\special{header=mdrlib.ps}

\def\build#1^#2_#3{\mathop{#1}\limits^{#2}_{#3}}
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\long\def\InsertFig#1 #2 #3 #4\EndFig{\par
\hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{" 
#3}}#4$}}

\long\def\InsertPSFile#1 #2 #3 #4 #5 #6\EndFig{\par
\hbox{\hskip #1mm$\vbox to#2mm{%
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angle #6}}#8$}}

\long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise2mm\hbox{#3}}}

% This file is a solution template for:
% - Giving a talk on some subject.
% - The talk is between 15min and 45min long.
% - Style is ornate.
% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice. 
\mode<presentation>
{
% \setbeamertemplate{background canvas}[vertical shading][bottom=red!10,
% top=blue!10]
  \usetheme{Warsaw}
  \usefonttheme[onlysmall]{structurebold}
}
% or whatever

\usepackage{amsmath,amssymb}
\usepackage[latin1]{inputenc}
\usepackage{colortbl}
\usepackage[english]{babel}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.

\catcode`\@=11
%%\font\eightrm=cmr10 at 8pt
%%\let\FRAMETITLE=\frametitle
%%\def\frametitle#1{\FRAMETITLE{#1\hfill{\eightrm\number\c@page/\pgt}}}
\def\pgn{{\number\c@page/\pgt}}
\catcode`\@=12

\title[\ \kern-190pt
\blank{Jean-Pierre Demailly, Stockholms Universitet, June 6, 2013\kern22pt
On the cohomology of pseudoeffective line bundles\kern15pt\pgn}]
% (optional, use only with long paper titles)
{On the cohomology of\vskip0pt pseudoeffective line bundles}

%% \subtitle{Presentation Subtitle} % (optional)

\author[] % (optional, use only with lots of authors)
{Jean-Pierre Demailly}

\institute[]{Institut Fourier, Universit\'e de Grenoble I, France\\
\&\ Acad\'emie des Sciences de Paris}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date[]% (optional)
{Conference ``Facets of Geometry''\\
a tribute to Torsten Ekedahl \&\ Mikael Passare\\
Stockholms Universitet, June 3--7, 2013}

%%\subject{Talks}
% This is only inserted into the PDF information catalog. Can be left
% out. 

% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:

\definecolor{ColClaim}{rgb}{0,0,0.8}
\definecolor{Alert}{rgb}{0.8,0,0}
\definecolor{Blank}{rgb}{1,1,1}
\def\claim#1{{\color{ColClaim}#1}}
\def\alert#1{{\color{Alert}#1}}
\def\blank#1{{\color{Blank}#1}}
\def\RGBColor#1#2{\special{color push rgb #1}#2\special{color pop}}

%%\def\\{\hfil\break}

\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\tors}{\operatorname{torsion}}
\newcommand{\rk}{\operatorname{rk}}
\newcommand{\reg}{\operatorname{reg}}
\newcommand{\sing}{\operatorname{sing}}

\newcommand{\bB}{{\mathbb B}}
\newcommand{\bC}{{\mathbb C}}
\newcommand{\bD}{{\mathbb D}}
\newcommand{\bK}{{\mathbb K}}
\newcommand{\bN}{{\mathbb N}}
\newcommand{\bP}{{\mathbb P}}
\newcommand{\bQ}{{\mathbb Q}}
\newcommand{\bR}{{\mathbb R}}
\newcommand{\bZ}{{\mathbb Z}}

\newcommand{\cA}{{\mathcal A}}
\newcommand{\cC}{{\mathcal C}}
\newcommand{\cD}{{\mathcal D}}
\newcommand{\cE}{{\mathcal E}}
\newcommand{\cF}{{\mathcal F}}
\newcommand{\cH}{{\mathcal H}}
\newcommand{\cI}{{\mathcal I}}
\newcommand{\cK}{{\mathcal K}}
\newcommand{\cM}{{\mathcal M}}
\newcommand{\cN}{{\mathcal N}}
\newcommand{\cO}{{\mathcal O}}
\newcommand{\cP}{{\mathcal P}}
\newcommand{\cX}{{\mathcal X}}

\newcommand{\dbar}{\overline\partial}
\newcommand{\ddbar}{\partial\overline\partial}
\newcommand{\ovl}{\overline}
\newcommand{\wt}{\widetilde}
\newcommand{\lra}{\longrightarrow}
\newcommand{\bul}{{\scriptscriptstyle\bullet}}

% mathematical operators
\renewcommand{\Re}{\mathop{\rm Re}\nolimits}
\renewcommand{\Im}{\mathop{\rm Im}\nolimits}
\newcommand{\Pic}{\mathop{\rm Pic}\nolimits}
\newcommand{\codim}{\mathop{\rm codim}\nolimits}
\newcommand{\Id}{\mathop{\rm Id}\nolimits}
\newcommand{\Sing}{\mathop{\rm Sing}\nolimits}
\newcommand{\Supp}{\mathop{\rm Supp}\nolimits}
\newcommand{\Vol}{\mathop{\rm Vol}\nolimits}
\newcommand{\rank}{\mathop{\rm rank}\nolimits}
\newcommand{\pr}{\mathop{\rm pr}\nolimits}

\newcommand{\NS}{\mathop{\rm NS}\nolimits}
\newcommand{\GG}{{\mathop{\rm GG}\nolimits}}
\newcommand{\NE}{\mathop{\rm NE}\nolimits}
\newcommand{\ME}{\mathop{\rm ME}\nolimits}
\newcommand{\SME}{\mathop{\rm SME}\nolimits}
\newcommand{\alg}{{\rm alg}}
\newcommand{\nef}{{\rm nef}}
\newcommand{\num}{\nu}
\newcommand{\ssm}{\mathop{\mathbb r}}
\newcommand{\smallvee}{{\scriptscriptstyle\vee}}

% figures inserted as PostScript files
\special{header=/home/demailly/psinputs/mathdraw/grlib.ps}
\long\def\InsertFig#1 #2 #3 #4\EndFig{\par
\hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{"
#3}}#4$}}
\long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise#2mm\hbox{#3}}}
\def\ovl{\overline}
\def\build#1^#2_#3{\mathrel{\mathop{\null#1}\limits^{#2}_{#3}}}
\def\bibitem[#1]#2#3{\medskip{\bf[#1]} #3}

\begin{document}

% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
%%\AtBeginSubsection[]
%%{
%% \begin{frame}<beamer>
%%    \frametitle{Outline}
%%    \tableofcontents[currentsection,currentsubsection]
%%  \end{frame}
%%}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command: 

%\beamerdefaultoverlayspecification{<+->}

\begin{frame}
  \pgfdeclareimage[height=2cm]{ujf-logo}{logo_ujf}
  \pgfuseimage{ujf-logo}
  \pgfdeclareimage[height=2cm]{acad-logo}{academie_logo2}
  \pgfuseimage{acad-logo}
  \vskip-5pt
  \titlepage
\end{frame}

%%\begin{frame}
%%  \frametitle{Outline}
%%  \tableofcontents
%% You might wish to add the option [pausesections]
%%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%%\section*{Basic concepts}
\def\pause{}

\begin{frame}
\frametitle{Basic concepts (1)}
\hbox{Let $X={}$compact K\"ahler manifold, $L\to X$ holomorphic line bundle,\kern-15pt}
$h$ a hermitian metric on $L$.\\\pause Locally $L_{|U}\simeq U\times\bC$ and
for \claim{$\xi\in L_x\simeq\bC$, $\Vert\xi\Vert_h^2=|\xi|^2e^{-\varphi(x)}$}.
\pause

Writing $h=e^{-\varphi}$ locally, one defines the \claim{curvature form} of 
$L$ to be the real $(1,1)$-form\vskip-18pt
\alert{
\begin{eqnarray*}
&&\Theta_{L,h}={i\over 2\pi}\ddbar\varphi=-dd^c\log h,\pause\\
&&c_1(L)=\big\{\Theta_{L,h}\big\}\in H^2(X,\bZ).\\
\end{eqnarray*}\pause}\vskip-32pt
Any subspace $V_m\subset H^0(X,L^{\otimes m})$ define a meromorphic map
\alert{
\begin{eqnarray*}
\Phi_{mL}:X\ssm Z_m&&\longrightarrow\quad\bP(V_m)~~~\hbox{(hyperplanes of $V_m$)}\\
x&&\longmapsto\quad H_x=\big\{\sigma\in V_m\,;\;\sigma(x)=0\big\}
\end{eqnarray*}}
where \claim{$Z_m={}$base locus~$B(mL)=\bigcap\sigma^{-1}(0)$}.
\end{frame}

\begin{frame}
\frametitle{Basic concepts (2)}
Given sections $\sigma_1,\ldots,\sigma_n\in H^0(X,L^{\otimes m})$, one gets
a \claim{singular hermitian metric} on $L$ defined by
\alert{$$
\vert\xi\vert_h^2={|\xi|^2\over\big(\sum|\sigma_j(x)|^2\big)^{1/m}},
$$}\pause %%
its weight is the \claim{plurisubharmonic (psh)} function
$$
\varphi(x)={1\over m}\log\Big(\sum|\sigma_j(x)|^2\Big)
$$\pause %%
and the curvature is \alert{$\Theta_{L,h}={1\over m}dd^c\log\varphi\ge 0$}\\
in the sense of currents, with \claim{logarithmic poles} along the base locus
\alert{$$
B=\bigcap\sigma_j^{-1}(0)=\varphi^{-1}(-\infty).
$$}\pause %%
One has
$$\alert{
(\Theta_{L,h})_{|X\ssm B}={1\over m}\Phi_{mL}^*\omega_{\rm FS}}~~\hbox{where}~~
\alert{\Phi_{mL}:X\ssm B\to\bP(V_m)\simeq\bP^{N_m}.}
$$
\end{frame}

\begin{frame}
\frametitle{Basic concepts (3)}
\begin{block}{Definition} 
\begin{itemize}
\item $L$ is pseudoeffective (\claim{psef}) if 
$\exists h=e^{-\varphi}$, $\varphi\in L^1_{\rm loc}$,\\
(possibly singular) such that \alert{$\Theta_{L,h}=-dd^c\log h\ge 0$
on $X$},\\ in the sense of currents.
\item $L$ is \claim{semipositive} if 
$\exists h=e^{-\varphi}$ smooth
such that\\ \alert{$\Theta_{L,h}=-dd^c\log h\ge 0$ on $X$}.
\item $L$ is \claim{positive} if 
$\exists h=e^{-\varphi}$ smooth
such that\\ \alert{$\Theta_{L,h}=-dd^c\log h>0$ on $X$}.
\end{itemize}
\end{block}

The well-known Kodaira embedding theorem states that\\
\alert{$L$ is positive if and only if $L$ is ample}, namely:\\
\claim{$Z_m=B(mL)=\emptyset$ and\vskip3pt
\centerline{$\Phi_{|mL|}:X\to \bP(H^0(X,L^{\otimes m}))$}\vskip3pt
is an embedding for $m\ge m_0$ large enough}.
\end{frame}

\begin{frame}
\frametitle{Positive cones}
\begin{block}{Definitions} Let $X$ be a compact K\"ahler manifold.
\begin{itemize}
\item The \claim{K\"ahler cone} is the (open) set 
\claim{$\cK\subset H^{1,1}(X,\bR)$} of cohomology classes $\{\omega\}$
of positive K\"ahler forms.\pause
\vskip2pt
\item The \claim{pseudoeffective} cone is the set
\claim{$\cE\subset H^{1,1}(X,\bR)$} of cohomology classes $\{T\}$ of 
closed positive $(1,1)$ currents.\\
This is a closed convex cone.\\
$($by weak compactness of bounded sets of currents$)$.\pause\vskip2pt
\item \claim{$\ovl\cK$ is
the cone of ``nef classes''.} One has \alert{$\ovl\cK\subset\cE$}.
\pause\vskip2pt
\item It may happen that \alert{$\ovl\cK\subsetneq\cE$}:\\
if $X$ is the surface
obtained by blowing-up $\bP^2$ in one point, then the exceptional
divisor $E\simeq\bP^1$ has a cohomology class $\{\alpha\}$ such that 
\alert{$\int_E\alpha= E^2=-1$}, hence $\{\alpha\}\notin\ovl\cK$, although
$\{\alpha\}=\{[E]\}\in\cE$.
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Ample / nef / effective / big divisors}
\vskip-10pt
Positive cones can be visualized as follows :
\InsertFig 39 72
{
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\LabelTeX -42 64 $H^{1,1}(X,\bR)$~~ (containing Neron-Severi space $\NS_\bR(X)$)\ELTX
\LabelTeX -26 53 $\RGBColor{1 0 0}{\cK}$\ELTX
\LabelTeX -3 46 $\RGBColor{1 0 0}{\cK_{\NS}}$\ELTX
\LabelTeX -24 34 $\RGBColor{0 0 1}{\cE}$\ELTX
\LabelTeX -19 26.5 $\RGBColor{0 0 1}{\cE_{\NS}}$\ELTX
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\LabelTeX -32 42 \RGBColor{1 0 0}{cone}\ELTX
\LabelTeX -32 28 \RGBColor{0 0 1}{psef}\ELTX
\LabelTeX -32 23 \RGBColor{0 0 1}{cone}\ELTX
\LabelTeX 32 54 \RGBColor{1 0 0}{ample divisors: $\cK_{\NS}$}\ELTX
\LabelTeX 32 47 \RGBColor{1 0 0}{nef divisors: $\ovl\cK_{\NS}$}\ELTX
\LabelTeX 37 34 \RGBColor{0 0 1}{big divisors: $\cE_{\NS}^\circ$}\ELTX
\LabelTeX 37 27 \RGBColor{0 0 1}{effective \& psef: $\cE_{\NS}$}\ELTX
\LabelTeX 22 17 $\cK_{\NS}=\cK\cap\NS_\bR(X)$\ELTX
\LabelTeX 22 12 $\cE_{\NS}=\cE\cap\NS_\bR(X)$\ELTX
\LabelTeX 32 6 where\ELTX
\LabelTeX 4 -1 $\NS_\bR(X)=(H^{1,1}(X,\bR)\cap H^2(X,\bZ))\otimes_\bZ\bR$\ELTX
\LabelTeX -38.5 2 $\NS_\bR(X)$\ELTX
\EndFig
\end{frame}

\begin{frame}
\frametitle{Approximation of currents, Zariski decomposition}
 \vskip-4pt
\begin{block}{Definition} On $X$ compact K\"ahler,
a \alert{K\"ahler current} $T$ is a closed
positive $(1,1)$-current $T$ such that $T\ge \delta\omega$ for some
smooth hermitian metric $\omega$ and a constant $\delta\ll 1$.
\end{block}\pause

\begin{block}{Easy observation} $\alpha\in\cE^\circ~~
\hbox{(interior of $\cE$)}~\Longleftrightarrow~\alpha=\{T\}$,
$T={}$a K\"ahler current.\vskip2pt
We say that $\cE^\circ$ is the cone of \alert{big $(1,1)$-classes}.
\end{block}\pause

\begin{block}{Theorem on approximate Zariski decomposition~ (D92)}
Any K\"ahler current can be written
$T=\lim T_m$ where $T_m\in \alpha=\{T\}$ has \alert{logarithmic poles}, i.e.\\
\alert{$\exists$ a modification \hbox{$\mu_m:\wt X_m\to X$} such that
$\mu_m^\star T_m=[E_m]+\beta_m$}\\
where $E_m$ is an effective $\bQ$-divisor on $\wt X_m$ with coefficients
in ${1\over m}\bZ$ and $\beta_m$ is a K\"ahler form on $\wt X_m$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Idea of proof of analytic Zariski decomposition (1)}
$\bullet$ Write locally \alert{$$T=i\ddbar\varphi$$}for some strictly 
plurisubharmonic psh potential $\varphi$ on $X$.\pause\vskip2pt
$\bullet$ Approximate $T$ (again locally) as
\vskip4pt
\alert{\centerline{$\displaystyle
T_m=i\ddbar\varphi_m,\qquad
\varphi_m(z)={1\over 2m}\log\sum_\ell |g_{\ell,m}(z)|^2
$}}
\vskip4pt
where $(g_{\ell,m})$ is a Hilbert basis of the space 
\vskip4pt
\alert{\centerline{$\displaystyle
\cH(\Omega,m\varphi)=\big\{
f\in\cO(\Omega)\,;\;\int_\Omega |f|^2 e^{-2m\varphi}dV<+\infty\big\}.
$}}\pause\vskip4pt
$\bullet$ The Ohsawa-Takegoshi $L^2$ extension theorem 
(extending from a single isolated point) implies that there
are enough such holomorphic functions, and 
thus \alert{$\varphi_m\geq\varphi-C/m$}.\pause\vskip3pt
$\bullet$ Further, \alert{$\displaystyle\varphi=\lim_{m\to+\infty}\varphi_m$} 
by the mean value inequality.
\end{frame}

\begin{frame}
\frametitle{Idea of proof of analytic Zariski decomposition (2)}
$\bullet$ 
The Hilbert basis $(g_{\ell,m})$ is a family of local generators of the
multiplier ideal sheaf $\cI(m\varphi)$. \hbox{Thanks to \claim{Hironaka}, 
the modification\kern-25pt}\\ $\mu_m:\wt X_m\to X$ is obtained by 
\claim{blowing-up $\cI(m\varphi)$}, with
\vskip3pt
\centerline{\alert{$\mu_m^\star\cI(mT)=\cO(-mE_m)$}.}
\vskip3pt
for some effective $\bQ$-divisor $E_m$ with normal crossings on $\wt X_m$.
\pause\\
$\bullet$ Now, we set 
\vskip3pt
\centerline{\alert{$T_m=i\ddbar\varphi_m$},\kern20pt
\alert{$\beta_m=\mu_m^*T_m-[E_m]$.}}\vskip2pt\pause
$\bullet$ Locally on $\wt X_m$ one has
\vskip3pt
\centerline{\alert{$\beta_m=i\ddbar\psi_m$}~ where~
\alert{$\psi_m={1\over 2m}\log\sum_\ell|g_{\ell,m}\circ\mu_m/h|^2$}}
\vskip4pt
and $h$ is a generator of $\cO(-mE_m)$, thus \claim{$\beta_m\ge 0$ smooth}.
\vskip4pt\pause
$\bullet$ The construction \claim{can be made global} by using a gluing technique, 
e.g.\ a partition of unity, and\pause\vskip3pt
$\bullet$ \claim{$\beta_m$ can be made K\"ahler} by a perturbation argument.
\end{frame}

\begin{frame}
\frametitle{Concept of volume (very important !)}

\claim{{\bf Definition} (\rm Boucksom 2002).\\ {\it
The \alert{volume} $($\alert{movable self-intersection}$)$ of a 
big class $\alpha\in\cE^\circ$ is
$$
\Vol(\alpha)=\sup_{T\in \alpha}\int_{\wt X}\beta^n>0
$$
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 $\mu:\wt X\to X$.}}
\medskip

If $\alpha\in\cK$, then $\Vol(\alpha)=\alpha^n=\int_X\alpha^n$.
\medskip

\claim{{\bf Theorem.} {\rm (Boucksom 2002)}. {\it If $L$ is a big line bundle
and $\mu_m^*(mL)=[E_m]+[D_m]$\\
$($where $E_m={}$fixed part, $D_m={}$moving part$)$, then
$$
\Vol(c_1(L))=\lim_{m\to+\infty}{n!\over m^n}h^0(X,mL)=
\lim_{m\to+\infty} D_m^n.
$$}}
\end{frame}

\begin{frame}
\frametitle{Approximate Zariski decomposition}
\def\srelbar{\vrule width0.6ex height0.65ex depth-0.55ex}
\def\merto{\mathrel{\srelbar\kern1.3pt\srelbar\kern1.3pt\srelbar
    \kern1.3pt\srelbar\kern-1ex\raise0.28ex\hbox{${\scriptscriptstyle>}$}}}

In other words, the volume measures the amount of sections and
the growth of the degree of the images of the rational maps
$$
\Phi_{|mL|}:X\merto \bP^n_\bC
$$
By Fujita 1994 and Demailly-Ein-Lazarsfeld 2000, one has
\medskip

\claim{{\bf Theorem.} {\it 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
\alert{$$\Vol(L) -\varepsilon\le \Vol(\beta) \le \Vol(L).$$}}}
\end{frame}

\begin{frame}
\frametitle{Movable intersection theory}

\claim{{\bf Theorem} {\rm (Boucksom 2002)} {\it Let $X$ be a compact K\"ahler
manifold and 
\alert{$$
H^{k,k}_{\ge 0}(X)=\big\{\{T\}\in H^{k,k}(X,\bR)\,;\;
\hbox{$T$ closed${}\ge 0$}\big\}.
$$}}}%
\pause

\begin{itemize}
\item \claim{\it $\forall k=1,2,\ldots,n$, $\exists$ canonical
``movable intersection product''
\alert{$$
\cE\times\cdots\times\cE\to H^{k,k}_{\ge 0}(X), \quad
(\alpha_1,\ldots,\alpha_k)\mapsto \langle\alpha_1\cdot\alpha_2\cdots
\alpha_{k-1}\cdot \alpha_k\rangle
$$}%
such that $\Vol(\alpha)=\langle\alpha^n\rangle$ whenever $\alpha$ is
a big class.}
\pause

\item \claim{\it 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 $\alpha_j\in\ovl{\cK}$ are nef classes.}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Movable intersection theory (continued)}
\begin{itemize}
\item \claim{\it For $k=1$, one gets a ``divisorial Zariski decomposition''
$$
\alpha=\{N(\alpha)\}+\langle\alpha\rangle
$$
where~:}
\begin{itemize}
\item<2>
\claim{\it $N(\alpha)$ 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~;}
\item<2>
\claim{\it $\langle\alpha\rangle$ is ``nef outside codimension $2$''.}
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Construction of the movable intersection product}
First assume that all classes $\alpha_j$ are big, i.e.\ 
$\alpha_j\in\cE^\circ$. Fix a smooth closed $(n-k,n-k)$ \emph{semi-positive}
form $u$ on $X$. We select K\"ahler currents $T_j\in\alpha_j$ with
logarithmic poles, and simultaneous \alert{more and more accurate} 
log-resolutions
$\mu_m:\wt X_m\to X$ such that 
\alert{$$
\mu_m^\star T_j=[E_{j,m}]+\beta_{j,m}.
$$}%
We define
\alert{$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle =
\mathop{\lim\uparrow}\limits_{m\to+\infty}
\{(\mu_m)_\star(\beta_{1,m}\wedge\beta_{2,m}\wedge\ldots\wedge\beta_{k,m})\}
$$}%
as a weakly convergent subsequence. The main point is to show that there
is actually convergence and that the \alert{limit is unique in cohomology}~;
this is based on ``monotonicity properties'' of the Zariski decomposition.\\
\phantom{\strut}
\end{frame}


\begin{frame} 
\frametitle{Generalized abundance conjecture}

\claim{{\bf Definition.} {\it For a class $\alpha\in H^{1,1}(X,\bR)$, 
the numerical dimension $\num(\alpha)$ is}}
\begin{itemize}
\item \claim{\it $\num(\alpha)=-\infty$ if $\alpha$ is not pseudoeffective,}
\item \claim{\it
$\num(\alpha)=\max\{p\in\bN\;;\;\langle\alpha^p\rangle\ne 0\}~~~{}\in
\{0,1,\ldots,n\}$\\
if $\alpha$ is pseudoeffective.}
\end{itemize}
\pause
\medskip

\claim{{\bf Conjecture} {\rm (``generalized abundance conjecture'')}. {\it
  For an arbitrary compact K\"ahler manifold~$X$, the Kodaira
  dimension should be equal to the numerical dimension~:
\alert{$$\kappa(X)=\num(c_1(K_X)).$$}}}
$\phantom{\strut}$\vskip-28pt$\phantom{\strut}$

{\bf Remark.} The generalized abundance conjecture holds true when
$\nu(c_1(K_X))=-\infty,\,0,\,n$ (cases $-\infty$ and $n$ being easy).
\end{frame}

\begin{frame}
\frametitle{Orthogonality estimate}

\claim{{\bf Theorem.} {\it Let $X$ be a projective manifold.\\
Let $\alpha=\{T\}\in\cE^\circ_{\NS}$ be a big class represented by
a K\"ahler current~$T$, and consider
an approximate Zariski decomposition
$$\mu_m^\star T_m = [E_m]+[D_m]$$
Then
\alert{$$
(D_m^{n-1}\cdot E_m)^2\le 20\,(C\omega)^n\big(\Vol(\alpha)-D_m^n\big)
$$}%
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$\,)$.}}
\medskip

By going to the limit, one gets
\medskip

\claim{{\bf Corollary.}~~ $\alpha\cdot\langle\alpha^{n-1}\rangle -
\langle\alpha^n\rangle=0$.}
\end{frame}


\begin{frame}
\frametitle{Schematic picture of orthogonality estimate}

The proof is similar to the case of projecting a point onto a convex
set, where the segment to closest point is orthogonal to tangent
plane.

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\end{frame}


\begin{frame}
\frametitle{Proof of duality between $\cE_{\NS}$ and $\cM_{\NS}$}

\claim{{\bf Theorem} {\rm (Boucksom-Demailly-P\v{a}un-Peternell 2004)}.\\
For $X$ projective, a class $\alpha$ is in $\cE_{\NS}$
$($pseudoeffective$)$ if and only if it is dual to the cone
$\cM_{\NS}$ of moving curves.}
\medskip

{\it Proof of the theorem.} 
We want to show that $\cE_{\NS}=\cM_{\NS}^\vee$. By obvious positivity of
the integral pairing, one has in any case
$$\cE_{\NS}\subset (\cM_{\NS})^\smallvee.$$ 
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 $\cN_{\NS}^\smallvee$. Hence
$$
\alpha\cdot \Gamma\ge\varepsilon\omega\cdot \Gamma\leqno(*)
$$
for every moving curve $\Gamma$, while $\langle\alpha^n\rangle=\Vol(\alpha)=0$.
\end{frame}

\begin{frame}
\frametitle{Schematic picture of the proof}
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\LabelTeX -8.5 24 $\alpha$\ELTX
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\LabelTeX -8.5 37 $\omega$\ELTX
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\EndFig
\medskip

Then use approximate Zariski decomposition of $\{\alpha+\delta\omega\}$ 
and orthogonality relation to contradict $(*)$ with 
$\Gamma=\langle\alpha^{n-1}\rangle$.
\end{frame}

\begin{frame} 
\frametitle{Characterization of uniruled varieties}
Recall that a projective variety is called \alert{uniruled} if it can be
covered by a family of rational curves $C_t\simeq\bP^1_\bC$.
\medskip

\claim{{\bf Theorem} {\rm (Boucksom-Demailly-Paun-Peternell 2004)}\\ {\it
A projective manifold $X$ is \alert{not uniruled} if and only if
$K_X$ is pseudoeffective, i.e.\ $K_X\in\cE_{\NS}$.}}
\medskip

{\it Proof (of the non trivial implication).} If $K_X\notin\cE_{\NS}$, 
the duality pairing shows that
there is a moving curve $C_t$ such that $K_X\cdot C_t<0$. The standard
\alert{``bend-and-break''} lemma of Mori then implies that there is family 
$\Gamma_t$ of \alert{rational curves} with $K_X\cdot\Gamma_t<0$, so $X$ 
is uniruled.

\end{frame}

\begin{frame}
\frametitle{Plurigenera and the Minimal Model Program}

\claim{\bf Fundamental question.} Prove that every birational class of non
uniruled algebraic varieties contains a ``minimal'' member $X$ with mild
singularities, where ``minimal'' is taken in the 
sense of avoiding unnecessary blow-ups; minimality actually means that 
$K_X$ is nef and not just pseudoeffective (pseudoeffectivity is known
by the above results).\medskip

This requires  performing certain birational
transforms known as \alert{flips}, and one would like to know whether\\
a) flips are indeed possible (\alert{``existence of flips''}),\\ 
b) the process terminates (\alert{``termination of flips''}).\\
Thanks to Kawamata 1992 and Shokurov (1987, 1996), this has been proved in 
dimension $3$ at the end of the 80's and more recently in dimension $4$ 
(C.~Hacon and J.~McKernan also introduced in 2005 a new induction 
procedure).
\end{frame}

\begin{frame}
\frametitle{Finiteness of the canonical ring}

\claim{{\bf Basic questions.}}\\
\begin{itemize}
\item \claim{\it \alert{Finiteness of the canonical ring:}\\
Is the \alert{canonical ring
$R=\bigoplus H^0(X,mK_X)$}
of a variety of general type always finitely generated~?}
\medskip

\claim{\it If true, Proj$(R)$ of this graded ring $R$ yields of course a 
``canonical model'' in the birational class of~$X$.}
\pause

\item \claim{\it \alert{Boundedness of pluricanonical embeddings:}\\
Is there a bound $r_n$ depending only on dimension $\dim X=n$, such that
the pluricanonical map \alert{$\Phi_{mK_X}$} of a variety of general
type yields a birational embedding in projective space for $m\ge r_n$~?}
\pause

\item \claim{\it\alert{Invariance of plurigenera:}\\ 
Are plurigenera
\alert{$p_m=h^0(X,mK_X)$} always invariant under deformation~?}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Recent results on extension of sections}

The following is a very slight extension of results
by M.~P\v{a}un (2005) and B.\ Claudon (2006), which are
themselves based on the ideas of Y.T.~Siu 2000 and S.\ Takayama~2005.
\medskip

\claim{{\bf Theorem.} {\it
Let $\pi: \cX \to \Delta$ be a family of projective manifolds over the unit 
disk, and  let $(L_j,h_j)_{0\le j\le m-1}$ be $($singular$)$ hermitian 
line bundles with semipositive curvature currents $i\Theta_{L_j,h_j}\ge 0$
on~$\cX$. Assume that}}

\begin{itemize}
\item \claim{\it the restriction of $h_j$ to the central fiber $X_0$ is well
defined $($i.e.\ not identically~$+\infty)$.}
\item \claim{\it additionally the multiplier ideal sheaf 
\alert{$\cI(h_{j|X_0})$ is trivial for $1\le j\le m-1$.}}
\end{itemize}

\claim{\it Then any section $\sigma$ of 
\alert{$\cO(mK_{\cX}+\sum L_j)_{|X_0}\otimes\cI(h_{0|X_0})$} over the central 
fiber $X_0$ extends to $\cX$.}

\end{frame}

\begin{frame}
\frametitle{Proof / invariance of plurigenera}

The proof relies on a clever iteration procedure based on the 
Ohsawa-Takegoshi $L^2$ extension theo\-rem, and a convergence process of
an analytic nature \alert{(no algebraic proof at present !)}
\medskip

The special case of the theorem obtained by taking all bundles $L_j$ trivial
tells us in particular that any pluricanonical section $\sigma$ of $mK_\cX$
over $X_0$ extends to $\cX$. By the upper semi-continuity of 
$t\mapsto h^0(X_t,mK_{X_t})$, this implies
\medskip

\claim{{\bf Corollary} {\rm (Siu 2000)}. {\it For any projective
family $t\mapsto X_t$ of algebraic varieties, the plurigenera 
\alert{$p_m(X_t)=h^0(X_t,mK_{X_t})$} do not depend on $t$.}}

\end{frame}
\end{document}

\section{Fundamental $L^2$ existence theorems}
$(E,h)$ hermitian holomorphic vector bundle of rank $r$ over a
complex manifold $X$. If $E_{|U}\simeq U\times\bC^r$ is a local
holomorphic trivialization, the hermitian product can be written as
$\langle u,v\rangle = {}^tu H(z)\ovl v$ where $H(z)$ is
the hermitian matric of $h$ and $u,v\in E_z$. It is well known
that there exists a unique ``Chern connection'' $D=D^{1,0}+D^{0,1}$ such
that $D^{0,1}=\dbar$ and such that $D$ is compatible with the hermitian metric;
in the given trivialization we have 
$D^{1,0}u=\dbar u+\Gamma^{1,0}\wedge u$ where
$\Gamma^{1,0}=\ovl H^{-1}\partial\ovl H$, and its curvature operator
$\Theta_{E,h}=D^2$ is the smooth section of 
$\Lambda^{1,1}T^*_X\otimes\Hom(E,E)$ given by
$\Theta_{E,h}=\dbar(\ovl H^{-1}\partial\ovl H)$. If $E$ is of rank $r=1$,
then it is customary to write $H(z)=e^{-\varphi(z)}$, and the curvature
tensor then takes the simple expression $\Theta_{E,h}=\ddbar\varphi$. In that
case, the \emph{first Chern class} of $E$ is the cohomology class
$c_1(E)=\{\frac{i}{2\pi}\Theta_{E,h}\}\in H^{1,1}(X,\bC)$, which is also
an integral class in $H^2(X,\bZ)$). 

In case $(X,\omega)$ is a K\"ahler manifold, the bundles $\Lambda^{p,q}T^*_X
\otimes E$ are equipped with the hermitian metric induced by
$\Lambda^{p,q}\omega \otimes h$, and we have a Hilbert space of global 
$L^2$ sections over $X$ by integrating with respect to the K\"ahler volume 
form $dV_\omega=\omega^n/n!$. If $A,\,B$ are differential operators 
acting on $L^2$ space of sections (in general, they are just closed
and densely defined operators), we denote by $A^*$ the formal adjoint
of $A$, and by $[A,B]=AB-(-1)^{\deg A\,\deg B}BA$ the usual commutator
bracket of operators. The fundamental operator $\Lambda_\omega$
of K\"ahler geometry is the adjoint of the wedge multiplication operator
$u\mapsto\omega\wedge u$.

In this context, we have the following fundamental existence theorems
for $\dbar$-equations, which is the culmination of several decades
of work by Bochner (1949), Kodaira (1954), Kohn (1963),
Andreotti-Vesentini (1965), H\"ormander (1965), Skoda 
(1978), Ohsawa-Takegoshi (1987) and followers. The proofs 
always proceed through differential 
geometric inequalities relating the Laplace-Beltrami operators with the 
curvature (Bochner-Kodaira identities and inequalities).

The most basic result is the $L^2$ existence theorem for solutions
of $\dbar$-equations.

\begin{theorem} {\rm Andreotti-Vesentini 1965, see also Demailly 1982)}
Let $(X,\omega)$ be a K\"ahler manifold which is ``complete'' in the sense
that it possesses a geodesically complete K\"ahler metric $\wt\omega$. 
Let $E$ be a hermitian holomorphic vector bundle of
rank $r$ over $X$, and assume that the curvature operator
$A^{p,q}_{E,h,\omega}=[i\Theta_{E,h},\Lambda_\omega]$ is positive
definite everywhere on $\Lambda^{p,q}T^\star_X\otimes E$, $q\ge 1$.
Then for any form $g\in L^2(X,\Lambda^{p,q}T^\star_X\otimes E)$
satisfying $\dbar g=0$ and $\int_X\langle(A^{p,q}_{E,h,\omega})^{-1}g,g\rangle
\,dV_\omega<+\infty$, there exists $f\in
L^2(X,\Lambda^{p,q-1}T^\star_X\otimes E)$ such that  $\dbar f=g$ and
$$\int_X|f|^2\,dV_\omega\le\int_X\langle 
(A^{p,q}_{E,h,\omega})^{-1}g,g\rangle\,dV_\omega.$$
\end{theorem}

It is thus of crucial importance to study conditions under which the operator
$A^{p,q}_{E,h,\omega}$ is positive definite. An easier case is when $E$ is
a line bundle. Then we denote by $\gamma_1(z)\leq\ldots\leq\gamma_n(z)$ the
eigenvalues of the real $(1,1)$-form $i\Theta_{E,h}(z)$ with respect to 
the metric $\omega(z)$ at each point. A straightforward calculation shows that
$$
\langle A^{p,q}_{E,h,\omega}u,u\rangle = \sum_{|J|=p,|K|=q}
\Big(\sum_{k\in K}\gamma_k-\sum_{j\in\complement J}\gamma_j\Big)|u_{JK}|^2.
$$
In particular, for $(n,q)$-forms the negative sum 
$-\sum_{j\in\complement J}\gamma_j$ disappears and we have
$$
\langle A^{n,q}_{E,h,\omega}u,u\rangle \geq(\gamma_1+\ldots+\gamma_q)|u|^2,
\quad
\langle (A^{n,q}_{E,h,\omega})^{-1}u,u\rangle \leq
(\gamma_1+\ldots+\gamma_q)^{-1}|u|^2
$$
provided the line bundle $(E,h)$ has positive definite curvature.
Therefore $\dbar$-equations can be solved for all $L^2$ $(n,q)$-forms
with $q\ge 1$, and this is the major reason why vanishing results for
$H^q$ cohomology groups are usually obtained for sections of the
``\emph{adjoint line bundle}'' $\wt E=K_X\otimes E$, where
$K_X=\Lambda^nT^*_X=\Omega^n_X$ is the ``\emph{canonical bundle}''
of~$X$, rather than for $E$ itself. Especially, if $X$ is compact (or
weakly pseudoconvex) and $i\Theta_{E,h}>0$, then $H^q(X,K_X\otimes E)=0$
for $q\ge 1$ (Kodaira), and more generally $H^{p,q}(X,E)=0$ for
$p+q\ge n+1$ (Kodaira-Nakano, take $\omega=i\Theta_{E,h}$, in
which case $\gamma_j\equiv 1$ for all $j$ and
$\sum_{k\in K}\gamma_k-\sum_{j\in\complement J}\gamma_j=p+q-n$).

As shown in Demailly 1982, 
Theorem on solutions of $\dbar$-equations still holds true in that case 
when $h$
is a \emph{singular hermitian metric}, i.e.\ a metric whose weights
$\varphi$ are arbitrary locally integrable functions, provided that
the curvature is $(E,h)$ is positive in the sense of currents
(i.e., the weights $\varphi$ are strictly plurisubharmonic).
This implies the well-known Nadel vanishing theorem (1989),
(Demailly 1989, 1993), a generalization of the Kawamata-Viehweg 
vanishing theorem (Kawamata 1982, Viehweg 1982).

\begin{theorem} {\rm (Nadel)}
Let $(X,\omega)$ be a compact $($or weakly
pseudoconvex$)$ K\"ahler manifold, and $(L,h)$ a singular hermitian line
bundle such that $\Theta_{L,h}\geq\varepsilon\omega$ for some
$\varepsilon>0$. Then $H^q(X,K_X\otimes L\otimes\cI(h))=0$ for $q\ge 1$, 
where $\cI(h)$ is the multiplier ideal sheaf of $h$, namely the
sheaf of germs of holomorphic functions $f$ on $X$ such that
$|f|^2e^{-\varphi}$ is locally integrable with respect to the
local weights $h=e^{-\varphi}$.
\end{theorem}

It is well known that Theorems
more specifically, its 
``singular hermitian'' version, implies almost all other fundamental 
vanishing or existence theorems of algebraic geometry, as well as their 
analytic counterparts in the framework of Stein manifolds (general solution
of the Levi problem by Grauert), see e.g.\ Demailly 2001 for 
a recent account. In particular, one gets as a consequence the
\emph{Kodaira embedding theorem} (1954).

Another fundamental existence theorem is the $L^2$-extension result by 
Ohsawa-Take\-goshi (1987). Many different versions and generalizations
have been given in recent years by
Ohsawa (1988, 1994, 95, 2001, 2003). Here is another one, due to Manivel 
1993, which is slightly less general but simpler to state.

\begin{theorem} {\rm (Ohsawa-Takegoshi 1987, Manivel 1993)}
Let $X$ be a compact or weakly pseudoconvex $n$-dimensional complex 
manifold equipped with a K\"ahler metric $\omega$, let $L$ $($resp.\ $E)$ be
a hermitian holomorphic line bundle $($resp.\ a hermitian holomorphic 
vector bundle of rank $r$ over~$X)$, and $s$ a global holomorphic 
section of~$E$. Assume that $s$ is generically transverse to the zero
section, and let
$$
Y=\big\{x\in X\,;\;s(x)=0, \Lambda^r ds(x)\not= 0\big\},\qquad
p=\dim Y=n-r.
$$
Moreover, assume that the $(1,1)$-form 
$i\Theta(L)+r\,i\,\ddbar\log|s|^2$ is semipositive and that there is
a continuous function $\alpha\ge 1$ such that the following two inequalities
hold everywhere on $X:$
\begin{itemize}
\item[\rm (i)]
$\displaystyle i\Theta(L)+r\,i\,\ddbar\log|s|^2\ge\alpha^{-1}
{\{i\Theta(E)s,s\}\over|s|^2}\,,$
\item[\rm (ii)] $|s|\le e^{-\alpha}$.
\end{itemize}
Then for every holomorphic section $f$ over $Y$ of the adjoint line bundle 
$\wt L=K_X\otimes L$ $($restricted to~$Y)$, such that
$\int_Y|f|^2|\Lambda^r(ds)|^{-2}dV_\omega<+\infty$, 
there  exists a holomorphic extension $F$ of $f$ over $X$, with values 
in $\wt L$, such that
$$
\int_X{|F|^2\over|s|^{2r}(-\log|s|)^2}\,dV_{X,\omega}
\le C_r\int_Y{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}\,,
$$
where $C_r$ is a numerical constant depending only on~$r$.
\end{theorem}

The proof actually shows that the extension theorem holds true as well
for $\dbar$-closed $(0,q)$-forms with values in $\wt L$, of which the
stated theorem is the special case $q=0$. Finally, Skoda's theorem 
1978 gives
a criterion for surjectivity of holomorphic bundle morphisms -- more
concretely, a Bezout type division theorem for holomorphic function.
Ohsawa also obtained it as a consequence of his extension theorem
(Ohsswa 2001).

\begin{theorem} {\rm(Skoda 1978, see also Demailly 1982)}
Let $X$ be a complete K\"ahler manifold equipped with
a K\"ahler metric $\omega$ on~$X$, let $g:E\to Q$ be
a surjective morphism of hermitian vector bundles and let
$L\to X$ be a hermitian holomorphic line bundle. We consider 
the adjoint morphism
$$
K_X\otimes L\otimes E\lra K_X\otimes L\otimes Q
$$
induced by $g$, and the problem of lifting holomorphic sections or
$\dbar$-closed $(0,q)$-forms with values
in $K_X\otimes L\otimes Q$. Denote by $\wt{gg^*}$ the comatrix of 
$gg^*$ in $\Hom(Q,Q)$ $($this is just the identity if rank$\,Q=1)$.
Define 
$$
r_E=\hbox{rank}\,E,\quad
r_Q=\hbox{rank}\,Q,\quad\hbox{and}\quad
m=\min\{n-q,r_E-r_Q\}
$$ 
$($where $q=0$ in the holomorphic case$)$, and assume that the curvature tensor
$i\Theta_{E,h}$ is Nakano semi-positive $($i.e.\ semi-positive as
a hermitian form on $T_X\otimes E)$, and that
$$
i\Theta_L-(m+\varepsilon)i\Theta_{\det Q}\ge 0
$$
for some $\varepsilon>0$. Then for every holomorphic section $($resp.\
$\dbar$-closed $(0,q)$-form$)$ $f$ with values in $K_X\otimes L\otimes Q$ 
such that 
$$
I=\int_X\langle\wt{gg^\star}f,f\rangle\,(\det gg^\star)^{-m-1-\varepsilon}\,
dV<+\infty,
$$
there exists a holomorphic section $($resp.\ $\dbar$-closed 
$(0,q)$-form$)$ $h$ with values in  $K_X\otimes L\otimes E$ such that 
$f=g\cdot h$ and 
$$
\int_X|h|^2\,(\det gg^\star)^{-m-\varepsilon}\,dV
\le(1+m/\varepsilon)\,I.\eqno\square
$$
\end{theorem}

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