% Ecole d'été de Mahdia, Juillet 2004
% "Analyse Complexe et Géométrie"
%
% Title:
% ``Analytic techniques in algebraic geometry"
% J.P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France

%%% Macros

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

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

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

\magnification=\magstep1
\vsize=24.2truecm 
\hsize=15.3truecm
\hoffset=3.3truemm
\parskip=2pt plus 2pt minus 0pt
\hfuzz=2pt
\tolerance=500
\abovedisplayskip=3 mm plus6pt minus 4pt
\belowdisplayskip=3 mm plus6pt minus 4pt
\abovedisplayshortskip=0mm plus6pt minus 2pt
\belowdisplayshortskip=2 mm plus4pt minus 4pt
\predisplaypenalty=0
\frenchspacing
\newdimen\oldparindent\oldparindent=15pt
\def\newline{\hfil\break}
\skewchar\ninei='177 \skewchar\sixi='177
\skewchar\ninesy='60 \skewchar\sixsy='60
\hyphenchar\ninett=-1

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

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

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

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

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

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

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

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

\def\titlebref#1{???}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\def\gm{{\goth m}}

\def\cHom{{\cal H}{\it om}}

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

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

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

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

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

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

\font\smallcap=cmcsc10
\font\smallcap=cmbx8
\font\bftwelve=cmbx10 at 12pt
\font\sytwelve=cmsy10 at 12pt

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

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

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

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

\def\miscell#1|#2|#3|#4|%
    {{\ifdigit\leftskip=7mm\noindent
    \hangindent=2mm\hangafter=1
     \llap{[#1]\hskip1.35em}{\smallcap #2}\pointdash{\it #3}, {\rm #4}.\par
\else\noindent
\advance\laenge by 4mm \hangindent=\laenge\advance\laenge by -4mm
\hangafter=1
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\titlea{}{Analytic techniques in algebraic geometry}
\leftrunningtitle{J.-P.\ Demailly,~~Analytic techniques in algebraic geometry}


\vskip20pt\noindent
\hskip2cm{\tbfontt %Lectures by
Jean-Pierre Demailly}
\vskip3pt\noindent
\hskip2cm{\tbfontt Universit\'e de Grenoble I, Institut Fourier}

\vskip25pt\noindent
Lectures given at the School held in Mahdia, Tunisia,
July 14 -- July 31, 2004\hfil\break
{\it Analyse Complexe et G\'eom\'etrie}
\bigskip

The purpose of this series of lectures is to explain some advanced techniques
of Complex Analysis which can be applied to obtain fundamental results
in algebraic geometry: vanishing of cohomology groups, embedding theorems,
description of the geometric structure of projective algebraic varieties.
\bigskip

\titlec{}{\phantom{$~$}\newline\tbfontt Contents}
0. Preliminary Material
   \dotfill \titlebref{0}\break
1. Holomorphic Vector Bundles, Connections and Curvature
   \dotfill \titlebref{1}\break
2. Bochner Technique and Vanishing Theorems
   \dotfill \titlebref{2}\break
3. $L^2$ Estimates and Existence Theorems
   \dotfill \titlebref{3}\break
4. Multiplier Ideal Sheaves
   \dotfill \titlebref{4}\break
5. Nef and Pseudoeffective Cones
   \dotfill \titlebref{5}\break
6. Numerical Characterization of the K\"ahler Cone
  \dotfill \titlebref{6}\break
7. Cones of Curves
  \dotfill \titlebref{7}\break  
8. Duality Results
  \dotfill \titlebref{8}\break
9. Approximation of psh functions by logarithms of holomorphic functions
  \dotfill \titlebref{9}\break
10. Zariski Decomposition and Movable Intersections
  \dotfill \titlebref{10}\break
11. The Orthogonality Estimate
  \dotfill \titlebref{11}\break
12. Proof of the Main Duality Theorem
  \dotfill \titlebref{12}\break
13. References
  \dotfill \titlebref{13}

\titleb{0.}{Preliminary material}

Let $X$ be a complex manifold and $n=\dim_\bC X$. The bundle of differential
forms of typr $(p,q)$ is denoted by $\Lambda^{p,q}T^*_X$. We are especially
interested in {\it closed positive currents} of type $(p,p)$
$$
T=i^{p^2}\sum_{|J|=|K|=p}T_{JK}(z)dz_J\wedge d\ol z_J,\qquad
dz_J=dz_{j_1}\wedge\ldots\wedge dz_{j_p},\quad
dT=0.
$$
Recall that a current is a differential form with distribution coefficients,
and that such a $(p,p)$ current is said to be positive (in the ``medium
positivity'' sense) if the distribution
$\sum\lambda_J\overline \lambda_K T_{JK}$ is a positive measure for all
complex numbers $\lambda_J$. The coefficients $T_{JK}$ are then
complex measures. Important examples of closed positive $(p,p)$-currents
are currents of integration over analytic cycles of codimension $p$~:
$$
Z=\sum c_jZ_j,\qquad [Z]=\sum c_j[Z_j]
$$
where the current $[Z_j]$ is defined by duality as 
$$
\langle[Z_j],u\rangle=\int_{Z_j}u_{|Z_j}
$$
for every $(n-p,n-p)$ test form $u$ on $X$. Another important
example of positive $(1,1)$-current is the Hessian form $T=i\ddbar\varphi$ 
of a plurisubharmonic
function on an open set $\Omega\subset X$. A~{\it K\"ahler metric} on
$X$ is a positive definite hermitian $(1,1)$-form
$$
\omega(z)=i\sum_{1\le j,k\le n}\omega_{jk}(z)dz_j\wedge d\ol z_k\qquad
\hbox{such that $d\omega=0$},
$$
with smooth coefficients. To every closed real $(1,1)$-form (or current) 
$\alpha$ is associated its De Rham cohomology class
$$
\{\alpha\}\in H^{1,1}(X,\bR)\subset H^2(X,\bR).
$$
We denote here by $H^k(X,\bC)$ (resp.\ $H^k(X,\bR)$) the complex (real)
De Rham cohomology group of degree $k$, and by $H^{p,q}(X,\bC)$ the 
subspace of classes which can be represented as closed $(p,q)$-forms,
$p+q=k$.

We will rely on the nontrivial fact that all cohomology groups involved 
(De Rham, Dolbeault, $\ldots$) can  be 
defined either in terms of smooth forms or in terms of currents. In fact, 
if we consider the associated complexes of sheaves,
forms and currents both provide acyclic resolutions of the same sheaf
(locally constant functions, resp.\ holomorphic sections), hence define
the same cohomology groups.

In the sequel, we are mostly interested in the geometry of {\it compact
complex manifolds}. The compactness assumption brings many 
interesting features such as finitess results for the cohomology
or the topology, Stokes theorem, intersection formulas of Bezout type, etc. 
A {\it projective algebraic manifold} is a closed submanifold $X$ of some
complex projective space $\bP^N=\bP_\bC^N$ defined by a finite collection of
homogeneous polynomial equations
$$
P_j(z_0,z_1,\ldots,z_N)=0,~~~~1\le j\le k
$$
(in such a way that $X$ is non singular). An important theorem due to Chow
states that every complex analytic submanifold of $\bP^N$ is in fact
automatically algebraic, i.e.\ defined as above by a finite collection of
polynomials. We will prove this in section~4.

However, we will sometimes need to study local situations, and in that
case it is also useful to consider the case of (pseudoconvex)
open sets in~$\bC^n$.

\claim{(0.1) Definition} \smallskip
\item{\rm a)} A hermitian manifold is a pair  $(X,\omega)$ where $\omega$
is a $\ci$ positive definite $(1,1)$-form on $X$.
\smallskip
\item{\rm b)} $X$ is said to be a K\"ahler manifold if $X$ carries at
least one K\"ahler metric $\omega$.\smallskip
\endclaim

Since $\omega$ is real, the conditions $d\omega=0$, $d'\omega=0$,
$d''\omega=0$ are all equivalent. In local coordinates we see that
$d'\omega=0$ if and only if
$${\partial \omega_{jk}\over\partial z_l}={\partial \omega_{lk}\over
\partial z_j}\quad,\quad 1\le j,k,l\le n.$$
A simple computation gives
$${\omega^n\over n!}
=\det(\omega_{jk})\bigwedge_{1\le j\le n}\Big(\ii dz_j\wedge d\ol z_j\Big)
=2^n\det(\omega_{jk})\,dx_1\wedge dy_1\wedge\cdots\wedge dx_n\wedge dy_n,$$
where $z_n=x_n+\ii y_n$. Therefore the $(n,n)$-form
$$dV={1\over n!}\omega^n\leqno(0.2)$$
is positive and coincides with the hermitian volume element of $X$. If $X$ is
compact, then $\int_X\omega^n=n!\,\Vol_\omega(X)>0$. This simple remark
already implies that compact K\"ahler manifolds must satisfy some restrictive
topological conditions:

\claim{(0.3) Consequence} \smallskip
\item{\rm a)} If $(X,\omega)$ is compact K\"ahler and if $\{\omega\}$
denotes the  cohomology class of $\omega$ in $H^2(X,\bR)$, then
$\{\omega\}^n\ne 0$.
\smallskip
\item{\rm b)} If $X$ is compact K\"ahler, then $H^{2k}(X,\bR)\ne 0$ for
$0\le k\le n$. In fact, $\{\omega\}^k$ is a non zero class in
$H^{2k}(X,\bR)$.
\vskip0pt
\endclaim

\claim{(0.4) Example} \rm The complex projective space $\bP^n$ is K\"ahler.
A natural K\"ahler metric $\omega_\FS$ on $\bP^n$, called the {\it Fubini-Study
metric}, is defined by
$$p^\star\omega_\FS={\ii\over 2\pi} d'd''\log\big(|\zeta_0|^2+|\zeta_1|^2
+\cdots+|\zeta_n|^2\big)$$
where $\zeta_0,\zeta_1\ld\zeta_n$ are coordinates of $\bC^{n+1}$ and
where \hbox{$p:\bC^{n+1}\setminus\{0\}\to\bP^n$} is the projection.
Let $z=(\zeta_1/\zeta_0\ld\zeta_n/\zeta_0)$ be non homogeneous coordinates
on $\bC^n\subset\bP^n$. Then a calculation shows that
$$\omega_\FS={\ii\over 2\pi}d'd''\log(1+|z|^2),
~~~~\int_{\bP^n}\omega_\FS^n=1.$$
It is also well-known from topology that $\{\omega_\FS\}\in H^2(\bP^n,\bZ)$ is 
a generator of the cohomology algebra $H^\bu(\bP^n,\bZ)$.
\endclaim

\claim{(0.5) Example} \rm A {\it complex torus} is a quotient
$X=\bC^n/\Gamma$ by a lattice $\Gamma$ of rank~$2n$. Then $X$ is a
compact complex manifold. Any positive definite hermitian form
$\omega=\ii\sum \omega_{jk}dz_j\wedge d\ol z_k$ with constant coefficients
defines a K\"ahler metric on~$X$.
\endclaim

\claim{(0.6) Example} \rm Every (complex) submanifold $Y$ of a
K\"ahler manifold  $(X,\beta)$ is K\"ahler with metric
$\omega=\beta_{\restriction Y}$.  Especially, all complex submanifolds of 
$X\subset \bP^N$  are K\"ahler with K\"ahler metric
$\omega=\omega_{\FS\restriction X}$. Since $\omega_\FS$ is
in $H^2(\bP,\bZ)$, the restriction $\omega$ is an 
integral class in $H^2(X,\bZ)$. Conversely, the Kodaira embedding
theorem [Kod54] states that every compact K\"ahler manifold $X$
possessing a K\"ahler metric $\omega$ with an integral cohomology
class $\{\omega\}\in H^2(X,\bZ)$ can be embedded in projective space
as a projective algebraic subvariety. We will prove this in
section~4.
\endclaim

\claim{(0.7) Example} \rm Consider the complex surface
$$X=(\bC^2\setminus\{ 0\})/\Gamma$$
where $\Gamma=\{\lambda^n~;~n\in\bZ\}$, $\lambda<1$, acts as a
group of homotheties. Since $\bC^2\setminus\{ 0\}$ is diffeomorphic
to  $\bR^\star_+\times S^3$, we have $X\simeq S^1\times S^3$. 
Therefore $H^2(X,\bR)=0$ by K\"unneth's formula, and
property 0.3 b) shows that $X$ is not K\"ahler. More generally,
one can take $\Gamma$ to be an infinite cyclic group generated by
a holomorphic contraction of $\bC^2$, of the form
$$\pmatrix{z_1\cr z_2\cr}\longmapsto\pmatrix{\lambda_1z_1\cr\lambda_2z_2\cr},
\qquad\hbox{\rm resp.}\quad\pmatrix{z_1\cr z_2\cr}\longmapsto
\pmatrix{\lambda z_1\cr\lambda z_2+z_1^p\cr},$$
where $\lambda,\lambda_1,\lambda_2$ are complex numbers such that
$0<|\lambda_1|\le|\lambda_2|<1$, \hbox{$0<|\lambda|<1$}, and $p$ a positive
integer. These non K\"ahler surfaces are called {\it Hopf surfaces}.
\endclaim

\titleb{1.}{Hermitian Vector Bundles, Connections and Curvature}
The goal of this section is to recall the most basic definitions of
hemitian differential geometry related to the concepts of connection,
curvature and first Chern class of a line bundle.

Let $F$ be a complex vector bundle of rank $r$ over a smooth
differentiable manifold~$M$. A {\it connection} $D$ on $F$ is a
linear differential operator of order $1$
$$D:C^\infty(M,\Lambda^qT^\star_M\otimes F) \to C^\infty(M,
\Lambda^{q+1}T^\star_M\otimes F)$$
such that
$$D(f\wedge u) = df\wedge u + (-1)^{\deg\,f}f\wedge Du\leqno(1.1)$$
for all forms $f\in C^\infty(M,\Lambda^pT^\star_M)$, $u\in C^\infty(X,
\Lambda^qT^\star_M\otimes F)$. On an open set $\Omega \subset M$ where $F$
admits a trivialization $\theta:F_{|\Omega }\buildo\lra^\simeq \Omega
\times \bC^r$, a connection $D$ can be written
$$Du\simeq_\theta du+\Gamma \wedge u$$
where $\Gamma \in C^\infty(\Omega ,\Lambda^1T^\star_M\otimes\Hom
(\bC^r,\bC^r))$ is an arbitrary matrix of $1$-forms and $d$ acts
componentwise. It is then easy to check that
$$D^2u\simeq_\theta (d\Gamma +\Gamma \wedge \Gamma )\wedge u\quad
\hbox{on~}\Omega.$$
Since $D^2$ is a globally defined operator, there is a global $2$-form
$$\Theta(D) \in C^\infty(M,\Lambda^2T^\star_M\otimes\Hom(F,F))\leqno(1.2)$$
such that $D^2u=\Theta(D)\wedge u$ for every form $u$ with values in~$F$.

Assume now that $F$ is endowed with a $C^\infty$ hermitian metric along
the fibers and that the isomorphism $F_{|\Omega }\simeq \Omega \times
\bC^r$ is given by a $C^\infty$ frame $(e_\lambda)$. We then have
a canonical sesquilinear pairing
$$\leqalignno{
C^\infty(M,\Lambda^pT^\star_M\otimes F)\times
C^\infty(M,\Lambda^qT^\star_M\otimes F)
&\longrightarrow C^\infty(M,\Lambda^{p+q}T^\star_M\otimes\bC)&(1.3)\cr
(u,v) &\longmapsto \{u,v\}\cr}$$
given by
$$\{u,v\}=\sum_{\lambda ,\mu }u_\lambda \wedge \overline v_\mu \langle
e_\lambda,e_\mu \rangle,\qquad
u=\sum u_\lambda \otimes e_\lambda,\quad 
v=\sum v_\mu \otimes e_\mu.$$
The connection $D$ is said to be {\it hermitian} if it satisfies the
additional property
$$d\{u,v\}=\{Du,v\}+(-1)^{{\rm deg~}u}\{u,Dv\}.$$
Assuming that $(e_\lambda )$ is orthonormal, one easily checks that $D$
is hermitian if and only if $\Gamma^\star=-\Gamma$. In this case
$\Theta(D)^\star=-\Theta(D)$, thus
$$\ii\Theta(D) \in C^\infty(M,\Lambda^2T^\star_M\otimes\Herm(F,F)).$$
\medskip

\claim{(1.4) Special case} \rm For a bundle $F$ of rank 1, the
connection form $\Gamma$ of a hermitian connection $D$ can be seen as
a $1$-form with purely imaginary coefficients $\Gamma=\ii A$ ($A$ real).
Then we have $\Theta(D)=d\Gamma=\ii dA$. In particular $\ii\Theta(F)$
is a closed 2-form. The {\it first Chern class} of $F$
is defined to be the cohomology class
$$c_1(F)_\bR=\Big\{{\ii\over 2\pi}\Theta(D)\Big\}\in H^2_\DR(M,\bR).$$
The cohomology class is actually independent of the connection, since
any other connection $D_1$ differs by a global 1-form, $D_1u=Du+B\wedge u$,
so that $\Theta(D_1)=\Theta(D)+dB$. It is well-known that $c_1(F)_\bR$
is the image in $H^2(M,\bR)$ of an integral class $c_1(F)\in
H^2(M,\bZ)\,$; by using the exponential exact sequence
$$0\to\bZ\to\cE\to\cE^\star\to 0,$$
$c_1(F)$ can be defined in \v Cech cohomology theory as the image
by the coboundary map $H^1(M,\cE^\star)\to H^2(M,\bZ)$ of the cocycle
$\{g_{jk}\}\in H^1(M,\cE^\star)$ defining $F\,$; see e.g.\ [GrH78] for
details.\qed
\endclaim

We now concentrate ourselves on the complex analytic case. If $M=X$ is a
complex manifold $X$, every connection $D$ on a complex $\ci$ vector
bundle $F$ can be splitted in a unique way as a
sum of a $(1,0)$ and of a $(0,1)$-connection, $D=D'+D''$. In a local
trivialization $\theta$ given by a $C^\infty$ frame, one can write 
$$\leqalignno{
D'u &\simeq_\theta d'u + \Gamma '\wedge u,&(1.5')\cr
D''u &\simeq_\theta d''u + \Gamma ''\wedge u,&(1.5'')\cr}$$
with $\Gamma =\Gamma '+\Gamma ''$. The connection is hermitian if and
only if $\Gamma '=-(\Gamma '')^\star$ in any orthonormal frame. Thus there
exists a unique hermitian connection $D$ corresponding to a prescribed
$(0,1)$ part $D''$. 

Assume now that the bundle $F$ itself has a {\it holomorphic} structure.
The unique hermitian connection for which $D''$ is the $d''$ operator
defined in \S~1 is called the {\it Chern connection} of $F$. In a local
holomorphic frame $(e_\lambda )$ of $E_{|\Omega }$, the metric is given
by the hermitian matrix $H=(h_{\lambda\mu})$, $h_{\lambda\mu}
=\langle e_\lambda,e_\mu \rangle$. We have
$$\{u,v\}=\sum_{\lambda,\mu}h_{\lambda\mu}u_\lambda \wedge
\ol v_\mu=u^\dagger\wedge H\ol v,$$
where $u^\dagger$ is the transposed matrix of $u$, and
easy computations yield
$$\eqalign{d\{u,v\}
&=(du)^\dagger \wedge H\ol v+(-1)^{\deg u} u^\dagger
\wedge (dH\wedge \ol v + H\ol{dv})\cr
&=\big(du +\ol H^{-1}d'\ol H \wedge u\big)^\dagger \wedge H\ol v
+(-1)^{\deg u}u^\dagger\wedge(\ol{dv+ \ol H^{-1}d'\ol H\wedge v)}\cr}$$
using the fact that $dH = d'H+\ol{d'\ol H}$ and
$\ol H^\dagger=H$. Therefore the Chern connection $D$ coincides with the
hermitian connection defined by
$$\left\{\eqalign{
Du&{}\simeq_\theta du+\ol H^{-1}d'\ol H \wedge u,\cr
D'&{}\simeq_\theta d'+\ol H^{-1} d'\ol H\wedge\bu=
\ol H^{-1} d'(\ol H\bu),~~~~D'' = d''.\cr}\right.\leqno(1.6)$$
It is clear from this relations that $D^{\prime 2}=D^{\prime\prime 2}=0$.
Consequently $D^2$ is given by to $D^2=D'D''+D''D'$, and the curvature 
tensor $\Theta(D)$ is of type~$(1,1)$. Since $d'd''+d''d'=0$, we get
$$\eqalign{
(D'D''+D''D')u&\simeq_\theta\ol H^{-1}d'\ol H\wedge d''u 
+d''(\ol H^{-1}d'\ol H\wedge u)\cr
&=d''(\ol H^{-1}d'\ol H)\wedge u.\cr}$$

\claim{(1.7) Proposition} The Chern curvature tensor $\Theta(F):=
\Theta(D)$ is such that
$$\ii\,\Theta(F) \in C^\infty(X,\Lambda^{1,1}T^\star_X\otimes\Herm(F,F)).$$
If $\theta: E_{\restriction\Omega}\to\Omega\times\bC^r$ is a holomorphic
trivialization and if $H$ is the hermitian matrix representing the
metric along the fibers of $F_{\restriction\Omega}$, then
$$\ii\,\Theta(F)\simeq_\theta
\ii\,d''(\ol H^{-1} d'\ol H)\quad\hbox{\rm on}~\Omega.\eqno\square$$
\endclaim

Let $(z_1\ld z_n)$ be holomorphic coordinates on $X$ and let
$(e_\lambda)_{1\le\lambda\le r}$ be an orthonormal frame of $F$.
Writing
$$\ii\Theta(F)=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}dz_j\wedge dz_k\otimes e_\lambda^\star\otimes e_\mu,$$
we can identify the curvature tensor to a hermitian form
$$\wt\Theta(F)(\xi\otimes v)=\sum_{1\le j,k\le n,\,1\le\lambda,\mu\le r}
c_{jk\lambda\mu}\xi_j\ol\xi_kv_\lambda\ol v_\mu\leqno(1.8)$$
on $T_X\otimes F$. This leads in a natural way to positivity concepts,
following definitions introduced by Kodaira [Kod53], Nakano [Nak55] and
Griffiths [Gri69].

\claim{(1.9) Definition} The hermitian vector bundle $F$ is said to be
\smallskip
\item{\rm a)} positive in the sense of Nakano if
$\wt\Theta(F)(\tau)>0$ for all non zero tensors $\tau=\sum
\tau_{j\lambda}\partial/\partial z_j\otimes e_\lambda\in T_X\otimes F$.
\smallskip
\item{\rm b)} positive in the sense of Griffiths if
$\wt\Theta(F)(\xi\otimes v)>0$ for all non zero decomposable tensors
$\xi\otimes v\in T_X\otimes F\,;$
\smallskip\noindent
Corresponding semipositivity concepts are defined by relaxing the strict
inequalities.
\endclaim

\claim{(1.10) Special case of rank 1 bundles} \rm Assume that
$F$ is a line bundle. The hermitian matrix $H=(h_{11})$ associated
to a trivialization $\theta:F_{\restriction\Omega}\simeq\Omega\times\bC$
is simply a positive function which we find convenient to denote by
$e^{-\varphi}$, \hbox{$\varphi\in C^\infty(\Omega,\bR)$}. In this case
the curvature form $\Theta(F)$ can be identified to the $(1,1)$-form
$2d'd''\varphi$, and
$${\ii\over 2\pi}\Theta(F)={\ii\over\pi}d'd''\varphi=dd^c\varphi$$
is a real $(1,1)$-form. Hence $F$ is semipositive (in either the Nakano or
Griffiths sense) if and only if $\varphi$ is psh, resp.\ positive if
and only if $\varphi$ is {\it strictly psh}. In this setting, the
Lelong-Poincar\'e equation can be generalized as follows: let
$\sigma\in H^0(X,F)$ be a non zero holomorphic section. Then
$$dd^c\log\|\sigma\|=[Z_\sigma]-{\ii\over 2\pi}\Theta(F).\leqno(1.11)$$
Formula (1.11) is immediate if we write $\|\sigma\|=|\theta(\sigma)|
e^{-\varphi}$ and if we apply (1.20) to the holomorphic function
$f=\theta(\sigma)$. As we shall see later, it is very important for
the applications to consider also singular hermitian metrics.
\endclaim

\claim{(1.12) Definition} A singular $($hermitian$)$ metric on a line
bundle $F$ is a metric which is given in any trivialization
$\theta:F_{\restriction\Omega}
\buildo \lra^\simeq \Omega\times\bC$ by
$$\|\xi\|=|\theta(\xi)|\,e^{-\varphi(x)},~~~~x\in\Omega,~\xi\in F_x$$
where $\varphi\in L^1_\loc(\Omega)$ is an arbitrary function, called the
weight of the metric with respect to the trivialization $\theta$.
\endclaim

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

\claim{(1.13) Example} \rm Let $D=\sum\alpha_jD_j$ be a divisor with 
coefficients $\alpha_j\in\bZ$ and let $F=\cO(D)$ be the associated
invertible sheaf of meromorphic functions $u$ such that
${\rm div}(u)+D\ge 0\,$; the corresponding line bundle can be
equipped with the singular metric defined by $\|u\|=|u|$. If $g_j$ is a
generator of the ideal of $D_j$ on an open set $\Omega\subset X$ then 
$\theta(u)=u\prod\smash{g_j^{\alpha_j}}$ defines a trivialization of
$\cO(D)$ over $\Omega$, thus our singular metric is associated to the
weight $\varphi=\sum\alpha_j\log|g_j|$. By the Lelong-Poincar\'e equation,
we find
$${\ii\over 2\pi}\Theta\big(\cO(D)\big)=dd^c\varphi=[D],$$
where $[D]=\sum\alpha_j[D_j]$ denotes the current of integration
over~$D$.\qed
\endclaim

\claim{(1.14) Example} \rm Assume that $\sigma_1\ld \sigma_N$ are non zero 
holomorphic sections of $F$. Then we can define a natural (possibly
singular) hermitian metric 
on $F^\star$ by
$$\|\xi^\star\|^2=\sum_{1\le j\le n}\big|\xi^\star.\sigma_j(x)\big|^2
~~~~\hbox{\rm for}~~\xi^\star\in F^\star_x.$$
The dual metric on $F$ is given by
$$\|\xi\|^2={|\theta(\xi)|^2\over|\theta(\sigma_1(x))|^2+\ldots+
|\theta(\sigma_N(x))|^2}$$
with respect to any trivialization $\theta$. The associated weight function is
thus given by $\varphi(x)=\log\big(\sum_{1\le j\le N}|\theta(\sigma_j(x))|^2
\big){}^{1/2}$. In this case $\varphi$ is a psh function, thus
$\ii\Theta(F)$ is a closed positive current. Let us denote by $\Sigma$
the linear system defined by $\sigma_1\ld \sigma_N$ and by
$B_\Sigma=\bigcap\sigma_j^{-1}(0)$ its base locus. We have a meromorphic
map
$$\Phi_\Sigma:X\ssm B_\Sigma\to\bP^{N-1},\qquad
x\mapsto(\sigma_1(x):\sigma_2(x):\ldots:\sigma_N(x)).$$
Then ${\ii\over 2\pi}\Theta(F)$ is equal to the pull-back over $X\ssm
B_\Sigma$ of the Fubini-Study metric $\omega_\FS={\ii\over 2\pi}
\log(|z_1|^2+\ldots+ |z_N|^2)$ of $\bP^{N-1}$ by $\Phi_\Sigma$.\qed
\endclaim

\claim{(1.15) Ample and very ample line bundles} A holomorphic line
bundle $F$ over a compact complex manifold $X$ is said to be
\smallskip
\item{\rm a)} very ample if the map $\Phi_{|F|}:X\to\bP^{N-1}$ associated
to the complete linear system $|F|=P(H^0(X,F))$ is a regular embedding
$($by this we mean in particular that the base locus is empty, i.e.\
$B_{|F|}=\emptyset)$.
\smallskip
\item{\rm b)} ample if some multiple $mF$, $m>0$, is very ample.
\endclaim

Here we use an additive notation for $\Pic(X)=H^1(X,\cO^\star)$, hence
the symbol $mF$ denotes the line bundle $F^{\otimes m}$. By
Example~1.14, every ample line bundle $F$ has a smooth hermitian metric
with positive definite curvature form; indeed, if the linear system
$|mF|$ gives an embedding in projective space, then we get a smooth
hermitian metric on $F^{\otimes m}$, and the $m$-th root yields a
metric on $F$ such that ${\ii\over 2\pi}\Theta(F)={1\over m}
\Phi_{|mF|}^\star\omega_\FS$. Conversely, the Kodaira embedding theorem
[Kod54] tells us that every positive line bundle $F$ is ample (see
(4.14) for a straightforward analytic proof of the Kodaira
embedding theorem).

\titleb{2.}{Bochner Technique and Vanishing Theorems}
We first recall briefly a few basic facts of Hodge theory. Assume for
the moment that $M$ is a differentiable manifold equipped with a Riemannian
metric $g=\sum g_{ij}dx_i\otimes dx_j$. Given a $q$-form $u$ on $M$ with
values in $F$, we consider the global $L^2$ norm
$$\|u\|^2 = \int_M |u(x)|^2dV_g(x)$$
where $|u|$ is the pointwise hermitian norm and $dV_g$ is the
Riemannian volume form. The Laplace-Beltrami operator associated to the
connection $D$ is
$$\Delta =DD^\star+D^\star D$$
where
$$D^\star:\ci(M,\Lambda^qT^\star_M\otimes F)\to
\ci(M,\Lambda^{q-1}T^\star_M\otimes F)$$
is the (formal) adjoint of $D$ with respect to the $L^2$ inner product.
Assume that $M$ is {\it compact}. Since
$$\Delta:\ci(M,\Lambda^qT^\star_M\otimes F)\to
\ci(M,\Lambda^qT^\star_M\otimes F)$$
is a self-adjoint elliptic operator in each degree, standard results of
PDE theory show that there is an orthogonal decomposition
$$\ci(M,\Lambda^qT^\star_M\otimes F)=\cH^q(M,F)\oplus\Im\Delta$$
where $\cH^q(M,F)=\Ker\Delta$ is the space of harmonic forms of degree~$q$;
$\cH^q(M,F)$ is a finite dimensional space.
Assume moreover that the connection $D$ is {\it integrable}, i.e.\ that
$D^2=0$. It is then easy to check that there is an orthogonal direct sum
$$\Im\Delta=\Im D\oplus\Im D^\star,$$
indeed $\langle Du,D^\star v\rangle=\langle D^2u,v\rangle=0$ for all $u,v$.
Hence we get an orthogonal decomposition
$$\ci(M,\Lambda^qT^\star_M\otimes F)=\cH^q(M,F)\oplus\Im D\oplus\Im D^\star,$$
and $\Ker D=(\Im D^*)^\perp$ is precisely equal to $\cH^q(M,F)\oplus\Im D$.
Especially, the $q$-th cohomology group $H^q_\DR(M,F):=\Ker D/\Im D$
is isomorphic to $\cH^q(M,F)$. In general, a nontrivial vector bundle
$F$ does not admit an integrable connection, but this is certainly the
case for the trivial bundle $F=M\times\bC$. This implies that the De 
Rham cohomology groups $H^q_\DR(M,\bC)$ can be computed in terms of
harmonic forms:

\claim{(2.1) Hodge Fundamental Theorem} If $M$ is a compact Riemannian
manifold, there is an isomorphism
$$H^q_\DR(M,\bC)\simeq\cH^q(M,\bC)$$
from De Rham cohomology groups onto spaces of harmonic forms.\qed
\endclaim

A rather important consequence of the Hodge fundamental theorem is a proof
of the {\it Poincar\'e duality theorem}. Assume that the Riemannian manifold
$(M,g)$ is oriented. Then there
is a (conjugate linear) Hodge star operator
$$\star:\Lambda^qT^\star_M\otimes\bC\to\Lambda^{m-q}T^\star_M\otimes\bC,
\qquad m=\dim_\bR M$$
defined by $u\wedge\star v=\langle u,v\rangle dV_g$ for any two complex
valued $q$-forms $u$, $v$. A standard computation shows
that $\star$ commutes with $\Delta$, hence $\star u$ is harmonic if and only
if $u$ is. This implies that the natural pairing
$$H^q_\DR(M,\bC)\times H^{m-q}_\DR(M,\bC),\qquad
(\{u\},\{v\})\mapsto\int_M u\wedge v\leqno(2.2)$$
is a nondegenerate duality, the dual of a class $\{u\}$ represented by a
harmonic form being $\{\star u\}$.

Let us now suppose that $X$ is a compact complex manifold
equipped with a hermitian metric $\omega=\sum\omega_{jk}dz_j\wedge d\ol
z_k$. Let $F$ be a holomorphic vector bundle on $X$ equipped with a
hermitian metric, and let $D=D'+D''$ be its Chern curvature form.
All that we said above for the Laplace-Beltrami operator $\Delta$
still applies to the complex Laplace operators
$$\Delta'=D'D^{\prime\star}+D^{\prime\star}D',\qquad
\Delta''=D''D^{\prime\prime\star}+D^{\prime\prime\star}D'',$$
with the great advantage that we always have 
$D^{\prime 2}=D^{\prime\prime 2}=0$. Especially, if $X$ is a compact
complex manifold, there are isomorphisms
$$H^{p,q}(X,F)\simeq\cH^{p,q}(X,F)\leqno(2.3)$$
between Dolbeault cohomology groups $H^{p,q}(X,F):=\Ker D''/\Im D''$ and 
spaces $\cH^{p,q}(X,F)$ of $\Delta''$-harmonic forms of bidegree $(p,q)$
with values in~$F\,$; indeed, as above, we have an orthogonal direct sum
$$
C^\infty(X,\Lambda^{p,q}T^*_X\otimes F)=\Ker\Delta''\oplus\Im\Delta''=
\cH^{p,q}(X,F)\oplus\Im D''\oplus \Im D^{\prime\prime *}
$$
and $\Ker D''=(\Im D^{\prime\prime *})^\perp=\cH^{p,q}(X,F)\oplus\Im D''$.
Now, there is a generalized Hodge star operator
$$\star:\Lambda^{p,q}T^\star_X\otimes F\to
\Lambda^{n-p,n-q}T^\star_X\otimes F^\star,\qquad n=\dim_\bC X,$$
such that $u\wedge\star v=\langle u,v\rangle dV_g$,
for any two $F$-valued $(p,q)$-forms, when the wedge product
$u\wedge\star v$ is combined with the pairing $F\times F^\star\to\bC$.
This leads to the {\it Serre duality theorem} [Ser55]: the bilinear pairing
$$H^{p,q}(X,F)\times H^{n-p,n-q}(X,F^\star),\qquad
(\{u\},\{v\})\mapsto\int_X u\wedge v\leqno(2.4)$$
is a nondegenerate duality. Combining this with the Dolbeault isomorphism,
we may restate the result in the form of the duality formula
$$H^q(X,\Omega^p_X\otimes\cO(F))^\star\simeq H^{n-q}(X,
\Omega^{n-p}_X\otimes\cO(F^\star)).\leqno(2.4')$$

We now proceed to explain the basic ideas of the Bochner technique used
to prove vanishing theorems. Great simplifications occur in the computations
if the hermitian metric on $X$ is supposed to be {\it K\"ahler}, i.e.\
if the associated {\it fundamental $(1,1)$-form}
$$\omega=\ii\sum\omega_{jk}dz_j\wedge d\ol z_k$$
satisfies $d\omega=0$. It can be easily shown that $\omega$ is K\"ahler
if and only if there are holomorphic coordinates $(z_1\ld z_n)$ centered
at any point $x_0\in X$ such that the matrix of coefficients $(\omega_{jk})$
is tangent to identity at order $2$, i.e.\ 
$$\omega_{jk}(z)=\delta_{jk}+O(|z|^2)\quad\hbox{at $x_0$}.$$
It follows that all order 1 operators $D$, $D'$, $D''$ and their
adjoints $D^\star$, $D^{\prime\star}$, $D^{\prime\prime\star}$ admit at $x_0$
the same expansion as the analogous operators obtained when all hermitian
metrics on $X$ or $F$ are constant. From this, the basic commutation
relations of K\"ahler geometry can be checked. If $A,B$ are differential
operators acting on the algebra $C^\infty(X,\Lambda^{\bu,\bu}T^\star_X\otimes
F)$, their graded commutator (or graded Lie bracket) is defined by
$$[A,B]= AB - (-1)^{ab} BA$$
where $a,b$ are the degrees of $A$ and $B$ respectively. If $C$ is another
endomorphism of degree $c$, the following purely formal {\it Jacobi identity}
holds:
$$(-1)^{ca} \big[A,[B,C]\big] + (-1)^{ab}\big[B,[C,A]\big]
+ (-1)^{bc} \big[C,[A,B]\big] = 0.$$

\claim{(2.5) Basic commutation relations} Let $(X,\omega)$ be a K\"ahler
manifold and let $L$ be the operators defined by $Lu=\omega\wedge u$ and
$\Lambda=L^\star$. Then
$$\eqalign{
[D^{\prime\prime\star},L]&=\ii D',\cr
[\Lambda,D'']&=-\ii D^{\prime\star},\cr}
\qquad
\eqalign{
[D^{\prime\star},L]&=-\ii D'',\cr
[\Lambda,D']&=\ii D^{\prime\prime\star}.\cr}$$
\endclaim

\begproof{(sketch).} The first step is to check the identity
$[d^{\prime\prime\star},L]=\ii d'$ for constant metrics on $X=\bC^n$ and
$F=X\times\bC$, by a brute force calculation. All three other identities
follow by taking conjugates or adjoints. The case of variable metrics follows
by looking at Taylor expansions up to order~1; essentially nothing changes
since the K\"ahler metric $\omega$ just introduces additional $O(|z|^2)$ 
terms which have zero derivative
at the center of the coordinate chart.\qed
\endproof

\claim{(2.6) Bochner-Kodaira-Nakano identity} If $(X,\omega)$ is
K\"ahler, the complex Laplace operators $\Delta'$ and $\Delta''$ acting
on $F$-valued forms satisfy the identity
$$\Delta''=\Delta'+[\ii\Theta(F),\Lambda].$$
\endclaim

\begproof{} The last equality in (2.5) yields $D^{\prime\prime\star}
=-\ii[\Lambda,D']$, hence
$$\Delta''=[D'',\delta'']=-\ii[D'',\big[\Lambda,D']\big].$$
By the Jacobi identity we get
$$\big[D'',[\Lambda,D']\big]=\big[\Lambda,[D',D'']]+\big[D',[D'',\Lambda]\big]
=[\Lambda,\Theta(F)]+\ii[D',D^{\prime\star}],$$
taking into account that $[D',D'']=D^2=\Theta(F)$. The formula follows.\qed
\endproof

\claim{(2.7) Corollary {\rm(Hodge decomposition)}} If $(X,\omega)$ is
a compact K\"ahler manifold, there is a canonical decomposition
$$
H^k(X,\bR)=\bigoplus_{p+q=k}H^{p,q}(X,\bC)
$$
of the De Rham cohomology groups in terms of the Dolbeault cohomology
groups.
\endclaim

\begproof{} If we apply the Bochner-Kodaira-Nakano identity to the
trivial bundle $F=X\times\bC$, we find $\Delta''=\Delta'$. Morever
$$
\Delta=[d'+d'',d^{\prime\star}+d^{\prime\prime\star}]=
\Delta'+\Delta''+[d',d^{\prime\prime\star}]+[d'',d^{\prime\star}].
$$
We claim that  $[d',d^{\prime\prime\star}]=[d'',d^{\prime\star}]=0$.
Indeed, we have 
$[d',d^{\prime\prime\star}]=-\ii\big[d',[\Lambda,d']\big]$ by (2.5), and 
the Jacobi identity implies
$$
-\big[d',[\Lambda,d']\big]+\big[\Lambda,[d',d']\big]+
\big[d',[d',\Lambda]\big]=0,
$$
hence $-2\big[d',[\Lambda,d']\big]=0$ and $[d',d^{\prime\prime\star}]=0$. 
The second identity is similar. As a consequence
$$
\Delta'=\Delta''={1\over 2}\Delta.
$$
We infer that $\Delta$ preserves the bidegree of forms and operates
``separately'' on each term $C^\infty(X,\Lambda^{p,q}T^*_X)$. Hence,
on the level of harmonic forms we have
$$
\cH^k(X,\bC)=\bigoplus_{p+q=k}\cH^{p,q}(X,\bC).
$$
The decomposition theorem (2.7) now follows from the Hodge isomorphisms for 
De Rham and Dolbeault groups. The decomposition is canonical since $H^{p,q}(X)$
coincides with the set of classes in $H^k(X,\bC)$ which can be represented
by $d$-closed $(p,q)$-forms.\qed
\endproof

Now, assume that $X$ is compact K\"ahler and that 
$u\in\ci(X,\Lambda^{p,q}T^\star
X\otimes F)$ is an arbitrary $(p,q)$-form. An integration by parts yields
$$\langle\Delta'u,u\rangle=\|D'u\|^2+\|D^{\prime\star}u\|^2\ge0$$
and similarly for $\Delta''$, hence we get the basic a priori inequality
$$\|D''u\|^2+\|D^{\prime\prime\star}u\|^2\ge\int_X
\langle[\ii\Theta(F),\Lambda]u,u\rangle dV_\omega.\leqno(2.8)$$
This inequality is known as the {\it Bochner-Kodaira-Nakano} inequality
(see [Boc48], [Kod53], [Nak55]). When $u$ is $\Delta''$-harmonic, we get
$$\int_X\big(\langle[\ii\Theta(F),\Lambda]u,u\rangle+
\langle T_\omega u,u\rangle\big)dV\le 0.$$
If the hermitian operator $[\ii\Theta(F),\Lambda]$ acting on
$\Lambda^{p,q}T^\star_X\otimes F$ is positive on each fiber, we infer
that $u$ must be zero, hence
$$H^{p,q}(X,F)=\cH^{p,q}(X,F)=0$$
by Hodge theory. The main point is thus to compute the curvature form
$\Theta(F)$ and find sufficient conditions under which the operator
$[\ii\Theta(F),\Lambda]$ is positive definite. Elementary (but somewhat
tedious) calculations yield the following formulae: if the curvature of
$F$ is written as in (1.8) and 
$u=\sum u_{J,K,\lambda}dz_I\wedge d\ol z_J\otimes e_\lambda$,
$|J|=p$, $|K|=q$, $1\le\lambda\le r$ is a $(p,q)$-form with values in~$F$,
then
$$\leqalignno{\langle[\ii\Theta(F),\Lambda]u,u\rangle=
&\phantom{{}+{}}\sum_{j,k,\lambda,\mu,J,S}{\,}
c_{jk\lambda\mu}\,u_{J,jS,\lambda}~\ol{u_{J,kS,\mu}}&(2.9)\cr
&{}+\sum_{j,k,\lambda,\mu,R,K}
c_{jk\lambda\mu}\,u_{kR,K,\lambda}~\ol{u_{jR,K,\mu}}\cr
&{}-{\,}\sum_{j,\lambda,\mu,J,K}\phantom{k}
c_{jj\lambda\mu}\,u_{J,K,\lambda}~\ol{u_{J,K,\mu}},\cr}$$
where the sum is extended to all indices $1\le j,k\le n$, $1\le\lambda,
\mu\le r$ and multiindices $|R|=p-1$, $|S|=q-1$ (here the notation
$u_{JK\lambda}$ is extended to non necessarily increasing multiindices by
making it alternate with respect to permutations). It is usually hard to
decide the sign of the curvature term (2.9), except in some special cases.

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

\claim{(2.10) Nakano vanishing theorem {\rm ([Nak55])}} Let $X$ be a
compact complex manifold and let $F$ be a Nakano positive vector
bundle on~$X$. Then
$$H^{n,q}(X,F)=H^q(X,K_X\otimes F)=0\qquad\hbox{for every $q\ge 1$.}
\eqno\square$$
\endclaim

Another tractable case is the case where $F$ is a line bundle ($r=1$).
Indeed, at each point $x\in X$, we may then choose a coordinate
system which diagonalizes simultaneously the hermitians forms
$\omega(x)$ and $\ii\Theta(F)(x)$, in such a way that
$$\omega(x)=\ii\sum_{1\le j\le n}dz_j\wedge d\ol z_j,\qquad
\ii\Theta(F)(x)=\ii\sum_{1\le j\le n}\gamma_jdz_j\wedge d\ol z_j$$
with $\gamma_1\le\ldots\le\gamma_n$. The curvature eigenvalues
$\gamma_j=\gamma_j(x)$ are then uniquely defined and depend continuously
on~$x$. With our previous notation, we have $\gamma_j=c_{jj11}$ and all
other coefficients $c_{jk\lambda\mu}$ are zero. For any $(p,q)$-form
$u=\sum u_{JK}dz_J\wedge d\ol z_K\otimes e_1$, this gives
$$\leqalignno{\langle[\ii\Theta(F),\Lambda]u,u\rangle
&=\sum_{|J|=p,\,|K|=q}\Big(\sum_{j\in J}\gamma_j+\sum_{j\in K}\gamma_j-
\sum_{1\le j\le n}\gamma_j\Big)|u_{JK}|^2\cr
&\ge(\gamma_1+\ldots+\gamma_q-\gamma_{n-p+1}-\ldots-\gamma_n)|u|^2.
&(2.11)\cr}$$
Assume that $\ii\Theta(F)$ is positive. It is then natural to make the
special choice $\omega=\ii\Theta(F)$ for the K\"ahler metric. Then
$\gamma_j=1$ for $j=1,2\ld n$ and we obtain
$\langle[\ii\Theta(F),\Lambda]u,u\rangle=(p+q-n)|u|^2$.
As a consequence:

\claim{(2.12) Akizuki-Kodaira-Nakano vanishing theorem {\rm ([AN54])}}
If $F$ is a positive line bundle on a compact complex manifold~$X$, then
$$H^{p,q}(X,F)=H^q(X,\Omega^p_X\otimes F)=0\qquad\hbox{for}~~p+q\ge n+1.
\eqno\square$$
\endclaim

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

\titleb{3.}{$L^2$ Estimates and Existence Theorems}
The starting point is the following $L^2$ existence theorem, which is
essentially due to H\"ormander [H\"or65, 66], and Andreotti-Vesentini
[AV65]. We will only  outline the main ideas, referring e.g.\ to [Dem82]
for a detailed exposition of the technical situation considered here.

\claim{(3.1) Theorem} Let $(X,\omega)$ be a K\"ahler manifold. Here $X$
is not necessarily compact, but we assume that the geodesic distance
$\delta_\omega$ is complete on~$X$. Let $F$ be a hermitian vector bundle of
rank $r$ over $X$, and assume that the curvature operator
$A=A^{p,q}_{F,\omega}=[\ii\Theta(F),\Lambda_\omega]$ is positive
definite everywhere on $\Lambda^{p,q}T^\star_X\otimes F$, $q\ge 1$.
Then for any form $g\in L^2(X,\Lambda^{p,q}T^\star_X\otimes F)$
satisfying $D''g=0$ and $\int_X\langle A^{-1}g,g\rangle
\,dV_\omega<+\infty$, there exists $f\in
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$ such that  $D''f=g$ and
$$\int_X|f|^2\,dV_\omega\le\int_X\langle A^{-1}g,g\rangle\,dV_\omega.$$
\endclaim

\begproof{} The assumption that $\delta_\omega$ is complete implies the
existence of cut-off functions $\psi_\nu$ with arbitrarily large compact
support such that $|d\psi_\nu|\le 1$ (take $\psi_\nu$ to be a function of
the distance $x\mapsto \delta_\omega(x_0,x)$, which is an almost
everywhere differentiable 1-Lipschitz function, and regularize if
necessary). From this, it follows that very form $u\in
L^2(X,\Lambda^{p,q}T^\star_X\otimes F)$ such that $D''u\in L^2$ and
$D^{\prime\prime\star}u\in L^2$ in the sense of distribution theory is a
limit of a sequence of smooth forms $u_\nu$ with compact support, in such
a way that $u_\nu\to u$, $D''u_\nu\to D''u$ and $D^{\prime\prime\star}u_\nu
\to D^{\prime\prime\star}u$ in~$L^2$ (just take $u_\nu$ to be a
regularization of $\psi_\nu u$). As a consequence, the basic a priori
inequality (2.8) extends to arbitrary forms $u$ such that
$u,~D''u,D^{\prime\prime\star}u\in L^2$ . Now, consider the Hilbert
space orthogonal decomposition
$$L^2(X,\Lambda^{p,q}T^\star_X\otimes F)=\Ker D''\oplus({\Ker D''})^\perp,$$
observing that $\Ker D''$ is weakly (hence strongly) closed. Let $v=v_1+v_2$
be the decomposition of a smooth form $v\in\cD^{p,q}(X,F)$ with compact
support according to this decomposition ($v_1$, $v_2$ do not have compact
support in general$\,$!).
Since $(\Ker D'')^\perp\subset\Ker D^{\prime\prime\star}$ by duality
and $g,v_1\in\Ker D''$ by hypothesis, we get
$D^{\prime\prime\star}v_2=0$ and
$$|\langle g,v\rangle|^2=|\langle g,v_1\rangle|^2\le
\int_X\langle A^{-1}g,g\rangle \,dV_\omega
\int_X\langle Av_1,v_1\rangle \,dV_\omega$$
thanks to the Cauchy-Schwarz inequality. The a priori inequality (2.8)
applied to $u=v_1$ yields
$$\int_X\langle Av_1,v_1\rangle \,dV_\omega\le
\|D''v_1\|^2+\|D^{\prime\prime\star}v_1\|^2=\|D^{\prime\prime\star}v_1\|^2=
\|D^{\prime\prime\star}v\|^2.$$
Combining both inequalities, we find
$$|\langle g,v\rangle|^2\le\Big(\int_X\langle A^{-1}g,g\rangle \,dV_\omega\Big)
\|D^{\prime\prime\star}v\|^2$$
for every smooth $(p,q)$-form $v$ with compact support. This shows that we
have a well defined linear form
$$w=D^{\prime\prime\star}v\longmapsto\langle v,g\rangle,\qquad
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)
\supset D^{\prime\prime\star}(\cD^{p,q}(F))\lra\bC$$
on the range of $D^{\prime\prime\star}$. This linear form
is continuous in $L^2$ norm and has norm $\le C$ with
$$C=\Big(\int_X\langle A^{-1}g,g\rangle \,dV_\omega\Big)^{1/2}.$$
By the Hahn-Banach theorem, there is an element
$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$ with $||f||\le C$, such that
$\langle v,g\rangle=\langle D^{\prime\prime\star}v,f\rangle$ for every $v$,
hence $D''f=g$ in the sense of distributions. The inequality $||f||\le C$
is equivalent to the last estimate in the theorem.\qed
\endproof

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

\claim{(3.2) Theorem} Let $(X,\omega)$ be a K\"ahler manifold,
$\dim X=n$. Assume that $X$ is weakly pseudoconvex. Let $F$ be a hermitian
line bundle and let
$$\gamma_1(x)\le\ldots\le\gamma_n(x)$$
be the curvature eigenvalues $($i.e.\ the eigenvalues of $\ii\Theta(F)$
with respect to the metric~$\omega)$ at every point. Assume that the
curvature is positive, i.e.\ $\gamma_1>0$ everywhere. Then for any form
$g\in L^2(X,\Lambda^{n,q}T^\star_X\otimes F)$ satisfying $D''g=0$ and
\hbox{$\int_X\langle(\gamma_1+\ldots+\gamma_q)^{-1}|g|^2\,dV_\omega<+\infty$},
there exists \hbox{$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$} such that
$D''f=g$ and
$$\int_X|f|^2\,dV_\omega\le\int_X
(\gamma_1+\ldots+\gamma_q)^{-1}|g|^2\,dV_\omega.$$
\endclaim

\begproof{} Indeed, for $p=n$, Formula~2.11 shows that
$$\langle Au,u\rangle\ge(\gamma_1+\ldots+\gamma_q)|u|^2,$$
hence $\langle A^{-1}u,u\rangle\ge(\gamma_1+\ldots+\gamma_q)^{-1}|u|^2$.\qed
\endproof

An important observation is that the above theorem still applies when the
hermitian metric on $F$ is a singular metric with positive curvature in
the sense of currents. In fact, by standard
regularization techniques (convolution of psh functions by smoothing
kernels), the metric can be made smooth and the solutions obtained by
(3.1) or (3.2) for the smooth metrics have limits satisfying the desired
estimates. Especially, we get the following

\claim{(3.3) Corollary} Let $(X,\omega)$ be a K\"ahler manifold,
$\dim X=n$. Assume that $X$ is weakly pseudoconvex. Let $F$ be a
holomorphic line bundle equipped with a singular metric whose local
weights are denoted $\varphi\in L^1_\loc$. Suppose that
$$\ii\Theta(F)=2\ii d'd''\varphi\ge\varepsilon\omega$$
for some $\varepsilon>0$. Then for any form $g\in L^2(X,\Lambda^{n,q}
T^\star_X\otimes F)$ satisfying $D''g=0$, there exists
$f\in L^2(X,\Lambda^{p,q-1}T^\star_X\otimes F)$ such that  $D''f=g$ and
$$\int_X|f|^2e^{-\varphi}\,dV_\omega\le{1\over q\varepsilon}\int_X
|g|^2e^{-\varphi}\,dV_\omega.\eqno\square$$
\endclaim

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

In order to apply Corollary 3.3 in a  fruitful way, it is usually necessary
to select $\varphi$ with suitable logarithmic poles along an analytic set.
The basic construction of such a function is provided by the following
Lemma.

\claim{(3.4) Lemma} Let $X$ be a compact complex manifold~$X$ 
equipped with a K\"ahler metric 
$\omega=i\sum_{1\le j,k\le n}\omega_{jk}(z)dz_j\wedge d\ol z_k$
and let $Y\subset X$ be an analytic subset of~$X$. Then there exist
globally defined quasi-plurisubharmonic potentials $\psi$ and 
$(\psi_\varepsilon)_{\varepsilon\in{}]0,1]}$ on $X$,
satisfying the following properties.
{\oldparindent=6.6mm
\smallskip
\item {\rm (i)} The function $\psi$ is smooth on $X\ssm Y$, satisfies
$i\ddbar\psi\ge -A\omega$ for some $A>0$, and $\psi$
has logarithmic poles along~$Y$, i.e., locally near $Y$
$$
\psi(z)\sim \log\sum_k|g_k(z)|+O(1)
$$
where $(g_k)$ is a local system of generators of the ideal sheaf $\cI_Y$
of $Y$ in $X$.
\smallskip
\item {\rm(ii)} We have 
$\psi=\lim_{\varepsilon\to 0}\downarrow\psi_\varepsilon$ 
where the $\psi_\varepsilon$ are $C^\infty$ and possess a uniform 
Hessian estimate
$$
i\ddbar\psi_\varepsilon\ge -A\omega\qquad\hbox{on~$X$}.
$$
\smallskip
\item {\rm(iii)} Consider the family of hermitian metrics
$$
\omega_\varepsilon := \omega + {1\over 2A} i\ddbar\psi_\varepsilon
\ge {1\over 2}\omega.
$$
For any point $x_0\in Y$ and any neighborhood $U$ of $x_0$, 
the volume element of $\omega_\varepsilon$ has
a uniform lower bound 
$$
\int_{U\cap V_\varepsilon}\omega_\varepsilon^n\ge\delta(U)>0,
$$
where $V_\varepsilon=\{z\in X\,;\;\psi(z)<\log\varepsilon\}$
is the ``tubular neighborhood'' of radius $\varepsilon$ around $Y$.
\smallskip
\item {\rm(iv)} For every integer $p\ge 0$, the family of positive 
currents $\omega_\varepsilon^p$ is bounded in mass. Moreover,
if $Y$ contains an irreducible component $Y'$ of codimension $p$,
there is a uniform lower bound
$$
\int_{U\cap V_\varepsilon}\omega_\varepsilon^p\wedge\omega^{n-p}\ge
\delta_p(U)>0
$$
in any neighborhood $U$ of a regular point $x_0\in Y'$. In particular, 
any weak limit $\Theta$ of $\omega_\varepsilon^p$ as $\varepsilon$
tends to $0$ satisfies $\Theta\ge\delta'[Y']$ for some
$\delta'>0$.
\vskip0pt
}
\endclaim

\begproof{} By compactness of $X$, there is a covering of $X$ by open
coordinate balls~$B_j$, $1\le j\le N$, such that $\cI_Y$ is generated by 
finitely many holomorphic functions $(g_{j,k})_{1\le k\le m_j}$ on a 
neighborhood of $\ol B_j$. We take a partition of unity
$(\theta_j)$ subordinate to $(B_j)$ such that $\sum\theta_j^2=1$ on $X$,
and define
$$
\eqalign{
\psi(z)&={1\over 2}\log\sum_j\theta_j(z)^2\sum_k|g_{j,k}(z)|^2,\cr
\psi_\varepsilon(z)&={1\over 2}\log(e^{2\psi(z)}+\varepsilon^2)
={1\over 2}\log\Big(\sum_{j,k}\theta_j(z)^2|g_{j,k}(z)|^2+\varepsilon^2
\Big).\cr}
$$
Moreover, we consider the family of $(1,0)$-forms with support in $B_j$
such that
$$
\gamma_{j,k}=\theta_j\partial g_{j,k}+2g_{j,k}\partial\theta_j.
$$
Straightforward calculations yield
$$
\leqalignno{
\dbar\psi_\varepsilon&={1\over 2}
{\sum_{j,k}\theta_jg_{j,k}\ol{\gamma_{j,k}}\over e^{2\psi}+
\varepsilon^2},\cr
i\ddbar\psi_\varepsilon&={i\over 2}\left(
{\sum_{j,k}\gamma_{j,k}\wedge\ol{\gamma_{j,k}}\over e^{2\psi}+
\varepsilon^2}-
{\sum_{j,k}\theta_j\ol{g_{j,k}}\gamma_{j,k}
\wedge \sum_{j,k}\theta_jg_{j,k}\ol{\gamma_{j,k}}
\over (e^{2\psi}+\varepsilon^2)^2}\right),&(3.5)\cr
&\qquad{}+i\;
{\sum_{j,k}|g_{j,k}|^2(\theta_j\ddbar\theta_j-\partial\theta_j\wedge
\dbar\theta_j)\over e^{2\psi}+\varepsilon^2}.\cr
}
$$
As $e^{2\psi}=\sum_{j,k}\theta_j^2|g_{j,k}|^2$, the first big sum in
$i\ddbar\psi_\varepsilon$ is nonnegative by the Cauchy-Schwarz inequality;
when viewed as a hermitian form, the value of this sum on a tangent 
vector $\xi\in T_X$ is simply
$$
{1\over 2}\left(
{\sum_{j,k}|\gamma_{j,k}(\xi)|^2\over e^{2\psi}+\varepsilon^2}-
{\big|\sum_{j,k}\theta_j\ol{g_{j,k}}\gamma_{j,k}(\xi)\big|^2\over
(e^{2\psi}+\varepsilon^2)^2}\right)\ge
{1\over 2}{\varepsilon^2\over(e^{2\psi}+\varepsilon^2)^2}
\sum_{j,k}|\gamma_{j,k}(\xi)|^2.\leqno(3.6)
$$
Now, the second sum involving $\theta_j\ddbar\theta_j-\partial\theta_j\wedge
\dbar\theta_j$ in (3.5) is uniformly bounded below by a fixed negative 
hermitian form $-A\omega$, $A\gg 0$, and therefore
\hbox{$i\ddbar\psi_\varepsilon\ge -A\omega$}. Actually, for every pair
of indices $(j,j')$ we have a bound
$$
C^{-1}\le \sum_k|g_{j,k}(z)|^2/\sum_k|g_{j',k}(z)|^2\le C\qquad
\hbox{on $\ol B_j\cap\ol B_{j'}$},
$$
since the generators $(g_{j,k})$ can be expressed as holomorphic
linear combinations of the $(g_{j',k})$ by Cartan's theorem A (and
vice versa). It follows easily that all terms $|g_{j,k}|^2$ are uniformly
bounded by $e^{2\psi}+\varepsilon^2$. In particular, $\psi$ and
$\psi_\varepsilon$ are quasi-plurisubharmonic, and we see that 
(i) and (ii) hold true. By construction, 
the real $(1,1)$-form $\omega_\varepsilon:=\omega+{1\over 2A}
i\ddbar\psi_\varepsilon$ satisfies $\omega_\varepsilon\ge {1\over 2}\omega$,
hence it is K\"ahler and its eigenvalues with respect to $\omega$ are at 
least equal to~$1/2$. 

Assume now that we are in a neighborhood $U$ of a regular point
$x_0\in Y$ where $Y$ has codimension~$p$.  Then
\hbox{$\gamma_{j,k}=\theta_j\partial g_{j,k}$} at $x_0$, hence the
rank of the system of $(1,0)$-forms $(\gamma_{j,k})_{k\ge 1}$ is at
least equal to $p$ in a neighborhood of~$x_0$. Fix a holomorphic
local coordinate system $(z_1,\ldots,z_n)$ such that 
\hbox{$Y=\{z_1=\ldots=z_p=0\}$} near $x_0$, and let $S\subset T_X$ 
be the holomorphic subbundle generated by $\partial/\partial z_1,\ldots,
\partial/\partial z_p$. This choice ensures that the rank of the system
of $(1,0)$-forms $(\gamma_{j,k|S})$ is everywhere equal to~$p$.
By (1,3) and the minimax principle applied to the $p$-dimensional subspace
$S_z\subset T_{X,z}$, we see that the $p$-largest eigenvalues of 
$\omega_\varepsilon$ are bounded below by 
$c\varepsilon^2/(e^{2\psi}+\varepsilon^2)^2$.
\smallskip
However, we can even restrict the form defined in (3.6) to the 
$(p-1)$-dimen\-sional subspace $S\cap\Ker\tau$ where 
$\tau(\xi):=\sum_{j,k}\theta_j\ol{g_{j,k}}\gamma_{j,k}(\xi)$,
to see that the $(p-1)$-largest eigenvalues of $\omega_\varepsilon$
are bounded below by $c/(e^{2\psi}+\varepsilon^2)$, $c>0$. The
$p$-th eigenvalue is then bounded by 
$c\varepsilon^2/(e^{2\psi}+\varepsilon^2)^2$ and the
remaining $(n-p)$-ones by~$1/2$. From this we infer
$$
\eqalign{
\omega_\varepsilon^n
&\ge c\,{\varepsilon^2\over(e^{2\psi}+\varepsilon^2)^{p+1}}\omega^n
\qquad\hbox{near $x_0$},\cr
\omega_\varepsilon^p
&\ge c\,{\varepsilon^2\over(e^{2\psi}+\varepsilon^2)^{p+1}}\Big
(i\sum_{1\le\ell\le p}\gamma_{j,k_\ell}\wedge
\ol{\gamma_{j,k_\ell}}\Big)^p
\cr}
$$
where $(\gamma_{j,k_\ell})_{1\le \ell\le p}$ is a suitable $p$-tuple
extracted from the $(\gamma_{j,k})$, such that\break 
$\bigcap_\ell\Ker\gamma_{j,k_\ell}$ is a smooth complex
(but not necessarily holomorphic) subbundle of codimension $p$ of $T_X$;
by the definition of the forms $\gamma_{j,k}$, this
subbundle must coincide with $T_Y$ along $Y$. From this, properties
(iii) and (iv) follow easily; actually, up to constants,
we have $e^{2\psi}+\varepsilon^2\sim|z_1|^2+\ldots+|z_p|^2+\varepsilon^2$
and
$$
i\sum_{1\le\ell\le p}\gamma_{j,k_\ell}\wedge
\ol{\gamma_{j,k_\ell}}\ge c\,i\ddbar(|z_1|^2+\ldots+|z_p|^2)-
O(\varepsilon)i\ddbar|z|^2\qquad\hbox{on $U\cap V_\varepsilon$},
$$
hence, by a straightforward calculation,
$$
\omega_\varepsilon^p\wedge\omega^{n-p}\ge
c\big(i\ddbar\log(|z_1|^2+\ldots+|z_p|^2+\varepsilon^2)\big)^p\wedge
\big(i\ddbar(|z_{p+1}|^2+\ldots+|z_n|^2)\big)^{n-p}
$$
on $U\cap V_\varepsilon$; notice also that
$\omega_\varepsilon^n\ge 2^{-(n-p)}\omega_\varepsilon^p\wedge\omega^{n-p}$,
so any lower bound for the volume of 
$\omega_\varepsilon^p\wedge\omega^{n-p}$ will also produce a bound
for the volume of $\omega_\varepsilon^n$. As it is well known, the $(p,p)$-form
$$
\Big({i\over 2\pi}\ddbar\log(|z_1|^2+\ldots+|z_p|^2+\varepsilon^2)\Big)^p
\qquad\hbox{on $\bC^n$}
$$
can be viewed as the pull-back to $\bC^n=\bC^p\times\bC^{n-p}$ of 
the Fubini-Study volume form of the complex $p$-dimensional
projective space of dimension $p$ containing $\bC^p$ as an affine
Zariski open set, rescaled by the dilation ratio~$\varepsilon$. Hence
it converges weakly to the current of integration on the $p$-codimensional
subspace $z_1=\ldots=z_p=0$. Moreover the volume contained in any
compact tubular cylinder
$$
\{|z'|\le C\varepsilon\}\times K''\subset \bC^p\times\bC^{n-p}
$$
depends only on $C$ and $K$ (as one sees after rescaling by $\varepsilon$).
The fact that $\omega_\varepsilon^p$ is uniformly bounded in mass can
be seen easily from the fact that
$$
\int_X \omega_\varepsilon^p\wedge\omega^{n-p}=\int_X\omega^n,
$$
as $\omega$ and $\omega_\varepsilon$ are in the same K\"ahler class.
Let $\Theta$ be any weak limit of $\omega_\varepsilon^p$. By what we
have just seen, $\Theta$ carries non zero mass on every 
$p$-codimensional component $Y'$ of $Y$, for instance near every
regular point. However, standard results of the theory of currents
(support theorem and Skoda's extension result) imply that
${\bf 1}_{Y'}\Theta$ is a closed positive current and that
${\bf 1}_{Y'}\Theta=\lambda[Y']$ is a nonnegative multiple
of the current of integration on $Y'$. The fact that the mass
of $\Theta$ on $Y'$ is positive yields~$\lambda>0$. 
Lemma~3.4 is proved.\qed

\titleb{4.}{Multiplier Ideal Sheaves}

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

\claim{(4.1) Definition} Let $\varphi$ be a
psh function on an open subset $\Omega\subset X\,$; to $\varphi$ is
associated the ideal subsheaf $\cI(\varphi)\subset\cO_\Omega$ of germs of
holomorphic functions $f\in\cO_{\Omega,x}$ 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. Of course,
such points occur only if $\varphi$ has logarithmic poles. This is made
precise as follows.

\claim{(4.2) Definition} A psh function $\varphi$ is said to have a
logarithmic pole of coefficient $\gamma$ at a point $x\in X$ if the Lelong
number
$$\nu(\varphi,x):=\liminf_{z\to x}{\varphi(z)\over\log|z-x|}$$
is non zero and if $\nu(\varphi,x)=\gamma$.
\endclaim

\claim{(4.3) Lemma {\rm (Skoda [Sko72a])}} Let $\varphi$ be a psh function
on an open set $\Omega$ and let $x\in\Omega$.
\smallskip
\item{\rm a)} If $\nu(\varphi,x)<2$, then $e^{-\varphi}$ is integrable in
a neighborhood of~$x$, in particular $\cI(\varphi)_x=\cO_{\Omega,x}$.
\smallskip
\item{\rm b)} If $\nu(\varphi,x)\ge 2(n+s)$ for some integer $s\ge 0$, then
$e^{-\varphi}\ge C|z-x|^{-2n-2s}$ in a neighborhood of $x$ and
$\cI(\varphi)_x\subset\gm_{\Omega,x}^{s+1}$, where $\gm_{\Omega,x}$ is the
maximal ideal of $\cO_{\Omega,x}$.
\smallskip
\item{\rm c)} The zero variety $V(\cI(\varphi))$ of $\cI(\varphi)$
satisfies 
$$E_{2n}(\varphi)\subset V(\cI(\varphi))\subset E_2(\varphi)$$
where $E_c(\varphi)=\{x\in X\,;\,\nu(\varphi,x)\ge c\}$ is the $c$-sublevel
set of Lelong numbers of~$\varphi$.
\vskip0pt
\endclaim

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

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

\claim{(4.3) 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

\begproof{} Since the result is local, we may assume that $\Omega$ is a
bounded pseudoconvex open set in $\bC^n$. 
By the strong noetherian property of coherent sheaves, the family of
sheaves generated by finite subsets of $\cH^2(\Omega,\varphi)$ has
a maximal element on each compact subset of $\Omega$, hence
$\cH^2(\Omega,\varphi)$ 
generates a coherent ideal sheaf $\cJ\subset\cO_\Omega$. It is clear that
$\cJ\subset\cI(\varphi)$; in order to prove the equality,
we need only check that $\cJ_x+\cI(\varphi)_x\cap
\gm^{s+1}_{\Omega,x}=\cI(\varphi)_x$ for every integer $s$, in
view of the Krull lemma. Let $f\in\cI(\varphi)_x$ be defined 
in a neighborhood $V$ of $x$ and let $\theta$ be a cut-off function
with support in $V$ such that $\theta=1$ in a neighborhood of $x$.
We solve the equation $d''u=g:=d''(\theta f)$ by means of H\"ormander's
$L^2$ estimates~3.3, where $F$ is the trivial line bundle
$\Omega\times\bC$ equipped with the strictly psh weight 
$$\wt\varphi(z)=\varphi(z)+2(n+s)\log|z-x|+|z|^2.$$ 
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

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

\claim{(4.5) Proposition} Let $\mu:X'\to X$ be a modification
of non singular complex manifolds $($i.e.\ a proper generically 1:1
holomorphic map$)$, and let $\varphi$ be a psh function on~$X$. Then 
$$\mu_\star\big(\cO(K_{X'})\otimes\cI(\varphi\circ\mu)\big)=
\cO(K_X)\otimes\cI(\varphi).$$
\endclaim

\begproof{} Let $n=\dim X=\dim X'$ and let $S\subset X$ be an analytic
set such that $\mu:X'\ssm S'\to X\ssm S$ is a biholomorphism. By
definition of multiplier ideal sheaves, $\cO(K_X)\otimes\cI(\varphi)$ is
just the sheaf of holomorphic $n$-forms $f$ on open sets $U\subset X$
such that
$\ii^{n^2}f\wedge\ol f\,e^{-\varphi}\in L^1_\loc(U)$. Since
$\varphi$ is locally bounded from above, we may even consider forms $f$
which are a priori defined only on $U\ssm S$, because $f$ will be
in $L^2_\loc(U)$ and therefore will automatically extend through~$S$.
The change of variable formula yields
$$\int_U\ii^{n^2}f\wedge\ol f\,e^{-\varphi}=
\int_{\mu^{-1}(U)}\ii^{n^2}\mu^\star f\wedge\ol{\mu^\star f}
\,e^{-\varphi\circ\mu},$$
hence $f\in\Gamma(U,\cO(K_X)\otimes\cI(\varphi))$ iff
$\mu^\star f\in\Gamma(\mu^{-1}(U),\cO(K_{X'})\otimes\cI(\varphi\circ
\mu))$. Proposition~4.5 is proved.\qed
\endproof

\claim{(4.6) Remark} \rm If $\varphi$ has analytic singularities, i.e.\
if there are holomorphic functions $(f_j)_{1\le j\le N}$ and a constant
$\alpha>0$ such that
$$
\varphi(z)={\alpha\over 2}\log(|f_1(z)|^2+\ldots+|f_N(z)|^2)+O(1)
$$
in a neighborhood of every point, the computation of $\cI(\varphi)$
can be reduced to a purely algebraic problem.

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

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

First, one computes a smooth modification $\mu:\wt X\to X$ of $X$ such
that $\mu^\star\cJ$ is an invertible sheaf $\cO(-D)$ associated with
a normal crossing divisor $D=\sum\lambda_jD_j$, where $(D_j)$ are the
components of the exceptional divisor of $\wt X$ (take the blow-up $X'$
of $X$ with respect to the ideal $\cJ$ so that the pull-back of $\cJ$
to $X'$ becomes an invertible sheaf $\cO(-D')$, then blow up again by
Hironaka [Hir64] to make $X'$ smooth and $D'$ have normal crossings).
Now, we have $K_{\wt X}=\mu^\star K_X+R$ where $R=\sum\rho_j
D_j$ is the zero divisor of the Jacobian function $J_\mu$ of the blow-up
map. By the direct image formula~4.5, we get
$$\cI(\varphi)=\mu_\star\big(\cO(K_{\wt X}-\mu^\star K_X)\otimes
\cI(\varphi\circ\mu)\big)=\mu_\star\big(\cO(R)\otimes
\cI(\varphi\circ\mu)\big).$$
Now, $(f_j\circ\mu)$ are generators of the ideal $\cO(-D)$, hence
$$\varphi\circ\mu\sim\alpha\sum\lambda_j\log|g_j|$$
where $g_j$ are local generators of $\cO(-D_j)$. We are thus reduced to
computing multiplier ideal sheaves in the case where the poles are
given by a $\bQ$-divisor with normal crossings $\sum\alpha\lambda_jD_j$.
We obtain $\cI(\varphi\circ\mu)=\cO(-\sum\lfloor\alpha\lambda_j\rfloor D_j)$,
hence
$$\cI(\varphi)=\mu_\star\cO_{\wt X}\big(\sum(\rho_j-\lfloor\alpha\lambda_j
\rfloor)D_j\big).\eqno\square$$
\endclaim

\claim{(4.7) Exercise} \rm Compute the multiplier ideal sheaf
$\cI(\varphi)$ associated with
$\varphi=\log(|z_1|^{\alpha_1}+\ldots+|z_p|^{\alpha_p})$
for arbitrary real numbers $\alpha_j>0$.\newline
{\it Hint:} using Parseval's formula and polar coordinates $z_j=r_j
e^{\ii\theta_j}$, show that the problem is equivalent to determining for
which $p$-tuples $(\beta_1\ld\beta_p)\in\bN^p$ the integral
$$\int_{[0,1]^p}{r_1^{2\beta_1}\ldots r_p^{2\beta_p}\,r_1dr_1\ldots r_pdr_p
\over r_1^{2\alpha_1}+\ldots+r_p^{2\alpha_p}}
=\int_{[0,1]^p}{t_1^{(\beta_1+1)/\alpha_1}\ldots t_p^{(\beta_p+1)/\alpha_p}
\over t_1+\ldots+t_p}\prod_{j=1}^p{1\over 2\alpha_j}{dt_j\over t_j}$$
is convergent. Conclude from this that $\cI(\varphi)$ is generated by
the monomials $z_1^{\beta_1}\ldots z_p^{\beta_p}$ such that
$\sum(\beta_p+1)/\alpha_p>1$. (This exercise shows that the analytic
definition of $\cI(\varphi)$ is sometimes also quite convenient for
computations).\qed
\endclaim

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

\claim{(4.8) Nadel vanishing theorem {\rm([Nad89], [Dem93b])}} Let
$(X,\omega)$ be a K\"ahler weakly pseudoconvex manifold, and let $F$ be a
holomorphic line bundle over $X$ equipped with a singular hermitian metric
$h$ of weight~$\varphi$. Assume that $\ii\Theta_h(F)\ge\varepsilon\omega$
for some continuous positive function $\varepsilon$ on~$X$. Then
$$H^q\big(X,\cO(K_X+F)\otimes\cI(h)\big)=0\qquad
\hbox{for all $q\ge 1$.}$$
\endclaim

\begproof{} Let $\cL^q$ be the sheaf of germs of $(n,q)$-forms 
$u$ with values in $F$ and with measurable coefficients, such that
both $|u|^2e^{-\varphi}$ and $|d''u|^2e^{-\varphi}$ are locally
integrable. The $d''$ operator defines a complex of sheaves
$(\cL^\bu,d'')$ which is a resolution of the sheaf
$\cO(K_X+F)\otimes\cI(\varphi)$: indeed, the kernel of $d''$ in degree
$0$ consists of all germs of holomorphic $n$-forms with values in $F$
which satisfy the integrability condition; hence the coefficient
function lies in $\cI(\varphi)$; the exactness in degree $q\ge 1$
follows from Corollary~3.3 applied on arbitrary small balls.
Each sheaf $\cL^q$ is a ${\cal C}^\infty$-module, so $\cL^\bu$ 
is a resolution by acyclic sheaves. Let $\psi$ be a smooth psh
exhaustion function on~$X$. Let us apply Corollary~3.3 globally on~$X$,
with the original metric of $F$ multiplied by the factor
$e^{-\chi\circ\psi}$, where $\chi$ is a convex increasing function of
arbitrary fast growth at infinity. This factor can be used to ensure
the convergence of integrals at infinity. By Corollary~3.3, we conclude
that $H^q\big(\Gamma(X,\cL^\bu)\big)=0$ for $q\ge 1$. The theorem
follows.\qed
\endproof 

\claim{(4.9) Corollary} Let $(X,\omega)$, $F$ and $\varphi$ be as in
Theorem~4.8 and let $x_1\ld x_N$ be isolated points in the zero
variety~$V(\cI(\varphi))$. Then there is a surjective map
$$H^0(X,K_X+F)\lraww\bigoplus_{1\le j\le N}\cO(K_X+L)_{x_j}\otimes
\big(\cO_X/\cI(\varphi)\big)_{x_j}.$$
\endclaim

\begproof{} Consider the long exact sequence of cohomology associated to the
short exact sequence $0\to\cI(\varphi)\to\cO_X\to\cO_X/\cI(\varphi)\to 0$
twisted by $\cO(K_X+F)$, and apply Theorem~4.8 to obtain the vanishing of
the first $H^1$ group. The asserted surjectivity property follows.\qed
\endproof

\claim{(4.10) Corollary} Let $(X,\omega)$, $F$ and $\varphi$ be as in
Theorem~4.8 and suppose that the weight function $\varphi$ is such that
$\nu(\varphi,x)\ge 2(n+s)$ at some point $x\in X$ which is an isolated point
of $E_2(\varphi)$. Then $H^0(X,K_X+F)$ generates all $s$-jets at~$x$.
\endclaim

\begproof{} The assumption is that $\nu(\varphi,y)<2$ for $y$ near $x$,
$y\ne x$. By Skoda's lemma 4.3~b), we conclude that $e^{-\varphi}$ is
integrable at all such points~$y$, hence $\cI(\varphi)_y=\cO_{X,y}$, whilst
$\cI(\varphi)_x\subset\gm_{X,x}^{s+1}$ by 4.3~a). Corollary 4.10 is thus
a special case of 4.9.\qed
\endproof

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

\claim{(4.11) Theorem} Let $X$ be a compact complex manifold, $E$ a holomorphic
vector bundle and $(F,h_F)$ a hermitian line bundle with a smooth metric
$h$ such that 
$$
\omega=\ii\Theta_{h_F}(F)>0.
$$ 
Let $\varphi$ be a quasi-psh function on $X$, i.e.\ a function
$\varphi$ such that $\ii d'd''\varphi\ge -C\omega$ for some constant
$C>0$. Then 
\smallskip
\item{\rm a)} There exists an integer $m_0$ such that
$$
H^q(X,E\otimes F^{\otimes m}\otimes\cI(\varphi))=0
$$
for $q\ge 1$ and $m\ge m_0$. 
\smallskip
\item{\rm b)} The restriction map
$$
H^0(X,E\otimes F^{\otimes m}\otimes\cI(\varphi))\lra
H^0(X,E\otimes F^{\otimes m}\otimes\cO_X/\cI(\varphi))
$$
is surjective for $m\ge m_0$.
\smallskip
\item{\rm b)} The vector bundle  $E\otimes F^{\otimes m}$ generates its
sections $($or jets of any order $s)$ for $m\ge m_0(s)$ large enough.
\vskip 0pt
\endclaim

\begproof{} a) Put an arbitrary smooth hermitian metric $h_E$ en $E$ and
consider the singular hermitian metric $h_E\cdot h_F^m\cdot e^{-\varphi}$
on $E\otimes F^{\otimes m}$. The $L^2$ holomorphic sections of
$E\otimes F^{\otimes m}$ are exactly the sections of the sheaf 
$E\otimes F^{\otimes m}\otimes\cI(\varphi)$. On the other hand, the 
curvature of the metric is
$$
\ii\Theta_{h_E}(E)+(m\,\ii\Theta_{h_F}(F)+\ii d'd''\varphi)\otimes\Id_E
$$
and therefore the curvature is Nakano${}>0$ for $m$ large. This implies
the vanishing of $H^q(X,E\otimes F^{\otimes m}\otimes\cI(\varphi))$. 
\smallskip\noindent
b) The vanishing of $H^1$ itself implies the surjevtivity statement on the
$H^0$ groups by the same argument as in Corollary (4.9).
\smallskip\noindent
c) Clearly, one can construct a quasi-psh function $\varphi$ with
a single logarithmic pole at a point $x\in X$ by taking
$$
\varphi(z)=\theta(z)(n+s-1)\log\sum|z_j-x_j|^2
$$
in some local coordinates near $x$, where $\theta$ is a cut-off function
with support in the coordinate open set. Then $\cI(\varphi)=\gm_x^s$ and
we conclude by b).\qed
\endproof

\claim{(4.12) Remark} \rm Assume that $X$ is compact and that $F$ is
a positive line bundle on~$X$. Let $\{x_1\ld x_N\}$ be a finite set.
Show that there are constants $C_1,C_2\ge 0$ depending only on $X$ and $E$, $F$
such that $H^0(X,E\otimes F^{\otimes m})$ interpolates given jets of order 
$s_j$ at $x_j$ for $m\ge C_1\sum s_j+C_2$. To see this,  we take a quasi-psh 
weight
$$
\varphi(z)=\sum\theta_j(z)(n+s_j-1)\log|w^{(j)}(z)|
$$
with respect to coordinate systems $(w^{(j)}_k(z))_{1\le k\le n}$ centered 
at $x_j$. The cut-off functions can be taken of a fixed radius (bounded away
from~$0$) with respect to a finite collection of coordinate patches 
covering~$X$. It is then easy to see that $\ii d'd''\varphi\ge -C(\sum s_j+1)
\omega$.\qed
\endclaim

\claim{(4.13) Theorem {\rm(Kodaira [Kod54])}} Let $X$ be a compact complex
manifold. A line bundle $L$ on $X$ is ample if and only if $L$ is positive.
In particular, a manifold $X$ posessing a positive line bundle is
projective, and can be embedded in projective space via the canonical
map $\Phi_{|mL|}:X\to\bP^N$ for $N$ large.
\endclaim

\begproof{} If the line bundle $L$ is ample, then by definition the
canonical map $\Phi_{|mL|}:X\to\bP^N$ is an embedding for $m$ large,
and $mL=\smash{\Phi_{|mL|}^{-1}}\cO(1)$. This implies that $L$ can be
equipped with a metric of positive curvature, as we saw in (1.15).
Conversely, if $L$ possesses a metric with positive curvature, then
by Theorem 4.11$\,$c), there exists an integer $m_0$ such that for $m\ge m_0$
sections in $H^0(X,mL)$ separate any pair of points $\{x,y\}\subset X$
and generate $1$-jets of sections at every point $x\in X$. However,
as is easily seen, separation of points is equivalent to the injectivity
of the map $\Phi_{|mL|}$, and the generation of $1$-jets is equivalent
to the fact that $\Phi_{|mL}$ is an immersion.\qed
\endproof

\claim{(4.14) Theorem {\rm(Chow [Chw49])}} Let $X\subset\bP^N$ be a 
$($closed$)$ complex analytic subset of $\bP^N$. Then $X$ is algebraic
and can be defined as the common zero set of a finite collection
of homogeneous polynomials $P_j(z_0,z_1,\ldots,z_N)=0$, $1\le j\le k$.
\endclaim

\begproof{} By Lemma 3.4 (i), there exists a quasi psh function $\psi$
with logarithmic poles along $X$. Then $\cI(2N\psi)\subset\cI_X$, since
$e^{-2N\psi}$ is certainly not integrable along~$Z$. For $x\in\bP^N\ssm X$, 
we consider a quasi-psh weight
$$
\varphi_x(z)=\psi(z)+\theta_x(z)(2N)\log|z-x|
$$
with a suitable cut-off function with support on a neighborhood of $x$
possessing holomorphic coordinates, in such a way that
$\cI(\varphi_x)_x=\gm_{\bP^N,x}$. We can arrange that $\ii\ddbar\varphi_x
\ge -C\omega$ uniformly for all $x$. Then we have vanishing of
$H^1(\bP^N,\cO(m)\otimes\cI(\varphi_x))$ for $m\ge m_0$ and therefore
we get a surjective map
$$
H^0(\bP^N,\cO(m))\to
H^0(\bP^N,\cO(m)\otimes\cO_{\bP^N}/\cI(\varphi_x)).
$$
As a consequence, we can find a homogeneous polynomial of degree $m$
which takes value $1$ at $x$ and vanishes on $X$ (as prescribed
by the ideal sheaf $\cI(\psi)$).\qed
\endproof

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

\claim{(4.15) Theorem {\rm(Chow [Chw49])}} Let $X\subset\bP^N$ be a complex
analytic submanifold of dimension $n$. Then $X$ is projective algebraic 
and can be defined
by a collection of homogeneous polynomials $P_j(z_0,z_1,\ldots,z_N)=0$,
$1\le j\le k$, such that the system of differentials $(dP_j)$ has rank 
equal to $\codim X$ at every point of $X$.
\endclaim

\begproof{} Put $r=\codim X=N-n$. There exists a quasi-psh function 
$\varphi$ which has logarithmic poles along $X$. To see this, just take
an open overing of $\bP^N$ by open sets $U_j$ where $X$ is defined by
$w^{(j)}_1=\ldots=w^{(j)}_r=0$ and $(w^{(j)}_k)_{1\le k\le N}$ is
a suitable coordinate system on $U_j$. As in the proof of Lemma 3.4,
the function
$$
\varphi(z)=\log\Big(\sum\theta_j^2(z)\big(
|w^{(j)}_1|^2+\ldots+|w^{(j)}_r|^2\big)\Big)
$$
is quasi-psh if the functions $\theta_j$ are cut-off functions with support 
in $U_j$ such that $\sum\theta_j^2=1$. An easy calculation also shows that
$\cI((r+1)\varphi)=\cI_X^2$ where $\cI_X$ is the reduced ideal sheaf of $X$ 
in $\bP^N$. Hence for $m\ge m_0$ large enough, we have
$$
H^q(\bP^N,\cO(m)\otimes\cI_X^2)=0
$$
for $q\ge 1$. The long-exact sequence associated with
$$
0\to \cI_X^2\to \cI_X\to\cI_X/\cI_X^2\to 0
$$
twisted by $\cO(m)$ implies the surjectivity of
$$
H^0(\bP^N,\cO(m)\otimes\cI_X)\to H^0(X,\cO(m)\otimes\cI_X/\cI_X^2).
$$
However, $\cI_X/\cI_X^2$ can be identified with the conormal bundle
$N_X^*$ (where $N_X=T_{\bP^N|X}/T_X$), and we infer from this that
$H^0(X,\cO(m)\otimes\cI_X/\cI_X^2)$ is generated by its global sections 
for $m\ge m_1$ large
enough by Theorem 4.11$\,$c). This means that we can generate any
$1$-differential of $N_X^*$ at any point $x\in X$ as the differential
of a section 
$$
P\in H^0(\bP^N,\cO(m)\otimes\cI_X)\subset H^0(\bP^N,\cO(m)),
$$
i.e.\ a homogeneous polynomial of degree $m$ vanishing on $X$,
for $m\ge\max(m_0,m_1)$.\qed
\endproof

\claim{(4.16) Corollary {\rm(Kodaira [Kod54])}} Let $X$ be a compact complex 
manifold and $\dim_\bC X=n$. The following conditions are equivalent.
\smallskip
\item{\rm a)} $X$ is projective algebraic, i.e.\ $X$ can be embedded as an
algebraic submanifold of the complex projective space $\bP^N$ for $N$ large.
\smallskip
\item{\rm b)} $X$ carries a positive line bundle $L$.
\smallskip
\item{\rm c)} $X$ carries a Hodge metric,
i.e.\ a K\"ahler metric $\omega$ with rational cohomology class
$\{\omega\}\in H^2(X,\bQ)$.\smallskip
\endclaim

\begproof{} a) $\Leftrightarrow$ b). This follows from Theorems 4.13 and 4.15
combined.
\medskip
\noindent b) $\Rightarrow$ c). Take $\omega={\ii\over 2\pi}\Theta(L)\,;$
then $\{\omega\}$ is the image of $c_1(L)\in H^2(X,\bZ)$.
\medskip
\noindent c) $\Rightarrow$ b). We can multiply $\{\omega\}$ by a common
denominator of its coefficients and suppose that $\{\omega\}$ is in the image
of $H^2(X,\bZ)$. Then a classical result due to A.~Weil shows that
there exists a hermitian line bundle $(L,h)$ such that 
${\ii\over 2\pi}\Theta_h(L)=\omega$. This bundle $L$ is then positive.
\qed
\endproof

\claim{(4.17) Exercise {\rm (solution of the Levi problem)}} \rm Show
that the following two properties are equivalent.
\smallskip
\item{\rm a)} $X$ is strongly pseudoconvex, i.e.\ $X$ admits a strongly psh
exhaustion function.
\smallskip
\item{\rm b)} $X$ is Stein, i.e.\ the global holomorphic functions
$H^0(X,\cO_X)$ separate points and yield local coordinates at any point,
and $X$ is holomorphically convex $($this means that for any discrete
sequence $z_\nu$ there is a function $f\in H^0(X,\cO_X)$ such that
$|f(z_\nu)|\to\infty)$.\qed
\vskip0pt
\endclaim

\titleb{5.}{Nef and Pseudo-Effective Cones}

We now introduce important concepts of positivity for
cohomology classes of type~$(1,1)$.

\claim{(5.1) Definition} Let $X$ be a compact K\"ahler manifold.
\smallskip
\item{\rm(i)} The K\"ahler cone is the set 
$\cK\subset H^{1,1}(X,\bR)$ of cohomology classes $\{\omega\}$
of K\"ahler forms. This is an \ul{open convex cone}.
\smallskip
\item{\rm(ii)}The pseudo-effective cone is the set
$\cE\subset H^{1,1}(X,\bR)$ of cohomology classes $\{T\}$ of closed positive 
currents of type $(1,1)$. This is a \ul{closed convex cone}.\vskip0pt
\endclaim

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

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

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

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

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\vskip8mm

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

\claim{(5.2) Theorem} Let $X$ be a projective manifold. Then
{\oldparindent=6.6mm
\smallskip
\item{\rm (i)} $\cE_{\NS}$ is the closure of the cone generated by classes
of \ul{effective} divisors, i.e.\ divisors $D=\sum c_jD_j$, $c_j\in\bR_+$.
\smallskip
\item{\rm (ii)} $\cK_{\NS}$ is the open cone generated by classes of
\ul{ample} $($or very ample$)$ divisors $A$ $($Recall that a divisor
$A$ is said to be very ample if the linear system $H^0(X,\cO(A))$
provides an embedding of $X$ in projective space$)$.
\smallskip
\item{\rm (iii)} The interior $\cE_{\NS}^\circ$ is the cone generated
by classes of \ul{big} divisors, namely divisors $D$ such that
$h^0(X,\cO(kD))\ge c\,k^{\dim X}$ for $k$ large.
\smallskip
\item{\rm (iv)} The closed cone $\ol\cK_{\NS}$ consists of the closure
of the cone generated by \ul{nef} divisors $D$ (or nef line bundles $L$), 
namely effective integral divisors $D$ such that $D\cdot C\ge 0$ for
every curve $C$.\vskip0pt
}
\endclaim

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

Similarly, if we take a
rational class $\{\alpha\}\in\cE_{\NS}^\circ$, then it is still in
$\cE$ by subtracting a small multiple $\varepsilon\omega$ of a
K\"ahler class, hence some multiple of $\{\alpha\}$ is the first Chern class
of a hermitian line bundle $(L,h)$ with curvature form
$$
T=\Theta_h(L):=-{i\over 2\pi}i\ddbar\log h \ge \varepsilon\omega.
$$
Let us apply Theorem 4.11 to a metric of the form $h^ke^{-m\psi}$ on 
$L^{\otimes k}$, where $\psi$ has logarithmic poles at given points $x_j$ 
in $X$. It is then easily shown that $L^{\otimes k}$ admits sections
which have given $m$-jets at the point $x_j$, provided that $k\ge Cm$,
$C\gg 1$, and the $x_j$ are chosen outside the Lelong sublevel sets
of $\log h$. From this we get $h^0(X,L^{\otimes k})\ge m^n/n!\ge ck^n$, 
hence the linear system $kL$ can be represented by a big divisor. 
This implies (iii) and also that $\cE_{\NS}^\circ\subset\cE_\alg$. Therefore 
$\cE_{\NS}\subset\cE_\alg$ by passing to the closure$\,$; (i) follows.
The statement (iv) about nef divisors follows
e.g.\ from [Kle66], [Har70], since every nef divisor is a limit 
of a sequence of ample rational divisors.
\qed

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

\titleb{6.}{Numerical Characterization of the K\"ahler Cone}

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

\claim{(6.1) Theorem} Let $X$ be a compact K\"ahler manifold. Let $\cP$ 
be the set of real $(1,1)$ cohomology classes $\{\alpha\}$
which are \ul{numerically} \ul{positive} on analytic cycles, i.e.\ such 
that $\int_Y\alpha^p>0$ for every irreducible analytic set $Y$ in $X$, 
$p=\dim Y$. Then the K\"ahler cone $\cK$ of $X$ is one of the 
connected components of $\cP$.
\endclaim

\claim{(6.2) Special case} If $X$ is projective algebraic, then $\cK=\cP$.
\endclaim

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

It turns out that the numerical conditions $\int_Y\alpha^p> 0$ also 
characterize arbitrary transcendental K\"ahler classes when  $X$ is 
projective$\,$: this is precisely the meaning of the special case 6.2.

\claim{(6.3) Example} {\rm The following example shows that the cone $\cP$
need not be connected (and also that the components of $\cP$ need not 
be convex, either). Let us consider for instance a complex torus 
$X=\bC^n/\Lambda$. It is well-known that a generic torus $X$ does not
possess any analytic subset except finite subsets and $X$ itself. 
In that case, the numerical positivity is expressed by the single
condition $\int_X\alpha^n>0$. However, on a torus, $(1,1)$-classes are 
in one-to-one correspondence with constant hermitian forms $\alpha$ on
$\bC^n$. Thus, for $X$ generic, $\cP$ is the set of hermitian forms on 
$\bC^n$ such that $\det(\alpha)>0$, and Theorem 6.1 just expresses the 
elementary result of linear algebra saying that the set $\cK$ of 
positive definite forms is one of the connected components of the open set
$\cP=\{\det(\alpha)>0\}$ of hermitian forms of positive determinant
(the other components, of course, are the sets of forms of signature 
$(p,q)$, $p+q=n$, $q$ even. They are not convex when $p>0$ and $q>0$).}
\endclaim

\noindent
{\it Sketch of proof of Theorems 6.1 and 6.2.} By definition a 
\ul{\it K\"ahler current} is a closed positive current $T$ of type $(1,1)$
such that $T\ge\varepsilon\omega$ for some smooth
K\"ahler metric $\omega$ and $\varepsilon>0$ small enough.
The crucial steps of the proof of Theorem~6.1 are contained in the 
following statements. 

\claim{(6.4) Proposition {\rm (Paun [Pau98a, 98b])}} Let $X$ be a compact 
complex manifold $($or more generally a compact complex space$)$. Then
\smallskip
\item{\rm (i)} The cohomology class of a closed positive 
$(1,1)$-current $\{T\}$ is nef if and only if the restriction 
$\{T\}_{|Z}$ is nef for every irreducible component $Z$ in any of
the Lelong sublevel sets~$E_c(T)$.

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

The proof of Proposition 6.4 is not extremely hard if we take for granted the
fact that K\"ahler currents can be approximated by K\"ahler currents
with logarithmic poles, a fact which was first proved in [Dem92] (see also 
section 9 below). Thus in (ii), we may assume that $T=
\alpha+\ii \ddbar\varphi$ is a current with analytic singularities,
where $\varphi$ is a quasi-psh function with logarithmic poles on some
analytic set $Z$, and $\varphi$ smooth on $X\ssm Z$. Now, we proceed by 
an induction on dimension (to do this, we have to consider analytic spaces
rather than with complex manifolds, but it turns out that this makes no
difference for the proof). Hence, by the induction hypothesis, there exists
a smooth potential $\psi$ on $Z$ such that 
$\alpha_{|Z}+\ii \ddbar\psi>0$ along $Z$. It is well known that
one can then find a potential $\wt\psi$ on $X$ such that
$\alpha+\ii \ddbar\wt\psi>0$ in a neighborhood $V$ of $Z$ (but possibly
non positive elsewhere). Essentially, it is enough to take an arbitrary
extension of $\psi$ to $X$ and to add a large multiple of the square of 
the distance to $Z$, at
least near smooth points; otherwise, we stratify $Z$ by its successive
singularity loci, and proceed again by induction on the dimension of these
loci. Finally, we use a
a standard gluing procedure~: the current $T=\alpha+
\ii \max_\varepsilon(\varphi,\wt\psi-C)$, $C\gg 1$, will be equal
to $\alpha+\ii \ddbar\varphi>0$ on $X\ssm V$, and to a smooth
K\"ahler form on $V$.\qed

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

\claim{(6.5) Theorem {\rm ([DP03])}} Let $X$ be a compact K\"ahler manifold
and let $\{\alpha\}$ be a nef class $($i.e.\ $\{\alpha\}\in\ol\cK)$.
Assume that $\int_X\alpha^n>0$. Then $\{\alpha\}$ contains a K\"ahler
current~$T$, in other words $\{\alpha\}\in\cE^\circ$.
\endclaim

\noindent
{\it Step 1}. The basic argument is to prove that for every irreducible 
analytic set $Y\subset X$ of
codimension $p$, the class $\{\alpha\}^p$ contains a closed positive
$(p,p)$-current $\Theta$ such that $\Theta\ge\delta[Y]$ for some $\delta>0$.
For this, we use in an essentail way the Calabi-Yau theorem [Yau78] on
solutions of Monge-Amp\`ere equations, which yields the following result
as a special case:

\claim{(6.6) Lemma {\rm([Yau78])}} Let $(X,\omega)$ 
be a compact K\"ahler manifold and $n=\dim X$. Then for any smooth
volume form $f>0$ such that $\int_X f = \int_X\omega^n$,
there exist a K\"ahler metric $\widetilde \omega = \omega+i\ddbar\varphi$
in the same K\"ahler class as $\omega$, such that 
$\widetilde \omega^n=f$.\qed
\endclaim

\noindent
We exploit this by observing that $\alpha+\varepsilon\omega$ is a
K\"ahler class, and by solving the Monge-Amp\`ere equation
$$
(\alpha+\varepsilon\omega+i\ddbar\varphi_\varepsilon)^n=C_\varepsilon
\omega_\varepsilon^n
\leqno(6.6\hbox{a})
$$
where $(\omega_\varepsilon)$ is the family of K\"ahler metrics on $X$
produced by Lemma 3.4~(iii), such that their volume is concentrated
in an $\varepsilon$-tubular neighborhood of $Y$.
$$
C_\varepsilon = {\int_X\alpha_\varepsilon^n\over\int_X\omega_\varepsilon^n}
={\int_X(\alpha+\varepsilon\omega)^n\over\int_X\omega^n}\ge
C_0={\int_X\alpha^n\over\int_X\omega^n}>0.
$$
Let us denote by 
$$
\lambda_1(z)\le\ldots\le\lambda_n(z)
$$
the eigenvalues of $\alpha_\varepsilon(z)$ with respect to
$\omega_\varepsilon(z)$, at every point $z\in X$ (these functions
are continuous with respect to $z$, and of course depend also 
on~$\varepsilon$). The equation (6.6a) is equivalent to the fact that
$$
\lambda_1(z)\ldots\lambda_n(z)=C_\varepsilon\leqno(6.6\hbox{b})
$$
is constant, and the most important observation for us is that the
constant $C_\varepsilon$ is bounded away from $0$, thanks to
our assumption $\int_X\alpha^n>0$.

Fix a regular point $x_0\in Y$ and a small neighborhood $U$ (meeting only
the irreducible component of $x_0$ in $Y$). By Lemma 3.4, we have a
uniform lower bound
$$
\int_{U\cap V_\varepsilon}\omega_\varepsilon^p\wedge\omega^{n-p}
\ge\delta_p(U)>0.\leqno(6.6\hbox{c})
$$
Now, by looking at the $p$ smallest (resp.\ $(n-p)$ largest) eigenvalues
$\lambda_j$ of $\alpha_\varepsilon$ with respect to $\omega_\varepsilon$, 
we find
$$
\leqalignno{
\alpha_\varepsilon^p&\ge \lambda_1\ldots\lambda_p\,\omega_\varepsilon^p,
&(6.6\hbox{d})\cr
\alpha_\varepsilon^{n-p}\wedge\omega_\varepsilon^p
&\ge {1\over n!}\lambda_{p+1}\ldots\lambda_n\,
\omega_\varepsilon^n,&(6.6\hbox{e})\cr
}
$$
The last inequality $(6.6\hbox{e})$ implies
$$
\int_X\lambda_{p+1}\ldots\lambda_n\,\omega_\varepsilon^n\le
n!\int_X\alpha_\varepsilon^{n-p}\wedge\omega_\varepsilon^p=
n!\int_X(\alpha+\varepsilon\omega)^{n-p}\wedge\omega^p\le M
$$
for some constant $M>0$ (we assume $\varepsilon\le 1$, say). 
In particular, for every $\delta>0$, the subset 
$E_\delta\subset X$ of points $z$ such that 
$\lambda_{p+1}(z)\ldots\lambda_n(z)>M/\delta$ satisfies
$\int_{E_\delta}\omega_\varepsilon^n\le\delta$, hence
$$
\int_{E_\delta}\omega_\varepsilon^p\wedge\omega^{n-p}\le 2^{n-p}
\int_{E_\delta}\omega_\varepsilon^n\le 2^{n-p}\delta.\leqno(6.6\hbox{f})
$$
The combination of (6.6c) and (6.6f) yields
$$
\int_{(U\cap V_\varepsilon)\ssm E_\delta}\omega_\varepsilon^p\wedge\omega^{n-p}
\ge \delta_p(U)-2^{n-p}\delta.
$$
On the other hand (6.6b) and (6.6d) imply
$$
\alpha_\varepsilon^p\ge{C_\varepsilon\over\lambda_{p+1}\ldots\lambda_n}
\omega_\varepsilon^p
\ge{C_\varepsilon\over M/\delta}\omega_\varepsilon^p\qquad
\hbox{on $(U\cap V_\varepsilon)\ssm E_\delta$}.
$$
From this we infer
$$
\int_{U\cap V_\varepsilon}\alpha_\varepsilon^p\wedge\omega^{n-p}\ge
{C_\varepsilon\over M/\delta}
\int_{(U\cap V_\varepsilon)\ssm E_\delta}
\omega_\varepsilon^p\wedge\omega^{n-p}\ge
{C_\varepsilon\over M/\delta}(\delta_p(U)-2^{n-p}\delta)>0
\leqno(6.6\hbox{g})
$$
provided that $\delta$ is taken small enough, e.g.\ 
$\delta=2^{-(n-p+1)}\delta_p(U)$. The family of $(p,p)$-forms
$\alpha_\varepsilon^p$ is uniformly bounded in mass since
$$
\int_X\alpha_\varepsilon^p\wedge\omega^{n-p}
=\int_X(\alpha+\varepsilon\omega)^p\wedge\omega^{n-p}\le\hbox{Const}.
$$
Inequality (6.6g) implies that any weak limit $\Theta$ of 
$(\alpha_\varepsilon^p)$
carries a positive mass on~$U\cap Y$. By Skoda's extension theorem [Sko82], 
${\bf 1}_Y\Theta$ is a closed positive current with support in $Y$, 
hence ${\bf 1}_Y\Theta=\sum c_j[Y_j]$ is a combination of the 
various components $Y_j$ of $Y$ with coefficients $c_j>0$. 
Our construction shows that $\Theta$ belongs to the cohomology class 
$\{\alpha\}^p$. Step 1 of Theorem 6.5 is proved.
\medskip

\noindent
{\it Step 2.} The second and final step consists in using
a ``diagonal trick'': for this, we apply Step 1 to 
$$
\wt X=X\times X,\qquad \wt Y=\hbox{diagonal}\,\Delta\subset \wt X,\qquad
\wt\alpha=\pr_1^*\alpha+\pr_2^*\alpha.
$$
It is then clear that $\wt\alpha$ is nef on $\wt X$ and that
$$
\int_{\wt X}(\wt\alpha)^{2n}=
{2n\choose n}\Big(\int_X\alpha^n\Big)^2>0.
$$
It follows by Step 1 that the
class $\smash{\{\wt\alpha\}}^n$ contains a K\"ahler current $\Theta$ of 
bidegree $(n,n)$ such that
$\Theta\ge\delta[\Delta]$ for some $\delta>0$. Therefore 
the push-forward
$$
T:=(\pr_1)_*(\Theta\wedge\pr_2^*\omega)
$$
is a positive $(1,1)$-current such that
$$
T\ge\delta(\pr_1)_*([\Delta]\wedge\pr_2^*\omega)=\delta\omega.
$$
It follows that $T$ is a K\"ahler current. On the other hand, $T$
is numerically equivalent to $(\pr_1)_*(\wt\alpha^n\wedge\pr_2^*\omega)$, 
which is the form given in coordinates by
$$
x\mapsto\int_{y\in X}\big(\alpha(x)+\alpha(y)\big)^n\wedge \omega(y)=
C\alpha(x)
$$
where $C=n\int_X\alpha(y)^{n-1}\wedge\omega(y)$. Hence $T\equiv C\alpha$,
which implies that $\{\alpha\}$ contains a K\"ahler current. 
Theorem 6.5 is proved.\qed
\medskip

\noindent
{\it End of Proof of Theorems 6.1 and 6.2}. Clearly the open cone $\cK$ 
is contained in $\cP$, hence in order to show that $\cK$ is one of the
connected components of $\cP$, we need only show that $\cK$ is \ul{closed}
in $\cP$, i.e.\ that $\ol\cK\cap\cP\subset\cK$. Pick a class
$\{\alpha\}\in\ol\cK\cap\cP$. In particular $\{\alpha\}$ is
nef and satisfies $\int_X\alpha^n>0$. By Theorem~6.5 we conclude that
$\{\alpha\}$ contains a K\"ahler current $T$. However, an induction on
dimension using the assumption $\int_Y\alpha^p$ for all analytic
subsets $Y$ (we also use resolution of singularities for $Y$ at this step)
shows that the restriction $\{\alpha\}_{|Y}$ is the class of a K\"ahler
current on~$Y$. We conclude that $\{\alpha\}$ is a K\"ahler class by
6.4~(ii), therefore $\{\alpha\}\in\cK$, as desired.\qed

The projective case 6.2 is a consequence of the following variant of
Theorem 6.1.

\claim{(6.7) Corollary} Let $X$ be a compact
K\"ahler manifold. A $(1,1)$ cohomology class $\{\alpha\}$ on $X$
is K\"ahler if and only if there exists a 
K\"ahler metric $\omega$ on $X$ such that $\int_Y\alpha^k\wedge\omega^{p-k}>0$
for all irreducible analytic sets $Y$ and all $k=1,2,\ldots,p=\dim Y$.
\endclaim

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

In the projective case, we can take $\omega=c_1(H)$ for a given very ample
divisor $H$, and the condition
 $\int_Y\alpha^k\wedge\omega^{p-k}>0$ is equivalent to
$$
\int_{Y\cap H_1\cap\ldots\cap H_{p-k}}\alpha^k>0
$$
for a suitable complete intersection $Y\cap H_1\cap\ldots\cap H_{p-k}$,
$H_j\in|H|$. This shows that algebraic cycles are sufficient to test
the K\"ahler property, and the special case 6.2 follows. On the other 
hand, we can pass to the limit in 6.7 by replacing $\alpha$ 
by $\alpha+\varepsilon\omega$,
and in this way we get also a characterization of nef classes.

\claim{(6.8) Corollary} Let $X$ be a compact
K\"ahler manifold. A $(1,1)$ cohomology class $\{\alpha\}$ on $X$
is nef if and only if there exists a 
K\"ahler metric $\omega$ on $X$ such that $\int_Y\alpha^k\wedge\omega^{p-k}
\ge 0$ for all irreducible analytic sets $Y$ and all $k=1,2,\ldots,p=\dim Y$.
\endclaim

By a formal convexity argument, one can derive from 6.7 or 6.8 the following
interesting consequence about the dual of the cone $\cK$. We will not 
give the proof here, because it is just a simple tricky argument which 
does not require any new analysis.

\claim{(6.9) Theorem} Let $X$ be a compact K\"ahler manifold. 
A $(1,1)$ cohomology class $\{\alpha\}$ on $X$ is nef if and only 
for every irreducible analytic set $Y$ in $X$, $p=\dim X$
and every K\"ahler metric $\omega$ on $X$ we have
$\int_Y\alpha\wedge\omega^{p-1}\ge 0$. In other words, the dual of the
nef cone $\smash{\ol\cK}$ is the closed convex cone 
in $H^{n-1,n-1}_\bR(X)$ generated by cohomology 
classes of currents of the form $[Y]\wedge\omega^{p-1}$ in 
$H^{n-1,n-1}(X,\bR)$, where $Y$ runs over the 
collection of irreducible analytic subsets of $X$ and $\{\omega\}$ over
the set of K\"ahler classes of~$X$.
\endclaim

Our main Theorem 6.1 has an important application to the deformation theory of
compact K\"ahler manifolds.

\claim{(6.10) Theorem} Let $\pi:\cX\to S$ be a deformation of compact K\"ahler
manifolds over an irreducible base~$S$. Then there exists a countable union 
$S'=\bigcup S_\nu$ of analytic subsets $S_\nu\subsetneq S$, such that 
the K\"ahler cones $\cK_t\subset H^{1,1}(X_t,\bC)$ of the fibers
$X_t=\pi^{-1}(t)$ are invariant over $S\ssm S'$ under 
parallel transport with respect to the $(1,1)$-projection $\nabla^{1,1}$ of
the Gauss-Manin connection $\nabla$ in the decomposition of
$$
\nabla=\pmatrix{
\nabla^{2,0} & * & 0\cr 
* & \nabla^{1,1} & * \cr
0 & * & \nabla^{0,2}\cr}
$$
on the Hodge bundle $H^2=H^{2,0}\oplus H^{1,1}\oplus H^{0,2}$.
\endclaim

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

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

\noindent
{\it Sketch of Proof of Theorem 6.10.} The result is local on the base,
hence we may assume that $S$ is contractible. Then the family is
differentiably trivial, the Hodge bundle $t\mapsto H^2(X_t,\bC)$ is
the trivial bundle and $t\mapsto H^2(X_t,\bZ)$ is a trivial lattice.
We use the existence of a relative cycle space $C^p(\cX/S)\subset C^p(\cX)$
which consists of all cycles contained in the fibres of $\pi:X\to S$.
It is equipped with a canonical holomorphic projection 
$$
\pi_p:C^p(\cX/S)\to S.
$$
We then define 
the $S_\nu$'s to be the images in $S$ of those connected components of 
$C^p(\cX/S)$ which do not project onto $S$. By the fact that the projection 
is proper on each component, we infer that $S_\nu$ is an analytic subset 
of $S$. The definition of the $S_\nu$'s imply that the cohomology classes
induced by the analytic cycles $\{[Z]\}$, $Z\subset X_t$, remain exactly 
the same for all $t\in S\ssm S'$. This result implies in its turn 
that the conditions defining the numerically positive cones $\cP_t$
remain the same, except for the fact that the spaces $H^{1,1}_\bR(X_t)\subset
H^2(X_t,\bR)$ vary along with the Hodge decomposition. At this point,
a standard calculation implies that the $\cP_t$ are invariant by parallel
transport under $\nabla^{1,1}$. This is done as follows.

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

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

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

\claim{(6.11) Theorem} A compact complex manifold carries a K\"ahler
current if and only if it is bimeromorphic to a K\"ahler manifold
$($or equivalently, dominated by a K\"ahler manifold$)$.
\endclaim

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

\claim{(6.12) Conjecture} Let $X$ be a compact complex manifold of
dimension~$n$. Assume that $X$ possesses a nef cohomology class $\{\alpha\}$
of type $(1,1)$ such that $\int_X\alpha^n>0$. Then $X$ is in the
Fujiki class~$\cC$. $[$Also, $\{\alpha\}$ would contain a K\"ahler 
current, as it follows from Theorem 6.5 if Conjecture 6.12 is proved$\,]$.
\endclaim

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

Shortly after the original [DP03] manuscript appeared 
in April 2001, Daniel Huybrechts [Huy01] informed us Theorem 6.1 can be 
used to calculate the K\"ahler cone of
a very general hyperk\"ahler manifold: the K\"ahler cone is then equal
to a suitable connected component of the positive cone defined by the
Beauville-Bogomolov quadratic form. In the case of an  arbitrary
hyperk\"ahler manifold, S.Boucksom [Bou02] later showed that a $(1,1)$ class
$\{\alpha\}$ is K\"ahler if and only if it lies in the positive part
of the Beauville-Bogomolov quadratic cone and moreover $\int_C\alpha>0$
for all \ul{\it rational curves} $C\subset X$ (see also [Huy99]).

\titleb{7.}{Cones of Curves}

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

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

\claim{(7.1) Definition} Let $X$ be a compact K\"ahler manifold.
{\oldparindent=6.6mm
\smallskip
\item {\rm (i)} We define $\cN$ to be the $($closed$)$ convex cone in 
$H^{n-1,n-1}_\bR(X)$ gene\-rated by classes of positive currents $T$ of type
\hbox{$(n-1,n-1)$} $($i.e., of bidimension $(1,1))$.

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

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

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

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

\claim{(7.2) Proposition} Let $X$ be a compact K\"ahler manifold. 
Consider the Poincar\'e duality pairing
$$
H^{1,1}(X,\bR)\times H^{n-1,n-1}_\bR(X)\lra \bR,\qquad
(\alpha,\beta)\longmapsto\int_X \alpha\wedge\beta.
$$
Then the duality pairing takes nonnegative values
\smallskip
\item{\rm (i)} for all pairs $(\alpha,\beta)\in\ol\cK\times\cN$;
\item{\rm (ii)} for all pairs $(\alpha,\beta)\in\cE\times\cM$.
\item{\rm (iii)} for all pairs $(\alpha,\beta)$ where $\alpha\in\cE$ and 
$\beta=[C_t]\in{\rm ME}(X)$ is the class of a movable curve.
\vskip0pt
\endclaim

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

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

Proposition 7.2 leads to the natural question whether the 
cones $(\cK,\cN)$ and $(\cE,\cM)$ are dual under Poincar\'e duality. 
This question is addressed in the next section. Before doing so,
we observe that the algebraic and transcendental cones
of $(n-1,n-1)$ cohomology classes are related by the following 
equalities.

\claim{(7.3) Theorem} Let $X$ be a projective manifold. Then
\smallskip
\item{\rm(i)} $\overline{\NE(X)} =\cN_{\NS}$.
\smallskip
\item{\rm(ii)} $\overline{{\rm SME}(X)}=\overline{{\rm ME}(X)} =\cM_{\NS}$.
\endclaim

\begproof{} (i) It is a standard result of algebraic geometry (see e.g.\
[Har70]), that the cone of effective cone $\NE(X)$ is dual to the cone
$\overline{\cK_{\NS}}$ of nef divisors, hence 
$$\cN_{\NS}\supset\overline{\NE(X)}=\cK^\vee.$$ 
On the other hand, (7.3)~(i) implies that $\cN_{\NS}\subset\cK^\vee$, so 
we must have equality and (i) follows.

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

\titleb{8.}{Main Duality Results}

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

\claim{(8.1) Theorem {\rm(Demailly-Paun, 2001)}} If $X$ is K\"ahler,
the cones $\overline{\cK}\subset H^{1,1}(X,\bR)$ and 
$\cN\subset H^{n-1,n-1}_\bR(X)$ are dual by Poincar\'e duality, and
$\cN$ is the closed convex cone generated by classes $[Y]\wedge\omega^{p-1}$
where $Y\subset X$ ranges over $p$-dimensional analytic subsets,
$p=1,2,\ldots, n$, and $\omega$ ranges over K\"ahler forms.
\endclaim

\begproof{} Indeed, Prop.~7.4 shows that the dual cone $\cK^\vee$ contains
$\cN$ which itself contains the cone $\cN'$ of all classes of the form
$\{[Y]\wedge\omega^{p-1}\}$. The main result of [DP03] conversely shows
that the dual of $(\cN')^\vee$ is equal to $\overline\cK$, so we must
have
$$\cK^\vee = \overline{\cN'} = \cN.\eqno\square$$
\endproof

The other important duality result is the following characterization of
pseudo-effective classes, proved in [BDPP03] (the ``only if'' part 
already follows from 7.4~(iii)).

\claim{(8.2) Theorem} If $X$ is projective, then a class $\alpha\in\NS_\bR(X)$
is pseudo-effective if $($and only if$\,)$ it is in the dual cone of the cone
$\SME(X)$ of strongly movable curves.
\endclaim

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

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

The K\"ahler analogue should be~:

\claim{(8.3) Conjecture} For an arbitrary compact 
K\"ahler manifold $X$, the cones $\cE$ and $\cM$ are dual.
\endclaim


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\vskip8mm

The relation between the various cones of movable curves and currents in 
(7.5) is now a rather direct consequence of Theorem 8.2. In fact, using 
ideas hinted in [DPS96], we can say a little bit more. Given 
an irreducible curve $C \subset X$, we consider its normal ``bundle''  
$N_C = \Hom(\cI/\cI^2,\cO_C)$, where $\cI$ is the ideal sheaf of $C$. 
If $C$ is a general member of a covering family $(C_t)$, then $N_C$ 
is nef. Now [DPS96] says that the dual cone of the pseudo-effective cone
of $X$ contains the closed cone spanned by curves with nef normal 
bundle, which in turn contains the cone of movable curves. In this
way we get~:

\claim{(8.4) Theorem} Let $X$ be a projective manifold. Then the following 
cones coincide.
{\oldparindent=6.6mm
\smallskip
\item {\rm (i)} the cone $\cM_{\NS}=\cM\cap N_1(X)\,;$
\smallskip
\item {\rm (ii)} the closed cone $\overline{\SME(X)}$ of strongly movable 
curves$\,;$
\smallskip
\item {\rm (iii)} the closed cone $\overline{\ME(X)}$ of movable curves$\,;$
\smallskip
\item {\rm (iv)} the closed cone $\overline{\ME_{\nef}(X)}$ of curves with nef 
normal bundle.\vskip0pt
}
\endclaim

\begproof{} We have already seen that 
$$\SME(X)\subset\ME(X)\subset\ME_{\nef}(X)\subset(\cE_{\NS})^\smallvee$$
and 
$$
\SME(X)\subset\ME(X)\subset\cM_{\NS}\subset(\cE_{\NS})^\smallvee
$$ 
by 7.4~(iii). Now Theorem 8.2 implies $(\cM_{\NS})^\smallvee =
\overline{\SME(X)}$, and 8.4 follows.\qed
\endproof

\claim{(8.5) Corollary} Let $X$ be a projective manifold and $L$ a line bundle 
on $X$.
\smallskip
\item {\rm (i)} $L$ is pseudo-effective if and only if $L \cdot C \geq 0$ 
for all curves $C$ with nef
normal sheaf $N_C$.
\smallskip
\item {\rm (ii)} If $L$ is big, then $L \cdot C > 0$ for all curves $C$ 
with nef normal sheaf $N_C$.\vskip0pt
\endclaim

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

The most important special case of Theorem 8.2 is

\claim{(8.6) Theorem} If $X$ is a projective manifold and is not uniruled,
then $K_X$ is pseudo-effective, i.e.\ $K_X\in\cE_{\NS}$.
\endclaim

\begproof{} If $K_X\notin\cE_{\NS}$, Theorem 7.2 shows that
there is a moving curve $C_t$ such that $K_X\cdot C_t<0$. The
``bend-and-break'' lemma then implies that there is family $\Gamma_t$ of 
\ul{rational curves} with $K_X\cdot\Gamma_t<0$, so $X$ is uniruled.\qed
\endproof

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

\claim{(8.7) Conjecture {\rm (special case of the ``abundance conjecture'')}}
If $K_X$ is pseudo-effective, then $\kappa(X)\ge 0$.
\endclaim

\titleb{9.}{Approximation of psh functions by logarithms of holomor\-phic
functions}
\counttitleb=9

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

\claim{(9.1) Theorem} Let $\Omega\subset\bC^n$ be a bounded
pseudoconvex domain, and let $Y\subset X$ be a nonsingular complex
submanifold defined by a section $s$ of some hermitian vector
bundle $E$ with {\rm bounded} curvature tensor on~$\Omega$. Assume
that $s$ is everywhere transverse to the zero section and that
$|s|\le e^{-1}$ on~$\Omega$. Then there is a constant $C>0$ $($depending 
only on $E)$, with the following property:
for every psh function $\varphi$ on $\Omega$, every holomorphic 
function $f$ on $Y$ with $\int_Y|f|^2|\Lambda^r(ds)|^{-2}e^{-\varphi}dV_Y
<+\infty$, there exists an extension $F$ of $f$ to $\Omega$ such that
$$
\int_\Omega{|F|^2\over|s|^{2r}(-\log|s|)^2}e^{-\varphi}dV_\Omega\le
C\int_Y{|f|^2\over|\Lambda^r(ds)|^2}e^{-\varphi}dV_Y.
$$
\endclaim

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

\claim{(9.2) Theorem} Let $\varphi$ be a plurisubharmonic function on a
bounded pseudoconvex open set~$\Omega\subset\bC^n$. For every $m>0$, let
$\cH_\Omega(m\varphi)$ be the Hilbert space of holomorphic functions $f$
on $\Omega$ such that $\int_\Omega|f|^2e^{-2m\varphi}d\lambda<+\infty$ and
let $\varphi_m={1\over 2m}\log\sum|\sigma_\ell|^2$ where $(\sigma_\ell)$
is an orthonormal basis of~$\cH_\Omega(m\varphi)$. Then there are
constants $C_1,C_2>0$ independent of $m$ such that
\item{\rm a)}$\displaystyle\varphi(z)-{C_1\over m}\le
\varphi_m(z)\le\sup_{|\zeta-z|<r}\varphi(\zeta)+{1\over m}
\log{C_2\over r^n}$\hfill\break
for every $z\in\Omega$ and $r<d(z,\partial\Omega)$. In particular,
$\varphi_m$ converges to $\varphi$ pointwise and in $L^1_{\rm loc}$ topology
on~$\Omega$ when $m\to+\infty$ and
\item{\rm b)}$\displaystyle\nu(\varphi,z)-{n\over m}\le\nu(\varphi_m,z)\le
\nu(\varphi,z)$~ for every $z\in\Omega$.
\endclaim

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

Theorem 9.2 implies in a straighforward manner a deep result of
[Siu74] on the analyticity of the Lelong number sublevel sets.

\claim{(9.3) Corollary} Let $\varphi$ be a plurisubharmonic function
on a complex manifold~$X$. Then, for every $c>0$, the Lelong number
sublevel set
$$
E_c(\varphi)=\big\{z\in X\,;\;\nu(\varphi,z)\ge c\big\}
$$
is an analytic subset of~$X$.
\endclaim

\begproof{} Since analyticity is a local property, it
is enough to consider the case of a psh function $\varphi$ on a 
pseudoconvex open set $\Omega\subset\bC^n$. The
inequalities obtained in 9.2~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{(9.4) Regularization theorem for currents} 
Let $X$ be a compact complex manifold equipped with a hermitian metric
$\omega$. Let $T=\alpha+i\ddbar\psi$ be a closed $(1,1)$-current on~$X$,
where $\alpha$ is smooth and $\psi$ is a quasi-plurisubharmonic function. 
Assume that $T\ge\gamma$ for some real $(1,1)$-form $\gamma$ on $X$ with 
real coefficients. Then there exists 
a sequence $T_k=\alpha+i\ddbar\psi_k$ of closed $(1,1)$-currents such that
{\oldparindent=6.6mm
\smallskip
\item{\rm (i)} $\psi_k$ $($and thus $T_k)$ is smooth on the complement
$X\ssm Z_k$ of an analytic set $Z_k$, and the $Z_k$'s form an increasing
sequence
$$
Z_0\subset Z_1\subset\ldots\subset Z_k\subset\ldots \subset X.
$$
\smallskip
\item{\rm (ii)} There is a uniform estimate $T_k\ge \gamma-\delta_k\omega$
with $\lim\downarrow\delta_k=0$ as $k$ tends to $+\infty$.
\smallskip
\item{\rm (iii)} The sequence $(\psi_k)$ is non increasing, and we 
have $\lim\downarrow\psi_k=\psi$. As a consequence, $T_k$ converges
weakly to $T$ as  $k$ tends to $+\infty$.
\smallskip
\item{\rm (iv)} Near $Z_k$, the potential $\psi_k$ has logarithmic poles,
namely, for every $x_0\in Z_k$, there is a neighborhood $U$ of $x_0$
such that $\psi_k(z)=\lambda_k\log\sum_\ell|g_{k,\ell}|^2+O(1)$ for 
suitable holomorphic functions $(g_{k,\ell})$ on $U$ and $\lambda_k>0$.
Moreover, there is a $($global$)$ proper modification $\mu_k:\wt X_k\to X$ 
of $X$, obtained as a sequence of blow-ups with smooth centers, such that 
$\psi_k\circ\mu_k$ can be written locally on $\wt X_k$ as
$$
\psi_k\circ\mu_k(w)=\lambda_k\big(\sum n_\ell\log|\wt g_\ell|^2+f(w)\big)
$$
where $(\wt g_\ell=0)$ are local generators of suitable $($global$)$
divisors $D_\ell$ on $\wt X_k$ such that $\sum D_\ell$ has normal crossings,
$n_\ell$ are positive integers, and the $f$'s are smooth functions 
on~$\wt X_k$.
\vskip0pt
}
\endclaim

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

\counttitleb=9
\titleb{10.}{Zariski Decomposition and Movable Intersections}
\counttitleb=10

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

\claim{(10.1) Theorem {\rm (Demailly [Dem92], [Bou02, 3.1.24]}} If $T$
is a K\"ahler current, then one can write $T=\lim T_m$ for a sequence of 
K\"ahler currents $T_m$ which have
logarithmic poles with coefficients in ${1\over m}\bZ$, i.e.\ there
are modifications $\mu_m:X_m\to X$ such that
$$\mu_m^\star T_m=[E_m]+\beta_m$$
where $E_m$ is an effective $\bQ$-divisor on $X_m$ with coefficients
in ${1\over m}\bZ$ $($the ``fixed part''$)$ and $\beta_m$ is a closed 
semi-positive form $($the ``movable part''$)$.
\endclaim

\begproof{} This is a direct consequence of the results of section~9.
Locally we can write $T=i\ddbar\varphi$ for some strictly 
plurisubharmonic potential $\varphi$. By the Bergman kernel trick and the
Ohsawa-Takegoshi $L^2$ extension theorem, we get local approximations
$$
\varphi=\lim\varphi_m,\qquad \varphi_m(z)={1\over 2m}\log\sum_\ell
|g_{\ell,m}(z)|^2
$$
where $(g_{\ell,m})$ is a Hilbert basis of the set of holomorphic
functions which are $L^2$ with respect to the weight $e^{-2m\varphi}$.
This Hilbert basis is also a family of local generators of the globally defined
multiplier ideal sheaf $\cI(mT)=\cI(m\varphi)$. Then $\mu_m:X_m\to X$
is obtained by blowing-up this ideal sheaf, so that 
$$\mu_m^\star\cI(mT)=\cO(-mE_m).$$
We should notice that by approximating $T-{1\over m}\omega$ instead of $T$,
we can replace $\beta_m$ by $\beta_m+{1\over m}\mu^\star\omega$ which is
a big class on $X_m\,$; by playing with the multiplicities of the components
of the exceptional divisor, we could even achieve that $\beta_m$ is a 
K\"ahler class on $X_m$, but this will not be needed here.\qed
\endproof

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

\claim{(10.2) Definition} We define the \ul{volume}, or \ul{movable
self-intersection} of a big class $\alpha\in\cE^\circ$ to be
$$
\Vol(\alpha)=\sup_{T\in \alpha}\int_{\wt X}\beta^n>0
$$
where the supremum is taken over all K\"ahler currents $T\in \alpha$
with logarithmic poles, and $\mu^\star T=[E]+\beta$ with
respect to some modification $\mu:\wt X\to X$.
\endclaim

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

\claim{(10.3) Proposition} Let $L$ be a big line bundle on the
projective manifold $X$.  Let $\epsilon > 0$. Then there exists a
modification $\mu: X_{\epsilon} \to X$ and a decomposition $\mu^*(L) =
E + \beta $ with $E$ an effective $\bQ$-divisor and $\beta$ a big and
nef $\bQ$-divisor such that
$$\Vol(L) -\varepsilon\le \Vol(\beta) \le \Vol(L).$$
\endclaim

It is very useful to observe that the supremum in Definition 10.2 is actually
achieved by a collection of currents whose singularities satisfy a filtering 
property.
Namely, if $T_1=\alpha+i\ddbar\varphi_1$ and $T_2=\alpha+i\ddbar\varphi_2$ 
are two K\"ahler currents with logarithmic poles in the class of $\alpha$, 
then 
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi=\max(\varphi_1,\varphi_2)\leqno(10.4)$$ 
is again a K\"ahler current with weaker 
singularities than $T_1$ and $T_2$. One could define as well 
$$T=\alpha+i\ddbar\varphi,\qquad
\varphi={1\over 2m}\log(e^{2m\varphi_1}+e^{2m\varphi_2}),\leqno(10.4')$$
where $m={\rm lcm}(m_1,m_2)$ is the lowest common multiple of the denominators
occuring in $T_1$, $T_2$. Now, take a simultaneous log-resolution
$\mu_m:X_m\to X$ for which the singularities of $T_1$ and $T_2$ are resolved 
as $\bQ$-divisors $E_1$
and $E_2$. Then clearly the associated divisor in the decomposition 
$\mu_m^\star T=[E]+\beta$ is given by $E=\min(E_1,E_2)$. By doing so, 
the volume $\int_{X_m}\beta^n$ gets increased, as we shall see in the
proof of Theorem 10.5 below.

\claim{(10.5) Theorem {\rm (Boucksom [Bou02])}} Let $X$ be a compact K\"ahler
manifold. We denote here by $H^{k,k}_{\ge 0}(X)$ the cone of cohomology classes
of type $(k,k)$ which have non-negative intersection with all closed
semi-positive smooth forms of bidegree $(n-k,n-k)$.
\smallskip
\item{\rm(i)} For each integer $k=1,2,\ldots,n$, there exists a canonical
``movable intersection product''
$$
\cE\times\cdots\times\cE\to H^{k,k}_{\ge 0}(X), \quad
(\alpha_1,\ldots,\alpha_k)\mapsto \langle\alpha_1\cdot\alpha_2\cdots
\alpha_{k-1}\cdot \alpha_k\rangle
$$
such that $\Vol(\alpha)=\langle\alpha^n\rangle$ whenever $\alpha$ is
a big class.
\smallskip
\item{\rm(ii)} The product is increasing, homogeneous of 
degree $1$ and superadditive in each argument, i.e.\
$$
\langle\alpha_1\cdots(\alpha'_j+\alpha''_j)\cdots \alpha_k\rangle\ge
\langle\alpha_1\cdots\alpha'_j\cdots \alpha_k\rangle+
\langle\alpha_1\cdots\alpha''_j\cdots \alpha_k\rangle.
$$
It coincides with the ordinary intersection
product when the $\alpha_j\in\overline{\cK}$ are nef classes.
\smallskip
\item{\rm(iii)} 
The movable intersection product satisfies the Teissier-Hovanskii inequalities
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_n\rangle\ge
(\langle\alpha_1^n\rangle)^{1/n}\ldots(\langle\alpha_n^n\rangle)^{1/n}\qquad
\hbox{$($with $\langle\alpha_j^n\rangle=\Vol(\alpha_j)\,)$}.
$$
\smallskip
\item{\rm(iv)} For $k=1$, the above ``product'' reduces to a
$($non linear$)$ projection operator 
$$
\cE\to\cE_1,\qquad \alpha\to\langle\alpha\rangle
$$
onto a certain convex subcone $\cE_1$ of $\cE$ such that
$\overline{\cK}\subset\cE_1\subset\cE$. Moreover, there is
a ``divisorial Zariski decomposition''
$$
\alpha=\{N(\alpha)\}+\langle\alpha\rangle
$$
where $N(\alpha)$ is a uniquely defined effective divisor which is
called the ``negative divisorial part'' of $\alpha$. The map
$\alpha\mapsto N(\alpha)$ is homogeneous and subadditive, and 
$N(\alpha)=0$ if and only if $\alpha\in\cE_1$.
\smallskip
\item{\rm(v)} The components of $N(\alpha)$ always consist of
divisors whose cohomology classes are linearly independent,
especially $N(\alpha)$ has at most $\rho=\rank_\bZ\NS(X)$ components.
\vskip0pt
\endclaim

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

\noindent
(i) First assume that all classes $\alpha_j$ are big, i.e.\ 
$\alpha_j\in\cE^\circ$. Fix a smooth closed $(n-k,n-k)$ {\it semi-positive}
form $u$ on $X$. We select K\"ahler currents $T_j\in\alpha_j$ with
logarithmic poles, and a simultaneous log-resolution
$\mu:\wt X\to X$ such that 
$$
\mu^\star T_j=[E_j]+\beta_j.
$$
We consider the direct image current
$\mu_\star(\beta_1\wedge\ldots\wedge\beta_k)$ (which is a closed
positive current of bidegree $(k,k)$ on $X$) and the corresponding
integrals
$$
\int_{\wt X}\beta_1\wedge\ldots\wedge\beta_k\wedge\mu^\star u\ge 0.
$$
If we change the representative $T_j$ with another current $T_j'$,
we may always take a simultaneous log-resolution such that
$\mu^\star T_j'=[E_j']+\beta_j'$, and by using $(10.4')$ we can always
assume that $E'_j\le E_j$. Then $D_j=E_j-E'_j$ is an effective
divisor and we find $[E_j]+\beta_j\equiv [E'_j]+\beta'_j$, hence
$\beta'_j\equiv\beta_j+[D_j]$. A substitution in the integral implies
$$
\eqalign{
\int_{\wt X}\beta_1'\wedge\beta_2
&{}\wedge\ldots\wedge\beta_k\wedge\mu^\star u\cr
&=\int_{\wt X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u+
\int_{\wt X}[D_1]\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u\cr
&\ge \int_{\wt X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge
\mu^\star u.\cr}
$$
Similarly, we can replace successively all forms $\beta_j$ by the $\beta'_j$,
and by doing so, we find
$$
\int_{\wt X}\beta'_1\wedge\beta'_2\wedge\ldots\wedge\beta'_k\wedge\mu^\star u
\ge\int_{\wt X}\beta_1\wedge\beta_2\wedge\ldots\wedge\beta_k\wedge\mu^\star u.
$$
We claim that the closed positive currents $\mu_\star(\beta_1\wedge\ldots
\wedge\beta_k)$ are uniformly bounded in mass. In fact, if $\omega$ is a 
K\"ahler metric in $X$, there exists a constant $C_j\ge 0$
such that $C_j\{\omega\}-\alpha_j$ is a K\"ahler class. Hence
$C_j\omega-T_j\equiv\gamma_j$ for some K\"ahler form $\gamma_j$ on $X$.
By pulling back with $\mu$, we find
$C_j\mu^\star \omega-([E_j]+\beta_j)\equiv\mu^\star \gamma_j$, hence 
$$\beta_j\equiv C_j\mu^\star \omega-([E_j]+\mu^\star \gamma_j).$$
By performing again a substitution in the integrals, we find
$$
\int_{\wt X}\beta_1\wedge\ldots\wedge\beta_k\wedge\mu^\star u
\le C_1\ldots C_k\int_{\wt X}\mu^\star\omega^k\wedge\mu^\star u
= C_1\ldots C_k\int_X\omega^k\wedge u
$$
and this is true especially for $u=\omega^{n-k}$. We can now arrange
that for each of the integrals associated with a countable dense family 
of forms $u$,  the supremum is achieved by a sequence of currents 
$(\mu_m)_\star(\beta_{1,m}\wedge\ldots\wedge\beta_{k,m})$ obtained
as direct images by a suitable sequence of modifications $\mu_m:
\smash{\wt X_m}\to X$.
By extracting a subsequence, we can achieve that this sequence is weakly 
convergent and we set
$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle =
\mathop{\lim\uparrow}\limits_{m\to+\infty}
\{(\mu_m)_\star(\beta_{1,m}\wedge\beta_{2,m}\wedge\ldots\wedge\beta_{k,m})\}
$$
(the monotonicity is not in terms of the currents themselves, but in terms
of the integrals obtained when we evaluate against a smooth closed
semi-positive form~$u$). By evaluating against a basis of positive classes
$\{u\}\in H^{n-k,n-k}(X)$, we infer by Poincar\'e duality that the class of
$\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle$ is uniquely defined
(although, in general, the representing current is not unique). 
\smallskip

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

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

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

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

\noindent
(v) Let $\alpha\in\cE^\circ$, and assume that $N(\alpha)$ contains
linearly dependent components $F_j$. Then already all currents
$T\in \alpha$ should be such that $\mu^\star T=[E]+\beta$ where 
$F=\mu_\star E$ contains those linearly dependent components. 
Write $F=\sum\lambda_jF_j$, $\lambda_j>0$ and assume that 
$$\sum_{j\in J}c_jF_j\equiv 0$$
for a certain non trivial linear combination. Then some of the
coefficients $c_j$ must be negative (and some other positive).
Then $E$ is numerically equivalent to
$$
E'\equiv E+t\mu^\star\Big(\sum\lambda_jF_j\Big),
$$
and by choosing $t>0$ appropriate, we obtain an effective divisor $E'$ 
which has a zero coefficient on one of the components $\mu^\star F_{j_0}$.
By replacing $E$ with $\min(E,E')$ via $(10.4')$, we eliminate the component
$\mu^\star F_{j_0}$. This is a contradiction since $N(\alpha)$
was supposed to contain $F_{j_0}$.\qed
\endproof

\claim{(10.6) Definition} For a class $\alpha\in H^{1,1}(X,\bR)$, 
we define the \ul{numerical dimension} $\num(\alpha)$ to be 
$\num(\alpha)=-\infty$ if $\alpha$ is not pseudo-effective, and
$$
\num(\alpha)=\max\{p\in\bN\;;\;\langle\alpha^p\rangle\ne 0\},\qquad
\num(\alpha)\in\{0,1,\ldots,n\}
$$
if $\alpha$ is pseudo-effective.
\endclaim

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

\claim{(10.7) Theorem} Let $X$ be a compact
K\"ahler manifold. Then the subset $\cD_0$ of irreducible divisors 
$D$ in $X$ such that $\num(D)=0$ is countable, and these divisors 
are rigid as well as their multiples. If $\alpha\in\cE$ is a 
pseudo-effective class of numerical dimension~$0$, then $\alpha$ 
is numerically equivalent to 
an effective $\bR$-divisor $D=\sum_{j\in J}\lambda_j D_j$,
for some finite subset $(D_j)_{j\in J}\subset\cD_0$ such that the 
cohomology classes $\{D_j\}$ are linearly independent and
some \hbox{$\lambda_j>0$}. If such a linear combination is of
numerical dimension~$0$, then so is any other linear combination of the
same divisors.
\endclaim

\begproof{} It is immediate from the definition that a pseudo-effective
class is of numerical dimension $0$ if and only if $\langle\alpha\rangle=0$,
in other words if $\alpha=N(\alpha)$. Thus $\alpha\equiv\sum\lambda_jD_j$
as described in 10.7, and since $\lambda_j\langle D_j\rangle\le 
\langle\alpha\rangle$, the divisors $D_j$ must themselves have numerical 
dimension $0$. There is at most one such divisor $D$ in any given cohomology
class in $NS(X)\cap\cE\subset H^2(X,\bZ)$, otherwise two such divisors 
$D\equiv D'$ would yield a blow-up $\mu:\wt X\to X$ resolving the 
intersection, and by taking
$\min(\mu^\star D,\mu^\star D')$ via $(10.4')$, we would find 
$\mu^\star D\equiv E+\beta$, $\beta\ne 0$, so that
$\{D\}$ would not be of numerical dimension $0$. This implies 
that there are at most countably many divisors of numerical 
dimension~$0$, and that these divisors are rigid as well as their
multiples.\qed
\endproof

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

\claim{(10.8) Generalized abundance conjecture} For an arbitrary compact 
K\"ahler manifold~$X$, the Kodaira dimension should be equal to the
numerical dimension~:
$$\kappa(X)=\num(X):=\num(c_1(K_X)).$$
\endclaim

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

\claim{(10.9) Remark} {\rm Using the Iitaka fibration, it is immediate to 
see that $\kappa(X)\leq\num(X)$.}
\endclaim

\claim{(10.10) Remark} {\rm It is known that abundance holds in the case
\hbox{$\num(X)=-\infty$} (if $K_X$ is not pseudo-effective, no
multiple of $K_X$ can have sections), or in the case \hbox{$\num(X)=n$}
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 [DP03]).

In the remaining cases, the most tractable situation is the case when 
$\num(X)=0$.
In fact Theorem 10.7 then gives $K_X\equiv\sum\lambda_j D_j$ for some
effective divisor with numerically independent components, $\num(D_j)=0$. 
It follows that the $\lambda_j$ are rational and therefore
$$
K_X\sim \sum\lambda_j D_j+F\qquad
\hbox{where $\lambda_j\in\bQ^+$, $\num(D_j)=0$ and $F\in\Pic^0(X)$.}
\leqno(*)
$$
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{$\num(X)=0$} and positive 
irregularity $q(X)>0$ has been solved by Campana-Peternell
[CP04], Corollary~3.7. It would be interesting to understand 
the K\"ahler  case as well.}
\endclaim

\titleb{11.}{The Orthogonality Estimate}

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

\claim{(11.1) Theorem} Let $X$ be a projective manifold, and
let $\alpha=\{T\}\in\cE^\circ_{\NS}$ be a big class represented by
a K\"ahler current~$T$. Consider
an approximate Zariski decomposition
$$\mu_m^\star T_m = [E_m]+[D_m]$$
Then
$$
(D_m^{n-1}\cdot E_m)^2\le 20\,(C\omega)^n\big(\Vol(\alpha)-D_m^n\big)
$$
where $\omega=c_1(H)$ is a K\"ahler form and $C\ge 0$ is a constant such that
$\pm\alpha$ is dominated by $C\omega$ $($i.e., $C\omega\pm\alpha$ is nef$\,)$.
\endclaim

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

\claim{(11.2) Lemma} For all nef $\bR$-divisors $A$, $B$ we have
$$\Vol(A-B)\ge A^n - nA^{n-1}\cdot B$$
as soon as the right hand side is positive.
\endclaim

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

\claim{(11.3) Remark} {\rm We hope that Lemma 11.2 also holds true on an
arbitrary K\"ahler mani\-fold for arbitrary nef (non neces\-sarily
integral) classes. This would follow from a generalization of
holomorphic Morse inequalities to non integral classes. However 
the proof of such a result seems technically much more involved than
in the case of integral classes.}
\endclaim
 
\claim{(11.4) Lemma} Let $\beta_1,\ldots,\beta_n$ and 
$\beta'_1,\ldots,\beta'_n$ be nef classes on a compact K\"ahler manifold
$\wt X$ such that each difference $\beta'_j-\beta_j$ is pseudo-effective. Then
the $n$-th intersection products satisfy
$$
\beta_1\cdots\beta_n \le \beta'_1\cdots\beta'_n.
$$ 
\endclaim

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

\noindent
{\it End of proof of Theorem 11.1.}
In order to exploit the lower bound of the volume, we write
$$
tE_m+D_m=A-B,\qquad
A=D_m+t\mu_m^\star(\alpha+C\omega),\quad B=t(D_m+C\mu_m^\star\omega).
$$
By our choice of the constant $C$, both $A$ and $B$ are nef. Lemma 11.2 and 
the binomial formula imply
$$\eqalign{
\Vol(tE_m+&D_m)\ge A^n - nA^{n-1}\cdot B\cr
&=D_m^n+nt\,D_m^{n-1}\cdot\mu_m^\star(\alpha+C\omega)+\sum_{k=2}^n
t^k{n\choose k}D_m^{n-k}\cdot\mu_m^\star(\alpha+C\omega)^k\cr
&\phantom{{}=D_m^n}-nt\,D_m^{n-1}\cdot(D_m+C\mu_m^\star\omega)\cr
&\phantom{{}=D_m^n}-nt^2\sum_{k=1}^{n-1}t^{k-1}{n-1\choose k}
D_m^{n-1-k}\cdot\mu_m^\star(\alpha+C\omega)^k\cdot(D_m+C\mu_m^\star\omega).
\kern-4pt\cr}
$$
Now, we use the obvious inequalities
$$
D_m\le \mu_m^\star(C\omega),\qquad
\mu_m^\star(\alpha+C\omega)\le 2\mu_m^\star(C\omega),\qquad
D_m+C\mu_m^\star\omega\le 2\mu_m^\star(C\omega)
$$
in which all members are nef (and where the inequality${}\le$ means 
that the difference of classes is pseudo-effective). We use Lemma~11.4
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 11.4), whence Theorem 11.1. 
Of course, the constant 20 is certainly not optimal.\qed

\claim{(11.5) Corollary} If $\alpha\in\cE_{\NS}$, then the divisorial
Zariski decomposition $\alpha=N(\alpha)+\langle\alpha\rangle$
is such that
$$
\langle\alpha^{n-1}\rangle\cdot N(\alpha)=0.
$$
\endclaim

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

\titleb{12.}{Proof of the Main Duality Theorem}

The proof is reproduced from [BDPP03]. We want to show that 
$\cE_{\NS}$ and $\SME(X)$ are dual (Theorem 8.2). By 7.4~(iii) we have 
in any case
$$\cE_{\NS}\subset (\SME(X))^\smallvee.$$ 
If the inclusion is strict, there is an element
$\alpha\in\partial\cE_{\NS}$ on the boundary of $\cE_{\NS}$ which is in
the interior of $\SME(X)^\smallvee$.


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\EndFig
\vskip8mm

Let $\omega=c_1(H)$ be an ample class. Since $\alpha\in\partial\cE_{\NS}$, 
the class $\alpha+\delta\omega$ is big for every $\delta>0$, and since
$\alpha\in((\SME(X))^{\smallvee})^\circ$ we still have 
$\alpha-\varepsilon\omega\in(\SME(X))^{\smallvee}$ for
$\varepsilon>0$ small. Therefore
$$
\alpha\cdot \Gamma\ge\varepsilon\omega\cdot \Gamma\leqno(12.1)
$$
for every movable curve $\Gamma$. We are going to contradict (12.1).
Since $\alpha+\delta\omega$ is big, we have an approximate Zariski
decomposition
$$
\mu_\delta^\star(\alpha+\delta\omega)=E_\delta+D_\delta.
$$
We pick $\Gamma=(\mu_\delta)_\star(D_\delta^{n-1})$. By the Hovanskii-Teissier
concavity inequality
$$
\omega\cdot \Gamma\ge (\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}.
$$
On the other hand
$$
\eqalign{
\alpha\cdot \Gamma&=\alpha\cdot(\mu_\delta)_\star(D_\delta^{n-1})\cr
&=\mu_\delta^\star\alpha\cdot D_\delta^{n-1}\le
\mu_\delta^\star(\alpha+\delta\omega)\cdot D_\delta^{n-1}\cr
&=(E_\delta+D_\delta)\cdot D_\delta^{n-1}
=D_\delta^n+D_\delta^{n-1}\cdot E_\delta.\cr
}
$$
By the orthogonality estimate, we find
$$\eqalign{
{\alpha\cdot \Gamma\over
\omega\cdot \Gamma}
&\le {D_\delta^n + \big(20(C\omega)^n
(\Vol(\alpha+\delta\omega)-D_\delta^n)\big)^{1/2}\over
(\omega^n)^{1/n}(D_\delta^n)^{(n-1)/n}}\cr
&\le C'(D_\delta^n)^{1/n}+C''{(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}\over
(D_\delta^n)^{(n-1)/n}}.\cr}
$$
However, since $\alpha\in\partial\cE_{\NS}$, the class $\alpha$ cannot be big 
so
$$\lim_{\delta \to 0}D_\delta^n=\Vol(\alpha)=0.$$ 
We can also take $D_\delta$ to approximate $\Vol(\alpha+\delta\omega)$ 
in such a way that
$(\Vol(\alpha+\delta\omega)-D_\delta^n)^{1/2}$ tends to $0$ much faster than
$D_\delta^n$. Notice that \hbox{$D_\delta^n\ge\delta^n\omega^n$}, so in fact
it is enough to take
$$
\Vol(\alpha+\delta\omega)-D_\delta^n\le \delta^{2n}.
$$
This is the desired contradiction by (12.1).\qed
\bigskip

\claim{(12.2) Remark} {\rm If holomorphic Morse inequalities were known also
in the K\"ahler case, we would infer by the same proof that
``$\alpha$ not pseudo-effective'' implies the existence of a blow-up
$\mu:\wt X\to X$ and a K\"ahler metric 
$\wt\omega$ on $\smash{\wt X}$ such that 
$\alpha\cdot\mu_\star(\wt\omega)^{n-1} < 0$. In the special case
when $\alpha=K_X$ is not pseudo-effective, we would expect the 
K\"ahler manifold $X$ to be covered by rational curves.
The main trouble is that characteristic $p$ techniques are no longer
available. On the other hand it is tempting to approach the
question via techniques of symplectic geometry~: } \endclaim

\claim{(12.3) Question} Let $(M,\omega)$ be a compact real symplectic manifold.
Fix an almost\break complex structure $J$ compatible with $\omega$, and
for this structure, assume that\break \hbox{$c_1(M)\cdot\omega^{n-1}>0$}. 
Does it follow that $M$ is covered by rational $J$-pseudoholomorphic curves~?
\endclaim
\vskip20pt


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}

\vskip30pt
\noindent(July 30, 2004, revised on November 26, 2007, printed on \today)
\vskip10pt
\noindent
Jean-Pierre Demailly\hfill\break
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, UMR 5582 du CNRS, BP$\,$74,\hfill\break
F-38402 Saint-Martin d'H\`eres, France

\noindent{\it e-mail address:}\/ demailly@fourier.ujf-grenoble.fr
\end