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% In principle, this file can be redistributed and/or modified under
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%
% However, this file is supposed to be a template to be modified
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\title[\ \kern-190pt\rlap{\blank{J.-P. Demailly, Virtual Conference CAG at UWO, May 7, 2020}}\kern183pt\rlap{\blank{Bergman bundles and applications to geometry}}\kern178pt\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{Bergman bundles and applications to the\\
geometry of compact complex manifolds}

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

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

\institute[]{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)
{Virtual Conference in\\
Complex Analysis and Geometry\\
hosted at Western University, London, Ontario\\
May 4 -- 24, 2020}

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\begin{document}
%%\def\pause{}

% 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{<+->}

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  \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}

%%\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.

\begin{frame}
\frametitle{Projective vs K\"ahler vs non K\"ahler varieties}
\vskip-5pt
\claim{\bf Goal.} Investigate positivity for general compact manifolds/$\bC$.
\pause\vskip2pt
Obviously, non projective varieties do not carry any
\hbox{\alert{ample line bundle}.\kern-20pt}
\vskip2pt\pause
In the K\"ahler case, a K\"ahler class $\{\omega\}\in H^{1,1}(X,\bR)$,
$\omega>0$, may sometimes be used as a substitute for a polarization.
\vskip2pt\pause
What for non K\"ahler compact complex manifolds?
\pause\vskip-3pt
\begin{block}{Surprising facts (?)}
-- Every compact complex manifold $X$ carries a \alert{``very ample''
complex Hilbert bundle}, produced by means of a natural Bergman
space construction.\pause\vskip3pt
-- The curvature of this bundle is strongly positive
in the sense of Nakano, and is given by a universal formula.\\
\end{block}
\pause\vskip-4pt
The aim of this lecture is to investigate further this construction and
explain potential applications to analytic geometry (invariance
of plurigenera, transcendental holomorphic Morse inequalities...)
\end{frame}

\begin{frame}
\frametitle{Tubular neighborhoods (thanks to Grauert)}
\vskip-6pt
Let $X$ be a compact complex manifold, $\dim_\bC X=n$.
\vskip3pt\pause
Denote by $\overline X$ its complex conjugate $(X,-J)$, so that
\alert{$\cO_{\overline X}=\overline{\cO_X}$}.
\pause\vskip3pt
The diagonal of $X\times\overline X$ is totally real, and by Grauert, we know
that it possesses a fundamental system of Stein tubular neighborhoods.
\pause\vskip3pt
Assume that $X$ is equipped with a real analytic hermitian metric~$\gamma$,
\pause and let \alert{$\exp:T_X\to X\times X$,
$(z,\xi)\mapsto(z,\exp_z(\xi))$, $z\in X$, $\xi\in T_{X,z}$} be
the associated geodesic exponential map.\vskip4pt
$\strut$\kern2cm\pgfdeclareimage[height=3.75cm]{FFig1}{FFig1}
\pgfuseimage{FFig1}
\end{frame}

\begin{frame}
\frametitle{Exponential map diffeomorphism and its inverse}  
\vskip-7pt
\begin{block}{Lemma}
Denote by \alert{$\exph$} the ``holomorphic'' part of $\exp$, so that
for $z\in X$ and $\xi\in T_{X,z}$\vskip2pt
\alert{\centerline{$\displaystyle
\exp_z(\xi)=\sum_{\alpha,\beta\in\bN^n}a_{\alpha\,\beta}(z)\xi^\alpha
\overline\xi^\beta,\quad
\exph_z(\xi)=\sum_{\alpha\in\bN^n}a_{\alpha\,0}(z)\xi^\alpha.$}}\pause\vskip2pt
Then $d_\xi\exp_z(\xi)_{\xi=0}=d_\xi\exph_z(\xi)_{\xi=0}=\Id_{T_X}$, and so
$\exph$ is a diffeomorphism from a neighborhood $V$ of the $0$ section
of $T_X$ to a neighborhood $V'$ of the diagonal in
$X\times X$.
\end{block}
\pause\vskip-6pt
\begin{block}{Notation} With the identification $\overline X\simeq_{\rm diff} X$, let $\logh:X\times\overline X\supset V'\to T_{\overline X}$ be the inverse 
diffeomorphism of $\exph$
and\vskip5pt
\alert{\centerline{$\displaystyle
U_\varepsilon=\{(z,w)\in V'\subset X\times\overline X\,;\;|\logh_z(w)|_\gamma
<\varepsilon\},~~~\varepsilon>0.$}}\pause\vskip5pt
Then, for $\varepsilon\ll 1$, $U_\varepsilon$ is Stein and
$\pr_1:U_\varepsilon\to X$ is a \alert{real analytic locally trivial
bundle} with fibers biholomorphic to complex balls.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Such tubular neighborhoods are Stein}  
$\strut$\kern2cm\pgfdeclareimage[height=5cm]{FFig2}{FFig2}
\pgfuseimage{FFig2}
\pause\vskip2pt
In the special case $X=\bC^n$, \hbox{$U_\varepsilon=
\{(z,w)\in\bC^n\times\bC^n\,;\;|\overline z-w|<\varepsilon\}$\kern-10pt}\break
is of course Stein since\vskip5pt
\alert{\centerline{$|\overline z-w|^2=|z|^2+|w|^2-2\Re\sum z_jw_j$}}\vskip5pt
and $(z,w)\mapsto\Re\sum z_jw_j$ is pluriharmonic.
\end{frame}

\begin{frame}
\frametitle{Bergman sheaves}  
\vskip-5pt
Let $U_\varepsilon=U_{\gamma,\varepsilon}\subset X\times\overline X$
be the ball bundle as above, and\vskip3pt
\centerline{\alert{$
p=(\pr_1)_{|U_\varepsilon}:U_\varepsilon\to X,\qquad
\overline p=(\pr_2)_{|U_\varepsilon}:U_\varepsilon\to\overline X
$}}\vskip3pt
the natural projections.\pause
\vskip7pt
$\strut$\kern2cm\pgfdeclareimage[height=5.75cm]{FFig3}{FFig3}
\pgfuseimage{FFig3}
\end{frame}

\begin{frame}
\frametitle{Bergman sheaves (continued)}
\vskip-6pt  
\begin{block}{Definition of the Bergman sheaf $\cB_\varepsilon$}
The Bergman sheaf $\cB_\varepsilon=\cB_{\gamma,\varepsilon}$ is
by definition the $L^2$ direct image\vskip5pt
\alert{\centerline{$
\cB_\varepsilon=p^{L^2}_*(\overline p^*\cO(K_{\overline X})),
$}}\pause\vskip5pt
i.e.\ the space of sections over an open subset $V\subset X$ defined by
$\cB_\varepsilon(V)={}$holomorphic sections $f$ of
$\overline p^*\cO(K_{\overline X})$ on $p^{-1}(V)$,\vskip5pt
\centerline{$
f(z,w)=f_1(z,w)\;dw_1\wedge\ldots\wedge dw_n,\quad z\in V,$}\pause\vskip5pt
that are
in $L^2(p^{-1}(K))$  for all compact subsets $K\compact V\;:$\vskip5pt
\alert{\centerline{$\displaystyle
\int_{p^{-1}(K)}i^{n^2}f(z,w)\wedge \overline{f(z,w)}\wedge \gamma(z)^n
<+\infty,\quad\forall K\compact V.
$}}\vskip3pt
(This $L^2$ condition is the reason we speak of ``$L^2$ direct image'').
\end{block}
\pause
Clearly, $\cB_\varepsilon$ is an $\cO_X$-module over $X$, but since it is
a space of functions in $w$, it is of infinite rank.
\end{frame}

\begin{frame}
\frametitle{Associated Bergman bundle and holom structure}  
\vskip-5pt
\begin{block}{Definition of the associated Bergman bundle $B_\varepsilon$}
We consider the vector bundle $B_\varepsilon\to X$ whose
fiber $B_{\varepsilon,z_0}\,$consists of all holomorphic functions
$f\,$on $p^{-1}(z_0)\,{\subset}\,U_\varepsilon$ \hbox{such that\kern-15pt}
\vskip4pt
\centerline{\alert{$\displaystyle
\Vert f(z_0)\Vert^2=\int_{p^{-1}(z_0)}i^{n^2}f(z_0,w)\wedge
\overline{f(z_0,w)}<+\infty.$}}
\end{block}
\pause\vskip-3pt
Then $B_\varepsilon$ is a \alert{real analytic} locally trivial Hilbert bundle
whose fiber $B_{\varepsilon,z_0}$ is isomorphic to the Hardy-Bergman space
$\cH^2(B(0,\varepsilon))$ of $L^2$ holomorphic $n$-forms on $p^{-1}(z_0)\simeq
B(0,\varepsilon)\subset\bC^n$.\pause\break
The Ohsawa-Takegoshi extension theorem
implies that every $f\in B_{\varepsilon,z_0}$ can be extended as a
germ $\tilde f$ in the sheaf $\cB_{\varepsilon,z_0}$.\pause\break
Moreover, for $\varepsilon'>\varepsilon$, there is a restriction
map \alert{$\cB_{\varepsilon',z_0}\to B_{\varepsilon,z_0}$}
such~that $B_{\varepsilon,z_0}$ is the \alert{$L^2$ completion of
$\cB_{\varepsilon',z_0}/\gm_{z_0}\cB_{\varepsilon',z_0}$}.
\pause\vskip-3pt

\begin{block}{Question}
Is there a ``complex structure'' on $B_\varepsilon$ such that
``$\cB_\varepsilon=\cO(B_\varepsilon)$''~?
\end{block}
\end{frame}

\begin{frame}
\frametitle{Bergman Dolbeault complex}
\vskip-5pt
For this, consider the ``Bergman Dolbeault'' complex
\hbox{$\dbar:\cF_\varepsilon^q\to \cF_\varepsilon^{q+1}$\kern-20pt}\\
over $X$, with
$\cF_\varepsilon^q(V)={}$smooth $(n,q)$-forms\vskip5pt
\alert{\centerline{$\displaystyle
f(z,w)\,{=}\sum_{|J|=q}f_J(z,w)\,dw_1\wedge...\wedge dw_n\wedge
d\overline z_J,~~
(z,w)\in U_\varepsilon\cap(V\times\overline X),$}}\vskip3pt
such that $f_J(z,w)$ is holomorphic in $w$, and for all $K\compact V$ one has
$$\alert{f(z,w)\in L^2(p^{-1}(K))}
~~\hbox{and}~~\alert{\dbar_zf(z,w)\in L^2(p^{-1}(K))}.$$\pause
An immediate consequence of this definition is:
\begin{block}{Proposition}
$\dbar=\dbar_z$ yields a complex of sheaves $(\cF_\varepsilon^\bullet,\dbar\,)$,
and the kernel $\Ker\dbar:\cF_\varepsilon^0\to\cF_\varepsilon^1$ coincides with $\cB_\varepsilon$.
\end{block}\pause
\noindent
If we define $\cO_{L^2}(B_\varepsilon)$ to be the sheaf of
$L^2_{\rm loc}$
sections $f$ of $B_\varepsilon$ such that $\dbar f=0$ in the sense
of distributions, then we exactly have
\alert{$\cO_{L^2}(B_\varepsilon)=\cB_\varepsilon$}
as a sheaf. 
\end{frame}

\begin{frame}
\frametitle{Bergman sheaves are ``very ample''}
\vskip-7pt  
\begin{block}{Theorem}
Assume that $\varepsilon>0$ is taken so small
that $\psi(z,w):=|\logh_z(w)|^2$ is strictly plurisubharmonic up to the
boundary on the compact set $\overline U_\varepsilon
\subset X\times\overline X$.\pause\ Then the complex of sheaves
$(\cF_\varepsilon^\bullet,\dbar)$ is a resolution of $\cB_\varepsilon$ by soft
sheaves over~$X$ $($actually, by $\cC^\infty_X$-modules$\,)$,
and for every holomorphic vector bundle
$E\to X$ we have\vskip5pt
\alert{\centerline{$
H^q(X,\cB_\varepsilon\otimes\cO(E))=H^q\big(\Gamma(X,
\cF_\varepsilon^\bullet\otimes\cO(E)),\dbar\,\big)=0,\quad\forall q\geq 1.
$}}\pause\vskip5pt
Moreover the fibers $B_{\varepsilon,z}\otimes E_z$ are always generated
by global sections of $H^0(X,\cB_\varepsilon\otimes\cO(E))$.
\end{block}
\vskip-2pt
In that sense, $B_\varepsilon$ is a \alert{``very ample holomorphic
vector bundle''}\\ (as a Hilbert bundle of infinite dimension).\pause\\
The proof is a direct consequence of H\"ormander's $L^2$ estimates.
\vskip-2pt
\begin{block}{Caution !!}
$B_\varepsilon$ is \alert{NOT} a locally trivial {\it holomorphic} bundle.
\end{block}  
\end{frame}

\begin{frame}
\frametitle{Embedding into a Hilbert Grassmannian}
\vskip-6pt
\begin{block}{Corollary of the very ampleness of Bergman sheaves}
Let $X$ be an arbitrary compact complex manifold, $E\to X$ a
holomorphic vector bundle (e.g.\ the trivial bundle).
Consider the Hilbert space $\bH=H^0(X,\cB_\varepsilon\otimes\cO(E))$.\pause\
Then one gets a ``holomorphic embedding'' into a Hilbert Grassmannian,\vskip4pt
\centerline{\alert{$\Psi:
X\to{\rm Gr}(\bH),\quad z\mapsto S_z,
$}}\vskip4pt
mapping every point $z\in X$ to the infinite codimensional
closed subspace $S_z$ consisting of sections $f\in\bH$ such
that $f(z)=0$ in~$B_{\varepsilon,z}$, i.e.\ $f_{|p^{-1}(z)}=0$.  
\end{block}
\vskip-2pt\pause
The main problem with this ``holomorphic embedding'' is that the
holomorphicity is to be understood in a weak sense, for instance
the map $\Psi$ is not even continuous with respect to the strong
metric topology of ${\rm Gr}(\bH)$, given by\break
$d(S,S')={}$ Hausdorff distance of the unit balls of $S$, $S'$.
\end{frame}

\begin{frame}
\frametitle{Chern connection of Bergman bundles}
Since we have a natural $\nabla^{0,1}=\dbar$ connection on $B_\varepsilon$,
and a natural hermitian metric as well, it follows from the usual
formalism that $B_\varepsilon$ can be equipped with a \alert{unique Chern
connection}.\pause\vskip5pt

\claim{\bf Model case: $X=\bC^n$, $\gamma={}$ standard hermitian metric.}\pause

Then one sees that a orthonormal frame of $B_\varepsilon$ is
given by\vskip5pt
\alert{\centerline{$\displaystyle
e_\alpha(z,w)=\pi^{-n/2}\varepsilon^{-|\alpha|-n}
\sqrt{{(|\alpha|+n)!\over\alpha_1!\ldots\alpha_n!}}\,(w-\overline z)^\alpha,
\quad \alpha\in\bN^n.$}}
\pause\vskip5pt
It is non holomorphic!\pause\
The $(0,1)$-connection $\nabla^{0,1}=\dbar$ is given by\vskip5pt
\alert{\centerline{$\displaystyle
\nabla^{0,1}e_\alpha=\dbar_ze_\alpha(z,w)=\varepsilon^{-1}
\sum_{1\leq j\leq n}
\sqrt{\alpha_j(|\alpha|+n)}\;d\overline z_j\otimes e_{\alpha-c_j}$}}\vskip3pt
where $c_j=(0,...,1,...,0)\in\bN^n$.
\end{frame}

\begin{frame}
\frametitle{Curvature of Bergman bundles}
\vskip-5pt
Let $\Theta_{B_\varepsilon,h}=\nabla^2$
be the curvature tensor of $B_\varepsilon$ with its natural Hilbertian
metric $h$.\pause\ Remember that\vskip4pt
\alert{\centerline{
$\displaystyle
\Theta_{B_\varepsilon,h}=\nabla^{1,0}\nabla^{0,1}+\nabla^{0,1}\nabla^{1,0}
\in C^\infty(X,\Lambda^{1,1}T^\star_X\otimes{\rm Hom}(
B_\varepsilon,B_\varepsilon)),$}}\vskip4pt
and that one gets an associated
quadratic Hermitian form on $T_X\otimes B_\varepsilon$ such that\vskip4pt
\alert{\centerline{
$\displaystyle
\widetilde\Theta_\varepsilon(v\otimes\xi)=
\langle\Theta_{B_\varepsilon,h}\sigma(v,Jv)\xi,\xi\rangle_h$}}\vskip4pt
for $v\in T_X$ and
$\xi=\sum_\alpha\xi_\alpha e_\alpha\in B_\varepsilon$.\pause\vskip5pt
\begin{block}{Definition}
One says that the curvature tensor is \alert{Griffiths positive} if
\vskip4pt
\alert{\centerline{
$\displaystyle
\widetilde\Theta_\varepsilon(v\otimes\xi)>0,\quad
\forall 0\ne v\in T_X,~~\forall 0\ne\xi\in B_\varepsilon,$}}\pause\vskip4pt
and \alert{Nakano positive} if
\vskip4pt
\alert{\centerline{
$\displaystyle
\widetilde\Theta_\varepsilon(\tau)>0,\quad
\forall 0\ne \tau\in T_X\otimes B_\varepsilon.$}}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Calculation of the curvature tensor for $X=\bC^n$}
\vskip-3pt  
A simple calculation of $\nabla^2$ in the orthonormal
frame $(e_\alpha)$ leads to:
\begin{block}{Formula}
In the model case $X=\bC^n$, the curvature tensor of the Bergman bundle
$(B_\varepsilon,h)$ is given by\vskip2pt
\alert{\centerline{
$\displaystyle
\widetilde\Theta_{\varepsilon}(v\otimes\xi)
=\varepsilon^{-2}\kern-3pt\sum_{\alpha\in\bN^n}\kern-3pt\Bigg(\bigg|
\sum_j\sqrt{\alpha_j}\;\xi_{\alpha-c_j}v_j\bigg|^2\kern-3pt{}
+\sum_{j}(|\alpha|+n)\;|\xi_\alpha|^2|v_j|^2\Bigg)\kern-3pt.$}}\vskip-16pt
\strut
\end{block}
\pause\vskip-3pt
\begin{block}{Consequence}
In $\bC^n$, the curvature tensor $\Theta_{\varepsilon}(v\otimes\xi)$
is Nakano positive.
\end{block}
\pause
On should observe that $\widetilde\Theta_{\varepsilon}(v\otimes\xi)$
is an \alert{unbounded} quadratic form on $B_\varepsilon$
with respect to the standard metric $\Vert\xi\Vert^2=
\sum_\alpha|\xi_\alpha|^2$.\vskip4pt\pause
However there is convergence for all
$\xi=\sum_\alpha\xi_\alpha e_\alpha\in \alert{B_{\varepsilon'}}$,
\alert{$\varepsilon'>\varepsilon$}, since then
$\sum_\alpha(\varepsilon'/\varepsilon)^{2|\alpha|}|\xi_\alpha|^2<+\infty$.
\end{frame}


\begin{frame}
\frametitle{Curvature of Bergman bundles (general case)}
\vskip-5pt
\begin{block}{Bergman curvature formula on a general hermitian manifold}
Let $X$ be a compact complex manifold equipped with a $C^\omega$
hermitian metric $\gamma$, and $B_\varepsilon=B_{\gamma,\varepsilon}$ the
associated Bergman bundle.\pause\break
Then its curvature is given by an asymptotic expansion
\alert{\centerline{
$\displaystyle
\widetilde\Theta_{\varepsilon}(z,v\otimes\xi)
=\sum_{p=0}^{+\infty}\varepsilon^{-2+p}Q_p(z,v\otimes\xi),~~
v\in T_X,~~\xi\in B_\varepsilon$}}\vskip1pt\pause
where $Q_0(z,v\otimes\xi)=Q_0(v\otimes\xi)$ is given by the
model case $\bC^n\,$:
\alert{\centerline{$\displaystyle
Q_0(v\otimes\xi)
=\varepsilon^{-2}\kern-3pt\sum_{\alpha\in\bN^n}\kern-3pt\Bigg(\bigg|
\sum_j\sqrt{\alpha_j}\;\xi_{\alpha-c_j}v_j\bigg|^2\kern-3pt{}
+\sum_{j}(|\alpha|+n)\;|\xi_\alpha|^2|v_j|^2\Bigg)\kern-3pt.$}}\vskip0pt\pause
The other terms $Q_p(z,v\otimes\xi)$ are real analytic; $Q_1$ and $Q_2$
depend respectively on the torsion and curvature tensor of $\gamma$.
\pause\break
In particular $Q_1=0$ is $\gamma$ is K\"ahler.
\end{block}
\pause\vskip-3pt
A consequence of the above formula is that $B_\varepsilon$ is strongly
Nakano positive for $\varepsilon>0$ small enough.
\end{frame}

\begin{frame}
\frametitle{Idea of proof of the asymptotic expansion}
\vskip-3pt
The formula is in principle a special case of a more general result proved
by \alert{Wang Xu}, expressing the curvature of
\alert{weighted Bergman bundles $\cH_t$}
attached to a \alert{smooth family $\{D_t\}$ of strongly pseudoconvex
domains}.\pause\
Wang's formula is however in integral form and not completely
explicit.\pause\vskip5pt
Here, one simply uses the real analytic Taylor expansion of
\alert{$\logh:X\times\overline X\to T_X$} ~(inverse
diffeomorphism of $\exph$)
$$
\logh_z(w)=w-\overline z+\sum z_ja_j(w-\overline z)+\sum\overline z_j
a'_j(w-\overline z)
$$\vskip-18pt
$$
\kern120pt{}+\sum z_jz_kb_{jk}(w-\overline z)
+\sum \overline z_j\overline z_kb'_{jk}(w-\overline z)
$$
$$
\kern65pt{}+\sum z_j\overline z_kc_{jk}(w-\overline z)+O(|z|^3),
$$
\pause\vskip-8pt
which is used to compute the difference with the model case
$\bC^n$, for which~ \alert{$\logh_z(w)=w-\overline z$}.

\end{frame}

\begin{frame}
\frametitle{Potential application: invariance of plurigenera for
polarized families of compact K\"ahler manifolds~?}

\begin{block}{Conjecture} Let $\pi:\cX\to S$ be a proper holomorphic
map defining a family of smooth compact K\"ahler manifolds over an
irreducible base~$S$.\pause\
Assume that the family \alert{admits a polarization},
i.e.\ a closed smooth $(1,1)$-form $\omega$ such that $\omega_{|X_t}$
is positive definite on each fiber $X_t:=\pi^{-1}(t)$.\pause\
Then the plurigenera\vskip4pt
\centerline{\alert{$p_m(X_t)=h^0(X_t,mK_{X_t})$
are independent of~$t$} for all $m\ge 0$.}
\end{block}\pause
The conjecture is known to be true for a \alert{projective family} $\cX\to S$:\\
$\bu$ Siu and Kawamata (1998) in the case of varieties of \alert{general type}\\
$\bu$ Siu (2000) and P\u{a}un (2004) in the arbitrary projective case
\vskip5pt\pause
No algebraic proof is known in the latter case; one deeply uses the
$L^2$ estimates of the \alert{Ohsawa-Takegoshi extension theorem}.
\end{frame}

\begin{frame}
\frametitle{Invariance of plurigenera: strategy of proof (1)}
It is enough to consider the case of a family $\cX\to\Delta$ over
the disc, such that there exists a \alert{relatively ample line bundle
$\cA$} over $\cX$.
\vskip5pt\pause
Given $s\in H^0(X_0,mK_{X_0})$, the point is to show that it extends
into $\widetilde s\in H^0(\cX,mK_\cX)$, and for this, one only needs
to produce a hermitian metric $h=e^{-\varphi}$ on $K_\cX$ such that:\\
$\bu$ \alert{$\Theta_h=i\ddbar\varphi\geq 0$ in the sense of currents}\\
$\bu$ $|s|^2_h=|s|^2e^{-\varphi}\leq 1$,
i.e.\ \alert{$\varphi\geq\log|s|$} on $X_0$.
\vskip5pt\pause
The Ohsawa-Takegoshi theorem then implies the
\alert{existence of $\widetilde s$}.\vskip5pt\pause
To produce $h=e^{-\varphi}$, one produces inductively (also by O-T~!)
sections of $\sigma_{p,j}$ of $\cL_p:=\cA+pK_\cX$ such that:\\
$\bu$ \alert{$(\sigma_{p,j})$ generates $\cL_p$} for $0\leq p<m$\\
$\bu$ \alert{$\sigma_{p,j}$ extends $(\sigma_{p-m,j}s)_{|X_0}$ to $\cX$}
for $p\geq m$\\
$\bu$ \alert{$\displaystyle
  \int_{\cX}{\sum_j|\sigma_{p,j}|^2\over\sum_j|\sigma_{p-1,j}|^2}\leq C$}
for $p\geq 1$.
\end{frame}

\begin{frame}
\frametitle{Invariance of plurigenera: strategy of proof (2)}
\vskip-4pt
By H\"older, the $L^2$ estimates imply
\alert{$\int_{\cX}\big(\sum_j|\sigma_{p,j}|^2\big)^{1/p}\leq C$} for all $p$,
and using the fact that \claim{$\lim{1\over p}\Theta_\cA=0$},
one can take\vskip3pt
\centerline{\alert{$\varphi=\limsup_{p\to+\infty}\varphi_p,\quad
\varphi_p:={1\over p}\log\sum_j|\sigma_{p,j}|^2$}.}\pause\vskip6pt

\claim{\bf Idea.} In the polarized K\"ahler case, use the Bergman
bundle
\hbox{$B_\varepsilon\,{\to}\,\cX$\kern-20pt}\\
instead of an ample line bundle $\cA\to\cX$.
This amounts to applying the Ohsawa-Takegoshi $L^2$ extension on Stein
tubular neighborhoods \alert{$U_\varepsilon\subset\cX\times\overline\cX$,
with projections $\pr_1:U_\varepsilon\to\cX$ and $\pi:\cX\to\Delta$}.\pause
\vskip-2pt
\begin{block}{Proposition}
In the polarized K\"ahler case $(\cX,\omega)$, shrinking from~$U_\varepsilon$
to $U_{\rho\varepsilon}$ with $\rho<1$, the $B_\varepsilon$ curvature estimate
gives\vskip3pt
\centerline{\alert{$\displaystyle
\varphi_p:={1\over p}\log\sum_j\Vert\sigma_{p,j}
\Vert_{U_{\rho\varepsilon}}^2~~\Rightarrow~~
i\ddbar\varphi_p\geq -{C\over \varepsilon^2\rho^2}(C'-\varphi_p)\omega.
$}}\vskip0pt\pause
This implies that $\varphi=\limsup\varphi_p$ satisfies
\hbox{$\psi:=-\log(C''-\varphi)$\kern-20pt}\break
quasi-psh, but yields invariance of
plurigenera only for \hbox{\alert{$\varepsilon\to+\infty$}.\kern-15pt}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Transcendental holomorphic Morse inequalities}
\vskip-5pt
\begin{block}{Conjecture} Let $X$ be a compact $n$-dimensional
complex manifold
and $\alpha\in H^{1,1}_{BC}(X,\bR)$ a Bott-Chern class, represented
by closed real $(1,1)$-forms modulo $\ddbar$ exact forms.\pause\
Set\vskip3pt
\alert{\centerline{$\displaystyle
\Vol(\alpha)=\sup_{T=\alpha+i\ddbar\varphi\geq 0}\int_X T_{ac}^n$,~~~$T\geq 0$
current.}}\pause\vskip-3pt
Then\vskip-3pt
\alert{\centerline{$\displaystyle
\Vol(\alpha)\geq\sup_{u\in\{\alpha\},~u\in C^\infty}\int_{X(u,0)}u^n$}}\vskip-3pt
where\vskip3pt
\alert{\centerline{$\displaystyle
X(u,0)=0\hbox{-index set of $u$}=
\big\{x\in X\,;\;u(x)~\hbox{positive definite}\big\}$.}}
\end{block}
\pause\vskip-4pt
\begin{block}{Conjectural corollary (fundamental volume estimate)}
Let $X$ be compact K\"ahler, $\dim X=n$, and
\alert{$\alpha,\beta\in H^{1,1}(X,\bR)$ be nef classes}. Then
\vskip-3pt
\alert{\centerline{$\displaystyle
\Vol(\alpha-\beta)\geq\alpha^n-n\alpha^{n-1}\cdot\beta$.}}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Transcendental Morse: known facts \&\ beyond}
\vskip-5pt
The conjecture on Morse inequalities is known to be true when $\alpha=c_1(L)$
is the class of a line bundle ([D-1985]), and the corollary
can be derived from this when $\alpha,\beta$ are integral classes
(by~[D-1993] and independently by [Trapani, 1993]).\pause\vskip3pt

Recently, the volume estimate for $\alpha$, $\beta$ transcendental
has been established by D.\ Witt-Nystr\"om when \alert{$X$ is projective},
and Xiao-Popovici even proved in general that \alert{$\Vol(\alpha-\beta)>0$
if $\alpha^n-n\alpha^{n-1}\cdot\beta>0$}.
\pause\vskip4pt

\claim{\bf Idea.} In the general case, one can find a sequence of
non holomorphic hermitian line bundles $(L_m,h_m)$ such that\vskip3pt
\centerline{\alert{$\displaystyle
m\alpha=\Theta_{L_m,h_m}+\gamma^{2,0}_m+\overline\gamma^{0,2}_m,\quad
\gamma_m\to 0$}.}\pause\vskip3pt
As $U_\varepsilon$ is Stein, $\overline\gamma^{0,2}_m=\dbar v_m$, $v_m\to 0$,
and $\pr_1^*L_m$ becomes a holomorphic line bundle  with curvature
form $\Theta_{\pr_1^*L_m}\simeq m\pr_1^*\alpha$.\pause\\
Then apply $L^2$ direct image $(\pr_1)_*^{L^2}$ and
\alert{use Bergman estimates instead of dimension counts in Morse
inequalities}.
\end{frame}

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

\end{frame}

\end{document}

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