% This file is a solution template for:
% - Giving a talk on some subject.
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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
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% package/program, I grant the extra permission to freely copy and
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\title[\ \kern-190pt\rlap{\blank{J.-P.\ Demailly (Grenoble), Helmut Hofer's 60th birthday, Z\"urich}}\kern183pt\rlap{\blank{Embeddings of compact almost complex structures}}\kern178pt\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{Algebro-differential embeddings\\
of compact almost complex structures}

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

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

\institute[]{Institut Fourier, Université de 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)
{Conference at ETH Z\"urich ``Analysis in the large,\\
Calculus of Variations, Dynamics, Geometry, ...''\\
in honour of Helmut Hofer, June 6--10, 2016}

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\begin{document}

% 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}
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  \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{Symplectic geometry and almost complex geometry}
\vskip-4pt
An important part of Helmut Hofer's work deals with symplectic geometry,
and the geometry of pseudoholomorphic curves.
\pause

The subject was given a strong impetus by Mikhail Gromov in his famous 
Inventiones paper from 1985.
\pause

 Since then, many interconnections between symplectic geometry, topology
and algebraic geometry have been developed, e.g.\ through the study of Gromov-Witten invariants.
\pause

\begin{block}{Basic question} Let $(M^{2n},\omega)$ be a \alert{
compact symplectic manifold\\
and $J$ a compatible almost complex structure}. Assume that
\vskip3pt
\centerline{$\displaystyle
\alert{\int_Mc_1(M,J)\wedge\omega^{n-1}>0}.
$}
\vskip3pt
Is it true that there exists a differentiable family of\\
\alert{mobile pseudoholomorphic curves $(f_t)_{t\in S}:\bP^1\to M$},\\
i.e.\ generically injective and covering an open set in $M$~?
\end{block}

\end{frame}

\begin{frame}
\frametitle{Existence of rational curves}
\vskip-4pt
\begin{block}{Related question} Let $(X^{n},\omega)$ be a \alert{
compact K\"ahler manifold.} Assume that that $c_1(K_X)\cdot \omega^{n-1}<0$
or more generally that \alert{$K_X$ is not pseudoeffective} (this means that
the class $c_1(K_X)$ does not contain any closed $(1,1)$-current $T\ge 0$.\\
Can one conclude that $X$ is \alert{covered by rational curves}~?
\end{block}
\pause
This would be crucial for the theory of compact K\"ahler manifolds.
\pause
\begin{block}{Theorem (BDPP = Boucksom - D - Peternell - Paun, 2002)}
The answer is positive when $X$ is a complex projective manifold.
\end{block}
\pause
The proof uses intersection theory of currents and characteristic $p$
techniques due to Mori.\vskip3pt\pause
It would be nice to have a \alert{``symplectic proof''}, especially in  the
K\"ahler case.
\end{frame}

\begin{frame}
\frametitle{A question raised by Fedor Bogomolov}
\begin{block}{Rough question}
Can one produce an arbitrary compact complex manifold $X$ /\\ 
an arbitrary compact K\"ahler manifold $X$ by means of a\\
\alert{``purely algebraic construction''} ?
\end{block}
\pause\vskip5pt

Let $Z$ be a projective algebraic manifold, $\dim_\bC Z=N$,
equipped with a subbundle (or rather subsheaf) $\cD\subset\cO_Z(T_Z)$.\vskip5pt
\pause
Assume that $X^{2n}$ is a compact $C^\infty$ real even dimensional 
manifold that is embedded in $Z$, as follows:\pause
\vskip5pt
(i) $f:X\hookrightarrow Z $ is a smooth (say $C^\infty$) embedding
\vskip2pt
(ii) $\forall x\in X,\quad f_*T_{X,x}\oplus\cD_{f(x)}=T_{Z,f(x)}.$
\vskip2pt
(iii) $f(X)\cap \cD_{\sing}=\emptyset$.
\vskip5pt
We say that \alert{$X\hookrightarrow(Z,\cD)$ is a transverse
embedding}.
\end{frame}

\begin{frame}
\frametitle{A conjecture of Bogomolov}
\pgfdeclareimage[height=3cm]{Fig1}{Fig1}
\pgfuseimage{Fig1}
\vskip-2cm\strut\kern4cm
$f_*T_{X,x}=T_{M,f(x)}\simeq T_{Z,f(x)}/\cD_{f(x)}$
\vskip1.3cm
\begin{block}{Observation 1}
If $\cD\subset T_Z$ is an \alert{algebraic foliation}, 
i.e.\ $[\cD,\cD]\subset\cD$,
then the almost complex structure $J_f$ on $X$ induced by $(Z,\cD)$ is
\alert{integrable}.
\end{block}
\vskip2mm
\pause
\pgfdeclareimage[height=2.8cm]{Fig2}{Fig2}
$\raise2.7cm\hbox{\bf Proof:}$
\vskip-3.1cm\strut
\kern3cm\pgfuseimage{Fig2}
\end{frame}

\begin{frame}
\frametitle{A conjecture of Bogomolov (2)}

\begin{block}{Observation 2}
If $\cD\subset T_Z$ is an algebraic foliation and $f_t:X\hookrightarrow
(Z,\cD)$ is an \alert{isotopy of transverse embeddings}, $t\in[0,1]$, 
then all complex structures $(X,J_{f_t})$ are \alert{biholomorphic}.
\end{block}
\vskip2mm
\pause
\pgfdeclareimage[height=4cm]{Fig3}{Fig3}
$\raise3.7cm\hbox{\bf Proof:}$
\vskip-3.1cm\strut
\kern2cm\pgfuseimage{Fig3}
\end{frame}

\begin{frame}
\frametitle{A conjecture of Bogomolov (3)}
\vskip-4pt
To each triple $(Z,\cD,\alpha)$ where\\
$\bullet$ $Z$ is a complex projective manifold\\
$\bullet$ $\cD\subset T_Z$ is an \alert{algebraic foliation}\\
$\bullet$ $\alpha$ is an \alert{isotopy class of transverse embeddings}
$f:X\hookrightarrow(Z,\cD)$\\
one can thus associate a \alert{biholomorphism class $(X,J_f)$}.\pause

\begin{block}{Conjecture (Bogomolov, 1995)}
One can construct in this way every compact complex manifold $X$.
\end{block}
\pause

\begin{block}{Additional question 1}
What if $(X,\omega)$ is K\"ahler ? Can one embed in such a way that
$\omega$ is the pull-back of a transversal K\"ahler structure on
$(Z,\cD)$~?
\end{block}
\pause

\begin{block}{Additional question 2}
Can one define moduli spaces of such embeddings, describing the non
injectivity of the ``Bogomolov fonctor'' ?
\end{block}
\end{frame}

\begin{frame}
\frametitle{There exist large classes of examples !}
\vskip-4pt
\begin{block}{Example 1 : tori}
If $Z$ is an Abelian variety and $N\ge 2n$,
 every $n$-dimensional compact complex torus
$X=\bC^n/\Lambda$ can be embedded transversally
to a linear codimension $n$ foliation $\cD$ on $Z$.
\end{block}
\pgfdeclareimage[height=2cm]{Fig4}{Fig4}
\pgfuseimage{Fig4}
\pause\vskip-3pt
\begin{block}{Example 2 : LVMB manifolds}
One obtains a rich class, named after
\alert{Lopez de Medrano, Verjovsky, Meersseman, Bosio}, by
considering foliations on $\bP^N$ given by a commutative
Lie subalgebra of the Lie algebra of ${\rm PGL}(N+1,\bC)$.\\
\pause
The corresponding transverse varieties produced include 
e.g. Hopf surfaces and the 
Calabi-Eckmann manifolds $S^{2p+1}\times S^{2q+1}$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{What about the almost complex case ?}
\vskip-3pt
\begin{block}{Easier question : drop the integrability assumption}
Can one realize every compact almost complex manifold $(X,J)$ by
a transverse embedding into a projective algebraic pair $(Z,\cD)$,
$\cD\subset T_Z$, so that \alert{$J=J_f$}~?
\end{block}
\pause
Not surprisingly, there are constraints, and $Z$ cannot be\\
``too small''. But how large exactly~?\vskip4pt
\pause
Let $\Gamma^\infty(X,Z,\mathcal{D})$ the Fr\'echet manifold of transverse
embeddings $f:X\hookrightarrow(Z,\cD)$  and $\cJ^\infty(X)$ the space of 
smooth almost complex structures on~$X$.

\begin{block}{Further question}
When is $\alert{f\mapsto J_f},\quad \Gamma^\infty(X,Z,\mathcal{D})\to
\cJ^\infty(X)$ a \alert{submersion}~?
\end{block}

Note: technically one has to consider rather Banach spaces of maps of
\alert{$C^{r+\alpha}$ H\"older regularity}.
\end{frame}

\begin{frame}
\frametitle{Variation formula for $J_f$}
First, the tangent space to the Fréchet manifold $\Gamma^\infty(X,Z,\mathcal{D})$
at a point $f$ consists of\vskip4pt
\centerline{
$C^\infty(X,f^*T_Z)=C^\infty(X,f^*\cD)\oplus C^\infty(X,T_X)$}
\vskip4pt
\begin{block}{Theorem (D - Gaussier, arxiv:1412.2899, 2014)}
The differential of the natural map $f\mapsto J_f$
along any infinitesimal variation $w=u+f_*v:X\to f^*T_Z=
f^*\mathcal {D}\oplus f_*TX$ of $f$ is given by
\vskip6pt
\centerline{\alert{%
$dJ_f(w)=2J_f\big(f_*^{-1}\theta(\overline\partial_{J_f}f,u)
+\overline\partial_{J_f}v\big)$}}
\vskip4pt
where 
\vskip4pt
\centerline{\alert{%
$\theta:\mathcal{D}\times\mathcal{D}\to TZ/\mathcal{D},\quad
(\xi,\eta)\mapsto[\xi,\eta]\mod\cD$}}
\vskip6pt
is the torsion tensor of the holomorphic distribution $\mathcal{D}$,
and $\overline\partial f=\overline\partial_{J_f} f$,
$\overline\partial v=\overline\partial_{J_f}v$
are computed with respect to the almost complex structure $(X,J_f)$.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Sufficient condition for submersivity}
\vskip-4pt
\begin{block}{Theorem (D - Gaussier, 2014)}
Let $f:X \hookrightarrow(Z,\cD)$ be a smooth transverse embedding.
Assume that $f$ and the torsion 
tensor $\theta$ of $\mathcal{D}$ satisfy the following additional 
conditions$\,:$
\vskip3pt
{\rm(ii)} $f$ is a totally real embedding, i.e.\ 
$\alert{\overline\partial f(x)}
\in\End_{\overline{\mathbb{C}}}(T_{X,x},T_{Z,f(x)})$ is 
\alert{injective} at every point $x\in X\,;$
\vskip3pt
{\rm(ii)} for every $x\in X$ and every
$\eta\in \End_{\overline{\mathbb{C}}}(T_X)$, there exists a vector
$\lambda\in\mathcal{D}_{f(x)}$ such that $\theta(\overline\partial f(x)\cdot
\xi,\lambda)=\eta(\xi)$ for all $\xi\in T_X$.
\vskip3pt
Then there is a neighborhood $\mathcal{U}$ of $f$ in 
$\Gamma^\infty(X,Z,\mathcal{D})$ and a neighborhood $\mathcal {V}$ of $J_f$ in 
$\mathcal{J}^\infty(X)$ such that\vskip3pt
\alert{\centerline{%
$\mathcal{U}\to\mathcal{V}$, $f\mapsto J_f$ is a submersion.}}
\end{block}

{\bf Remark.} A necessary condition for (ii) to be possible is that
$\rank\cD=N-n\ge n^2=\dim\End(T_X)$, i.e.\ \alert{$N\ge n+n^2$}.
\end{frame}

\begin{frame}
\frametitle{Existence of universal embedding spaces}
\vskip-4pt
\begin{block}{Theorem (D - Gaussier, 2014)}
For all integers $n\ge 1$ and $k\ge 4n$, there exists a complex affine 
algebraic manifold $Z_{n,k}$ of dimension \alert{$N=2k+2(k^2+n(k-n))$}
possessing a real structure $($i.e.\ an anti-holomorphic algebraic 
involution$)$ and an algebraic distribution $\cD_{n,k}\subset T_{Z_{n,k}}$ of 
codimension $n$, with the following property:\vskip2pt\pause
for every compact $n$-dimensional almost complex manifold $(X,J)$ admits
an embedding $f:X\hookrightarrow Z^{\mathbb{R}}_{n,k}$ transverse to 
$\cD_{n,k}$ and contained in the real part of $Z_{n,k}$,
such that \alert{$J=J_f$}.
\end{block}
\pause

The choice $k=4n$ yields the explicit embedding dimension 
$N=38n^2 + 8n$ (and a quadratic bound $N=O(n^2)$ is optimal
by what we have seen previously).\pause\vskip3pt

\alert{\bf Hint.} $Z_{n,k}$ is 
produced by a fiber space construction mixing Grassmannians
and twistor spaces ...
\end{frame}

\begin{frame}
\frametitle{Symplectic embeddings}
\vskip-4pt
Consider the case of a \alert{compact almost complex symplectic manifold
$(X,J,\omega)$} 
where the symplectic form $\omega$ is assumed to be $J$-compatible, 
i.e.\ $J^*\omega=\omega$ and \hbox{$\omega(\xi,J\xi)>0$}. 
\pause
\begin{block}{Definition} We say that a closed semipositive $(1,1)$-form
$\beta$ on $Z$ is a transverse K\"ahler structure to $\cD\subset T_Z$
if the kernel of $\beta$ is contained in $\cD$, i.e., if 
$\beta$ induces a K\"ahler form on germs of complex submanifolds 
transverse to $\cD$.
\end{block}
\pause

\begin{block}{Theorem (D - Gaussier, 2014)}
There also exist universal embedding spaces for compact almost complex
symplectic manifolds, i.e. a certain
triple $(Z,\cD,\beta)$ as above, such that every $(X,J,\omega)$,
$\dim_\bC X=n$, $\{\omega\}\in H^2(X,\bZ)$,
embeds transversally by $f:X\hookrightarrow(Z,\cD,\beta)$ such that\vskip3pt
\centerline{\alert{$J=J_f$ and $\omega=f^*\beta$.}}
\end{block}
\end{frame}
\begin{frame}
\frametitle{Integrability condition}
\vskip-5pt
Recall that\vskip2pt
\alert{%
$N_J(\zeta,\eta)=4\Re\,[\zeta^{0,1},\eta^{0,1}]^{1,0}=[\zeta,\eta]-[J\zeta,J\eta]+J[\zeta,J\eta]+J[J\zeta,\eta].$}
\begin{block}{Nijenhuis tensor formula}
If $\theta$ denotes the torsion of $(Z,\cD)$, the Nijenhuis tensor of 
the almost complex structure $J_f$ induced by a transverse embedding 
$f:X\hookrightarrow(Z,\cD)$ is given by
$\forall z \in X$, $\forall \zeta,\eta \in T_zX$
\vskip4pt
\centerline{\alert{%
$N_{J_f}(\zeta,\eta) = 4\,\theta(\overline{\partial}_{J_f}f(z)\cdot \zeta, \overline{\partial}_{J_f}f(z)\cdot \eta)$.}}
\end{block}
\vskip-5pt\pause
\begin{block}{Weak solution to the Bogomolov conjecture}
There exist universal embeddings spaces $(Z,\cD,\cS)$ where
$\cS\subset\cD\subset T_Z$ are algebraic subsheaves satisfying the partial
integrability condition $[\cS,\cS]\subset\cD$, such that every compact
complex manifold $(X,J)$ of given dimension $n$ embeds transversally by
$f:X\hookrightarrow(Z,\cD)$, i.e. $J=J_f$, with the additional constraint 
\alert{$\Im(\overline\partial f)\subset\cS$}. [Note: our construction
yields $\dim Z=O(n^4)$].
\end{block}
\end{frame}

\begin{frame}
\frametitle{What about Bogomolov's original conjecture ?}
\vskip-7.5pt
\begin{block}{Proposition (reduction of the conjecture to another one !)} 
Assume that holomorphic foliations can be approximated by \alert{Nash 
algebraic foliations} uniformly on compact subsets of any polynomially 
convex open subset of $\bC^N$.\\
Then every compact complex manifold can be approximated by compact 
complex manifolds that are embeddable in the sense of Bogomolov 
in foliated projective manifolds.
\end{block}\vskip-3pt
\pause
The proof uses the Grauert technique of embedding \alert{$X$}
as a totally real submanifold of $X\times\overline X$, and taking a 
Stein neighborhood \hbox{$U\supset\Delta$.\kern-15pt}
\vskip1mm
\pause
\pgfdeclareimage[height=2.5cm]{Fig5}{Fig5}
$\raise2.8cm\hbox{\bf Proof:}$
\vskip-3.2cm\strut
\kern1.5cm\pgfuseimage{Fig5}\vskip-1.2cm
\strut\kern9cm\claim{$\Phi(U)$ Runge}\vskip0.6cm
$\exists\Phi:U\to Z$ holomorphic embedding into $Z$ \alert{affine algebraic}
\hbox{(Stout).\kern-20pt}\\
\strut
\end{frame}

\begin{frame}
\frametitle{The end}
\strut\vskip3mm
\centerline{\huge\bf Happy birthday Helmut!}
\pgfdeclareimage[height=6.5cm]{CalabiYau}{CalabiYau}
\strut\kern2cm\pgfuseimage{CalabiYau}

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\end{document}