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\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}}}
\let\framenumbering=\detailedframenumbering

\title[\kern-190pt\rlap{\
\blank{Jean-Pierre Demailly~~--~~Stockholms Universitet, June 6, 2013}}
\kern180pt\rlap{\ 
\blank{On the cohomology of pseudoeffective line bundles}}
\kern180pt
\llap{\blank{\framenumbering~}}]
% (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}}

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\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}
  \strut\vskip-14pt
  \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{Goals}
\begin{itemize}
\item Study sections and cohomology of holomorphic line bundles $L\to X$ on \claim{compact K\"ahler manifolds},\\
\alert{without assuming any strict positivity of the curvature}\pause
\item Generalize the \claim{Nadel vanishing theorem}\\
(and therefore Kawamata-Viehweg)\pause
\item Several known results already in this direction:\\
-- \claim{Skoda division theorem} (1972)\pause\\
-- \claim{Ohsawa-Takegoshi $L^2$ extension theorem} (1987)\pause\\
-- more recent work of \alert{Yum-Tong Siu}:\\
\claim{invariance of plurigenera} (1998 $\to$ 2000),\\
analytic version of \claim{Shokurov's non vanishing theorem},\\
\claim{finiteness of the canonical ring} (2007),\\
study of the \claim{abundance conjecture} (2010) ...\pause\\
-- \claim{solution of MMP} (BCHM 2006), D-Hacon-P\u{a}un (2010)
\end{itemize}
\end{frame}

\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.\pause
\item $L$ is \claim{semipositive} if 
$\exists h=e^{-\varphi}$ smooth
such that\\ \alert{$\Theta_{L,h}=-dd^c\log h\ge 0$ on $X$}.\pause
\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}\pause

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 :
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\LabelTeX 32 6 where\ELTX
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\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 \claim{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 \claim{big $(1,1)$-classes}.
\end{block}\pause

\begin{block}{Theorem on approximate Zariski decomposition~ (D, '92)}
Any K\"ahler current can be written
$T=\lim T_m$ where $T_m\in\{T\}$ has \alert{analytic singularities} \&\ 
\alert{logarithmic poles},\\
i.e.\ \alert{$\exists$ 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{Schematic picture of Zariski decomposition}
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\EndFig
\end{frame}

\begin{frame}
\frametitle{Idea of proof of analytic Zariski decomposition}
$\bullet$ 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{``Movable'' intersection of currents}

\strut\rlap{Let\ }\strut\alert{\centerline{$\cP(X)={}$ closed positive $(1,1)$-currents on $X$}
\vskip4pt
\centerline{$H^{k,k}_{\ge 0}(X)=\big\{\{T\}\in H^{k,k}(X,\bR)\,;\;
\hbox{$T$ closed${}\ge 0$}\big\}.$}}
\vskip3pt\pause

\begin{block}{Theorem (Boucksom PhD 2002, Junyan Cao PhD 2012)}
$\forall k=1,2,\ldots,n$, $\exists$ canonical
``movable intersection product''\vskip3pt
\alert{\centerline{$\displaystyle
\cP\times\cdots\times\cP\to H^{k,k}_{\ge 0}(X), \quad
(T_1,\ldots,T_k)\mapsto \langle T_1\cdot T_2\cdots T_k\rangle$}}
\end{block}\pause
\claim{Method}. $T_j=\lim_{\varepsilon\to 0}T_j+\varepsilon\omega$,
can assume $T_j$ K\"ahler.\\
Approximate each $T_j$ by K\"ahler currents $T_{j,m}$ with
logarithmic poles,take a \claim{simultaneous log-resolution}
$\mu_m:\wt X_m\to X$ such that 
\alert{\centerline{$\mu_m^\star T_j=[E_{j,m}]+\beta_{j,m}.$}}\vskip2pt
and define\vskip3pt
\alert{\centerline{$
\langle T_1\cdot T_2\cdots T_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})\}.
$}}
\end{frame}

\begin{frame}
\frametitle{Volume and numerical dimension of currents}
\alert{Remark.} The limit exists a weak limit of currents thanks to uniform boundedness in mass.\pause\\
Uniqueness comes from monotonicity ($\beta_{j,m}$ ``increases'' with~$m$)\kern-5pt\pause
\vskip6pt
\alert{Special case.} The \claim{volume} of a class $\alpha\in H^{1,1}(X,\bR)$ is
\vskip3pt
\alert{\centerline{$\displaystyle
\Vol(\alpha)=\sup_{T\in\alpha}\langle T^n\rangle\quad
\hbox{if $\alpha\in\cE^\circ$ (big class)},$}}
\vskip3pt
\alert{\centerline{$\displaystyle
\Vol(\alpha)=0\kern2.6cm
\hbox{if $\alpha\not\in\cE^\circ$},$}}\pause
\begin{block}{Numerical dimension of a current}
$$\nd(T)=\max\big\{p\in\bN\,;\;\langle T^p\rangle\ne 0\quad
\hbox{in $H^{p,p}_{\ge 0}(X)\big\}$.}$$
\end{block}
\pause
\begin{block}{Numerical dimension of a hermitian line bundle $(L,h)$}
$$\nd(L,h)=\nd(\Theta_{L,h}).$$
\end{block}
\end{frame}


\begin{frame} 
\frametitle{Generalized abundance conjecture}
\vskip-6pt
\begin{block}{Numerical dimension of a class $\alpha\in H^{1,1}(X,\bR)$}
If $\alpha$ is \alert{not pseudoeffective}, set \alert{$\nd(\alpha)=-\infty$}, 
otherwise\vskip3pt
\alert{\centerline{$\displaystyle\nd(\alpha)=\max\big\{p{\in}\bN\,;\,\exists T_\varepsilon{\in}\{\alpha{+}\varepsilon\omega\},\,\lim_{\varepsilon\to 0}
\langle T_\varepsilon^p\rangle\wedge\omega^{n-p}\ge C{>}0\big\}$.}}
\end{block}
\pause

\begin{block}{Numerical dimension of a pseudo-effective line bundle}
\centerline{$\nd(L)=\nd(c_1(L)).$}
$L$ is said to be \claim{abundant} if $\kappa(L)=\nd(L)$.
\end{block}\pause

\alert{Subtlety !}
Let $E$ be the rank 2 v.b.${}={}$non trivial extension $0\to\cO_C\to E\to\cO_C\to 0$ on $C={}$elliptic curve, let $X=\bP(E)$\\
(ruled surface over $C$) and $L=\cO_{\bP(E)}(1)$. Then
\alert{$\nd(L)=1$} but\\
$\exists\,!$ positive current $T=[\sigma(C)]
\in c_1(L)$ and \alert{$\nd(T)=0$}~!!
\pause

\begin{block}{Generalized abundance conjecture} 
For $X$ compact K\"ahler, \claim{$K_X$ is
abundant}, i.e.\ $\kappa(X)=\nd(K_X)$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Hard Lefschetz theorem with pseudoeffective coefficients}
Let $(L,h)$ be a pseudo-effective line bundle
on a compact K\"ahler manifold $(X,\omega)$ of dimension~$n$, and
for $h=e^{-\varphi}$, let $\cI(h)=\cI(\varphi)$ be the \claim{multiplier
ideal sheaf}:
\alert{$$
\cI(\varphi)_x:=\big\{f\in\cO_{X,x}\,;\,\exists V\ni x,\,\int_V|f|^2e^{-\varphi}
dV_\omega<+\infty\big\}.
$$}\pause
The \claim{Nadel vanishing theorem} claims that 
\alert{$$
\Theta_{L,h}\ge\varepsilon\omega~~\Longrightarrow~~
H^q(X,K_X\otimes L\otimes\cI(h)=0~~\hbox{for $q\ge 1$}.
$$}\pause\vskip-15pt

\begin{block}{Hard Lefschetz theorem (D-Peternell-Schneider 2001)}
Assume merely $\Theta_{L,h}\ge 0$. Then, the
Lefschetz map${}:$\\
$u\mapsto \omega^q\wedge u$ induces
a \alert{surjective morphism}$\,:$\vskip4pt
\alert{\centerline{$\Phi^q_{\omega,h}:
H^0(X,\Omega_X^{n-q}\otimes L\otimes\cI(h))\longrightarrow
H^q(X,\Omega_X^n\otimes L\otimes\cI(h)).$}}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Idea of proof of Hard Lefschetz theorem}
\claim{Main tool. ``Equisingular approximation theorem'':}
\alert{$$\varphi=\lim\downarrow\varphi_\nu~~\Rightarrow~~h=\lim h_\nu$$}
with:
\begin{itemize}
\item
\alert{$\varphi_\nu\in C^\infty(X\ssm Z_\nu)$}, where $Z_\nu$ is an 
increasing sequence of analytic sets,
\item \alert{$\cI(h_\nu)=\cI(h)$,~~ $\forall\nu$},
\item \alert{$\Theta_{L,h_\nu}\ge -\varepsilon_\nu\omega$}.
\end{itemize}
(Again, the proof uses in several ways the Ohsawa-Takegoshi theorem).
\pause\vskip4pt
Then, use the fact that $X\ssm Z_\nu$ is K\"ahler complete, so one can apply
(non compact) \claim{harmonic form theory} on $X\ssm Z_\nu$, and pass to the limit
to get rid of the errors $\varepsilon_\nu$.
\end{frame}

\begin{frame}
\frametitle{Generalized Nadel vanishing theorem}
\vskip-6pt
\begin{block}{Theorem (Junyan Cao, PhD 2012)}
Let $X$ be compact K\"ahler, and let $(L,h)$ be pseudoeffective on~$X$. Then
\vskip3pt
\alert{\centerline{$H^q(X,K_X\otimes L\otimes\cI_+(h))=0~~
\hbox{for $q\ge n-\nd(L,h)+1$},
$}}\vskip3pt
where\vskip0pt
\centerline{$\cI_+(h)=\lim_{\varepsilon\to 0}\cI(h^{1+\varepsilon})=
\lim_{\varepsilon\to 0}\cI((1+\varepsilon)\varphi)$}
\vskip4pt
is the \claim{``upper semicontinuous regularization''} of $\cI(h)$.
\end{block}\pause

\claim{Remark 1}. Conjecturally \alert{$\cI_+(h)=\cI(h)$}.
This might \hbox{follow~from\kern-5pt}
recent work by Bo Berndtsson on the openness
conjecture.\vskip3pt\pause

\claim{Remark 2}. In the projective case, one can use a
hyperplane section argument, provided one first shows that \alert{$\nd(L,h)$}
coincides with H.~Tsuji's \alert{algebraic definition}~($\dim Y=p$)~:
\alert{$$
\nd(L,h)=\max\big\{p{\in}\bN\,;\,\exists Y^p{\subset}X,\,
h^0(Y,(L^{\otimes m}\otimes\cI(h^m))_{|V})\ge c m^p\big\}.
$$}
\end{frame}

\begin{frame}
\frametitle{\strut\kern-12pt Proof of generalized Nadel vanishing$\,$\hbox{(projective case)\kern-15pt}}
\claim{Hyperplane section argument (projective case)}. Take $A={}$very ample
divisor, $\omega=\Theta_{A,h_A}>0$, and $Y=A_1\cap\ldots\cap A_{n-p}$, 
$A_j\in|A|$. Then\vskip3pt
\alert{\centerline{$\displaystyle
\langle \Theta_{L,h}^p\rangle\cdot Y
=\int_X\langle\Theta_{L,h}^p\rangle\cdot Y=\int_X
\langle\Theta_{L,h}^p\rangle\wedge\omega^{n-p}>0.$}}\vskip6pt
From this one concludes that $(\Theta_{L,h})_{|Y}$ is big.\pause

\begin{block}{Lemma (J.\ Cao)}
When $(L,h)$ is big, i.e.\ $\langle \Theta_{L,h}^n\rangle>0$, 
there exists a metric $\widetilde h$ such that 
\alert{$\cI(\widetilde h)=\cI_+(h)$}
with \alert{$\Theta_{L,\widetilde h}\ge\varepsilon\omega$}~~~[Riemann-Roch].
\end{block}
Then \alert{Nadel $\Rightarrow$ $H^q(X,K_X\otimes L\otimes\cI_+(h))=0$
for $q\ge 1$}.\pause\vskip3pt
Conclude by \alert{induction on $\dim X$} and the exact cohomology sequence for
the restriction to a \alert{hyperplane section}.
\end{frame}

\begin{frame}
\frametitle{Proof of generalized Nadel vanishing \hbox{(K\"ahler case)\kern-15pt}}
\claim{K\"ahler case}. Assume $c_1(L)$ nef for simplicity. Then \alert{$c_1(L)+\varepsilon\omega$} K\"ahler. By Yau's theorem, solve \claim{Monge-Amp\`ere equation}:\vskip3pt
\alert{\centerline{$\exists h_\varepsilon~\hbox{on $L$},~~~
(\Theta_{L,h_\varepsilon}+\varepsilon\omega)^n=C_\varepsilon\omega^n.
$}}\vskip3pt
Here $C_\varepsilon\ge{n\choose p}\langle\Theta_{L,h}^p\rangle \cdot(\varepsilon\omega)^{n-p}\sim C\varepsilon^{n-p}$, $p=\nd(L,h)$.
\pause\vskip4pt
\claim{Ch. Mourougane argument (PhD 1996)}.
Let $\lambda_1\le\ldots\le\lambda_n$ be the eigenvalues of $\Theta_{L,h}+\varepsilon\omega$ w.r.to $\omega$. Then
\vskip3pt
\alert{\centerline{%
$\lambda_1\ldots\lambda_n=C_\varepsilon\ge \hbox{Const}~\varepsilon^{n-p}$}}
\vskip3pt and\vskip3pt
\alert{\centerline{$\displaystyle\int_X\lambda_{q+1}\ldots\lambda_n\;\omega^n=
\int_X\Theta_{L,h}^{n-q}\wedge\omega^q\le\hbox{Const},~~~\forall q\ge 1,
$}}\vskip3pt\pause
so $\lambda_{q+1}\ldots\lambda_n\le C$ on a large open set $U\subset X$ and
\vskip3pt
\centerline{$\displaystyle\alert{
\lambda_q^q\ge \lambda_1\ldots\lambda_q\ge c\varepsilon^{n-p}}~~\Rightarrow~~
\alert{\lambda_q\ge c\varepsilon^{(n-p)/q}~~
\hbox{on $U$}},$}
\vskip3pt
\alert{\centerline{$\sum_{j=1}^q(\lambda_j-\varepsilon)\ge\lambda_q-q\varepsilon
\ge c\varepsilon^{(n-p)/q}-q\varepsilon>0~~
\hbox{for $q>n-p$}.
$}}
\end{frame}

\begin{frame}
\frametitle{Final step: use Bochner-Kodaira formula}
$\lambda_j={}$eigenvalues of $(\Theta_{L,h_\varepsilon}{+}\varepsilon\omega)$
$\Rightarrow$
\hbox{(eigenvalues of $\Theta_{L,h_\varepsilon})=\alert{\lambda_j-\varepsilon}$.\kern-30pt}%
\pause\vskip4pt
Bochner-Kodaira formula yields
\alert{$$
\Vert \partial u\Vert^2_\varepsilon+\Vert \partial^* u\Vert^2_\varepsilon\ge
\int_X\Big(\sum_{j=1}^q(\lambda_j-\varepsilon)\Big)|u|^2e^{-\varphi_\varepsilon}
dV_\omega.$$}\pause
Then one has to show that one can take the limit by assuming integrability
with $e^{-(1+\delta)\varphi}$, thus introducing $\cI_+(h)$.
\end{frame}

\begin{frame}
\frametitle{Application to K\"ahler geometry}
\begin{block}{Definition (Campana)}A compact K\"ahler manifold is said to be
\claim{simple} if there are no positive dimensional analytic sets $A_x\subset X$ through a very generic point $x\in X$.
\end{block}\pause
\begin{block}{Well-known fact}A complex torus $X=\bC^n/\Lambda$ defined by
a sufficiently generic lattice $\Lambda\subset\bC^n$ \alert{is simple}, and
in fact has no positive dimensional analytic subset $A\subsetneq X$ at all.
\end{block}
\pause
In fact $[A]$ would define a non zero $(p,p)$-cohomology class with 
integral periods, and there are no such classes in general.\pause\vskip4pt
It is expected that simple compact K\"ahler manifolds are either \alert{generic  complex tori}, \alert{generic hyperk\"ahler manifolds} and their \alert{finite quotients}, up to modification.
\end{frame}

\begin{frame}
\frametitle{On simple K\"ahler 3-folds}
\vskip-7pt
\begin{block}{Theorem (Campana - D - Verbitsky, 2013)} Let $X$ be a compact K\"ahler 3-fold without any positive dimensional analytic subset $A\subsetneq X$.
Then\\
\alert{$X$ is a complex 3-dimensional torus}.
\end{block}\pause
\claim{Sketch of proof}
\begin{itemize}
\item Every pseudoeffective class is nef, i.e.\ 
\alert{$\overline\cK=\cE$} (D, '90)\pause
\item \alert{$K_X$ is pseudoeffective}: otherwise $X$ would be covered by rational curves (Brunella 2008), hence in fact nef.
\item All multiplier ideal sheaves $\cI(h)$ are \alert{trivial}\pause
\item $H^0(X,\Omega_X^{n-q}\otimes K_X^{\otimes m-1})\to
H^q(X,K_X^{\otimes m})$ is \alert{surjective}\pause
\item Hilbert polynomial $P(m)=\chi(X,K_X^{\otimes m})$ is bounded, hence
\alert{$\chi(X,\cO_X)=0$}.\pause
\item Albanese map \alert{$\alpha:X\to{\rm Alb}(X)$ is a biholomorphism}.
\end{itemize}
\end{frame}

\strut\vskip-7pt
\claim{\huge\bf References}
\setbibliopages
\vskip7pt

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