% beamer-slides pdfmode
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% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
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% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
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\title[\ \kern-190pt\rlap{\blank{J.-P. Demailly, virtual conference Geom.\ \&\ TACoS, July 7, 2020}}\kern183pt\rlap{\blank{Cohomology of quasi holomorphic line bundles}}\kern178pt\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{On the approximate cohomology\\
of quasi holomorphic line bundles}

%% \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\\
Geometry and TACoS\\
hosted at Universit\`a di Firenze\\
July 7 -- 21, 2020}

%%\subject{Talks}
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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{<+->}

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

%%\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{Quasi holomorphic line bundles}
\vskip-5pt
Let $X$ be a compact complex manifold, and let\vskip2pt
\alert{\centerline{$\displaystyle
H^{p,q}_\BC(X,\bC)
=\frac{\Ker\partial\cap\Ker\dbar}{\Im\ddbar}\qquad\hbox{in bidegree
$(p,q)$}$}}\vskip5pt
be the corresponding Bott-Chern cohomology groups.\vskip-2pt\pause
\begin{block}{Basic observation {\rm (cf.\ Laurent Laeng, PhD thesis 2002)}}
Given a \alert{class $\gamma\in H^{1,1}_\BC(X,\bR)$} and a $(1,1)$-form
$u$ representing $\gamma$, there exists an infinite subset $S\subset\bN$ and
$C^\infty$ Hermitian line bundles \alert{$(L_k,h_k)_{k\in S}$} equipped
with Hermitian connections $\nabla_k$,\pause\ such that the curvature $2$-forms
$\theta_k={i\over 2\pi}\nabla_k^2$ satisfy
\alert{$\theta_k = ku+\beta_k$} and\vskip4pt
\alert{\centerline{$\displaystyle
\beta_k=O(k^{-1/b_2}),\qquad b_2=b_2(X)$.}}\vskip5pt
\end{block}\vskip-3pt\pause

\claim{\bf Proof}. This is a consequence of Kronecker's approximation theorem
applied to the lattice $H^2(X,\bZ)\hookrightarrow H^2_\DR(X,\bR)$.\pause\\
In fact $\beta_k$~can be chosen in a finite dimensional space of $C^\infty$\\
closed $2$-forms isomorphic to $H^2_\DR(X,\bR)$.
\end{frame}

\begin{frame}
\frametitle{Approximate holomorphic structure}
\strut\kern2cm\pgfdeclareimage[height=4.4cm]{lattice}{lattice}
\pgfuseimage{lattice}
\vskip-3mm
\InsertLegend
\LabelTeX 48 24.5 $0$\ELTX
\LabelTeX 56.3 25.3 \RGBColor{0 0.7 0}{$\theta_1$}\ELTX
\LabelTeX 71 41 \RGBColor{0 0.7 0}{$\theta_5$}\ELTX
\LabelTeX 82 21 \RGBColor{0 0.7 0}{$\{\theta_k\}\in H^2(X,\bZ)$}\ELTX
\LabelTeX 80 42.5 \RGBColor{1 0 0}{$ku,~~k=1,2,3,4,5$}\ELTX
\EndLegend\pause\vskip-2pt
\begin{block}{Consequence}
Let $\nabla_k=\nabla_k^{1,0}+\nabla_k^{0,1}$. Then
$\theta_k = ku+\beta_k$ implies\vskip6pt
\alert{\centerline{$\displaystyle
(\nabla_k^{0,1})^2=\theta_k^{0,2}=\beta_k^{0,2}=O(k^{-1/b_2}).$}}\vskip2pt
\end{block}\pause\vskip-2pt
Thus the $L_k$ are ``closer and closer'' to be holomorphic
as \hbox{$k\to+\infty$.\kern-10pt}
\end{frame}

\begin{frame}
\frametitle{Spectrum of the Laplace-Beltrami operator}
\vskip-3pt  
Let \alert{$\ovl\square_k=\dbar_k\dbar_k^*+\dbar_k^*\dbar_k$} be the complex
Laplace-Beltrami operator of $(L_k,h_k,\nabla_k)$ with respect to some
Hermitian metric $\omega$ on $X$.\pause\\
Let \alert{$\ovl\square_{k,E}^{p,q}$} the operator
acting on $C^\infty(X,\Lambda^{p,q}T^*_X\otimes L_k\otimes E)$, where
\alert{$(E,h_E)$} is a \alert{holomorphic} Hermitian vector bundle
of rank $r$.\vskip3pt\pause
We are interested in analyzing the (discrete) spectrum of the
elliptic operator $\ovl\square_{k,E}^{p,q}$.\pause\ Since the curvature is
$\theta_k\simeq ku$, it is better to renormalize and to
consider instead ${1\over 2\pi k}\ovl\square_{k,E}^{p,q}$.\pause\
For $\lambda\in\bR$, we define\vskip6pt
\alert{\centerline{$\displaystyle
N_k^{p,q}(\lambda)=\dim\bigoplus\hbox{eigenspaces of
$\displaystyle{1\over 2\pi k}\ovl\square_{k,E}^{p,q}$
of eigenvalues~$\le\lambda.$}$}}\pause\vskip5pt
Let $u_j(x)$, $1\le j\le n$, be the eigenvalues of
$u(x)$ with respect to $\omega(x)$ at any point $x\in X$, ordered so that
if $s={}$rank$(u(x))$, then
$|u_1(x)|\ge\cdot{\cdot}\cdot\ge |u_s(x)|>|u_{s+1}(x)|=\cdot{\cdot}\cdot
=|u_n(x)|=0$.\pause\vskip5pt
For a multi-index $J=\{j_1<j_2<\ldots<j_q\}\subset\{1,\ldots,n\}$,
set \vskip3pt
\alert{\centerline{$\displaystyle
u_J(x)=\sum\nolimits_{j\in J}u_j(x)$, $\quad x\in X$.}}
\end{frame}

\begin{frame}
\frametitle{Fundamental spectral theory results}
\vskip-5pt  
Consider the ``spectral density functions'' $\nu_u$, $\ovl\nu_u$ defined
by\vskip5pt
\alert{\centerline{$\displaystyle
{\displaystyle \nu_u(\lambda)\atop\displaystyle\ovl\nu_u(\lambda)}\bigg\}
=\frac{2^{s-n}\,|u_1|\,{\cdot}{\cdot}{\cdot}\,|u_s|}{\Gamma(n-s+1)}
\!\!\sum_{(p_1,\ldots,p_s)\in\bN^s}\Big[\lambda-\sum(2p_j+1)|u_j|\Big]_+^{n-s}.$}}
\vskip0pt(where $0^0=0$ for $\nu_u$, resp.\ $0^0=1$ for $\ovl\nu_u$).
\pause\vskip-4pt
\begin{block}{Theorem {\rm ([D] 1985)}} The spectrum of
$\frac{1}{2\pi k}\ovl\square_k^{p,q}$ on $C^\infty(X,\Lambda^{p,q}T^*_X\otimes
L_k\otimes E)$ has an asymptotic distribution of eigenvalues
such that $\forall\lambda\in\bR$\vskip4pt
\alert{\centerline{$\displaystyle
r{n \choose p}\sum_{|J|=q}\int_X
\nu_u(2\lambda+u_{\complement J}-u_J)\,dV_\omega\le
\liminf_{k\to+\infty}k^{-n}N_k^{p,q}(\lambda)\le{}\quad$}
\centerline{$\displaystyle
\quad{}\le\limsup_{k\to+\infty}k^{-n}N_k^{p,q}(\lambda) \le
r{n \choose p}\sum_{|J|=q}\int_X
\ovl\nu_u(2\lambda+u_{\complement J}-u_J)\,dV_\omega$}}\vskip2pt
where $r={}$rank$(E)$.\pause\ By monotonicity, as
$\ovl\nu_u(\lambda)=\lim_{\lambda\to 0_+}\nu_u(\lambda)$,
all~four terms are equal for $\lambda\in\bR\ssm\cD$ with $\cD$ countable.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Approximate cohomology lower bounds}
\vskip-2pt  
\claim{\bf Proof.} One first estimates the spectrum of the total
Laplacian $\Delta_{k,E}=\nabla_{k,E}\nabla_{k,E}^*+\nabla_{k,E}^*\nabla_{k,E}$
(harmonic oscillator with magnetic and electric fields),\pause\ and then
one uses a Bochner formula to relate $\ovl\square_{k,E}$ and 
$\Delta_{k,E}$ ($\ovl\square_{k,E}\simeq\frac{1}{2}\Delta_{k,E}+{}$curvature~terms)
for each $(p,q)$.\pause\vskip3pt
\begin{block}{Important special case $\lambda=0$ (harmonic forms)}
\alert{\centerline{$\displaystyle
\sum_{|J|=q}\ovl\nu_u(u_{\complement J}-u_J)\,dV_\omega
=(-1)^q\frac{u^n}{n!}\,.$}}
\end{block}\pause
\begin{block}{Corollary (Laurent laeng, 2002)} For $\lambda_k\to 0$
slowly enough, i.e.\ with $k^{2+2/b_2}\lambda_k\to +\infty$, one has\vskip3pt
\centerline{\alert{$\displaystyle
\liminf_{k\to+\infty}k^{-n}N^{0,0}_{k,E}(\lambda_k)\ge
{r\over n!}\bigg(\int_{X(u,0)}u^n+\int_{X(u,1)}u^n\bigg)\quad$} where}\vskip4pt
$\alert{X(u,q)=\hbox{$q$-index set}}=
\big\{x\in X/\;u(x)~\hbox{has signature $(n{-}q,q)$}\big\}$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Proof of the lower bound}
\vskip-4pt  
\claim{\bf Proof.} One uses the fact that for $\delta'>\delta>0$ and $k\gg 1$,
the composition $\Pi\circ\dbar_k$ with an eigenspace projection
yields an injection
\vskip5pt
\alert{\centerline{$\displaystyle
\bigoplus_{\lambda\in{}\,]\lambda_k,\delta]}\hbox{eigenspace}_\lambda^{0,0}
\hookrightarrow
\bigoplus_{\lambda\in\,{}]0,\delta']}\hbox{eigenspace}_\lambda^{0,1}.$}}\vskip2pt\pause
In fact, in the holomorphic case $\dbar_k^2=0$ implies 
$\dbar_k\ovl\square_k^{0,0}=\ovl\square_k^{0,1}\dbar_k$, hence $\dbar_k$
maps the $(0,0)$-eigenspaces to the $(0,1)$-eigenspaces for the same
eigenvalues, and one can even take $\lambda_k=0$, $\delta'=\delta$.\pause\vskip3pt
In the quasi holomorphic case
$\dbar_k^2=O(k^{-1/b_2})$, one can show that
$\ovl\square_k^{0,1}\dbar_k-\dbar_k\ovl\square_k^{0,0}=\dbar_k^*\dbar_k^2$
yields a small ``deviation'' of the \hbox{eigenvalues\kern-15pt}\\
to $[\lambda_k-\varepsilon,
\delta+\varepsilon]$ with $\varepsilon<\min(\lambda_k,\delta'-\delta)$,
whence the injectivity.\pause\vskip5pt
This implies
\vskip2pt
\alert{\centerline{$\displaystyle
N^{0,1}_{k,E}(\delta')\ge N^{0,0}_{k,E}(\delta)- N^{0,0}_{k,E}(\lambda_k)$}}
\vskip-3pt thus\vskip2pt
\centerline{\alert{$\displaystyle
N^{0,0}_{k,E}(\lambda_k)\ge N^{0,0}_{k,E}(\delta)- N^{0,1}_{k,E}(\delta')$,}
\kern20pt\hbox{QED\kern-50pt}}
\end{frame}

\begin{frame}
\frametitle{Transcendental holomorphic Morse inequalities}
\vskip-9pt
\begin{block}{Conjecture on Morse inequalities}
Let $\gamma\in H^{1,1}_\BC(X,\bR)$. Then
\vskip-2pt
\alert{\centerline{$\displaystyle
\Vol(\gamma)\ge\sup_{u\in\gamma,\,u\in\smash{C^\infty}}\int_{X(u,\le 1)}u^n.$}}\vskip1pt
(One could even suspect \alert{equality}, an even stronger conjecture~!).
\end{block}\pause\vskip-5pt
If~one sets by definition\vskip2pt
\alert{\centerline{$\displaystyle
\Vol(\gamma)=\sup_{u\in\gamma}~~\lim_{\lambda\to 0_+}~~
\liminf_{k\to+\infty}N^{0,0}_k(\lambda)$}}\vskip0pt
for the eigenspaces of the sequence $(L_k,h_k,\nabla_k)$ approximating $ku$, 
then the above expected lower bound \alert{is a theorem!}\pause\vskip3pt
There is however a stronger \&\ more usual definition of the volume.\vskip-4pt
\begin{block}{Definition} For $\gamma\in H^{1,1}_\BC(X,\bR)$, set
$\Vol(\gamma)=0$ if $\gamma\not\ni$ any current $T\ge 0$,\\
and otherwise set~
\alert{$\displaystyle
\Vol(\gamma)=\sup_{T\in\gamma,\,\smash{T=u_0+i\ddbar\varphi\ge 0}}~~
\int_X T_{\rm ac}^n\,$},~~$u_0\in C^\infty$.\vskip2pt
\end{block}
\end{frame}

\begin{frame}
\frametitle{Transcendental holomorphic Morse inequalities (2)}
\vskip-5pt
The conjecture on Morse inequalities is known to be true when $\gamma=c_1(L)$
is an integral class ([D-1985]). In fact, one then gets a Hermitian holomorphic
line bundle $(L,h)$ and its multiples $L^{\otimes k}$. The spectral estimates
provide many holomorphic sections $\sigma_{k,\ell}\,$, and one gets positive
currents right away by putting\vskip4pt
\alert{\centerline{
$\displaystyle
T_k=\frac{i}{2k\pi}\ddbar\log\sum_{\ell}|\sigma_{k,\ell}|^2_h
+\frac{i}{2\pi}\Theta_{L,h}\ge 0$}}\pause\vskip2pt
(the volume estimate can be derived from there by Fujita).\vskip3pt\pause 
In the ``quasi-holomorphic'' case, one only gets eigenfunctions
$\sigma_{k,\ell}$ with small eigenvalues, and \alert{the positivity of $T_k$
is a priori lost}.\pause

\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 cohomology classes}. Then
\vskip3pt
\alert{\centerline{$\displaystyle
\Vol(\alpha-\beta)\geq\alpha^n-n\alpha^{n-1}\cdot\beta$.}}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Known results on holomorphic Morse inequalities}
\vskip-4pt
The conjectural corollary is derived from the main conjecture
by an elementary symmetric function argument. In fact, one has
a pointwise inequality of forms\vskip4pt
\alert{\centerline{$\displaystyle{\bf 1}_{X(\alpha-\beta,\leq 1)}(\alpha-\beta)^n
\geq\alpha^n-n\alpha^{n-1}\cdot\beta$.}}\vskip4pt\pause
Again, the corollary is known for $\gamma=\alpha-\beta$ when
$\alpha,\beta$ are integral classes  (by [D-1993] and independently
[Trapani, 1993]).\pause\vskip4pt
Recently (2016), the volume estimate for $\gamma=\alpha-\beta$
\hbox{transcendental\kern-10pt}\\
has been established by D.\ Witt-Nystr\"om when \alert{$X$ is
projective}, using deep facts on Monge-Ampère operators and
upper envelopes.\pause\vskip4pt
Xiao and Popovici also proved in the K\"ahler case that\vskip3pt
\alert{$\alpha^n-n\alpha^{n-1}\cdot\beta>0~~\Rightarrow~~
\Vol(\alpha-\beta)>0$\\
\strut\kern4.6cm and $\alpha-\beta$ contains a
K\"ahler current}.\pause\vskip3pt
(The proof is short, once the Calabi-Yau theorem is taken for granted).
\end{frame}

\begin{frame} 
\frametitle{Projective vs K\"ahler vs non K\"ahler varieties}
\vskip-5pt
\claim{\bf Problem.} 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 \alert{strongly positive
in the sense of Nakano}, and is given by a universal formula.\\
\end{block}
\pause\vskip-2pt
In the sequel of this lecture, we aim to investigate this construction and
look for potential applications, especially to
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}
\pause\break
It 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\vskip7pt

\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\vskip8pt
This frame 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{
$\displaystyle
\Theta_{B_\varepsilon,h}=\nabla^{1,0}\nabla^{0,1}+\nabla^{0,1}\nabla^{1,0}
\in C^\infty(X,\Lambda^{1,1}T^\star_X\otimes{\rm Hom}(
B_\varepsilon,B_\varepsilon)),$}}\pause\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{Back to holomorphic Morse inequalities}
\vskip-4pt
\claim{\bf Idea for the general case.} Let $\gamma\in H^{1,1}_\BC(X,\bR)$
and $u\in\gamma$ a smooth form. As we have seen, one can find a sequence of
Hermitian line bundles $(L_k,h_k,\nabla_k)$ such that\vskip3pt
\centerline{\alert{$\displaystyle
\theta_k=\frac{i}{2\pi}\nabla_k^2=ku+\beta_k,\quad
\beta_k=O(k^{-1/b_2})$}.}\pause\vskip3pt
Then $d\theta_k=0\Rightarrow \dbar\beta_k^{0,2}=0$, and as
$U_\varepsilon$ is Stein,
$\pr_1^*\beta_k^{0,2}=\dbar \eta_k$ with a $C^\infty$ $(0,1)$-form 
$\eta_k=O(k^{-1/b_2})$.\pause\ This shows
that $\tilde L_k:=pr_1^*L_k$ becomes a \alert{holomorphic line bundle}
when equipped with the
connection $\tilde\nabla_k=\pr_1^*\nabla_k-\eta_k$, which has a curvature
form $\Theta_{\tilde L_k,\tilde \nabla_k}=k\pr_1^*u+O(k^{-1/b_2})$.\pause\
Two possibilities emerge:
\begin{itemize}
\item correct the small eigenvalue eigenfunctions $\pr_1^*\sigma_{k,\ell}$
given by
Laeng's method to actually get holomorphic sections of $\tilde L_k$
on \hbox{$U_\varepsilon$.\kern-15pt}\pause
\item directly deal with the Hilbert Dolbeault complex of
$(\pr_1)_*^{L^2}(\cO_{U_\varepsilon}(\tilde L_k))$, and
\alert{use Bergman estimates instead of dimension counts in Morse
inequalities}.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Other potential target: invariance of plurigenera for
polarized families of compact K\"ahler \hbox{manifolds?\kern-10pt}}
\vskip-9pt
\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\vskip-4pt
The conjecture is known to be true for a \alert{projective family}
$\cX\to S$:\pause\\
$\bu$ Siu and Kawamata (1998) in the case of varieties of
\alert{general type}\pause\\
$\bu$ Siu (2000) and P\u{a}un (2004) in the arbitrary projective case
\vskip5pt\pause\vskip-4pt
The proof is based on an iterated application
of the \alert{Ohsawa-Takegoshi $L^2$ extension theorem} w.r.t.\ an
ample line bundle $\cA$ on $\cX$:\pause\ \alert{replace $\cA$ by
a Bergman bundle in the K\"ahler \hbox{case~?\kern-10pt}}
\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}

\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}

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