\documentclass[12pt]{beamer}

% 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{pgf,pgfnodes,pgfautomata,pgfheaps,pgfshade}
\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.

\definecolor{ColClaim}{rgb}{0,0,0.8}
\def\claim#1{{\color{ColClaim}#1}}

\definecolor{Alert}{rgb}{0.8,0,0}
\def\alert#1{{\color{Alert}#1}}


\title[\ \kern-190pt Jean-Pierre Demailly (Grenoble I), 18/12/2008\kern53pt
Morse inequalities and volume of (1,1) classes]
% (optional, use only with long paper titles)
{Holomorphic Morse inequalities and volume of (1,1) cohomology classes}

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

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

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

\date[]% (optional)
{CMI, Chennai, December 18, 2008
\vskip12pt
\alert{RMS-SMF-IMSc-CMI Conference\\
held in Chennai, December 15--19, 2008}}

%%\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:

\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\tors}{\operatorname{torsion}}

\newcommand{\bC}{{\mathbb C}}
\newcommand{\bN}{{\mathbb N}}
\newcommand{\bP}{{\mathbb P}}
\newcommand{\bQ}{{\mathbb Q}}
\newcommand{\bR}{{\mathbb R}}
\newcommand{\bZ}{{\mathbb Z}}

\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{\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{\Bbb 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}

\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=1cm]{ujf-logo}{logo_ujf}
  \pgfuseimage{ujf-logo}
  \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{}


%%\section{Positive cones}

\begin{frame}
\frametitle{Positive cones}
\claim{{\bf Definition.} Let $X$ be a compact K\"ahler manifold.}
\begin{itemize}
\item \claim{The \alert{K\"ahler cone} is the set 
\alert{$\cK\subset H^{1,1}(X,\bR)$} of cohomology classes $\{\omega\}$
of K\"ahler forms. This is an open convex cone.}
\pause
\smallskip
\item \claim{The \alert{pseudo-effective} cone is the set
\alert{$\cE\subset H^{1,1}(X,\bR)$} 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
\item \claim{Always true: \alert{$\ovl\cK\subset\cE$}.}
\pause
\item \claim{One can have: \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{$\int_E\alpha^2= E^2=-1$},\\
hence $\{\alpha\}\notin\ovl\cK$, although
$\{\alpha\}=\{[E]\}\in\cE$.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{K\"ahler (red) cone and pseudoeffective (blue) cone}
\InsertFig 50 67
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\LabelTeX -3 48 $\cK$\ELTX
\LabelTeX -19 29 $\cE$\ELTX
\LabelTeX -38.5 55 $H^{1,1}(X,\bR)$\ELTX
\EndFig
\end{frame}

\begin{frame}
\frametitle{Neron Severi parts of the cones}
In case $X$ is projective, it is interesting to consider the
``algebraic part'' of our ``transcendental cones'' $\cK$ and $\cE$.
which consist of suitable integral divisor classes. 
\vskip3pt
Cohomology classes of algebraic divisors live in $H^2(X,\bZ)$.
\pause

\begin{itemize}
\item
\claim{{\bf Neron-Severi lattice and Neron-Severi space}}
\alert{%
\begin{eqnarray*}
\NS(X)&:=& H^{1,1}(X,\bR)\cap \big(H^2(X,\bZ)/\{\tors\}\big),\\
\NS_\bR(X)&:=&\NS(X)\otimes_\bZ\bR.
\end{eqnarray*}
\pause
\item
\claim{{\bf Algebraic parts of $\cK$ and $\cE$}}
\begin{eqnarray*}
\cK_{\NS}&:=&\cK\cap \NS_\bR(X),\\
\cE_{\NS}&:=&\cE\cap \NS_\bR(X).
\end{eqnarray*}}
The rest we refer to as the ``transcendental part''
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Neron Severi parts of the cones}
\InsertFig 58 67
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\LabelTeX -3 46 $\cK_{\NS}$\ELTX
\LabelTeX -19 27 $\cE_{\NS}$\ELTX
\LabelTeX -38.5 55 $\NS_\bR(X)$\ELTX
\LabelTeX -58 65 $H^{1,1}(X,\bR)$\ELTX
\EndFig
\end{frame}

\begin{frame}
\frametitle{ample / nef / effective / big divisors}
{\it
\claim{{\bf Theorem} {\rm (Kodaira+successors, D90)}. Assume $X$ projective.
\vskip-4pt
\begin{itemize}
\item \claim{$\cK_{\NS}$ is the open cone generated by
\alert{ample} $($or \alert{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$)$.}\pause
\item \claim{The closed cone $\ovl\cK_{\NS}$ consists of the closure
of the cone of \alert{nef divisors} $D$ (or nef line bundles $L$), 
namely effective integral divisors $D$ such that $D\cdot C\ge 0$ for
every curve $C$.}\pause
\item \claim{$\cE_{\NS}$ is the closure of the cone of
\alert{effective divisors}, i.e.\ divisors $D=\sum c_jD_j$, $c_j\in\bR_+$.}
\pause
\item \claim{The interior $\cE_{\NS}^\circ$ is the cone of
\alert{big divisors}, namely divisors $D$ such that
$h^0(X,\cO(kD))\ge c\,k^{\dim X}$ for $k$ large.}\vskip0pt
\end{itemize}}}
\pause
Proof: $L^2$ estimates for $\dbar$ / Bochner-Kodaira technique
\end{frame}

\begin{frame}
\frametitle{ample / nef / effective / big divisors}
\InsertFig 45 67
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\LabelTeX -3 46 $\cK_{\NS}$\ELTX
\LabelTeX -19 27 $\cE_{\NS}$\ELTX
\LabelTeX -38.5 55 $\NS_\bR(X)$\ELTX
\LabelTeX 32 54 ample\ELTX
\LabelTeX 32 44 nef\ELTX
\LabelTeX 42 34 big\ELTX
\LabelTeX 42 24 (pseudo)\ELTX
\LabelTeX 42 19 effective\ELTX
\EndFig
\end{frame}


\begin{frame} 
\frametitle{Characterization of the K\"ahler cone}
\claim{{\bf Theorem} {\rm (Demailly-Paun 2004)}.\\
{\it Consider the ``numerically positive cone'' 
\alert{$$
\cP=\big\{\alpha\in H^{1,1}(X,\bR)\,;\;\int_Y\alpha^p>0\big\}
$$}
where $Y\subset X$ irreducible analytic subset, $\dim Y=p$.\\
The K\"ahler cone $\cK$ is one of the \alert{connected components
of $\cP$}.}}
\medskip
\pause

\claim{{\bf Corollary} {\rm (DP2004)}. {\it
Let $X$ be a compact K\"ahler manifold. 
\alert{
$\displaystyle
\alpha\in H^{1,1}(X,\bR)~~\hbox{\it is nef}~(\alpha\in\ovl\cK)
\Leftrightarrow{}$\\
$\displaystyle
\int_Y\alpha\wedge\omega^{p-1}\ge 0$,
$\forall \omega$ K\"ahler, $\forall Y\subset X$ irreducible, $\dim Y=p$.}}}
\pause
\medskip

\claim{{\bf Re-interpretation.} {\it \alert{the dual of the
nef cone $\smash{\ovl\cK}$} is the closed convex cone 
in $H^{n-1,n-1}_\bR(X)$ generated by cohomology 
classes of currents of the form \alert{$[Y]\wedge\omega^{p-1}$ in 
$H^{n-1,n-1}(X,\bR)$.}}}\\
\phantom{\strut}
\end{frame}


\begin{frame}
\frametitle{Duality theorem for $\cK$}
\InsertFig 23.5 60
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\LabelTeX 1 52 $\cK$\ELTX
\LabelTeX -5 46 $\cK_{\NS}$\ELTX
\LabelTeX 15 31 $\cE$\ELTX
\LabelTeX -19 26 $\cE_{\NS}$\ELTX
\LabelTeX -26 4 $\NS_\bR(X)$\ELTX
\LabelTeX 55 53 \claim{$???????$}\ELTX
\LabelTeX 47 40 $\cN=\langle\{[Y]\cap\omega^{p-1}\}\rangle$\ELTX
\LabelTeX 47 25 $\cN_{\NS}=\langle\hbox{curves}\rangle$\ELTX
\LabelTeX 70 4 $\NS^{n-1}_\bR(X)$\ELTX
\LabelTeX 25 47 duality\ELTX
\LabelTeX -8 -6 (divisors)\kern1.95cm $\alpha\cdot\Gamma$
\kern 1.95cm(curves)\ELTX
\LabelTeX -8 -11 $H^{1,1}(X,\bR)$~~$\leftarrow$ Serre duality $\to$~~ $H^{n-1,n-1}(X,\bR)$\ELTX
\EndFig
\end{frame}

\begin{frame} 
\frametitle{Variation of complex structure}
Suppose $\pi:\cX\to S$ is a \alert{deformation} of compact K\"ahler 
manifolds. Put $X_t=\pi^{-1}(t)$, $t\in S$ and let
\alert{$$
\nabla=\begin{pmatrix}
\nabla^{2,0} & * & 0\cr 
* & \nabla^{1,1} & * \cr
0 & * & \nabla^{0,2}\cr
\end{pmatrix}
$$}
be the Gauss-Manin connection on the Hodge bundle $t\mapsto H^2(X_t,\bC)$,
relative to the decomposition $H^2=H^{2,0}\oplus H^{1,1}\oplus H^{0,2}$.
\pause
\medskip

\claim{{\bf Theorem} {\rm (Demailly-P\v{a}un 2004)}. {\it
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 \alert{$\cK_t\subset H^{1,1}(X_t,\bC)$ of the fibers
$X_t=\pi^{-1}(t)$ are $\nabla^{1,1}$-invariant} over $S\ssm S'$ under 
parallel transport with respect to $\nabla^{1,1}$.}}
\end{frame}

\begin{frame}
\frametitle{Approximation of currents, Zariski decomposition}
\begin{itemize}
\item
\claim{{\bf Definition.} {\it 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 $\delta\ll 1$.}}
\pause
\medskip

\item 
\claim{{\bf Theorem.} \alert{\it $\alpha\in\cE^\circ\Leftrightarrow \alpha\ni
T$, a K\"ahler current.}}
\medskip

We say that $\cE^\circ$ is the cone of \alert{big $(1,1)$-classes}.
\pause
\medskip

\item
\claim{{\bf Theorem} (D92). {\it
Any K\"ahler current $T$ can be written
$$T=\lim T_m$$ 
where $T_m\in \alpha=\{T\}$ has \alert{logarithmic poles, i.e.\\
$\exists$ a modification \hbox{$\mu_m:\wt X_m\to X$} such that
$$\mu_m^\star T_m=[E_m]+\gamma_m$$}
where $E_m$ is an effective $\bQ$-divisor on $\wt X_m$ with coefficients
in ${1\over m}\bZ$ and $\gamma_m$ is a K\"ahler form on $\wt X_m$.}}
\vskip0pt
$\phantom{\strut}$
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Idea of proof of analytic Zariski decomposition (1)}
Locally one can write $T=i\ddbar\varphi$ for some strictly 
plurisubharmonic potential $\varphi$ on $X$. The approximating potentials
$\varphi_m$ of $\varphi$ are defined as
\alert{%
$$
\varphi_m(z)={1\over 2m}\log\sum_\ell |g_{\ell,m}(z)|^2
$$}%
where $(g_{\ell,m})$ is a Hilbert basis of the space 
\alert{%
$$
\cH(\Omega,m\varphi)=\big\{
f\in\cO(\Omega)\,;\;\int_\Omega |f|^2 e^{-2m\varphi}dV<+\infty\big\}.
$$}%
The Ohsawa-Takegoshi $L^2$ extension theorem 
(applied to extension from a single isolated point) implies that there
are enough such holomorphic functions, and thus $\varphi_m\geq\varphi-C/m$. 
On the other hand $\varphi=\lim_{m\to+\infty}\varphi_m$ by a Bergman kernel 
trick and by the mean value inequality.
\end{frame}

\begin{frame}
\frametitle{Idea of proof of analytic Zariski decomposition (2)}
The Hilbert basis $(g_{\ell,m})$ is a family of local generators of the
multiplier ideal sheaf $\cI(mT)=\cI(m\varphi)$. The
modification $\mu_m:\wt X_m\to X$ is obtained by blowing-up this ideal
sheaf, with
\alert{$$\mu_m^\star\cI(mT)=\cO(-mE_m).$$}%
for some effective $\bQ$-divisor $E_m$ with normal crossings on $\wt X_m$. 
Now, we set 
$T_m=i\ddbar\varphi_m$ and $\gamma_m=\mu_m^*T_m-[E_m]$.
Then $\gamma_m=i\ddbar\psi_m$ where 
$$
\psi_m={1\over 2m}\log\sum_\ell|g_{\ell,m}\circ\mu_m/h|^2
\quad\hbox{locally on $\wt X_m$}
$$
and $h$ is a generator of $\cO(-mE_m)$, and we see that $\gamma_m$ is a smooth
semi-positive form on $\wt X_m$. The construction can be made global
by using a gluing technique, e.g.\ via partitions of unity, and $\gamma_m$ can
be made K\"ahler by a perturbation argument.\\
$\phantom{\strut}$
\end{frame}

\begin{frame} 
\frametitle{Algebraic analogue}
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
\alert{%
$$
\mu_m^\star L\sim E_m+D_m,\qquad E_m~~\hbox{effective},~~D_m~~\hbox{free}.
$$}%
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 $L$.
\medskip

We will also use the terminology of \alert{``approximate Zariski
decomposition''} for the above decomposition of K\"ahler currents
with logarithmic poles.
\end{frame}

\begin{frame}
\frametitle{Analytic Zariski decomposition}
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\LabelTeX -3 46 $\cK_{\NS}$\ELTX
\LabelTeX -19 27 $\cE_{\NS}$\ELTX
\LabelTeX -38.5 55 $\NS_\bR(\wt X_m)$\ELTX
\LabelTeX -11 17 $\wt\alpha$\ELTX
\LabelTeX -15 6 $[E_m]$\ELTX
\LabelTeX -2.3 8.8 $\gamma_m$\ELTX
\LabelTeX 21 17 $\wt\alpha=\mu_m^\star\alpha=[E_m]+\gamma_m$\ELTX
\EndFig
\end{frame}


\begin{frame}
\frametitle{Fujita approximation / concept of volume}

\claim{{\bf Theorem.} {\rm (Fujita 1994)} {\it If $L$ is a big line bundle
and $\mu_m^*(mL)=[E_m]+[D_m]$
$(E_m{=}$fixed part, $D_m{=}$moving part$)$
\alert{$$
\lim_{m\to+\infty}{n!\over m^n}h^0(X,mL)=
\lim_{m\to+\infty} D_m^n.
$$}}}
\vskip-12pt

This quantity will be called \alert{$\Vol(c_1(L))$}. More generally~:
\medskip
\pause

\claim{{\bf Definition} (\rm Boucksom 2002). {\it Let
$\alpha\in\cE^\circ$ be a big class\\
The \alert{volume} $($\alert{movable self-intersection}$)$ of $\alpha$ is
\alert{$$
\Vol(\alpha)=\sup_{T\in \alpha}\int_{\wt X}\gamma^n>0
$$}
with K\"ahler currents $T\in \alpha$ with log poles,\\
and $\mu^\star T=[E]+\gamma$ where $\mu:\wt X\to X$ modification.}}
\vskip4pt
\pause
If $\alpha\in\cK$, then $\Vol(\alpha)=\alpha^n=\int_X\alpha^n$.
\vskip4pt
\pause

\claim{{\bf Theorem} (\rm Boucksom 2002). \alert{$\alpha$ contains $T_{\min}$ 
and
$\Vol(\alpha)=\lim_{m\to+\infty}\int_X\gamma_m^n$ for the approximation
of $T_{\min}$.}}

\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{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{$$
\mu_m^\star T_j=[E_{j,m}]+\gamma_{j,m}.
$$}%
We define
\alert{$$
\langle\alpha_1\cdot\alpha_2\cdots \alpha_k\rangle =
\mathop{\lim\uparrow}\limits_{m\to+\infty}
\{(\mu_m)_\star(\gamma_{1,m}\wedge\gamma_{2,m}\wedge\ldots\wedge\gamma_{k,m})\}
$$}%
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{Transcendental Holomorphic Morse inequalities}

\claim{{\bf Conjecture.} {\it For any class $\alpha\in H^{1,1}(X,\bR)$
and $\theta\in\alpha$ smooth
\alert{
$$
\Vol(\{\alpha\})\ge\int_{X(\theta,\le 1)}\theta^n
$$}
where $\Vol(\alpha):=0$ if $\alpha\notin\cE^\circ$ and
\alert{
\begin{eqnarray*}
X(\theta,q)&=&\big\{x\in X\,;\;\theta(x)~\hbox{has signature $(n-q,q)$}\big\}\\
X(\theta,\le q)&=&\textstyle\bigcup_{0\le j\le q}X(\theta,j).
\end{eqnarray*}}}}
\vskip-8pt
\pause

\claim{{\bf Theorem (D 1985)} {\rm (Holomorphic Morse inequalities)}\\
The above is true when $\alpha=c_1(L)$ is integral. Then, with 
$\theta={i\over 2\pi}\Theta_{L,h}\in\alpha$
\alert{$$
H^0(X,L^{\otimes k})\ge {k^n\over n!}\int_{X(\theta,\le 1)}\theta^n-o(k^n)
$$}}

(and more generally, bounds for all $H^q(X,L^{\otimes k})$ hold true).

\end{frame}

\begin{frame}
\frametitle{Three equivalent properties}

\claim{{\bf Lemma.} {\it $A$, $B$ nef divisors on $X$ projective. Then
\alert{
$$
\Vol(A-B)\ge A^n-nA^{n-1}\cdot B.
$$}}}
\vskip-15pt
Elementary / easy corollary of Morse inequalities.
\pause
\medskip

\claim{{\bf Theorem.} {\it Let $X$ be compact K\"ahler. We have~~~
$\Longleftrightarrow$
\vskip3pt
{\rm (1)} $\forall \alpha,\beta\in\ovl\cK$, \alert{$\Vol(\alpha-\beta)\ge
\alpha^n-n\alpha^{n-1}\cdot\beta$.} {\rm (Weak Morse)}
\vskip3pt
{\rm (2)} $\forall \alpha,\beta\in\cE$, 
\alert{$\Vol(\alpha-\beta)\ge
\Vol(\alpha)-n\int_0^1\langle \alpha-t\beta\rangle^{n-1}\cdot\beta\,dt$.}
\vskip3pt
{\rm (3)} \alert{Orthogonality property~:}
Let $\alpha=\{T\}\in\cE^\circ$ big, and\\
$\mu_m^\star T_m = [E_m]+\gamma_m$ approximate Zariski decomposition.\\
Then \alert{$\gamma_m^{n-1}\cdot E_m\to 0$ as $\Vol(\gamma_m)\to\Vol(\alpha)$.}
}}
\vskip5pt

{\bf Proof}. $(2)\Rightarrow(1)$ obvious.\\
What remains to show is : $(1)\Rightarrow(3)$, $(3)\Rightarrow(2)$.
\end{frame}

\begin{frame}
\frametitle{Morse implies orthogonality}

$(1)\Rightarrow (3)$.
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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\EndFig
\end{frame}

\begin{frame}
\frametitle{Orthogonality implies differential estimate}
$(3)\Rightarrow (2)$: Take a parametrized) 
approximate Zariski Decomposition~:
$$\mu^*(\alpha-t\beta)\ni [E_t]+\gamma_t$$
where $E_t=\sum c_j(t)E_j$. Take $d/dt$~:
$$
-\mu^*\beta\ni\sum \dot c_j(t) E_j+\dot\gamma_t
$$
while
$$
\Vol(\alpha-t\beta)\simeq\int_{\wt X}\gamma_t^n,~~~
{d\over dt}\Vol(\alpha-t\beta)\simeq
n\int_{\wt X}\gamma_t^{n-1}\dot\gamma_t.
$$
Since $\int_{\wt X}\gamma_t^{n-1}\cdot E_j$ small (by orthogonality), we get
$$
{d\over dt}\Vol(\alpha-t\beta)\simeq
n\int_{\wt X}\gamma_t^{n-1}\cdot(-\mu^*\beta)=
-n\int_{\wt X}\mu_*(\gamma_t^{n-1})\cdot\beta\Rightarrow
$$
$$
{d\over dt}\Vol(\alpha-t\beta)\simeq
-n\int_{\wt X}\langle(\alpha-t\beta)^{n-1}\rangle\cdot\beta.
$$
\end{frame}

\begin{frame}
\frametitle{Positive cones in $H^{n-1,n-1}(X)$ and Serre duality}

\claim{{\bf Definition.} {\it Let $X$ be a compact K\"ahler manifold.}}
\begin{itemize}
\item \claim{\it Cone of $(n-1,n-1)$ positive currents\\
\alert{$\cN=\ovl{\hbox{cone}}\big\{\{T\}\in H^{n-1,n-1}(X,\bR)\,;\;
\hbox{$T$ closed${}\ge 0$}\big\}.$}}
\smallskip

\item \claim{\it Cone of effective curves
$\phantom{\strut}$\vskip-25pt$\phantom{\strut}$
\alert{\begin{align}
\cN_{\NS}
&=\cN\cap \NS_\bR^{n-1,n-1}(X),\cr
&=\ovl{\hbox{cone}}\big\{\{C\}\in H^{n-1,n-1}(X,\bR)\,;\;
\hbox{$C$ effective curve}\big\}.
\nonumber
\end{align}}
$\phantom{\strut}$\vskip-36pt$\phantom{\strut}$}%

\item \claim{\it Cone of movable curves : with $\mu:\wt X\to X$, let\\
\alert{$\cM_{\NS}=
\ovl{\hbox{cone}}\big\{\{C\}\in H^{n-1,n-1}(X,\bR)\,;\;
[C]=\mu_\star(H_1\cdots H_{n-1})\big\}$\kern-40pt}
where $H_j={}$ample hyperplane section of $\wt X$.}
\smallskip

\item \claim{\it Cone of movable currents : with $\mu:\wt X\to X$, let\\
\alert{$\cM=
\ovl{\hbox{cone}}\big\{\{T\}\in H^{n-1,n-1}(X,\bR)\,;\;
T=\mu_\star(\wt\omega_1\wedge\ldots\wedge\wt\omega_{n-1})\big\}$\kern-20pt}\\
where $\wt\omega_j={}$K\"ahler metric on $\wt X$.}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Main duality theorem}
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\LabelTeX 1 52 $\cK$\ELTX
\LabelTeX -5 46 $\cK_{\NS}$\ELTX
\LabelTeX 15 32 $\cE$\ELTX
\LabelTeX -19 27 $\cE_{\NS}$\ELTX
\LabelTeX -26 4 $\NS_\bR(X)$\ELTX
\LabelTeX 57 46 $\cM_{\NS}$\ELTX
\LabelTeX 63 52 $\cM$\ELTX
\LabelTeX 77 32 $\cN$\ELTX
\LabelTeX 43 27 $\cN_{\NS}$\ELTX
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\LabelTeX 25 47 duality\ELTX
\LabelTeX -8 -11 $H^{1,1}(X,\bR)$~~$\leftarrow$ Serre duality $\to$~~ $H^{n-1,n-1}(X,\bR)$\ELTX
\EndFig
\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 $\cE_{\NS}$
$($pseudo-effective$)$ 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 65 38 $\cM_{\NS}$\ELTX
\LabelTeX -9 15 $\alpha-\varepsilon\omega$\ELTX
\LabelTeX -8.5 24 $\alpha$\ELTX
\LabelTeX -8.5 27.5 $\alpha+\delta\omega$\ELTX
\LabelTeX -8.5 37 $\omega$\ELTX
\LabelTeX 59 24.5 $\Gamma$\ELTX
\LabelTeX 68 4 $\cN^{n-1}_{\NS}(X)$\ELTX
\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 $X$ has \alert{$K_X$ pseudo-effective}, 
i.e.\ $K_X\in\cE_{\NS}$,
if and only if $X$ is \alert{not uniruled}.}}
\medskip
\pause

{\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.
\vskip8pt

\claim{{\bf Conjecture.} {\rm (BDPP 2004)} The same is expected to be
true for \alert{$X$ compact K\"ahler}.}

\end{frame}

\begin{frame} 
\frametitle{Weak K\"ahler Morse inequalities (new approach)}

\claim{{\bf Theorem} {\rm (D 2008)} Let $X$ be compact K\"ahler,
$\gamma$ a K\"ahler class on $X$ and $E=\sum c_j E_j\ge 0$ a divisor
with normal crossings. Then, if $\Vol_{X|Y}$ denotes the ``restricted''
volume on~$Y$ (``sections'' on $Y$ which extend to $X$)
\vskip4pt
\alert{
$\displaystyle
\Vol\big(\gamma+\sum c_jE_j\big)\ge\Vol(\gamma)
+n\sum_j\int_0^{c_j}\Vol_{X|E_j}(\gamma+t E_j)\,dt$\\
$\displaystyle~~~{}+n(n-1)\sum_{j<k}\int_0^{c_j}\int_0^{c_k}
\Vol_{X|E_j\cap E_k}(\gamma+t_j E_j+t_kE_k)\,dt_j dt_k$\\
$\displaystyle~~~{}+n(n-1)(n-2)\sum_{j<k<\ell}\int_0^{c_j}\int_0^{c_k}\int_0^{c_\ell}
\Vol_{X|E_j\cap E_k\cap E_\ell}~\ldots$}}
\vskip4pt
\pause

The proof relies on pluripotential theory
\alert{(glueing psh functions)}.\kern-1cm
\medskip

This should imply the orthogonality estimate in the K\"ahler case,
and therefore also the duality theorem (work in progress).
\end{frame}

\end{document}