire curves}
\noindent
{\twelvebf Jean-Pierre Demailly}
\vskip3pt\noindent
{\bf Universit\'e de Grenoble I, Institut Fourier}
\vskip3mm
\noindent
{\bf July 2011}
\vskip10mm

\noindent
{\bf Abstract.}
These lectures are devoted to the study of various contemporary
problems of algebraic geometry, using fundamental tools from complex
potential theory, namely plurisubharmonic functions, positive currents
and Monge-Amp\`ere operators. Since their inception by Oka and Lelong in 
the mid 1940's, plurisubharmonic functions have been used extensively in
many areas of algebraic and analytic geometry, as they are the function 
theoretic counterpart of pseudoconvexity, the complexified version of
convexity. One such application is the theory of $L^2$ estimates via
the Bochner-Kodaira-H\"ormander technique, which provides
very strong existence theorems for sections of holomorphic vector
bundles with positive curvature. One can mention here
the foundational work achieved by Bochner, Kodaira, Nakano, Morrey, Kohn, 
Andreotti-Vesentini, Grauert, H\"ormander, Bombieri, Skoda and Ohsawa-Takegoshi
in the course of more than 4 decades. Another development is the theory 
of holomorphic Morse inequa\-lities (1985), which relate certain curvature
integrals with the asymptotic cohomology of large tensor powers of
line or vector bundles, and bring a useful complement to
the Riemann-Roch formula.

We describe here the main techniques involved in the proof of
holomorphic Morse inequalities (chapter I) and their link with
Monge-Amp\`ere operators and intersection theory. Chapter~II,
especially, gives a fundamental approximation theorem for closed
$(1,1)$-currents, using a Bergman kernel technique in combination with
the Ohsawa-Takegoshi theorem. As an application, we study the
geometric properties of positives cones of an algebraic variety (nef
and pseudo-effective cone), and derive from there some results about
asymptotic cohomology functionals in chapter~III. 
\medskip

\noindent
{\bf Acknowledgment.}
The author expresses his warm thanks to the GAGC organizers for their 
invitation and the opportunity to deliver a substantial part of these 
lectures to an audience of young researchers. These notes are also
an expansion of a course given at the CIME School in Pluripotential 
Theory held in Cetraro in July 2011, organized by Filippo  Bracci and 
John$\,$Erik Forn{\ae}ss.
\vfill\eject

\section{-1}{Contents}
\vskip10pt
\opentocfile
\def\titlebreak{\break}



\chapter{1}{Holomorphic Morse inequalities}

Holomorphic Morse inequalities provide asymptotic bounds for
the cohomology of tensor powers of holomorphic line bundles. They 
are a very useful complement to the Riemann-Roch formula in many
circumstances.  They were first introduced in [Dem85], and were 
largely motivated by Siu's solution [Siu84, Siu85] of the 
Grauert-Riemen\-schneider 
conjecture, which we reprove here as a special case of a
stronger statement. The basic tool is a spectral theorem which describes
the eigenvalue distribution of complex Laplace-Beltrami operators.
The original proof of [Dem85] was based partly on Siu's techniques and
partly on an extension of Witten's analytic proof of 
standard Morse inequalities [Wit82]. Somewhat later Bismut [Bis87] and
Getzler [Get89] gave new proofs, both relying on an analysis of
the heat kernel in the spirit of the Atiyah-Bott-Patodi proof of the
Atiyah-Singer index theorem [ABP73]. Although the basic idea is simple,
Bismut used deep results arising from probability theory (the Malliavin
calculus), while Getz\-ler relied on his supersymmetric symbolic calculus
for spin pseudodifferential operators [Get83].

We present here a slightly more elementary and self-contained proof which was
suggested to us  by Mohan Ramachadran on the occasion of a visit to 
Chicago in 1989. The~reader is referred to [Dem85, Dem91] for more details.

\section{0}{Introduction}

\subsection{0.A}{Real Morse inequalities}
Let $M$ be a compact $C^\infty$ manifold, $\dim_\bR M=m$, and $h$ a
Morse function, i.e.\ a function such that all critical points are non
degenerate. The standard (real) Morse inequalities relate the Betti numbers
$b_q=\dim H^q_{DR}(M,\bR)$ and the numbers
$$s_q = \# \hbox{~critical points of index~} q~,$$
where the index of a critical point is the number of negative
eigenvalues of the Hessian form $(\partial^2h/\partial x_i\partial
x_j)$. Specifically, the following ``strong Morse inequalities" hold~:
$$b_q-b_{q-1}+\cdots +(-1)^qb_0\le s_q-s_{q-1}+\cdots
+(-1)^qs_0\leqno(0.1)$$
for each integer $q\ge 0$. As a consequence, one recovers the ``weak
Morse inequalities" $b_q\le s_q$ and the expression of the
Euler-Poincar\'e characteristic
$$\chi (M) = b_0-b_1+\cdots + (-1)^mb_m = s_0-s_1+\cdots
+(-1)^ms_m~.\leqno(0.2)$$
These results are purely topological. They are obtained by showing
that $M$ can be reconstructed from the structure of the Morse function 
by attaching cells according to the index of the critical points;
real Morse inequalities are then obtained as a consequence of the 
Mayer-Vietoris exact sequence (see [Mil63]).

\subsection{0.B}{Dolbeault cohomology}
Instead of looking at De Rham cohomology, we want to investigate here 
Dolbeault cohomology, i.e.\ cohomology of the $\dbar$-complex.
Let $X$ be a compact complex manifold, $n=\dim_\bC X$ and $E$
be a holomorphic vector bundle over $X$ with $\rank E=r$.
Let us recall that there is a canonical $\dbar$-operator 
$$\dbar:C^\infty(X,\Lambda^{p,q}T_X^*\otimes E)
\longrightarrow C^\infty(X,\Lambda^{p,q+1}T_X^*\otimes E)
\leqno(0.3)
$$
acting on spaces of $(p,q)$-forms with values in $E$. By the Dolbeault
isomorphism theorem, there is an isomorphism 
$$
H^{p,q}_{\dbar}(X,E):=H^q_{\dbar}(C^\infty(X,\Lambda^{p,\bu}T_X^*\otimes E))\simeq
H^q(X,\Omega^p_X\otimes\cO(E))\leqno(0.4)
$$
from the cohomology of the $\dbar$-complex onto the cohomology 
of the sheaf of holomorphic $p$-forms with values in $E$. In particular,
we have
$$
H^{0,q}_{\dbar}(X,E)\simeq H^q(X,\cO(E)),\leqno(0.5)
$$
and we will denote as usual $h^q(X,E) = \dim H^q(X,\cO(E))$.

\subsection{0.C}{Connections and curvature}
Leut us consider first a $C^\infty$ complex vector bundle $E\to M$ on a 
real differential manifold $M$ (without necessarily any holomorphic structure 
at this point). A \emph{connection} $D$ on $E$ is a linear differential 
operator
$$D:C^\infty(M,\Lambda^q T^*_M\otimes E) \to 
C^\infty(M,\Lambda^{q+1}T^*_M\otimes E)\leqno(0.6)$$
satisfying the Leibniz rule
$$D(f\wedge s) = df\wedge s + (-1)^{{\rm deg}~f}f\wedge Ds\leqno(0.7)$$
for all forms $f\in C^\infty(X,\Lambda^pT^*_M)$, $s\in C^\infty(X,\Lambda^q T^*_M
\otimes E)$. On an open set $U \subset M$ where $E$ is trivial, 
$E_{|U }\simeq U \times \bC^r$, the Leibniz rule shows that a connection 
$D$ can be written in a unique way
$$Ds\simeq ds+\Gamma \wedge s\leqno(0.8)$$
where $\Gamma\in C^\infty(U,\Lambda^1 T^*_M\otimes\Hom(\bC^r,\bC^r))$
is an arbitrary $r\times r$ matrix of $1$-forms and $d$ acts 
componentwise. It is then easy to check that
$$D^2s\simeq (d\Gamma +\Gamma\wedge\Gamma)\wedge s\qquad\hbox{on}\quad U.
\leqno(0.9)$$
Therefore $D^2s=\theta_D\wedge s$ for some global $2$-form 
$\theta_D\in C^\infty(M,\Lambda^2T^*_M\otimes\Hom(E,E))$, given by 
$\theta_D\simeq d\Gamma_U+\Gamma_U\wedge\Gamma_U$ on any trivializing open 
set $U$ with a connection matrix $\Gamma_U$.

\claim{(0.10) Definition} {The $($normalized$)$ curvature tensor of $D$ 
is defined to be $\Theta_D = {\ii\over 2\pi}\theta_D$, in other words 
$${\ii\over 2\pi}D^2s=\Theta_D\wedge s$$ 
for any section $s\in C^\infty(M,\Lambda^qT^*_M\otimes E)$.}
\endclaim

The main reason for the introduction of the factor ${\ii\over 2\pi}$ is
the well known formula for the expression of the Chern classes
in the ring of differential forms of even degree: one has
$$
\det(\Id+\lambda\Theta_D)=1+\lambda \gamma_1(D)+\lambda^2 \gamma_2(D)+
\ldots+\lambda^r\gamma_r(D),
$$
where $\gamma_j(D)$ is a $d$-closed differential form of degree $2j$. Moreover,
$\gamma_j(D)$ has integral periods, i.e.\ the De~Rham cohomology class 
$\{\gamma_j(D)\} \in H^{2j}(M,\bR)$ is the image of an integral class,
namely the $j$-th Chern class $c_j(E)\in H^{2j}(M,\bZ)$.
\medskip

\subsection{0.D}{Hermitian connections}
Assume now that the fibers of $E$ are endowed with a $C^\infty$ 
Hermitian metric $h$, and that the isomorphism $E_{|U}\simeq U \times
\bC^r$ is given by a $C^\infty$ frame $(e_\lambda)$. Then we have a canonical
sesquilinear pairing
$$\matrix{
C^\infty(M,\Lambda^pT^*_M\otimes E) \times C^\infty(M,\Lambda^qT^*_M\otimes E) 
&\longrightarrow  
&C^\infty(M,\Lambda^{p+q}T^*_M)\cr
(u,v) &\longmapsto &\{u,v\}_h\cr}$$
given by
$$\{u,v\}_h = \sum_{\lambda ,\mu }u_\lambda \wedge \overline v_\mu \langle
e_\lambda ,e_\mu \rangle_h\qquad\hbox{for}\quad 
u=\sum u_\lambda \otimes e_\lambda,~~v=\sum v_\mu \otimes e_\mu.$$
The connection $D$ is said to be \emph{Hermitian} (or compatible with
the Hermitian metric $h$) if it satisfies the
additional property
$$d\{u,v\}_h = \{Du,v\}_h + (-1)^{{\rm deg~}u}\{u,Dv\}_h.\leqno(0.11)$$
Assuming that $(e_\lambda)$ is $h$-orthonormal, one easily checks that $D$
is Hermitian if and only if the associated connection matrix $\Gamma$ 
is skew-symmetric, i.e.\ $\Gamma^*=-\Gamma $. In this case 
$\theta_D=d\Gamma+\Gamma\wedge\Gamma$ also satisfies
$\theta_D^*=-\theta_D$, thus
$$\Theta_D={\ii\over 2\pi}\theta_D\in C^\infty(M,\Lambda^2T^*_M\otimes
\Herm(E,E)).\leqno(0.12)$$
\medskip

\noindent{\bf (0.13) Special case.} For a bundle $E$ of rank $r=1$, the
connection matrix $\Gamma $ of a Hermitian connection $D$ can be more 
conveniently written
$\Gamma =-iA$ where $A$ is a \emph{real} $1$-form. Then we have 
$$\Theta_D = {\ii\over 2\pi}d\Gamma  = {1\over 2\pi}dA.$$
Frequently, especially in physics,
the real $2$-form $B=dA=2\pi\Theta_D\in C^\infty(M,\Lambda^2T^*_M)$
is referred to as the \emph{magnetic field},
and the $1$-form $A$ as its potential.
A phase change $\tilde s(x)=s(x)e^{i\alpha(x)}$ in the isomorphism $E_{|U }\simeq
U \times \bC$ replaces $A$ with the new connection form $\tilde A=A+d\alpha $.

\subsection{0.E}{Connections on a Hermitian holomorphic vector bundle}
If $M=X$ is a complex manifold, every connection $D$ can be split in 
a unique way as the sum $D=D'+D''$ of a $(1,0)$-connection $D'$ and a 
$(0,1)$-connection $D''$~:
$$\eqalign{
D'&:C^\infty(M,\Lambda^{p,q}T_X^*\otimes E)\longrightarrow
C^\infty(M,\Lambda^{p+1,q}T_X^*\otimes E),\cr 
D''&:C^\infty(M,\Lambda^{p,q}T_X^*\otimes E)\longrightarrow
C^\infty(M,\Lambda^{p,q+1}T_X^*\otimes E).\cr}$$
In a local trivialization given by a $C^\infty$ frame, one can write 
$$\eqalign{
D'u &= d'u + \Gamma'\wedge u~,\cr
D''u &=d''u + \Gamma''\wedge u~,\cr}$$
with $\Gamma =\Gamma'+\Gamma''$ and $d'=\partial$, $d''=\dbar$. 
If $(E,h)$ is a $C^\infty$ Hermitian 
structure, the connection is Hermitian if and
only if $\Gamma'=-(\Gamma'')^*$ in any $h$-orthonormal frame. Thus there
exists a unique Hermitian connection corresponding to a prescribed
$(0,1)$ part $D''$. 

Assume now that the Hermitian bundle $(E,h)$ has a \emph{holomorphic} 
structure. The unique Hermitian connection $D$ for which $D''=\overline
\partial$ is called the \emph{Chern connection} of $(E,h)$. In a local
holomorphic frame $(e_\lambda)$ of $E_{|U}$~, the metric $h$ is given
by some Hermitian matrix $H=(h_{\lambda \mu })$ where $h_{\lambda \mu }
= \langle e_\lambda ,e_\mu \rangle_h$. Standard computations yield the
expression of the Chern connection~:
$$\left\{
\eqalign{
D's&=\partial s+\overline H^{-1}\partial \overline H \wedge s,\cr
D''s&=\overline \partial s,\cr
\theta_D\wedge s&=\smash{D^2s = (D'D''+D''D')s=-\dbar(\overline H^{-1}
\partial \overline H)\wedge s}.\cr}\right.$$

\claim{(0.14) Definition} {The Chern curvature tensor of $(E,h)$ is the
curvature tensor of its Chern connection, denoted
$$
\theta_{E,h}=D'D''+D''D'=-\dbar(\overline H^{-1}\partial \overline H).
$$}
\endclaim
\vskip-12pt

\noindent
In the special case of a rank 1 bundle $E$, the matrix $H$ is simply a 
positive function, and it is convenient to introduce its weight $\varphi$
such that $H=(e^{-\varphi })$ where $\varphi \in C^\infty (U,\bR)$ 
depends on the given trivialization $E_{|U}\simeq U\times\bC$. We have
in this case
$$\Theta_{E,h} = {\ii\over 2\pi}\theta_{E,h}=
{\ii\over 2\pi}\partial \overline \partial \varphi
\qquad\hbox{on}\quad U,\leqno(0.15)$$
and therefore $\Theta_{E,h}$ is a closed real $(1,1)$-form.\medskip

\subsection{0.F}{Fundamental facts of Hodge theory}
Assume here that $M$ is a Riemannian
manifold with metric $g=\sum g_{ij} dx_i\otimes dx_j$. Given $q$-forms
$u$, $v$ on $M$ with values in $E$~, we consider the global $L^2$ norm and
inner product
$$\|u\|^2 = \int_M |u(x)|^2d\sigma (x),\qquad
\Langle u,v\Rangle = \int_M \langle u(x),v(x)\rangle\, d\sigma (x),
\leqno(0.16)$$
where $|u|$ is the pointwise Hermitian norm and  $d\sigma $ the
Riemannian volume form. The Laplace Beltrami operator associated with the
connection $D$ is
$$\Delta =DD^*+D^*D,$$
acting on any of the spaces $C^\infty(M,\Lambda^qT^*_M\otimes E)$;
here 
$$D^*:C^\infty(M,\Lambda^qT^*_M\otimes E)\longrightarrow
C^\infty(M,\Lambda^{q-1}T^*_M\otimes E)\leqno(0.17)$$
is the (formal) $L^2$ adjoint of $D$. The complex Laplace operators
$\Delta'=D'D^{\prime *}+D^{\prime *}D'$ and $\Delta''=
D''D^{\prime\prime *}+D^{\prime\prime *}D''$ are defined similarly
when $M=X$ is a complex manifold. In degree $0$ we simply have $\Delta =D^*D$.
A well-known calculation shows that the principal symbol of $\Delta$ is 
$\sigma_\Delta(x,\xi)=-|\xi|^2\Id$ (while
$\sigma_{\Delta'}(x,\xi)=\sigma_{\Delta''}(x,\xi)=-{1\over 2}|\xi|^2\Id$).
As a consequence $\Delta$, $\Delta'$, $\Delta''$ are always
\emph{elliptic operators}. 

When $M$ is \emph{compact}, the operator $\Delta$ acting on any of
the spaces $C^\infty(M,\Lambda^qT^*_M\otimes E)$ has a discrete
spectrum
$$\lambda_1\le \lambda_2\le \cdots \le \lambda_j\le \cdots$$
and corresponding eigenfunctions $\psi_j\in C^\infty(M,\Lambda^qT^*_M
\otimes E)$, depending of course on~$q$.

Our main goal is to obtain asymptotic formulas for the eigenvalues. For
this, we will make an essential use of the \emph{heat operator} 
$e^{-t\Delta }$. In the above
setting, the heat operator is the bounded Hermitian operator associated
to the \emph{heat kernel}
$$K_t(x,y) =\sum^{+\infty}_{\nu=1} e^{-\lambda_\nu t} \psi_\nu(x) \otimes
\psi^*_\nu(y),\leqno(0.18)$$
i.e.
$$
\Langle u,e^{-t\Delta}v\Rangle = \int_{M\times M} \langle u(x),K_t(x,y)\cdot 
v(y)\rangle\,d\sigma(x)\,d\sigma(y).
$$
Standard results of the theory of elliptic operators show that 
$$K_t \in C^\infty(\,{}]0,+\infty[{}\times M\times M,\Hom(E,E))$$
and that $K_t(x,y)$ is the solution of the differential equation
$$
{\partial\over\partial t}K_t(x,y)=-\Delta_xK_t(x,y),\qquad 
\lim_{t\to 0_+}K_t(x,y)=\delta_y(x)
\quad\hbox{(Dirac at $y$)},\leqno(0.19)
$$
as follows formally from the fact that 
${\partial\over\partial t}e^{-t\Delta}=-\Delta e^{-t\Delta}$ and
$e^{-0\Delta}=\Id$.
The asymptotic distribution of eigenvalues can be recovered from the
straightforward formula
$$\sum^{+\infty}_{\nu=1} e^{-\lambda_\nu t} = \int_M {\rm tr}_E K_t(x,x)
d\sigma (x)~.\leqno(0.20)$$
In the sequel, we are especially interested in the $0$-eigenspace:

\claim{(0.21) Definition} {The space of $\Delta$-harmonic forms is defined
to be
$$
\cH^q_\Delta(M,E)=\Ker\Delta=\big\{u\in C^\infty(M,\Lambda^q T^*_M\otimes E)\,;\;
\Delta u =0\big\}.
$$}
\endclaim

\vskip-20pt
\noindent When $M$ is compact, an integration by part shows that
$$
\Langle \Delta u,u\Rangle=\|Du\|^2+\|D^*u\|^2,
$$
hence $u$ is $\Delta$-harmonic if and only if $Du=D^*u=0$. Moreover,
as $\Delta$ is a self-ajoint operator, standard elliptic theory implies
that
$$
C^\infty(M,\Lambda^q T^*_M\otimes E)=\Ker\Delta\oplus\Im\Delta=
\cH^q_\Delta(M,E)\oplus\Im\Delta,\leqno(0.22)
$$
and $\Ker\Delta=\cH^q_\Delta(M,E)$, $\Im\Delta$ are orthogonal with respect 
to the $L^2$ inner product. Clearly $\Im\Delta\subset \Im D+\Im D^*$, and 
both images $\Im D$, $\Im D^*$ are orthogonal to the space of 
harmonic forms by what we have just seen. As a consequence, we 
have 
$$\Im\Delta=\Im D+\Im D^*.\leqno(0.23)$$
\medskip

\noindent{\bf(0.24) Hodge isomorphism theorem.} Assume that $M$ is compact
and that $D$ is an \emph{integrable connection}, i.e.\ $D^2=0$ 
(or $\theta_D=0$). Then $D$ defines on spaces of sections
$C^\infty(M,\Lambda^q T^*_M\otimes E)$ a differential complex which can be
seen as a generalization of the De Rham complex. The condition 
$D^2=0$ immediately implies that $\Im D\perp \Im D^*$ and we conclude
from the above discussion that there is an orthogonal direct sum
$$
C^\infty(M,\Lambda^q T^*_M\otimes E)=
\cH^q_\Delta(M,E)\oplus\Im D\oplus \Im D^*.\leqno(0.25)
$$
If we put $u=h+Dv+D^* w$ according to this decomposition, then $Du=DD^* w=0$
if and only if $\|D^*w\|=\Langle DD^* w,w\Rangle=0$, thus 
$$
\Ker D=\cH^q_\Delta(M,E)\oplus\Im D.
$$
This implies the \emph{Hodge isomorphism theorem}
$$
H^q_\DR(M,E):=\Ker D/\Im D\simeq \cH^q_\Delta(M,E).\leqno(0.26)
$$
In case $M=X$ is a compact complex manifold, $(E,h)$ a Hermitian holomorphic 
vector bundle and $D=D'+D''$ the Chern connection, the integrability condition
\hbox{$D^{\prime\prime 2}=\smash{\dbar}^2=0$} is always satisfied. Thus 
we get an analogous isomorphism
$$
H^q(X,\cO(E))\simeq H^{0,q}_\dbar(X,E)\simeq \cH^{0,q}_{\Delta''}(M,E),
\leqno(0.27)_{0,q}
$$
and more generally
$$
H^q(X,\Omega^p_X\otimes\cO(E))\simeq H^{p,q}_\dbar(X,E)\simeq 
\cH^{p,q}_{\Delta''}(M,E),
\leqno(0.27)_{p,q}
$$
where $\cH^{p,q}_{\Delta''}(M,E)$ is the space of $\Delta''$-harmonic forms
of type $(p,q)$ with values in~$E$.

\claim{(0.28) Corollary {\rm (Hodge decomposition theorem)}} If $(X,\omega)$
is a compact K\"ahler mani\-fold and $(E,h)$ is a {\rm flat} Hermitian vector
bundle over $X$ $($i.e.\ $D_{E,h}^2=0)$, then there is an isomorphism
$$
H^k_\DR(M,E)\simeq\bigoplus_{p+q=k}H^{p,q}_\dbar(X,E).
$$
\vskip-8pt
\endclaim

\noindent 
In fact, under the condition that $\omega$ is K\"ahler, i.e.\ $d\omega=0$, 
well-known identities of K\"ahler geometry imply
$\Delta'=\Delta''={1\over 2}\Delta$, and as a consequence
$$
\cH^k_{\Delta}(M,E)=\bigoplus_{p+q=k}\cH^{p,q}_{\Delta''}(X,E).
$$

\section{1}{Holomorphic Morse inequalities}

\subsection{1.A}{Main statements}

Let $X$ be a compact complex $n$-dimensional manifold, $L\to X$ a holomorphic 
line bundle and $E\to X$ a holomorphic vector bundle of rank
$r=\rank E$. We assume that $L$ is equipped with a smooth Hermitian metric
$h$ and denote accordingly $\Theta_{L,h}$ its curvature form; by
definition this is a closed real $(1,1)$-form and its cohomology class
$c_1(L)_\bR=\{\Theta_{L,h}\}\in H^2_\DR(X,\bR)$ is the first Chern class 
of $L$.

\claim{(1.1) q-index sets} {We define the $q$-index sets and
$\{\le q\}$-index sets of $(L,h)$ to be
$$\eqalign{
X(L,h,q)&=\left\{x\in X\,;\;\Theta_{L,h}(x) \hbox{~has~}
{\displaystyle q\atop\displaystyle n-q}~
{\hbox{negative eigenvalues}\atop\hbox{positive eigenvalues}}\right\}\cr
X(L,h,\le q) &= \bigcup_{1\le j\le q} X(L,h,j)~.\cr}$$}
\endclaim

\noindent
Clearly $X(L,h,q)$ and $X(L,h,\le q)$ are open subsets of $X$, and 
we have a partition into ``chambers'' $X=S~\cup~\bigcup_{0\le q\le n}X(L,h,q)$
where $S=\{x\in X\,;\;\Theta_{L,h}(x)=0\}$ is the degeneration set.
The following theorem was first proved in [Dem85].

\claim{(1.2) Main Theorem} {The cohomology groups of tensor powers 
$E\otimes L^k$ satisfy the following asymptotic estimates as $k\to +\infty\,:$

\noindent $(1.2)_{\rm WM}$ Weak Morse inequalities$\,:$
$$h^q(X,E\otimes L^k)\le r {k^n\over n!}\int_{X(L,h,q)} (-1)^q \Theta_{L,h}^n + o(k^n)~.$$

\noindent $(1.2)_{\rm SM}$ Strong Morse inequalities$\,:$
$$\sum_{0\le j\le q} (-1)^{q-j}h^j(X,E\otimes L^k) \le r {k^n\over n!}
\int_{X(L,h,\le q)}(-1)^q\Theta_{L,h}^n+o(k^n)~.$$

\noindent $(1.2)_\RR$ Asymptotic Riemann-Roch formula$\,:$
$$\chi(X,E\otimes L^k) := \sum_{0\le j\le n} (-1)^jh^j(X,E\otimes L^k)
= r{k^n\over n!}\int_X \Theta_{L,h}^n + o(k^n)~.$$}
\vskip-12pt
\endclaim

The weak Morse form $(1.2)_{\rm WM}$ follows from strong Morse
$(1.2)_{\rm SM}$ by adding conse\-cutive inequalities for the indices
$q-1$ and $q$, since the signs $(-1)^{q-j}$ and $(-1)^{q-1-j}$ are
opposite. Also, $(1.2)_\RR$ is just a weaker formulation of
the existence of the Hilbert polynomial, and as such, is a consequence of 
the Hirzebruch-Riemann-Roch formula; it~follows formally 
from $(1.2)_{\rm SM}$  with $q=n$ and $q=n+1$, since $h^{n+1}=0$ 
identically and the signs are reversed. Now, by adding $(1.2)_{\rm SM}$ for the 
indices of opposite parity $q+1$ and $q-2$, we find
$$
h^{q+1}(X,E\otimes L^k)-h^q(...)+h^{q-1}(...)\le r {k^n\over n!}
\int_{X(L,h,\{q-1,q,q+1\})}(-1)^{q+1}\Theta_{L,h}^n+o(k^n),$$
where $X(L,h,\{q-1,q,q+1\})$ is meant for the union of chambers of indices
$q-1$, $q$, $q+1$. As a consequence, we get lower bounds for the
cohomology groups:
$$
h^q(X,E\otimes L^k)\ge h^q-h^{q+1}-h^{q-1} \ge r {k^n\over n!}
\int_{X(L,h,\{q-1,q,q+1\})}(-1)^q\Theta_{L,h}^n-o(k^n).\leqno(1.3)$$
Another important special case is $(1.2)_{\rm SM}$ for $q=1$, which yields the
lower bound
$$
h^0(X,E\otimes L^k)\ge h^0-h^1\ge r {k^n\over n!}
\int_{X(L,h,\le 1)}\Theta_{L,h}^n-o(k^n).\leqno(1.4)$$
As we will see later in the applications, this lower bound provides a very 
useful criterion to prove the existence of sections of large tensor
powers of a line bundle.\qed
\endproof

\subsection{1.B}{Heat kernel and eigenvalue distribution}

We introduce here a basic heat equation technique, from which all asymptotic
eigenvalue estimates can be derived via an explicit formula, known as
Mehler's formula.

We start with a compact Riemannian manifold $(M,g)$ with $\dim_\bR M=m$, 
and denote by $d\sigma$ its Riemannian volume form.
Let $(L,h_L)$ (resp.\ $(E,h_E)$) be a Hermitian complex line 
(resp.\ vector bundle) on $M$, equipped with a Hermitian connection
$D_L$ (resp.\ $D_E$). 

We denote by $D_k=D_{E\otimes L^k}$ the associated connection on $E\otimes L^k$,
and by $\Delta_k = D^*_kD_k$ the Laplace-Beltrami operator acting on sections
of $E\otimes L^k$ (i.e.\ forms of degree $0$). As in (0.13), we 
introduce the (local) connection form $\Gamma_L=-iA$ of $L$ and the 
corresponding (global) curvature $2$-form $B=dA\in C^\infty(M,\Lambda^2T_M^*)$,
i.e.\ the ``magnetic field'' ($\Gamma_E$ and the corresponding curvature tensor 
$\Theta_E$ of $D_E$ will not play a significant role here). Finally, 
we assume that 
an additional section $V\in C^\infty(M,\Herm(E,E))$ is given
(``electric field'')~; for simplicity of notation, we still denote 
by $V$ the operator $V\otimes \Id_{L^k}$ acting on~$E\otimes L^k$.

If $\Omega \subset M$ is a smoothly bounded open subset of~$M$, we 
consider for $u$ in the Sobolev space 
$W^1_0(\Omega,E\otimes L^k)$ the quadratic form
$$Q_{k,\Omega }(u) = \int_\Omega  {1\over k} |D_ku|^2 - 
\langle Vu,u\rangle.\leqno(1.5)$$
Here $W^1_0(\Omega ,E\otimes L^k)$ is the closure of the space of 
smooth sections with compact support in~$\Omega$, taken
in the Hilbert space $W^1_{\rm loc}(M,E\otimes L^k)$ of sections that have
$L^2_{{\rm loc}}$ coefficients as well as their first derivatives. In other
words, we consider the densily defined self adjoint operator
$$\square_k = {1\over k} D^*_k D_k - V\leqno(1.6)$$
acting in the Hilbert space $W^1_0(\Omega,E\otimes L^k)$,
i.e.\ with Dirichlet boundary conditions. Again, $\square_k$ acting
on  $W^1_0(\Omega,E\otimes L^k)$ has a discrete spectrum whenever $\Omega$
is relatively compact (and also sometimes when $\Omega$ is unbounded,
according to the behavior of $B$ and $V$ at infinity; except otherwise stated,
we will assume that we are in this case later on). Then, there is an 
associated ``localized'' heat kernel
$$
K_{t,k,\Omega}(x,y) =\sum^{+\infty}_{\nu=1} e^{-\lambda_{\nu,k,\Omega}t} \psi_{\nu,k,\Omega}(x) \otimes\psi^*_{\nu,k,\Omega}(y)\leqno(1.7)
$$
where $\psi_{\nu,k,\Omega}\in W^1_0(\Omega,E\otimes L^k)$ are the eigenfunctions
and $\lambda_{\nu,k,\Omega}$ their eigenvalues.

We want to study the asymptotic eigenvalue distribution of $\square_k$
as $k\to +\infty$, and more precisely get an asymptotic formula for 
the corresponding heat kernel $e^{-t\square_k}$. The basic idea is to
decompose the proof in three steps~:

\item{$(\alpha)$} convince ourselves that the asymptotic estimates can be 
``localized'', up to lower order error terms.

\item{$(\beta)$} show that the local estimates can be obtained by freezing the
coefficients of the operators involved at any given point.

\item{$(\gamma)$} compute explicitly the heat kernel in the case of
connections with constant curvature, assuming moreover that $\Omega\simeq
\bR^m$ with the flat euclidean metric.
\smallskip

\noindent$(\alpha)$ 
In order to see that the situation can be localized, we fix a partition 
of unity $(\tau_j)$ relative to an arbitrarily fine finite covering 
$(\Omega_j)$ of $\overline \Omega $, such that $\sum \tau^2_j = 1$ near 
$\overline\Omega $. We~consider the continuous injection
$$
I_{\Omega,\Omega_j}:W^1_0(\Omega ,E\otimes L^k)\to\bigoplus_j
W^1_0(\Omega\cap\Omega_j,E\otimes L^k),\qquad u\mapsto(\tau_ju)_j,
$$
the inverse of which is $(u_j)\mapsto u=\sum\tau_j u_j$. As
$\sum \tau_jd\tau_j = 0$ on $\Omega$, we find
$$
\sum_j Q_{k,\Omega_j}(\tau_ju) - Q_{k,\Omega }(u) = {1\over k}
\int_\Omega \Big(\sum |d\tau_j|^2\Big)|u|^2
\le O\Big({1\over k}\Big) |u|^2.\leqno(1.8)$$
By the minimax principle, it follows that the eigenvalues of
$\smash{\bigoplus Q_{k,\Omega_j|\Im I_{\Omega,\Omega_j}}}$ and those of 
$Q_{k,\Omega}$ differ by at most $O(1/k)$ as $k\to +\infty$. This explains
why a localization process is possible, at least as far as the eigenvalue
distribution is concerned. For the related heat kernels on small geodesic
balls, one can use the following localization principle.

\claim{(1.9) Proposition} Let $\Omega_\rho=B(x^0,\rho)$ be a geodesic ball of $(M,g)$ of radius $\rho$ where $\rho<{}$injectivity radius. Then there exist constants $C_1$ and $\varepsilon_1>0$ such that for all
$t\in{}]0,\min(k\varepsilon_1,k\rho^2/2m)]$ and every $x_0\in M$ we have
$$
\big|K_{t,k,M}(x^0,x^0)-K_{t,k,\Omega_\rho}(x^0,x^0)\big|\le
C_1\Big({k\over t}\Big)^{m/2}\exp\Big(-{k\rho^2\over 4t}+2t\sup_{\Omega_\rho}\Vert V\Vert\Big).
$$
\endclaim

\vskip-12pt
\noindent A proof of this technical result is given in Thierry Bouche's 
PhD thesis (cf.\ [Bou90]). It relies on a use of Kato's inequality
(cf.\ [HeSU80]), which amounts to say that we get an upper bound for
$K_{t,k,M}$ in the case when the curvature is trivial; one can then use 
the calculations given below to get the explicit bound, see e.g.\ $(1.10')$.

\medskip
\noindent$(\beta)$
Now, let $x^0\in M$ be a given point. We choose coordinates 
$(x_1,\ldots,x_m)$ centered at $x^0$ such that $(\partial/\partial x_1,\ldots,
\partial/\partial x_m)$ is orthonormal at $x^0$ with respect to the 
Riemannian metric~$g$. By changing the orthonormal frame of $L$ as 
in (0.13), we can adjust
the connection  form $\Gamma_L=-iA$ of $L$ to be given by 
any local potential $A(x)=\sum_jA_j(x)\,dx_j$ such that $B=dA$, 
and we can therefore arrange that $A(x^0)=0$. Similarly, we can fix
a unitary frame of $E$ such that $\Gamma_E(x^0)=0$. Set $x^0=0$ 
for simplicity. The first term of our Laplace operator 
$\square_k={1\over k}D_k^*D_k-V$ is the square of the first order 
operator
$$
\eqalign{
k^{-1/2}D_ku(x)&=k^{-1/2}\big(du(x)+k\Id_E\otimes \Gamma_L(x)\cdot u(x)+
\Id_{L^k}\otimes\Gamma_E(x)\cdot u(x)\big)\cr
&=k^{-1/2}\sum_j\Big({\partial u\over \partial x_j}-ik^{1/2}A_j(x)u(x)\Big)
dx_j+k^{-1/2}\Id_{L^k}\otimes\Gamma_E(x)\cdot u(x).\cr}
$$
If we use a rescaling $x=k^{-1/2}\wt x$ and set $\wt u(\wt x) = u(x) =
u(k^{-1/2}\wt x)$,
this operator takes the form
$$
\wt D_k\wt u(\wt x)= 
\sum_j\Big({\partial\wt u\over \partial \wt x_j}-ik^{1/2}A_j(k^{-1/2}\wt x)\,
\wt u(\wt x)\Big)dx_j+O(k^{-1/2}|\wt x|)\,\wt u(\wt x)\,dx.
$$
As $A_j(0)=0$, the term $k^{1/2}A_j(k^{-1/2}\wt x)$ converges modulo
$O(k^{-1/2}|\wt x|^2)$ terms to the linea\-rized part $\wt A_j(\wt x)
=\sum_{i,j}{\partial A_j
\over\partial x_i}(0)\,\wt x_i$. Observe also that the connection form
$\Gamma_E$ of $E$ only contributes for terms of the form $O(k^{-1/2}|\wt x|)$ 
(and thus will be negligible in the end, together with the quadratic terms 
of~$A_j$). Our initial operator 
$\square_k={1\over k}D_k^*D_k-V$ becomes
$$\wt\square_k=\wt D_k^*\wt D_k-\wt V$$
where $\wt V(\wt x)=V(k^{-1/2}\wt x)$ and where the ajoint is computed
with respect to the rescaled metric $\wt g(x)=
\sum g_{ij}(k^{-1/2}\wt x)\, d\wt x_jd\wt x_j\,$; here
$\wt g\to \sum (d\wt x_j)^2$ as $k\to +\infty$ thanks to the
assumption that $g_{ij}(0)=\delta_{ij}$. Modulo lower order 
terms $O(k^{-1/2}|\wt x|^2)$, $\wt D_k$ is given by a linear connection form
$$
\wt A(\wt x)=\sum B_{ij}\,\wt x_i\,d\wt x_j
$$
assciated with the constant magnetic field $B(x^0)=\sum_{i,j}B_{ij}\,dx_i\wedge 
dx_j$ frozen at~$x^0=0$. We can moreover choose orthonormal coordinates
so that $B(x^0)$ takes the standard form
$$B(x^0) = \sum^s_{j=1} B_j\,dx_j\wedge dx_{j+s}$$
where $2s\le m$ is the rank of the alternate $2$-form $B(x^0)$ and
$B_j$ the curvature eigenvalues with respect to $g(x^0)$. The corresponding
linearized potential is
$$
\wt A(\wt x)=\sum_{j=1}^s B_j\,\wt x_j\,d\wt x_{j+s}.
$$
The intuition from Physics is that the eigenfunctions represent ``waves'' of
heat propagation of a certain typical wave length $\lambda$ in the
coordinates $\wt x$, and of a corresponding (much shorter) 
wave length $\lambda\,k^{-1/2}$ in the original coordinates. At that scale,
our space behaves as if the metrics were flat and the curvature constant.

\medskip
\noindent$(\gamma)$
Let us consider the operators obtained by ``freezing" the coefficients
at any point~$x^0$, as explained at step $(\beta)$, although we will not
perform the rescaling here. More specifically, we assume that

\item{$\bullet$} $L$ has constant curvature $B=\sum^s_{j=1}B_jdx_j\wedge
dx_{j+s}$. Then there is a local trivialization in which
$$D_Lu = du - iA\wedge u,\qquad A = \sum^s_{j=1} B_jx_jdx_{j+s}.$$

\item{$\bullet$} $\Omega \simeq \bR^m$ and the metric $g$ is flat~:
$g=\sum dx_j\otimes dx_j$.

\item{$\bullet$} $E\simeq\Omega\times\bC^r$ is a
trivial (flat) Hermitian bundle.

\item{$\bullet$} the Hermitian form $V$ is constant. We choose an
orthonormal frame of $E$ in which $V$ is diagonal, i.e.\
$$\langle Vu,u\rangle = \sum_{1\le \lambda \le r} V_\lambda|u_\lambda|^2.$$
\par\noindent
In this ideal situation, the connection $D_k$ on $E\otimes L^k$ can 
be written
$D_ku=du-ikA\wedge u$ and the quadratic form $Q_{k,\Omega }$ is given by
\ifnum 0=1
$$\eqalign{
Q_{k,\Omega }(u) = \int_{\bR^m} {1\over k} \left(
\sum_{{1\le j\le s}\atop {1\le \lambda \le r}}\Bigg(\Big|{\partial
u_\lambda \over \partial x_j}\Big|^2+\Big|{\partial u_\lambda \over
\partial x_{j+s}}-ikB_jx_ju_\lambda \Big|^2\Bigg)\right.&+
\left.\sum_{{j>2s}\atop
{1\le \lambda \le r}} \Big|{du_\lambda \over dx_j}\Big|^2\right)\cr
&-\sum_{1\le \lambda \le r} V_\lambda |u_\lambda |^2.\cr}$$
\else
$$
Q_{k,\Omega }(u) = \int_{\bR^m} {1\over k}\!\left(\!
\sum_{{1\le j\le s}\atop {1\le \lambda \le r}}\!\!\!\Bigg(\Big|{\partial
u_\lambda \over \partial x_j}\Big|^2\!{+}\Big|{\partial u_\lambda \over
\partial x_{j+s}}-ikB_jx_ju_\lambda \Big|^2\!\Bigg){+}\!\!\!
\sum_{{j>2s}\atop
{1\le \lambda \le r}} \Big|{du_\lambda \over dx_j}\Big|^2\right)\!{-}\!\!
\sum_{1\le \lambda \le r} V_\lambda |u_\lambda |^2.$$
\fi
In this situation, $Q_{k,\Omega }$ is a direct sum of quadratic forms
acting on each component $u_\lambda $ and the computation of
$e^{-t\square_k}$ is reduced to the following model cases $(1.10)$, $(1.11)$
in dimension $1$ or $2\,$:
$$Q(f) = \int_{\bR} \Big|{df\over dx}\Big|^2~,~~~\square f= - {d^2f\over
dx^2}\leqno(1.10)$$
As is well known (and although the spectrum is not discrete in that case)
the kernel of the ``elementary'' heat operator $e^{-t\square}$ is given by
$$K_{t,\bR}(x,y) = {1\over \sqrt {4\pi t}} e^{-(x-y)^2/4t},\leqno(1.10')$$
as follows from solving equation (0.19). The second model case is$\,$:
$$Q(f) = \int_{\bR^2} \Big|{df\over dx_1}\Big|^2 + \Big|{df\over
dx_2}-iax_1f\Big|^2.\leqno (1.11)$$
A partial Fourier transform $\wh f(x_1,\xi_2)={1\over\sqrt{2\pi}}
\int_\bR f(x_1,x_2)\,e^{-ix_2\xi_2}\,dx_2$ gives 
$$Q(f) = \int_{\bR^2} \Big|{d\wh f\over dx_1}(x_1,\xi
_2)\Big|^2+a^2\Big(x_1-{\xi_2\over a}\Big)^2|\wh f(x_1,\xi_2)|^2$$
and the change of variables $x'_1=x_1-\xi_2/a$, $x'_2 = \xi_2$ leads (after
dropping the second variable $x'_2$) to
the so called ``harmonic oscillator'' energy functional
$$
q(g) = \int_{\bR}\Big|{dg\over dx}\Big|^2+a^2x^2|g|^2~,\qquad
\square = - {d^2\over dx^2} + a^2x^2.\leqno(1.12)$$
The heat kernel of this operator is given by \emph{Mehler's formula}$\,$:
$$k_{t,\bR}(x,y) = \sqrt{a\over 2\pi \sinh 2at}\, \exp \Big(-{a\over 2}(\coth
2at)(x-y)^2 - a(\tanh at)xy\Big),\leqno(1.12')$$
which actually reduces to $(1.10')$ when $a\to 0$.
One way of obtaining this relation is to observe that the unitary 
eigenfunctions of $\square$ are
$$\left(2^pp!\sqrt{{\pi \over a}}\,\right)^{-1/2} \Phi_p(\sqrt a
x),\qquad p=0,1,2,\ldots~,$$
with associated eigenvalues $(2p+1)a$, where $(\Phi_p)$ is the sequence
of functions associated with Hermite polynomials:
$$\Phi_p(x) = e^{x^2/2}{d^p\over dx^p} (e^{-x^2}).$$
In fact, for $a=1$, easy calculations bearing on derivatives of $e^{x^2/2}$ 
show that
$$
\Big(- {d^2\over dx^2} + x^2\Big)\Phi_p(x)
=-e^{x^2/2}{d^{p+2}\over dx^{p+2}}(e^{-x^2})
-2x\,e^{x^2/2}{d^{p+1}\over dx^{p+1}}(e^{-x^2})
-e^{x^2/2}{d^p\over dx^p}(e^{-x^2}).
$$
We can now replace the first term by
$e^{x^2/2}{d^{p+1}\over dx^{p+1}}(2x\cdot e^{-x^2})$ and use the Leibniz formula
for the differentiation of the product to see that 
$\square\Phi_p(x)=(2p+1)\Phi_p(x)$. Therefore
$$k_{t,\bR}(x,y) = \sqrt{a\over \pi } e^{a(x^2+y^2)/2} \sum^{+\infty}_{p=0}
{e^{-(2p+1)at}\over 2^pp!a^p} {d^p\over dx^p} (e^{-ax^2}) {d^p\over dy^p}
(e^{-ay^2}).$$
The above summation $\Sigma(x,y)=\sum_{p=0}^{+\infty}...$ can be computed via
its Fourier transform
$$\eqalign{
\widehat \Sigma (\xi ,\eta) &= 
{1\over 2a}\,e^{-at}\sum^{+\infty}_{p=0}{1\over p!}\Big({e^{-2at}\over 2a}\Big)^p
(i\xi)^p(i\eta)^pe^{-\xi^2/4a}e^{-\eta^2/4a}\cr
&={1\over 2a}\,e^{-at} \exp\Big(-{1\over 4a} (\xi^2+\eta^2+2\,e^{-2at}\xi\eta)
\Big),\cr}$$
thus
$$
\Sigma(x,y)= {e^{-at}\over \sqrt{1-e^{-4at}}}
\exp\Big(-{a\over 1-e^{-4at}} (x^2+y^2-2\,e^{-2at}xy)\Big).
$$
and Mehler's formula $(1.12')$ follows. Through our change of variables,
the heat operator of $Q$ is given by
$$\wh{K_{t,\bR^2}f}(x_1,\xi_2) = \int_{\bR} k_{t,\bR}\Big(x_1-{\xi_2\over a},
y_1-{\xi_2\over a}\Big) \wh f(y_1,\xi_2)dy_1.$$
By an inverse partial Fourier transform left to the reader, we obtain 
the desired heat kernel expression
$$\leqalignno{
K_{t,\bR^2}(x_1,x_2;y_1,y_2) = {a\over 4\pi\sinh at} \exp \Big(-{a\over
4}(\coth at)\big(&(x_1-y_1)^2 + (x_2-y_2)^2\big)\Big)\cr
&\times \exp
\Big({\ii\over 2}a(x_1+y_1)(x_2-y_2)\Big).&(1.11')\cr}$$
The heat kernel associated with a sum of (pairwise commuting) operators
$\square_1,\ldots, \square_m$ acting on disjoint sets of variables is the
product of the corresponding heat kernels $e^{-t\square_j}$. Let 
$K^\lambda_{t,k,\Omega}$ be 
the heat kernel of the component of $Q_{k,\Omega }$ acting on each
single entry $u_\lambda$. The factor
in the heat kernel corresponding to each pair of variables
$(x_j,x_{j+s})$, $1\le j\le s$, is obtained by substituting $kB_j$ to
$a$ and $t/k$ to $t$ (the latter rescaling comes from the initial factor
${1\over k}$ in the expression of $Q_{k,\Omega}$). For the other
coordinates $j>2s$ where $B$ has no coefficients, the kernel falls back
to the ``elementary'' heat kernel $(1.10')$. Finally, the constant term
$-V_\lambda|u_\lambda|^2$ contributes to multiplying the heat kernel by
$e^{tV_\lambda}$. Therefore we get for the global heat kernel on 
$\Omega=\bR^n$ the explicit formula
$$\leqalignno{
&K^\lambda_{t,k,\bR^n}(x,y) = \prod^s_{j=1} {kB_j\over 4\pi  \sinh B_jt} \exp
\Big(-{kB_j\over 4} (\coth B_jt)\big((x_{2j-1}-y_{2j-1})^2 +
(x_{2j}-y_{2j})^2\big)\cr
& \kern4cm{}+ {\ii\over 2} kB_j (x_{2j-1}+y_{2j-1})(x_{2j}-y_{2j})\Big)\cr
& \kern5cm{} \times e^{tV_\lambda }\times {1\over (4\pi t/k)^{m-2s}/2} \exp
\big(-k\sum_{j>2s} (x_j-y_j)^2/4t\big).&(1.13)\cr}$$
On the diagonal of $\bR^n\times\bR^n$, the global heat kernel 
$K_{t,k,\bR^n}$ is thus given by
the rather simple ($\Herm(E)\otimes\Id_{L^k}$)-valued tensor depending only
on $B$, $V$ and $t/k\,$:
$$K_{t,k,\bR^n}(x,x) = \Big({k\over 4\pi t}\Big)^{m/2} e^{tV}\prod^s_{j=1}
{B_jt\over \sinh B_jt}.\leqno(1.14)$$

\claim{(1.15) Theorem} {Consider the general $($variable coefficient$)$ case. 
For $\delta>0$ small, the~heat kernel of $\square_k$ over $M$ admits an 
asymptotic estimate
$$K_{t,k,M}(x,x) = \Big({k\over 4\pi t}\Big)^{m/2}
e^{tV(x)}\prod^s_{j=1} {B_j(x)\,t\over \sinh B_j(x)\,t}\,\big(1+O(k^{-1/2+\delta})\big)$$
as $k\to +\infty$, where $O(k^{-1/2+\delta})$ is uniform with respect to 
$x\in M$ and $t$ in a bounded interval $]0,T] \subset{}]0,+\infty[$ 
$($moreover, for every open set $\Omega\subset M$, a similar estimate is 
valid for  $K_{t,k,\Omega}$ on relatively compact subsets of~$\Omega)$.}
\endclaim

\proof{Proof} Notice first that
$(t,x)\mapsto \prod^s_{j=1} {B_j(x)\,t\over \sinh B_j(x)\,t}$
extends as a smooth positive function on $[0,+\infty[{}\times M$, equal to
$1$ when $t=0\,$: this is in fact the inverse of the square root of
the determinant of the positive definite symmetric matrix 
$${\sin(tb(x))\over tb(x)}=\sum_{p=0}^{+\infty}
{t^{2p}(-b(x)^2)^p\over (2p+1)!}{}\ge \Id,$$
where $b(x)$ is the antisymmetric endomorphism of $T_M$ associated with the
alternate \hbox{$2$-form} $B(x)$ and $-b(x)^2=b(x)^\dagger b(x)\ge 0$.

The only thing one has still to get convinced of is that
the kernel of $e^{-t\square_k} - e^{-t\square^0_k}$ is $(k/t)^{m/2}
O(k^{-1/2+\delta})$ 
uniformly along the diagonal at any point $(x^0,x^0)\in M\times M$, 
where $\square^0_k$~is the operator $\square_k$ ``freezed" at $x^0$. 
We can do this in a canonical way by using normal coordinates from the 
Riemannian exponential mapping 
$$\exp_{x^0}:\bR^m\simeq T_{M,x^0}\to M,$$
and trivializations of $E$ and $L$ produced by parallel transport along
geodesics from $x^0$ to any point $x\in B(x^0,\rho_0)$, where
$\rho_0={}$injectivity radius of $M$. In this way, we actually get
automatically that $\Gamma_L(x^0)=\Gamma_E(x^0)=0$. When
\hbox{$\Supp u\subset \Omega_\rho:=B(x^0,\rho)$}, a Taylor expansion yields 
$D_ku-D^0_ku=O(|x|+k|x|^2)\cdot u$ and we get the estimates
$$\eqalign{
Q_{k,\Omega_\rho}(u)-Q^0_{k,\Omega_\rho}(u)&=
\int_M{1\over k}\big(|D_ku|^2-|D^0_ku|^2\big)-\langle(V-V^0)u,u\rangle\cr
&=O\Big(\int_M{1\over k}\big((\rho+k\rho^2)|D_k^0u||u|+(\rho+k\rho^2)^2
|u|^2\big)+\rho|u|^2\Big)\cr
&=O\Big(\int_M{\varepsilon\over k}|D_k^0u|^2+\Big(
{(\rho+k\rho^2)^2\over k\varepsilon}+\rho\Big)|u|^2\Big),\cr
&=O\Big(\varepsilon\, Q^0_{k,\Omega_\rho}(u)+\Big(
{(\rho+k\rho^2)^2\over k\varepsilon}+\rho+\varepsilon\Big)|u|^2\Big)\cr}$$
whenever $\varepsilon<1$, hence there is a constant $C_{\rho,k,\varepsilon}=
O\big({(\rho+k\rho^2)^2\over k\varepsilon}+\rho+\varepsilon\big)$ 
such that
$$
(1-\varepsilon)Q^0_{k,\Omega_\rho}(u)-C_{\rho,k,\varepsilon}|u|^2\le
Q_{k,\Omega_\rho}(u)\le (1+\varepsilon)Q^0_{k,\Omega_\rho}(u)+C_{\rho,k,\varepsilon}
|u|^2.$$
From this, we conclude that $e^{-t\square_k}$ is squeezed (as a positive
bounded self-adjoint operator) between 
$e^{-C_{\rho,k,\varepsilon}t}e^{-t(1+\varepsilon)\square^0_k}$ and
$e^{C_{\rho,k,\varepsilon}t}e^{-t(1-\varepsilon)\square^0_k}$. By definition of
the heat kernel we have
$$
\eqalign{
K_{t,k,\Omega_\rho}(x^0,x^0)&=
\lim_{\nu\to+\infty}\int_{\Omega_\rho\times \Omega_\rho}K_{t,k,\Omega_\rho}(x,y)u_\nu(x)\overline{u_\nu(y)}\,d\sigma(x)\,d\sigma(y)\cr
&=\lim_{\nu\to+\infty}\Langle e^{-t\square_k}u_\nu,u_\nu\Rangle\cr}
$$
when $u_\nu\!\!\buildu L^1\under\longrightarrow\delta_{x^0}$ (Dirac measure),
thus
$$
\eqalign{
e^{-C_{\rho,k,\varepsilon}T}K^0_{(1+\varepsilon)t,k,\Omega_\rho}(x^0,x^0)
&-K^0_{t,k,\Omega_\rho}(x^0,x^0)
\le K_{t,k,\Omega_\rho}(x^0,x^0)-K^0_{t,k,\Omega_\rho}(x^0,x^0)\cr
&\le e^{C_{\rho,k,\varepsilon}T}
K^0_{(1-\varepsilon)t,k,\Omega_\rho}(x^0,x^0)-K^0_{t,k,\Omega_\rho}(x^0,x^0).\cr}
$$
We take here $\rho=\varepsilon=k^{-1/2+\delta}$, so that $C_{\rho,k,\varepsilon}
=O(k^{-1/2+\delta})$. The expected
uniform bounds are then obtained by an application of Proposition~1.9, where
the choice \hbox{$\rho=k^{-1/2+\delta}\gg k^{-1/2}$} ensures that the 
relative errors
$$K_{t,k,M}-K_{t,k,\Omega_\rho}\quad\hbox{and}\quad
K^0_{t,k,\bR^m}-K^0_{t,k,\Omega_\rho}$$
are very small, namely of the order of magnitude $O(\exp(-k^\delta/4T))$.\qed
\endproof

As a consequence, we obtain the following estimate for the eigenvalues~:

\claim{(1.16) Corollary} {The eigenvalues $\lambda_{\nu,k,\Omega }$ of
$Q_{k,\Omega }$ satisfy for every $t>0$ the estimate
$$\sum^{+\infty}_{\nu=1} e^{-t\lambda_{\nu,k,\Omega }}=(1+O(k^{-1/2}))
\Big({k\over 4\pi t}\Big)^{m/2}
\int_\Omega \tr (e^{tV(x)})\prod^s_{j=1}
{B_j(x)\,t\over \sinh B_j(x)\,t}\,d\sigma(x).$$}
\vskip-12pt
\endclaim

\noindent
This result can be also interpreted in terms of the counting function
$$N_{k,\Omega }(\lambda) = \# \{\nu\,;\;\lambda_{\nu,k,\Omega}\le\lambda\}$$
and of the spectral density measure (a sum of Dirac measures on the real line)
$$\mu_{k,\Omega } = k^{-m/2} {d\over d\lambda } N_{k,\Omega }(\lambda).$$
Notice that the measures $\mu_{k,\Omega}$ are all supported in the fixed
interval $[-v_0,+\infty[$, where $v_0$ is an upper bound for the eigenvalues
of $V(x)$, $x\in M$. In these notations, Corollary~1.16 can be restated~:
$$\lim_{k\to +\infty} \int^{+\infty}_{-\infty} e^{-t\lambda } d\mu
_{k,\Omega }(\lambda) = {1\over (4\pi t)^{m/2}} 
\int_\Omega \tr (e^{tV(x)})
 \prod^s_{j=1} {B_j(x)\,t\over \sinh B_j(x)\,t} d\sigma
(x).$$
We thus see that the sequence of measures $\mu_{k,\Omega }$ converges
weakly to a measure $\mu_\Omega $ whose Laplace transform is given by
the right hand side. Inverting the formula, one obtains~:

\vbox{%
\claim{(1.17) Corollary} {For almost all $\lambda \in \bR$
$$\lim_{k\to +\infty} k^{-m/2} N_{k,\Omega }(\lambda) = \mu_\Omega
(]-\infty,\lambda ]) = \int_\Omega \sum^r_{j=1} \nu_{B(x)}
(V_j(x)+\lambda)d\sigma (x)\leqno(1.18)$$
where $\nu_{B(x)}(\lambda)$ is the function on $M\times \bR$ defined
by
$$\nu_B(\lambda) = {2^{s-m}\pi^{-m/2}\over \Gamma ({m\over 2}-s+1)}
B_1\cdots B_s \sum_{(p_1,\ldots, p_s)\in \bN^s} \Big[\lambda -\sum
(2p_j+1)B_j\Big]^{{m\over 2}-s}_+\,.\leqno(1.19)$$}
\vskip-12pt
\endclaim}

\proof{Proof} We leave as an exercise to the reader to check that
the Laplace transform
$$
\int_{-\infty}^{+\infty}e^{-t\lambda}d\nu_B(v+\lambda)=
e^{tv}\int_{-\infty}^{+\infty}e^{-t\lambda}d\nu_B(\lambda)
$$
is actually equal to
$$
 {e^{tv}\over (4\pi t)^{m/2}} \prod^s_{j=1}
{B_j(x)\,t\over \sinh B_j(x)\,t}.
$$
\endproof

\subsection{1.C}{Proof of the holomorphic Morse inequalities}

Let $X$ be a compact complex manifold, $L$ and $E$ holomorphic Hermitian
vector bundles of rank $1$ and $r$ over $X$. If $X$ is endowed with a
Hermitian metric $\omega $, Hodge theory shows that the Dolbeault
cohomology group $H^q(X,E\otimes L^k)$ can be identified with the space
of harmonic $(0,q)$-forms with respect to the Laplace-Beltrami operator
$\Delta''_k = \overline \partial_k \overline\partial_k^* +
\overline\partial^*_k\overline \partial_k$ acting on $E\otimes L^k$. We
thus have to estimate the zero-eigenspace of~$\Delta''_k$.

In order to apply corollary 1.17, we first have to compute $\Delta''_k$
in terms of the Hermitian connection $\nabla_k$ on $E\otimes L^k\otimes
\Lambda^{0,q}T_X^*$ deduced from the Chern connections of $L,E,T_X$. What
plays now the role of $E$ is the (non holomorphic) 
bundle $E\otimes \Lambda^{0,q}T_X^*$.

The relation between $\Delta''_k$ and $\nabla_k$ is most easily
obtained by means of the Bochner-Kodaira-Nakano identity. In order to
simplify the exposition, we assume here that the metric $\omega $ on $X$
is \emph{K\"ahler}. For any Hermitian holomorphic line bundle $G$ on $X$,
the operators $\Delta'$ and $\Delta''$ associated with the Chern connection
$D=D_G$ are related by the B-K-N identity (cf.\ [Boc48], [Kod53], [AN54],
[Nak55])
$$\Delta'' = \Delta' + [i\theta_G,\Lambda ]\leqno(1.20)$$
where $\theta_G=D_G^2\in C^\infty(X,\Lambda^{1,1}T_X^*\otimes\Hom(G,G))$ is the
curvature tensor and
$\Lambda=L^*$ is the adjoint of the Lefschetz operator $Lu=\omega\wedge u$.

The Leibniz rule implies $\theta_{E\otimes L^k} = 
k\theta_L \otimes \Id_E + \theta_E\otimes \Id_{L^k}$ (omitting the Hermitian
metrics for simplicity of notation), thus 
$$\Delta''_k = \Delta'_k + k[i\theta_L,\Lambda ] + [i\theta_E,\Lambda ].$$
At a given point $z^0\in X$~, we can find a coordinate system $(z_1,\ldots,
z_n)$ such that $(\partial/\partial z_j)$ is an orthonormal basis of
$T_X$ diagonalizing $i\theta_L(z^0)$, in such a way that
$$\omega(z^0)={\ii\over 2}\sum_{1\le j\le n} dz_j\wedge d\overline z_j,\qquad
i\theta_L(z^0) = {\ii\over 2}\sum_{1\le j\le n} \alpha_jdz_j\wedge d\overline
z_j$$
where $\alpha_1,\ldots, \alpha_n$ are the curvature eigenvalues of 
$i\theta_L(z^0)$. A standard formula gives the expression of the curvature term
$[i\theta_L,\Lambda ]u$ for any $(p,q)$-form $u$. In fact, for
$u=\sum u_{I,J,\lambda}dz_I\wedge d\overline z_J\otimes e_\lambda $, we have
$$\langle [i\theta_L,\Lambda ]u,u\rangle = \sum_{I,J,\lambda }(\alpha
_J-\alpha_{\complement I})|u_{I,J,\lambda }|^2$$
where $\alpha_J = \sum_{j\in J}\alpha_j$. In the case of a
$(0,q)$-form $u=\sum u_{J,\lambda }d\overline z_J \otimes e_\lambda $ we
simply have $\Delta'_ku= D'^*_k D'_ku=\nabla^{\prime *}_k \nabla'_ku$
and
$$\Delta''_k=\nabla^{\prime *}_k\nabla'_k - kV'+[i\theta_E,\Lambda
]~,\leqno(1.21')$$
$$\langle V'u,u\rangle = \sum_{J,\lambda }\alpha_{\complement J}
|u_{J,\lambda }|^2 ~\hbox{~~~(here~} I=\emptyset).$$
This is not yet what was needed, since only the $(1,0)$ part $\nabla'_k$
appears. To get the $(0,1)$ component, we consider $u$ as a $(n,q)$ form
with values in $E\otimes L^k\otimes \Lambda^nT_X$. We then get
$\Delta'_ku = D'_k D^{\prime *}_ku$ where
$$D^{\prime *}_ku= - \sum \partial u_{I,J,\lambda }/\partial \overline
z_jdz_1\wedge \cdots \widehat{dz_j}\cdots \wedge dz_n\wedge d\overline
z_J\otimes e_\lambda $$
in normal coordinates. Thus $\Delta'_ku =
\nabla^{\prime\prime*}_k \nabla''_k u$ and
$$\Delta''_k= \nabla^{\prime\prime *}_k\nabla''_k + kV'' + [i\theta_{E\otimes
\Lambda^nT_X},\Lambda ]~,\leqno (1.21'')$$
$$\langle V''u,u\rangle = \sum_{J,\lambda } \alpha_J|u_{J,\lambda }|^2
~~\hbox{~~(here~} I=\{1,\ldots, n\}).$$
If the metric $\omega $ is non K\"ahler, we get additional torsion
terms, but these terms are independent of $k$. A combination of $(1.21')$
and $(1.21'')$ yields
$${2\over k} \Delta''_k = {1\over k} \nabla^*_k\nabla_k - V + {1\over
k}W \leqno(1.22)$$
where $W$ is a Hermitian form independent of $k$ and
$$\langle Vu,u\rangle = \sum_{J,\lambda } (\alpha_{\complement J} -
\alpha_J)|u_{J,\lambda }|^2.$$
Now apply Theorem~1.15 and observe that $W$ does not give any
significant contribution to the heat kernel as $k\to +\infty$. We write
here $z_j = x_j+iy_j$ and the ``magnetic field''
$$B=i\theta_L = \sum_{1\le j\le n} \alpha_jdx_j\wedge dy_j.$$
The curvature eigenvalues are given by $B_j=|\alpha_j|$. We denote
$s=s(x)$ the rank of $B(x)$ and order the eigenvalues so that
$$|\alpha_1|\ge \cdots \ge |\alpha_s| >0 = \alpha_{s+1}=\cdots =
\alpha_n.$$
The eigenvalues of $V$ acting on $E\otimes \Lambda^nT_X^*$ are the
coefficients $\alpha_{\complement J} - \alpha_J$, counted with
multiplicity $r$. Therefore

\claim{(1.23) Theorem} {The heat kernel associated with 
$e^{-{2t\over k}\Delta''_k}$ in bidegree $(0,q)$ satisfies
$$K^k_t(x,x) \sim k^n {r\sum_{|J|=q}e^{t(\alpha_{\complement
J}(x)-\alpha_J(x))}\over (4\pi)^nt^{n-s}} \prod^s_{j=1} {|\alpha
_j(x)|\over \sinh |\alpha_j(x)|t}$$
as $k\to +\infty$. In particular, if $\lambda^{k,q}_1 \le \lambda
^{k,q}_2\le \cdots$ are the eigenvalues of ${1\over k}\Delta''_k$ in
bidegree $(0,q)$, we have
$$\sum^{+\infty}_{\nu=1} e^{-2t\lambda^{k,q}_\nu} \sim rk^n
\sum_{|J|=q}\int_X {e^{t(\alpha_{\complement J}(x)-\alpha_J(x))}\over
(4\pi)^nt^{n-s}} \prod^s_{j=1} {|\alpha_j(x)|\over \sinh|\alpha_j(x)|t}$$
for every $t>0$.}
\endclaim

At this point, the main idea is to use the eigenspaces to construct
a finite dimensional subcomplex of the Dolbeault complex possessing
the same cohomology groups. This was already the basic idea in Witten's
analytic proof of the standard Morse inequalities~[Wit82]. We denote by
$$\cH^{k,q}_\lambda,~~~~\hbox{resp.}~~\cH^{k,q}_{\le \lambda }$$
the $\lambda $-eigenspace of ${1\over k} \Delta''_k$ acting on
$C^\infty(X,\Lambda^{0,q}T_X^*\otimes E\otimes L^k)$, resp.\ the direct
sum of eigenspaces
corresponding to all eigenvalues $\le \lambda $. As $\overline \partial
_k$ and $\Delta''_k$ commute, we see that $\overline \partial 
(\cH^{k,q}_\lambda) \subset \cH^{k,q+1}_\lambda $, thus 
$\cH^{k,\bullet}_\lambda $ and $\cH^{k,\bullet}_{\le \lambda }$ are
finite dimensional subcomplexes of the Dolbeault complex
$$\overline\partial~: C^\infty(X,\Lambda^{0,\bullet}T_X^*E\otimes L^k).$$
Since $\overline \partial_k\overline \partial^*_k + \overline
\partial^*_k \overline \partial_k = \Delta''_k = k\lambda \Id$ on
$\cH^{k,\bullet}_\lambda $~, we see that $\cH^{k,\bullet}_\lambda $
has trivial cohomology for $\lambda \ne 0$.
Since $\cH^{k,\bullet}_0$ is the space of harmonic forms, we see
that
$\cH^{k,\bullet}_{\le \lambda }$ has the same cohomology as the
Dolbeault complex for $\lambda >0$. We will call this complex the
Witten $\overline \partial$-complex. We need an elementary lemma of
linear algebra.

\claim{(1.24) Lemma} {Set $h^q_k = \dim H^q(X,E\otimes L^k)$. Then for every
$t>0$ 
$$h^q_k-h^{q-1}_k+\cdots+(-1)^qh^0_k \le
\sum^q_{\ell=0}(-1)^{q-\ell}\sum^{+\infty}_{j=1}e^{-t\lambda
^{k,\ell}_j}.$$}
\endclaim

\proof{Proof} The left hand side is the contribution of the $0$
eigenvalues in the right hand side. All we have to check is that the
contribution of the other eigenvalues is $\ge 0$. The contribution of
the eigenvalues such that $\lambda^{k,\ell}_j = \lambda >0$ is
$$e^{-t\lambda }\sum^q_{\ell=0} (-1)^{q-\ell} \dim \cH^{k,\ell}_\lambda.$$
As $\cH^{k,\bullet}_\lambda $ is exact, one easily sees that the last
sum is equal to the dimension of $\overline \partial\cH^{k,q}_\lambda
\subset \cH^{k,q+1}_\lambda $, hence $\ge 0$.\qed
\endproof

Combining Theorem 1.23 with Lemma 1.24, we get
$$\displaylines{
h^q_k-h^{q-1}_k +\cdots + (-1)^q h^0_k \le o(k^n) +\hfill\cr
\hfill rk^n \sum^q_{\ell=0} (-1)^{q-\ell}\sum_{|J|=\ell} \int_X
{\prod_{j\le s}|\alpha_j|\cdot e^{t(\alpha_{\complement J}-\alpha
_J-\sum|\alpha_j|)}\over
2^{2n-s}\pi^nt^{n-s} \prod_{j\le s} (1-e^{-2t|\alpha_j|})}.\cr}$$
This inequality is valid for any $t>0$, so we can let $t$ tend to
$+\infty$. It is clear that $\alpha_{\complement J}-\alpha_J -
\sum|\alpha_j|$ is always $\le 0$, thus the integrand tends to $0$ at
every point where $s<n$. When $s=n$, we have $\alpha_{\complement J}(x)
- \alpha_Jx) -\sum|\alpha_j(x)|=0$ if and only if $\alpha_j(x) >0$
for every $j\in \complement J$ and $\alpha_j(x)<0$ for every $j\in J$.
This implies $x\in X(L,h,\ell)$~; in this case there is only one
multi-index $J$ satisfying the above conditions and the limit is 
$$(2\pi)^{-n}|\alpha_1\cdots \alpha_n|=(2\pi)^{-n}|(i\theta_{L,h})^n|=
|\Theta_{L,h}^n|,$$
as $\Theta_{L,h}={\ii\over 2\pi}\theta_{L,h}$ by definition. By the monotone 
convergence theorem, our sum of integrals converges to
$$\sum^q_{\ell=0} (-1)^{q-\ell}\int_{X(L,h,\ell)}(2\pi)^{-n} |\alpha
_1\cdots \alpha_n| d\sigma  = {1\over n!}\int_{X(L,h,\le q)} (-1)^q
\Theta_{L,h}^n\,.$$
The Main Theorem 1.2 follows.\qed
\endproof

\section{2}{Applications to algebraic geometry}

\subsection{2.A}{Solution of the Grauert-Riemenschneider conjecture}

Let $L$ be a holomorphic line bundle over a compact connected complex
manifold $X$ of dimension $n$ and $V_k = H^0(X,L^k)$. Denote by $Z(V_k)$
the set of common zeroes of all sections in $V_k$, and
fix a basis $(\sigma_0,\ldots,\sigma_N)$ of $V_k$. There is a
canonical holomorphic map
$$\Phi_{kL}\,:~X\ssm Z(V_k) \longrightarrow \bP(V_k),\qquad
x\mapsto[\sigma_0(x):\ldots:\sigma_N(x)]\leqno(2.1)$$
sending a point $x\in X\ssm Z(V_k)$ to the hyperplane
$H\subset V_k$ of sections $\sigma=\sum\lambda_j\sigma_j \in V_k$ such 
that $\sigma (x)=\sum\lambda_j\sigma_j(x)=0$; it is therefore given by 
$x\mapsto [\sigma_0(x):\ldots:\sigma_N(x)]$ in projective coordinates
on $\bP(V_k)\simeq\bP^N$.
The pull-back $\Phi_{kL}^*\cO(d)$ can be identified with the
restriction of $L^{kd}$ to $X\ssm Z(V_k)$; indeed, to any homogeneous
polynomial $P(w_0,\ldots,w_N)\in H^0(\bP^N,\cO(d))$
of degree $d$, one can associate a section 
$$s=P(\sigma_0,\ldots,\sigma_N)\in H^0(X,L^{kd}).\leqno(2.2)$$

When $L$ possesses a smooth Hermitian metric $h$ with $\Theta_{L,h}>0$, 
one can construct many sections of high tensor powers
$L^k$ (e.g.\ by H\"ormander's $L^2$ estimates [H\"or65], [AV65] for~$\dbar$).
For $k\ge k_0$ large
enough, the ``base locus" $Z(V_k)$ is empty, the sections in $V_k$
separate any two points of $X$ and generate all $1$-jets at any point.
Then $\Phi_{kL}$ gives an embedding of $X$ in some projective space $\bP^N$,
for $N=N(k)$ and $k\ge k_0$. In this way, the theory of $L^2$ estimates
implies the \emph{Kodaira embedding theorem}$\,$: a compact
complex manifold $X$ is projective algebraic if and only if $X$ possesses
a Hermitian line bundle $(L,h)$ with $C^\infty$ positive curvature.

The Grauert-Riemenschneider conjecture [GR70] is an attempt to
characterize the more general class of Moishezon varieties in terms of
semi-positive line bundles. Let us first recall a few definitions. The
algebraic dimension $a(X)$ is the transcendence degree
of the field $\cM(X)$ of meromorphic functions on $X$. A well-known
theorem of Siegel [Sie55] asserts that $0\le a(X)\le n$
(see Corollary~2.6 below). A compact manifold or variety $X$ is said to be 
\emph{Moishezon} if $a(X)=n$.

By definition, the \emph{Kodaira dimension} $\kappa (L)$ is the supremum of the
dimension of the images $Y_k=\Phi_{kL}(X\ssm Z(V_k))\subset \Bbb
P(V^*_k)$ for all integers $k>0$ [one defines $\kappa(L)=-\infty$ when
$V_k=0$ for all $k$, in which case we always have $Y_k=\emptyset$]. 
Since the field of meromorphic
functions on $X$ obtained by restriction of rational functions of $\Bbb
P(V^*_k)$ to $Y_k$ has transcendence degree at least equal to~$\dim Y_k$,
we infer that
$$-\infty\le \kappa (L) = \sup\dim Y_k \le a(X)\le n.\leqno(2.3)$$

\claim{(2.4) Definition} {The line bundle $L\to X$ is said to be {\rm big}
if $\kappa(L)$ is maximal, i.e.\ $\kappa(L)=n=\dim X$.}
\endclaim

\noindent The following standard lemma is needed (cf.\ [Ser54], [Sie55]).

\claim{(2.5) Lemma {\rm (Serre-Siegel)}} {For every line bundle $L\to X$, there
exist constants \hbox{$C\ge c>0$} and $k_0\in\bN^*$ such that
$$\eqalign{
\dim H^0(X,L^k) &\le C \,k^{\kappa (L)}\qquad\hbox{for all $k\ge 1$},\cr
\dim H^0(X,L^k) &\ge {}\,c\,\,k^{\kappa (L)}\qquad\hbox{for all $k\ge 1$ multiple
of $k_0$}.\cr}
$$}
\vskip-12pt
\endclaim

\proof{Proof} The lower bound is obtained by taking $k_0$ such that
$p:=\dim Y_{k_0}=\kappa(L)$. Then, by the rank theorem, there exists
a point $x_0\in X\ssm Z(V_{k_0})$ and a basis $(\sigma_0,\ldots,\sigma_N)$
of $H^0(X,L^{k_0})$ such that $\sigma_0(x_0)\ne 0$ and 
$\big(d(\sigma_1/\sigma_0)\wedge\ldots\wedge  d(\sigma_p/\sigma_0)\big)
(x_0)\ne 0$.
Then by taking $s=P(\sigma_0,\ldots,\sigma_p,0,\ldots,0)$ in (2.2),
we obtain an injection of the space of homogeneous polynomials of degree 
$d$ in $p+1$ variables into $H^0(X,L^{k_0d})$, whence
$$h^0(X,L^{k_0d})\ge {d+p\choose p}\ge d^p/p!.$$

The proof of the upper bound proceeds as follows~: select a Hermitian 
metric $h$, on $L$ and a finite
family of coordinate balls $B_j = B(z_j,r_j)$ such that $B'_j=B(z_j,r_j/2)$
cover~$X$, and $L_{|B_j}$ is trivial for each~$j$. By moving a little bit
the points $z_j$, we may assume that $\Phi_{kL}$ has maximal rank at all
points $z_j$ for all $k$ (the bad set is at most a countable union
of analytic sets, so it is nowhere dense). If $L^k$ has many 
sections, one can solve a linear system in many unknowns to get a 
section $s$ vanishing at a high order $m$ at all centers~$z_j$. 
Then the Schwarz lemma gives
$$\|s\|_{h,\infty} = \sup_j\|s\|_{h,B'_j} \le 2^{-m} C(h)^k \sup_j\|s\|_{h,B_j} \le
2^{-m}C(h)^k\|s\|_{h,\infty}$$
where $C(h)$ is a bound for the oscillation of the metric $h$ on
$B_j$, which we may assume to be finite after possibly
shrinking~$B_j$. Thus $m\le k\log C(h)/\log 2$ if $s\ne 0$. Since the
sections of $L^k$ are constant along the fibers of $\Phi_{kL}$, only
$m^{\dim Y_k}\# \{z_j\}$ equations transversally to the fibers are
needed to make $s$ vanish at order~$m$. Therefore we can choose
\hbox{$m\approx (h^0(X,L^k)/\#\{z_j\})^{1/\dim Y_k}$} and still get a
non zero section, so that
$$h^0(X,L^k) \approx \#\{z_j\}\cdot m^{\dim Y_k} \le C\,k^{\kappa (L)}.\eqno\qed$$

\claim{(2.6) Corollary {\rm(Siegel)}} For every compact complex manifold~$X$
$$a(X):=\tr\deg_\bC\cM(X)\le n.$$
\endclaim

\proof{Proof} Fix $s$ algebraically independent elements $f_1,\ldots,f_s\in
\cM(X)$ and let $D$ be the sup of the pole divisors of the~$f_j$'s. To 
every polynomial $P(f_1,\ldots,f_s)$ of degree${}\le k$ corresponds 
injectively a section $\sigma_P=P(f_1,\ldots,f_s)\in H^0(X,\cO(kD))$.
A dimension count implies
$$
{k^s\over s!}\le {k+s\choose s} \le C\,k^{\kappa(\cO(D))}\le C\,k^n
$$
by Lemma~2.5. Therefore $s\le n$.\qed
\endproof

\noindent
Now, the Grauert-Riemenschneider conjecture [GR70] can be stated as follows.

\claim{(2.7) Grauert-Riemenschneider conjecture} {A compact complex 
variety $Y$ is Moi\-shezon if and only if there is
a proper non singular modification $X\to Y$ and a Hermitian line bundle 
$(L,h)$ over $X$ such that the curvature form $\Theta_{L,h}$ is $>0$ on 
a dense open subset of~$X$.}
\endclaim

\proof{Proof} When $Y$ is Moishezon, it is well known that there exists 
a projective algebraic modification $X$~; therefore we can even take $L$ 
to be ample and then there exists $h$ such that $\Theta_{L,h}>0$ 
everywhere on~$X$. 

The converse statement was proved by Siu in [Siu84, Siu85], assuming only
$\Theta_{L,h}\ge 0$ every\-where and $\Theta_{L,h}>0$ in at least 
one point. Morse inequalities provide in fact a much stronger criterion, 
requiring only the positivity of some curvature integral:

\claim{(2.8) Theorem} {If a Hermitian line bundle $(L,h)$ on $X$ satisfies 
the integral condition $$\int_{X(L,h,\le 1)}(\Theta_{L,h})^n>0,$$ 
then $\kappa (L)=n$, in particular $X$ is Moishezon.}
\endclaim

In fact, the lower bound (1.4) applied with $E=\cO_X$ implies immediately that
$h^0(X,L^k)\ge c\,k^n$, hence $\kappa(L)=n$. Now, if $X$ is a modification of
$Y$, we have $\cM(Y) \simeq \cM(X)$, so $a(X)=a(Y)$, and $Y$ has to be 
Moishezon.\qed
\medskip

\subsection{2.B}{Cohomology estimates for nef line bundles}

On a projective algebraic manifold $X$, a line bundle $L$ is said to be
\emph{nef} if $L\cdot C\ge 0$ for every algebraic curve~$C\subset X$. 
If $\omega$
is a given K\"ahler or Hermitian $(1,1)$-form on $X$, it can be
shown (cf.\ [Dem90]) that $L$ is nef if and only if for every 
$\varepsilon>0$ there exists a smooth Hermitian metric $h_\varepsilon$
such that $\Theta_{L,h_\varepsilon}\ge-\varepsilon\omega$ on~$X\,$; in
fact, the latter property clearly implies 
$$L\cdot C=\int_C \Theta_{L,h_\varepsilon}\ge-\varepsilon\int_C\omega
~~\Longrightarrow~~L\cdot C\ge 0$$
for every curve $C$. Conversely, if $L\cdot C\ge 0$ for every curve $C$,
the well-known Kleiman criterion (cf.\ [Har70]) implies that $kL+A$ is ample
for every ample divisor $A$. Hence there exists a smooth Hermitian metric 
$h_k$ on $L$ such that
$$
\Theta_{kL+A}=k\Theta_{L,h_k}+\Theta_{A,h_A}>0~~\Longrightarrow~~
\Theta_{L,h_k}\ge -{1\over k}\omega,~~\hbox{where}~~\omega=\Theta_{A,h_A}>0.
$$
Therefore, one can introduce
the following \emph{definition of nefness} on an arbitrary compact 
complex manifold.

\claim{(2.9) Definition} {Let $X$ be a compact complex manifold and $\omega$ 
a given smooth positive $(1,1)$-form on~$X$. A line bundle $L\to X$
is said to be {\rm nef} if for every 
$\varepsilon>0$ there exists a smooth Hermitian metric $h_\varepsilon$ on $L$
such that $\Theta_{L,h_\varepsilon}\ge-\varepsilon\omega$ everywhere on~$X$.}
\endclaim

\claim{(2.10) A consequence of holomorphic Morse inequalities} {If $X$ 
is compact K\"ahler and $L$ is nef, for every holomorphic vector bundle 
$E$ on $X$ one has
$$
h^q(X,\cO(E)\otimes\cO(kL)) = o(k^n)\qquad\hbox{for all $q\ge 1$}.
$$}
\vskip-12pt
\endclaim


\proof{Proof} Let $\omega$ be a K\"ahler metric. The nefness of $L$ implies
that there exists a smooth Hermitian metric $h_\varepsilon$ on $L$ such that
$\Theta_{L,h_\varepsilon}\ge-\varepsilon\omega$. On
$X(L,h_\varepsilon,1)$ we have exactly $1$~negative eigenvalue $\lambda_1$
which is belongs to $[-\varepsilon,0[$ and the other ones $\lambda_j$
($j\ge 2$) are positive. The product $\lambda_1\cdots\lambda_n$
satisfies $|\lambda_1\cdots\lambda_n|\le\varepsilon\prod_{j\ge 2}
(\lambda_j+\varepsilon)$, hence
$${1\over n!}\big|\Theta_{L,h_\varepsilon}^n\big|\le
{1\over (n-1)!}\varepsilon\omega\wedge(
\Theta_{L,h_\varepsilon}+\varepsilon\omega)^{n-1}\qquad
\hbox{on $X(L,h_\varepsilon,1)$.}
$$
By integrating, we find
$$
\int_{X(L,h_\varepsilon,1)}
\Theta_{L,h_\varepsilon}^n\le
n\varepsilon\int_X\omega\wedge(c_1(L)+\varepsilon\omega)^{n-1}
$$
and the result follows.\qed
\endproof

\claim{(2.11) Note} {\rm When $X$ is non K\"ahler, D.~Popovici
[Pop08] has announced bounds for the Monge-Amp\`ere masses of
$\Theta_{L,h_\varepsilon}$ which
still imply the result, but the proof is much harder in that case. On 
the other hand, when $X$ is projective algebraic, an elementary 
hyperplane section argument and an induction on dimension easily implies
the stronger upper bounds
$$
h^q(X,\cO(E)\otimes\cO(kL)) = O(k^{n-q})\qquad\hbox{for all $q\ge 0$}.
\leqno(2.12)
$$
\emph{Hint.} By Serre duality, it is enough to show that
$$
h^q(X,\cO(F)\otimes\cO(-kL)) = O(k^q)
\qquad\hbox{for every $q\ge 0$}
$$
and every holomorphic vector bundle~$F$.
Choose a very ample line bundle $A$ so big that
$F'=F^*\otimes \cO(A)$ is Nakano positive, and
apply the Nakano vanishing theorem and Serre duality to see that
$H^q(X,\cO(F)\otimes\cO(-A)\otimes \cO(-kL))=0$ for all $k$ and $q\ge 1$. Use
the exact sequence $0\to \cO_X(-A)\to\cO_X\to\cO_A\to 0$, take the
tensor product with $\cO(F)\otimes \cO(-kL)$ and apply induction.
\qed}
\endclaim

\noindent It is unknown whether the accurate bound (2.12) holds true 
on a general compact complex manifold, even when $X$ is assumed to 
be K\"ahler.\medskip

\subsection{2.C}{Distortion inequalities for asymptotic Fubini-Study metrics}

Another application of the heat kernel estimates is a generalization of 
G.~Kempf's distortion inequalities ([Kem89], [Ji89]) to all projective
algebraic manifolds. In this generality, the result was obtained by
Th.\ Bouche [Bou90], and in less generality (but with somewhat
stronger estimates) by G.~Tian [Tia90].

Let $L$ be a positive Hermitian line bundle over a projective manifold
$X$, equipped with a Hermitian metric $\omega $. Then $V_k=H^0(X,L^k)$
has a natural Hermitian metric given by the global $L^2$ norm of
sections.
For $k\ge k_0$ large enough, $\Phi_{kL}$ is an embedding and $L^k$ can be
identified to the pull-back $\Phi^*_k O(1)$. We want to compare the
original metric $|\,{\scriptscriptstyle\bullet}\,|$ of $L$ and the metric 
$|\,{\scriptscriptstyle\bullet}\,|_\FS$ induced by the
Fubini-Study metric of $O(1)$.

Let $(s_1,\ldots,s_N)$ be an orthonormal basis of $H^0(X,L^k)$. It is not
difficult to check that
$$|\xi |^2_\FS = {|\xi |^2\over |s_1(x)|^2+\cdots +|s_N(x)|^2}
\hbox{~for~} \xi \in L^k_x~,$$
thus all that we need is to get an estimate of $\sum |s_j(x)|^2$.
However, this sum is the contribution of the $0$ eigenvalue in the heat
kernel
$$K^k_t(x,x) = \sum^{+\infty}_{j=1} e^{-2t\lambda^k_j} |\psi_j(x)|^2$$
associated to ${2\over k}\square''_k$ in bidegree $(0,0)$. We observe
that non zero eigenvalues $\lambda^k_j$ are also eigenvalues in
bidegree $(0,1)$, since $\overline \partial$  is injective on the
corresponding eigenspaces. The associated eigenfunctions are $\overline
\partial \psi_j/\sqrt{k\lambda^k_j}$, for 
$$\|\overline
\partial \psi_j\|^2 = \Langle \Delta''_k\psi_j,\psi_j\Rangle = k\lambda^k_j.$$
Thus the summation
$$\sum^{+\infty}_{j=1} e^{-2t\lambda^k_j}|\overline
\partial \psi_j(x)|^2$$
is bounded by the heat kernel in bidegree $(0,1)$, which is itself
bounded by $k^ne^{-ct}$ with $c>0$ (note that $\alpha_{\complement
J}-\alpha_J - \sum|\alpha_j|<0$ on $X$ for $|J|=1$). Taking
$t=k^\varepsilon $ with $\varepsilon $ small, one can check that all
estimates remain uniformly valid and that the contribution of the non
zero eigenfunctions in $K^k_t(x,x)$ becomes negligible in $C^0$ norm.
Then theorem~1.23 shows that
$$\sum|s_j(x)|^2 \sim K^k_t(x,x) \sim k^n(2\pi)^{-n}|\alpha_1(x)\cdots
\alpha_n(x)|$$
as $t=k^\varepsilon \to +\infty$. For $\xi \in L^k_x$ we get therefore
the $C^0$ uniform estimate
$${|\xi|^2\over |\xi |^2_\FS} \sim \Big({k\over 2\pi }\Big)^n|\alpha
_1(x)\cdots \alpha_n(x)|\qquad \hbox{as}\quad k\to +\infty.\leqno(2.13)$$
As a consequence, the Fubini-Study metric on $L$ induced by $\Phi_{kL}$
converges uniformly to the original metric. G.~Tian [Tia90] proved that 
this last convergence statement holds in norm $C^4$. It is now
known that there is in fact an asymptotic expansion in $1/k$, and therefore
$C^\infty$ convergence; this holds true even in the almost complex
setting, see [BU00] and [SZ02].\medskip

\subsection{2.D}{Algebraic counterparts of the holomorphic Morse inequalities}

\noindent One difficulty in the application of the analytic form of the
inequalities is that the curvature integral is in general quite uneasy
to compute, since it is neither a topological nor an algebraic
invariant. However, the Morse inequalities can be reformulated in a
more algebraic setting in which only algebraic invariants are
involved. We give here two such reformulations -- after they were found
via analysis in [Dem94], F.~Angelini [Ang96] gave a purely algebraic proof
(see  also [Siu93] and [Tra95] for related ideas).

\claim{(2.14) Theorem} Let $L=F-G$ be a holomorphic line bundle over a
compact K\"ahler manifold~$X$, where $F$ and $G$ are numerically effective
line bundles. Then for every $q=0,1,\ldots,n=\dim X$, there is an asymptotic
strong Morse inequality
$$\sum_{0\le j\le q}(-1)^{q-j}h^j(X,kL)\le{k^n\over n!}
\sum_{0\le j\le q}(-1)^{q-j}{n\choose j}F^{n-j}\cdot G^j+o(k^n).$$
\endclaim
%
\proof{Proof} By adding $\varepsilon$ times a K\"ahler metric $\omega$ to the
curvature forms of $F$ and $G$, $\varepsilon>0$ one can write $
\Theta_L=\tilde\Theta_{F,\varepsilon}-\tilde\Theta_{G,\varepsilon}$ where
$\tilde\Theta_{F,\varepsilon}={\ii\over 2\pi}\Theta_F+\varepsilon\omega$ and
$\tilde\Theta_{G,\varepsilon}={\ii\over 2\pi}\Theta_G+\varepsilon\omega$ are
positive definite. Let $\lambda_1\ge\cdots\ge\lambda_n>0$ be the
eigenvalues of $\tilde\Theta_{G,\varepsilon}$ with respect to $\tilde\Theta_{F,\varepsilon}$.
Then the eigenvalues of ${\ii\over 2\pi}\Theta_L$ with respect
to $\tilde\Theta_{F,\varepsilon}$ are the real numbers $1-\lambda_j$ and the
set $X(L,h,\le q)$ is the set $\{\lambda_{q+1}<1\}$ of points $x\in X$ such
that $\lambda_{q+1}(x)<1$. The strong Morse inequalities yield
$$\sum_{0\le j\le q}(-1)^{q-j}h^j(X,kL)\le{k^n\over n!}
\int_{\{\lambda_{q+1}<1\}}\!\!\!(-1)^q\!\!
\prod_{1\le j\le n}(1-\lambda_j)\tilde\Theta_{F,\varepsilon}^n+o(k^n).$$
On the other hand we have
$${n\choose j}\tilde\Theta_{F,\varepsilon}^{n-j}\wedge\tilde\Theta_{G,\varepsilon}^j=
\sigma_n^j(\lambda)\,\tilde\Theta_{F,\varepsilon}^n,$$
where $\sigma_n^j(\lambda)$ is the $j$-th elementary symmetric function in
$\lambda_1,\ldots,\lambda_n\,$, hence
$$\sum_{0\le j\le q}(-1)^{q-j}{n\choose j}F^{n-j}\cdot G^j=
\lim_{\varepsilon\to 0}\int_X\sum_{0\le j\le q}(-1)^{q-j}
\sigma_n^j(\lambda)\,\tilde\Theta_{F,\varepsilon}^n.$$
Thus, to prove the lemma, we only have to check that
$$\sum_{0\le j\le n}(-1)^{q-j}\sigma_n^j(\lambda)-
\bOne_{\{\lambda_{q+1}<1\}}(-1)^q\prod_{1\le j\le n}(1-\lambda_j)\ge 0$$
for all $\lambda_1\ge\cdots\ge\lambda_n\ge 0$, where $\bOne_
{\{\ldots\}}$ denotes the characteristic function of a set.
This is easily done by induction on~$n$ (just split apart the parameter
$\lambda_n$ and write $\sigma_n^j(\lambda)=
\sigma_{n-1}^j(\lambda)+\sigma_{n-1}^{j-1}(\lambda)\,\lambda_n$).\qed
\endproof

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

\claim{(2.15) Consequence}
$h^0(X,kL)-h^1(X,kL)\ge{k^n\over n!}(F^n-nF^{n-1}\cdot G)-o(k^n)$.\\
Therefore some multiple $kL$ has a section as soon as $F^n-nF^{n-1}\cdot
G>0$.
\endclaim
%
\claim{(2.16) Remark} \rm The weaker inequality
$$h^0(X,kL)\ge{k^n\over n!}(F^n-nF^{n-1}\cdot G)-o(k^n)$$
is easy to prove if $X$ is projective algebraic. Indeed, by adding a
small ample $\bQ$-divisor to $F$ and $G$, we may assume that
$F$, $G$ are ample. Let $m_0G$ be very ample and let $k'$ be the smallest
integer $\ge k/m_0$. Then $h^0(X,kL)\ge h^0(X,kF-k'm_0G)$. We select
$k'$ smooth members $G_j$, $1\le j\le k'$ in the linear system $|m_0G|$
and use the exact sequence
$$0\to H^0(X,kF-\sum G_j)\to H^0(X,kF)\to\bigoplus H^0(G_j,kF_{|G_j}).$$
Kodaira's vanishing theorem yields $H^q(X,kF)=0$ and $H^q(G_j,kF_{|G_j})=0$
for $q\ge 1$ and $k\ge k_0$. By the exact sequence combined with
Riemann-Roch, we get
$$\eqalign{
h^0(X,kL)&\ge h^0(X,kF-\sum G_j)\cr
&\ge{k^n\over n!}F^n-O(k^{n-1})-
\sum\Big({k^{n-1}\over(n-1)!}F^{n-1}\cdot G_j-O(k^{n-2})\Big)\cr
&\ge{k^n\over n!}\Big(F^n-n{k'm_0\over k}F^{n-1}\cdot G\Big)-O(k^{n-1})\cr
&\ge{k^n\over n!}\Big(F^n-n\,F^{n-1}\cdot G\Big)-O(k^{n-1}).\cr}$$
(This simple proof is due to F.~Catanese.)\qed
\endclaim

\claim{(2.17) Corollary} Suppose that $F$ and $G$ are nef and that $F$ is
big. Some multiple of $mF-G$ has a section as soon as
$$m>n\,{F^{n-1}\cdot G\over F^n}.$$
\endclaim
%
In the last condition, the factor $n$ is sharp: this is easily seen by taking
$X=\bP_1^n$ and $F=\cO(a,\ldots,a)$ and $G=\cO(b_1,\ldots,b_n)$ over
$\bP_1^n\,$; the condition of the corollary is then
$m>\sum b_j/a$, whereas $k(mF-G)$ has a section if and only if
$m\ge\sup b_j/a$; this shows that we cannot replace $n$ by
$n(1-\varepsilon)$.\medskip

\section{3}{Morse inequalities on $q$-convex varieties}

Thierry Bouche [Bou89] has obtained an extension of holomorphic Morse
inequalities to the case of strongly $q$-convex manifolds. We explain
here the main ideas involved.

A complex (non compact) manifold $X$ of dimension $n$ is strongly
$q$-convex in the sense of Andreotti and Grauert [AG62] if there exists a
$C^\infty$ exhaustion function $\psi$ on $X$ such that $i\partial
\overline\partial\psi$ has at least $n-q+1$ positive eigenvalues outside a
compact subset of $X$. In this case, the Andreotti-Grauert theorem shows
that all cohomology groups $H^m(X,\cF)$ with values in a coherent
analytic sheaf are finite dimensional for $m\ge q$.

\claim{(3.1) Theorem} {Let $L$, $E$ be holomorphic vector bundles over $X$
with rank$\,L=1$, rank$\,E=r$. Assume that $X$ is strongly $q$-convex 
and that $L$ has a Hermitian metric $h$ for which $\Theta_{L,h}$ has at 
least $n-p+1$ nonnegative eigenvalues outside a compact subset
$K\subset X$. Then for all $m\ge p+q-1$ the
following strong Morse inequalities hold~:
$$\sum^n_{\ell=m} (-1)^{\ell-m} \dim H^\ell(X,E\otimes L^k) \le
r{k^n\over n!} \int_{X(L,h,\ge m)} (-1)^m \Theta_{L,h}^n
+ o(k^n).$$}
\endclaim

\proof{Proof} For every $c\in\bR$, we consider the sublevel sets
$$X_c=\{x\in X~;~\psi(x)<c\}.$$
Select $c_0$ such that $i\partial\overline\partial\psi$ has $n-q+1$
positive eigenvalues on $X\ssm X_c$. One can choose a Hermitian
metric $\omega_0$ on $X$ in such a way that the eigenvalues $\gamma
^0_1\le \cdots \le \gamma^0_n$ of $i\partial\overline\partial\psi$ with
respect to $\omega_0$ satisfy
$$-{1\over n} \le \gamma^0_1 \le \cdots \le \gamma^0_{q-1}\le 1
\hbox{~~~and~~~} \gamma^0_q = \cdots = \gamma^0_n = 1 \hbox{~on~} X\ssm
X_{c_0}~;\leqno(3.2)$$
this can be achieved by taking $\omega_0$ equal to
$i\partial\overline\partial\psi$ on a $C^\infty$ subbundle of $T_X$ of
rank $n-q+1$ on which $i\partial\overline\partial\psi$ is positive, and
$\omega_0$ very large on the orthogonal complement. We set $\omega
=e^\rho \omega_0$ where $\rho $ is a function increasing so fast at
infinity that $\omega $ will be complete.

More important, we multiply the metric of $L$ by a weight $e^{-\chi
\circ \psi }$ where $\chi $ is a convex increasing function. The
resulting Hermitian line bundle is denoted $(L_\chi,h_\chi)$. For any $(0,m)$
form $u$ with values in $E\otimes L^k$, viewed as an $(n,m)$ form with
values in $E\otimes L^k\otimes \Lambda^nT_X$, the Bochner-Kodaira-Nakano
formula implies an inequality
$$\Langle \Delta''_ku,u\Rangle \ge \int_X k\langle [i\theta_{L_\chi,h_\chi}),
\Lambda]u,u\rangle + \langle W u,u\rangle$$
where $W$ depends only on the curvature of $E\otimes \Lambda^nT_X$
and the torsion of $\omega $. By the formulas of \S1.C, we have
$$\langle [i\theta_{L_\chi,h_\chi}),\Lambda ]u,u\rangle \ge 
(\alpha_1+\cdots+\alpha_m)|u|^2$$
where $\alpha_1\le \cdots \le \alpha_n$ are the eigenvalues of
$$i\theta_{L_\chi,h_\chi} = i\theta_{L,h} + i\partial\overline\partial(\chi \circ \psi) \ge
i\theta_{L,h} + (\chi'\circ \psi) i\partial\overline\partial \psi.$$
If $\beta $ is the lowest eigenvalue of $i\theta_{L,h}$ with respect
to $\omega$, we find
$$\eqalign{
\alpha_j &\ge \beta +(\chi'\circ \psi)\gamma^0_j/e^\rho ~,\cr
\alpha_1+\cdots + \alpha_m &\ge m\beta  + (\chi'\circ \psi)(\gamma
^0_1+\cdots + \gamma^0_m)/e^\rho ~,\cr}$$
and by (3.2) we get for all $m\ge q$~:
$$\alpha_1+\cdots + \alpha_m \ge m\beta  + {1\over n} e^{-\rho }\chi
'\circ \psi  \hbox{~on~} X\ssm X_{c_0}.$$
It follows that one can choose $\chi $ increasing very fast in such a
way that the Bochner inequality becomes
$$\langle \Delta''_ku,u\rangle \ge k\int_{X\ssm X_{c_0}}A(x)|u(x)|^2 - 
C_1\int_X|u(x)|^2\leqno(3.3)$$
where $A\ge 1$ is a function tending to $+\infty$ at infinity on $X$ and
$C_1\ge 0$. Now, Rellich's lemma easily shows that
$\Delta''_k$ has a compact resolvent. Hence the spectrum of $\Delta
''_k$ is discrete and its eigenspaces are finite dimensional. Standard
arguments also show the following~:

\claim{(3.4) Lemma} {When $\chi $ increases sufficiently fast at infinity, the
space 
$\cH^m(X,L^k_\chi  \otimes E)$ of $L^2$-harmonic forms of bidegree
$(0,m)$ for $\Delta''_k$ is isomorphic to the cohomology group
$H^m(X,E\otimes L^k)$ for all $k\in \bN$ and $m\ge q$.}
\endclaim

For a domain $\Omega \subset\!\subset X$, we consider the quadratic form
$$Q^{k,m}_\Omega (u) = {1\over k} \int_\Omega  |\overline \partial_ku|^2
+ |\overline\partial^*_ku|^2$$
with Dirichlet boundary conditions on $\partial \Omega $. We denote by
$\cH^{k,m}_{\le \lambda ,\Omega }$ the direct sum of all
eigenspaces of $Q^{k,m}_\Omega $ corresponding to eigenvalues $\le
\lambda $ (i.e.\ $\le k\lambda $ for $\Delta''_k$).

\claim{(3.5) Lemma} {For every $\lambda \ge 0$ and $\varepsilon >0$, there
exists a domain $\Omega \subset \!\subset X$ and an integer $k_0$ such
that
$$\dim\cH^{k,m}_{\le \lambda ,\Omega } \le \dim\cH^{k,m}_{\le \lambda ,X} 
\le \dim\cH^{k,m}_{\le \lambda
+\varepsilon ,\Omega } \hbox{~for~} k\ge k_0.$$}
\endclaim

\proof{Proof} The left hand inequality is a straightforward consequence
of the minimax principle, because the domain of the global quadratic
form $Q^{k,m}_\Omega $ is
contained in the domain of $Q^{k,m}_X$.

For the other inequality, let $u\in \cH^{k,m}_{\le \lambda ,X}$.
Then (3.3) gives
$$k\int_{X\ssm X_{c_0}} A|u|^2-C_1\int_{X_{c_0}}|u|^2\le k\lambda
\int_X |u|^2.$$
Choose $c_2>c_1>c_0$ so that $A(x) \ge a$ on $X\ssm X_{c_1}$ and a
cut-off function $\varphi $ with compact support in $X_{c_2}$ such that
$0\le \varphi \le 1$ and $\varphi =1$ on $X_{c_1}$. Then we find
$$\int_{X\ssm X_{c_1}} |u|^2 \le {C_1+k\lambda \over ka} \int_X
|u|^2.$$
For a large enough, we get 
$\int_{X\ssm X_{c_1}}|u|^2\le \varepsilon \|u\|^2$. Set $\Omega
=X_{c_2}$. Then
$$\eqalign{
Q^{k,m}_\Omega (\varphi u) 
&= {1\over k} \int_\Omega  |\overline \partial \varphi \wedge u +
\varphi \overline \partial_ku|^2+|\varphi \overline\partial
^*_ku-\partial \varphi \ort u|^2\cr
&\le (1+\varepsilon)Q^{k,m}_X(u) + {C_2\over k}\Big(1+{1\over \varepsilon
}\Big)\|u\|^2\cr
&\le (1+\varepsilon)(\lambda +{C_2\over k\varepsilon })\|u\|^2.\cr}$$
As $\|\varphi u\|^2\ge \int_{X_{c_1}}|u|^2 \ge (1-\varepsilon
)\|u\|^2$~, we infer
$$Q^{k,m}_\Omega (\varphi u) \le {1+\varepsilon \over 1-\varepsilon
}\Big(\lambda +{C_2\over k\varepsilon }\Big)\|\varphi u\|^2.$$
If $\varepsilon $ is replaced by a suitable smaller number and $k$ taken
large enough, we obtain $Q^{k,m}_\Omega (v) \le (\lambda +\varepsilon
)\|v\|^2$ for all $v\in \varphi \cH^{k,m}_{\le \lambda ,X}$. Then
the right hand inequality in lemma 3.5 follows by the minimax principle.
\qed

Now, Corollary 1.17 easily computes the counting function $N^{k,m}_\Omega
$ for the eigenvalues~:
$$\lim_{\lambda \to 0_+} \lim_{k\to +\infty} k^{-n} N^{k,m}_\Omega
(\lambda) = {r\over n!}\int_{X(L_\chi,h_\chi,m)}(-1)^m \Big({\ii\over 2\pi }
\theta_{L_\chi,h_\chi})\Big)^n.$$
Applying this to the Witten complex $\cH^{k,\bullet}_{\le \lambda
,X}$, we easily infer the inequality of theorem 3.1, except that $c(L)$
is replaced by $c(L_\chi)$. However, up to now, the inequality is valid
for all $m\ge q$. Take the convex function $\chi $ equal to $0$ on
$]-\infty,c_0]$. Then
$$\Theta_{L_\chi,h_\chi} = {\ii\over 2\pi}\theta_{L_\chi,h_\chi} = 
\Theta_{L,h} + {\ii\over 2\pi}\ddbar (\chi \circ \psi)$$
coincides with $\Theta_{L,h}$ on $X_{c_0}$ and has at most $(p-1)+(q-1)$
negative eigenvalues on $X\ssm X_{c_0}$. Hence $X(L_\chi,h_\chi,m)=
X(L,h,m)$ for $m\ge p+q-1$ and $\Theta_{L_\chi,h_\chi} = \Theta_{L,h}$ 
on these sets. Theorem 3.1 is proved.\qed
\endproof

As a corollary, one obtains a general a priori estimate for the
Monge-Amp\`ere operator $(i\ddbar)^n$ on $q$-convex manifolds.

\claim{(3.6) Corollary: calculus inequalities} {Let $X$ be a strongly 
$q$-convex manifold and
$\varphi $ a $C^\infty$ function on $X$, weakly $p$-convex outside a
compact subset of $X$. For $\ell = 0,1,\ldots,n$, let $G_\ell$ be the open
set of points where $i\ddbar\varphi $ is non degenerate and admits
$\ell$ negative eigenvalues. Then for all $m\ge p=q-1$
$$\sum^n_{\ell=m} \int_{G_\ell}(i\ddbar\varphi)^m \hbox{~has the
sign of~} (-1)^m.$$}
\endclaim
\vskip-12pt

This result has been first obtained by Y.T.\ Siu [Siu90] for $q$-convex
domains in a Stein manifold. At that time, the $q$-convex case of the
inequalities was not yet available and Siu had to rely on a rather
sophisticated approximation argument of Stein manifolds by algebraic
varieties~; the proof could then be reduced to the compact case.

The general statement given above is in fact a direct consequence of
Theorem 3.1~: take for $L$ the trivial bundle $L=\cO_X$ equipped
with the metric defined by the weight $e^{-\varphi }$ and $E=\cO_X$. 
Since $H^m(X,L^k) = H^m(X,\cO_X)$ is independent of $k$ and
finite dimensional, Theorem 3.1 implies
$$k^n\sum^n_{\ell=m} \int_{G_\ell} (-1)^m (i\ddbar\varphi)^n\ge
\hbox{~constant~}- o(k^n)$$
for all $k\ge k_0$ and $m\ge p+q-1$, whence the result.
\qed 

\long\def\holomvect{%
\section{4}{Holomorphic Morse inequalities for vector bundles}

A natural question arising in connection with our Morse inequalities is
whether one can extend the inequalities to high tensor powers of a
vector bundle $W$ of rank${}\ge 2$. Since $W^{\otimes k}$ is decomposable
for $k\ge 2$ (e.g.\ $W^{\otimes 2}=S^2W\oplus \Lambda^2W)$ we are led to
consider only irreducible tensor powers of $W$, i.e.\ the irreducible
representations of the linear group $\GL(W)$. This is done by Getzler
[Get87], in the general framework of Lie group theory and
representations. As we are only dealing with the case of the full
linear group, we will give here an elementary presentation. We first
recall some ideas from the Borel-Weil theory and a special case of Bott's
formula [Bot57].

Let $V$ be a complex vector space of dimension $r$ and $F(V)$ the flag
manifold of $V$, i.e.\ the set of all $(r+1)$-tuples $z=(V_0,V_1,\ldots,V_r)$
with $V=V_0\supset V_1\supset \cdots \supset V_r = \{0\}$ and
codim$V_j=j$. On $F(V)$ we have canonical line bundles $Q_j$ such that
$$Q_{j,z} = V_{j-1}/V_j,~~1\le j\le r^.$$
For any $r$-tuple $(a_1,\ldots,a_r) \in \bZ^r$, we set
$$Q^a = Q^{a_1}_1\otimes \cdots \otimes Q^{a_r}_r.$$
As $\GL(V)$ acts equivariantly on $Q^a \to F(V)$, the spaces of
sections
$$\Gamma^aV = H^0\big(F(V),Q^a\big)\leqno(4.1)$$
are equipped with a natural $\GL(V)$ action. Observe that $Q^{(1,\ldots,
1)}$ is isomorphic to the trivial bundle $F(V)\times \det V$ (but of
course the action of $\GL(V)$ on $\det V$ is non trivial). To describe
$\Gamma^aV$, we can therefore assume that all $a_j$ are nonnegative.
Then any section $\sigma \in \Gamma^aV$ can be viewed as a polynomial
$P_\sigma (\xi_1,\ldots,\xi_r)$ on $(V^*)^r$ as follows~: if $\xi_1,\ldots,
\xi_r\in V^*$ are linearly independent, one can associate to $(\xi_j)$
the flag $z=(V_j)$ defined by $V_j = \xi^{-1}_1(0) \cap \cdots \cap \xi
^{-1}_j(0)$. Then $\xi_j$ induces a well defined linear form $\tilde
\xi_j$ on $Q_{j,z} = V_{j-1}/V_j$ and we set
$$P_\sigma (\xi_1,\ldots,\xi_r) = (\tilde \xi^{a_1}_1\otimes \cdots
\otimes \tilde \xi^{a_r}_r)\cdot \sigma (z).$$
It is clear that $P_\sigma $ remains locally bounded on a neighborhood
of the hypersurface $\det(\xi_1,\ldots,\xi_r) = 0\,$; therefore $P_\sigma $
extends to a polynomial on $(V^*)^r$ that is homogeneous of degree $a_j$
in the variable $\xi_j$. Also, neither the flag $z$ nor the linear
forms
$\tilde \xi_j$ are modified if we replace $\xi_j$ by $\xi
_j+\sum_{k<j} \lambda_{jk}\xi_k$. It follows that $P_\sigma $
satisfies the relation
$$P_\sigma (\xi_j+\sum_{k<j}\lambda_{jk}\xi_k) = P_\sigma (\xi_1,\ldots,
\xi_r),~~\forall \lambda_{jk}\in \bC~,$$
and conversely any polynomial $P$ of multidegree $(a_1,\ldots,a_r)$
satisfying this condition yields a (unique) section $\sigma \in \Gamma
^aV$. Hence $\Gamma^aV$ is the subspace of tensors in
$S^{a_1}V\otimes\cdots \otimes S^{a_r}V$ enjoying the above additional
antisymmetry properties. In particular we have
$$\eqalign{
S^kV &= \Gamma^{(k,0,\ldots,0)}V~,\cr
\Lambda^kV  &=\Gamma^{(1,\ldots,1,0,\ldots,0)}V,~~(k\hbox{~first
integers~}=1).\cr}$$
We will see soon that $\Gamma^aV=\{0\}$ unless $a_1\ge a_2\ge \cdots
\ge a_r$. The spaces $(\Gamma^aV)_{a_1\ge \cdots \ge a_r}$ can be seen to
be \emph{irreducible} representations of $\GL(V)$. As is well known in
representation theory, $(\Gamma^aV)$ is in fact the complete list of
irreducible representations of $\GL(V)$ up to isomorphism.

Assume now that $V$ is equipped with a Hermitian metric. Then any flag
$z^0\in F(V)$ is represented by an orthonormal basis $(v_1,\ldots,v_r)$ such
that $V^0_j = {\rm Vect}(v_{j+1},\ldots,v_r)$. Now $z^0$ is contained in the
affine chart of points $z=(V_j)$ with
$$V_j = {\rm Vect}(v_{j+1},\ldots,v_r),~~v_k = v_k+\sum_{j<k} z_{jk}v_j$$
where $(z_{jk})\in \bC^{n(n-1)/2}$ are the affine coordinates of $z$.
The canonical metric on $Q^a$ induced by $V$ has curvature
$$c(Q^a)_{z^0} = \sum_{1\le j<k\le n} (a_j-a_k)dz_{jk}\wedge d\overline
z_{jk}\leqno(4.2)$$
(we leave this computation as an exercise to the reader). By homogeneity, we
see that $Q^a$ is positive as soon as $a_1>a_2>\cdots >a_r$. On the
other hand, when $a_{j-1}<a_j$~, we see that $Q^a$ is negative along the
$\bP^1$ line in $F(V)$ obtained by fixing all $V_k = V^0_k$ except~$V_j$.
Therefore $\Gamma^aV=H^0\big(F(V),Q^a\big) = \{0\}$ in this
case.

Assume from now on that $a_1\ge \cdots \ge a_r$, and more specifically
that
$$a_1 = \cdots = a_{s_1} > a_{s_1+1} = \cdots = a_{s_2} > \cdots >
a_{s_{m-1}+1} = \cdots = a_{s_m}~,$$
where $s_m = r$. As $Q_{j+1,z} \otimes \cdots \otimes Q_{k,z} \simeq
\det(V_j/V_k)$, we see that $Q^a$ is the pull back of the bundle
$$Q^a_s = \det(V/V_{s_1})^{a_{s_1}}\otimes\cdots\otimes
\det(V_{s_{m-1}}/V_{s_m})^{a_{s_m}}$$
over the manifold $F_s(V)$ of partial flags
$$V\supset V_{s_1} \supset \cdots \supset V_{s_m} = \{0\}~,$$
via the obvious projection $\pi_s : F(V)\to F_s(V)$. On $F_s(V)$ we
have a formula completely analogous to (4.2), where the only indices
$(j,k)$ involved are those for which $a_j>a_k$. Thus $Q^a_s$ is ample
and $Q^a=\pi^*_sQ^a_s$, in particular
$$H^0\big(F_s(V),Q^a_s\big) = H^0\big(F(V),Q^a\big)=\Gamma^aV.\leqno(4.3)$$

Now let $E$, $W$ be holomorphic vector bundles over a compact manifold $X$
and let
$$n=\dim_\bC X,~~r=\rank E,~~r'=\rank W.$$
We want to get asymptotic estimates for the dimension of cohomology
groups 
$$H^q(X,\Gamma^{ka}W\otimes E)\qquad\hbox{as $k\to +\infty$}.
$$
For this, we introduce the flag bundle $$F_s(W) \to X~,$$
where $s=(s_1,\ldots,s_m)$ is
defined as above, and we consider the universal line bundle $Q^a_s$ over
$F_s(W)$. As $Q^a_s$ is ample along the fibers of $\pi_s~: F_s(W) \to X$~,
the higher direct images
$$R^q(\pi_s)_*  (Q^{ka}_s \otimes \pi^*_s E),~~~~q\ge 1$$
vanish for $k\ge k_0.$ By (4.3) we get
$$(\pi_s)_*  (Q^{ka}_s \otimes \pi^*_sE) = \Gamma^{ka}W\otimes E.$$
The Leray spectral sequence gives the isomorphism
$$H^q(X,\Gamma^{ka}W\otimes E) \simeq H^q(Y,Q^{ka}_s \otimes \pi^*_s
E),\qquad Y = F_s(W),\leqno(4.4)$$
and we are reduced to applying Morse inequalities to tensor powers of
the line
bundle \hbox{$L=Q^a_s$}. We still need a formula for the curvature of $Q^a_s$
with the metric induced by a given Hermitian metric on $W$.
Let $z^0\in F_s(W_{x^0})$ be a point in $F_s(W)$. Choose a holomorphic
frame $(w_1,\ldots,w_r)$ of $W$ such that the flag $z^0$ is given by the
basis $(w_1(x^0),\ldots,w_r(x^0))$ (supposed to be orthonormal). Assume also
$(w_\lambda)$ chosen such
that $Dw_\lambda (x^0)=0$ and consider the curvature tensor
$$c(W)_{x^0}=\sum_{{1\le j,k\le n}\atop{1\le\lambda,\mu\le r}}
c_{jk\lambda\mu}dx_j\wedge dx_k\otimes w_\lambda^*\otimes w_\mu.$$
It can be shown that the associated curvature of $Q_s^a$ is
$$c(Q_s^a)_{z^0} = \sum_{{1\le j,k\le n}\atop{1\le \lambda \le r}}
a_\lambda c_{jk\lambda \lambda }dx_j\wedge dx_k + \sum_{a_\lambda >a_\mu}
(a_\lambda -a_\mu)\,dz_{\lambda \mu }\wedge d\overline z_{\lambda \mu}$$
where $(z_{\lambda \mu })$ are the affine coordinates along the fiber
$F_s(W_{x^0})$. Finally, let $N(s)$ be the dimension of the fibers
$F_s(W_x)$. Using the isomorphism (4.4), the strong Morse inequality
becomes
$$\sum^q_{m=0}(-1)^{q-m} \dim H^m(X,\Gamma^{ka}W\otimes E) \le
r{k^{n+N(s)}\over (n+N(s))!} \int_{Y(Q^a_s,\le q)}
\Theta_{Q^a_s}^{n+N(s)}+o(k^{n+N(s)}).$$
The most interesting case is the case of symmetric powers $S^kW$. Then
we simply have $F_s(W) =\bP(W):=P(W^*)$, $N(s) = r-1$, $Q^a_s = O_{\bP(W)}(1)$.}
\ifnum \doextras=1 \holomvect \fi

\chapter{2}{Approximation of currents and intersection theory}

\section{0}{Introduction}

Many concepts described in this chapter (e.g.\ pseudo-effectivity) are
quite general and make sense on an arbitrary compact complex 
manifold~$X$ -- no projective or K\"ahler assumption is needed. 
In this general context, it is better to
work with $\ddbar$-cohomology classes instead of De Rham cohomology classes:
we define the {\it Bott-Chern cohomology} of~$X$ to be
$$H_\BC^{p,q}(X,\bC)=\big\{d\hbox{-closed $(p,q)$-forms}\}/
\big\{\ddbar\hbox{-exact $(p,q)$-forms}\}.\leqno(0.1)$$
It is easily shown that these cohomology groups are finite dimensional 
and can be computed either with spaces of smooth forms or with currents$\,$; 
in fact, they can be computed by certain complexes of sheaves of forms or
currents that both provide fine resolutions of the same sheaves of holomorphic
or anti-holomorphic forms. Our statement therefore follows formally from 
general results of sheaf theory. Also, finiteness can be obtained by the usual
Cartan-Serre proof based on Montel's theorem for \v{C}ech cohomology. In both cases,
the quotient topology of $\smash{H_\BC^{p,q}(X,\bC)}$ induced by
the Fr\'echet topology of smooth forms or by the weak topology of
currents is Hausdorff. Clearly, $H_\BC^\bu(X,\bC)$ is a bigraded
algebra, and it is trivial by definition that there are always
canonical morphisms
$$
H^{p,q}_\BC(X,\bC)\to H^{p,q}_\dbar(X,\bC),\qquad
\bigoplus_{p+q=k}H^{p,q}_\BC(X,\bC)\to H^k_\DR(X,\bC).
\leqno(0.2)
$$
By Hodge decomposition and by the well-known $\ddbar$-lemma of K\"ahler 
geometry, these morphisms are isomorphisms when $X$ is K\"ahler; especially,
we get a canonical algebra isomorphism
$$H_\DR^\bu(X,\bC)\simeq\bigoplus_{p,q}H^{p,q}_\dbar(X,\bC)\qquad
\hbox{if $X$ is K\"ahler}.\leqno(0.3)$$ 
We will see in Section~5 (Remark~5.15) that this is true more generally if $X$ is in 
the Fujiki  class $\cC$, i.e., the class of manifolds bimeromorphic
to K\"ahler manifolds.

\section{1}{Pseudo-effective line bundles and singular Hermitian metrics}

Let $L$ be a holomorphic line bundle on a compact complex manifold~$X$. It is
important for many applications to allow singular Hermitian metrics.

\claim{(1.1) Definition} {A singular Hermitian metric $h$ on $L$ is a Hermitian
metric such that, for any trivialisation $L_{|U}\simeq U\times\bC$, the metric is
given by $h=e^{-\varphi}$, $\varphi\in L^1_\loc(U)$.}
\endclaim

\noindent The curvature tensor 
$$\Theta_{L,h}={\ii\over 2\pi}\ddbar\varphi=-{\ii\over 2\pi}\ddbar\log h
\leqno(1.2)
$$
can then be computed in the sense of distributions, and defines in this way
a (global) closed $(1,1)$-current on~$X$. It defines a (real) cohomology
class $\{\Theta_{L,h}\}\in H^{1,1}_\BC(X,\bC)$ which is mapped to the first
Chern class $c_1(L)$ by the canonical morphisms $(0.2)$. We will therefore
still denote this Bott-Chern class by $c_1(L)$.
The positive case is of special interest.

\claim{(1.3) Definition} We say that $L$ pseudo-effective if
$c_1(L)\in H_\BC^{1,1}(X,\bC)$ is the cohomology class of some closed
positive current~$T$, i.e.\ if $L$ can be equipped with a singular
Hermitian metric $h$ with $T=\Theta_{L,h}\ge 0$ as a
current, in other words, if the weight functions $\varphi$ can be
chosen to be plurisubharmonic on each trivialization open set~$U$.
\endclaim

\noindent The locus where $h$ has singularities turns out to be 
extremely important. One way is to introduce multiplier ideal sheaves
following A.~Nadel [Nad89]. The main idea actually goes back to the
fundamental works of Bombieri [Bom70] and H.~Skoda [Sko75].

\claim{(1.4) Definition} Let $\varphi$ be a psh $($plurisubharmonic$)$
function on an open subset \hbox{$\Omega\subset X$}. To $\varphi$ we
associate the ideal subsheaf $\cI(\varphi)\subset\cO_\Omega$ of germs of
holomorphic functions $f\in\cO_{\Omega,x}$ such that $|f|^2e^{-\varphi}$
is integrable with respect to the Lebesgue measure in some local
coordinates near~$x$.
\endclaim

The zero variety $V(\cI(\varphi))$ is thus the set of points in
a neighborhood of which $e^{-\varphi}$ is non integrable. The following
result implies that this is always an analytic set.

\claim{(1.5) Proposition {\rm([Nad89])}} For any psh function
$\varphi$ on~$\Omega\subset X$, the sheaf
$\cI(\varphi)$ is a coherent sheaf of ideals over~$\Omega$. Moreover,
if $\Omega$ is a bounded Stein open set, the sheaf $\cI(\varphi)$ is
generated by any Hilbert basis of the $L^2$~space $\cH^2(\Omega,\varphi)$
of holomorphic functions $f$
on $\Omega$ such that $\int_\Omega|f|^2e^{-\varphi}\,d\lambda<+\infty$.
\endclaim
%
\proof{Proof} Since the result is local, we may assume that $\Omega$ is a
bounded pseudoconvex open set in $\bC^n$.
By the strong noetherian property of coherent sheaves, the family of
sheaves generated by finite subsets of $\cH^2(\Omega,\varphi)$ has
a maximal element on each compact subset of~$\Omega$, hence
$\cH^2(\Omega,\varphi)$
generates a coherent ideal sheaf $\cJ\subset\cO_\Omega$. It is clear that
$\cJ\subset\cI(\varphi)$; in order to prove the equality,
we need only check that $\cJ_x+\cI(\varphi)_x\cap
\gm^{s+1}_{\Omega,x}=\cI(\varphi)_x$ for every integer $s$, in
view of the Krull lemma. Let $f\in\cI(\varphi)_x$ be defined
in a neighborhood $V$ of $x$ and let $\theta$ be a cut-off function
with support in $V$ such that $\theta=1$ in a neighborhood of $x$.
We solve the equation $\dbar u=g:=\dbar(\theta f)$ by means of H\"ormander's
$L^2$ estimates [H\"or65, AV65], applied with the strictly psh weight
$$\wt\varphi(z)=\varphi(z)+(n+s)\log|z-x|^2+|z|^2.$$
We get a solution $u$ such that
$\int_\Omega |u|^2e^{-\varphi}|z-x|^{-2(n+s)}d\lambda<\infty$, thus
$F=\theta f-u$ is holomorphic, $F\in \cH^2(\Omega,\varphi)$ and
$f_x-F_x=u_x\in\cI(\varphi)_x\cap\gm_{\Omega,x}^{s+1}$.
This proves the coherence. Now, $\cJ$ is generated by any Hilbert basis
of $\cH^2(\Omega,\varphi)$, because it is well-known that the space of
sections of any coherent sheaf is a Fr\'echet space, therefore
closed under local $L^2$ convergence.\qed
\endproof

Another important way of measuring singularities is via Lelong numbers --
a natural generalization of the concept of multiplicity to psh functions.
Recall that the Lelong number of a function $\varphi\in\Psh(\Omega)$ 
at a point $x_0$ is defined to be
$$
\nu(\varphi,x_0)=\liminf_{z\to x_0}{\varphi(z)\over\log|z-x_0|}=
\lim_{r\to 0_+}{\sup_{B(x_0,r)}\varphi\over \log r}.\leqno(1.6)
$$
In particular, if $\varphi=\log|f|$ with $f\in\cO(\Omega)$, then
$\nu(\varphi,x_0)$ is equal to
the vanishing order
$$\ord_{x_0}(f)=\sup\{k\in\bN\,;D^{\alpha}f(x_0)=0,~
\forall|\alpha|<k\}.$$
The link with multiplier ideal sheaves is provided by the following standard
result due to Skoda [Sko72].

\claim{(1.7) Lemma} Let $\varphi$ be a psh function
on an open set $\Omega$ and let $x\in\Omega$.
\smallskip
\item{\rm (a)} If $\nu(\varphi,x)<2$, then $e^{-\varphi}$ is Lebesgue integrable on
a neighborhood of~$x$, in particular $\cI(\varphi)_x=\cO_{\Omega,x}$.
\smallskip
\item{\rm (b)} More generally, if $\nu(\varphi,x)\ge 2(n+s)$ for some integer $s\ge 0$, then
$$e^{-\varphi}\ge c|z-x|^{-2n-2s},\qquad c>0$$ 
in a neighborhood of $x$, and
$\cI(\varphi)_x\subset\gm_{\Omega,x}^{s+1}$, where $\gm_{\Omega,x}$ is the
maximal ideal of $\cO_{\Omega,x}$. In particular $e^{-\varphi}$ is non integrable at $x$ if
$\nu(\varphi,x)\ge 2n$.
\smallskip
\item{\rm (c)} The zero variety $V(\cI(\varphi))$ of $\cI(\varphi)$
satisfies
$$V_{2n}(\varphi)\subset V(\cI(\varphi))\subset E_2(\varphi)$$
where $E_c(\varphi)=\{x\in X\,;\,\nu(\varphi,x)\ge c\}$ is the $c$-upperlevel
set of Lelong numbers of~$\varphi$.
\vskip0pt
\endclaim

\noindent
The only non trivial part is 1.7~(a); the proof relies on the Bochner-Martinelli 
representation formula for $T={\ii\over\pi}\ddbar\varphi$ (see [Sko72]).
One should observe that 1.7~(a) (resp.\ (b)) is optimal, as one can see by taking 
$\varphi(z)=\lambda\log|z_1|$, resp.\ $\varphi(z)=\lambda\log|z|$, on $\Omega=\bC^n$.

\section{2}{Hermitian metrics with minimal singularities and 
analytic Zariski decomposition}

We show here by a general ``abstract'' method that a pseudo-effective
line bundle always has a Hermitian metric $h_{\min}$ with minimal
singularities among those with nonnegative curvature $\Theta_{L,h}\ge 0$
in the sense of currents. The following definition was introduced in [DPS01].

\claim{(2.1) Definition} Let $L$ be a pseudo-effective line bundle
on a compact complex manifold~$X$. Consider two Hermitian metrics
$h_1$, $h_2$ on $L$ with curvature \hbox{$\Theta_{L,h_j}\ge 0$}
in the sense of currents.
\smallskip
\item{\rm(a)} We will write $h_1\preccurlyeq h_2$, and say that $h_1$
is less singular than $h_2$, if there exists a constant $C>0$ such that
$h_1\le C h_2$.
\smallskip
\item{\rm(b)} We will write $h_1\sim h_2$, and say that $h_1$, $h_2$ are
equivalent with respect to singularities, if there exists a constant
$C>0$ such that $C^{-1}h_2\le h_1\le C h_2$.
\vskip0pt
\endclaim
%
\noindent

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

\claim{(2.2) Theorem} For every pseudo-effective line bundle $L$ over
a compact complex manifold $X$, there exists up to equivalence of
singularities a unique class of Hermitian metrics $h$ with minimal
singularities such that $\Theta_{L,h}\ge 0$.
\endclaim
%
\proof{Proof} The proof is almost trivial. We fix once for all a smooth
metric $h_\infty$ (whose curvature is of random sign and signature),
and we write singular metrics of $L$ under the form $h=h_\infty e^{-\psi}$.
The condition $\Theta_{L,h}\ge 0$ is equivalent to
${\ii\over2\pi}\ddbar\psi\ge -u$ where $u=\Theta_{L,h_\infty}$. This
condition implies that $\psi$ is plurisubharmonic up to the addition
of the weight $\varphi_\infty$ of $h_\infty$, and therefore locally
bounded from above.  Since we are concerned with metrics only up to
equivalence of singularities, it is always possible to adjust
$\psi$ by a constant in such a way that $\sup_X\psi=0$. We now set
$$
h_{\min}=h_\infty e^{-\psi_{\min}},\qquad \psi_{\min}(x)=\sup_\psi\psi(x)
$$
where the supremum is extended to all functions $\psi$ such that
$\sup_X\psi=0$ and ${\ii\over2\pi}\ddbar\psi\ge -u$. By standard
results on plurisubharmonic functions (see Lelong [Lel69]),
$\psi_{\min}$ still satisfies ${\ii\over2\pi}\ddbar\psi_{\min}\ge -u$
(i.e.\ the weight $\varphi_\infty+\psi_{\min}$ of $h_{\min}$ is
plurisubharmonic), and $h_{\min}$ is obviously the metric with
minimal singularities that we were looking for. [In~principle one
should take the upper semicontinuous regularization $\psi_{\min}^*$
of $\psi_{\min}$ to really get a plurisubharmonic weight, but since
$\psi_{\min}^*$ also participates to the upper envelope, we obtain
here $\psi_{\min}=\psi_{\min}^*$ automatically].
\qed
\endproof

\claim{(2.3) Remark} \rm In general, the supremum
$\psi=\sup_{j\in I}\psi_j$ of a
locally dominated family of plurisubharmonic functions $\psi_j$ is
not plurisubharmonic strictly speaking, but its ``upper semi-continuous
regularization'' $\psi^*(z)=\limsup_{\zeta\to z}\psi(\zeta)$ is
plurisubharmonic and coincides almost everywhere with $\psi$, with
$\psi^*\ge\psi$. However, in the context of (2.3), $\psi^*$ still
satisfies $\psi^*\le 0$ and ${\ii\over2\pi}\ddbar\psi\ge -u$, hence
$\psi^*$ participates to the upper envelope. As a consequence, we have
$\psi^*\le \psi$ and thus $\psi=\psi^*$ is indeed plurisubharmonic.
Under a strict
positivity assumption, namely if $L$ is a big line bundle (i.e.\ the
curvature can be taken to be strictly positive in the sense of
currents, see Definition (3.3~d) and Theorem (3.4~b), then $h_{\min}$ can be shown
to possess some regularity properties. The reader may consult
[BmD09] for a rather general (but certainly non trivial) proof that
$\psi_{\min}$ possesses locally bounded
second derivatives $\partial^2\psi_{\min}/\partial z_j\partial\overline z_k$
outside an analytic set $Z\subset X\,$; in other words, $\Theta_{L,h_{\min}}$
has locally bounded coefficients on~$X\ssm Z$.\qed
\endclaim

\claim{(2.4) Definition} Let $L$ be a pseudo-effective line bundle.
If $h$ is a singular Hermitian metric such
that $\Theta_{L,h}\ge 0$ and
$$
H^0(X,mL\otimes\cI(h^{\otimes m}))\simeq H^0(X,mL)\qquad
\hbox{for all $m\ge 0$},
$$
we say that $h$ is an {\it analytic Zariski decomposition} of~$L$.
\endclaim
%
In other words, we require that $h$ has singularities so mild that
the vanishing conditions prescribed by the multiplier ideal sheaves
$\cI(h^{\otimes m})$ do not kill any sections of $L$ and its multiples.

\claim{(2.5) Exercise} {\rm A special case is when there is an
isomorphism $pL=A+E$ where $A$ and $E$ are effective divisors such
that $H^0(X,mpL)=H^0(X,mA)$
for all $m$ and $\cO(A)$ is generated by sections. Then $A$ possesses
a smooth Hermitian metric~$h_A$, and this metric defines a singular Hermitian
metric $h$ on $L$ with poles ${1\over p}E$ and curvature
${1\over p}\Theta_{A,h_A}+{1\over p}[E]$. Show that this metric $h$
is an analytic Zariski decomposition.\\
{\it Note}: when $X$ projective and there is a decomposition $pL=A+E$
with $A$ nef (see (I~2.9)), $E$ effective and
$H^0(X,mpL)=H^0(X,mA)$ for all $m$, one says that the $\bQ$-divisor
equality $L={1\over p}A+{1\over p}E$ is an
{\it algebraic Zariski decomposition} of~$L$. It can be shown that Zariski
decompositions exist in dimension $2$, but in higher dimension they do not
exist in general.}\qed
\endclaim

\claim{(2.6) Theorem} The metric $h_{\min}$ with minimal singularities
provides an analytic Zariski decomposition.
\endclaim
\vskip-6pt

\noindent It follows that an analytic Zariski decomposition always exists
(while algebraic decompositions do not exist in general, especially in
dimension $3$ and more).

\proof{Proof} Let $\sigma\in H^0(X,mL)$ be any section.
Then we get a singular metric
$h$ on $L$ by putting $|\xi|_h=|\xi/\sigma(x)^{1/m}|$ for $\xi\in L_x$, and
it is clear that $|\sigma|_{h^m}=1$ for this metric. Hence
$\sigma\in H^0(X,mL\otimes\cI(h^{\otimes m}))$, and a fortiori
$\sigma\in H^0(X,mL\otimes\cI(h^{\otimes m}_{\min}))$ since $h_{\min}$ is
less singular than~$h$.\qed
\endproof

\section{3}{Description of the positive cones (K\"ahler and projective 
cases)\kern-2pt}

Let us recall that an integral cohomology class in
$H^2(X,\bZ)$ is the first Chern class of a holomorphic (or algebraic)
line bundle if and only if it lies in the
{\it Neron-Severi} group
$$\NS(X)=\Ker\big(H^2(X,\bZ)\to H^2(X,\cO_X)\big)\leqno(3.1)$$
(this fact is just an elementary consequence of the exponential exact
sequence\bbreak $0\to\bZ\to\cO\to\cO^*\to 0$). If $X$ is compact
K\"ahler, as we will suppose from now on in this section,
this is the same as saying that the class is of type $(1,1)$
with respect to Hodge decomposition.

Let us consider the real vector space $\smash{\NS_\bR(X)=\NS(X)\otimes_\bZ\bR}$,
which can be viewed as a subspace of the space $H^{1,1}(X,\bR)$ of real
$(1,1)$ cohomology classes. Its dimension is by definition the Picard number
$$\rho(X)=\rank_\bZ \NS(X)=\dim_\bR \NS_\bR(X).\leqno(3.2)$$
We thus have $0\le\rho(X)\le h^{1,1}(X)$, and the example of complex tori shows
that all intermediate values can occur when $n=\dim X\ge 2$.

The positivity concepts for line bundles considered in section I~2.B and
II~1 possess in fact natural generalizations to
$(1,1)$ classes which are not necessarily integral or rational~-- and
this works at least in the category of compact K\"ahler manifolds
(in fact, by using Bott-Chern cohomology, one could even extend these
concepts to arbitrary compact complex manifolds).

\claim{(3.3) Definition} Let $(X,\omega)$ be a compact K\"ahler manifold.
\vskip3pt
{\plainitemindent=7mm
\item{\rm(a)} The K\"ahler cone is the set
$\cK\subset H^{1,1}(X,\bR)$ of cohomology classes $\{\omega\}$
of K\"ahler forms. This is an {\rm open convex cone}.
\vskip3pt
\item{\rm(b)} The closure $\overline{\cK}$ of the K\"ahler cone consists
of classes $\{\alpha\}\in H^{1,1}(X,\bR)$ such that for every
$\varepsilon>0$ the sum $\{\alpha+\varepsilon\omega\}$
is K\"ahler, or equivalently, for every $\varepsilon>0$, there exists a
smooth function $\varphi_\varepsilon$ on $X$ such that
$\alpha+i\ddbar\varphi_\varepsilon\ge -\varepsilon\omega$.
We say that $\overline{\cK}$ is the cone of {\rm nef} $(1,1)$-classes.
\vskip3pt
\item{\rm(c)}The pseudo-effective cone is the set
$\cE\subset H^{1,1}(X,\bR)$ of cohomology classes $\{T\}$ of closed positive
currents of type $(1,1)$. This is a {\rm closed convex cone}.
\vskip3pt
\item{\rm(d)}The interior $\cE^\circ$ of $\cE$ consists of classes
which still contain a closed positive current after one subtracts
$\varepsilon\{\omega\}$ for $\varepsilon>0$ small, in other words,
they are classes of closed $(1,1)$-currents $T$ such that
$T\ge \varepsilon\omega$. Such a current will be called a {\rm K\"ahler
current}, and we say that $\{T\}\in H^{1,1}(X,\bR)$ is a {\rm big
$(1,1)$-class}.
\vskip0pt}
\endclaim
%
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[closure of $\cK$]}$\ELTX
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[closed]}$\ELTX
\LabelTeX 27 10 $\cE^\circ=\hbox{big cone in $H^{1,1}(X,\bR)$
[interior of $\cE$]}$\ELTX
\EndFig
\vskip8mm

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

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

In case $X$ is projective, all Chern classes $c_1(L)$ of line bundles
lie by definition in $\NS(X)$, and likewise,
all classes of real divisors $D=\sum c_jD_j$, $c_j\in\bR$, lie
in $\NS_\bR(X)$. In order to deal with such {\it algebraic classes},
we therefore introduce the intersections
$$\cK_{\NS} = \cK\cap \NS_\bR(X),\qquad \cE_{\NS}=\cE\cap \NS_{\bR}(X),$$
and refer to classes of $H^{1,1}(X,\bR)$ not contained in $\NS_\bR(X)$
as {\it transcendental classes}.

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\vskip8mm

A very important fact is that all four cones $\cK_{\NS}$, $\cE_{\NS}$,
$\overline\cK_{\NS}$, $\cE^\circ_{\NS}$ have simple algebraic
interpretations.

\claim{(3.4) Theorem} Let $X$ be a projective manifold. Then
{\plainitemindent=7mm
\vskip3pt
\item{\rm (a)} $\cK_{\NS}$ is equal to the open cone $\Amp(X)$
generated by classes of
{\rm ample} $($or very ample$)$ divisors $A$ $($recall that a divisor
$A$ is said to be very ample if the linear system $H^0(X,\cO(A))$
provides an embedding of $X$ in projective space$)$.
\vskip3pt
\item{\rm (b)} The interior $\cE_{\NS}^\circ$ is the cone $\Bigdiv(X)$
generated by classes of {\rm big} divisors, namely divisors $D$ such that
$h^0(X,\cO(kD))\ge c\,k^{\dim X}$ for $k$ large.
\vskip3pt
\item{\rm (c)} $\cE_{\NS}$ is the closure $\overline{\Eff(X)}$
of the cone generated by classes
of {\rm effective} divisors, i.e.\ divisors $D=\sum c_jD_j$, $c_j\in\bR_+$.
\vskip3pt
\item{\rm (d)} The closed cone $\overline\cK_{\NS}$ consists of the closure
$\overline{\Nef(X)}$
of the cone generated by {\rm nef} divisors $D$ $($or nef line bundles $L)$,
namely effective integral divisors $D$ such that \hbox{$D\cdot C\ge 0$} for
every curve $C$, also equal to $\overline{\Amp(X)}$.\vskip0pt
}
\endclaim
%
In other words, the terminology ``nef'', ``big'',
``pseudo-effective'' used for classes of the full transcendental cones
appear to be a natural extrapolation of the algebraic case.

\proof{Proof} First notice that since all of our cones $\cC$ have non
empty interior in $\NS_\bR(X)$ (which is a rational vector space in terms
of a basis of elements in $H^2(X,\bQ)$), the rational points $\cC_\bQ:=
\cC\cap \NS_{\bQ}(X)$, $\NS_{\bQ}(X)=\NS(X)\otimes_\bZ\bQ$,
are dense in each of them.
\smallskip
\noindent
(a) is therefore just Kodaira's embedding theorem when we look at
rational points, and properties (b) and (d) are obtained easily by
passing to the closure of the open cones. We will now give details of
the proof only for (b) which is possibly slightly more involved.

By looking at points of $\cE^\circ_\bQ=\cE^\circ\cap\NS_{\bQ}(X)$ and
multiplying
by a denominator, it is enough to check that a line bundle $L$ such that
$c_1(L)\in\cE^\circ$ is big. However, this means that $L$ possesses a
singular Hermitian metric $h_L$ such that $\Theta_{L,h_L}\ge\varepsilon
\omega$ for some K\"ahler metric~$\omega$. For some integer $p_0>0$, we can then
produce a singular Hermitian metric with positive curvature and with a
given logarithmic pole
$h_L^{p_0}e^{-\theta(z)\log|z-x_0|^2}$ in a neighborhood of every point $x_0\in X$
(here $\theta$ is a smooth cut-off function supported on a neighborhood
of~$x_0$). Then H\"ormander's $L^2$ existence theorem [H\"or65, AV65] can 
be used to produce sections of $L^k$ which generate all jets of order
$(k/p_0)-n$ at points~$x_0$, so that $L$ is big.

Conversely, if $L$ is big and $A$ is a (smooth) very ample divisor,
the exact sequence $0\to\cO_X(kL-A)\to\cO_X(kL)\to\cO_A(kL_{\restriction A})
\to 0$ and the estimates $h^0(X,\cO_X(kL))\ge ck^n$,
$h^0(A,\cO_A(kL_{\restriction A}))=O(k^{n-1})$ imply that
$\cO_X(kL-A)$ has a section for $k$ large, thus $kL-A\equiv E$ for some
effective divisor~$E$. This means that there exists a singular metric
$h_L$ on $L$ such that
$$\Theta_{L,h_L}={1\over k}
\Big(\Theta_{A,h_A}+[E]\Big)\ge{1\over k}\omega$$
where $\omega=\Theta_{A,h_A}$, hence
$c_1(L)\in\cE^\circ$.\hfill\qed
\endproof

\claim{(3.5) Corollary} If $L$ is nef, then $L$ is big $($i.e.\
$\kappa(L)=n)$ if and only if $L^n>0$. Moreover, if $L$ is nef and big,
then for every $\delta>0$, $L$ has a singular metric $h=e^{-\varphi}$
such that $\,\max_{x\in X}\nu(\varphi,x)\le\delta$ and
$\ii\Theta_{L,h}\ge\varepsilon\,\omega$ for some $\varepsilon>0$.
The metric $h$ can be chosen to be smooth on the complement of a fixed
divisor $E$, with logarithmic poles along~$E$.
\endclaim
%
\proof{Proof} By (I~2.10) and the Riemann-Roch formula, we have
$$h^0(X,kL)=\chi(X,kL)+o(k^n)=k^nL^n/n!+o(k^n),$$
whence the first statement. By the proof of Theorem 3.4~(b), there exists
a singular metric $h_1$ on $L$ such that
$${\ii\over 2\pi}\Theta_{L,h_1}={1\over k}
\Big({\ii\over2\pi}\Theta_{A,h_A}+[E]\Big)\ge{1\over k}\omega,
\qquad \omega={\ii\over2\pi}\Theta_{A,h_A}.$$
Now, for every $\varepsilon>0$, there is a smooth metric $h_\varepsilon$
on $L$ such that ${\ii\over2\pi}\Theta_{L,h_\varepsilon}\ge-\varepsilon
\omega$. The convex combination
of metrics $h'_\varepsilon=h_1^{k\varepsilon}h_\varepsilon^{1-k\varepsilon}$
is a singular metric with poles along $E$ which satisfies
$${\ii\over2\pi}\Theta_{L,h'_\varepsilon}\ge \varepsilon(\omega+[E])-
(1-k\varepsilon)\varepsilon\omega\ge k\varepsilon^2\omega.$$
Its Lelong numbers are $\varepsilon\nu(E,x)$ and they can be made
smaller than $\delta$ by choosing $\varepsilon>0$ small.\qed
\endproof
%
We still need a few elementary facts about the numerical dimension of nef
line bundles.

\claim{(3.6) Definition} Let $L$ be a nef line bundle on a compact
K\"ahler manifold~$X$. One defines the numerical dimension of $L$ to be
$$\nd(L)=\max\big\{k=0,\ldots,n\,;\,c_1(L)^k\ne0~\hbox{{\rm in}}~
H^{2k}(X,\bR)\big\}.$$
\endclaim
Notice that if $L$ is nef, each power $c_1(L)^k$ can be represented
by a closed positive current $\Theta_k\in c_1(L)^k$ obtained as a weak 
limit of powers of smooth positive forms
$$\Theta_k=\lim_{m\to+\infty}\Big(\alpha+{1\over m}\omega
+\ddbar\varphi_m\Big)^k,\qquad \alpha\in c_1(L).$$
Such a weak limit exists since $\int_X\big(\alpha+{1\over m}\omega
+\ddbar\varphi_m\big)^k\wedge\omega^{n-k}$ is uniformly bounded
as $m\to+\infty$. Then we see that
$$
\int_X c_1(L)^k\wedge\omega^{n-k}=
\int_X \Theta_k\wedge\omega^{n-k}>0~~\Longleftrightarrow~~
\Theta_k\ne 0~~\Longleftrightarrow~~c_1(L)^k\ne 0.
$$
By Corollary 3.5, we have $\kappa(L)=n$ if and only if $\nd(L)=n$.
In general, we merely have an inequality.

\claim{(3.7) Proposition} If $L$ is a nef line bundle on a compact
K\"ahler manifold $(X,\omega)$, then $\kappa(L)\le\nd(L)$.
\endclaim
%
\proof{Proof} We consider arbitrary irreducible analytic subsets 
$Z\subset X$ and prove by induction on $p=\dim Z$ that
$\kappa(L_{|Z})\le\nd(L_{|Z})$ where $\nd(L_{|Z})$ is the supremum
of all integers $k$ such that $c_1(L_{|Z})^k\ne 0$, i.e.\ 
$\int_X [Z]\wedge c_1(L)^k\wedge\omega^{p-k}>0$. This will prove our
statement when $Z=X$, $p=n$.
The statement is trivial if~$p=0$, so we suppose now that $p>0$. 
We can also assume that $r=\kappa(L_{|Z})>0$, otherwise there is nothing 
to prove. This implies that a sufficient large 
multiple $m_0L$ has at least two independent sections $\sigma_0$, 
$\sigma_1$ on~$Z$.
Consider the linear system $|a_0\sigma_0+a_1\sigma_1|$, $a=[a_0:a_1]\in
\bP^1_\bC$, and take
$Y=Y_a\subset Z$ to be an irreducible component of the divisor of
$\sigma_a:=a_0\sigma_0+a_1\sigma_1$
which is not a fixed component when $a$ varies. For $m$ sufficiently
divisible, $\Phi_{mL_{|Z}}$ has rank $r$ at a generic (smooth) point of~$Z$, 
hence the rank of
$\smash{(\Phi_{mL_{|Z}})_{|Y}}$ is${}\ge r':=\min(r,p-1)$ if $a\in\bP^1_\bC$ is 
itself generic. A fortiori $\rank(\Phi_{mL_{|Y}})\ge r'$
(we may even have sections on~$Y$ which do not extend to~$Z$).
By the induction hypothesis we find
$$
\int_X [Y]\wedge c_1(L)^{r'}\wedge\omega^{p-1-r'}>0.
$$
Now, we use the fact that $[Z]\wedge c_1(m_0L)
-[Y]$ can be represented by an effective cycle (the sum of all
components${}\ne Y$ in the divisor of our generic section $\sigma_a$).
This implies
$$
\int_X [Z]\wedge c_1(L)^{r'+1}\wedge\omega^{p-1-r'}\ge
{1\over m_0}\int_X [Y]\wedge c_1(L)^{r'}\wedge\omega^{p-1-r'}>0.
$$
If $r=p$, we have $r'=p-1$, hence $r'+1=r$ and we are done. If $r<p$, we have
$r'=r$ and then we use the obvious inequality $\alpha\le C_0\omega$ for
some representative $\alpha\in c_1(L)$ and some $C_0>0$ to conclude that
$$
\int_X [Z]\wedge c_1(L)^r\wedge\omega^{p-r}\ge
{1\over C_0}\int_X [Z]\wedge c_1(L)^{r+1}\wedge\omega^{p-1-r}>0.
\eqno\square
$$
\endproof
%
\claim{(3.8) Remark} \rm It may happen that $\kappa(L)<\nd(L)$: take e.g.
$$L\to X=X_1\times X_2$$
equal to the total tensor product of an ample line bundle $L_1$ on a
projective manifold $X_1$ and of a unitary flat line bundle $L_2$ on an
elliptic curve $X_2$ given by a representation $\pi_1(X_2)\to U(1)$ such
that no multiple $kL_2$ with $k\ne0$ is trivial. Then
$H^0(X,kL)=H^0(X_1,kL_1)\otimes H^0(X_2,kL_2)=0$ for $k>0$, and thus
$\kappa(L)=-\infty$. However $c_1(L)=\pr_1^* c_1(L_1)$ has
numerical dimension equal to $\dim X_1$. The same example shows
that the Kodaira dimension may increase by restriction to a
subvariety (if $Y=X_1\times\{{\rm point}\}$, then
$\kappa(L_{\restriction Y})=\dim Y$).\qed
\endclaim

\section{4}{Approximation of plurisubharmonic functions 
via Berg\-man kernels}

We prove here, as an application of the Ohsawa-Takegoshi $L^2$ 
extension theorem [OT87], 
that every psh function on a pseudoconvex open set $\Omega\subset\bC^n$
can be approximated very accurately by functions of the form
$c\log|f|$, where $c>0$ and $f$ is a holomorphic function. The main idea
is taken from [Dem92]. For other applications to algebraic geometry,
see [Dem93] and Demailly-Koll\'ar [DK01]. We first recall the statement of
the generalized $L^2$ extension theorem; its proof relies on a subtle
enhancement of the Bochner-Kodaira technique, and we refer to the litterature
for details.

\claim{(4.1) Theorem {\rm(Ohsawa-Takegoshi [OT87], Manivel [Man93])}}
Let $X$ be a complex \hbox{$n$-dimensional} manifold possessing
a smooth plurisubharmonic exhaustion function\break
$($''weakly pseudoconvex'' or ``weakly $1$-con\-vex'' manifold$)$,
and 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
$$
Y=\big\{x\in X\,;\;s(x)=0, \Lambda^r ds(x)\not= 0\big\},\qquad
p=\dim Y=n-r.
$$
Finally, let $\varphi$ be an arbitrary plurisubharmonic function on $X$.
Assume that the $(1,1)$-form
$\Theta_L+r\,{i\over 2\pi}\,\ddbar(\log|s|^2+\varphi)$ is semi-positive and 
that there is
a continuous function $\alpha\ge 1$ such that the following two inequalities
hold everywhere on $X:$
\vskip3pt
\item{\rm (a)}
$\displaystyle\Theta_L+r\,{i\over 2\pi}\,\ddbar(\log|s|^2+\varphi)
\ge\alpha^{-1}{\{\Theta_Es,s\}\over|s|^2}\,,$
\vskip3pt
\item{\rm (b)} $|s|\le e^{-\alpha}$.
\vskip7pt\noindent
Then for every holomorphic section $f_Y$ of 
the line bundle $\Lambda^nT_X^*\otimes L$ over $Y$ such that
$\int_Y|f_Y|^2e^{-\varphi}|\Lambda^r(ds)|^{-2}dV_\omega<+\infty$,
there  exists a holomorphic extension $f_X$ of $f_Y$ over~$X$
such that
$$
\int_X{|f_X|^2e^{-\varphi}\over|s|^{2r}(-\log|s|)^2}\,dV_{X,\omega}
\le C_r\int_Y{|f_Y|^2e^{-\varphi}\over|\Lambda^r(ds)|^2}dV_{Y,\omega}\,,
$$
where $C_r$ is a numerical constant depending only on~$r$.
\endclaim

\claim{(4.2) Theorem} Let $\varphi$ be a plurisubharmonic function on a
bounded pseudoconvex open set~$\Omega\subset\bC^n$. For every $m>0$, let
$\cH_\Omega(m\varphi)$ be the Hilbert space of holomorphic functions $f$
on $\Omega$ such that $\int_\Omega|f|^2e^{-2m\varphi}d\lambda<+\infty$ and
let $\varphi_m={1\over 2m}\log\sum|\sigma_\ell|^2$ where $(\sigma_\ell)$
is an orthonormal basis of~$\cH_\Omega(m\varphi)$. Then there are
constants $C_1,C_2>0$ independent of $m$ such that
{\plainitemindent=8mm
\item{\rm (a)}$\displaystyle\varphi(z)-{C_1\over m}\le
\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over m}
\log{C_2\over r^n}$ for every $z\in\Omega$ and $r<d(z,\partial\Omega)$. In particular,
$\varphi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$ and
\item{\rm (b)}$\displaystyle\nu(\varphi,z)-{n\over m}\le\nu(\varphi_m,z)\le
\nu(\varphi,z)$~ for every $z\in\Omega$.}
\endclaim
%
\proof{Proof} (a) Note that $\sum|\sigma_\ell(z)|^2$ is the square of
the norm
of the evaluation linear form $\ev_z:f\mapsto f(z)$ on $\cH_\Omega(m\varphi)$,
since $\sigma_\ell(z)=\ev_z(\sigma_\ell)$ is the $\ell$-th coordinate of $\ev_z$
in the orthonormal basis $(\sigma_\ell)$. In other words, we have
$$\sum|\sigma_\ell(z)|^2=\sup_{f\in B(1)}|f(z)|^2$$
where $B(1)$ is the unit ball of $\cH_\Omega(m\varphi)$ (The sum
is called the {\it Bergman kernel} associated with $\cH_\Omega(m\varphi)$).
As $\varphi$ is locally bounded from 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, by what we just
explained, we have
$$\varphi_m(z)=\sup_{f\in B(1)}{1\over m}\log|f(z)|.$$
For $z_0\in\Omega$ and $r<d(z_0,\partial\Omega)$, the mean value
inequality applied to the psh function $|f|^2$ implies
$$\eqalign{
|f(z_0)|^2&\le{1\over\pi^nr^{2n}/n!}\int_{|z-z-0|<r}
|f(z)|^2d\lambda(z)\cr
&\le{1\over\pi^nr^{2n}/n!}\exp\Big(2m\sup_{|z-z_0|<r}\varphi(z)\Big)
\int_\Omega|f|^2e^{-2m\varphi}d\lambda.\cr}$$
If we take the supremum over all $f\in B(1)$ we get
$$
\varphi_m(z_0)\le\sup_{|z-z_0|<r}\varphi(z)+{1\over 2m}
\log{1\over\pi^nr^{2n}/n!}
$$
and the second inequality in (a) is proved -- as we see, this is an easy
consequence of the mean value inequality. Conversely, the
Ohsawa-Takegoshi $L^2$ extension theorem~4.1 applied
to the $0$-dimensional subvariety $\{z_0\}\subset\Omega$ and to the
trivial bundles $L=\Omega\times\bC$ and $E=\Omega\times\bC^n$, 
with the section $s(z)=z-z_0$ of $E$, shows that for any
$a\in\bC$ there is a holomorphic function $f$ on $\Omega$ such that
$f(z_0)=a$ and
$$\int_\Omega|f|^2e^{-2m\varphi}d\lambda\le C_3|a|^2e^{-2m\varphi(z_0)},$$
where $C_3$ only depends on $n$ and~${\rm diam}\,\Omega$. We fix $a$
such that the right hand side is~$1$. Then $\Vert f\Vert \le 1$ and so
we get
$$\varphi_m(z_0)\ge{1\over m}\log|f(z_0)|=
{1\over m}\log|a|=\varphi(z)-{\log C_3\over 2m}.$$
The inequalities given in (a) are thus proved. Taking $r=1/m$, we find
that 
$$\lim_{m\to+\infty}\sup_{|\zeta-z|<1/m}\varphi(\zeta)=\varphi(z)$$
by the upper semicontinuity of $\varphi$, and thus
$\lim\varphi_m(z)=\varphi(z)$, since $\lim{1\over m}\log (C_2m^n)=0$.
\medskip

\noindent
(b) The above estimates imply
$$
\sup_{|z-z_0|<r}\varphi(z)-{C_1\over  m}\le
\sup_{|z-z_0|<r}\varphi_m(z)\le\sup_{|z-z_0|<2r}\varphi(z)+
{1\over m}\log{C_2\over r^n}.$$
After dividing by $\log r<0$ when $r\to 0$, we infer
$$
{\sup_{|z-z_0|<2r}\varphi(z)+{1\over m}\log{C_2\over r^n}\over \log r}\le
{\sup_{|z-z_0|<r}\varphi_m(z)\over \log r}\le
{\sup_{|z-z_0|<r}\varphi(z)-{C_1\over  m}\over\log r},
$$
and from this and definition (1.6), it follows immediately that
$$
\nu(\varphi,x)-{n\over m}\le\nu(\varphi_m,z)\le\nu(\varphi,z).
\eqno\square
$$
\endproof

Theorem 4.2 implies in a straightforward manner the deep result of
[Siu74] on the analyticity of the Lelong number upperlevel sets.

\claim{(4.3) Corollary {\rm([Siu74])}} Let $\varphi$ be a plurisubharmonic
function on a complex manifold~$X$. Then, for every $c>0$, the Lelong number
upperlevel set
$$
E_c(\varphi)=\big\{z\in X\,;\;\nu(\varphi,z)\ge c\big\}
$$
is an analytic subset of~$X$.
\endclaim
%
\proof{Proof} Since analyticity is a local property, it
is enough to consider the case of a psh function $\varphi$ on a
pseudoconvex open set $\Omega\subset\bC^n$. The
inequalities obtained in Theorem 4.2~(b) imply that
$$
E_c(\varphi)=\bigcap_{m\ge m_0}E_{c-n/m}(\varphi_m).
$$
Now, it is clear that $E_c(\varphi_m)$ is the analytic set defined by the
equations $\sigma_\ell^{(\alpha)}(z)=0$ for all multi-indices $\alpha$
such that~$|\alpha|<mc$. Thus $E_c(\varphi)$ is analytic as a (countable)
intersection of analytic sets.\qed
\endproof

\claim{(4.4) Remark} {\rm It can be easily shown that the Lelong numbers
of any closed positive $(p,p)$-current coincide (at least locally) with the
Lelong numbers of a suitable plurisubharmonic potential $\varphi$
(see [Sko72]). Hence Siu's theorem also holds true for the
Lelong number upperlevel sets $E_c(T)$ of any closed positive
$(p,p)$-current~$T$.}
\endclaim

\noindent Theorem 4.2 motivates the following definition.

\claim{(4.5) Definition} A plurisubharmonic function $\varphi$ on a complex manifold $X$ 
is said to have analytic singularities if it can be written locally near every point
$x_0\in X$ as
$$
\varphi(z)=c\log\sum_{1\le j\le N}|g_j(z)|^2+O(1),\qquad\hbox{i.e.\ up
to equivalence of singularities},
$$
with a family of holomorphic functions $(g_j)$ defined near $x_0$ and $c>0$. Also, a closed 
positive $(1,1)$ current $T$ is said to have analytic 
singularities if its plurisubharmonic potential has analytic singularities. We also refer
to this situation by saying that $\varphi$ or $T$ have logarithmic poles. When $X$ is 
algebraic, we say that the
singularities are algebraic if~$c\in\bQ_+$ and the $(g_j)$ are sections of some algebraic
line bundle $\cO(D)$, $x_0\notin\Supp D$.
\endclaim

\noindent Notice that by Noetherianity, a convergent series $\log\sum_{j\in\bN}|g_j|^2$ 
can be replaced by a finite sum up to equivalence of singularities, thus Theorem~4.2
always produces plurisubharmonic functions $\varphi_m$ with analytic singularities.

\section{5}{Global approximation of closed (1,1)-currents on 
a compact complex manifold}

We take here $X$ to be an arbitrary compact complex manifold (no K\"ahler
assumption is needed). Now, let $T$ be a closed $(1,1)$-current
on~$X$. We assume that $T$ is {\it quasi-positive}, i.e.\ that there exists
a $(1,1)$-form $\gamma$ with continuous coefficients such
that~$T\ge\gamma\;$; the case of positive currents ($\gamma=0$) is of course
the most important.

\claim{(5.1) Lemma} There exists a smooth closed $(1,1)$-form $\alpha$
representing the same
$\ddbar$-cohomology class as $T$ and a {\rm quasi-psh} function $\varphi$
on $X$ such that $T=\alpha+{\ii\over\pi}\ddbar\varphi$. $($We say that
a function $\varphi$ is quasi-psh if its complex Hessian is bounded below
by a $(1,1)$-form with locally bounded coefficients, that is, if
$\ii\ddbar\varphi$ is quasi-positive$)$.
\endclaim
%
\proof{Proof} Select
an open covering $(U_j)$ of $X$ by coordinate balls such that
$T=\smash{{\ii\over\pi}}
\ddbar\varphi_j$ over $U_j$, and construct a global function
$\varphi=\sum\theta_j \varphi_j$ by means of a partition of unity $\{\theta_j\}$
subordinate to~$U_j$. Now, we observe that $\varphi-\varphi_k$ is smooth
on $U_k$  because all differences $\varphi_j-\varphi_k$ are smooth in
the intersections $U_j\cap U_k$, and we have the equality
$\varphi-\varphi_k=\sum\theta_j(\varphi_j-\varphi_k)$.
Therefore $\alpha:=T-{\ii\over\pi}\ddbar\varphi$ is smooth.\qed
\endproof

By replacing $T$ with $T-\alpha$ and $\gamma$ with
$\gamma-\alpha$, we can assume without loss of
generality that $\{T\}=0$, i.e. that $T={\ii\over\pi}\ddbar\varphi$
with a quasi-psh function~$\varphi$ on~$X$ such that
${\ii\over\pi}\ddbar\varphi\ge\gamma$.

Our goal is to approximate $T$ in the weak topology by currents
$T_m={\ii\over\pi}\ddbar\varphi_m$ such their potentials $\varphi_m$
have analytic singularities in the sense of Definition~4.5, more
precisely, defined on a neighborhood $V_{x_0}$ of any point $x_0\in X$
in the form
$\varphi_m(z)=c_m\log\sum_j|\sigma_{j,m}|^2+O(1)$,
where $c_m>0$ and the $\sigma_{j,m}$ are holomorphic functions on $V_{x_0}$.

We select a finite covering $(W_\nu)$ of $X$ with open
coordinate charts, and shrink them a little to be on the safe side. 
Given $\delta>0$, we take in each $W_\nu$ a
maximal family of points with (coordinate) distance to the boundary
$\ge 3\delta$ and mutual distance $\ge\delta/2$. In this way, we get for
$\delta>0$ small a finite covering of $X$ by open balls $U'_j$ of
radius~$\delta$ (actually every point is even at
\hbox{distance${}\le\delta/2$} of
one of the centers, otherwise the family of points would not be maximal),
such that the concentric ball $U_j$ of radius
$2\delta$ is relatively compact in the corresponding chart~$W_\nu$. Let
$\tau_j:U_j\lra B(a_j,2\delta)$ be the isomorphism given by the
coordinates of~$W_\nu\,$; by taking $\delta>0$ small enough, we can assume 
that the coordinates of $U_j$ extend to $U_j\cup U_k$ whenever 
$U_j\cap U_k\ne \emptyset$. Let $\varepsilon(\delta)$ be a modulus
of continuity for $\gamma$ on the sets $U_j$, such that
$\lim_{\delta\to 0}\varepsilon(\delta)=0$ and
$\gamma_x-\gamma_{x'}\le{1\over 2}\varepsilon(\delta)\,\omega_x$ for all
$x,x'\in U_j$. We denote by $\gamma_j$ the $(1,1)$-form with constant
coefficients on $B(a_j,2\delta)$ such that $\tau_j^*\gamma_j$
coincides with $\gamma-\varepsilon(\delta)\,\omega$ at
$\tau_j^{-1}(a_j)$. Then we have
$$0\le\gamma-\tau_j^*\gamma_j\le 2\varepsilon(\delta)\,\omega~~~~
\hbox{\rm on}~~U_j\leqno(5.2)$$
for $\delta>0$ small. We set $\varphi_j=\varphi\circ\tau_j^{-1}$ on
$B(a_j,2\delta)$ and let $q_j$ be the homogeneous quadratic
function in $z-a_j$ such that ${\ii\over\pi}\ddbar q_j=
\gamma_j$ on $B(a_j,2\delta)$. Then $\varphi_j-q_j$ is plurisubharmonic
on $B(a_j,2\delta)$ since
$${\ii\over\pi}\ddbar((\varphi_j-q_j)\circ\tau_j)=T-\tau_j^*\gamma_j
\ge\gamma-\tau_j^*\gamma_j\ge 0. \leqno(5.3)$$
We let $U'_j\compact U''_j\compact U_j$ be the
concentric balls of radii $\delta$, $1.5\,\delta$, $2\delta$ respectively.
On each open set $U_j$ the function
$\psi_j:=\varphi-q_j\circ\tau_j=(\varphi_j-q_j)\circ\tau_j$ is plurisubharmonic, 
so Theorem~4.2 applied with $\Omega=U_j\simeq B(a_j,2\delta)$ produces functions
$$\psi_{j,m}={1\over 2m}\log\sum_\ell|\sigma_{j,\ell}|^2,~~~~
(\sigma_{j,\ell})=\hbox{\rm basis of}~\cH_{U_j}(m\psi_j).\leqno(5.4)$$
The functions $\psi_{j,m}+q_j\circ\tau_j$ on $U_j$ then have
to be glued together by a partition of unity technique. For this,
we rely on the following ``discrepancy'' lemma, estimating the variation
of the approximating functions on overlapping balls.

\claim{(5.5) Lemma} There is a constant $C$
independent of $m$ and $\delta$ such that the quasi-psh functions
$w_{j,m}=2m(\psi_{j,m}+q_j\circ\tau_j)$, i.e.\
$$w_{j,m}(x)=2m\,q_j\circ\tau_j(x)+
\log\sum_\ell\big|\sigma_{j,\ell}(x)\big|^2,~~~~
x\in U''_j,$$
satisfy
$$|w_{j,m}-w_{k,m}|\le C\big(\log\delta^{-1}+
m\varepsilon(\delta)\delta^2\big)
~~~~\hbox{\rm on}~~U''_j\cap U''_k.$$
\endclaim
%
\proof{Proof} The details will be left as an exercise to the reader.
The main idea is the following: for any holomorphic function
$f_j\in \cH_{U_j}(m\psi_j)$,
a $\dbar$ equation $\dbar u = \dbar(\theta f_j)$ can be solved on $U_k$,
where $\theta$ is a cut-off function with support in $U''_j\cap U''_k$,
on a ball of radius${}<\delta/4$,
equal to $1$ on the ball of radius $\delta/8$ centered at a given
point $x_0\in U''_j\cap U''_k$, with $|\dbar\theta|=O(\delta^{-1})$. We apply
the $L^2$ estimate with respect to the weight
\hbox{$(n+1)\log|x-x_0|^2+2m\psi_k$}, where the first term is picked up so as
to force the solution $u$ to vanish at $x_0$, in such a way that
$F_k=u-\theta f_j$ is holomorphic and $F_k(x_0)=f_j(x_0)$.
The discrepancy between the weights on $U''_j$ and $U''_k$~is given by
$$
\psi_j-\psi_k=-\big(q_j\circ\tau_j-q_k\circ\tau_k\big).
$$
By re-centering the quadratic functions at $\tau_j(x_0)$, resp.\ $\tau_k(x_0)$,
we can write 
$$q_j\circ\tau_j-q_k\circ\tau_k=\Re G_{jk}+R_{jk}$$
where $G_{jk}$ is holomorphic on $U_j\cup U_k$ [equal to a difference of linear 
forms in the coordinates of 
$B(a_j,2\delta)$ and $B(a_k,2\delta)$], 
$G_{jk}(x_0)=q_j\circ\tau_j(x_0)-q_k\circ\tau_k(x_0)$ and 
\hbox{$R_{jk}=O(\varepsilon(\delta)\delta^2)$}
is a remainder term coming from the change of coordinates and the 
slight discrepancy between $\ddbar(q_j\circ\tau_j)$ and 
$\ddbar(q_k\circ\tau_k)$ at the common point~$x_0$, with $R_{jk}(x_0)=0$.
In this way, we get 
$$|e^{mG_{jk}}|^2e^{-m\psi_k}=e^{-m\psi_j-2mR_{jk}},$$
so that we have a uniform control of the $L^2$ norm of the
solution $f_k=e^{mG_{jk}}F_k=e^{mG_{jk}}(u-\theta f_j)$ of the form
$$\int_{U_k}|f_k|^2e^{-2m\psi_k}\le C\delta^{-2n-4}
e^{m O(\varepsilon(\delta)\delta^2)} \int_{U_j}|f_j|^2e^{-2m\psi_j}.$$
The required estimate follows, using the equality
$$
e^{2m\psi_{j,m}(x)}=\sum_\ell|\sigma_{j,\ell}(x)|^2=
\sup_{f\in\cH_{U_j}(m\psi_j),\,\Vert f\Vert\le 1}|f(x)|^2~~~~\hbox{on $U_j$},
$$
and the analogous equality on $U_k$.\qed
\endproof

Now, the actual glueing of our quasi-psh functions is performed
using the following elementary partition of unity calculation.

\claim{(5.6) Lemma} Let $U'_j\compact U''_j$ be locally finite
open coverings
of a complex manifold $X$ by relatively compact open sets, and let
$\theta_j$ be smooth nonnegative functions with support in $U''_j$,
such that $\theta_j\le 1$ on $U''_j$ and $\theta_j=1$ on $U'_j$.
Let $A_j\ge 0$ be such that
$$\ii(\theta_j\ddbar\theta_j-\partial\theta_j\wedge\dbar\theta_j)\ge
-A_j\omega~~~~\hbox{\rm on}~~U''_j\ssm U'_j$$
for some positive $(1,1)$-form~$\omega$.
Finally, let $w_j$ be quasi-psh functions on $U_j$ with
the property that $\ii\partial\smash{\dbar}w_j\ge\gamma$
for some real $(1,1)$-form $\gamma$ on~$M$, and let $C_j$ be constants
such that
$$w_j(x)\le C_j+\sup_{k\ne j,\,U'_k\ni x}w_k(x)~~~~\hbox{\rm on}~~
U''_j\ssm U'_j.$$
Then the function $w=\log\big(\sum\theta_j^2 e^{w_j}\big)$
is quasi-psh and satisfies
$$\ii\ddbar w\ge\gamma-2\Big(\sum_j\bOne_{U''_j\ssm U'_j}A_je^{C_j}\Big)
\omega.$$
\endclaim
%
\proof{Proof} If we set $\alpha_j=\theta_j\partial w_j+2\partial\theta_j$,
a straightforward computation shows that
$$\eqalign{
\partial w&={\sum(\theta_j^2\partial w_j+2\theta_j\partial\theta_j)e^{w_j}
\over\sum\theta_j^2e^{w_j}}={\sum\theta_je^{w_j}\alpha_j
\over\sum\theta_j^2e^{w_j}},\cr
\ddbar w&={\sum\!\big(\alpha_j\wedge\overline\alpha_j{+}\theta_j^2\ddbar w_j
{+}2\theta_j\ddbar\theta_j{-}2\partial\theta_j{\wedge}\dbar\theta_j\big)
e^{w_j}\over\sum\theta_j^2e^{w_j}}
-{\sum_{j,k}\theta_je^{w_j}\theta_ke^{w_k}\alpha_j{\wedge}
\overline\alpha_k\over\big(\sum\theta_j^2e^{w_j}\big)^2}\cr
&={\sum_{j<k}\!\big|\theta_j\alpha_k{-}\theta_k\alpha_j\big|^2e^{w_j}e^{w_k}
\over\big(\sum\theta_j^2e^{w_j}\big)^2}{+}{\sum\theta_j^2e^{w_j}\ddbar w_j
\over\sum\theta_j^2e^{w_j}}{+}{\sum\!\big(2\theta_j\ddbar\theta_j{-}
2\partial\theta_j{\wedge}\dbar\theta_j\big)e^{w_j}\over\sum\theta_j^2e^{w_j}}\cr}$$
by using the Legendre identity. The first term in the last line is nonnegative
and the second one is $\ge\gamma$. In the third term, if $x$ is in the support
of $\theta_j\ddbar\theta_j-\partial\theta_j\wedge\dbar\theta_j$, then
$x\in U''_j\ssm U'_j$ and so $w_j(x)\le C_j+w_k(x)$ for some $k\ne j$
with $U'_k\ni x$ and $\theta_k(x)=1$. This gives
$$i{\sum\big(
2\theta_j\ddbar\theta_j-2\partial\theta_j\wedge\dbar\theta_j\big)e^{w_j}
\over\sum\theta_j^2e^{w_j}}\ge
-2\sum_j\bOne_{U''_j\ssm U'_j}e^{C_j}A_j\omega.$$
The expected lower bound follows.\qed
\endproof

We apply Lemma~5.6 to functions $\wt w_{j,m}$ which are just
slight modifications of the functions $w_{j,m}=2m(\psi_{j,m}+q_j\circ\tau_j)$
occurring in Lemma~5.5~:
$$\eqalign{
\wt w_{j,m}(x)
&=w_{j,m}(x)+2m\Big({C_1\over m}+
C_3\varepsilon(\delta)(\delta^2/2-|\tau_j(x)|^2)\Big)\cr
&=2m\Big(\psi_{j,m}(x)+q_j\circ\tau_j(x)+{C_1\over m}+
C_3\varepsilon(\delta)(\delta^2/2-|\tau_j(x)|^2)\Big)\cr}$$
where $x\mapsto z=\tau_j(x)$ is a local coordinate identifying $U_j$
to $B(0,2\delta)$, $C_1$ is the constant occurring in Lemma~5.5 and $C_3$ is
a sufficiently large constant.
It is easy to see that we can take $A_j=C_4\delta^{-2}$ in Lemma~5.6.
We have
$$
\wt w_{j,m}\ge w_{j,m}+2C_1+m{C_3\over 2}\varepsilon(\delta)\delta^2
~~~~\hbox{on $B(x_j,\delta/2)\subset U'_j$},
$$
since $|\tau_j(x)|\le\delta/2$ on $B(x_j,\delta/2)$, while
$$
\wt w_{j,m}\le w_{j,m}+2C_1-mC_3\varepsilon(\delta)\delta^2
~~~~\hbox{on $U''_j\ssm U'_j$}.
$$
For $m\ge m_0(\delta)=(\log\delta^{-1}/(\varepsilon(\delta)\delta^2)$,
Lemma~5.5 implies $|w_{j,m}-w_{k,m}|\le C_5m\varepsilon(\delta)\delta^2$ on
$U''_j\cap U''_k$. Hence, for $C_3$ large enough, we get
$$\wt w_{j,m}(x)\le \sup_{k\ne j,\,B(x_k,\delta/2)\ni x}w_{k,m}(x)
\le \sup_{k\ne j,\,U'_k\ni x}w_{k,m}(x)~~~~\hbox{\rm on}~~
U''_j\ssm U'_j,$$
and we can take $C_j=0$ in the hypotheses of Lemma~5.6.
The associated function $w=\log\big(\sum\theta_j^2e^{\wt w_{j,m}}\big)$
is given by
$$w=\log\sum_j\theta_j^2\exp\Big(2m\big(\psi_{j,m}+q_j\circ\tau_j+
{C_1\over m}+C_3\varepsilon(\delta)(\delta^2/2-|\tau_j|^2)\big)
\Big).$$
If we define $\varphi_m={1\over 2m}w$, we get
$$
\varphi_m(x):={1\over 2m}w(x)\ge \psi_{j,m}(x)+
q_j\circ\tau_j(x)+{C_1\over m}+{C_3\over 4}\varepsilon(\delta)\delta^2
>\varphi(x)
$$
in view of Lemma~5.5, by picking an index $j$ such that $x\in B(x_j,\delta/2)$.
In the opposite
direction, the maximum number $N$ of overlapping
balls~$U_j$ does not depend on $\delta$, and we thus get
$$w\le\log N+2m\Big(
\max_j\big\{\psi_{j,m}(x)+q_j\circ\tau_j(x)\big\}+
{C_1\over m}+{C_3\over 2}\varepsilon(\delta)\delta^2\Big).$$
By definition of $\psi_j$ we have $\sup_{|\zeta-x|<r}\psi_j(\zeta)\le
\sup_{|\zeta-x|<r}\varphi(\zeta)-q_j\circ\tau_j(x)+C_5r$ thanks to
the uniform Lipschitz continuity of $q_j\circ\tau_j$, thus by
Lamme~5.5 again we find
$$\varphi_m(x)\le
{\log N\over 2m}+\sup_{|\zeta-x|<r}\varphi(\zeta)+
{C_1\over m}+{1\over m}\log{C_2\over r^n}+
{C_3\over 2}\varepsilon(\delta)\delta^2+C_5 r.$$
By taking for instance $r=1/m$ and $\delta=\delta_m\to 0$, we see that
$\varphi_m$ converges to $\varphi$.
On the other hand $(5.2)$ implies
${\ii\over\pi}\ddbar q_j\circ\tau_j(x)=\tau_j^*\gamma_j\ge
\gamma-2\varepsilon(\delta)\omega$, thus
$${\ii\over\pi}\ddbar\wt w_{j,m}\ge
2m\big(\gamma-C_6\varepsilon(\delta)\omega\big).$$
Lemma~5.6 then produces the lower bound
$${\ii\over\pi}\ddbar w
\ge 2m\big(\gamma-C_6\varepsilon(\delta)\omega\big)-C_7\delta^{-2}\omega,$$
whence
$${\ii\over\pi}\ddbar\varphi_m\ge\gamma-C_8\varepsilon(\delta)\omega$$
for $m\ge m_0(\delta)=(\log\delta^{-1})/(\varepsilon(\delta)\delta^2)$.
We can fix $\delta=\delta_m$ to be the smallest value of $\delta>0$ such that
$m_0(\delta)\le m$, then $\delta_m\to 0$ and we have
obtained a sequence of quasi-psh functions $\varphi_m$ satisfying
the following properties.

\claim{(5.7) Theorem}
Let $\varphi$ be a quasi-psh function on a compact complex
manifold $X$ such that ${\ii\over\pi}\ddbar\varphi\ge\gamma$ for some
continuous $(1,1)$-form $\gamma$. Then there is a sequence of
quasi-psh functions $\varphi_m$ such that $\varphi_m$ has the
same singularities
as a logarithm of a sum of squares of holomorphic functions and
a decreasing sequence $\varepsilon_m>0$ converging to $0$ such that
{\plainitemindent=7mm
\item{\rm (a)}$\displaystyle\varphi(x)<\varphi_m(x)\le\sup_{|\zeta-x|<r}
\varphi(\zeta)+C\Big({|\log r|\over m}+r+\varepsilon_m\Big)$\\
with respect to coordinate open sets covering $X$. In particular,
$\varphi_m$ converges to $\varphi$ pointwise and in $L^1(X)$ and
\item{\rm (b)}$\displaystyle\nu(\varphi,x)-{n\over m}\le\nu(\varphi_m,x)\le
\nu(\varphi,x)$~ for every $x\in X\,;$
\item{\rm (c)}$\displaystyle{\ii\over\pi}\ddbar\varphi_m\ge
\gamma-\varepsilon_m\omega$.}
\endclaim
%
In particular, we can apply this to an arbitrary positive or
quasi-positive closed
$(1,1)$-current $T=\alpha+{\ii\over\pi}\ddbar\varphi$.

\claim{(5.8) Corollary}
Let $T$ be a quasi-positive  closed
$(1,1)$-current on a compact complex
manifold $X$ such that $T\ge\gamma$ for some
continuous $(1,1)$-form $\gamma$. Then there is a sequence of
currents $T_m$ whose local potentials have the
same singularities as $1/m$ times a logarithm of a sum of squares
of holomorphic functions and
a decreasing sequence $\varepsilon_m>0$ converging to $0$ such that
{\plainitemindent=7mm
\item{\rm (a)} $T_m$ converges weakly to $T$,
\item{\rm (b)}$\displaystyle\nu(T,x)-{n\over m}\le\nu(T_m,x)\le
\nu(T,x)$~ for every $x\in X\,;$
\item{\rm (c)}$T_m\ge\gamma-\varepsilon_m\omega$.
\vskip0pt
}
\noindent
We say that our currents $T_m$ are approximations of $T$ with logarithmic poles.
\endclaim
%
By using blow-ups of $X$, the structure of the currents $T_m$ can be
better understood.  In fact, consider the coherent ideals $\cJ_m$
generated locally by the holomorphic functions $(\sigma^{(k)}_{j,m})$
on $U_k$ in the local approximations
$$
\varphi_{k,m}={1\over 2m}\log\sum_j|\sigma^{(k)}_{j,m}|^2+O(1)
$$
of the potential $\varphi$ of~$T$ on~$U_k$. These ideals are in fact
globally defined, because the local ideals
$\cJ_m^{(k)}=(\sigma^{(k)}_{j,m})$ are integrally closed, and they
coincide on the intersections $U_k\cap U_\ell$ as they have the same
order of vanishing by the proof of Lemma~5.5.  By Hironaka
[Hir64], we can find a composition of blow-ups with smooth centers
$\mu_m:\wt X_m\to X$ such that $\mu_m^*\cJ_m$ is an invertible ideal
sheaf associated with a normal crossing divisor $E_m$. Now, we can
write
$$
\mu_m^*\varphi_{k,m}=\varphi_{k,m}\circ\mu_m=
{1\over m}\log|s_{E_m}|+\wt\varphi_{k,m}
$$
where $s_{E_m}$ is the canonical section of $\cO(-E_m)$ and
$\wt\varphi_{k,m}$ is a smooth potential. This implies
$$
\mu_m^*T_m={1\over m}[E_m]+\beta_m\leqno(5.9)
$$
where $[E_m]$ is the current of integration over $E_m$ and $\beta_m$
is a smooth closed $(1,1)$-form which satisfies the lower bound
$\beta_m\ge\mu_m^*(\gamma-\varepsilon_m\omega)$. (Recall that
the pull-back of a closed $(1,1)$-current by a holomorphic map $f$
is always well-defined, by taking a local plurisubharmonic potential
$\varphi$ such that $T=\ii\ddbar\varphi$ and writing
$f^*T=\ii\ddbar(\varphi\circ f)$). In the remainder of
this section, we derive from this a rather important geometric consequence,
first appeared in~[DP04]). We need two related definitions.

\claim{(5.10) Definition} A K\"ahler current on a compact
complex space $X$ is a closed positive current $T$ of bidegree $(1,1)$
which satisfies $T\ge\varepsilon\omega$ for some $\varepsilon>0$ and
some smooth positive Hermitian form $\omega$ on $X$.
\endclaim
%
\claim{(5.11) Definition} A compact complex manifold is said to be
in the {\rm Fujiki class $\cC$} if it is bimeromorphic to a K\"ahler manifold
$($or equivalently, using Hironaka's desingularization theorem, if
it admits a proper K\"ahler modification$)$.
\endclaim
%
\claim{(5.12) Theorem} A compact complex manifold $X$
is bimeromorphic to a K\"ahler mani\-fold $($i.e.\ $X\in\cC)$
if and only if it admits a K\"ahler current.
\endclaim
%
\proof{Proof} If $X$ is bimeromorphic to a K\"ahler manifold $Y$,
Hironaka's desingularization theorem implies that there exists
a blow-up $\widetilde Y$ of $Y$ (obtained by a sequence of blow-ups with
smooth centers) such that the bimeromorphic map from
$Y$ to $X$ can be resolved into a modification $\mu:\widetilde Y\to X$. Then
$\widetilde Y$ is K\"ahler and the push-forward $T=\mu_*\wt\omega$
of a K\"ahler form $\wt\omega$ on $\widetilde Y$ provides a K\"ahler
current on~$X$. In fact, if $\omega$ is a smooth Hermitian form on~$X$,
there is a constant $C$ such that $\mu^*\omega\le C\wt\omega$ (by compactness
of $\wt Y$), hence
$$
T=\mu_*\wt\omega\ge \mu_*(C^{-1}\mu^*\omega)=C^{-1}\omega.
$$
Conversely, assume that $X$ admits a K\"ahler current $T\ge\varepsilon\omega$.
By Theorem 5.8~(c), there exists a K\"ahler current
$\wt T=T_m\ge{\varepsilon\over 2}\omega$ (with $m\gg 1$ so large
that $\varepsilon_m\le\varepsilon/2$) in
the same $\ddbar$-cohomology class as~$T$, possessing logarithmic poles.
Observation (5.9) implies the existence of a composition of blow-ups
$\mu:\wt X\to X$ such that
$$
\mu^*\wt T=[\wt E]+\wt\beta\qquad\hbox{on $\wt X$},
$$
where $\wt E$ is a $\bQ$-divisor with normal crossings and $\wt\beta$
a smooth closed $(1,1)$-form such that $\wt\beta\ge{\varepsilon\over 2}
\mu^*\omega$. In particular $\wt\beta$ is positive outside the exceptional
locus of~$\mu$. This is not enough yet to produce a K\"ahler
form on $\wt X$, but we are not very far. Suppose that $\wt X$
is obtained as a tower of blow-ups
$$
\wt X=X_N\to X_{N-1}\to\cdots \to X_1\to X_0=X,
$$
where $X_{j+1}$ is the blow-up of $X_j$ along a smooth center $Y_j\subset X_j$.
Denote by $S_{j+1}\subset X_{j+1}$ the exceptional divisor, and let
$\mu_j:X_{j+1}\to X_j$ be the blow-up map. Now, we use the following
simple
\endproof

\claim{(5.13) Lemma} For every K\"ahler current
$T_j$ on $X_j$, there exists $\varepsilon_{j+1}>0$ and a smooth form $u_{j+1}$
in the $\ddbar$-cohomology class of $[S_{j+1}]$ such that
$$
T_{j+1}=\mu_j^* T_j-\varepsilon_{j+1}u_{j+1}
$$
is a K\"ahler current on~$X_{j+1}$.
\endclaim
%
\proof{Proof} The line bundle
$\cO(-S_{j+1})|S_{j+1}$ is equal to $\cO_{P(N_j)}(1)$ where $N_j$
is the normal bundle to $Y_j$ in $X_j$. Pick an arbitrary smooth Hermitian
metric on $N_j$, use this metric to get an induced Fubini-Study metric on
$\cO_{P(N_j)}(1)$, and finally extend this metric as a smooth Hermitian
metric on the line bundle $\cO(-S_{j+1})$. Such a metric has positive
curvature along tangent vectors of $X_{j+1}$ which are tangent to the
fibers of $S_{j+1}=P(N_j)\to Y_j$. Assume furthermore that
$T_j\ge\delta_j\omega_j$ for some Hermitian form $\omega_j$ on $X_j$
and a suitable $0<\delta_j\ll 1$. Then
$$
\mu_j^* T_j-\varepsilon_{j+1}u_{j+1}\ge
\delta_j\mu_j^*\omega_j-\varepsilon_{j+1}u_{j+1}
$$
where $\mu_j^*\omega_j$ is semi-positive on $X_{j+1}$, positive definite on
$X_{j+1}\ssm S_{j+1}$, and also positive definite on tangent vectors of
$T_{X_{j+1}|S_{j+1}}$ which are not tangent to the fibers of $S_{j+1}\to Y_j$.
The statement is then easily proved by taking $\varepsilon_{j+1}\ll\delta_j$
and by using an elementary compactness argument on the unit sphere bundle
of~$T_{X_{j+1}}$ associated with any given Hermitian metric.\qed
\endproof

\noindent
{\it End of proof of Theorem {\rm 5.12.}}
If $\widetilde u_j$ is the pull-back of $u_j$ to the final blow-up
$\widetilde X$, we conclude inductively that $\mu^*
\wt T-\sum\varepsilon_j\widetilde u_j$ is a K\"ahler current. Therefore
the smooth form
$$
\wt\omega:=\wt\beta-\sum\varepsilon_j\widetilde u_j
=\mu^* \wt T-\sum\varepsilon_j\widetilde u_j-[\wt E]
$$
is K\"ahler and we see that $\wt X$ is a K\"ahler manifold.\qed
\medskip

\claim{(5.14) Remark} {\rm A special case of Theorem 5.12 is the following
characterization of Moishezon varieties (i.e.\ manifolds which are
bimeromorphic to projective algebraic varieties or, equivalently, whose
algebraic dimension is equal to their complex dimension):
\smallskip
\noindent
{\it A compact complex manifold $X$ is Moishezon if and only if $X$
possesses a K\"ahler current $T$ such that the De Rham cohomology class
$\{T\}$ is rational, i.e.\ $\{T\}\in H^2(X,\bQ)$}.
\smallskip
\noindent
In fact, in the above proof, we get an integral current $T$ if we take
the push forward $T=\mu_*\wt\omega$ of an integral ample class
$\{\wt\omega\}$ on $Y$, where $\mu:Y\to X$ is a projective model
of~$Y$. Conversely, if $\{T\}$ is rational, we can take
the $\varepsilon_j$'s to be rational in Lemma~5.13. This produces
at the end a K\"ahler metric $\wt\omega$ with rational De Rham cohomology
class on $\wt X$. Therefore $\wt X$ is projective by the Kodaira
embedding theorem. This result was already observed in [JS93]
(see also [Bon93, Bon98] and Section~III~6 for a more general perspective based
on a singular holomorphic Morse inequalities).}
\endclaim

\claim{(5.15) Remark} {\rm Hodge decomposition also holds true for 
manifolds $X\in\cC$. In fact let $\mu:\wt X\to X$ be a modification such that
$\wt X$ is K\"ahler. Then there are natural morphisms
$$
\mu^*:H^{p,q}_\dbar(X,\bC)\to H^{p,q}_\dbar(\wt X,\bC),\qquad
\mu_*:H^{p,q}_\dbar(\wt X,\bC)\to H^{p,q}_\dbar(X,\bC)
$$
induced respectively by the pull-back of smooth forms (resp.\ the direct image
of currents). Clearly, $\mu_*\circ\mu^*=\Id$, therefore $\mu^*$ is injective and
$\mu_*$ surjective, and similar results hold true for Bott-Chern cohomology or
De Rham cohomology. It follows easily from this that the $\ddbar$-lemma still 
holds true for $X\in\cC$, and that there are isomorphisms
$$
H^{p,q}_\BC(X,\bC)\to H^{p,q}_\dbar(X,\bC),\qquad
\bigoplus_{p+q=k}H^{p,q}_\BC(X,\bC)\to H^k_\DR(X,\bC).
$$\vskip-30pt}\endclaim

\section{6}{Zariski decomposition and mobile intersections}

Let $X$ be compact K\"ahler and let $\alpha\in\cE^\circ$ be in the
{\it interior} of the pseudo--effective cone. In analogy with the
algebraic context such a class $\alpha$ is called ``big'', and it can
then be represented by a {\it K\"ahler current} $T$, i.e.\ a closed
positive $(1,1)$-current $T$ such that $T\ge \delta\omega$ for some
smooth Hermitian metric $\omega$ and a constant $\delta\ll 1$. We first
need a variant of the approximation theorem proved in Section~5.

\claim{(6.1) Regularization theorem for currents}
Let $X$ be a compact complex manifold equipped with a Hermitian metric
$\omega$. Let $T=\alpha+\ii\ddbar\varphi$ be a closed $(1,1)$-current on~$X$,
where $\alpha$ is smooth and $\varphi$ is a quasi-plurisubharmonic function.
Assume that $T\ge\gamma$ for some real $(1,1)$-form $\gamma$ on $X$ with
real coefficients. Then there exists
a sequence $T_m=\alpha+\ii\ddbar\varphi_m$ of closed $(1,1)$-currents such that
{\plainitemindent=7mm
\smallskip
\item{\rm (a)} $\varphi_m$ $($and thus $T_m)$ is smooth on the complement
$X\ssm Z_m$ of an analytic set $Z_m$, and the $Z_m$'s form an increasing
sequence
$$
Z_0\subset Z_1\subset\cdots\subset Z_m\subset\cdots \subset X.
$$
\smallskip
\item{\rm (b)} There is a uniform estimate $T_m\ge \gamma-\delta_m\omega$
with $\lim\downarrow\delta_m=0$ as $m$ tends to $+\infty$.
\smallskip
\item{\rm (c)} The sequence $(\varphi_m)$ is non increasing, and we
have $\lim\downarrow\varphi_m=\varphi$. As a consequence, $T_m$ converges
weakly to $T$ as  $m$ tends to $+\infty$.
\smallskip
\item{\rm (d)} Near $Z_m$, the potential $\varphi_m$ has logarithmic poles,
namely, for every $x_0\in Z_m$, there is a neighborhood $U$ of $x_0$
such that $\varphi_m(z)=\lambda_m\log\sum_\ell|g_{m,\ell}|^2+O(1)$ for
suitable holomorphic functions $(g_{m,\ell})$ on $U$ and $\lambda_m>0$.
Moreover, there is a $($global$)$ proper modification $\mu_m:\wt X_m\to X$
of $X$, obtained as a sequence of blow-ups with smooth centers, such that
$\varphi_m\circ\mu_m$ can be written locally on $\wt X_m$ as
$$
\varphi_m\circ\mu_m(w)=\lambda_m\big(\sum n_\ell\log|\wt g_\ell|^2+f(w)\big)
$$
where $(\wt g_\ell=0)$ are local generators of suitable $($global$)$
divisors $E_\ell$ on $\wt X_m$ such that $\sum E_\ell$ has normal crossings,
$n_\ell$ are positive integers, and the $f$'s are smooth functions
on~$\wt X_m$.
\vskip0pt
}
\endclaim
%
\noindent
{\it Sketch of proof.} We essentially repeat the proofs of Theorems 4.2
and 5.7 with additional considerations. One fact that does not
follow readily from these proofs is the
monotonicity of the sequence $\varphi_m$ (which we will not really need
anyway -- it can be obtained by applying Theorem~$4.2$ with $2^m$ instead
of~$m$, and by using the Ohsawa-Takegoshi $L^2$ extension theorem~4.1 for 
potentials $2^m\varphi(x)+2^{m}\varphi(y)$ on the diagonal of $X\times X$,
so that the restriction is $2^{m+1}\varphi(x)$ on the diagonal; we refer e.g.\ 
to [DPS01] for details). The map $\mu_m$ is obtained by blowing-up the
(global) ideals $\cJ_m$ defined by the holomorphic functions $(g_{j,m})$
in the local approximations $\varphi_m\sim{1\over 2m}\log\sum_j|g_{j,m}|^2$.
By Hironaka [Hir64], we can achieve that $\mu_m^*\cJ_m$ is an invertible
ideal sheaf associated with a normal crossing divisor.\qed

\claim{(6.2) Corollary} If $T$
is a K\"ahler current, then one can write $T=\lim T_m$ for a sequence of
K\"ahler currents $T_m$ which have
logarithmic poles with coefficients in ${1\over m}\bZ$, i.e.\ there
are modifications $\mu_m:X_m\to X$ such that
$$\mu_m^* T_m=[E_m]+\beta_m$$
where $E_m$ is an effective $\bQ$-divisor on $X_m$ with coefficients
in ${1\over m}\bZ$ $($the ``fixed part''$)$ and $\beta_m$ is a closed
semi-positive form $($the ``mobile part''$)$.
\endclaim
%
\proof{Proof} We apply Theorem 6.1 with $\gamma=\varepsilon\omega$ and
$m$ so large that $\delta_m\le\varepsilon/2$. Then $T_m$ has analytic
singularities and $T_m\ge{\varepsilon\over 2}\omega$, so we get
a composition of blow-ups $\mu_m:X_m\to X$ such
$$\mu_m^*T_m=[E_m]+\beta_m,$$
where $E_m$ is an effective $\bQ$-divisor and
$\beta_m\ge{\varepsilon\over 2}\mu_m^*\omega$. In particular,
$\beta_m$ is strictly positive outside the exceptional divisors, by playing
with the multiplicities of the components of the exceptional divisors in
$E_m$, we could even achieve that $\beta_m$ is a
K\"ahler class on $X_m$. Notice also that by construction, $\mu_m$
is obtained by blowing-up the multiplier ideal sheaves $\cI(mT)=
\cI(m\varphi)$ associated to a potential $\varphi$ of $T$.\qed
\endproof

The more familiar algebraic analogue would be to take \hbox{$\alpha=c_1(L)$}
with a big line bundle $L$ and to blow-up the base locus of $|mL|$,
$m\gg 1$, to get a $\bQ$-divisor decomposition
$$
\mu_m^* L\sim E_m+D_m,\qquad E_m~~\hbox{effective},~~D_m~~\hbox{base point free}.
\leqno(6.3)
$$
(One says that $D_m$ is base point free if $H^0(X,\cO(D_m)$ is generated by sections,
in other words  if $D_m$ is entirely ``mobile'' in the linear system $|D_m|$).
Such a blow-up is usually referred to as a ``log resolution'' of the
linear system $|mL|$, and we say that $E_m+D_m$ is an approximate
Zariski decomposition of $L$. We will also use this terminology for K\"ahler
currents with logarithmic poles.

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\vskip4mm

\claim{(6.4) Definition} We define the {\rm volume}, or {\rm mobile
self-intersection} of a class $\alpha\in H^{1,1}(X,\bR)$ to be
$$
\Vol(\alpha)=\sup_{T\in \alpha}\int_{X\ssm\Sing(T)}T^n=
\sup_{T\in \alpha}\int_{\wt X}\beta^n>0,
$$
where the supremum is taken over all K\"ahler currents $T\in \alpha$
with logarithmic poles, and $\mu^* T=[E]+\beta$ with
respect to some modification $\mu:\wt X\to X$. Correspondingly, we set
$$
\Vol(\alpha) = 0\qquad\hbox{if $\alpha\notin \cE^\circ$.}
$$
\endclaim
%
\noindent In the special case where $\alpha=c_1(L)$ is an integral class, we have the
following interpretation of the volume.

\claim{(6.5) Theorem} If $L$ is a big line bundle and $\mu_m^*L\sim E_m+D_m$ is
a log resolution of~$|mL|$, we have
$$
\Vol(c_1(L))=\lim_{m\to+\infty}D_m^n=\lim_{m\to+\infty}{n!\over m^n}h^0(X,mL),
$$
\endclaim
%
\proof{Sketch of proof} Given a K\"ahler current $T\in c_1(L)$ with logarithmic pole, 
we can always take a blow-up $\mu:\wt X\to X$ so that
$\mu^* T=[E]+\beta$ where $E$ is an effective $\bR$-divisor and $\beta\ge 0$. By using 
a perturbation technique as in Lemma~5.13, we can always assume that $E$ is a $\bQ$-divisor
and that $\beta$ is K\"ahler. Then $\{\beta\}=\mu^*c_1(L)-\{[E]\}$ is a rational
class and therefore $\beta$ is the first Chern class $c_1(A)$ of an
ample $\bQ$-divisor on $\smash{\wt X}$. When $m$ is a multiple of a suitable 
denominator $m_0$ and $m=qm_0+r$, $0\le r<m_0$, we get by the elementary Riemann-Roch
formula
$$
h^0(X,mL)\ge h^0(\wt X,m\mu^*L-m_0[m/m_0]E)=h^0(\wt X,m_0[m/m_0]A+r\mu^*L)
\sim {m^n\over n!}\int_{\wt X}\beta^n,
$$
hence $\liminf{n!\over m^n}h^0(X,mL)\ge \Vol(c_1(L))$ by taking the supremum over all
such currents~$T$. In the other direction, the inequality 
$\limsup{n!\over m^n}h^0(X,mL)\le \Vol(c_1(L))$ is obtained by subtracting a small
rational multiple $\varepsilon A$ of an ample line bundle $A$. One shows
that multiples of $L-\varepsilon A$ roughly have the same number of sections
as those of $L$ by an exact sequence argument similar to what was done in the 
proof of 3.4~(b).
By a result of Fujita [Fuj94] (cf.\ also [DEL00]), the volume of the base point free
part $D_{m,\varepsilon}$ in a log resolution of  $|m(L-\varepsilon A)|$ approximates
$\limsup{n!\over m^n}h^0(X,m(L-\varepsilon A))$, so we get 
$\mu_{m,\varepsilon}^*L=E_{m,\varepsilon}+
(D_{m,\varepsilon}+\varepsilon A)$ where
$D_{m,\varepsilon}+A$ is ample. The positive $(1,1)$-current 
$T_{m,\varepsilon}=(\mu_{m,\varepsilon})_*
\Theta_{D_{m,\varepsilon}+\varepsilon A}$ is a K\"ahler current with logarithmic
poles and its volume approaches $\limsup{n!\over m^n}h^0(X,mL)$ when $\varepsilon\ll 1$ and
$m$ is large.
\qed
\endproof

\noindent
In these terms, we get the following statement.

\claim{(6.6) Proposition} Let $L$ be a big line bundle on the
projective manifold $X$.  Let $\varepsilon > 0$. Then there exists a
modification $\mu: X_\varepsilon \to X$ and a decomposition $\mu^*(L) =
E + \beta $ with $E$ an effective $\bQ$-divisor and $\beta$ a big and
nef $\bQ$-divisor such that
$$\Vol(L) -\varepsilon\le \Vol(\beta) \le \Vol(L).$$
\endclaim
%
It is very useful to observe that the supremum in Definition 6.4 is actually
achieved by a collection of currents whose singularities satisfy a filtering
property.
Namely, if $T_1=\alpha+\ii\ddbar\varphi_1$ and $T_2=\alpha+\ii\ddbar\varphi_2$
are two K\"ahler currents with logarithmic poles in the class of $\alpha$,
then
$$T=\alpha+\ii\ddbar\varphi,\qquad
\varphi=\max(\varphi_1,\varphi_2)\leqno(6.7)$$
is again a K\"ahler current with weaker
singularities than $T_1$ and $T_2$. One could define as well
$$T=\alpha+\ii\ddbar\varphi,\qquad
\varphi={1\over 2m}\log(e^{2m\varphi_1}+e^{2m\varphi_2}),\leqno(6.7')$$
where $m={\rm lcm}(m_1,m_2)$ is the lowest common multiple of the denominators
occuring in $T_1$, $T_2$. Now, take a simultaneous log-resolution
$\mu_m:X_m\to X$ for which the singularities of $T_1$ and $T_2$ are resolved
as $\bQ$-divisors $E_1$
and $E_2$. Then clearly the associated divisor in the decomposition
$\mu_m^* T=[E]+\beta$ is given by $E=\min(E_1,E_2)$. By doing so,
the volume $\int_{X_m}\beta^n$ gets increased, as we shall see in the
proof of Theorem 6.8 below.

\claim{(6.8) Theorem {\rm (Boucksom [Bck02])}} Let $X$ be a compact K\"ahler
manifold. We denote here by $H^{k,k}_{\ge 0}(X)$ the cone of cohomology classes
of type $(k,k)$ which have non-negative intersection with all closed
semi-positive smooth forms of bidegree $(n-k,n-k)$.
{\plainitemindent=7mm
\smallskip
\item{\rm(a)} For each integer $k=1,2,\ldots,n$, there exists a canonical
``mobile intersection product''
$$
\cE\times\cdots\times\cE\to H^{k,k}_{\ge 0}(X), \quad
(\alpha_1,\ldots,\alpha_k)\mapsto \langle\alpha_1\cdot\alpha_2.\cdots.
\alpha_{k-1}\cdot \alpha_k\rangle
$$
such that $\Vol(\alpha)=\langle\alpha^n\rangle$ whenever $\alpha$ is
a big class.
\smallskip
\item{\rm(b)} The product is increasing, homogeneous of
degree $1$ and superadditive in each argument, i.e.\
$$
\langle\alpha_1\cdots(\alpha'_j+\alpha''_j)\cdots \alpha_k\rangle\ge
\langle\alpha_1\cdots\alpha'_j\cdots \alpha_k\rangle+
\langle\alpha_1\cdots\alpha''_j\cdots \alpha_k\rangle.
$$
It coincides with the ordinary intersection
product when the $\alpha_j\in\overline{\cK}$ are nef classes.
\smallskip
\item{\rm(c)}
The mobile intersection product satisfies the Hovanskii-Teissier
inequalities {\rm([Hov79], [Tei79, Tei82])} 
$$
\langle\alpha_1\cdot\alpha_2.\cdots. \alpha_n\rangle\ge
(\langle\alpha_1^n\rangle)^{1/n}\cdots(\langle\alpha_n^n\rangle)^{1/n}\qquad
\hbox{$($with $\langle\alpha_j^n\rangle=\Vol(\alpha_j)\,)$}.
$$
\smallskip
\item{\rm(d)} For $k=1$, the above ``product'' reduces to a
$($non linear$)$ projection operator
$$
\cE\to\cE_1,\qquad \alpha\to\langle\alpha\rangle
$$
onto a certain convex subcone $\cE_1$ of $\cE$ such that
$\overline{\cK}\subset\cE_1\subset\cE$. Moreover, there is
a ``divisorial Zariski decomposition''
$$
\alpha=\{N(\alpha)\}+\langle\alpha\rangle
$$
where $N(\alpha)$ is a uniquely defined effective divisor which is
called the ``negative divisorial part'' of $\alpha$. The map
$\alpha\mapsto N(\alpha)$ is homogeneous and subadditive, and
$N(\alpha)=0$ if and only if $\alpha\in\cE_1$.
\smallskip
\item{\rm(e)} The components of $N(\alpha)$ always consist of
divisors whose cohomology classes are linearly independent,
especially $N(\alpha)$ has at most $\rho=\rank_\bZ\NS(X)$ components.
\vskip0pt}
\endclaim
%
\proof{Proof} We essentially repeat the arguments developped
in~[Bck02], with some simplifications arising from the fact that $X$
is supposed to be K\"ahler from the beginning.
\smallskip

\noindent

(a) First assume that all classes $\alpha_j$ are big, i.e.\
$\alpha_j\in\cE^\circ$. Fix a smooth closed $(n-k,n-k)$ {\it semi-positive}
form $u$ on $X$. We select K\"ahler currents $T_j\in\alpha_j$ with
logarithmic poles, and a simultaneous log-resolution
$\mu:\wt X\to X$ such that
$$
\mu^* T_j=[E_j]+\beta_j.
$$
We consider the direct image current
$\mu_*(\beta_1\wedge\cdots\wedge\beta_k)$ (which is a closed
positive current of bidegree $(k,k)$ on $X$) and the corresponding
integrals
$$
\int_{\wt X}\beta_1\wedge\cdots\wedge\beta_k\wedge\mu^* u\ge 0.
$$
If we change the representative $T_j$ with another current $T_j'$,
we may always take a simultaneous log-resolution such that
$\mu^* T_j'=[E_j']+\beta_j'$, and by using $(6.7')$ we can always
assume that $E'_j\le E_j$. Then $D_j=E_j-E'_j$ is an effective
divisor and we find $[E_j]+\beta_j\equiv [E'_j]+\beta'_j$, hence
$\beta'_j\equiv\beta_j+[D_j]$. A substitution in the integral implies
$$
\eqalign{
\int_{\wt X}\beta_1'\wedge\beta_2
&{}\wedge\cdots\wedge\beta_k\wedge\mu^* u\cr
&=\int_{\wt X}\beta_1\wedge\beta_2\wedge\cdots\wedge\beta_k\wedge\mu^* u+
\int_{\wt X}[D_1]\wedge\beta_2\wedge\cdots\wedge\beta_k\wedge\mu^* u\cr
&\ge \int_{\wt X}\beta_1\wedge\beta_2\wedge\cdots\wedge\beta_k\wedge
\mu^* u.\cr}
$$
Similarly, we can replace successively all forms $\beta_j$ by the $\beta'_j$,
and by doing so, we find
$$
\int_{\wt X}\beta'_1\wedge\beta'_2\wedge\cdots\wedge\beta'_k\wedge\mu^* u
\ge\int_{\wt X}\beta_1\wedge\beta_2\wedge\cdots\wedge\beta_k\wedge\mu^* u.
$$
We claim that the closed positive currents $\mu_*(\beta_1\wedge\cdots
\wedge\beta_k)$ are uniformly bounded in mass. In fact, if $\omega$ is a
K\"ahler metric in $X$, there exists a constant $C_j\ge 0$
such that $C_j\{\omega\}-\alpha_j$ is a K\"ahler class. Hence
$C_j\omega-T_j\equiv\gamma_j$ for some K\"ahler form $\gamma_j$ on $X$.
By pulling back with $\mu$, we find
$C_j\mu^* \omega-([E_j]+\beta_j)\equiv\mu^* \gamma_j$, hence
$$\beta_j\equiv C_j\mu^* \omega-([E_j]+\mu^* \gamma_j).$$
By performing again a substitution in the integrals, we find
$$
\int_{\wt X}\beta_1\wedge\cdots\wedge\beta_k\wedge\mu^* u
\le C_1\cdots C_k\int_{\wt X}\mu^*\omega^k\wedge\mu^* u
= C_1\cdots C_k\int_X\omega^k\wedge u
$$
and this is true especially for $u=\omega^{n-k}$. We can now arrange
that for each of the integrals associated with a countable dense family
of forms $u$,  the supremum is achieved by a sequence of currents
$(\mu_m)_*(\beta_{1,m}\wedge\cdots\wedge\beta_{k,m})$ obtained
as direct images by a suitable sequence of modifications $\mu_m:
\smash{\wt X_m}\to X$.
By extracting a subsequence, we can achieve that this sequence is weakly
convergent and we set
$$
\langle\alpha_1\cdot\alpha_2.\cdots. \alpha_k\rangle =
\mathop{\lim\uparrow}\limits_{m\to+\infty}
\{(\mu_m)_*(\beta_{1,m}\wedge\beta_{2,m}\wedge\cdots\wedge\beta_{k,m})\}
$$
(the monotonicity is not in terms of the currents themselves, but in terms
of the integrals obtained when we evaluate against a smooth closed
semi-positive form~$u$). By evaluating against a basis of positive classes
$\{u\}\in H^{n-k,n-k}(X)$, we infer by Serre duality that the class of
$\langle\alpha_1\cdot\alpha_2.\cdots. \alpha_k\rangle$ is uniquely defined
(although, in general, the representing current is not unique).
\smallskip

\noindent

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

\noindent

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

\noindent

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

Since $N(\alpha)=\alpha-\langle\alpha\rangle$ we easily see that
$N(\alpha)$ is subadditive and that $N(\alpha)=0$ if $\alpha$ is the
class of a smooth semi-positive form. When $\alpha$ is no longer a big
class, we define
$$
\langle\alpha\rangle=\lim_{\delta\downarrow 0}\downarrow\langle\alpha+
\delta\omega\rangle,\qquad
N(\alpha)=\lim_{\delta\downarrow 0}\uparrow N(\alpha+\delta\omega)
$$
(the subadditivity of $N$ implies
$N(\alpha+(\delta+\varepsilon)\omega)\le N(\alpha+\delta\omega)$).
The divisorial Zariski decomposition follows except maybe for the fact
that $N(\alpha)$ might be a convergent countable sum of divisors.
However, this will be ruled out when (e) is proved. As
$N({\scriptscriptstyle\bullet})$ is subadditive
and homogeneous, the set $\cE_1=\{\alpha\in\cE\;;\;N(\alpha)=0\}$ is a
closed convex cone, and we find that $\alpha\mapsto\langle\alpha\rangle$
is a projection of $\cE$ onto $\cE_1$ (according to [Bck02], $\cE_1$
consists of those pseudo-effective classes which are ``nef in
codimension~$1$'').
\smallskip

\noindent

(e) Let $\alpha\in\cE^\circ$, and assume that $N(\alpha)$ contains
linearly dependent components $F_j$. Then already all currents
$T\in \alpha$ should be such that $\mu^* T=[E]+\beta$ where
$F=\mu_* E$ contains those linearly dependent components.
Write $F=\sum\lambda_jF_j$, $\lambda_j>0$ and assume that
$$\sum_{j\in J}c_jF_j\equiv 0$$
for a certain non trivial linear combination. Then some of the
coefficients $c_j$ must be negative (and some other positive).
Then $E$ is numerically equivalent to
$$
E'\equiv E+t\mu^*\Big(\sum\lambda_jF_j\Big),
$$
and by choosing $t>0$ appropriate, we obtain an effective divisor $E'$
which has a zero coefficient on one of the components $\mu^* F_{j_0}$.
By replacing $E$ with $\min(E,E')$ via $(6.7')$, we eliminate the component
$\mu^* F_{j_0}$. This is a contradiction since $N(\alpha)$
was supposed to contain $F_{j_0}$.\qed
\endproof

\claim{(6.9) Definition} For a class $\alpha\in H^{1,1}(X,\bR)$,
we define the {\rm numerical dimension} $\nd(\alpha)$ to be
$\nd(\alpha)=-\infty$ if $\alpha$ is not pseudo-effective, and
$$
\nd(\alpha)=\max\{p\in\bN\;;\;\langle\alpha^p\rangle\ne 0\},\qquad
\nd(\alpha)\in\{0,1,\ldots,n\}
$$
if $\alpha$ is pseudo-effective.
\endclaim
%
By the results of [DP04], a class is big ($\alpha\in\cE^\circ$) if and only
if $\nd(\alpha)=n$. Classes of numerical dimension $0$ can be
described much more precisely, again following Boucksom [Bck02].

\claim{(6.10) Theorem} Let $X$ be a compact
K\"ahler manifold. Then the subset $\cD_0$ of irreducible divisors
$D$ in $X$ such that $\nd(D)=0$ is countable, and these divisors
are rigid as well as their multiples. If $\alpha\in\cE$ is a
pseudo-effective class of numerical dimension~$0$, then $\alpha$
is numerically equivalent to
an effective $\bR$-divisor $D=\sum_{j\in J}\lambda_j D_j$,
for some finite subset $(D_j)_{j\in J}\subset\cD_0$ such that the
cohomology classes $\{D_j\}$ are linearly independent and
some \hbox{$\lambda_j>0$}. If such a linear combination is of
numerical dimension~$0$, then so is any other linear combination of the
same divisors.
\endclaim
%
\proof{Proof} It is immediate from the definition that a pseudo-effective
class is of numerical dimension $0$ if and only if $\langle\alpha\rangle=0$,
in other words if $\alpha=N(\alpha)$. Thus $\alpha\equiv\sum\lambda_jD_j$
as described in 6.10, and since $\lambda_j\langle D_j\rangle\le
\langle\alpha\rangle$, the divisors $D_j$ must themselves have numerical
dimension $0$. There is at most one such divisor $D$ in any given cohomology
class in $NS(X)\cap\cE\subset H^2(X,\bZ)$, otherwise two such divisors
$D\equiv D'$ would yield a blow-up $\mu:\wt X\to X$ resolving the
intersection, and by taking
$\min(\mu^* D,\mu^* D')$ via $(6.7')$, we would find
$\mu^* D\equiv E+\beta$, $\beta\ne 0$, so that
$\{D\}$ would not be of numerical dimension $0$. This implies
that there are at most countably many divisors of numerical
dimension~$0$, and that these divisors are rigid as well as their
multiples.\qed
\endproof

\claim{(6.11) Remark} {\rm If $L$ is an arbitrary holomorphic line bundle,
we define its numerical dimension to be $\nd(L)=\nd(c_1(L))$. Using
the canonical maps $\Phi_{|mL|}$ and pulling-back the Fubini-Study metric
it is immediate to see that $\nd(L)\ge\kappa(L)$.}
\endclaim
%
The above general concept of numerical dimension leads to a very natural
formulation of the abundance conjecture for K\"ahler varieties.

\claim{(6.12) Generalized Abundance Conjecture} Let $X$ be an
arbitrary compact K\"ahler manifold~$X$.
\smallskip
\item{\rm(a)} The Kodaira dimension of $X$ should be equal to its
numerical dimension{\rm :} $\kappa(K_X)=\nd(K_X)$.
\smallskip
\item{\rm(b)} More generally, let $\Delta$ be a $\bQ$-divisor which is
klt $($Kawamata log terminal, i.e.\ such that $c_X(\Delta)>1)$.
Then $\kappa(K_X+\Delta)=\nd(K_X+\Delta)$.\vskip0pt
\endclaim

\claim{(6.13) Remark} {\rm It is obvious that abundance holds in the case
\hbox{$\nd(K_X)=-\infty$} (if $L$ is not pseudo-effective, no
multiple of $L$ can have sections), or in the case \hbox{$\nd(K_X)=n$}
which implies $K_X$ big (the latter property follows e.g.\ from
the solution of the Grauert-Riemenschneider conjecture in the form proven
in [Dem85], see also [DP04]).

In the remaining cases, the most tractable situation is the case when
$\nd(K_X)=0$.
In fact Theorem 6.10 then gives $K_X\equiv\sum\lambda_j D_j$ for some
effective divisor with numerically independent components, $\nd(D_j)=0$.
It follows that the $\lambda_j$ are rational and therefore
$$
K_X\sim \sum\lambda_j D_j+F\qquad
\hbox{where $\lambda_j\in\bQ^+$, $\nd(D_j)=0$ and $F\in\Pic^0(X)$.}
\leqno(*)
$$
If we assume additionally that $q(X)=h^{0,1}(X)$ is zero, then
$mK_X$ is linearly equivalent
to an integral divisor for some multiple $m$, and it follows
immediately that $\kappa(X)=0$. The case of a general projective
manifold with \hbox{$\nd(K_X)=0$} and positive
irregularity $q(X)>0$ has been solved by Campana-Peternell
[CP04], Proposition~3.7. It would be interesting to understand
the K\"ahler  case as well.}
\endclaim
%
\section{7}{The orthogonality estimate}

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

\claim{(7.1) Theorem} Let $X$ be a projective manifold, and
let $\alpha=\{T\}\in\cE^\circ_{\NS}$ be a big class represented by
a K\"ahler current~$T$. Consider
an approximate Zariski decomposition
$$\mu_m^* T_m = [E_m]+[D_m]$$
Then
$$
(D_m^{n-1}\cdot E_m)^2\le 20\,(C\omega)^n\big(\Vol(\alpha)-D_m^n\big)
$$
where $\omega=c_1(H)$ is a K\"ahler form and $C\ge 0$ is a constant such that
$\pm\alpha$ is dominated by $C\omega$ $($i.e., $C\omega\pm\alpha$ is nef$\,)$.
In other words, $E_m$ and $D_m$ become ``more and more orthogonal'' as
$D_m^n$ approaches the volume.
\endclaim
%
\proof{Proof} For every $t\in[0,1]$, we have
$$
\Vol(\alpha)=\Vol(E_m+D_m)\ge \Vol(tE_m+D_m).
$$
Now, by our choice of $C$, we can write $E_m$ as a difference of two
nef divisors
$$E_m=\mu^*\alpha-D_m=\mu_m^*(\alpha+C\omega)-(D_m+C\mu_m^*
\omega).\eqno\qed$$
\endproof

\claim{(7.2) Lemma} For all nef $\bR$-divisors $A$, $B$ we have
$$\Vol(A-B)\ge A^n - nA^{n-1}\cdot B$$
as soon as the right hand side is positive.
\endclaim
%
\proof{Proof} In case $A$ and $B$ are integral divisors, this
is a consequence of holomorphic Morse inequalities (cf.\ (I~2.15)).
If $A$ and $B$ are $\bQ$-divisors, we conclude by the homogeneity of the volume. The
general case of $\bR$-divisors follows by approximation (actually, as it is
defined to be a supremum, the volume function can easily be shown to be lower 
semi-continuous, but it is in fact even continuous, cf.\ [Bck02, 3.1.26]).\qed
\endproof

\claim{(7.3) Remark} {\rm We hope that Lemma 7.2 also holds true on an
arbitrary K\"ahler mani\-fold for arbitrary nef (non neces\-sarily
integral) classes. This would follow from Conjecture (III~2.11) generalizing
holomorphic Morse inequalities to non integral classes, exactly by the
same proof as Theorem~(I~2.14).}
\endclaim

\claim{(7.4) Lemma} Let $\beta_1,\ldots,\beta_n$ and
$\beta'_1,\ldots,\beta'_n$ be nef classes on a compact K\"ahler manifold
$\wt X$ such that each difference $\beta'_j-\beta_j$ is pseudo-effective. Then
the $n$-th intersection products satisfy
$$
\beta_1\cdots\beta_n \le \beta'_1\cdots\beta'_n.
$$
\endclaim
%
\proof{Proof} We can proceed step by step and replace just one $\beta_j$
by $\beta'j\equiv\beta_j+T_j$ where $T_j$ is a closed positive
$(1,1)$-current and the other classes $\beta'_k=\beta_k$, $k\ne j$ are
limits of K\"ahler forms. The inequality is then obvious.\qed
\endproof

\noindent
{\it End of proof of Theorem {\rm 7.1.}}
In order to exploit the lower bound of the volume, we write
$$
tE_m+D_m=A-B,\qquad
A=D_m+t\mu_m^*(\alpha+C\omega),\quad B=t(D_m+C\mu_m^*\omega).
$$
By our choice of the constant $C$, both $A$ and $B$ are nef. Lemma 7.2 and
the binomial formula imply
$$\eqalign{
\Vol(tE_m+&D_m)\ge A^n - nA^{n-1}\cdot B\cr
&=D_m^n+nt\,D_m^{n-1}\cdot\mu_m^*(\alpha+C\omega)+\sum_{k=2}^n
t^k{n\choose k}D_m^{n-k}\cdot\mu_m^*(\alpha+C\omega)^k\cr
&\phantom{{}=D_m^n}-nt\,D_m^{n-1}\cdot(D_m+C\mu_m^*\omega)\cr
&\phantom{{}=D_m^n}-nt^2\sum_{k=1}^{n-1}t^{k-1}{n-1\choose k}
D_m^{n-1-k}\cdot\mu_m^*(\alpha+C\omega)^k\cdot(D_m+C\mu_m^*\omega).
\kern-4pt\cr}
$$
Now, we use the obvious inequalities
$$
D_m\le \mu_m^*(C\omega),\qquad
\mu_m^*(\alpha+C\omega)\le 2\mu_m^*(C\omega),\qquad
D_m+C\mu_m^*\omega\le 2\mu_m^*(C\omega)
$$
in which all members are nef (and where the inequality${}\le$ means
that the difference of classes is pseudo-effective). We use Lemma~7.4
to bound the last summation in the estimate of the volume, and in this
way we get
$$
\Vol(tE_m+D_m)\ge D_m^n+nt D_m^{n-1}\cdot E_m-nt^2\sum_{k=1}^{n-1}2^{k+1}
t^{k-1}{n-1\choose k}(C\omega)^n.
$$
We will always take $t$ smaller than $1/10n$ so that the last summation is
bounded by $4(n-1)(1+1/5n)^{n-2}<4ne^{1/5}<5n$. This implies
$$
\Vol(tE_m+D_m)\ge D_m^n+nt\,D_m^{n-1}\cdot E_m-5n^2t^2(C\omega)^n.
$$
Now, the choice $t={1\over 10n}(D_m^{n-1}\cdot E_m)((C\omega)^n)^{-1}$
gives by substituting
$$
\eqalign{
{1\over 20}{(D_m^{n-1}\cdot E_m)^2\over (C\omega)^n}
&\le \Vol(E_m+D_m)-D_m^n\le \Vol(\alpha)-D_m^n\cr}
$$
(and we have indeed $t\le {1\over 10n}$ by Lemma 7.4), whence Theorem 7.1.
Of course, the constant 20 is certainly not optimal.\qed

\claim{(7.5) Corollary} If $\alpha\in\cE_{\NS}$, then the divisorial
Zariski decomposition $\alpha=N(\alpha)+\langle\alpha\rangle$
is such that
$$
\langle\alpha^{n-1}\rangle\cdot N(\alpha)=0.
$$
\endclaim
%
\proof{Proof} By replacing $\alpha$ with $\alpha+\delta c_1(H)$, one
sees that it is sufficient to consider the case where $\alpha$ is big. Then
the orthogonality estimate implies
$$
\eqalign{
(\mu_m)_*(D_m^{n-1})\cdot (\mu_m)_* E_m
&= D_m^{n-1}\cdot(\mu_m)^* (\mu_m)_* E_m\cr
&\le D_m^{n-1}\cdot E_m \le C (\Vol(\alpha)-D_m^n)^{1/2}.\cr}
$$
Since $\langle\alpha^{n-1}\rangle = \lim (\mu_m)_*(D_m^{n-1})$,
$N(\alpha)=\lim (\mu_m)_* E_m$ and $\lim D_m^n=\Vol(\alpha)$,
we get the desired conclusion in the limit.\qed
\endproof

\section{8}{Dual of the pseudo-effective cone}

We consider here the Serre duality pairing
$$H^{1,1}(X,\bR)\times H^{n-1,n-1}(X,\bR)\lra \bR,\qquad
(\alpha,\beta)\longmapsto \alpha\cdot\beta=\int_X \alpha\wedge\beta.\leqno(8.1)$$
When restricted to real vector subspaces generated by integral classes, 
it defines a perfect pairing
$$\NS_\bR\times \NS^{n-1,n-1}_\bR(X)\lra \bR\leqno(8.2)
$$
where $\NS_\bR\subset H^{1,1}(X,\bR)$ and $\NS^{n-1,n-1}_\bR(X)\subset  H^{n-1,n-1}(X,\bR)$.
Next, we introduce the concept of mobile curves.

\claim{(8.3) Definition} Let $X$ be a smooth projective variety.
{\plainitemindent=7mm
\smallskip
\item{\rm (a)} One defines $\NE(X)\subset \NS^{n-1,n-1}_\bR(X)$ to be the
convex cone generated by cohomology classes of all effective curves
in $H^{n-1,n-1}(X,\bR)$.
\smallskip
\item{\rm (b)} We say that $C$ is a {\rm mobile curve} if $C=C_{t_0}$
is a member of an analytic family $\{C_t\}_{t\in S}$ such that
$\bigcup_{t\in S}C_t=X$
and, as such, is a reduced irreducible $1$-cycle. We define the
mobile cone $\ME(X)$, to be the convex cone generated by all
mobile curves.
\smallskip
\item{\rm (c)} If $X$ is projective, we say that an effective $1$-cycle $C$
is  a {\rm strongly mobile} if we have
$$
C=\mu_*(\wt A_1\cap\cdots\cap \wt A_{n-1})
$$
for suitable very ample divisors $\wt A_j$ on $\wt X$, where
$\mu:\wt X\to X$ is a modification. We let $\ME^s(X)$
be the convex cone generated by all strongly mobile effective
$1$-cycles $($notice that by taking $\wt A_j$ general enough these classes can
be represented by reduced irreducible curves; also, by
Hironaka, one could just restrict oneself to compositions of blow-ups
with smooth centers$)$.\vskip0pt}
\endclaim
%

\noindent 
Clearly, we have
$$
\ME^s(X)\subset\ME(X)\subset\NE(X)\subset\NS^{n-1,n-1}_\bR(X).\leqno(8.4)
$$
Another simple observation is:

\claim{(8.5) Proposition} 
One has $\alpha\cdot C\ge 0$ whenever $\{\alpha\}\in\cE$ and $\{C\}\in\ME(X)$. In other 
words $\cE_{\NS}=\cE\cap\NS_\bR(X)$ is contained in the dual cone $(\ME(X))^\smallvee$.
\endclaim

\proof{Proof} If the class $\{\alpha\}$ is represented by a closed positive current
$T$ and $C=C_{t_0}$ belongs to a covering family $(C_t)_{t\in S}$, it is easy to see that
$T_{|C_t}$ is locally well defined and nonnegative as soon as $C_t$ is not contained in
the set of poles of a local potential $\varphi$ of $T$. However, this occurs only when
$t$ belongs to a pluripolar set $P\subset S$, hence for $t\in S\ssm P$ we~have
$$\alpha\cdot C=\int_{C_t}T_{|C_t}\ge 0.\eqno\square$$
\endproof

\noindent The following statement was first proved in [BDPP04].

\claim{(8.6) Theorem} If $X$ is projective, the cones
$\cE_{\NS}=\overline{\Eff(X)}$ and $\overline{\ME^s(X)}$ are dual with respect to
Serre duality, and we have $\overline{\ME^s(X)}=\overline{\ME(X)}$.
\endclaim
%
In other words, a line bundle $L$ is pseudo-effective if (and
only if) $L\cdot C\ge 0$ for all {\it mobile curves}, i.e., $L\cdot C\ge 0$
for every very generic curve $C$ (not contained in a countable union of
algebraic subvarieties). In fact, by definition of $\ME^s(X)$, it is enough
to consider only those curves $C$ which are images of generic complete
intersection of very ample divisors on some variety $\wt X$, under
a modification \hbox{$\mu:\wt X\to X$}.
By a standard blowing-up argument, it also follows that a line bundle
$L$ on a normal Moishezon variety is pseudo-effective if and only if
$L \cdot C \ge 0$ for every mobile curve~$C$.

\proof{Proof} By (8.5) we have $\cE_{\NS}\subset (\ME(X))^\smallvee$ and 
(8.4) implies $(\ME(X))^\smallvee\subset(\ME^s(X))^\smallvee$,
therefore
$$\cE_{\NS}\subset (\ME^s(X))^\smallvee.\leqno(8.7)$$
If we show that $\cE_{\NS}=(\ME^s(X))^\smallvee$, we get at the same time
$(\ME^s(X))^\smallvee=(\ME(X))^\smallvee$, and therefore by biduality (Hahn-Banach theorem) 
we will infer $\overline{\ME^s(X)}=\overline{\ME(X)}$.
Now, if the inclusion were strict in (8.7), there would be an element
$\alpha\in\partial\cE_{\NS}$ on the boundary of $\cE_{\NS}$ which is in
the interior of $\ME^s(X)^\smallvee$.

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\vskip8mm

\noindent
Let $\omega=c_1(H)$ be an ample class. Since $\alpha\in\partial\cE_{\NS}$,
the class $\alpha+\delta\omega$ is big for every $\delta>0$, and since
$\alpha\in((\ME^s(X))^{\smallvee})^\circ$ we still have
$\alpha-\varepsilon\omega\in(\ME^s(X))^{\smallvee}$ for
$\varepsilon>0$ small. Therefore
$$
\alpha\cdot \Gamma\ge\varepsilon\omega\cdot \Gamma\leqno(8.8)
$$
for every strongly mobile curve $\Gamma$, and therefore for every
$\Gamma\in\overline{\ME^s(X)}$. We are going to contradict (8.8).
Since $\alpha+\delta\omega$ is big, we have an approximate Zariski
decomposition
$$
\mu_\delta^*(\alpha+\delta\omega)=E_\delta+D_\delta.
$$
We pick $\Gamma=(\mu_\delta)_*(D_\delta^{n-1})\in\overline{\ME^s(X)}$.
By the Hovanskii-Teissier concavity inequality
$$
\omega\cdot \Gamma\ge (\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}.
$$
On the other hand
$$
\eqalign{
\alpha\cdot \Gamma&=\alpha\cdot(\mu_\delta)_*(D_\delta^{n-1})\cr
&=\mu_\delta^*\alpha\cdot D_\delta^{n-1}\le
\mu_\delta^*(\alpha+\delta\omega)\cdot D_\delta^{n-1}\cr
&=(E_\delta+D_\delta)\cdot D_\delta^{n-1}
=D_\delta^n+D_\delta^{n-1}\cdot E_\delta.\cr
}
$$
By the orthogonality estimate, we find
$$\eqalign{
{\alpha\cdot \Gamma\over
\omega\cdot \Gamma}
&\le {D_\delta^n + \big(20(C\omega)^n
(\Vol(\alpha+\delta\omega)-D_\delta^n)\big)^{1/2}\over
(\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}}\cr
&\le C'(D_\delta^n)^{1/n}+C''{(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}\over
(D_\delta^n)^{(n-1)/n}}.\cr}
$$
However, since $\alpha\in\partial\cE_{\NS}$, the class $\alpha$ cannot be big
so
$$\lim_{\delta \to 0}D_\delta^n=\Vol(\alpha)=0.$$
We can also take $D_\delta$ to approximate $\Vol(\alpha+\delta\omega)$
in such a way that
$(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}$ tends to $0$ much faster than
$D_\delta^n$. Notice that \hbox{$D_\delta^n\ge\delta^n\omega^n$}, so in fact
it is enough to take
$$
\Vol(\alpha+\delta\omega)-D_\delta^n\le \delta^{2n},
$$
which gives $(\alpha\cdot \Gamma)/(\omega\cdot \Gamma)\le (C'+C'')\delta$.
This contradicts (8.8) for $\delta$ small.\qed
\endproof

\chapter{3}{Asymptotic cohomology functionals\\ and Monge-Amp\`ere operators}
\chapterrunning{3}{Asymptotic cohomology functionals
and Monge-Amp\`ere operators}
\vskip-5mm

The goal of this chapter is to show that there are strong relations
between  certain Monge-Amp\`ere integrals appearing in holomorphic Morse
inequa\-lities, and asymptotic cohomology estimates for tensor
powers of holomorphic line bundles. Especially, we prove that these
relations hold without restriction for projective surfaces, and in
the special case of the volume, i.e.\ of asymptotic $0$-cohomology,
for all projective manifolds. These results can be seen as a 
partial converse to the Andreotti-Grauert vanishing theorem.

\section{0}{Introduction and main definitions}

Throughout this chapter, $X$ denotes a compact complex manifold, 
\hbox{$n=\dim_\bC X$} its complex dimension and $L\to X$ a holomorphic
line bundle. In order to estimate the growth of cohomology groups, it 
is interesting to consider appropriate ``asymptotic cohomology functions''. 
Following partly notation and concepts introduced by A.~K\"uronya 
[K\"ur06, FKL07], we introduce

\claim{(0.1) Definition} Let $X$ be a compact complex manifold and let
$L\to X$ be a holomorphic line bundle.
{\plainitemindent=6.5mm
\item{\rm (a)} The $q$-th asymptotic cohomology functional is defined as
$$
\widehat h^q(X,L):=
\limsup_{k\to+\infty}~{n!\over k^n}h^q(X,L^{\otimes k}).
$$
\item{\rm (b)} The $q$-th asymptotic holomorphic Morse sum of $L$ is
$$
\widehat h^{\,\leq q}(X,L):=
\limsup_{k\to+\infty}~{n!\over k^n}\sum_{0\le j\le q}(-1)^{q-j}h^j(X,L^{\otimes k}).
$$\vskip-5pt}
\endclaim

\noindent
When the $\limsup$'s are limits, we have the obvious relation 
$$
\widehat h^{\,\leq q}(X,L)=\sum_{0\le j\le q}(-1)^{q-j}\widehat h^j(X,L).
$$
\noindent
Clearly, Definition 0.1 can also be given for a $\bQ$-line bundle $L$
or a $\bQ$-divisor $D$, and in the case $q=0$ one gets by (II~6.5) what is
called the volume of $L$ (see also [DEL00], [Bck02], [Laz04]):
$$
\Vol(X,L):=\widehat h^0(X,L)=
\limsup_{k\to+\infty}~{n!\over k^n}h^0(X,L^{\otimes k}).\leqno(0.2)
$$

\section{1}{Extension of the functionals to real cohomology classes}
We are going  to show
that the $\smash{\widehat h}^q$ functional induces a continuous map
$$
\DNS_\bR(X)\ni \alpha\mapsto \widehat h^q_{\DNS}(X,\alpha),\leqno(1.1)
$$
which is defined on the ``divisorial N\'eron-Severi space'' $\DNS_\bR(X)
\subset H^{1,1}_\BC(X,\bR)$, i.e.\ the vector space spanned by
real linear combinations of classes of divisors in the real Bott-Chern 
cohomology group of bidegree $(1,1)$. Here $H^{p,q}_\BC(X,\bC)$ is defined
as the quotient of $d$-closed $(p,q)$-forms by $\smash{\ddbar}$-exact 
$(p,q)$-forms, and there is a natural conjugation 
$H^{p,q}_\BC(X,\bC)\to H^{q,p}_\BC(X,\bC)$ which allows us to speak
of real classes when $q=p$. Notice that $H^{p,q}_\BC(X,\bC)$ coincides with
the usual Dolbeault cohomology group $H^{p,q}(X,\bC)$ when $X$ is K\"ahler,
and that $\DNS_\bR(X)$ coincides with the usual N\'eron-Severi space
$$
\NS_\bR(X)=\bR\otimes_\bQ\big(H^2(X,\bQ)\cap H^{1,1}(X,\bC)\big)
\leqno(1.2)
$$
when $X$ is projective (the inclusion can be strict in general, 
e.g.\ on complex 2-tori which only have indefinite
integral $(1,1)$-classes, cf.\ [BL04]).

For $\alpha\in\NS_\bR(X)$ (resp.\ $\alpha\in\DNS_\bR(X)$), we set
$$
\leqalignno{
\widehat h^q_{\NS}(X,\alpha)~~\hbox{
$\Big($resp.\ $\widehat h^q_{\DNS}(X,\alpha)\Big)$}
&=
\limsup_{k\to+\infty, {1\over k}c_1(L)\to \alpha}~{n!\over k^n}h^q(X,L)\cr
&=\inf_{\varepsilon>0,\,k_0>0}~~
\sup_{k\ge k_0,\Vert{1\over k}c_1(L)-\alpha\Vert\le\varepsilon}
~{n!\over k^n}h^q(X,L).&(1.3)\cr}
$$
when the pair $(k,L)$ runs over $\bN^*\times\Pic(X)$, resp.\ 
over $\bN^*\times\Pic_D(X)$ where $\Pic_D(X)\subset\Pic(X)$ is 
the subgroup generated by ``divisorial
line bundles'', i.e.\ line bundles of the form $\cO_X(D)$. 
Similar definitions
can be given for the Morse sum functionals $\widehat h^{\le q}_{\NS}(X,\alpha)$
and $\widehat h^{\le q}_{\DNS}(X,\alpha)$.
Clearly $\widehat h^{\le q}_{\DNS}(X,\alpha)\le \widehat h^{\le q}_{\NS}
(X,\alpha)$ on $\DNS_\bR(X)$, but we do not know at this point whether
this is always an equality. From the very definition, 
$\smash{\widehat h^q_{\NS}}$ , $\smash{\widehat h^{\le q}_{\NS}}$ 
(and likewise $\smash{\widehat h^q_{\DNS}}$ , $\smash{\widehat h^{\le q}_{\DNS}}$)
are upper semi-continuous functions which are positively homogeneous 
of degree $n$, namely
$$
\widehat h^q_{\NS}(X,\lambda\alpha)=
\lambda^n\widehat h^q_{\NS}(X,\alpha)\leqno(1.4)
$$
for all $\alpha\in\NS_\bR(X)$ and all $\lambda\ge 0$. Notice
that $\widehat h^q_{\NS}(X,\alpha)$ and $\widehat h^{\le q}_{\NS}(X,\alpha)$
are always finite thanks to holomorphic Morse inequalities
(see below).

\claim{(1.5) Proposition} 
{\plainitemindent=6.5mm
\item{\rm(a)} For $L\in\Pic_D(X)$, one has
$\widehat h^q(X,L){=}\widehat h^q(X,c_1(L))$ and
$\widehat h^{\le q}(X,L){=}\widehat h^{\le q}_{\DNS}(X,c_1(L))$,
in particular
asymptotic cohomology depends only on the numerical class of~$L$.
\smallskip
\item{\rm(b)} The map $\alpha\mapsto\widehat h^q_{\DNS}(X,\alpha)$ is
$($locally$)$ Lipschitz continuous on $\DNS_\bR(X)$.
\smallskip
\item{\rm(c)} When $q=0$, $\widehat h^0_{\DNS}(X,\alpha)$ and
$\widehat h^0_{\NS}(X,\alpha)$ coincide on $\DNS_\bR(X)$ and the
limsups are limits.\vskip0pt}
\endclaim

The proof is derived from arguments quite similar to those already
developed in [K\"ur06] (see also [Dem10a] for the non projective situation).
If $D=\sum p_jD_j$ is
an integral divisor, we define its norm to be $\Vert D\Vert=\sum |p_j|
\Vol_\omega(D_j)$, where the volume of an irreducible divisor is computed
by means of a given Hermitian metric $\omega$ on~$X$; in other words, this
is precisely the mass of the current of integration $[D]$ with
respect to~$\omega$.
Clearly, since $X$ is compact, we get equivalent norms for all
choices of Hermitian metrics $\omega$ on~$X$. We can also use $\omega$
to fix a normalized metric on $H^{1,1}_\BC(X,\bR)$. Elementary properties
of potential theory show that $\Vert c_1(\cO(D))\Vert\le C\Vert D\Vert$
for some constant $C>0$ (but the converse inequality is of course wrong
in most cases). Proposition 1.5 is a simple consequence of the
more precise cohomology estimates (1.9) which will be obtained below.
The special case $q=0$ is easier, in fact, one can get non zero
values for $\widehat h^0(X,L)$ only when $L$ is big, i.e.\ when
$X$ is Moishezon (so that we are always reduced to the divisorial
situation); the fact that limsups are limits was proved in II~(6.5).
We postpone the proof to section~19, which will provide stronger results
based on approximate Zariski decomposition.

\claim{(1.6) Lemma} Let $X$ be a compact complex $n$-fold.
Then for every coherent sheaf $\cF$ on~$X$, there is a constant $C_{\cF}>0$
such that for every holomorphic line bundle $L$ on~$X$
we have
$$
h^q(X,\cF\otimes\cO_X(L))\le C_\cF(\Vert c_1(L)\Vert+1)^p
$$
where $p=\dim\Supp\cF$.
\endclaim

\proof{Proof} We prove the result by induction on $p\,$; it is indeed clear
for $p=0$ since we then have cohomology only in degree $0$ and the
dimension of $H^0(X,\cF\otimes\cO_X(L))$ does not depend on~$L$
when $\cF$ has finite support.
Let us consider the support $Y$ of $\cF$ and a resolution of singularity
$\mu:\widehat Y\to Y$ of the corresponding (reduced) analytic space.
Then $\cF$ is an $\cO_Y$-module for some non
necessarily reduced complex structure $\cO_Y=\cO_X/\cJ$ on~$Y$. We can
look at the reduced structure $\cO_{Y,\red}=\cO_X/\cI$, $\cI=\sqrt{\cJ}$,
and filter $\cF$ by $\cI^k\cF$, $k\ge 0$. Since $\cI^k\cF/\cI^{k+1}\cF$ is a
coherent $\cO_{Y,\red}$-module, we can easily reduce the situation to the
case where $Y$ is reduced and $\cF$ is an $\cO_Y$-module. In that case
the cohomology
$$
H^q(X,\cF\otimes\cO_X(L))=H^q(Y,\cF\otimes\cO_Y(L_{|Y}))
$$
just lives on the reduced space~$Y$.

Now, we have an injective sheaf morphism
$\cF\to\mu_*\mu^*\cF$ whose cokernel $\cG$ has support in dimension~$<p$.
By induction on $p$, we conclude from the exact sequence that
$$
\big|h^q(X,\cF\otimes\cO_X(L))-h^q(X,\mu_*\mu^*\cF\otimes\cO_X(L))\big|
\le C_1(\Vert c_1(L)\Vert+1)^{p-1}.
$$
The functorial morphisms
$$
\eqalign{
\mu^*&{}:H^q(Y,\cF\otimes\cO_Y(L_{|Y}))\to H^q(\widehat Y,\mu^*\cF
\otimes\cO_{\widehat Y}(\mu^*L)_{|Y}),\cr
\mu_*&{}:H^q(\widehat Y,\mu^*\cF
\otimes\cO_{\widehat Y}(\mu^*L)_{|Y})
\to H^q(Y,\mu_*\mu^*\cF\otimes\cO_Y(L_{|Y}))\cr}
$$
yield a composition
$$\mu_*\circ\mu^*:H^q(Y,\cF\otimes\cO_Y(L_{|Y}))\to
H^q(Y,\mu_*\mu^*\cF\otimes\cO_Y(L_{|Y}))
$$
induced by the natural injection $\cF\to\mu_*\mu^*\cF$. This implies
$$
h^q(Y,\cF\otimes\cO_Y(L_{|Y}))\le
h^q(\widehat Y,\mu^*\cF\otimes\cO_{\widehat Y}(\mu^*L_{|Y}))
+C_1(\Vert c_1(L)\Vert+1)^{p-1}.
$$
By taking a suitable modification $\mu':Y'\to Y$ of the
desingularization $\widehat Y$, we can assume that $(\mu')^*\cF$ is
locally free modulo torsion. Then we are reduced to the case where
$\cF'=(\mu')^*\cF$ is a locally free sheaf on a smooth manifold $Y'$,
and $L'=(\mu')^*L_{|Y}$.
In this case, we apply Morse inequalities to conclude
that $h^q(Y',\cF'\otimes\cO_{Y'}(L'))\le C_2(\Vert c_1(L')\Vert+1)^p$.
Since $\Vert c_1(L')\Vert\le C_3\Vert c_1(L)\Vert$ by pulling-back,
the statement follows easily.\qed

\claim{(1.7) Corollary} For every irreducible divisor $D$ on $X$, there
exists a constant $C_D$ such that
$$
h^q(D,\cO_D(L_{|D}))\le C_D(\Vert c_1(L)\Vert+1)^{n-1}
$$
\endclaim

\proof{Proof} It is enough to apply Lemma~1.6 with $\cF=(i_D)_*\cO_D$ where
$i_D:D\to X$ is the injection.\qed

\claim{(1.8) Remark} {\rm It is very likely that one can get an ``elementary''
proof of Lemma~1.6 without invoking resolutions of singularities, e.g.\
by combining the Cartan-Serre finiteness argument along with the standard
Serre-Siegel proof based ultimately on the Schwarz lemma. In this
context, one would invoke $L^2$ estimates to get explicit bounds for
the homotopy
operators between \v{C}ech complexes relative to two coverings
$\cU=(B(x_j,r_j))$, $\cU'=(B(x_j,r_j/2))$ of $X$ by concentric balls.
By exercising enough care in the estimates, it is likely that one could
reach an explicit dependence $C_D\le C'\Vert D\Vert$ for the constant
$C_D$ of Corollary~1.7. The proof would of course become much more
technical than the rather naive brute force approach we have used.}
\endclaim

\claim{(1.9) Theorem} Let $X$ be a compact complex manifold. Fix a finitely
generated subgroup $\Gamma$ of the group of $\bZ$-divisors on~$X$. Then
there are constants $C$, $C'$ depending only on~$X$, its Hermitian metric
$\omega$ and the subgroup $\Gamma$, satisfying the following properties.
\smallskip
{\plainitemindent=6.5mm
\item{\rm(a)} Let $L$ and $L'=L\otimes\cO(D)$ be holomorphic line bundles
on $X$, where $D\in\Gamma$ is an integral divisor. Then
$$
\big| h^q(X,L')-h^q(X,L)\big|
\le C(\Vert c_1(L)\Vert+\Vert D\Vert)^{n-1}\Vert D\Vert.
$$
\item{\rm(b)} On the subspace $\DNS_\bR(X)$, the asymptotic $q$-cohomology
function $\widehat h^q_{\DNS}$ satisfies a global estimate
$$
\big|\widehat h^q_{\DNS}(X,\beta)-\widehat h^q_{\DNS}(X,\alpha)\big|\le C'
(\Vert \alpha\Vert+\Vert\beta\Vert)^{n-1}\Vert \beta-\alpha\Vert.
$$}
\vskip-\baselineskip\noindent
In particular $($without any further assumption on $X)$, $\widehat h^q_{\DNS}$
is locally Lipschitz continuous on $\DNS_\bR(X)$.
\endclaim

\proof{Proof} (a) We want to compare the cohomology of
$L$ and $L'=L\otimes\cO(D)$ on $X$.
For this we write $D=D_+-D_-$, and compare the cohomology of the pairs
$L$ and $L_1=L\otimes\cO(-D_-)$ one one hand, and of $L'$ and
$L_1=L'\otimes\cO(-D_+)$ on the other hand. Since
$\Vert c_1(\cO(D))\Vert\le C\Vert D\Vert$ by elementary potential theory,
we see that is is enough to consider the case of a negative divisor,
i.e.\ $L'=L\otimes\cO(-D)$, $D\ge 0$. If~$D$ is an irreducible
divisor, we use the exact sequence
$$
0\to L\otimes\cO(-D) \to L \to \cO_D\otimes L_{|D}\to 0
$$
and conclude by Corollary 1.7 that
$$
\eqalign{
\big|h^q(X,L\otimes\cO(-D))-h^q(X,L)\big|&\le
h^q(D,\cO_D\otimes L_{|D})+h^{q-1}(D,\cO_D\otimes L_{|D})\cr
&\le 2C_D(\Vert c_1(L)\Vert+1)^{n-1}.\cr}
$$
For $D=\sum p_jD_j\ge 0$, we easily get by induction
$$
\big|h^q(X,L\otimes\cO(-D))-h^q(X,L)\big|\le
2\sum_j p_jC_{D_j}\Big(\Vert c_1(L)\Vert+
\sum_k p_k\Vert\nabla_k\Vert+1\Big)^{n-1}.
$$
If we knew that $C_D\le C'\Vert D\Vert$ as expected in Remark~1.6, then
the argument would be complete without any restriction on $D$.
The trouble disappears if we fix $D$ in
a finitely generated subgroup $\Gamma$ of divisors, because only finitely
many irreducible components appear in that case, and so we have to deal
with only finitely many constants $C_{D_j}$. Property 1.9~(a) is proved.

\medskip\noindent
~~~~(b) Fix once for all a finite set of divisors $(\Delta_j)_{1\le j\le t}$
providing a basis of $\DNS_\bR(X){\subset}$ $H^{1,1}_\BC(X,\bR)$.
Take two elements $\alpha$ and $\beta$ in $\DNS_\bR(X)$, and fix
$\varepsilon>0$. Then $\beta-\alpha$ can be $\varepsilon$-approximated
by a $\bQ$-divisor $\sum\lambda_jD_j$, $\lambda_j\in \bQ$,
and we can find a pair $(k,L)$ with $k$ arbitrary large
such that ${1\over k}c_1(L)$ is $\varepsilon$-close to $\alpha$ and
${n!/k^n}h^q(X,L)$ approaches $\widehat h^q_{\DNS}(X,\alpha)$ by $\varepsilon$.
Then ${1\over k}L+\sum \lambda_j\Delta_j$ approaches $\beta$ as closely
as we want. When approximating $\beta-\alpha$, we can arrange that
$k\lambda_j$ is an integer by taking $k$ large enough. Then
$\beta$ is approximated by ${1\over k}c_1(L')$ with
$L'=L\otimes\cO(\sum k\lambda_j\Delta_j)$. Property (a) implies
$$
\eqalign{
h^q(X,L')-h^q(X,L)
&\ge
-C\Big(\Vert c_1(L)\Vert+\Big\Vert\sum k\lambda_j\Delta_j\Big\Vert\Big)^{n-1}
\Big\Vert\sum k\lambda_j\Delta_j\Big\Vert\cr
&\ge -Ck^n\big(\Vert\alpha\Vert+\varepsilon+\Vert\beta-\alpha\Vert+\varepsilon)^{n-1}
(\Vert\beta-\alpha\Vert+\varepsilon).\cr}
$$
We multiply the previous inequality by $n!/k^n$ and get in this way
$$
{n!\over k^n}h^q(X,L')\ge \widehat h^q_{\DNS}(X,\alpha)-\varepsilon
 -C'\big(\Vert\alpha\Vert+\Vert\beta\Vert+\varepsilon)^{n-1}
(\Vert\beta-\alpha\Vert+\varepsilon).
$$
By taking the limsup and letting $\varepsilon\to 0$, we finally obtain
$$
\widehat h^q_{\DNS}(X,\beta)-\widehat h^q_{\DNS}(X,\alpha)\ge
-C'\big(\Vert\alpha\Vert+\Vert\beta\Vert)^{n-1}\Vert\beta-\alpha\Vert.
$$
Property 1.9~(b) follows by exchanging the roles of $\alpha$ and $\beta$.\qed
\endproof

\section{2}{Transcendental asymptotic cohomology functions}

Our ambition is to extend the function $\smash{\widehat h^q_{\NS}}$
in a natural way to the full cohomology group $H^{1,1}_\BC(X,\bR)$. The main
trouble, already when $X$ is projective algebraic, is that the Picard
number $\rho(X)=\dim_\bR\NS_\bR(X)$ may be much smaller than
$\dim_\bR H^{1,1}_\BC(X,\bR)$, namely, there can be rather few integral classes
of type~$(1,1)$ on~$X$. It is well known for instance that $\rho(X)=0$ for
a generic complex torus of dimension $n\ge 2$, while
$\dim_\bR H^{1,1}_\BC(X,\bR)=n^2$. However, if we look at the natural
morphism
$$
H^{1,1}_\BC(X,\bR)\to H^2_\DR(X,\bR)\simeq H^2(X,\bR)
$$
to de Rham cohomology, then $H^2(X,\bQ)$ is dense in $H^2(X,\bR)$. Therefore,
given a class $\alpha\in H^{1,1}_\BC(X,\bR)$ and a smooth $d$-closed
$(1,1)$-form $u$ in $\alpha$,
we can find an infinite sequence ${1\over k}L_k$ ($k\in S\subset\bN$) of
topological $\bQ$-line bundles, equipped with Hermitian metrics $h_k$ and
compatible connections $\nabla_k$ such that the curvature forms
${1\over k}\Theta_{\nabla_k}$ converge to $u$. By using Kronecker's approximation
with respect to the integral lattice $H^2(X,\bZ)/{\rm torsion}
\subset H^2(X,\bR)$, we can even achieve a fast diophantine approximation
$$\Vert  \Theta_{\nabla_k}-ku\Vert\le Ck^{-1/b_2}\leqno(2.1)$$
for a suitable infinite subset $k\in S\subset\bN$ of multipliers. Then
in particular
$$
\Vert\Theta_{\nabla_k}^{0,2}\Vert=\Vert\Theta_{\nabla_k}^{0,2}-
k\,u^{0,2}\Vert\le Ck^{-1/b_2},\leqno(2.2)
$$
and we see that $(L_k,h_k,\nabla_k)$ is a $C^\infty$ Hermitian
line bundle which is extremely close to being holomorphic, since
$(\nabla_k^{0,1})^2=\Theta_{\nabla_k}^{0,2}$ is very small. We fix a
Hermitian metric $\omega$ on $X$ and introduce
the complex Laplace-Beltrami operator
$$
\overline{\bigsquare}_{k,q}
=(\nabla_k^{0,1})(\nabla_k^{0,1})^*+(\nabla_k^{0,1})^*(\nabla_k^{0,1})
\quad\hbox{acting on $L^2(X,\Lambda^{0,q}T_X^*\otimes L_k)$.}
$$
We look at its eigenspaces with respect to the $L^2$ 
metric induced by $\omega$ on $X$ and $h_k$ on~$L_k$. In the
holomorphic case, Hodge theory tells us that the $0$-eigenspace is
isomorphic to $H^q(X,\cO(L_k))$, but in the ``almost holomorphic case''
the $0$-eigenvalues deviate from~$0$, essentially by a
shift of the order of magnitude of
$\Vert\Theta_{\nabla_k}^{0,2}\Vert\sim k^{-1/b_2}$
(see also the PhD thesis of L.~Laeng [Lae02], Chapter~4, for more details).
It is thus natural to introduce in this case

\claim{(2.3) Definition} Let $X$ be a compact complex manifold and
$\alpha\in H^{1,1}_\BC(X,\bR)$ an arbitrary Bott-Chern $(1,1)$-class.
We define the ``transcendental'' asymptotic $q$-cohomology functions
to be
{\plainitemindent=6.5mm
\item{\rm(a)}\kern19pt
$\displaystyle
\widehat h^q_\tr(X,\alpha)=\inf_{u\in\alpha}~~
\limsup_{\varepsilon\to 0,\,k\to+\infty,\,L_k,\,h_k,\,\nabla_k,
{1\over k}\Theta_{\nabla_k}\to u}~~{n!\over k^n}
N(\overline{\bigsquare}_{k,q},\le k\varepsilon)$\smallskip
\item{\rm(b)}\kern15pt
$\displaystyle
\widehat h^{\le q}_\tr(X,\alpha)=\inf_{u\in\alpha}~~
\limsup_{\varepsilon\to 0,\,k\to+\infty,\,L_k,\,h_k,\,\nabla_k,
{1\over k}\Theta_{\nabla_k}\to u}~~{n!\over k^n}
\sum_{0\le j\le q}(-1)^{q-j}
N(\overline{\bigsquare}_{k,j},\le k\varepsilon)$\vskip4pt\noindent
where the $\limsup$ runs over all $5$-tuples $(\varepsilon,k,L_k,h_k,\nabla_k)$,
and where $N(\overline{\bigsquare}_{k,q},k\varepsilon)$ denotes the sum of
dimensions of all eigenspaces of eigenvalues at most equal to $k\varepsilon$
for the Laplace-Beltrami operator
$\overline{\bigsquare}_{k,q}$ on $L^2(X,\Lambda^{0,q}T_X^*\otimes L_k)$
associated with $(L_k,h_k,\nabla_k)$ and the base Hermitian metric~$\omega$.}
\endclaim

The word ``transcendental'' refers here to the fact that we deal with classes
$\alpha$ of type $(1,1)$ which are not algebraic or even analytic. Of course,
in the definition, we could have restricted the limsup to families satisfying
a better approximation property $\Vert{1\over k}\Theta_{\nabla_k}-u\Vert\le C
k^{-1-1/b_2}$ for some large constant $C$ (this would lead a priori to
a smaller limsup, but there is enough stability in the parameter
dependence of the spectrum for making such a change irrelevant).
The minimax principle easily shows that Definition 2.1 does not
depend on $\omega$, as the eigenvalues
are at most multiplied or divided by constants under a change of base metric.
When $\alpha\in\NS_\bR(X)$, by restricting our families
$\{(\varepsilon,k,L_k,h_k,\nabla_k)\}$ to the case
of holomorphic line bundles only, we get the obvious inequalities
$$
\leqalignno{
\widehat h^q_{\NS}(X,\alpha)&\le \widehat h^q_\tr(X,\alpha),\qquad
\forall\alpha\in\NS_\bR(X),&(2.4{\rm a})\cr
\widehat h^{\le q}_{\NS}(X,\alpha)&\le \widehat h^{\le q}_\tr(X,\alpha),
\kern16pt
\forall\alpha\in\NS_\bR(X).&(2.4{\rm b})\cr}
$$
It is natural to raise the question whether these inequalities are always 
equalities. Hopefully, the calculation of the quantities
 $\lim_{k\to+\infty}{n!\over k^n}
N(\overline{\bigsquare}_{k,q},\le k\varepsilon)$ is a problem of spectral theory
which is completely understood thanks to Chapter~I (see also [Dem85, 91]).
In fact, by Corollary I~(1.13), the above limit can be evaluated explicitly 
for any value of $\varepsilon\in\bR$, except possibly for a countable 
number of values of $\varepsilon$ for which jumps occur; one only has 
to take care that the 
non-integrability of $\dbar$ due to the diophantine approximation does
not contribute asymptotically to the eigenvalue distribution, a fact which 
follows immediately from (2.2) (cf.\ [Lae02]).

\claim{(2.5) Theorem} With the above notations and assumptions, let
us introduce at each point $x$ in $X$ the ``spectral density function'',
defined as a finite sum
$$
\nu_u(\lambda)={n!\,(4\pi)^{s-n}\over (n-s)!}|u_1|\ldots |u_s|
\sum_{(p_1,\ldots,p_s)\in\bN^s}\Big(\lambda-\sum_{j=1}^s
(2p_j+1)|u_j|\Big)_+^{n-s}
$$
where $s=s(x)$ is the rank of the real $(1,1)$-form $u$ at $x$, and 
$u_j$, $1\le j\le s$, its non zero eigenvalues with respect to the
base Hermitian metric~$\omega$, and $u_{s+1}=\ldots=u_n=0$. 
For each multi-index $J\subset\{1,2,\ldots,n\}$, let us set
$u_J=\sum_{j\in J}u_j$. Then the asymptotic spectrum of 
$\smash{\overline{\bigsquare}_{k,q}}$ admits the estimate
$$\lim_{k\to+\infty}
{n!\over k^n}\,N(\overline{\bigsquare}_{k,q},\le k\lambda)=
\int_X\sum_{|J|=q}\nu_u(\lambda+u_{\complement J}-u_J)\,dV_\omega
$$
except possibly for a countable number of values of $\lambda$
which are discontinuities of the right hand integral as an increasing
integral of $\lambda$.
\endclaim

\claim{(2.6) Corollary} We have $($as a limit rather than just a
$\limsup\,)$ the spectral estimate
$$
\lim_{\varepsilon\to 0,\,k\to+\infty,\,L_k,\,h_k,\,\nabla_k,
{1\over k}\Theta_{\nabla_k}\to u}~~{n!\over k^n}
N(\overline{\bigsquare}_{k,q},\le k\varepsilon)=
\int_{X(u,q)}(-1)^qu^n.
$$
\endclaim

\vskip-12pt\noindent
Coming back to the transcendental asymptotic cohomology functions, we get
the following fundamental result, which gives in some sense an explicit formula
for $\widehat h^q_\tr(X,\alpha)$ and $\smash{\widehat h^{\le q}_\tr(X,\alpha)}$
in terms of Monge-Amp\`ere operators.

\claim{(2.7) Theorem} The $\limsup$'s defining $\widehat h^q_\tr(X,\alpha)$
and $\widehat h^{\le q}_\tr(X,\alpha)$ are limits, and we have
\vskip5pt
{\plainitemindent=6.5mm
\item{\rm(a)}\kern19pt
$\displaystyle
\widehat h^q_\tr(X,\alpha)=\inf_{u\in\alpha}\int_{X(u,q)}(-1)^qu^n\qquad
(\hbox{$u$ smooth}).
$
\smallskip
\item{\rm(b)}\kern15pt
$\displaystyle
\widehat h^{\le q}_\tr(X,\alpha)=\inf_{u\in\alpha}\int_{X(u,\le q)}(-1)^qu^n
\quad~(\hbox{$u$ smooth}).
$}
\endclaim

\noindent
Now, if $L\to X$ is a holomorphic line bundle, we have by definition
$$
\wh h^{\le q}(X,L)\le \wh h^{\le q}_\DNS(X,c_1(L))\le \wh h^
{\le q}_\NS(X,c_1(L))
\le \inf_{u\in c_1(L)}\int_{X(u,\le q)}(-1)^qu^n
\leqno(2.8)
$$
($u$ smooth), where the last inequality is a consequence of holomorphic
Morse inequalities. We hope for the following conjecture which would
imply that we always have equalities.

\claim{(2.9) Conjecture} For every holomorphic line bundle $L\to X$ on
a compact complex manifold $X$, we have
{\plainitemindent=6.5mm
\item{\rm(a)}\kern17pt
$\displaystyle
\widehat h^q(X,L)=\inf_{u\in\alpha}\int_{X(u,q)}(-1)^qu^n,~~~
\hbox{$u$ smooth},$\smallskip
\item{\rm(b)}\kern17pt
$\displaystyle
\widehat h^{\le q}(X,L)=\inf_{u\in\alpha}\int_{X(u,\le q)}(-1)^qu^n,~~~
\hbox{$u$ smooth}.$\smallskip}
\endclaim

\noindent
Since the right hand side is easily seen to depend continuously 
on~$\alpha\in H^{1,1}_\BC(X,\bC)$, one would get~:

\claim{(2.10) Corollary of the conjecture} If $(2.9)$ holds true, then
$$\widehat h^q_{\NS}(X,\alpha)=\widehat h^q_{\tr}(X,\alpha)\qquad\hbox{and}\qquad
\hbox{\rm (b)}\qquad\widehat h^{\le q}_{\NS}(X,\alpha)=\widehat h^{\le q}_{\tr}(X,\alpha)
\leqno\hbox{\rm(a)}$$
for all classes $\alpha\in \NS_\bR(X)$.
\endclaim

In general, equalities 2.9~(a,$\,$b) seem rather hard to prove. In some 
sense, they would stand as an asymptotic converse of the 
Andreotti-Grauert theorem
[AG62]~: under a suitable $q$-convexity assumption, the latter asserts
the vanishing of related cohomology groups in degree~$q$; here,
conversely, assuming a known growth of these groups in degree~$q$, we
expect to be able to say something about the $q$-index sets of
suitable Hermitian metrics on the line bundles under consideration.
The only cases where we have a positive answer to Question 2.8 are
when $X$ is projective and $q=0$ or $\dim X\le 2$ (see 
Theorems~4.1 and 5.1 below). In~the general setting of compact
complex manifolds, we also hope for the following ``transcendental''
case of holomorphic Morse inequalities.

\claim{(2.11) Conjecture} Let $X$ be a compact complex $n$-fold and
$\alpha$ an arbitrary cohomology class in~$H^{1,1}_\BC(X,\bR)$. Then
the volume, defined as the supremum
$$
\Vol(\alpha)
:=\sup_{0<T\in\alpha}\int_{X\ssm\Sing(T)}T^n,\leqno(2.12)
$$
extended to all K\"ahler currents $T\in\alpha$ with analytic singularities
$($see Definition {\rm II~(4.4)}$)$, satisfies
$$
\Vol(\alpha)\ge\sup_{u\in\alpha}\int_{X(u,0)\cup X(u,1)}u^n\leqno(2.13)
$$
where $u$ runs over all smooth closed  $(1,1)$ forms.
In particular, if the right hand side is positive, then $\alpha$
contains a K\"ahler current.
\endclaim

By the holomorphic Morse inequalities, Conjecture 2.11 holds true in
case $\alpha$ is an integral class. Our hope is that the general case
can be attained by the diophantine approximation technique described
earlier; there are however major hurdles, see [Lae02] for a few hints
on these issues.

\section{3}{Invariance by modification}

We end this section by the observation that the asymptotic cohomology 
functions are invariant by modification, namely that for every 
modification $\mu:\wt X\to X$ and every line bundle $L$ we have~e.g.
$$
\wh h^q(X,L)=\wh h^q(\wt X,\mu^*L).\leqno(3.1)
$$
In fact the Leray spectral sequence provides an $E_2$ term
$$
E_2^{p,q}=H^p(X,R^q\mu_*\cO_{\wt X}(\mu^*L^{\otimes k}))=
H^p(X,\cO_X(L^{\otimes k})\otimes R^q\mu_*\cO_{\wt X}).
$$
Since $R^q\mu_*\cO_{\wt X}$ is equal to $\cO_X$ for $q=0$ and is supported
on a proper analytic subset of $X$ for $q\ge 1$, one infers that
$h^p(X,\cO_X(L^{\otimes k}\otimes R^q\mu_*\cO_{\wt X})=O(k^{n-1})$ for all
$q\ge 1$. The spectral sequence implies that
$$
h^q(X,L^{\otimes k})-\wh h^q(\wt X,\mu^*L^{\otimes k})=O(k^{n-1}).
$$
We claim that the Morse integral infimums are also invariant by modification.

\claim{(3.2) Proposition} Let $(X,\omega)$ be a compact K\"ahler manifold,
$\alpha\in H^{1,1}(X,\bR)$ a real cohomology class and 
$\mu:\wt X\to X$ a modification. Then
$$
\leqalignno{
&\strut\kern20pt\inf_{u\in \alpha}\int_{X(u,q)}(-1)^qu^n
=\inf_{v\in \mu^*\alpha}\int_{X(v,q)}(-1)^qv^n,
&\hbox{\rm(a)}\cr
&\strut\kern20pt\inf_{u\in \alpha}\int_{X(u,\le q)}(-1)^qu^n
=\inf_{v\in \mu^*\alpha}\int_{X(v,\le q)}(-1)^qv^n.
&\hbox{\rm(b)}\cr}
$$
\endclaim

\proof{Proof} Given $u\in\alpha$ on $X$, we obtain Morse integrals with the same 
values by taking $v=\mu^*u$ on~$\wt X$, hence the infimum on $\wt X$ 
is smaller or equal to what is on~$X$. Conversely, we have to show 
that given a smooth representative 
$v\in\mu^*\alpha$ on $\smash{\wt X}$, one can find a smooth representative 
$u\in X$ such that the Morse integrals do not differ much. We can
always assume that $\wt X$ itself is K\"ahler, since by Hironaka [Hir64]
any modification $\smash{\wt X}$ is dominated by a composition of blow-ups
of $X$. Let us fix some $u_0\in\alpha$ and write
$$
v=\mu^*u_0+dd^c\varphi
$$
where $\varphi$ is a smooth function on $\wt X$. We adjust $\varphi$ by a
constant in such a way that $\varphi\ge 1$ on $\wt X$. There exists an analytic
set $S\subset X$ such that $\mu:\wt X\ssm\mu^{-1}(S)\to X\ssm S$ is a
biholomorphism, and a quasi-psh function $\psi_S$ which is smooth on
$X\ssm S$ and has $-\infty$ logarithmic poles on $S$ (see e.g.\
[Dem82]). We define
$$
\wt u=
\mu^*u_0+dd^c\max\nolimits_{\varepsilon_0}(\varphi+\delta\,\psi_S\circ\mu,~~0)
=v+dd^c\max\nolimits_{\varepsilon_0}(\delta\,\psi_S\circ\mu,~~-\varphi)
\leqno(3.3)
$$
where $\max\nolimits_{\varepsilon_0}$, $0<\varepsilon_0<1$, is a 
regularized max function and 
$\delta>0$ is very small. By construction $\wt u$ coincides with $\mu^*u_0$ in
a neighborhood of $\mu^{-1}(S)$ and therefore $\wt u$ descends to a smooth 
closed $(1,1)$-form $u$ on $X$ which coincides with $u_0$ near~$S$, so that 
$\wt u=\mu^* u$. Clearly $\wt u$ converges uniformly to $v$ on every compact 
subset of $\wt X\ssm \mu^{-1}(S)$ as $\delta\to 0$, so we only
have to show that the Morse integrals are small (uniformly in $\delta$) 
when restricted to a
suitable small neighborhood of the exceptional set $E=\mu^{-1}(S)$.
Take a sufficiently large K\"ahler metric $\wt\omega$ on $\wt X$ such that
$$
-{1\over 2}\wt\omega\le v\le{1\over 2}\wt\omega,\quad
-{1\over 2}\wt\omega\le dd^c\varphi\le{1\over 2}\wt\omega,\quad
-\wt\omega\le dd^c\psi_S\circ\mu.
$$
Then $\wt u\ge -\wt\omega$ and 
$\wt u\le \wt\omega+\delta\,dd^c\psi_S\circ\mu$ everywhere on $\wt X$.
As a consequence
$$
|\wt u^n|\le \big(\wt\omega+\delta(\wt\omega+dd^c\psi_S\circ\mu)\big)^n
\le\wt\omega^n+n\delta(\wt\omega+dd^c\psi_S\circ\mu)\wedge
\big(\wt\omega+\delta(\wt\omega+dd^c\psi_S\circ\mu)\big)^{n-1}
$$
thanks to the inequality $(a+b)^n\le a^n+nb(a+b)^{n-1}$. For any neighborhood
$V$ of $\mu^{-1}(S)$ this implies
$$
\int_V|\wt u^n|\le \int_V\wt\omega^n+n\delta(1+\delta)^{n-1}\int_{\wt X}
\wt\omega^n
$$
by Stokes formula. We thus see that the integrals are small if $V$ and
$\delta$ are small. The reader may be concerned that Monge-Amp\`ere
integrals were used with an unbounded potential $\psi_S$, but in fact,
for any given $\delta$, all the above formulas and estimates are still
valid when we replace $\psi_S$ by 
$\max_{\varepsilon_0}(\psi_S, -(M+2)/\delta)$ with
$M=\max_{\wt X}\varphi$, especially formula (3.3) shows that
the form $\wt u$ is unchanged.
Therefore our calculations can be handled by using
merely smooth potentials.\qed

\section{4}{Proof of the infimum formula for the volume}

We prove here
\claim{(4.1) Theorem} Let $L\to X$ be a holomorphic line bundle on a projective
algebraic manifold~$X$. Then
$$
\Vol(X,L)=\inf_{u\in c_1(L)}\int_{X(u,0)}u^n.
$$
\endclaim

\noindent
It is enough to show the inequality
$$\inf_{u\in c_1(L)}\int_{X(u,0)}u^n\le \Vol(X,L)\leqno(4.1')
$$
and for this, we have to construct metrics approximating the volume.
Let us first assume that $L$ is a big line bundle, i.e.\ 
that $\Vol(X,L)>0$. We have seen in II~(6.4--6.5) (cf.\ also [Bck02]) that
$\Vol(X,L)$ is obtained as the supremum
of $\smash{\int_{X\ssm \Sing(T)}T^n}$ for K\"ahler currents 
$T=-{\ii\over 2\pi}\ddbar h$ with analytic
singularities in $c_1(L)$; this means that locally $h=e^{-\varphi}$
where $\varphi$ is a strictly plurisubharmonic function which has the
same singularities as $c\log\sum|g_j|^2$ where $c>0$ and the
$g_j$ are holomorphic functions. By [Dem92], there exists a
blow-up $\mu:\widetilde X\to X$ such that $\mu^*T=[E]+\beta$ where $E$ is a
normal crossing divisor on $\widetilde X$ and $\beta\ge 0$
smooth. Moreover, by [BDPP04] we have the orthogonality estimate
$$
[E]\cdot \beta^{n-1}=\int_{E}\beta^{n-1}\le 
C\big(\Vol(X,L)-\beta^n\big)^{1/2},\leqno(4.2)
$$
while
$$
\beta^n=\int_{\wt X}\beta^n=\int_{X\ssm \Sing(T)}T^n\quad
\hbox{approaches $\Vol(X,L)$}.\leqno(4.3)
$$
In other words, $E$ and $\beta$ become ``more and more orthogonal'' as
$\beta^n$ approaches the volume (these properties are summarized
by saying that $\mu^*T=[E]+\beta$ defines an approximate Zariski 
decomposition of $c_1(L)$, cf.\ also [Fuj94]). By subtracting to $\beta$
a small linear combination of the exceptional divisors and increasing
accordingly the coefficients of $E$, we can achieve that the
cohomology class $\{\beta\}$ contains a positive definite form
$\beta'$ on $\wt X$ (i.e.\ the fundamental 
form of a K\"ahler metric); we refer e.g.\ to ([DP04], proof of
Lemma~3.5) for details. This means that we can replace $T$ by
a cohomologous current such that the corresponding form $\beta$ is 
actually a K\"ahler metric, and we will assume for simplicity of notation
that this situation occurs right away for~$T$. Under this assumption,
there exists a smooth closed $(1,1)$-form $v$ belonging to the 
Bott-Chern cohomology class of $[E]$, such that we have identically
$(v-\delta\beta)\wedge\beta^{n-1}=0$ where
$$
\delta={[E]\cdot\beta^{n-1}\over \beta^n}\le C'(\Vol(X,L)-\beta^n\big)^{1/2}
\leqno(4.4)
$$
for some constant $C'>0$. In fact, given an arbitrary smooth
representative $v_0\in\{[E]\}$, the existence of $v=v_0+i\ddbar\psi$ amounts 
to solving a Laplace equation $\Delta\psi=f$ with respect to the K\"ahler 
metric $\beta$, and the choice of $\delta$ ensures that we have 
$\int_X f\,\beta^n=0$ and hence that the equation is solvable.
Then $\wt u:=v+\beta$ is a smooth closed $(1,1)$-form in the cohomology
class $\mu^* c_1(L)$, and its eigenvalues with respect to $\beta$ are
of the form $1+\lambda_j$ where $\lambda_j$ are the eigenvalues of $v$. The
Laplace equation is equivalent to the identity $\sum\lambda_j=n\delta$. 
Therefore
$$
\sum_{1\le j\le n}\lambda_j\le C''(\Vol(X,L)-\beta^n\big)^{1/2}.
\leqno(4.5)
$$
The inequality between arithmetic means and geometric means implies
$$
\prod_{1\le j\le n}(1+\lambda_j)\le 
\Big(1+{1\over n}\sum_{1\le j\le n}\lambda_j\Big)^n
\le 1+C_3(\Vol(X,L)-\beta^n\big)^{1/2}
$$
whenever all factors $(1+\lambda_j)$ are nonnegative. By 2.2~(i) we get
$$
\eqalign{
\inf_{u\in c_1(L)}\int_{X(u,0)}u^n&\le\int_{\wt X(\wt u,0)}\wt u^n\cr
&\le 
\int_{\wt X}\beta^n\big(1+C_3(\Vol(X,L)-\beta^n\big)^{1/2}\big)\cr
&\le \Vol(X,L)+C_4(\Vol(X,L)-\beta^n\big)^{1/2}.\cr}
$$
As $\beta^n$ approches $\Vol(X,L)$, this implies inequality (4.1).
\medskip

We still have to treat the case when $L$ is not big, i.e.\ $\Vol(X,L)=0$. 
Let $A$ be an ample
line bundle and let $t_0\ge 0$ be the infimum of real numbers such that
$L+tA$ is a big $\bQ$-line bundle for $t$ rational, $t>t_0$. The
continuity of the volume function implies that $0<\Vol(X,L+tA)\le\varepsilon$ 
for $t>t_0$ sufficiently close to~$t_0$. By what we have just proved, there
exists a smooth form $u_t\in c_1(L+tA)$ such that $\int_{X(u_t,0)}u_t^n\le
2\varepsilon$. Take a K\"ahler metric $\omega\in c_1(A)$ and define
$u=u_t-t\omega$. Then clearly
$$
\int_{X(u,0)}u^n\le \int_{X(u_t,0)}u_t^n\le 2\varepsilon,
$$
hence
$$
\inf_{u\in c_1(L)}\int_{X(u,0)}u^n=0.
$$
Inequality (4.1) is now proved in all cases.\qed

\section{5}{Estimate of the first cohomology group on a projective surface}

Our goal here is to show the following result.

\claim{(5.1) Theorem} Let $L\to X$ be a holomorphic line bundle on
a complex projective surface. Then both weak and strong 
inequalities {\rm(1.3)~(i)} and {\rm(1.3)~(ii)} 
are equalities for $q=0,\,1,\,2$, and the $\limsup$'s involved in
$\smash{\widehat h}^q(X,L)$ and $\smash{\widehat h}^{\,\leq q}(X,L)$
are limits.
\endclaim

We start with a projective non singular variety $X$ of arbitrary dimension
$n$, and will later restrict ourselves to the case when $X$ is a surface.
The proof again consists of using (approximate) Zariski decomposition,
but now we try to compute more explicitly the resulting curvature forms 
and Morse integrals; this will turn out to be much easier on surfaces.

Assume first that $L$ is a {\it big} line bundle on~$X$. As in section 3, we 
can find an approximate Zariski decomposition,
i.e.\ a blow-up $\mu:\wt X\to X$ and a current $T\in c_1(L)$ such
$\mu^*T=[E]+\beta$, where $E$ an effective divisor and $\beta$ a K\"ahler 
metric on $\wt X$ such that
$$
\Vol(X,L)-\eta<\beta^n<\Vol(X,L),\qquad\eta\ll 1.\leqno(5.2)
$$
(On a projective surface, one could even get exact Zariski decomposition, 
but we want to remain general as long as possible).
By blowing-up further, we may assume that $E$ is a normal crossing 
divisor. We select a Hermitian metric $h$ on $\cO(E)$ and take
$$
u_\varepsilon={\ii\over 2\pi}\ddbar\log(|\sigma_E|_h^2+\varepsilon^2)+
\Theta_{\cO(E),h}+\beta~~\in~~\mu^*c_1(L)
\leqno(5.3)
$$
where $\sigma_E\in H^0(\widetilde X,\cO(E))$ is the canonical section
and $\Theta_{\cO(E),h}$ the Chern curvature form. Clearly, by the
Lelong-Poincar\'e equation, $u_\varepsilon$ converges to $[E]+\beta$ in 
the weak topology as $\varepsilon\to 0$. Straightforward calculations yield
$$
u_\varepsilon={\ii\over 2\pi}{\varepsilon^2D^{1,0}_h\sigma_E\wedge 
\overline{D^{1,0}_h\sigma_E}\over(\varepsilon^2+|\sigma_E|^2)^2}
+{\varepsilon^2\over\varepsilon^2+|\sigma_E|^2}\Theta_{E,h}+\beta.
$$
The first term converges to $[E]$ in the weak topology, while the second,
which is close to $\Theta_{E,h}$ near~$E$, converges pointwise everywhere to $0$
on $\smash{\widetilde X}\ssm E$. A simple asymptotic analysis shows that 
$$
\Big({\ii\over 2\pi}{\varepsilon^2D^{1,0}_h\sigma_E\wedge 
\overline{D^{1,0}_h\sigma_E}\over(\varepsilon^2+|\sigma_E|^2)^2}
+{\varepsilon^2\over\varepsilon^2+|\sigma_E|^2}\Theta_{E,h}\Big)^p\to
[E]\wedge\Theta_{E,h}^{p-1}
$$
in the weak topology for $p\ge 1$, hence
$$
\lim_{\varepsilon\to 0}u_\varepsilon^n=
\beta^n+\sum_{p=1}^n{n\choose p}[E]\wedge \Theta_{E,h}^{p-1}
\wedge\beta^{n-p}.\leqno(5.4)
$$
In arbitrary dimension, the signature of $u_\varepsilon$ is hard to evaluate,
and it is also non trivial to decide the sign of the limiting measure
$\lim u_\varepsilon^n$. However, when $n=2$, we get the simpler formula
$$
\lim_{\varepsilon\to 0}u_\varepsilon^2=\beta^2+2[E]\wedge\beta
+[E]\wedge\Theta_{E,h}.
$$
In this case, $E$ can be assumed to be an exceptional divisor (otherwise
some part of it would be nef and could be removed from the poles of $T$).
Hence the matrix $(E_j\cdot E_k)$ is negative definite and we can find 
a smooth Hermitian
metric $h$ on $\cO(E)$ such that $(\Theta_{E,h})_{|E}<0$, i.e.\
$\Theta_{E,h}$ has one negative eigenvalue everywhere along~$E$. 

\claim{(5.5) Lemma} One can adjust the metric $h$ of $\cO(E)$ in such a
way that $\Theta_{E,h}$ is negative definite 
on a neighborhood of the support $|E|$ of the exceptional divisor,
and $\Theta_{E,h}+\beta$ has signature $(1,1)$ there.
$($We do not care about the signature far away from $|E|)$.
\endclaim

\proof{Proof} At a given point $x_0\in X$, let us fix coordinates and a positive
quadratic form $q$ on $\bC^2$. If we put
$\psi_\varepsilon(z)=\varepsilon\chi(z)\log(1+\varepsilon^{-1}q(z))$
with a suitable cut-off function~$\chi$,
then the Hessian form of $\psi_\varepsilon$ is equal to $q$ at $x_0$ and
decays rapidly to $O(\varepsilon\log\varepsilon)|dz|^2$ away from $x_0$. In 
this way, after multiplying $h$ with $e^{\pm \psi_\varepsilon(z)}$, we can replace
the curvature $\Theta_{E,h}(x_0)$ with $\Theta_{E,h}(x_0)\pm q$ without
substantially modifying the form away from~$x_0$. This allows to adjust
$\Theta_{E,h}$ to be equal to (say) $-{1\over 4}\beta(x_0)$ at any singular
point $x_0\in E_j\cap E_k$ in the support of $|E|$, while keeping
$\Theta_{E,h}$ negative definite along~$E$. In order to adjust the
curvature at smooth points $x\in|E|$, we replace the metric $h$ with
$h'(z)= h(z)\exp(-c(z)|\sigma_E(z)|^2)$. Then the curvature form 
$\Theta_{E,h}$ is
replaced by $\Theta_{E,h'}(x)=\Theta_{E_h}(x)+c(x)|d\sigma_E|^2$ at 
$x\in|E|$ (notice that $d\sigma_E(x)=0$ if $x\in\Sing|E|$), and we can
always select a real function $c$ so that 
$\Theta_{E,h'}$ is negative definite with one negative
eigenvalue between $-1/2$ and~$0$ at any point of $|E|$. Then 
$\Theta_{E,h'}+\beta$ has signature $(1,1)$ near~$|E|$.\qed

With this choice of the metric, we see that for $\varepsilon>0$ small,
the sum
$$
{\varepsilon^2\over\varepsilon^2+|\sigma_E|^2}\Theta_{E,h}+\beta
$$
is of signature $(2,0)$ or $(1,1)$ (or degenerate of signature $(1,0)$),
the non positive definite points being concentrated in a neighborhood
of~$E$. In particular the index set $X(u_\varepsilon,2)$ is empty, and 
also
$$
u_\varepsilon\le
{\ii\over 2\pi}{\varepsilon^2D^{1,0}_h\sigma_E\wedge 
\overline{D^{1,0}_h\sigma_E}\over(\varepsilon^2+|\sigma_E|^2)^2}+\beta
$$
on a neighborhood $V$ of $|E|$, while $u_\varepsilon$ converges uniformly
to $\beta$ on $\wt X\ssm V$. This implies that
$$
\beta^2\le\liminf_{\varepsilon\to 0}\int_{X(u_\varepsilon,0)}u_\varepsilon^2
\le\limsup_{\varepsilon\to 0}\int_{X(u_\varepsilon,0)}u_\varepsilon^2
\le \beta^2+2\beta\cdot E.
$$
Since $\int_{\wt X}u_\varepsilon^2=L^2=\beta^2+2\beta\cdot E+E^2$ we conclude
by taking the difference that
$$
-E^2-2\beta\cdot E
\le\liminf_{\varepsilon\to 0}\int_{X(u_\varepsilon,1)}-u_\varepsilon^2
\le\limsup_{\varepsilon\to 0}\int_{X(u_\varepsilon,1)}-u_\varepsilon^2
\le -E^2.
$$
Let us recall that $\beta\cdot E\le C(\Vol(X,L)-\beta^2)^{1/2}=0(\eta^{1/2})$ 
is small by (5.3) and the orthogonality estimate. The asymptotic 
cohomology is given here
by $\wh h^2(X,L)=0$ since $h^2(X,L^{\otimes k})=H^0(X,K_X\otimes L^{\otimes -k})=
0$ for $k\ge k_0$, and we have by Riemann-Roch
$$
\wh h^1(X,L)=\wh h^0(X,L)-L^2=\Vol(X,L)-L^2=-E^2-\beta\cdot E+O(\eta).
$$
Here we use the fact that ${n!\over k^n}h^0(X,L^{\otimes k})$ converges to 
the volume when $L$ is big. All this shows that equality occurs in the Morse 
inequalities~(1.3) when we pass to the infimum. By taking limits in
the Neron-Severi space $\NS_\bR(X)\subset H^{1,1}(X,\bR)$, we
further see that equality occurs as soon as $L$ is pseudo-effective, and the 
same is true if $-L$ is pseudo-effective by Serre duality.
\medskip
It remains to treat the case when neither $L$ nor $-L$ are pseudo-effective.
Then $\wh h^0(X,L)=\wh h^2(X,L)=0$, and asymptotic cohomology appears
only in degree~$1$, with $\wh h^1(X,L)=-L^2$ by Riemann-Roch.
Fix an ample line bundle $A$ and let $t_0>0$ be the infimum of real
numbers such that $L+tA$ is big  for $t$~rational, $t>t_0$, 
resp.\ let $t'_0>0$ be the infimum of real
numbers $t'$ such that $-L+t'A$ is big  for $t'>t'_0$. Then for
$t>t_0$ and $t'>t'_0$, we can find a modification 
$\mu:\wt X\to X$ and currents $T\in c_1(L+tA)$,
$T'\in c_1(-L+t'A)$ such that
$$
\mu^*T=[E]+\beta,\qquad \mu^*T'=[F]+\gamma
$$
where $\beta$, $\gamma$ are K\"ahler forms and $E$, $F$ normal crossing
divisors. By taking a suitable linear combination $t'(L+tA)-t(-L+t'A)$
the ample divisor $A$ disappears, and we get
$$
{1\over t+t'}\Big(t'[E]+t'\beta-t[F]-t\gamma\Big)\in \mu^*c_1(L).
$$
After replacing $E$, $F$, $\beta$, $\gamma$ by suitable multiples, we
obtain an equality
$$
[E]-[F]+\beta-\gamma\in \mu^*c_1(L).
$$
We may further assume by subtracting that the divisors $E$, $F$ have no 
common components.
The  construction shows that $\beta^2\le\Vol(X,L+tA)$ can be taken 
arbitrarily small (as well of course as $\gamma^2$), and the orthogonality
estimate implies that we can assume $\beta\cdot E$ and $\gamma\cdot F$
to be arbitrarily small. Let us introduce metrics $h_E$ on 
$\cO(E)$ and $h_F$ on $\cO(F)$ as in Lemma~5.5, and consider the forms
$$
\eqalign{
u_\varepsilon=
&+{\ii\over 2\pi}{\varepsilon^2D^{1,0}_{h_E}\sigma_E\wedge 
\overline{D^{1,0}_{h_E}\sigma_E}\over(\varepsilon^2+|\sigma_E|^2)^2}
+{\varepsilon^2\over\varepsilon^2+|\sigma_E|^2}\Theta_{E,h_E}+\beta\cr
&-{\ii\over 2\pi}{\varepsilon^2D^{1,0}_{h_F}\sigma_F\wedge 
\overline{D^{1,0}_{h_F}\sigma_F}\over(\varepsilon^2+|\sigma_F|^2)^2}
-{\varepsilon^2\over\varepsilon^2+|\sigma_F|^2}\Theta_{F,h_F}-\gamma
~~\in~~\mu^*c_1(L).\cr}
$$
Observe that $u_\varepsilon$ converges uniformly to $\beta-\gamma$ outside
of every neighborhood of $|E|\cup|F|$. Assume that
$\Theta_{E,h_E}<0$ on $V_E=\{|\sigma_E|<\varepsilon_0\}$ and
$\Theta_{F,h_F}<0$ on $V_F=\{|\sigma_F|<\varepsilon_0\}$. On 
$V_E\cup V_F$ we have
$$
u_\varepsilon\le 
{\ii\over 2\pi}{\varepsilon^2D^{1,0}_{h_E}\sigma_E\wedge 
\overline{D^{1,0}_{h_E}\sigma_E}\over(\varepsilon^2+|\sigma_E|^2)^2}
-{\varepsilon^2\over\varepsilon^2+|\sigma_F|^2}\Theta_{F,h_F}+\beta
+{\varepsilon^2\over\varepsilon_0^2}\Theta_{E,h_E}^+
$$
where $\Theta_{E,h_E}^+$is the positive part of $\Theta_{E,h_E}$ with respect
to $\beta$. One sees immediately that this term is negligible. The first
term is the only one which is not uniformly bounded, and actually it
converges weakly to the current $[E]$. By squaring, we find
$$
\limsup_{\varepsilon\to 0}\int_{X(u_\varepsilon,0)}u_\varepsilon^2\le
\int_{X(\beta-\gamma,0)}(\beta-\gamma)^2+2\beta\cdot E.
$$
Notice that the term $-\smash{\varepsilon^2\over\varepsilon^2+|\sigma_F|^2}\,
\Theta_{F,h_F}$ does not contribute to the limit as
it converges boun\-dedly almost everywhere to $0$, the exceptions
being points of~$|F|$, but this set is
of measure zero with respect to the current $[E]$. Clearly we have
$\int_{X(\beta-\gamma,0)}(\beta-\gamma)^2\le \beta^2$ and therefore
$$
\limsup_{\varepsilon\to 0}\int_{X(u_\varepsilon,0)}u_\varepsilon^2\le
\beta^2+2\beta\cdot E.
$$
Similarly, by looking at $-u_\varepsilon$, we find
$$
\limsup_{\varepsilon\to 0}\int_{X(u_\varepsilon,2)}u_\varepsilon^2\le
\gamma^2+2\gamma\cdot F.
$$
These $\limsup$'s are small and we conclude that the essential part of
the mass is concentrated on the $1$-index set, as desired.\qed

\claim{(5.6) Remark} {\rm It is interesting to put these results in perspective
with the algebraic version (I~2.14) of holomorphic Morse inequalities. When
$X$ is projective, the algebraic Morse inequalities
used in combination with the birational invariance of the Morse integrals
imply the inequalities
{\plainitemindent=6.5mm
\vskip5pt
\item{\rm (a)}\kern20pt
$\displaystyle\inf_{u\in c_1(L)}\int_{X(u,q)}(-1)^qu^n
\leq \inf_{\mu^*(L)\simeq\cO(F-G)}{n\choose q}F^{n-q}\cdot G^q\,,$
\vskip5pt
\item{\rm (b)}\kern20pt
$\displaystyle\inf_{u\in c_1(L)}\int_{X(u,\le q)}(-1)^qu^n
\leq \inf_{\mu^*(L)\simeq\cO(F-G)}\sum_{0\le j\le q}(-1)^{q-j}
{n\choose j}F^{n-j}\cdot G^j\,,$\vskip5pt}
\noindent
where the infimums on the right hand side are taken over all
modifications $\mu:\smash{\widetilde X}\to X$ and all decompositions
$\mu^*L=\cO(F-G)$ of $\mu^*L$ as a difference of two nef $\bQ$-divisors 
$F,\,G$ on~$\smash{\widetilde X}$. Again, a natural question is to know 
whether these infimums derived from algebraic intersection numbers
are equal to the asymptotic cohomology functionals 
$\smash{\widehat h}^q(X,L)$ and $\smash{\widehat h}^{\,\leq q}(X,L)$.
A positive answer would of course automatically yield a positive answer 
to the equality cases in 2.9~(a) and (b). However, the Zariski 
decompositions involved in our proofs of equality for $q=0$ or $n\le 2$
produce certain effective exceptional divisors which are not nef. It
is unclear how to write those effective divisors as a difference of nef 
divisors. This fact raises a lot of doubts upon the sufficiency of taking
merely differences of nef divisors in the infimums 5.6~(a) and 
5.6~(b), and it is likely that one needs a more subtle formula.\qed}
\endclaim

\section{6}{Singular holomorphic Morse inequalities}

The goal of this short section is to extend holomorphic Morse
inequalities to the case of singular Hermitian metrics, following
Bonavero's PhD thesis~[Bon93] (cf.\ also [Bon98]).We always assume
that our Hermitian metrics $h$ are given by quasi-psh\break
weights~$\varphi$. By Theorem (II~5.7), one can always approximate the
weight by an arbitrary close quasi-psh weight $\varphi$ with analytic
singularities, modulo smooth functions.

\claim{(6.1) Theorem} Let $(L,h)$ be a holomorphic line bundle on a compact 
complex $n$-fold~$X$, and let $E$ be an arbitrary holomorphic vector bundle
of rank~$r$. Assume that locally $h=e^{-\varphi}$ has 
analytic singularities, and that $\varphi$ is quasi-psh of the form
$$
h=c\log\sum|g_j|^2~~\mod C^\infty,\qquad c>0,
$$
in such a way that for a suitable modification $\mu:\wt X\to X$ one has
$\mu^*\Theta_{L,h}=[D]+\beta$ where $D$ is an effective divisor and
$\beta$ a smooth form on $\smash{\wt X}$. Let $S=\mu(\Supp D)$ be the singular
set of~$h$. Then we have the following asymptotic estimates for the cohomology
twisted by the appropriate multiplier ideal sheaves~:
{\plainitemindent=6.5mm
\vskip5pt
\item{\rm (a)}
$\strut\kern70.2pt\displaystyle h^q(X,E\otimes L^k
\otimes\cI(h^k))\le r {k^n\over n!}
\int_{X(L,h,q)\ssm S} (-1)^q \Theta_{L,h}^n + o(k^n)~.$
\vskip5pt
\item{\rm (b)}
$\strut\quad\displaystyle \sum_{0\le j\le q}
(-1)^{q-j}h^j(X,E\otimes L^k
\otimes\cI(h^k))\le r {k^n\over n!}
\int_{X(L,h,\le q)\ssm S} (-1)^q \Theta_{L,h}^n + o(k^n)~.$}
\endclaim

\proof{Proof} For this, we observe that the Morse integrals are given by 
$$\int_{\wt X(\beta,q)}(-1)^q\beta^n,$$
thanks to a change of variable $z=\mu(x)$. In fact, by our assumption
$\Theta_{L,h}$ is smooth on $X\ssm S$, and its pull-back
$\mu^*\Theta_{L,h}$ coincides with the smooth form $\beta$ on the
complement $\smash{\wt X}\ssm\Supp D$ (and $\Supp D$ is a negligible
set with respect to the integration of the smooth $(n,n)$
form~$\beta^n$ on $\smash{\wt X}$.) Now, a straightforward $L^2$
argument in the change of variable (cf.\ [Dem01]) yields the direct
image formula
$$
K_X\otimes \cI(h^k)=\mu_*\big(K_{\wt X}\otimes\cI(\mu^*h^k)\big).\leqno(6.2)
$$
Let us introduce the relative canonical sheaf 
$K_{\wt X/X}=K_{\wt X}\otimes\mu^* K_X^{-1}=\cO(\div(\Jac\mu))$ and let us put 
$$\wt L=\mu^*L,\quad \wt h=\mu^*h,\quad \wt E=\mu^*E\otimes K_{\wt X/X}.$$
Then $\wt h$ has divisorial singularities and therefore $\cI(\wt
h^k)=\cO(-\lfloor kD\rfloor)$ where $\lfloor...\rfloor$ means the
integral part of a divisor. The projection formula for direct images
yields
$$\eqalign{
\mu_*\big(\wt E\otimes \wt L^k\otimes \cI(\wt h^k)\big)&=E\otimes
L^k\otimes \cI(h^k),\cr R^q\mu_*\big(\wt E\otimes \wt L^k\otimes
\cI(\wt h^k)\big)&=E\otimes L^k\otimes K_X^{-1}\otimes
R^q\mu_*\big(K_{\wt X}\otimes \cI(\wt h^k)\big).\cr}$$ However, for
$k\ge k_0$ large enough, the multiplicities of $\lfloor kD\rfloor$ are
all${}>0$ for each of the components of~$D$, hence
$\cI(h^k)=\cO(-\lfloor kD\rfloor)$ is relatively ample with respect to
the morphism $\mu:\wt X\to X$. From this, e.g.\ by an application of
H\"ormander's $L^2$ estimates (see [Bon93] for more details), we
conclude that $R^q\mu_*\big(K_{\wt X}\otimes \cI(\wt h^k)\big)=0$ for
$k\ge k_0$.  The Leray spectral sequence then implies
$$H^q\big(X,E\otimes L^k\otimes \cI(h^k)\big)\simeq
H^q\big(\wt X,\wt E\otimes \wt L^k\otimes\cI(\wt h^k)\big).\leqno(6.3)
$$
This reduces the proof to the case of divisorial singularities. Let us
next assume that $D$ is a $\bQ$-divisor. Let $a$ be a denominator for
$D$, and put $k=a\ell+b$, $0\le b\le a-1$. Then
$$
\wt E\otimes \wt L^k\otimes\cI(\wt h^k)=\wt E\otimes \wt L^{a\ell+b}\otimes\cO(-a\ell D-\lfloor bD\rfloor)
=F_b\otimes G^\ell
$$
where
$$
F_b=\wt E\otimes  \wt L^b\otimes\cO(-\lfloor bD\rfloor),\qquad 
G=\wt L^a\otimes\cO(-aD).
$$
By construction, we get a smooth Hermitian metric $h_G$ on $G$ such
that $\Theta_{G,h_G}=a\beta$. In this case, the proof is reduced to
the standard case of holomorphic Morse inequalities, applied to the
smooth Hermitian line bundle $(G,h_G)$ on $\smash{\wt X}$ and the
finite family of rank~$r$ vector bundles $F_b$, $0\le b\le a-1$.  The
result is true even when $D$ is a real divisor. In fact, we can then
perturb the coefficients of $D$ by small $\varepsilon$'s to get a
rational divisor $D_\varepsilon$, and we then have to change the smooth 
part of $\smash{\Theta_{\wt L,\wt h}}$ to $\beta_\varepsilon=\beta+O(\varepsilon)$ 
(again smooth); actually $\beta_\varepsilon-\beta$ can be taken to be a
linear combination by
coefficients $O(\varepsilon)$ of given smooth forms representing the
Chern classes $c_1(\cO(D_j))$ of the components of~$D$. The Morse
integrals are then perturbed by $O(\varepsilon)$.  On the other hand,
Theorem~1.9 shows that the cohomology groups in the right
hand side of (6.3) are perturbed by $\varepsilon k^n$. The result
follows as $\varepsilon\to 0$, thanks to the already settled rational case.\qed
\endproof

Using singular holomorphic Morse inequalities, we can easily convert
Theorem~2.8 from Chapter~I into a necessary and sufficient condition
characterizing Moishezon manifolds.\medskip

\vbox{%
\claim{(6.4) Theorem} {A compact $n$-dimensional complex manifold is 
Moishezon if and only if it possesses a Hermitian line bundle $(L,h)$ 
whose metric has at most analytic singularities, satisfying
the integral condition $$\int_{X(L,h,\le 1)\ssm Z}(\Theta_{L,h})^n>0$$ 
in the complement of the set $Z\subset X$ of poles of~$h$.}
\endclaim}

\proof{Proof} The necessity comes from the fact that $X$ has a projective
desingularization $\mu:\wt X\to X$. The image $D=\mu_*(A)$ of an ample
devisor on $\wt X$ yields a big line bundle $L=\cO(D)$ on $X$, and the
direct image $h=\mu_*h_A$ of a metric such that $\Theta_{A,h_A}>0$ satisfies
$\Theta_{L,h}>0$ outside the set of poles $Z=\mu(E)\subset X$, if $E$ is the
exceptional divisor. The sufficiency is obtained exactly in the same way 
as Theorem~2.8, if we use Theorem~6.1 instead of the regular holomorphic
Morse inequalities.\qed

\long\def\cohomestimates{%
\section{7}{Cohomology estimates for effective divisors}

To state the precise result, we need the notion of nefness for a real
$(1,1)$ cohomology class which does not necessarily belong to
$\NS_\bR(X)$: we say that $\{u\}\in H^{1,1}(X,\bR)$ is nef if $\{u\}$
belongs to the closure of the convex cone generated by classes of
K\"ahler forms (the so-called \emph{K\"ahler cone} of
$H^{1,1}(X,\bR)$).

\claim{(7.1) Theorem} {Suppose that there is a nef cohomology class 
$\{u\}$ in
$H^{1,1}(X,\bR)$ such that $c_1\big(\cO_{T_X}(1)\big)+\pi^*\{u\}$ is nef
over the hyperplane bundle $\bP(T_X):=P(T^*X)$. Suppose 
moreover that $L$ is equipped with a singular metric $h$ such that
$T=\Theta_{L,h}\ge 0$. For $p=1,2,\ldots,n,n+1$ set
$$b_p=\inf\{c>0\,;\,\codim E_c(T)\ge p\},$$
with $b_{n+1}=\max_{x\in X}\nu(T,x)$. Then for any holomorphic
vector bundle $E$ of rank $r$ over~$X$ we have
$$h^q\big(X,E\otimes\cO(kL)\big)\le A_qr\,k^n+o(k^n)$$
where $A_q$ is the cup product
$$A_q={1\over q!\,(n-q)!}\big(b_{n-q+1}\{u\}\big)^q\cdot
\big(c_1(L)+b_{n-q+1}\{u\}\big)^{n-q}$$
in $H^{2n}(X,\bR)$, identified to a positive number.}
\endclaim

\claim{(7.2) Remark} {\rm When $X$ is projective algebraic and $\kappa(L)=n$,
the proof of (6.2b) shows  that $mL\simeq\cO(A+D)$ with $A$ ample and $D$
effective, for some $m\ge 1$. Then we can choose a singular metric $h$ on $L$
such that $T=\Theta_{L,h}=\omega+m^{-1}[D]$, 
where $\omega=m^{-1}\Theta_{A,h_A})$ is a K\"ahler metric. 
As $\nu(T,x)=m^{-1}\nu(D,x)$
at each point, the constants $b_j$ of theorem~7.1 are obtained
by counting the multiplicities of the singular points of $D\,$; 
for example, if $D$ only has isolated singularities, then $b_1=0$,
$b_2=\ldots=b_n=1/m$. Observe moreover that the nefness
assumption on $\cO_{T_X}(1)$ is satisfied with $\{u\}=c_1(G)$ if $G$ is a nef
$\bQ$-divisor such that $\cO(T_X)\otimes\cO(G)$ is nef, e.g.\ if
$\cO(S^mT_X)\otimes\cO(mG)$ is spanned by sections for some $m\ge 1$.}
\endclaim

\proof{Proof of theorem {\rm 7.1}} By definition, we have
$0=b_1\le b_2\le\ldots\le
b_n\le b_{n+1}$, and for $c\in{}]b_p,b_{p+1}]$,
$E_c(T)$ has codimension $\ge p$ with some component(s) of codimension~$p$
exactly. Let $\omega$ be a fixed K\"ahler metric on $X$. By adding
$\varepsilon\omega$ to $u$ if necessary, we may assume that $u\ge 0$
and that $\cO_{T_X}(1)$ has a smooth Hermitian metric such that
$c\big(\cO_{T_X}(1)\big)+\pi^* u\ge 0$.

Under this assumption, the main approximation theorem of [Dem91] shows
that the metric of $L$ can be approximated by a sequence of smooth metrics
such that the associated curvature forms $T_j$ satisfy the uniform lower bound 
$$T_j\ge -\lambda_j(x)\,u(x)-\varepsilon_j\,
\omega(x)\leqno(7.3)$$
where ${\lim\!\downarrow\,}_{j\to+\infty}\varepsilon_j=0$ and 
$(\lambda_j)_{j>0}$ is a decreasing sequence of
continuous functions on $X$ such that $\lim_{j\to+\infty}
\lambda_j(x)=\nu(T,x)$ at each point.

Morse inequalities cannot be used directly with $T=\Theta_{L,h}$ because wedge
products of currents do not make sense in general. Therefore, we
replace $T$ with its approximations $T_j$ and try to find
an upper bound for the limit. 

\claim{(7.4) Lemma} {Let $U_j=X(T_j,q)$ be the $q$-index set 
associated to $T_j$ and let $c$ be a positive number. On the open 
set $\Omega_{c,j}=\{x\in X\,;\,\lambda_j(x)<c\}$
we have
$$(-1)^q\bOne_{U_j}T_j^n\le{n!\over q!\,(n-q)!}
\big(c\,u+\varepsilon_j\,\omega\big)^q\wedge\big(T_j+c\,u
+\varepsilon_j\,\omega\big)^{n-q}.$$}
\endclaim

\proof{Proof} Write $v=c\,u+\varepsilon_j\,\omega>0$ and let 
$\alpha_{1,j}\le\ldots
\le\alpha_{n,j}$ be the eigenvalues of $T_j$ with
respect to $v$ at each point. Then $T_j^n=
\alpha_{1,j}\ldots\alpha_{n,j}\,v^n$ and
$$v^q\wedge(T_j+v)^{n-q}={q!\,(n-q)!\over n!}
\sum_{1\le i_1<\ldots<i_{n-q}\le n}(1+\alpha_{i_1,j})\ldots
(1+\alpha_{i_{n-q},j})\,v^n.$$
On $\Omega_{c,j}$ we get $T_j\ge -v$ 
by inequality (7.3), thus $\alpha_{i,j}\ge -1$; moreover, we 
have $\alpha_1\le\ldots\le\alpha_q<0$ and $0<\alpha_{q+1}\le\ldots\le
\alpha_n$ on $U_j$. On $\Omega_{c,j}$ we thus find
$$0\le(-1)^q\bOne_{U_j}\alpha_{1,j}\ldots
\alpha_{n,j}\le
\bOne_{U_j}\alpha_{q+1,j}\ldots\alpha_{n,j}\le
(1+\alpha_{q+1,j})\ldots(1+\alpha_{n,j}),$$
therefore $(-1)^q\bOne_{U_j}T_j^n\le
\big(n!/q!\,(n-q)!\big)\,v^q\wedge(T_j+v)^{n-q}$.\qed
\endproof

\proof{End of the proof of theorem {\rm 7.1}} Set $\Lambda=
\max_X\lambda_1(x)$. By lemma 7.4 applied with an arbitrary
$c>\Lambda$ we have
$$(-1)^q\bOne_{U_j}T_j^n\le{n!\over q!(n-q)!}
(\Lambda\,u+\varepsilon_1\omega)^q\wedge(T_j+\Lambda\,u+\varepsilon_1
\omega)^{n-q}~~~~\hbox{\rm on}~~X.$$
Then holomorphic Morse inequalities imply
$$\leqalignno{
h^q\big(X,E\otimes\cO(kL)\big)
&\le r{k^n\over n!}\int_X(-1)^q\bOne_{U_j}T_j^n+o(k^n)\cr
&\le{r\,k^n\over q!\,(n-q)!}\Big(\int_{\Omega_{c,j}}
(c\,u+\varepsilon_j\,\omega)^q\wedge(T_j+c\,u+\varepsilon_j\,\omega)^{n-q}\cr
&+\int_{X\ssm\Omega_{c,j}}
(\Lambda\,u+\varepsilon_1\omega)^q\wedge(T_j+\Lambda\,u+\varepsilon_1
\omega)^{n-q}\Big)+o(k^n).&(7.5)\cr}$$
Since $\lambda_j(x)$ decreases to $\nu(T,x)$ as 
$j\to+\infty$, the set $X\ssm\Omega_{c,j}$ decreases
to $E_c(T)$. Now, $T_j+\Lambda\,u+\varepsilon_1\omega$ is a closed
positive $(1,1)$-form belonging to a fixed cohomology class, so the mass of
any wedge power $(T_j+\Lambda\,u+\varepsilon_1\omega)^p$ with respect
to $\omega$ is constant. By weak compactness, there is a subsequence
$(j_\nu)$ such that
$(T_{j_\nu}+\Lambda\,u+\varepsilon_1\omega)^p$ converges 
weakly to a closed positive current $\Theta_p$ of bidegree $(p,p)$,
for each $p=1,\ldots,n$.
For $c>b_{p+1}$, we have $\codim E_c(T)\ge p+1$, hence
$\bOne_{E_c(T)}\Theta_p=0$. It follows that the integral
over $X\ssm\Omega_{c,j}$ in (7.5) converges to $0$ when 
$c>b_{n-q+1}$. For the same reason the integral over 
$\Omega_{c,j}$ converges to the same limit as
its value over $X$: observe that 
$(T_j+c\,u+\varepsilon_j\,\omega)^{n-q}$ can be
expressed in terms of powers of $u,\omega$ and of the positive forms
$(T_j+\Lambda\,u+\varepsilon_1\omega)^p$ with $p\le n-q\,$; thus
the limit is a linear combination with smooth coefficients of the 
currents $\Theta_p$, which carry no mass on $E_c(T)$. In the limit,
we obtain
$$h^q\big(X,E\otimes\cO(kL)\big)\le
{r\,k^n\over q!\,(n-q)!}(c\{u\})^q\cdot\big(c_1(L)+c\{u\}\big)^{n-q}+o(k^n),$$
and since this is true for every $c>b_{n-q+1}$, Theorem~7.1
follows.\qed}
\ifnum\doextras=1 \cohomestimates \fi

\chapter{0}{References}
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\end
