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% 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>
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% \setbeamertemplate{background canvas}[vertical shading][bottom=red!10,
% top=blue!10]
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% or whatever




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

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\title[\RGBColor{1 1 1}{\
\kern-190pt Kyoto Symposium SCV XIV, 20/07/2011\kern63pt
~~From the Ohsawa-Takegoshi theorem to asymptotic cohomology estimates}]
% (optional, use only with long paper titles)
{From the Ohsawa-Takegoshi theorem to asymptotic cohomology estimates}

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\def\alert#1{{\color{Alert}#1}}

%% \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)
{\alert{dedicated to Prof. Takeo Ohsawa and Tetsuo Ueda}\\
\alert{on the occasion of their sixtieth birthday}\vskip7mm
July 20, 2011 / Kyoto Symposium SCV XIV}

%%\subject{Talks}
% This is only inserted into the PDF information catalog. Can be left
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% mathematical operators
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\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}\kern10mm
  \pgfdeclareimage[height=2cm]{acad-logo}{academie_logo2}
  \pgfuseimage{acad-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{}

\begin{frame}
\frametitle{The Ohsawa-Takegoshi extension theorem}

\begin{block}{Theorem {\rm (Ohsawa-Takegoshi 1987), (Manivel 1993)}}
Let $(X,\omega)$ be a Kähler manifold, which is
\alert{compact or weakly pseudoconvex},
$n=\dim_\bC X$, $L\to X$ a hermitian line bundle, $E$ a hermitian 
holomorphic vector bundle, and $s\in H^0(X,E)$ s.$\,$t.\break
$Y=\big\{x\in X\,;\;s(x)=0, \Lambda^r ds(x)\not= 0\big\}$ is dense
in \alert{$\overline Y=\{s(x)=0\}$},  so that $p=\dim\overline Y=n-r$.
Assume that $\exists\alpha(x)\ge 1$
continuous s.$\,$t.\vskip3pt
{\rm (i)}~~~
\alert{$\displaystyle i\Theta_L+r\,i\,\ddbar\log|s|^2\ge\max\Big(0,
\alpha^{-1}{\{i\Theta_Es,s\}\over|s|^2}\Big)$}\vskip-4pt
{\rm (ii)}~~ \alert{$|s|\le e^{-\alpha}$.}\kern1cm\pause\claim{Then}\vskip6pt
$\forall f\in H^0(Y,(K_X\otimes L)_{|Y})$ s.$\,$t.
$\int_Y|f|^2|\Lambda^r(ds)|^{-2}dV_{Y,\omega}<+\infty$,\kern-15pt\break
$\exists F\in H^0(X,K_X\otimes L)$ s.$\,$t.\ $F_{|Y}=f$ and
\vskip-9pt
\alert{$$
\int_X{|F|^2\over|s|^{2r}(-\log|s|)^2}\,dV_{X,\omega}
\le C_r\int_Y{|f|^2\over|\Lambda^r(ds)|^2}dV_{Y,\omega}.
$$}
\end{block}
\end{frame}

\begin{frame}
\frametitle{The Ohsawa-Takegoshi extension theorem (II)}

\begin{block}{Theorem {\rm (Ohsawa-Takegoshi 1987)}}
Let $X=\Omega\compact\bC^n$ be a \alert{bounded pseudoconvex set},\\
$\varphi$ a \alert{plurisubharmonic} function on $\Omega$
and $Y=\Omega\cap S$ where $S$ is an affine linear subspace
of~$\bC^n$ of any codimension $r$.\vskip3pt\pause
For every $f\in H^0(Y,\cO_Y)$ such that
$\int_Y|f|^2e^{-\varphi}dV_Y<+\infty$,\kern-15pt\break
there exists $F\in H^0(\Omega,\cO_\Omega)$ s.$\,$t.\ $F_{|Y}=f$ and
\vskip3pt
\alert{\centerline{$\displaystyle
\int_\Omega|F|^2e^{-\varphi}\,dV_{\Omega}
\le C_r(\diam \Omega)^{2r}\int_Y|f|^2e^{-\varphi}dV_Y.$}}
\end{block}
\pause
Even the case when $Y=\{z_0\}$ is highly non trivial,
thanks to the $L^2$ estimate~: 
$\forall z_0\in \Omega$, $\exists F\in H^0(\Omega,\cO_\Omega)$, such that
\alert{$F(z_0)=C_n^{-1/2}(\diam \Omega)^{-n}e^{\varphi(z_0)/2}$} and
\alert{$$
\Vert F\Vert^2=\int_\Omega|F|^2e^{-\varphi}\,dV_{\Omega}\le 1.
$$}
\end{frame}

\begin{frame}
\frametitle{Local approximation of plurisubharmonic functions}
Let $\Omega\compact\bC^n$ be a bounded pseudoconvex set,\\
$\varphi$ a \alert{plurisubharmonic} function on $\Omega$. Consider
the Hilbert space\kern-10pt\vskip-10pt
\alert{$$
\cH(\Omega,m\varphi)=\big\{
f\in\cO(\Omega)\,;\;\int_\Omega |f|^2 e^{-2m\varphi}dV<+\infty\big\}.
$$}
\vskip-8pt\pause\noindent
One defines an ``approximating sequence'' of $\varphi$ by putting
\alert{
$$
\varphi_m(z)={1\over 2m}\log\sum_{j\in\bN}|g_{j,m}(z)|^2
$$}
where $(g_{j,m})$ is a Hilbert basis of $\cH(\Omega,m\varphi)$
\claim{(Bergman kernel procedure)}.\vskip3pt
\pause
If $\ev_z:\cH(\Omega,m\varphi)\to\bC$ is the evaluation linear form,
one also has\vskip-9pt
\alert{
$$
\varphi_m(z)={1\over m}\log\Vert\ev_z\Vert=
{1\over m}\sup_{f\in\cH(\Omega,m\varphi),\,\Vert f\Vert \le 1}\log|f(z)|.
$$}
\end{frame}

\begin{frame}
\frametitle{Local approximation of psh functions (II)}
The Ohsawa-Takegoshi approximation theorem implies
\alert{$$
\varphi_m(z)\ge \varphi(z)-{C_1\over m}
$$}
\pause
In the other direction, the mean value inequality gives\vskip2pt
\alert{\centerline{$\displaystyle
\varphi_m(z)\le\sup_{B(z,r)}\varphi+{n\over m}\log{C_2\over r},\qquad
\forall B(z,r)\subset \Omega$}}
\pause
\begin{block}{Corollary 1 {\rm (``strong psh approximation'')}}
One has \alert{$\lim\varphi_m=\varphi$} and the Lelong-numbers satisfy\vskip3pt
\alert{\centerline{$\displaystyle
\nu(\varphi,z)-{n\over m}\le\nu(\varphi_m,z)\le\nu(\varphi,z).$}}
\end{block}

\begin{block}{Corollary 2 {\rm (new proof of Siu's Theorem, 1974)}}
The Lelong-number sublevel sets 
\alert{$E_c(\varphi)=\{z\in \Omega\,;\;\nu(\varphi,z)\ge c\}$, $c>0$}
are analytic subsets.
\end{block}

\end{frame}


\begin{frame}
\frametitle{Approximation of global closed (1,1) currents}
Let $(X,\omega)$ be a compact Kähler manifold and 
\alert{$\{\alpha\}\in H^{1,1}(X,\bR)$}\kern-12pt\break
a~cohomology class
given by a smooth representative $\alpha$.\pause\vskip3pt
Let $T\in\{\alpha\}$ be an \alert{almost positive current},
i.e.\ a closed $(1,1)$-current such that
\vskip-12pt
\alert{$$T=\alpha+i\ddbar\varphi,\qquad T\ge\gamma$$}
\vskip-15pt
where $\gamma$ is a continuous $(1,1)$-form (e.g.\ $\gamma=0$ in case
$T\ge 0$).\kern-20pt\vskip3pt\pause%
One can write $T=\alpha+i\ddbar\varphi$ for some quasi-psh
potential $\varphi$ on $X$, with $i\ddbar\varphi\ge\gamma-\alpha$. 
Then use a finite covering $(B_j)_{1\le j\le N}$\kern-15pt\break
of $X$ by coordinate balls, a partition of unity $(\theta_j)$, and set
\vskip-13pt
\alert{$$
\varphi_m(z)=\sum_{j=1}^N\theta_j(z)\Big(\psi_{j,m}+
\sum_{k=1}^n\lambda_{j,k}|z_k^{(j)}|^2\Big)
$$}
\vskip-8pt
where \alert{$\psi_{j,m}$} are Bergman approximations of 
\alert{$\psi_j(z):=\varphi(z)-
\sum\lambda_{j,k}|z_k^{(j)}|^2$} (coordinates $z^{(j)}$ and coefficients
$\lambda_{j,k}$ are chosen so that $\psi_j$ is psh on $B_j$).
\end{frame}

\begin{frame}
\frametitle{Approximation of global closed (1,1) currents (II)}
\null\vskip-43pt\null
\begin{block}{Approximation thorem {\rm ([D -- 1992])}} Let $(X,\omega)$
be a compact Kähler manifold and $T=\alpha+i\ddbar\varphi\ge \gamma$ 
a quasi-positive closed $(1,1)$-currents. Then 
\alert{$T=\lim T_m$ weakly} where\vskip2pt
(i) \alert{
$T_m=\alpha+i\ddbar\varphi_m\ge\gamma-\varepsilon_m\omega,\qquad
\varepsilon_m\to 0$}\vskip2pt
(ii) \alert{$\displaystyle
\nu(T,z)-{n\over m}\le\nu(T_m,z)\le \nu(T,z)$}\vskip2pt
(iii) the potentials $\varphi_m$ have only analytic singularities of
the form \alert{${1\over 2m}\log\sum_j|g_{j,m}|^2+C^\infty$}\vskip2pt
(iv) The local coherent ideal sheaves $(g_{j,m})$ glue together into
a global ideal
\alert{$\cJ_m=\hbox{multiplier ideal sheaf}~\cI(m\varphi)$.}
\end{block}
\vskip-1pt\pause%
The OT theorem implies that $(\varphi_{2^m})$ is decreasing, i.e.\ that 
the singularities of $\varphi_{2^m}$ increase to those of 
$\varphi$ by ``subadditivity'':\vskip2pt
\alert{\centerline{$
\cI(\varphi+\psi)\subset\cI(\varphi)+\cI(\psi)~~\Rightarrow~~
\cI(2^{m+1}\varphi)\subset(\cI(2^m\varphi))^2.$}}
\end{frame}

\begin{frame}
\frametitle{K\"ahler (red) cone and pseudoeffective (blue) cone}
\InsertFig 40 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
\LabelTeX 31 35 $\hbox{Kähler classes:}$\ELTX
\LabelTeX 31 30 \alert{$\cK=\big\{~\{\alpha\}\ni \omega\big\}$}\ELTX
\LabelTeX 31 25 $\hbox{(open convex cone)}$\ELTX
\LabelTeX 31 15 $\hbox{pseudoeffective classes:}$\ELTX
\LabelTeX 31 10 \alert{$\cE=\big\{~\{\alpha\}\ni T\ge 0\big\}$}\ELTX
\LabelTeX 31 5 $\hbox{(closed convex cone)}$\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. 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
\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,\\
\cK_{\NS}&:=&\cK\cap \NS_\bR(X)=\hbox{cone of ample divisors},\\
\cE_{\NS}&:=&\cE\cap \NS_\bR(X)=\overline{\hbox{cone of effective divisors}}.
\end{eqnarray*}}
The interior \alert{$\cE^\circ$} is by definition the cone of
\alert{big classes}.
\end{frame}

\begin{frame}
\frametitle{Neron Severi parts of the cones}
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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{Approximation of K\"ahler currents}

\begin{block}{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 $\delta\ll 1$.}
\end{block}
\pause
\medskip

\begin{block}{Observation} 
{\it $\alpha\in\cE^\circ\Leftrightarrow \alpha=\{T\}$,~
$T={}$ a K\"ahler current.}
\end{block}
\pause
\medskip

\begin{block}{Consequence of approximation theorem} {\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
\alert{$\mu_m^\star T_m=[E_m]+\beta_m$}
where~: $E_m$ effective $\bQ$-divisor and $\beta_m$ K\"ahler form on $\wt X_m$.}
\end{block}
\end{frame}


\begin{frame}
\frametitle{Proof of the consequence}
Since $T\ge \delta\omega$, the main approximation theorem implies\vskip-15pt
\alert{
$$T_m=i\ddbar{1\over 2m}\log\sum_j|g_{j,m}|^2\hbox{(mod $C^\infty$)}
\ge {\delta\over 2}\omega,~~m\ge m_0$$}
\vskip-13pt
and \alert{$\cJ_m=\cI(m\varphi)$} is a global coherent sheaf. The
modification $\mu_m:\wt X_m\to X$ is obtained by blowing-up this ideal
sheaf, so that\kern-15pt\vskip-11pt
\alert{$$\mu_m^\star\cJ_m=\cO(-mE_m)$$}
\vskip-12pt
for some effective $\bQ$-divisor $E_m$ with normal crossings on $\wt X_m$. 
If $h$ is a generator of $\cO(-mE_m)$, and we see that 
\vskip-14pt
\alert{$$\beta_m=\mu_m^*T_m-[E_m]
={1\over 2m}\log\sum_j|g_{j,m}\circ\mu_m/h|^2
\quad\hbox{locally on $\wt X_m$}
$$}\vskip-13pt
hence $\beta_m$ is a smooth semi-positive form on $\wt X_m$ which is${}>0$
on\kern-10pt\break%
\hbox{$\wt X_m\ssm \Supp E_m$. By a
perturbation argument using transverse~nega-}\kern-35pt\break
tivity of exceptional divisors, $\beta_m$ can easily be made K\"ahler.
\end{frame}

\begin{frame}
\frametitle{Analytic Zariski decomposition}
\null\vskip-40pt\null
\begin{block}{Theorem} {\it
For every class $\{\alpha\}\in\cE$, there exists a positive current 
$T_{\min}\in\{\alpha\}$ with \alert{minimal singularities}.}
\end{block}
{\it Proof.} Take $T=\alpha+i\ddbar\varphi_{\min}$ where\\
\alert{\centerline{$\varphi_{\min}(x)=
\max\{\varphi(x)\,;\;\varphi\le 0~\hbox{and}~\alpha+i\ddbar\varphi\ge 0\}.$}}

\begin{block}{Theorem} {\it Let $X$ be compact Kähler and
let $\{\alpha\}\in\cE^\circ$ be a big class and $T_{\min}\ge 0$ be a 
current with minimal singularities. Then $T_{\min}=\lim T_m$ where
$T_m$ are K\"ahler currents such that\vskip2pt
(i) $\exists$~modification $\mu_m:\wt X_m\to X$ with
\alert{$\mu_m^\star T_m = [E_m]+\beta_m$}, where $E_m$ is 
a $\bQ$-divisor and $\beta_m$ a Kähler form on $\wt X_m$.
\vskip2pt
(ii) $\int_{\wt X_m}\beta_m^n$ is an increasing sequence converging to
\vskip-20pt
\alert{$$\Vol(X,\{\alpha\}):=\int_X(T_{\min})_{\rm ac}^n=
\sup_{T\in\{\alpha\},{\rm anal.sing}}\int_{X\ssm\Sing(T)}T^n.$$}}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Orthogonality estimate}
\null\vskip-36pt\null
\begin{block}{Theorem {\rm (Boucksom-Demailly-P\v{a}un-Peternell 2004)}}
{\it Assume $X$ projective and $\{\alpha\}\in\cE_{\NS}^\circ$. Then
$\beta_m=[D_m]$ is an ample $\bQ$-divisor such that
\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 fixed polarization and $C\ge 0$ is a constant 
such that
$\pm\alpha$ is dominated by $C\omega$ $($i.e., $C\omega\pm\alpha$ 
nef$\,)$.\kern-10pt}
\end{block}

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\LabelTeX 22 49 $\mu_m^*T_m$\ELTX
\LabelTeX 20 22 $D_m$\ELTX
\LabelTeX  5 40 $E_m$\ELTX
\LabelTeX 45 44 $\hbox{Proof similar to projection of a}$\ELTX
\LabelTeX 45 39 $\hbox{point onto a convex set, using}$\ELTX
\LabelTeX 45 34 $\hbox{elementary case of Morse inequalities:}$\ELTX
\LabelTeX 45 27 \alert{$\Vol(\beta-\gamma)\ge \beta^n-n\beta^{n-1}\cdot\gamma$}\ELTX
\LabelTeX 45 22 \alert{$\forall \beta,\,\gamma~\hbox{ample classes}$}\ELTX
\EndFig

\begin{block}{Corollary} $\alpha\cdot\langle\alpha^{n-1}\rangle -
\langle\alpha^n\rangle=0$.
\end{block}
\end{frame}


\begin{frame}
\frametitle{Duality between $\cE_{\NS}$ and $\cM_{\NS}$}
\null\vskip-36pt\null
\begin{block}{Theorem {\rm (BDPP, 2004)}}
{\it For $X$ projective, a class $\alpha$ is in $\cE_{\NS}$
$($pseudo-effective$)$ if and only if $\alpha\cdot C_t\ge 0$
for all \alert{mobile curves}, i.e.\ algebraic curves which can be
deformed to fill the whole of $X$.\\
In other words, $\cE_{\NS}$ is the \alert{dual cone} of the cone 
$\cM_{\NS}$ of mobile curves with respect to Serre duality.}
\end{block}
\medskip

{\it Proof.} We want to show that $\cE_{\NS}=\cM_{\NS}^\vee$.
By obvious positivity of
the integral pairing, one has in any case
\vskip-12pt
\alert{$$\cE_{\NS}\subset (\cM_{\NS})^\smallvee.$$}
\vskip-14pt\pause
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
\vskip-12pt
\alert{$$\alpha\cdot \Gamma\ge\varepsilon\omega\cdot \Gamma$$}
\vskip-14pt
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 1 52 $\cE$\ELTX
\LabelTeX -5 46 $\cE_{\NS}$\ELTX
\LabelTeX 14 31 $\cM^\smallvee$\ELTX
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\LabelTeX -28 4 $\NS_\bR(X)$\ELTX
\LabelTeX 10 4 ${}\subset H^{1,1}(X,\bR)$\ELTX
\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

\begin{block}{Theorem {\rm (Boucksom-Demailly-Paun-Peternell 2004)}} {\it
A projective manifold $X$ has its canonical bundle $K_X$ pseudo-effective, 
i.e.\ $K_X\in\cE_{\NS}$, if and only if $X$ is \alert{not uniruled}.}
\end{block}
\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.
\pause\vskip3pt
\claim{Note:} Mori's proof uses characteristic $p$, so it is hard
to extend to the Kähler case~!

\end{frame}

\begin{frame} 
\frametitle{Asymptotic cohomology functionals}
\begin{block}{Definition} {\it Let $X$ be a compact complex manifold and let
$L\to X$ be a holomorphic line bundle.\vskip2pt
{\rm (i)}~ \alert{$\widehat h^q(X,L):=
\limsup_{k\to+\infty}~{n!\over m^n}h^q(X,L^{\otimes m})$}
\vskip2pt
{\rm (ii)} (asymptotic Morse partial sums)\\
\alert{$\widehat h^{\,\leq q}(X,L):=
\limsup_{m\to+\infty}~{n!\over m^n}\sum_{0\le j\le q}(-1)^{q-j}h^j(X,L^{\otimes m})$.}}
\end{block}

\pause

\begin{block}{Conjecture} {\it $\widehat h^q(X,L)$ and 
$\widehat h^{\,\leq q}(X,L)$ depend only on $c_1(L)\in H^{1,1}_{\BC}(X,\bR)$.}
\end{block}
\pause

\begin{block}{Theorem {\rm (K\"uronya, 2005), (D, 2010)}} {\it This is true
if $c_1(L)$ belongs to the \alert{``divisorial Neron-Severi group'' 
$\DNS_\bR(X)$} generated by divisors.}
\end{block}
\end{frame}

\begin{frame} 
\frametitle{Holomorphic Morse inequalities}
\null\vskip-38pt\null
\begin{block}{Theorem {\rm (D, 1985)}} {\it Let $L\to X$ be a holomorphic line bundle on a compact complex manifold. Then\vskip2pt
{\rm (i)}~ \alert{$\widehat h^q(X,L)\le 
\inf_{u\in c_1(L)}\int_{X(u,q)}(-1)^qu^n$}
\vskip2pt
{\rm (ii)} 
\alert{$\widehat h^{\,\leq q}(X,L)\le
\inf_{u\in c_1(L)}\int_{X(u,\le q)}(-1)^qu^n$}
\vskip2pt
where $X(u,q)$ is the $q$-index set of the $(1,1)$-form $u$ and
\alert{$X(u,\le q)=\bigcup_{0\le j\le q}X(u,j)$}.}
\end{block}
\pause
\vskip-5pt
\begin{block}{Question (or Conjecture !)} {\it Are these inequalities always
equalities~?}
\end{block}
\pause

If the answer is \alert{yes}, then $\widehat h^q(X,L)$ and
$\widehat h^{\,\leq q}(X,L)$ actually only depend only on $c_1(L)$ and
can be extended to $H^{1,1}_{\BC}(X,\bR)$, e.g.\vskip-11pt
\alert{$$
h^{\le q}_{\rm tr}(X,\alpha):=\inf_{u\in \alpha}\int_{X(u,\le q)}(-1)^qu^n,~~
\forall\alpha\in H^{1,1}_{\BC}(X,\bR)
$$}
\end{frame}

\begin{frame}
\frametitle{Converse of Andreotti-Grauert theorem}
\null\vskip-38pt\null
\begin{block}{Theorem {\rm (D, 2010) / related result S.-I.\ Matsumura, 2011}} 
{\it Let $X$ be a projective variety.
Then\vskip2pt
{\rm (i)} the conjectures are true for \alert{$q=0$}:\\
\alert{$\widehat h^0(X,L)=\Vol(X,c_1(L))=
\inf_{u\in c_1(L)}\int_{X(u,0)}u^n$}
\vskip2pt
{\rm (ii)} The conjectures are true for \alert{$\dim X\le 2$}\vskip2pt\pause%
The limsup's are \alert{limits} in all of these cases.}
\end{block}

\pause
\claim{Observation 1}. The question is invariant by Serre duality :\vskip-11pt
\alert{$$\widehat h^q(X,L)=\widehat h^{n-q}(X,-L)$$}
\pause\vskip-15pt

\claim{Observation 2}. (Birational invariance). If $\mu:\wt X\to X$ is a modification, then \alert{$\widehat h^q(X,L)=\widehat h^q(\wt X,\mu^*L)$} by the
Leray spectral sequence and\vskip-13pt
\alert{$$
\inf_{u\in \alpha}\int_{X(u,\le q)}(-1)^qu^n=
\inf_{\wt u\in \mu^*\alpha}\int_{\wt X(\wt u,\le q)}(-1)^q\wt u^n.
$$}
\end{frame}

\begin{frame}
\frametitle{Main idea of the proof}
It is enough to consider the case of a \alert{big line bundle} $L$.
Then use \alert{approximate Zariski decomposition}:
$$
\forall\delta>0,~\exists \mu=\mu_\delta:\wt X\to X,\quad 
\alert{\mu^*L=E+A}
$$
where $E$ is $\bQ$-effective and $A$ $\bQ$-ample, and
\alert{$$
\Vol(X,L)-\delta\le A^n\le\Vol(X,L),\quad E\cdot A^{n-1}\le C\delta^{1/2},
$$}%
the latter inequality by the orthogonality estimate.\pause
\vskip3pt
Take $\omega\in c_1(A)$
a K\"ahler form and a metric $h$ on $\cO(E)$ such that 
\alert{$$
\Theta_{\cO(E),h}\wedge\omega^{n-1}=c_\delta\omega^n,\qquad
c_\delta=O(\delta^{1/2}).
$$}%
The last line is obtained simply by solving a Laplace equation, thanks to the
orthogonality estimate.
\end{frame}

\begin{frame}
\frametitle{End of the proof}
$$\mu^*L=E+A~~\Rightarrow~~
\wt u=\Theta_{\cO(E),h}+\omega\in c_1(\mu^*L).
$$
If $\lambda_1\le\ldots\le\lambda_n$ are the eigenvalues of $\Theta_{\cO(E),h}$
with respect to $\omega$, then 
\alert{$\sum \lambda_i=\hbox{trace}\le C\delta^{1/2}$}. We have
$$
\wt u^n=\prod(1+\lambda_i)\omega^n\le\Big(1+{1\over n}\sum\lambda_i\Big)^n\omega^n
\le (1+O(\delta^{1/2})\omega^n,
$$
therefore
$$
\int_{\wt X(u,0)}\wt u^n\le (1+O(\delta^{1/2})\int_X\omega^n
\le (1+O(\delta^{1/2})\Vol(X,L).
$$
\pause
As $\delta\to 0$ we find
$$\alert{
\inf_{u\in c_1(L)}\int_{X(u,0)}u^n=\inf_\mu
\inf_{\wt u\in c_1(\mu^*L)}\int_{\wt X(u,0)}\wt u^n\le \Vol(X,L).}\quad
\hbox{QED}
$$
\end{frame}

\end{document}