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% In principle, this file can be redistributed and/or modified under
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%
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\title[\ \kern-190pt\rlap{\blank{J.-P.~Demailly (Grenoble), 24${}^{\hbox{\fivebf \`eme}}$ Colloque de la SMT, Sousse}}\kern183pt\rlap{\blank{On the curvature of Bergman bundles [March 19, 2019]}}\kern178pt\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{On Bergman bundles and\\
their curvature properties}

%% \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)
{24${}^{\hbox{\sevenbf ème}}$ Colloque de la Société Mathématique Tunisienne\\
Hotel Iberostar Kantaoui Bay\\
Sousse, 18--22 mars 2019}

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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{Curvature tensor of a holomorphic vector bundle}  
\vskip-5pt
Let $X$ be a complex manifold, $n=\dim_\bC X$, and
$(E,h)$ a holomorphic vector bundle of rank $r$
equipped with a hermitian metric $h$. With respect to
a local holomorphic frame $(e_\lambda)_{1\leq\lambda\leq r}$
\alert{$$
\langle u,v\rangle=\sum h_{\lambda\mu}(z)u_\lambda\overline v_\mu,~~~
u,v\in E_z.
$$}\pause
The Chern curvature tensor of $(E,h)$ is defined to be the global
$(1,1)$-form
\alert{$\Theta_{E,h}\in C^\infty(X,\Lambda^{1,1}T^*_X\otimes\End(E))$}\vskip4pt
\centerline{\alert{$\displaystyle
\Theta_{E,h}={i\over 2\pi}\sum_{1\leq j,k\leq n,\,1\leq\lambda,\mu\leq r}
c_{jk\lambda\mu}(z)\,dz_j\wedge d\overline z_k\otimes
e_\lambda^*\otimes e_\mu
$}}\vskip-2pt
locally computed as the matrix \alert{$-{i\over 2\pi}
\overline\partial(\overline H^{-1}\partial\overline H)$} where
\hbox{$H=(h_{\lambda\mu})$.\kern10pt}\\\pause One has an
associated hermitian form
\alert{$$
\widetilde\Theta_{E,h}(\tau)=\sum c_{jk\lambda\mu}(z)\,
\tau_{j\lambda}\overline\tau_{k\mu},~~~\tau\in T_X\otimes E,
$$}%
and one says that \alert{$\Theta_{E,h}>0$ (in the sense of Nakano)} if
\alert{$\widetilde\Theta_{E,h}(\tau)>0$} for all nonzero tensors
$\tau\in T_X\otimes E$.
\end{frame}

\begin{frame}
\frametitle{Kodaira embedding theorem}
\vskip-7pt  
The special case of a holomorphic hermitian line bundle $(L,h)$ is very
interesting. Then one usually write the hermitian metric as
$h=e^{-\varphi}$ locally on a trivializing open set $U\subset X$, so
that\vskip3pt
\centerline{\alert{$\displaystyle
\Theta_{L,h}={i\over 2\pi}\partial\overline\partial\varphi=
{i\over 2\pi}\sum_{j,k}{\partial^2\varphi\over\partial z_j
\overline\partial z_k}\,dz_j\wedge\overline z_k,
$}}
and $\Theta_{L,h}>0$ means that $\varphi$ is \alert{strictly plurisubharmonic}.
\pause\vskip-4pt

\begin{block}{Theorem (Kodaira 1953 - main reason for his Fields medal!)}
For $X$ a compact complex manifold, TFAE~:\vskip2pt
(i) \alert{$L>0$}, i.e.\ $L$ possesses a smooth hermitian metric s.t.\
$\Theta_{L,h}>0\,$;\\
(ii) \alert{$L$ is ample}, i.e. there exists a tensor power $L^{\otimes m}$ and
sections $\sigma_0,\ldots,\sigma_N\in H^0(X,L^{\otimes m})$ such that\vskip3pt
\alert{$\displaystyle
X\to\bP^N,~~x\mapsto [\sigma_0(x):\sigma_1(x):...:\sigma_N(x)]\in\bP^N
$} is an embedding.
\end{block}
\pause\vskip-4pt
Then $X$ is in fact an algebraic submanifold $\{P_1=\ldots=P_q=0\}$ of $\bP^N$,
and one says that $X$ is a \alert{projective algebraic manifold}.
\end{frame}

\begin{frame}
\frametitle{Projective vs K\"ahler vs non K\"ahler varieties}
\vskip-5pt
By Kodaira, non projective varieties do not have
\hbox{\alert{ample line bundles}.\kern-20pt}
\vskip4pt\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.
\vskip4pt\pause
What for non K\"ahler compact complex manifolds?
\pause\vskip-2pt
\begin{block}{Surprising fact (?)}
Every compact complex manifold $X$ carries a \alert{``very ample''
complex Hilbert bundle}, produced by means of a natural Bergman
space construction; the curvature of this bundle is strongly positive
and is given by a universal formula.\\
\pause
In particular, $X$ can be embedded holomorphically in a\\
\alert{``Hilbert Grassmannian''} of infinite dimension and codimension.
\end{block}
\pause\vskip-2pt
Our goal is to investigate further this construction and explain
potential applications to analytic geometry (K\"ahler invariance
of plurigenera, transcendental holomorphic Morse inequalities...)
\end{frame}

\begin{frame}
\frametitle{Tubular Stein neighborhoods}
\vskip-6pt
Let $X$ be a compact complex manifold, $\dim_\bC X=n$.
\vskip3pt
Denote by $\overline X$ its complex conjugate $(X,-J)$, so that
\alert{$\cO_{\overline X}=\overline{\cO_X}$}.
\pause\vskip5pt
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\vskip8pt
\strut\kern2cm\pgfdeclareimage[height=5.3cm]{FFig1}{FFig1}
\pgfuseimage{FFig1}

\end{frame}

\begin{frame}
\frametitle{Tubular Stein neighborhoods (continued)}  
\vskip-5pt
In the special case $X=\bC^n$,
$U_\varepsilon=\{(z,w)\,;\;|\overline z-w|<\varepsilon\}$ is of course Stein
since\vskip3pt
\alert{\centerline{$|\overline z-w|^2=|z|^2+|w|^2-2\Re\sum z_jw_j$}}\vskip4pt
and $(z,w)\mapsto\Re\sum z_jw_j$ is pluriharmonic.\pause\vskip-3pt
\begin{block}{Technical lemma}Let $\exp:T_X\to X\times X$,
$(z,\xi)\mapsto(z,\exp_z(\xi))$ be the exponential map
associated with a real analytic hermitian metric $\gamma$ on $X$,\\
and $\exph$ its ``holomorphic'' part, so that\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
Let $\logh:X\times X\supset W\to T_X$ be the inverse of $\exph$
and\vskip5pt
\alert{\centerline{$\displaystyle
U_\varepsilon=\{(z,w)\in X\times\overline X\,;\;|\logh_z(w)|_\gamma
<\varepsilon\},~~~\varepsilon>0.$}}\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{Bergman sheaves}  
Let $U_\varepsilon=U_{\gamma,\varepsilon}\subset X\times\overline X$
be the ball bundle as above, and\vskip5pt
\centerline{$
p=(\pr_1)_{|U_\varepsilon}:U_\varepsilon\to X,\qquad
\overline p=(\pr_2)_{|U_\varepsilon}:U_\varepsilon\to\overline X
$}\vskip5pt
the natural projections.

\begin{block}{Definition}
The ``Bergman sheaf'' $\cB_\varepsilon=\cB_{\gamma,\varepsilon}$ is the
$L^2$ direct image\vskip5pt
\alert{\centerline{$
\cB_\varepsilon=p^{L^2}_*(\overline p^*\cO(K_{\overline X})),
$}}\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,$}\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\wedge \overline f\wedge \gamma^n
<+\infty,\quad\forall K\compact V.
$}}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Associated Bergman bundle and holom structure}  
\vskip-7pt
Then $\cB_\varepsilon$ is an $\cO_X$-module, and by the
Ohsawa-Takegoshi extension theorem
applied to the subvariety $p^{-1}(z)\subset U_\varepsilon$,
its fiber $B_{\varepsilon,z}=\cB_{\varepsilon,z}/\gm_z\cB_{\varepsilon,z}$ is
isomorphic to the Hardy-Bergman space
$\cH^2(B(0,\varepsilon))$ of $L^2$ holomorphic $n$-forms on $p^{-1}(z)\simeq
B(0,\varepsilon)\subset\bC^n$.\pause\vskip-3pt

\begin{block}{Question}
By putting \alert{$\displaystyle
\Vert f(z)\Vert^2=\int_{p^{-1}(z)}i^{n^2}f\wedge \overline f$},
we get a (real analytic)\vskip3pt
locally trivial Hilbert bundle $B_\varepsilon\to X$.
\pause\\
Is there a ``complex structure'' on $B_\varepsilon$ such that
$\cB_\varepsilon=\cO(B_\varepsilon)$~?\pause
\end{block}

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\vskip1pt
\alert{\centerline{$\displaystyle
f(z,w)\,{=}\sum_{|J|=q}f_J(z,w)\,dw_1\wedge...\wedge dw_n\wedge
d\overline z_J,~~
(z,w)\in U_\varepsilon\cap(V\times\overline X),$}}\vskip0pt
$f_J(z,w)$ holomorphic in $w$ and
\hbox{all $\dbar_zf(z,w)\in L^2(p^{-1}(K))$, $K\compact V$.\kern-50pt}
\end{frame}

\begin{frame}
\frametitle{Very ampleness of Bergman bundles}
\vskip-5pt
By construction, $\dbar$ yields a complex of sheaves $(\cF^\bullet,\dbar\,)$
and the kernel $\Ker\dbar:\cF^0\to\cF^1$ coincides with $\cB_\varepsilon$.

\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 $\overline U_\varepsilon
\subset X\times\overline X$. Then the complex of sheaves
$(\cF^\bullet,\dbar)$ 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
\centerline{\alert{$
$$
H^q(X,\cB_\varepsilon\otimes\cO(E))=H^q\big(\Gamma(X,
\cF^\bullet_\varepsilon\otimes\cO(E)),\dbar\big)=0,\quad\forall q\geq 1.
$}}\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}
In other words, $B_\varepsilon$ is a \alert{``very ample holomorphic
vector bundle''}\\ (as a Hilbert bundle of infinite dimension).\pause\\
But it is \alert{NOT} holomorphically locally trivial.
\end{frame}

\begin{frame}
\frametitle{Chern connection of Bergman bundles}
Since we have a natural $\nabla^{0,1}=\dbar$ connection, and a
natural hermitian metric on the Bergman bundle, it follows 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 (non holomorphic) 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
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 $\Theta_{B_\varepsilon,h}=\nabla^{1,0}\nabla^{0,1}+\nabla^{0,1}\nabla^{1,0}$
be the curvature tensor of $B_\varepsilon$ with its natural Hilbertian
metric $h$, and\vskip4pt
\alert{\centerline{
$\displaystyle
\widetilde\Theta_\varepsilon(v\otimes\xi)=
\langle\Theta_{B_\varepsilon,h}\sigma(v,Jv)\xi,\xi\rangle_h$}}\vskip4pt
the associated quadratic form with $v\in T_X$,
$\xi=\sum_\alpha\xi_\alpha e_\alpha\in B_\varepsilon$.

\begin{block}{Formula}
In the model case $X=\bC^n$, the curvature tensor of the Bergman bundle
$(B_\varepsilon,h)$ is given by\vskip4pt
\alert{\centerline{
$\displaystyle
\widetilde\Theta_{\varepsilon}(v\otimes\xi)
=\varepsilon^{-2}\kern-3pt\sum_{\alpha\in\bN^n}\kern-3pt\Bigg(\bigg|
\sum_j\sqrt{\alpha_j}\;\xi_{\alpha-c_j}v_j\bigg|^2\kern-3pt{}
+\sum_{j}(|\alpha|+n)\;|\xi_\alpha|^2|v_j|^2\Bigg)\kern-3pt.$}}\vskip0pt\pause
\end{block}
Observe that $\widetilde\Theta_{\varepsilon}(v\otimes\xi)$
is a positive but \alert{unbounded} quadratic form on $B_\varepsilon$
with respect to the standard norm $\Vert\xi\Vert^2=
\sum_\alpha|\xi_\alpha|^2$.\vskip4pt\pause
However there is convergence for all
$\xi=\sum_\alpha\xi_\alpha e_\alpha\in \alert{B_{\rho\varepsilon}}$,
\alert{$\rho>1$}, since then $\sum_\alpha\rho^{2|\alpha|}|\xi_\alpha|^2<+\infty$.
\end{frame}

\begin{frame}
\frametitle{Curvature of Bergman bundles (general case)}
\vskip-3pt
\begin{block}{Bergman curvature formula on a general hermitian manifold}
Let $X$ be a compact complex manifold equipped with a hermitian metric
$\gamma$, and $B_\varepsilon=B_{\gamma,\varepsilon}$ the corresponding
Bergman bundle. Then its curvature is given by an asymptotic expansion
\alert{\centerline{
$\displaystyle
\widetilde\Theta_{\varepsilon}(z,v\otimes\xi)
=\sum_{p=0}^{+\infty}\varepsilon^{-2+p}Q_p(z,v\otimes\xi)$}}\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 and depend on the
torsion and curvature tensor of $\gamma$, especially $Q_1$, $Q_2$.\pause
\end{block}
A consequence of the above formula is that $B_\varepsilon$ is strongly
Nakano positive for $\varepsilon>0$ small enough.
\end{frame}

\begin{frame}
\frametitle{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~$S$. 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 $X_t:=\pi^{-1}(t)$. Then the
plurige\-nera\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 uses deeply the
\alert{Ohsawa-Takegoshi $L^2$ 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}\\
$\bu$ $|s|^2_h\leq 1$, 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 defines inductively 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 $(\sigma_{p-m,j}s^m)_{|X_0}$ 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-3pt
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{$\varphi=\limsup_{p\to+\infty}{1\over p}
\log\sum_j|\sigma_{p,j}|^2$}.}\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

\begin{block}{Proposition}
In the polarized K\"ahler case $(\cX,\omega)$, shrinking from
$U_{\rho\varepsilon}$, $\rho>1$, to~$U_\varepsilon$, one gets\vskip0pt
\centerline{\alert{$\displaystyle
i\ddbar\Big(\sum_j\Vert\sigma_{p,j}\Vert_{U_\varepsilon}^2\Big)^{\lambda/p}
\geq -\varepsilon^{-2}(\log\rho)^{-1}\rho^{n\lambda/p}e^{C\lambda}\omega\quad
\forall\lambda>0.
$}}\vskip0pt\pause
This is enough to imply the invariance of plurigenera if $\varepsilon>0$ can
be taken \alert{arbitrarily large}.
\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. 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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