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% - 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
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\title[\kern-190pt\rlap{\
\blank{Jean-Pierre Demailly~--~CAG-XXIII, Levico Terme, 13/06/2017}}
\kern180pt\rlap{\ 
\blank{Structure of compact K\"ahler manifolds with $-K_X$ nef}}
\kern180pt
\llap{\blank{\framenumbering~}}]
% (optional, use only with long paper titles)
{On the structure of\\
compact K\"ahler manifolds\\
with nef anticanonical bundles}

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

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

\institute[]{Institut Fourier, Universit\'e Grenoble Alpes, France\\
\&\ Acad\'emie des Sciences de Paris}
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% - Keep it simple, no one is interested in your street address.

\date[]% (optional)
{Complex Analysis and Geometry -- XXIII, June 13, 2017}

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% mathematical operators
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\newcommand{\Pic}{\mathop{\rm Pic}\nolimits}
\newcommand{\codim}{\mathop{\rm codim}\nolimits}
\newcommand{\Id}{\mathop{\rm Id}\nolimits}
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\newcommand{\Supp}{\mathop{\rm Supp}\nolimits}
\newcommand{\Vol}{\mathop{\rm Vol}\nolimits}
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\begin{document}

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%% \begin{frame}<beamer>
%%    \frametitle{Outline}
%%    \tableofcontents[currentsection,currentsubsection]
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%%\begin{frame}
%%  \frametitle{Outline}
%%  \tableofcontents
%% You might wish to add the option [pausesections]
%%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%%\section*{Basic concepts}
%%\def\pause{}

\def\adjustitem{%%
\setlength{\itemindent}{\dimexpr 26pt-.6in}
\setlength{\parindent}{\dimexpr 26pt-.6in}}

\begin{frame}
\frametitle{Goals / main positivity concepts}
\strut\vskip-19pt
\begin{itemize}\adjustitem
\item Analyze the structure of \alert{projective} or
\hbox{\alert{compact K\"ahler manifolds}\kern-15pt}\\\protect%%
\alert{$X$ with $-K_X$ nef}.\\\pause%%
\item As is well known since the beginning of the XX${}^{\rm th}$ century at
\\\protect%
least, the geometry depends on the sign of the curvature\\\protect%
of the canonical line bundle\\\protect%
\alert{\strut\kern20pt$K_X=\Lambda^nT^*_X,\kern8pt n=\dim_\bC X.$}\\\protect\pause
\item $L\to X$ is \claim{pseudoeffective} (\claim{psef}) if 
$\exists h=e^{-\varphi}$, $\varphi\in L^1_{\rm loc}$, s.t.\\\protect%
\strut\kern20pt\alert{$\Theta_{L,h}=-dd^c\log h\ge 0$
on $X$} in the sense of currents\\\protect%
\ecolo{${}\kern-12pt{}\Leftrightarrow{}$(for $X$ projective)~ $c_1(L)\in\overline{\rm Eff}$}.\\\protect\pause
\item $L\to X$ is \claim{semi-positive} if 
$\exists h=e^{-\varphi}$ smooth ($C^\infty$) such that\\\protect%
\strut\kern20pt
\alert{$\Theta_{L,h}=-dd^c\log h\geq 0$~~on $X$}.\\\protect%
\ecolo{${}\kern-12pt{}\Leftarrow{}$(for $X$ projective)~
$L^{\otimes m}=G\otimes H$, $G$ semi-ample,
\hbox{$H\in\Pic^0(X)$.\kern-15pt}}\pause
\item $L$ is \claim{nef} if $\forall\varepsilon>0$, 
$\exists h_\varepsilon=e^{-\varphi_\varepsilon}$ smooth
such that\\\protect%
\strut\kern20pt\alert{$\Theta_{L,h_\varepsilon}=-dd^c\log h_\varepsilon\ge
  -\varepsilon\omega$ on $X$}\\\protect%
\ecolo{${}\kern-12pt{}\Leftrightarrow{}$(for $X$ projective)~
$L\cdot C\ge 0,~\forall C$ algebraic curve}.\par
\end{itemize}
\end{frame}

\begin{frame}
  \frametitle{Complex curves ($n=1$) : genus and curvature}
  \pgfdeclareimage[height=7.8cm]{curves}{curves}
  \pgfuseimage{curves}
  \vskip-15pt
  \hbox{\alert{$K_X=\Lambda^nT^*_X,~~\deg(K_X)=2g-2$}}
  \vskip22pt$~$
\end{frame}

\begin{frame}
\frametitle{Comparison of positivity concepts}
Recall that for a line bundle\vskip3pt
\centerline{\alert{$\hbox{positive}~\ecolo{\Leftrightarrow}~
\hbox{ample}\Rightarrow\hbox{semi-ample}\Rightarrow\hbox{semi-positive}
\Rightarrow\hbox{nef}\Rightarrow\hbox{psef}$}}\vskip3pt
but none of the reverse implications in red holds true.
\pause

\begin{block}{Example} 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}}$
\end{block}
\end{frame}

\begin{frame}
\frametitle{Rationally connected manifolds}
\strut\vskip-22pt
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
If $G$ is a \alert{torsion point} in $\Pic^0(C)$, then one can
show that $-K_X$ is semi-ample, but otherwise \alert{\hbox{it is not
semi-ample.\kern-10pt}}\vskip3pt\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 that otherwise it may not be semi-positive (although nef).\vskip5pt\pause
$\bP^2\,\#\,9$ points is an example of rationally 
\hbox{connected manifold:\kern-10pt}\pause

\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

\claim{Remark.} $X={\bP^2}$ blown-up in${}\ge 10$ points is RC but
\hbox{$-K_X$ not nef.\kern-20pt}
\end{frame}

\begin{frame}
\frametitle{Ex.\ of compact K\"ahler manifolds with $-K_X\ge 0\kern-15pt$}
(\claim{Recall:} By Yau, \alert{$-K_X\ge 0\Leftrightarrow\exists\omega$
K\"ahler with Ricci$(\omega)\ge 0$.)}\vskip6pt\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\vskip4pt\pause
-- \alert{Calabi-Yau manifolds} $Y$: $\pi_1(Y)$ finite and some multiple 
of $K_Y$ is trivial (may assume \alert{$\pi_1(Y)=1$ and $K_Y$ trivial}
by passing to some finite étale cover\pause)\\
\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}
\vskip6pt
\claim{Main result.} Essentially, this is a complete list !
\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 $\UU(n''_\ell))$.\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{Criterion for rational connectedness}

\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)}] For every invertible subsheaf 
$\cF\subset\Omega^p_X:=\cO(\Lambda^pT^*_X)$, $1\le p\le n$, 
$\cF$ is \alert{not psef}.\pause
\item[{\rm (c)}] For every 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
$$
\alert{H^0(X,(T^*_X)^{\otimes m}\otimes A^{\otimes k})=0}\quad
\hbox{$\forall m,\,k\in\bN^*$ with $m\ge C_Ak$.}
$$
\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.
\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{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{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{$$z\mapsto\alpha_X(z)=\Big(\smash{\int_{z_0}^z}u_j\Big)_{1\le j\le q}~~
\hbox{\rm mod period subgroup}~\Lambda\subset\bC^q,$$}
where $(u_1,\ldots,u_q)$ is a basis of $H^0(X,\Omega^1_X)$.\pause
\begin{block}{Theorem [Qi Zhang, 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-6pt
\begin{block}{Definition} Let $X$ compact K\"ahler manifold, $\cE\to X$ torsion free sheaf.
\begin{enumerate}
\item[{\rm (a)}] $\cE$ is \alert{generically nef with respect to a K\"ahler class $\omega$} if\vskip4pt
\centerline{$\mu_{\omega}(\cS)=\hbox{$\omega$-slope of $\cS$}:={\displaystyle
\frac{\int_Xc_1(\cS)\wedge\omega^{n-1}}{\hbox{\rm rank}\,\cS}}\geq 0 $}\vskip4pt
for all torsion free quotients $\cE \to \cS \to 0.$\\
If $\cE$ is $\omega$-generically nef for all $\omega,$ we simply say that $\cE$ is \alert{generically nef}.\pause 
\item[{\rm (b)}] 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 $\cE_{i+1}/\cE_i$ are 
$\omega$-stable subsheaves of $\cE/\cE_i$ 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}
It is a standard fact that refined HN-filtrations always exist, moreover
$$ \mu_{\omega}(\cE_i/\cE_{i-1}) \geq \nu_{\omega}(\cE_{i+1}/\cE_i)$$
for all $i$.

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

\claim{Proof.} 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 [${}\Rightarrow \alpha_X$ 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 [J. 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 isotrivial}, 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 $-K_F$ 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 [DPS95], 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} (Simpson, Deng Ya).
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, 2 days ago!] }
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 isotrivial 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 Campana).
\end{itemize}
\end{block}\pause

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

\end{frame}

\begin{frame}
\frametitle{\blank{Further problems (II)}}
\vskip-4pt
\begin{block}{Definition}
A compact K\"ahler manifold $X=X^{2p}$ is a 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 the zero
divisor $E=\hbox{div}(\sigma^p)$ has numerical dimension~$0$ (so that
$K_{X'}=E$ again).
\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 taking suitable blow-ups to make it a 
genuine morphism). Then $K_Y$ 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)}}
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.\pause
\vskip5pt
They might possibly include all rationally connected varieties.
\end{frame}

\begin{frame}
\frametitle{The end}
\strut\vskip3mm
\centerline{\hugebf Thank you for your attention!}
\pgfdeclareimage[height=6.5cm]{CalabiYau}{CalabiYau}
\strut\kern2cm\pgfuseimage{CalabiYau}

\end{frame}

\end{document}