% 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. 

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%%\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:

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\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb}
\usepackage{colortbl}
\usepackage[english]{babel}
% Or whatever. Note that the encoding and the font should match. If T1
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\title[\ \kern-190pt\rlap{\blank{J.-P.\ Demailly (Grenoble), TSIMF, Sanya, Dec 18-22, 2017}}\kern181pt\rlap{\blank{
Ricci curvature and geometry of compact Kähler varieties}}\kern181pt
\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{Ricci curvature and geometry\\
of compact Kähler varieties}
%% \subtitle{Presentation Subtitle} % (optional)

\author[] % (optional, use only with lots of authors)
{\strut\vskip-20pt Jean-Pierre Demailly}

\institute[]{\strut\vskip-20pt
Institut Fourier, Université Grenoble Alpes\ \ \&\ \ Académie 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)
{\strut\vskip-20pt Master lectures delivered at TSIMF\\
Workshop: Global Aspects of Projective and K\"ahler Geometry\\
\vskip7pt Sanya, December 18-22, 2017}

%%\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:

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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]{uga-logo}{logo_UGA}
  \pgfuseimage{uga-logo}
  \pgfdeclareimage[height=2cm]{acad-logo}{academie_logo2}
  \pgfuseimage{acad-logo}
  \vskip-12pt
  \titlepage
\end{frame}
%%\def\pause{}

\begin{frame}
  \frametitle{General plan of the K\"ahler main lectures}
\vskip-6pt
\alert{\bf Ricci curvature and geometry of compact Kähler varieties}

\begin{itemize}
\item \claim{Lecture 1: Positivity concepts in Kähler geometry}\\
  -- definitions and characterizations of the concept of ample, nef,
  big and pseudoeffective line bundles and \hbox{(1,1)-classes\kern-20pt}\\
  -- Numerical characterization of the K\"ahler cone\\
  -- Approximate analytic Zariski decomposition and abundance
   
\item \claim{Lecture 2: Uniruledness, rational connectedness and $-K_X$ nef}\\
  -- Orthogonality estimates and duality of positive cones\\
  -- Criterion for uniruledness and rational connectedness\\
  -- Examples of compact K\"ahler mflds $X$ with $-K_X\ge 0$ or nef.

\item \claim{Lecture 3: Holonomy and main structure theorems}\\
  -- concept of holonomy of euclidean \&\ hermitian vector \hbox{bundles\kern-20pt}\\
  -- De Rham splitting theorem and Berger's classification of holonomy groups\\
  -- Generalized holonomy principle and structure theorems\\
  -- investigation of the case when $-K_X$ is nef (Cao, H\"oring)
\end{itemize}

\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}

\begin{frame}
\frametitle{Lecture 1}
\begin{block}{Positivity concepts in Kähler geometry}
\strut\vskip8pt
A brief survey of the main positivity concepts in algebraic and analytic
geometry.
\end{block}

\end{frame}
   
\begin{frame}
  \frametitle{Complex manifolds / $(p,q)$-forms}
%% \framesubtitle{Subtitles are optional.}

  \begin{itemize}
  \item
    Goal : study the \alert{geometric / topological / cohomological 
    properties of compact K\"ahler manifolds}
   \pause
  \item
    A complex $n$-dimensional manifold is given by coordinate charts equipped
    with\\ 
   \alert{local holomorphic coordinates $(z_1,z_2,\ldots,z_n)$.}
   \pause
  \item
    A differential form $u$ of type $(p,q)$ can be written as a sum
\alert{%
$$
u(z)=\sum_{|J|=p,|K|=q}u_{JK}(z)\,dz_J\wedge d\ovl z_K
$$}
   where 
   $J=(j_1,\ldots,j_p)$, $K=(k_1,\ldots,k_q)$,
\alert{$$dz_J=dz_{j_1}\wedge\ldots\wedge dz_{j_p},\quad
   d\ovl z_K=d\ovl z_{k_1}\wedge\ldots\wedge d\ovl z_{k_q}.$$}
  \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Complex manifolds / Currents}
%% \framesubtitle{Subtitles are optional.}

  \begin{itemize}
  \item A current is a differential form with \alert{distribution 
  coefficients
$$T(z)=i^{pq}\sum_{|J|=p,|K|=q}T_{JK}(z)\,dz_J\wedge d\ovl z_K$$}
  \pause
  \item The current $T$ is said to be \alert{positive} if the distribution
$\sum\lambda_j\ovl \lambda_k T_{JK}$ is a positive real measure for all
$(\lambda_J)\in\bC^N$ (so that $T_{KJ}=\ovl T_{JK}$, hence $\ovl T=T$).
  \item
The coefficients $T_{JK}$ are then
\alert{complex measures} -- and the diagonal ones $T_{JJ}$ are 
\alert{positive real measures.}
  \pause
  \item
$T$  is said to be \alert{closed} if $dT=0$ in the sense of distributions. 
   \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Complex manifolds / Basic examples of Currents}
  \begin{itemize}
  \item \alert{The current of integration over a codimension 
  $p$ analytic cycle}
$A=\sum c_jA_j$ is defined by duality as $[A]=\sum c_j[A_j]$ with
\alert{%
$$
\langle[A_j],u\rangle=\int_{A_j}u_{|A_j}
$$}
for every $(n-p,n-p)$ test form $u$ on $X$. 
\pause
  \item Hessian forms of plurisubharmonic functions :
\alert{$$\varphi~~\hbox{plurisubharmonic}\Leftrightarrow
\Big({\partial^2\varphi\over \partial z_j\partial\ovl z_k}\Big)\ge 0$$}
then
\alert{$$T=i\ddbar\varphi\qquad
\hbox{is a closed positive $(1,1)$-current.}$$}
   \end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Complex manifolds / K\"ahler metrics}
  \begin{itemize}
  \item A \alert{K\"ahler metric} is a smooth 
\alert{positive definite $(1,1)$-form}
\alert{%
$$
\omega(z)=i\sum_{1\le j,k\le n}\omega_{jk}(z)dz_j\wedge d\ovl z_k\qquad
\hbox{such that $d\omega=0$}.
$$}%
$\phantom{\strut}$\vskip-32pt$\phantom{\strut}$
\item The manifold $X$ is said to be \alert{K\"ahler} (or of 
\alert{K\"ahler type}) if it possesses at least one K\"ahler metric $\omega$
$[${\rm K\"ahler 1933}$]$
\pause
\item Every complex analytic and locally closed
submanifold  \alert{$X\subset\bP_\bC^N$} in projective space is K\"ahler
when equipped with the restriction of the 
\alert{Fubini-Study metric 
$$\omega_{FS}=\frac{i}{2\pi}
\log(|z_0|^2+|z_1|^2+\ldots+|z_N|^2).$$}%
$\phantom{\strut}$\vskip-32pt$\phantom{\strut}$
\item Especially projective algebraic varieties are K\"ahler.
\vskip0pt
  \end{itemize}
\end{frame}

%%\section*{Cohomology of complex manifolds} 
\begin{frame}
  \frametitle{Sheaf / De Rham / Dolbeault / cohomology}
  \begin{itemize} 
  \item \alert{Sheaf cohomology} $H^q(X,\cF)$\\ 
especially when $\cF$ is a \alert{coherent analytic sheaf}.
   \pause
  \item Special case : cohomology groups $H^q(X,R)$ with values in 
constant coefficient sheaves $R=\bZ,\,\bQ,\,\bR,\,\bC,\,\ldots$\\
\alert{${}={}$De Rham cohomology groups.}
\pause
\item $\Omega^p_X=\cO(\Lambda^p T^*_X)={}$ sheaf of holomorphic 
$p$-forms on~$X$. 
\pause
\item Cohomology classes [forms / currents yield same groups]
\alert{%
\begin{eqnarray*}
&&\hbox{$\alpha$ $d$-closed $k$-form/current to $\bC$ $\longmapsto 
\{\alpha\}\in H^k(X,\bC)$}\\
&&\hbox{$\alpha$ $\dbar$-closed $(p,q)$-form/current to $F$ $\longmapsto 
\{\alpha\}\in H^{p,q}(X,F)$}
\end{eqnarray*}}
Dolbeault isomorphism (Dolbeault - Grothendieck 1953)
 \alert{\begin{eqnarray*}
          &&H^{0,q}(X,F)\simeq H^q(X,\cO(F)),\\
          &&H^{p,q}(X,F)\simeq H^q(X,\Omega^p_X\otimes\cO(F))
        \end{eqnarray*}}
  \end{itemize}
\end{frame}

\begin{frame}
\frametitle{Bott-Chern and Aeppli cohomology}
\vskip-6pt  
The Bott-Chern cohomology groups are defined as
\alert{
\begin{eqnarray*}
&&H^{p,q}_\BC(X,\bC):=\{(p,q)-\hbox{forms}~u~\hbox{such that}~\partial u=\dbar u=0\}~/\\
&&\strut\kern80pt\{(p,q)-\hbox{forms}~u=\ddbar v\}
\end{eqnarray*}}\pause%
The Aeppli cohomology groups are defined as
\alert{
\begin{eqnarray*}
&&H^{p,q}_\A(X,\bC):=\{(p,q)-\hbox{forms}~u~\hbox{such that}~\ddbar u=0\}~/\\
&&\strut\kern80pt\{(p,q)-\hbox{forms}~u=\partial v+\dbar w\}
\end{eqnarray*}}
\pause%
These groups are dual each other via Serre duality:
\alert{$$
H^{p,q}_\BC(X,\bC)\times H^{n-p,n-q}_A(X,\bC)\to\bC,\quad
(\alpha,\beta)\mapsto\int_X\alpha\wedge\beta
$$}\pause%
One always has morphisms
\alert{$$
H^{p,q}_\BC(X,\bC)\to H^{p,q}(X,\bC)\to H^{p,q}_\A(C,\bC).
$$}
They are not always isomorphisms, \claim{but are if $X$ is K\"ahler}.
\end{frame}

\begin{frame}
\frametitle{Hodge decomposition theorem}
\vskip-6pt  
\begin{itemize} 
\item \claim{{\bf Theorem.} {\it If $(X,\omega)$ is compact K\"ahler, then}
$$
H^k(X,\bC)=\bigoplus_{p+q=k}H^{p,q}(X,\bC).
$$}\strut\vskip-33pt\strut
\item \claim{\it Each group $H^{p,q}(X,\bC)$ is isomorphic to the 
space of $(p,q)$
\alert{harmonic forms} $\alpha$ with respect to $\omega$, i.e.\ 
\alert{$\Delta_\omega\alpha=0$}.}
\end{itemize}
\strut\vskip-32pt\strut\pause
\begin{block}{Hodge Conjecture {\bf (a millenium problem!$\,$)}.}
If $X$ is a projective algebraic manifold,\\
\claim{Hodge $(p,p)$-classes${}=H^{p,p}(X,\bC)\cap H^{2p}(X,\bQ)$}\\
are generated by \alert{classes of algebraic cycles of codimension $p$
with $\bQ$-coefficients}.
\end{block}
\vskip-5pt\pause
\begin{block}{Theorem {\bf(Claire Voisin, 2001)}}
There exists a $4$-dimensional complex
torus $X$ possessing a non trivial Hodge class of type $(2,2)$, such that
every coherent analytic sheaf $\cF$ on $X$ satisfies $c_2(\cF)=0$.
\end{block}
\end{frame}

%%\section*{Kodaira embedding theorem}

\begin{frame}
\frametitle{Kodaira embedding theorem}
\vskip-6pt
\begin{block}{Theorem {\bf (Kodaira 1953)}}
Let $X$ be a compact complex $n$-dimensional 
manifold. Then the following properties are equivalent.
\begin{itemize} 
\item \claim{\alert{$X$ can be embedded in some projective space 
$\bP^N_\bC$} as a closed analytic submanifold
$($and such a submanifold is automatically algebraic by Chow's thorem$)$.}
\item \claim{
\alert{$X$ carries a hermitian holomorphic line bundle $(L,h)$ with positive
definite smooth curvature form $i\Theta_{L,h}>0$}.\\
For $\xi\in L_x\simeq\bC$, $\Vert\xi\Vert_h^2=|\xi|^2e^{-\varphi(x)}$,
\alert{
\begin{eqnarray*}
&&i\Theta_{L,h}=i\ddbar\varphi=-i\ddbar\log h,\\
&&c_1(L)=\Big\{{i\over 2\pi}\Theta_{L,h}\Big\}.\\
\noalign{\vskip-40pt}
\end{eqnarray*}}}
\item \claim{
$X$ possesses a Hodge metric, i.e., a \alert{K\"ahler metric $\omega$ such that
$\{\omega\}\in H^2(X,\bZ)$}.}
\end{itemize}
\end{block}
\end{frame}

%%\section{Positive cones}

\begin{frame}
\frametitle{Positive cones}
\begin{block}{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.}
\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$)$.}
\item \claim{Always true: \alert{$\ovl\cK\subset\cE$}.}
\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 
$\int_E\alpha= E^2=-1$, hence $\{\alpha\}\notin\ovl\cK$, although
$\{\alpha\}=\{[E]\}\in\cE$.
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{K\"ahler (red) cone and pseudoeffective (blue) cone}
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\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),\\
\cE_{\NS}&:=&\cE\cap \NS_\bR(X).
\end{eqnarray*}}
\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{ample / nef / effective / big divisors}
\vskip-7pt
\begin{block}{Theorem {\rm (Kodaira + successors, D 90)}}
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$)$.}
\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 $C$.}
\item \claim{$\cE_{\NS}$ is the closure of the cone of
\alert{effective divisors}, i.e.\ divisors $D=\sum c_jD_j$, $c_j\in\bR_+$.}
\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}
\end{block}
Proof: $L^2$ estimates for $\dbar$ / Bochner-Kodaira technique
\end{frame}

\begin{frame}
\frametitle{ample / nef / effective / big divisors}
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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 effective\ELTX
\EndFig
\end{frame}

\begin{frame}
\frametitle{Approximation of currents, Zariski decomposition}
\vskip-5pt
\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 Proposition.}
{\it $\alpha\in\cE^\circ\Leftrightarrow \alpha=\{T\}$,~
$T={}$ a K\"ahler current.}}
\medskip

We say that $\cE^\circ$ is the cone of \alert{big $(1,1)$-classes}.
\pause
\medskip
\end{itemize}
\strut\vskip-36pt\strut

\begin{block}{Theorem {\rm (D-- 92)}}
Any K\"ahler current $T$ can be written\vskip3pt
\centerline{\alert{$T=\lim T_m$}}\vskip3pt
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
\vskip3pt  
\centerline{$\mu_m^\star T_m=[E_m]+\beta_m,$}}\vskip3pt
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 $\wt X_m$.\vskip0pt
\end{block}
\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 $\beta_m=\mu_m^*T_m-[E_m]$.
Then $\beta_m=i\ddbar\psi_m$ 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 $\beta_m$ 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 $\beta_m$ 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 $L$. We will also use this terminology for K\"ahler
currents with logarithmic poles.
\end{frame}

\begin{frame}
\frametitle{Analytic Zariski decomposition}
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\LabelTeX -3 46 $\cK_{\NS}$\ELTX
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\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 $\beta_m$\ELTX
\LabelTeX 21 17 $\wt\alpha=\mu_m^\star\alpha=[E_m]+\beta_m$\ELTX
\EndFig
\end{frame}

\begin{frame}
\frametitle{Characterization of the Fujiki class $\cC$}
\begin{block}{Theorem {\rm (Demailly-P\v{a}un 2004)}} A compact complex 
manifold $X$ is bimeromorphic to a K\"ahler mani\-fold $\wt X$
$($or equivalently, dominated by a K\"ahler manifold $\wt X)$
if and only if it carries a K\"ahler current $T$.
\end{block}

{\it Proof}. If $\mu:\wt X\to X$ is a modification and
$\wt\omega$ is a K\"ahler metric on~$\wt X$, then $T=\mu_\star\wt\omega$
is a K\"ahler current on $X$.
\medskip

Conversely, if $T$ is a K\"ahler current, we take $\wt X=\wt X_m$ and
$\wt\omega=\beta_m$ for $m$ large enough.

\begin{block}{Definition} The class of compact complex manifolds $X$
bimeromorphic to some K\"ahler manifold $\wt X$ is called\\ \alert{the Fujiki
class $\cC$}.\\ Hodge decomposition still holds true in $\cC$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Numerical characterization of the K\"ahler cone}
\begin{block}{Theorem {\rm (Demailly-P\v{a}un 2004)}}
Let $X$ be a compact K\"ahler manifold. Let 
\alert{%
$$
\cP=\big\{\alpha\in H^{1,1}(X,\bR)\,;\; \int_Y\alpha^p>0,\;
\forall Y\subset X,\;\dim Y=p\big\}.
$$}%
``cone of numerically positive classes''.\pause
\alert{Then the K\"ahler cone $\cK$ is\\
  one of the connected components of $\cP$}.
\end{block}
\pause

\begin{block}{Corollary {\rm (Projective case)}}
If $X$ is projective algebraic, then \alert{$\cK=\cP$}.
\end{block}

{\it Note:} this is a ``transcendental version'' of the Nakai-Moishezon
criterion.
The proof relies in an essential way on Monge-Amp\`ere equations
(Calabi-Yau theorem).
\end{frame}

\begin{frame} 
\frametitle{Example (non projective) for which $\cK\subsetneq\cP$.}

Take $X={}$generic complex torus $X=\bC^n/\Lambda$. 
\medskip

Then $X$ \alert{does not possess any analytic subset} except finite subsets 
and $X$ itself. 
\medskip

Hence $\cP=\{\alpha\in H^{1,1}(X,\bR)\,;\;\int_X\alpha^n>0\big\}$. 
\medskip

Since $H^{1,1}(X,\bR)$ is in one-to-one correspondence with constant 
hermitian forms,  $\cP$ is the set of hermitian forms on 
$\bC^n$ such that $\det(\alpha)>0$, i.e.\\ 
\alert{possessing an even number of negative eigenvalues}.
\medskip

$\cK$ is the component with all eigenvalues${}>0$.
\end{frame}

\begin{frame}
\frametitle{Proof of the theorem : use Monge-Amp\`ere} 

Fix $\alpha\in\ovl\cK$ so that $\int_X\alpha^n>0$.
\medskip

If $\omega$ is K\"ahler, then $\{\alpha+\varepsilon\omega\}$ is a K\"ahler
class $\forall\varepsilon>0$.
\medskip

Use the \alert{Calabi-Yau theorem} (Yau 1978) to solve the
Monge-Amp\`ere equation
\alert{%
$$
(\alpha+\varepsilon\omega+i\ddbar\varphi_\varepsilon)^n=f_\varepsilon
$$}
where $f_\varepsilon>0$ is a suitably chosen volume form.
\medskip

Necessary and sufficient condition :
\alert{%
$$
\int_X f_\varepsilon = (\alpha+\varepsilon\omega)^n
\quad\hbox{in $H^{n,n}(X,\bR)$}.
$$}%
In other terms, the infinitesimal volume form of the K\"ahler metric
$\alpha_\varepsilon=\alpha+\varepsilon\omega+ i\ddbar\varphi_\varepsilon$ 
can be distributed \alert{randomly} on $X$.
\end{frame}

\begin{frame}
\frametitle{Proof of the theorem : concentration of mass} 
In particular, the mass of the right hand side $f_\varepsilon$ can be
distributed in an $\varepsilon$-neighborhood $U_\varepsilon$ of any given
subvariety $Y\subset X$.
\medskip

If $\codim Y=p$, on can show that
\alert{%
$$
(\alpha+\varepsilon\omega+i\ddbar\varphi_\varepsilon)^p\to \Theta\quad
\hbox{weakly}
$$
where $\Theta$ positive $(p,p)$-current and $\Theta\ge\delta[Y]$ for some
$\delta>0$.}\medskip

Now, ``diagonal trick'': apply the above result 
to 
$$
\wt X=X\times X,\qquad \wt Y=\hbox{diagonal}\subset \wt X,\qquad
\wt\alpha=\pr_1^*\alpha+\pr_2^*\alpha.
$$
As $\wt\alpha$ is nef on $\wt X$ and
$\int_{\wt X}(\wt\alpha)^{2n}>0$, it follows by the above that the
class $\smash{\{\wt\alpha\}}^n$ contains a K\"ahler current $\Theta$ such that
$\Theta\ge\delta[\smash{\wt Y}]$ for some $\delta>0$. Therefore 
$$
T:=(\pr_1)_*(\Theta\wedge\pr_2^*\omega)
$$
is numerically equivalent to a multiple of $\alpha$ and dominates
$\delta\omega$, and we see that $T$ is a K\"ahler current.
\end{frame}

\begin{frame}
\frametitle{Generalized Grauert-Riemenschneider result}
\vskip-4pt
This implies the following result.

\begin{block}{Theorem {\rm (Demailly-P\v{a}un, Annals of Math.\ 2004)}}
Let $X$ be a compact K\"ahler manifold
  and consider a class\\
\alert{$\{\alpha\}\in\ovl\cK$ such that $\int_X\alpha^n>0$}.\\
Then \alert{$\{\alpha\}$ contains a K\"ahler
  current $T$}, i.e.\ $\{\alpha\}\in\cE^\circ$.
\end{block}

Illustration:
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\LabelTeX -1 19 $\cK$\ELTX
\LabelTeX 16 19 $\alpha$\ELTX
\LabelTeX  6 35.7 $\alpha$\ELTX
\LabelTeX 23 19 $\displaystyle\int_X\alpha^n=0$\ELTX
\LabelTeX 19 34 $\displaystyle\int_X\alpha^n>0$\ELTX
\EndFig
\end{frame}

\begin{frame}
\frametitle{Final step of proof}
Clearly the open cone $\cK$ 
is contained in $\cP$, hence in order to show that $\cK$ is one of the
connected components of $\cP$, we need only show that $\cK$ is closed
in $\cP$, i.e.\ that $\ovl\cK\cap\cP\subset\cK$. Pick a class
$\{\alpha\}\in\ovl\cK\cap\cP$. In particular $\{\alpha\}$ is
nef and satisfies $\int_X\alpha^n>0$. Hence
$\{\alpha\}$ contains a K\"ahler current $T$. 
\medskip

Now, an induction on
dimension using the assumption $\int_Y\alpha^p>0$ for all analytic
subsets $Y$ (we also use resolution of singularities for $Y$ at this step)
shows that the restriction $\{\alpha\}_{|Y}$ is the class of a K\"ahler
current on~$Y$.
\medskip

We conclude that $\{\alpha\}$ is a K\"ahler class by
results of Paun (PhD~1997), therefore $\{\alpha\}\in\cK$.
\end{frame}

\begin{frame} 
\frametitle{Variants of the main theorem}
\claim{{\bf Corollary 1} {\rm (DP 2004)}. {\it
Let $X$ be a compact K\"ahler manifold. 
\alert{$$
\{\alpha\}\in H^{1,1}(X,\bR)~~\hbox{\it is K\"ahler}\Leftrightarrow
\exists \omega~\hbox{\it K\"ahler s.t.}~\int_Y\alpha^k\wedge\omega^{p-k}>0
$$}
for all $Y\subset X$ irreducible and all $k=1,2,\ldots,p=\dim Y$.}}
\medskip

{\it Proof.} Argue with $(1-t)\alpha+t\omega$, $t\in[0,1]$.
\medskip

\claim{{\bf Corollary 2} {\rm (DP 2004)}. {\it
Let $X$ be a compact K\"ahler manifold. 
\alert{$$
\{\alpha\}\in H^{1,1}(X,\bR)~~\hbox{\it is nef}~(\alpha\in\ovl\cK)
\Leftrightarrow
\forall \omega~\hbox{\it K\"ahler}~\int_Y\alpha\wedge\omega^{p-1}\ge 0
$$}
for all $Y\subset X$ irreducible and all $k=1,2,\ldots,p=\dim Y$.}}
\medskip

\begin{block}{Consequence} \alert{The dual of the
nef cone $\smash{\ovl\cK}$} is the closed convex cone 
in $H^{n-1,n-1}(X,\bR)$ generated by cohomology 
classes of currents of the form \alert{$[Y]\wedge\omega^{p-1}\in 
H^{n-1,n-1}(X,\bR)$}.
\end{block}
\end{frame}

\begin{frame} 
\frametitle{Theorem on deformation stability of K\"ahler cones}

\begin{block}{Theorem {\rm (Demailly-P\v{a}un 2004)}}
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 the $(1,1)$-projection $\nabla^{1,1}$ of
the Gauss-Manin connection $\nabla$ in the decomposition of
\alert{%
$$
\nabla=\begin{pmatrix}
\nabla^{2,0} & * & 0\cr 
* & \nabla^{1,1} & * \cr
0 & * & \nabla^{0,2}\cr
\end{pmatrix}
$$}%
on the Hodge bundle $H^2=H^{2,0}\oplus H^{1,1}\oplus H^{0,2}$.
\end{block}

\end{frame}

\begin{frame}
\frametitle{Positive cones in $H^{n-1,n-1}(X)$ and Serre duality}
\vskip-8pt
\begin{block}{Definition} Let $X$ be a compact K\"ahler manifold.
\begin{itemize}
\item \claim{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{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{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{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{block}
\end{frame}

\begin{frame}
\frametitle{Main duality theorem}
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\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 -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{Precise duality statement}

Recall that the Serre duality pairing is
$$
(\alpha^{(p,q)},\beta^{(n-p,n-q)})\longmapsto \int_X\alpha\wedge\beta.
$$
\pause
$\phantom{\strut}$\vskip-36pt$\phantom{\strut}$

\begin{block}{Theorem {\rm(Demailly-P\v{a}un 2001)}}
If $X$ is compact K\"ahler, then
\alert{$\cK$ and $\cN$ are dual cones.}\\
$($well known since a long time~: $\cK_{\NS}$ and $\cN_{\NS}$ are dual$)$.
\end{block}\pause

\begin{block}{Theorem {\rm(Boucksom-Demailly-Paun-Peternell 2004)}}
If $X$ is projective algebraic, then
\alert{$\cE_{\NS}$ and $\cM_{\NS}$ are dual cones.}
\end{block}\pause

\begin{block}{Conjecture {\rm(Boucksom-Demailly-Paun-Peternell 2004)}}
If $X$ is K\"ahler, then
\alert{$\cE$ and $\cM$ should be dual cones.}
\end{block}

\end{frame}

\begin{frame}
\frametitle{Concept of volume (very important !)}
\vskip-5pt
\begin{block}{Definition (\rm Boucksom 2002)}
The \alert{volume} $($\alert{movable self-intersection}$)$ of a 
big class $\alpha\in\cE^\circ$ is\vskip5pt
\alert{\centerline{$\displaystyle
\Vol(\alpha)=\sup_{T\in \alpha}\int_{\wt X}\beta^n>0
$}}\vskip5pt
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 $\mu:\wt X\to X$.
\end{block}

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

\begin{block}{Theorem {\rm (Boucksom 2002)}} 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
\vskip5pt
\alert{\centerline{$\displaystyle
\Vol(c_1(L))=\lim_{m\to+\infty}{n!\over m^n}h^0(X,mL)=
\lim_{m\to+\infty} D_m^n.
$}}
\end{block}
\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

\begin{block}{Theorem} 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{$$\Vol(L) -\varepsilon\le \Vol(\beta) \le \Vol(L).$$\vskip-18pt\strut}
\end{block}
\end{frame}

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

\begin{block}{Theorem {\rm (Boucksom, PhD 2002)}} Let $X$ be a compact K\"ahler
manifold and\vskip8pt
\alert{\centerline{$
H^{k,k}_{\ge 0}(X)=\big\{\{T\}\in H^{k,k}(X,\bR)\,;\;
\hbox{$T$ closed${}\ge 0$}\big\}.
$}}\vskip6pt
\pause\vskip-16pt\strut

\begin{itemize}
\item \claim{$\forall k=1,2,\ldots,n$, $\exists$ canonical
``movable intersection product''\vskip8pt
\alert{\centerline{$
\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
$}}\vskip8pt
such that $\Vol(\alpha)=\langle\alpha^n\rangle$ whenever $\alpha$ is
a big class.}
\pause

\item \claim{The product is increasing, homogeneous of 
degree $1$ and superadditive in each argument, i.e.\
\vskip8pt
\alert{\centerline{$
\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.
$}}\vskip8pt
It coincides with the ordinary intersection
product when the $\alpha_j\in\ovl{\cK}$ are nef classes.}
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
  \frametitle{Divisorial Zariski decomposition}
Using the above intersection product, one can easily derive the
following divisorial Zariski decomposition result.

\begin{block}{Theorem {\rm (Boucksom, PhD 2002)}}  
\begin{itemize}
\item For $k=1$, one gets a ``divisorial Zariski decomposition''
\alert{$$
\alpha=\{N(\alpha)\}+\langle\alpha\rangle
$$}
where~:
\item
$N(\alpha)$ 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
\alert{$\langle\alpha\rangle$ is ``nef in codimension $1$''.}
\end{itemize}
\end{block}
\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}

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

\begin{block}{Conjecture {\rm (``generalized abundance conjecture'')}}
For an arbitrary compact K\"ahler manifold~$X$, the Kodaira
dimension should be equal to the numerical dimension~:\vskip5pt
\alert{\centerline{$\kappa(X)=\num(c_1(K_X)).$}}
\end{block}

{\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{Lecture 2}
\begin{block}{Uniruledness, rational connectedness and $-K_X$ nef}
\strut\vskip8pt  
We start by proving an orthogonality estimate, which in its turn
identifies the dual of the cone of (pseudo)-effective divisors.\vskip8pt
From there, we derive necessary and sufficient conditions
for uniruledness and rational connectedness.\vskip8pt
We conclude this lecture by presenting examples of compact K\"ahler
manifolds $X$ such that $-K_X$ is semipositive or nef.
\end{block}
\end{frame}


\begin{frame}
\frametitle{Orthogonality estimate}
\vskip-5pt
\begin{block}{Theorem} 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\vskip5pt
\centerline{$\mu_m^\star T_m = [E_m]+[D_m]$.}\vskip5pt
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$\,)$.
\end{block}

By going to the limit, one gets
\medskip

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

\begin{frame}
\frametitle{Proof of the orthogonality estimate}

The argument for the ``almost'' orthogonality of the two parts in
$\mu_m^*T_m=E_m+D_m$ is similar to the one used for projections from
Hilbert space onto a closed convex set, where the segment to closest point is
orthogonal to tangent plane.

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\EndFig
\end{frame}

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

\begin{block}{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.
\end{block}

{\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 -28 4 $\NS_\bR(X)$\ELTX
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\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 $C_t\simeq\bP^1_\bC$ (the family is
assumed to be algebraic, and ``covered'' means that a Zariski open set
of the variety at least is covered).

\begin{block}{Theorem {\rm (Boucksom-Demailly-Paun-Peternell 2004)}}
A projective manifold $X$ is \alert{not uniruled} if and only if
$K_X$ is pseudo-effective, i.e.\ $K_X\in\cE_{\NS}$.
\end{block}

{\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{Criterion for rational connectedness}

\begin{block}{Definition} Recall that a compact complex manifold is
said to be \alert{rationally connected} (or RC for short) if any 
2 points can be joined by a chain of rational curves.
\end{block}\pause

\begin{block}{Criterion}
Let $X$ be a projective algebraic $n$-dimensional manifold. The 
following properties are equivalent.
\begin{itemize}
\item[{\rm (a)}] $X$ is \alert{rationally connected}.\pause
\item[{\rm (b)}] $\forall$ invertible subsheaf 
$\cF\subset\cO(\Lambda^pT^*_X)$, $p\ge 1$, 
$\cF$ is \alert{not psef}.\pause
\item[{\rm (c)}] $\forall$ invertible subsheaf 
$\cF\subset\cO((T^*_X)^{\otimes p})$, $p\ge 1$, 
$\cF$~is \alert{not psef}.\pause
\item[{\rm (d)}] For some $($resp.\ for any$)$ ample line bundle $A$ on $X$, 
there exists a constant $C_A>0$ such that\vskip-22pt
$$
\alert{H^0(X,(T^*_X)^{\otimes m}\otimes A^{\otimes k})=0}\quad
\hbox{$\forall m,\,k\in\bN^*$ with $m\ge C_Ak$.}
$$
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Proof of the RC criterion}
\claim{Proof} (essentially from Peternell 2006)\vskip3pt
(a) $\Rightarrow$ (d) is easy (RC implies there are many rational curves
on which $T_X$, so $T^*_X<0$), (d) $\Rightarrow$ (c) and
(c) $\Rightarrow$ (b) are~trivial.\vskip3pt\pause
Thus the only thing left to complete the proof is 
\alert{(b) $\Rightarrow$ (a)}.\vskip3pt\pause

Consider the \alert{MRC quotient} $\pi:X\to Y$, given by the 
``equivalence relation
$x\sim y$ if $x$ and $y$ can be joined by a chain of rational curves
(one may have to blow up $X$ to get a genuine morphism).
\vskip3pt\pause
Then (by definition) the fibers are RC, maximal, and a result of 
Graber-Harris-Starr (2002) implies that \alert{$Y$ is not uniruled}.
\vskip3pt\pause
By BDPP (2004), \alert{$Y$ not uniruled${}\Rightarrow K_Y$ psef}. Then 
$\pi^* K_Y\hookrightarrow\Omega^p_X$ where $p=\dim Y$,
which is a contradiction unless $p=0$, and therefore $X$ is RC.
\end{frame}

\begin{frame}
\frametitle{Structure of projective/compact Kähler varieties}
As is well known since the beginning of the XX${}^{\rm th}$ century at
the geometry of \alert{projective} or \alert{compact K\"ahler manifolds $X$}
depends on the sign of the curvature of the canonical line bundle~$K_X$.
\vskip6pt
\hbox{\strut\kern2cm  \pgfdeclareimage[height=5.4cm]{curves}{curves}
  \pgfuseimage{curves}}
\hbox{\alert{$K_X=\Lambda^nT^*_X,~~\deg(K_X)=2g-2$}}
\end{frame}

\begin{frame}
\frametitle{\strut\kern-8pt
Goal: K\"ahler manifolds with $-K_X\ge 0$ or $-K_X$ nef}
\claim{Recall:} By the Calabi-Yau theorem,\\
\alert{$-K_X\ge 0\Leftrightarrow\exists\omega$
K\"ahler with Ricci$(\omega)\ge 0$,}\\
\alert{$-K_X$ nef${}\Leftrightarrow\forall\varepsilon>0,~
\exists\omega_\varepsilon
=\omega+i\ddbar\varphi_\varepsilon$ such that
\hbox{Ricci$(\omega_\varepsilon)\ge -\varepsilon\omega_\varepsilon$.\kern-20pt}}\\
\vskip3pt\pause
\begin{itemize}\adjustitem
\item Ricci flat manifolds\vskip4pt
-- \alert{Complex tori} $T=\bC^q/\Lambda$\vskip4pt\pause
-- \alert{Holomorphic symplectic manifolds} $S$ (also called
\hbox{\alert{hyperk\"ahler}):\kern-15pt}\\\protect%
\strut\kern8pt$\exists\sigma\in H^0(S,\Omega^2_S)$ symplectic.
One can assume $\pi_1(S)=1$.\vskip4pt\pause
-- \alert{Calabi-Yau manifolds} $Y$: \alert{$\pi_1(Y)=1$, $K_Y$ is trivial
and $H^0(Y,\Omega^k_Y=0)$ for $0<k<\dim Y$.}
\item the rather large class of rationally connected manifolds $Z$\\\protect%
with $-K_Z\ge 0$\pause\\
\item all products $T\times\prod S_j\times\prod Y_k\times\prod Z_\ell$.
\end{itemize}
\vskip3pt
Let us first give examples of varieties in each category.
\end{frame}

\begin{frame}
\frametitle{Example of Ricci flat manifolds}
\vskip-5pt  
-- \alert{Examples of holomorphic symplectic manifolds}:\\
Hilbert schemes $X=S^{[n]}$ of length $n$ subschemes of a K3 surface
and similar ``Kummer varieties'' $X=A^{[n+1]}/A$ associated with
a complex 2-dimensional torus. Some ``sporadic'' examples have been
constructed by O'Grady.\pause

-- \alert{Examples of Calabi-Yau  manifolds}:\\
Smooth hypersurface of degree $n+2$ in $\bP^{n+1}$, suitable
complete intersections in (weighted) projective space.\pause\vskip4pt

Following work by Bogomolov and Fujiki, Beauville has shown:

\begin{block}{Beauville-Bogomolov decomposition theorem (1983)}
Every compact Kähler manifold $X$ with $c_1(X)=0$ admits a finite
étale cover $\widetilde X$ such that\vskip5pt
\centerline{\alert{$
\widetilde X\simeq T\times\prod S_j\times\prod Y_k$} (isometrically)}\vskip5pt
where $T$ is a torus, $S_j$ holomorphic symplectic and $Y_k$ Calabi-Yau.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Examples of RC manifolds with $-K_X\geq 0$ or nef}
\vskip-5pt
Let $X$ be the rational surface obtained
by blowing up $\bP^2$ in 9 distinct points $\{p_i\}$ on a smooth (cubic)
elliptic curve $C\subset\bP^2$, $\mu:X\to\bP^2$ and $\hat C$ the
strict transform of $C$. \pause
Then\vskip3pt
\centerline{$K_X=\mu^*K_{\bP^2}\otimes \cO(\sum E_i)
\Rightarrow
-K_X=\mu^*\cO_{\bP^2}(3)\otimes \cO(-\sum E_i),$}\vskip3pt
thus\vskip3pt
\alert{\centerline{$-K_X=\mu^*\cO_{\bP^2}(C)\otimes \cO(-\sum E_i)=\cO_X(\hat C).$}}\pause
One has\vskip3pt 
$\strut\quad\strut-K_X\cdot \Gamma = \hat C\cdot\Gamma\ge 0\qquad\hbox{if
$\Gamma\ne\hat C$},$
\vskip3pt
$\strut\quad\strut
-K_X\cdot\hat C=(-K_X)^2=(\hat C)^2=C^2-9=0~~\Rightarrow~~\alert{\hbox{$-K_X$ 
nef.\kern-15pt}}$\vskip4pt
In fact\vskip4pt
\centerline{%
$G:=(-K_X)_{|\hat C}\simeq \cO_{\bP^2|C}(3)\otimes \cO_C(-\sum p_i)
\in\Pic^0(C)$}\vskip4pt\pause
Brunella has shown that $-K_X$ is $C^\infty$ semi-positive if
$c_1(G)$ satisfies a diophantine condition found by T.~Ueda, 
but otherwise it need not be semi-positive (although nef).\vskip5pt\pause
\end{frame}

\begin{frame}
\frametitle{Lecture 3}
\begin{block}{Holonomy principle and main structure theorems}
\strut\vskip8pt
We describe here a structure theorem for compact K\"ahler manifolds
with $-K_X\geq 0$. It depends in an essential way on the concept
of holonomy and its implications on the geometry of the manifold.
\strut\vskip8pt
Then, following work of Junyan Cao and Andreas H\"oring, we discuss
some results describing 
the structure of compact K\"ahler manifolds with $-K_X$~nef.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Structure theorem for manifolds with $-K_X\ge 0$}
\begin{block}{Theorem [Campana, D., Peternell, 2012]}\strut\vskip-12pt
Let $X$ be a compact K\"ahler manifold with
$-K_X\ge 0$. Then:
\begin{itemize}
\item[{\rm (a)}] $\exists$ holomorphic and isometric splitting
in irreducible factors\vskip4pt
\alert{\centerline{$\widetilde X=\hbox{universal cover of}~X
\simeq\bC^q\times\prod Y_j\times\prod S_k\times\prod Z_\ell$}}\vskip4pt
where $Y_j={}$Calabi-Yau $($holonomy $\SU(n_j))$, $S_k={}$holomorphic
symplectic $($holonomy $\Sp(n'_k/2))$, and $Z_\ell={}$ RC with
$-K_{Z_\ell}\ge 0$ $($holonomy $\UU(n''_\ell)$ or compact
symmetric space$)$.\pause
\item[{\rm (b)}] There exists a finite \'etale Galois cover $\widehat X\to X$
such that the Albanese map \hbox{$\alpha:\widehat X\to\Alb(\widehat X)$} is an
$($isometrically$)$ locally trivial holomorphic fiber bundle whose fibers 
are products $\prod Y_j\times\prod S_k\times\prod Z_\ell$, as
described in {\rm(a)}.\pause
\item[{\rm (c)}] $\pi_1(\widehat X)\simeq\bZ^{2q}\rtimes \Gamma$,
$\Gamma$ finite (``\alert{almost abelian}'' group).
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Concept of holonomy / restricted holonomy}
\vskip-4pt  
Let $(E, h)$ be a Euclidean vector bundle over $X$, and $\nabla_h$ a compatible
connection. For every path $\gamma:[0,1]\to X$ joining $p,q\in X$,
one considers the (metric preserving) parallel transport operator
$\tau_{p,q}:E_p\to E_q$, $v(0)\mapsto v(1)$ where
$\frac{\nabla_h}{dt}(\frac{dv}{dt})=0$.
\vskip3pt\strut\kern1.4cm
\pgfdeclareimage[height=1.6cm]{parallel-transport}{parallel_transport}
\pgfuseimage{parallel-transport}
\pause

\begin{block}{Theorem} The holonomy group $\Hol(E,h)_p$ (resp. restricted
holonomy group $\Hol^\circ(E,h)_p$) of a Euclidean vector bundle
$E\to X$ is the subgroup of $\SO(E_p)$ generated by parallel
transport operators $\tau_{p,p}$ over loops based at $p$ (resp.\ contractible loops).
\vskip4pt\pause
It is independent of $p$, up to conjugation. In the hermitian case,
$\Hol^\circ(E,h)_p$ is contained in the unitary group $\UU(E_p)$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{The De Rham splitting theorem}

In the important case when $E=T_X$, we have

\begin{block}{De Rham splitting theorem}
If $(X,h)$ is complete and the holonomy representation of $H=\Hol^\circ(T_X,h)_p$ splits into irreductible representations $T_{X,p}=S_1\oplus\ldots\oplus S_k$,
then the universal cover $\widetilde X$ \alert{splits metrically} as
\alert{$$\widetilde X=X_1\times\ldots\times X_k$$}
where the holonomy of $X_j$ yields the irreducible representation
on $S_j\subset T_{X_j,p}$.  \\
This means in particular that the pull-back metric $\widetilde h$
splits as a direct sum of metrics $h_1\oplus\ldots\oplus h_k$ on the
factors~$X_j$.
\end{block}
\end{frame}  

\begin{frame}
\frametitle{Berger's classification of holonomy groups}
The Berger classification of holonomy groups of non locally
symmetric Riemannian manifolds stands as follows:\vskip5pt
\pgfdeclareimage[height=4.5cm]{tableau}{tableau}
\pgfuseimage{tableau}\pause
  
A simple proof of the Berger holonomy classification has been obtained
by Carlos E.\ Olmos in 2005, by showing that ${\rm Hol}(M,g)$ acts
transitively on the unit sphere if $M$ is not locally
symmetric.\vskip6pt\pause The complex cases are shown by the
\alert{red lines}.
\end{frame}

\begin{frame}
\frametitle{Generalized holonomy principle}
\strut\vskip-24pt
\begin{block}{Generalized holonomy principle}
Let $(E,h)\to X$ be a hermitian holomorphic vector bundle of rank $r$ 
over $X$ compact/$\bC$. Assume that\vskip6pt
\alert{\centerline{$
\Theta_{E,h}\wedge\frac{\omega^{n-1}}{(n-1)!}=
B\,\frac{\omega^n}{n!},\quad B\in\Herm(E,E),\quad B\ge 0~~\hbox{on $X$}.
$}}\vskip6pt\pause
Let $H$ the restricted holonomy group of $(E,h)$. Then
\begin{itemize}
\item[{\rm (a)}]If there exists a psef invertible sheaf $\cL\subset
\cO((E^*)^{\otimes m})$, then \alert{$\cL$ is flat} and invariant under parallel 
transport by the connection of $(E^*)^{\otimes m}$ induced by the Chern connection
$\nabla$ of~$(E,h)\,;$ moreover, \alert{$H$ acts trivially on~$\cL$}.\pause
\item[{\rm (b)}] If $H$ satisfies $H=\UU(r)$, then none of the 
invertible sheaves $\cL\subset\cO((E^*)^{\otimes m})$ can be psef
for $m\ge 1$.
\end{itemize}
\end{block}\pause
\claim{Proof.} $\cL\subset \cO((E^*)^{\otimes m})$ which has trace of 
curvature${}\le 0$ while $\Theta_{\cL}\ge 0$, use Bochner formula.\qed
\end{frame}

\begin{frame}
\frametitle{Proof of the generalized holonomy principle}
\vskip-5pt
Assume that we have an invertible sheaf $\cL\subset \cO((E^*)^{\otimes m})$
that is pseudoeffective. For a local non zero section $f$ of $\cL$,
one considers
\alert{$$
\psi=\frac{|f|^2_{h^{*m}}}{|f|^2_{h_\cL}}.
$$}%
Writing $|f|_{h_\cL}^2=e^{-\varphi}$, a standard Bochner type inequality yields
\alert{$$
\Delta_\omega\psi\ge |f|^2_{h^{*m}}~e^\varphi(\Delta_\omega\varphi+m\lambda_1)+
|\nabla^{1,0}_hf+f\partial\varphi|^2_{\omega,h^{*m}}~e^\varphi
$$}%
where $\lambda_1(z)\ge 0$ is the lowest eigenvalue of the hermitian
endomorphism $B=\Tr_\omega\Theta_{E,h}$ at point $z\in X$.\pause\vskip5pt
By a conformal change, one can arrange $\omega$ to be a Gauduchon
metric ($\ddbar(\omega^{n-1})=0$), and then observe by Stokes' theorem that
\alert{$$
\int_X\Delta_\omega\psi\,\omega^n=\int_X i\ddbar\psi\wedge\omega^{n-1}=0.
$$}
Then in particular \alert{$\nabla^{1,0}_hf+f\partial\varphi=0$} and
the theorem follows.
\end{frame}

\begin{frame}
\frametitle{Proof of the structure theorem for $-K_X\geq 0$}
\vskip-6pt
\begin{block}{Cheeger-Gromoll theorem (J.\ Diff.\ Geometry 1971)}
Let $(X,g)$ be a complete Riemannian manifold of nonnegative Ricci curvature.
Then the universal cover $\widetilde X$ splits as\vskip4pt
\centerline{\alert{$
\widetilde X=\bR^q\times Z
$}}\vskip4pt
where $Z$ contains no lines and still has nonnegative Ricci curvature.\\
\pause
Moreover, if $X$ is compact, then $Z$ itself is compact.
\end{block}\pause

\claim{Proof of the structure theorem for $-K_X\geq 0$, using the generalized
holonomy principle.} Let $(X,\omega)$ be
compact K\"ahler with $-K_X\geq 0$. By the De Rham and Cheeger-Gromoll theorems,
write $\widetilde X$ as a product of manifolds with irreducible holonomy
\centerline{\alert{$
\widetilde X
\simeq\bC^q\times\prod Y_j\times\prod S_k\times\prod Z_\ell$}}
\vskip4pt
where $\Hol^\circ(Y_j)=\SU(n_j))$ (Calabi-Yau), $\Hol^\circ(S_k)=\Sp(n'_k/2))$
(holomorphic symplectic), and $Z_\ell$ either compact hermitian symmetric,
or $\Hol^\circ(Z_\ell)=\UU(n''_\ell)
\Rightarrow~Z_\ell$
rationally \hbox{connected~\alert{(H.P.)}\kern-20pt}
\end{frame}
  
\begin{frame}
\frametitle{Surjectivity of the Albanese morphism}
\vskip-6pt
Recall that if $X$ is a compact K\"ahler manifold, the
\hbox{Albanese map\kern-15pt}%%
\alert{$$\alpha_X:X\to\Alb(X):=\bC^q/\Lambda$$}
is the holomorphic map given by
\alert{$$z\mapsto\alpha_X(z)=\Big(\smash{\int_{z_0}^z}u_j\Big)_{1\le j\le q}~~
\hbox{mod period subgroup}~\Lambda\subset\bC^q,$$}
where $(u_1,\ldots,u_q)$ is a basis of $H^0(X,\Omega^1_X)$.\pause
\begin{block}{Theorem [Qi Zhang, 1996, 2005]} If $X$ is projective and
\alert{$-K_X$ is nef},
then \alert{$\alpha_X$ is surjective}.
\end{block}\pause
\vskip-3pt
\claim{Proof.} Based on characteristic $p$ techniques.
\begin{block}{Theorem [M. P\u{a}un, 2012]} If $X$ is compact K\"ahler
and \alert{$-K_X$ is nef}, then \alert{$\alpha_X$ is surjective}.
\end{block}\pause
\vskip-3pt
\claim{Proof.} Based on variation arguments for twisted
K\"ahler-Einstein metrics.
\end{frame}

\begin{frame}
\frametitle{\strut\kern-6pt Approach via generically nef vector bundles
  \hbox{(J.Cao)\kern-20pt} }
\vskip-9pt
\begin{block}{Definition} Let $X$ be a compact K\"ahler manifold,
$\cE\to X$ a torsion free sheaf.
\begin{enumerate}
\item[{\rm (a)}] $\cE$ is \alert{stable with respect to a K\"ahler
class $\omega$} if\vskip2pt
\centerline{$\mu_{\omega}(\cS)=\hbox{$\omega$-slope of $\cS$}:={\displaystyle
\frac{\int_Xc_1(\cS)\wedge\omega^{n-1}}{\hbox{\rm rank}\,\cS}}$}\vskip4pt
is such that $\mu_\omega(\cS)<\mu_\omega(\cE)$ for all subsheaves
$0\subsetneq \cS\subsetneq \cE$.\pause
\item[{\rm (b)}] $\cE$ is \alert{generically nef with respect to $\omega$}
if $\mu_\omega(\cE/\cS)\geq 0$ for all subsheaves $\cS\subset\cE$. If
$\cE$ is $\omega$-generically nef for all $\omega,$ we simply say that $\cE$ is \alert{generically nef}.\pause 
\item[{\rm (c)}] Let\vskip-22pt
$$ 0  = \cE_0 \subset  \cE_1 \subset \ldots \subset \cE_s = \cE$$
be a filtration of $\cE$ by torsion free coherent subsheaves such that the quotients $\cE_{j}/\cE_{j-1}$ are 
$\omega$-stable subsheaves of $\cE/\cE_{j-1}$ of maximal rank.
We call such a sequence a \alert{refined Harder-Narasimhan (HN)
filtration w.r.t. $\omega.$}
\end{enumerate} 
\end{block} 
\end{frame}

\begin{frame}
\frametitle{Characterization of generically nef vector bundles}
\vskip-5pt
It is a standard fact that refined HN-filtrations always exist. Bando and
Siu have proved that the graded pieces $G_j:=\cE_j/\cE_{j-1}$ then possess
a Hermite-Einstein metric $h_j$ such that
\vskip4pt\centerline{\alert{
$\Tr_\omega\Theta_{G_j,h_j}=\mu_\omega(G_j)\cdot\Id_{G_j}$,
}}\vskip4pt
and that $h_j$ is smooth outside of the codim 2 locus where
$G_j:=\cE_j/\cE_{j-1}$ is not locally free. Moreover one always has
\vskip4pt\centerline{
\alert{$\mu_{\omega}(\cE_j/\cE_{j-1}) \geq \mu_{\omega}(\cE_{j+1}/\cE_j)$,
~~$\forall j$.}}

\begin{block}{Proposition} 
Let $(X,\omega)$ be a compact K\"ahler manifold and $\cE$ a torsion free sheaf on $X.$ 
Then $\cE$ is $\omega$-generically nef if and only if
\vskip4pt\centerline{\alert{
$\mu_{\omega}(\cE_j)/\cE_{j-1}) \geq 0 $}}
\vskip4pt
for some refined HN-filtration. 
\end{block} 

\claim{Proof.} This is done by easy arguments on filtrations.\qed
\end{frame}

\begin{frame}
\frametitle{A result of J.\ Cao about manifolds with $-K_X$ nef}
\begin{block}{Theorem} (Junyan Cao, 2013) \label{cao} Let $X$ be a compact K\"ahler manifold with $-K_X$ nef.
Then the tangent bundle $T_X$ is $\omega$-generically nef for all K\"ahler classes $\omega$. 
\end{block} 

\claim{Proof.} use the fact that $\forall\varepsilon>0$, $\exists$ K\"ahler metric with Ricci$(\omega_\varepsilon)\ge-\varepsilon\,\omega_\varepsilon$ (Yau, DPS 1995).\vskip5pt\pause

From this, one can deduce 

\begin{block}{Theorem} \label{tensorcao} Let $X$ be a compact K\"ahler manifold with nef anticanonical bundle. 
Then the bundles $T_X^{\otimes m} $ are $\omega$-generically nef for all K\"ahler classes $\omega$ and all positive integers $m.$ 
In particular, the bundles $S^kT_X$ and $\bigwedge^pT_X$ are $\omega$-generically nef. 
\end{block} 
\end{frame}

\begin{frame}
\frametitle{A lemma on sections of contravariant tensors}

\begin{block}{Lemma} Let $(X,\omega)$ be a compact K\"ahler manifold with 
$-K_X$ nef and\vskip4pt
\centerline{$ \eta \in H^0(X,(\Omega^1_X)^{\otimes m} \otimes \cL)$}\vskip4pt
where $\cL$ is a \alert{numerically trivial} line bundle on $X$.\pause\\
Then the filtered parts of $\eta$ w.r.t.\ the refined HN filtration are
\alert{parallel} w.r.t.\ the Bando-Siu metric in the $0$ slope parts,
and the${}<0$ slope parts vanish.
\end{block} 
\pause

\claim{Proof.} By Cao's theorem there exists a refined HN-filtration\vskip4pt
\centerline{$ 0  = \cE_0 \subset  \cE_1 \subset \ldots \subset \cE_s = T_X ^{\otimes m}$}\vskip4pt
with $ \omega$-stable quotients $\cE_{i+1}/\cE_i$ such that 
$\mu_{\omega}(\cE_{i+1}/\cE_i) \geq 0$ for all~$i$. Then we use the fact that
any section in a (semi-)negative slope reflexive sheaf
$\cE_{i+1}/\cE_i \otimes \cL$
is parallel w.r.t.\ its Bando-Siu metric (resp.\ vanishes).\qed
\end{frame}

\begin{frame}
\frametitle{Smoothness of the Albanese morphism (after Cao)\kern-15pt}
\vskip-8pt
\begin{block}{Theorem (Junyan Cao 2013)} Non-zero holomorphic $p$-forms on a compact K\"ahler manifold $X$ with $-K_X$ nef
\alert{vanish only on the singular locus of the refined 
HN filtration of $T^*X$.} 
\end{block}\pause

This already implies the following result.

\begin{block}{Corollary}\label{albanese}
Let $X$ be a compact K\"ahler manifold with nef anticanonical bundle.
Then the Albanese map $\alpha_X:X\to\Alb(X)$ is 
a~\alert{submersion} on the complement of
the HN filtration singular locus in $X$\vskip0pt
[${}\Rightarrow \alpha_X$ surjects onto $\Alb(X)\,$].
\end{block}\pause

\claim{Proof.} The differential $d\alpha_X$ is given by
$(du_1,\ldots,du_q)$ where  $(u_1,\ldots,u_q)$ is a basis of
$1$-forms, $q=\dim H^0(X,\Omega^1_X)$.\vskip5pt\pause

Cao's thm $\Rightarrow$ rank of $(du_1,\ldots,du_q)$ is${}=q$ generically.\qed
\end{frame} 

\begin{frame}
\frametitle{\blank{Isotriviality of the Albanese map}}
\vskip-5pt
\begin{block}{Theorem [Junyan Cao, arXiv:1612.05921] }
Let $X$ be a projective manifold with nef anti-canonical bundle.
Then the Albanese map $\alpha_X:X\to Y=\Alb(X)$ is
\alert{locally trivial}, i.e.,
for any small open set $U \subset Y$, $\alpha_X^{-1}(U)$ is biholomorphic
to the product $U \times F$, where $F$ is the generic fiber of $\alpha_X$.\\
\pause Moreover \alert{$-K_F$ is again nef}.
\end{block}
\pause

\claim{Proof.} Let $A$ be a (large) ample line bundle on $X$ and
$E=(\alpha_X)_*A$ its direct image. Then \alert{$E=(\alpha_X)_*(mK_{X/Y}+L)$}
with $L=A-mK_{X/Y}=A-mK_X$ nef. By results of Berndtsson-P\u{a}un
on direct images, one can show that \alert{$\det E$ is pseudoeffective}.\pause

Using arguments of [DPS 1994], one can infer that
$E'=E\otimes (\det E)^{-1/r}$, $r=\hbox{rank}(E)$, is \alert{numerically flat,
hence a locally constant coefficient system} (C.\ Simpson, Deng Ya 2017).
However, if $A\gg 0$, $E$ provides equations of the fibers.\qed
\end{frame}

\begin{frame}
\frametitle{\blank{The simply connected case}}

The above results reduce the study of projective manifolds with
$-K_X$ nef to the case when $\pi_1(X)=0$.\pause

\begin{block}{Theorem [Junyan Cao, Andreas H\"oring, arXiv:1706.08814] }
Let $X$ be a projective manifold such that $-K_X$ is nef and $\pi_1(X)=0$.
Then $X=W\times Z$ with $K_W\sim 0$ and $Z$ is a rationally
connected manifold.
\end{block}\pause

\begin{block}{Corollary [Junyan Cao, Andreas H\"oring]}
Let $X$ be a projective manifold such that $-K_X$ is nef.
Then after replacing $X$ with a finite étale cover, the Albanese map
$\alpha_X$ is locally trivial and its fibers are of the form $\prod S_j\times
\prod Y_k\times\prod Z_\ell$ with $S_j$ holomorphic symplectic,
$Y_k$~Calabi-Yau and $Z_\ell$ rationally connected.
\end{block}
\end{frame}

\begin{frame}
\frametitle{\blank{Further problems (I)}}
\vskip-6pt
\begin{block}{Partly solved questions}
\begin{itemize}
\item
Develop further the theory of singular Calabi-Yau and singular
holomorphic symplectic manifolds (work of Greb-Kebekus-Peternell).\pause
\item
Show that the ``slope $\pm\varepsilon$'' part corresponds to
blown-up tori, singular Calabi-Yau and singular holomorphic symplectic 
manifolds (as fibers and targets).\pause
\item
The rest of $T_X$ (slope${}<0$) should
yield a general type orbifold quotient
(according to conjectures of F.~Campana).
\end{itemize}
\end{block}\pause

\begin{block}{Possible general definition of singular Calabi-Yau manifolds}
A compact K\"ahler manifold $X$ is a \alert{singular Calabi-Yau manifold}
if~$X$ has a non singular model $X'$ satisfying $\pi_1(X')=0$ and\\
\alert{$K_{X'}=E$ for an effective divisor $E$ of numerical dimension $0$}
(an exceptional divisor), and \alert{$H^0(X',\Omega^p_{X'})=0$ for $0<p<\dim X$.}
\end{block}

\end{frame}

\begin{frame}
\frametitle{\blank{Further problems (II)}}
\vskip-4pt
\begin{block}{Possible general definition of singular hyperk\"ahler manifolds}
A compact K\"ahler manifold $X=X^{2p}$ is a
\alert{singular hyperk\"ahler manifold}
if $X$ has a non singular model $X'$ satisfying $\pi_1(X')=0$ and possessing
a section $\sigma\in H^0(X',\Omega^2_{X'})$ such that \alert{the zero
divisor $E=\hbox{div}(\sigma^p)$ has numerical dimension~$0$},\\
hence, as a consequence, $K_{X'}=E$ is purely exceptional.
\end{block}\pause

\begin{block}{Conjecture (known by BDPP for $X$ projective!)}
Let $X$ be compact K\"ahler, and let $X\to Y$ be
the MRC fibration (after replacing $X$ by a suitable blow-up to make
$X\to Y$ a genuine morphism). Then \alert{$K_Y$ is psef}.
\end{block}\pause

\claim{Proof ?} Take the part of slope${}>0$ in the HN filtration of 
$T_X$, w.r.t. to classes in the dual of the psef cone, show that this
corresponds to the MRC fibration, and apply duality.
\end{frame}

\begin{frame}
\frametitle{\blank{Further problems (III)}} 
\vskip-5pt
\begin{itemize}
\item According to F.~Campana, one should be able
to factorize ``special subvarieties'' of $Y$ (i.e.\ essentially the
RC, singular Calabi-Yau and hyperk\"ahler subvarieties) to get a
morphism $Y\to Z$, along with a ramification divisor $\Delta\subset Z$
of that morphism, in such a way that the pair $(Z,\Delta)$ is of
general type, i.e.\  \alert{$K_Z+\Delta$ is big}.\pause
\item
An interesting class of manifolds is the larger class of compact K\"ahler
manifolds such that \alert{$K_X=E-D$}, where $D$ is a pseudoeffective
divisor and $E$ an effective divisor of numerical dimension $0$.\pause
\vskip5pt
This class is obviously birationally invariant (while the condition
$-K_X$ nef was not~!).\pause
\vskip5pt
One can hopefully expect similar decomposition theorems for varieties
in this class. They might possibly include all rationally connected varieties.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{The end}
\strut\vskip-12pt  
\centerline{\hugebf Thank you for your attention!}
\pgfdeclareimage[height=6.5cm]{CalabiYau}{CalabiYau}
\strut\kern2cm\pgfuseimage{CalabiYau}
\vskip-5mm
\centerline{A representation of the real points of a quintic
 Calabi-Yau manifold}
\end{frame}

\end{document}

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