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% Since this a solution template for a generic talk, very little can
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% 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{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{Hodge decomposition theorem}
  \begin{itemize} 
  \item \claim{{\bf Theorem.} {\it If $(X,\omega)$ is compact K\"ahler, then}
$$
H^k(X,\bC)=\bigoplus_{p+q=k}H^{p,q}(X,\bC).
$$}
\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{$\Delta_\omega\alpha=0$}.}
\pause
\item \claim{{\bf Hodge Conjecture {\rm $[$a millenium problem!$\,]$}.}\\ 
{\it If $X$ is a projective algebraic manifold,\\
Hodge $(p,p)$-classes${}=H^{p,p}(X,\bC)\cap H^{2p}(X,\bQ)$\\
are generated by}} {\it \alert{classes of algebraic cycles of codimension $p$
with $\bQ$-coefficients.}}\\
\pause
\item \claim{{\bf(Claire Voisin, 2001)} {\it $\exists$ $4$-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{itemize}
\end{frame}

\begin{frame}
  \frametitle{Idea of proof of Claire Voisin's counterexample}
The idea is to show the existence of a $4$-dimensional complex torus 
$X=\bC^4/\Lambda$ which does not contain any analytic subset of 
positive dimension, and such
that the Hodge classes of degree $4$ are perpendicular to $\omega^{n-2}$
for a suitable choice of the K\"ahler metric $\omega$. 
\medskip

The lattice  $\Lambda$ is explicitly found via a number theoretic 
\alert{construction of Weil}
based on the number field $\bQ[i]$, also considered by S.~Zucker. 
\medskip

The theorem of existence of \alert{Hermitian Yang-Mills connections}
for stable bundles combined with L\"ubke's inequality then implies
$c_2(\cF)=0$ for every coherent sheaf $\cF$ on the torus.
\end{frame}

%%\section*{Kodaira embedding theorem}

\begin{frame}
  \frametitle{Kodaira embedding theorem}
\claim{{\bf Theorem.} {\it $X$ a compact complex $n$-dimensional 
manifold. Then the following properties are equivalent.}
\begin{itemize} 
\item \claim{\it\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{\it
\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-30pt}
\end{eqnarray*}}}
\item \claim{\it
$X$ possesses a Hodge metric, i.e., a \alert{K\"ahler metric $\omega$ such that
$\{\omega\}\in H^2(X,\bZ)$}.}
\end{itemize}}
\end{frame}

%%\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.}
\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{frame}

\begin{frame}
\frametitle{K\"ahler (red) cone and pseudoeffective (blue) cone}
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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. 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}
{\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$)$.}
\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}}}
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}
\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.} {\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

\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
$$\mu_m^\star T_m=[E_m]+\beta_m$$}
where $E_m$ is an effective $\bQ$-divisor on $\wt X_m$ with coefficients
in ${1\over m}\bZ$ and $\beta_m$ is a K\"ahler form on $\wt X_m$.}}
\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 $\beta_m=\mu_m^*T_m-[E_m]$.
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
\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 $\beta_m$\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 $\cC$}
\claim{{\bf Theorem {\rm (Demailly-P\v{a}un 2004)}.} {\it 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 $T$.}}%
\medskip

{\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.
\medskip

\claim{{\bf Definition.} {\it 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{frame}

\begin{frame}
\frametitle{Numerical characterization of the K\"ahler cone}
\claim{{\bf Theorem} {\rm (Demailly-P\v{a}un 2004)}.\\ 
{\it 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''.\\
\alert{Then the K\"ahler cone $\cK$ is\\
one of the connected components of $\cP$}.}}
\medskip
\pause

\claim{{\bf Corollary} {\rm (Projective case)}.\\ {\it
If $X$ is projective algebraic, then \alert{$\cK=\cP$}.}}%
\medskip

{\it Note:} this is a ``transcendental version'' of the Nakai-Moishezon
criterion.
\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{%
$$
\int_X f_\varepsilon = (\alpha+\varepsilon\omega)^n
\quad\hbox{in $H^{n,n}(X,\bR)$}.
$$}%
Otherwise, the volume form of the K\"ahler metric
$\alpha_\varepsilon=\alpha+\varepsilon\omega+ i\ddbar\varphi_\varepsilon$ 
can be spread \alert{randomly}.
\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
spread 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}
\claim{{\bf Main conclusion} {\rm (Demailly-P\v{a}un 2004).}\\
{\it Let $X$ be a compact K\"ahler manifold
and let $\{\alpha\}\in\ovl\cK$ such that $\int_X\alpha^n>0$.\\
Then $\{\alpha\}$ contains a K\"ahler
current $T$, i.e.\ $\{\alpha\}\in\cE^\circ$.}}
\InsertFig 45 46
{
1 mm unit
0.3 setlinewidth
0 20 moveto
20 circle stroke
5 20 moveto
15 circle stroke
20 20 moveto
0.7 disk
8 34.8 moveto
0.7 disk
}
\LabelTeX -16 19 $\cE$\ELTX
\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 (DP2004)}. {\it
Let $X$ be a compact K\"ahler manifold. 
$$
\{\alpha\}\in H^{1,1}(X,\bR)~~\hbox{\it is K\"ahler}\Leftrightarrow
\exists \omega~\hbox{\it K\"ahler s.t.}~\int_Y\alpha^k\wedge\omega^{p-k}>0
$$
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 (DP2004)}. {\it
Let $X$ be a compact K\"ahler manifold. 
$$
\{\alpha\}\in H^{1,1}(X,\bR)~~\hbox{\it is nef}~(\alpha\in\ovl\cK)
\Leftrightarrow
\forall \omega~\hbox{\it K\"ahler}~\int_Y\alpha\wedge\omega^{p-1}\ge 0
$$
for all $Y\subset X$ irreducible and all $k=1,2,\ldots,p=\dim Y$.}}
\medskip

\claim{{\bf Consequence.} {\it 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 $[Y]\wedge\omega^{p-1}$ in 
$H^{n-1,n-1}(X,\bR)$.}}\\
\phantom{\strut}
\end{frame}

\begin{frame}
\frametitle{Deformations of compact K\"ahler manifolds}
A \alert{deformation of compact complex manifolds} is a proper 
holomorphic map 
$$\pi:\cX\to S\qquad
\hbox{with smooth fibers $X_t=\pi^{-1}(t)$}.
$$

\claim{{\bf Basic question} {\rm (Kodaira $\sim$ 1960)}. Is every compact
K\"ahler manifold $X$ a limit of projective manifolds :
$$
X\simeq X_0=\lim X_{t_\nu},~~t_\nu\to 0,~~X_{t_\nu}~\hbox{projective}~?
$$}%

\claim{{\bf Recent results by Claire~Voisin (2004)}} {\it
\begin{itemize}
\item \claim{In any dimension${}\ge 4$, $\exists X$ compact K\"ahler
  manifold which does not have the homotopy type $($or even the
  homology ring$)$ of a complex projective manifold.}
\item 
  \claim{In any dimension${}\ge 8$, $\exists X$ compact K\"ahler
  manifold such that no compact bimeromorphic model $X'$ of $X$
  has the homotopy type of a projective manifold.}\\
\phantom{\strut}
\end{itemize}}
\end{frame}

\begin{frame}
\frametitle{Conjecture on deformation stability of the K\"ahler property}
\claim{{\bf Theorem} {\rm (Kodaira and Spencer 1960)}.\\
\alert{The K\"ahler property is open with 
respect to deformation}~:\\
if $X_{t_0}$ is K\"ahler for some $t_0\in S$, then 
the nearby fibers $X_t$ are also K\"ahler (where ``nearby'' is in
metric topology).}
\medskip

We expect much more.
\medskip

\claim{{\bf Conjecture}. {\it Let $\cX\to S$ be a deformation with 
irreducible base
space $S$ such that \alert{some fiber $X_{t_0}$ is K\"ahler}. Then
there should exist a countable union of analytic strata $S_\nu\subset S$, $S_\nu\neq S$, such that
\begin{itemize}
\item \alert{$X_t$ is K\"ahler for $t\in S\ssm\bigcup S_\nu$.}
\item \alert{$X_t$ is bimeromorphic to a K\"ahler manifold $($i.e.\ has
a K\"ahler current$)$ for $t\in \bigcup S_\nu$.}
\vskip0pt
\end{itemize}}}
\end{frame}

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

\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 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{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}
\InsertFig 23.5 60
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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
\LabelTeX 70 4 $\NS^{n-1}_\bR(X)$\ELTX
\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{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$\phantom{\strut}$

\claim{{\bf Theorem} {\rm(Demailly-P\v{a}un 2001)}\\ {\it
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$)$}}
\pause
\medskip

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

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

\end{frame}

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

\claim{{\bf Definition} (\rm Boucksom 2002).\\ {\it
The \alert{volume} $($\alert{movable self-intersection}$)$ of a 
big class $\alpha\in\cE^\circ$ is
$$
\Vol(\alpha)=\sup_{T\in \alpha}\int_{\wt X}\beta^n>0
$$
where the supremum is taken over all K\"ahler currents $T\in \alpha$
with logarithmic poles, and $\mu^\star T=[E]+\beta$ with
respect to some modification $\mu:\wt X\to X$.}}
\medskip

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

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

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

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

\claim{{\bf Theorem.} {\it Let $L$ be a big line bundle on the
projective manifold $X$.  Let $\epsilon > 0$. Then there exists a
modification $\mu: X_{\epsilon} \to X$ and a decomposition $\mu^*(L) =
E + \beta $ with $E$ an effective $\bQ$-divisor and $\beta$ a big and
nef $\bQ$-divisor such that
\alert{$$\Vol(L) -\varepsilon\le \Vol(\beta) \le \Vol(L).$$}}}
\end{frame}

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

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

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

\item \claim{\it The product is increasing, homogeneous of 
degree $1$ and superadditive in each argument, i.e.\
$$
\langle\alpha_1\cdots(\alpha'_j+\alpha''_j)\cdots \alpha_k\rangle\ge
\langle\alpha_1\cdots\alpha'_j\cdots \alpha_k\rangle+
\langle\alpha_1\cdots\alpha''_j\cdots \alpha_k\rangle.
$$
It coincides with the ordinary intersection
product when the $\alpha_j\in\ovl{\cK}$ are nef classes.}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Movable intersection theory (continued)}
\begin{itemize}
\item \claim{\it For $k=1$, one gets a ``divisorial Zariski decomposition''
$$
\alpha=\{N(\alpha)\}+\langle\alpha\rangle
$$
where~:}
\begin{itemize}
\item<2>
\claim{\it $N(\alpha)$ is a uniquely defined effective divisor which is
called the ``negative divisorial part'' of $\alpha$. The map
$\alpha\mapsto N(\alpha)$ is homogeneous and subadditive~;}
\item<2>
\claim{\it $\langle\alpha\rangle$ is ``nef outside codimension $2$''.}
\end{itemize}
\end{itemize}
\end{frame}

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


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

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

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

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

\begin{frame}
\frametitle{Orthogonality estimate}

\claim{{\bf Theorem.} {\it Let $X$ be a projective manifold.\\
Let $\alpha=\{T\}\in\cE^\circ_{\NS}$ be a big class represented by
a K\"ahler current~$T$, and consider
an approximate Zariski decomposition
$$\mu_m^\star T_m = [E_m]+[D_m]$$
Then
\alert{$$
(D_m^{n-1}\cdot E_m)^2\le 20\,(C\omega)^n\big(\Vol(\alpha)-D_m^n\big)
$$}%
where $\omega=c_1(H)$ is a K\"ahler form and $C\ge 0$ is a constant such that
$\pm\alpha$ is dominated by $C\omega$ $($i.e., $C\omega\pm\alpha$ 
is nef$\,)$.}}
\medskip

By going to the limit, one gets
\medskip

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


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

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

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\LabelTeX 22 49 $\mu_m^*T_m$\ELTX
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\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 1 52 $\cE$\ELTX
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\LabelTeX -28 4 $\NS_\bR(X)$\ELTX
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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 $X$ is \alert{not uniruled} if and only if
$K_X$ is pseudo-effective, i.e.\ $K_X\in\cE_{\NS}$.}}
\medskip

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

\end{frame}

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

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

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

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

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

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

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

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

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

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

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

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

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

\end{frame}

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

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

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

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

\end{frame}
\end{document}

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

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

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

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

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

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

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

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

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

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

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

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

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

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