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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
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\title[\ \kern-190pt\rlap{\blank{J.-P.\ Demailly (Grenoble), CIME 2018 on non K\"ahler geometry}}\kern183pt\rlap{\blank{Embeddings of complex and almost complex structures}}\kern178pt\llap{\blank{\framenumbering~}}\kern-10pt]
% (optional, use only with long paper titles)
{Algebraic embeddings of complex\\
and almost complex structures}

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

\author[] % (optional, use only with lots of authors)
{Jean-Pierre Demailly\\
(based on joint work with Hervé Gaussier)}

\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)
{CIME School on Complex non-K\"ahler Geometry\\
Cetraro, Italy, July 9--13, 2018}

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\begin{document}
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% 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}
%%}


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% the following command: 

%\beamerdefaultoverlayspecification{<+->}

\begin{frame}
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  \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{A question raised by Fedor Bogomolov}
\begin{block}{Rough question}
Can one produce an arbitrary compact complex manifold $X$,\\
resp.\ 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{Construction of an almost complex structure}
\pgfdeclareimage[height=3cm]{FFig1}{FFig1}
\pgfuseimage{FFig1}
\vskip-2.5cm\strut\kern6.2cm
\vbox{\hbox{$f_*T_{X,x}=T_{M,f(x)}\simeq T_{Z,f(x)}/\cD_{f(x)}$\kern-15pt}
is in a natural way\\
a complex vector space\\
$\Rightarrow$ almost complex structure $J_f$}
\pause\vskip4mm
\begin{block}{Observation 1 (André Haefliger)}
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]{FFig2}{FFig2}
$\smash{\rlap{\hbox{\vbox{{\bf Proof:}
Any 2 charts\\
yield a holomorphic\\
transition map $U\to V$\\
$\Rightarrow$ holomorphic atlas\vskip12pt\strut}}}}$
\strut
\kern4.6cm\pgfuseimage{FFig2}
\end{frame}

\begin{frame}
\frametitle{Invariance by transverse isotopies}

\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]{FFig3}{FFig3}
$\raise3.7cm\hbox{\bf Proof:}$
\vskip-3.1cm\strut
\kern2cm\pgfuseimage{FFig3}
\end{frame}

\begin{frame}
\frametitle{A conjecture of Bogomolov}
\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 (from RIMS preprint of 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 describe the non
injectivity of the ``Bogomolov functor'' \alert{$(Z,D,\alpha)\mapsto (X,J_f)$},
i.e.\ moduli spaces of such embeddings~?
\end{block}
\end{frame}

\begin{frame}
\frametitle{There exist large classes of examples !}
\vskip-6pt
\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]{FFig4}{FFig4}
\pgfuseimage{FFig4}
\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{
$C^\infty(X,f^*T_Z)=C^\infty(X,f^*\cD)\oplus C^\infty(X,T_X)$}
\pause\vskip4pt
\begin{block}{Theorem (D - Gaussier, arxiv:1412.2899, 2014, JEMS 2017)}
Let $[{\scriptstyle\bullet},{\scriptstyle\bullet}]$ be the Lie bracket of
vector fields in $T_Z$,
\vskip4pt
\centerline{\alert{%
$\theta:\mathcal{D}\times\mathcal{D}\to T_Z/\mathcal{D},\quad
(\xi,\eta)\mapsto[\xi,\eta]\mod\cD$}}
\vskip6pt
be the torsion tensor of the holomorphic distribution $\mathcal{D}$,
and \alert{$v\mapsto \overline\partial_{J_f}v$} the 
$\overline\partial$ operator
of the almost complex structure $(X,J_f)$.\pause\\
Then the differential of the natural map $f\mapsto J_f$
along any infinitesimal variation $w=u+f_*v:X\to f^*T_Z=
f^*\mathcal {D}\oplus f_*T_X$ 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
\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{Construction of the universal embedding}

\alert{\bf First observation.} There exists a $C^\infty$ embedding
$\varphi:X\hookrightarrow\bR^{2k}$, $k\geq 4n$, by the Whitney
embedding theorem, and one can assume
$N_{\varphi(X)}=(T_{\varphi(X)})^\perp$ to carry a complex structure
for $k\geq 8n$; otherwise take
$\Phi=\varphi\times\varphi:X\hookrightarrow \bR^{2k}\times\bR^{2k}$ and observe
that\vskip4pt
\alert{\centerline{$N_{\Phi(X)}\simeq N_X\oplus N_X\oplus T_X\simeq
(\bC\otimes_\bR\!N_X)\oplus (T_X,J).$}}\pause\vskip8pt

\alert{\bf Second step.} Assuming $(N_X,J')$ almost complex, let $Z^\bR_{n,k}$ be
the set of triples $(x,S,J)$ such that $S\in\Gr^\bR(2k,2n)$, $\codim S=2n$,
$J\in\End(\bR^{2k})$, $J^2=-\Id$, $J(S)\subset S$. Define\vskip4pt
\alert{\centerline{$f:X\to Z^\bR_{n,k},~~
x\mapsto (\varphi(x),N_{\varphi(X),\varphi(x)},\widetilde J(x))$}}
where $\widetilde J$ is induced by $J(x)\oplus J'(x)$ on
$\varphi_*T_X\oplus N_X$.\pause\vskip8pt

\alert{\bf Third step.} Complexify $Z^\bR_{n,k}$ as a variety
$Z_{n,k}=Z_{n,k}^\bC$ and define an algebraic distribution
$\cD_{n,k}\subset T_{Z_{n,k}}$.
\end{frame}

\begin{frame}
\frametitle{Definition of $Z_{n,k}$ and $\cD_{n,k}$}

We let $Z_{n,k}=Z_{n,k}^\bC$ be the set of triples\vskip4pt
\alert{\centerline{$(z,S,J)\in\bC^{2k}\times \Gr^\bC(2k,2n)\times
\End(\bC^{2k})$}}\vskip4pt
with $J^2=-\Id$, $J(S)=S$. Moreover we assume that we have ``balanced''
decompositions\vskip4pt
\alert{\centerline{\hbox{\strut\kern4cm\vbox{
      $S=S'\oplus S'',~~~~~\dim S'=\dim S''=n$,\\
      $\bC^{2k}=\Sigma'\oplus\Sigma'',~~
\dim\Sigma'=\dim\Sigma''=k$}}}}\vskip4pt
for the $i$ and $-i$ eigenspaces of $J_{|S}$ and $J$,
$S'\subset\Sigma'$, $S''\subset\Sigma''$.\vskip4pt\pause

Finally, if $\pi=\pr_1:Z_{n,k}\to\bC^{2k}$ is the first projection, we take
$\cD_{n,k}$ at point $w=(z,S,J)$ to be\vskip4pt
\alert{\centerline{$\cD_{n,k,\,w}:=(d\pi)^{-1}(S'\oplus \Sigma'')$.}}
\vskip4pt\pause

Since $\bC^{2k}=\Sigma'\oplus\Sigma''$, we have\vskip4pt
\alert{\centerline{
$(T_{Z_{n,k}}/\cD_{n,k})_w\simeq \Sigma'/S'$,}}\vskip4pt
which on real points, is isomorphic to $(S^\bR)^\perp$.


\end{frame}

\begin{frame}
\frametitle{Symplectic embeddings}
\vskip-6pt
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{$\omega(\xi,J\xi)>0$}.
\vskip-4pt
\pause
\begin{block}{Definition} We say that a closed
semipositive$\,(1{,}1)$-form$\,\beta\,$on $Z$ is a \hbox{transverse\kern-40pt}\\
K\"ahler structure to $\cD\subset T_Z$
if $\Ker\beta\subset\cD$, i.e., if 
$\beta$ induces a K\"ahler form on germs of complex submanifolds 
transverse to $\cD$.
\end{block}
\vskip-6pt
\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 $f:X\hookrightarrow(Z,\cD,\beta)$, in such
a way that\vskip3pt
\centerline{\alert{$J=J_f$ and $\omega=f^*\beta$.}}
\end{block}
\vskip-4pt
\pause
{\bf Proof.} Use the Tischler symplectic embedding
$X\hookrightarrow(\bP^{2n+1},\omega_{\rm FS})$.
\end{frame}


\begin{frame}
\frametitle{
Integrability condition for\\
an almost complex structure}
\vskip-5pt
Recall that the Nijenhuis tensor of an almost complex structure $J$ is\vskip4pt
\alert{%
$N_J(\zeta,\eta)=4\Re\,[\zeta^{0,1},\eta^{0,1}]^{1,0}=[\zeta,\eta]-[J\zeta,J\eta]+J[\zeta,J\eta]+J[J\zeta,\eta].$}\pause\vskip4pt
The \claim{Newlander-Nirenberg} theorem states that\\
\alert{$(X,J)$ is complex
analytic if and only if $N_J\equiv 0$}.\vskip4pt\pause

In fact, we have the following relation between the torsion form $\theta$ of
a distribution and the Nijenhuis tensor of the related transverse structure:

\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}
\end{frame}

\begin{frame}
\frametitle{Solution of a weak Bogomolov conjecture}
\vskip-6pt
\begin{block}{Theorem (D - Gaussier, 2014)}
There exist universal embedding spaces \hbox{$(W\!,\cE{,}\cS)\,{=}\,
(W_{n,k}{,}\cE_{n,k}{,}\cS_{n,k})$\kern-15pt}\\
where $\dim W_{n,k}<\dim Z_{n,k}+n(\dim Z_{n,k}-2n)=O(nk^2)=O(n^3)$,
and $\cS\subset\cE\subset T_W$ are algebraic
subsheaves satisfying $[\cS,\cS]\subset\cE$ (partial integrability),
such that every compact $\bC$-manifold $(X,J)$ of given
dimension $n$ embeds transversally by $f:X\hookrightarrow(W_{n,k},\cE_{n,k})$,
i.e.\ $J=J_f$, with the additional constraint
\alert{$\Im(\overline\partial f)\subset\cS_{n,k}$}.
\end{block}\pause

{\bf Proof.} By the Nijenhuis tensor formula, since
$\overline\partial_{J_f}f$ is injective with values in $\cD_{n,k}$, we see that
$S=\overline\partial_{J_f}f(T_{X,x})\subset \cD_{n,k,f(x)}$ must be an
$n$-dimensional complex subspace of $\cD_{n,k,\,x}\subset T_{Z,f(x)}$
that is totally isotropic
for $\theta$, i.e.\ $\theta_{|S\times S}\equiv 0$.\vskip3pt\pause

We let $W_{n,k}\subset \Gr(\cD_{n,k},n)$ be the subvariety
of the Grassmannian bundle consisting of the $\theta$-isotropic
$n$-subspaces, and
lift \hbox{$\cD_{n,k}{\subset}\,T_{Z_{n,k}}$\kern-15pt}\\
to $\cE_{n,k}\subset T_{W_{n,k}}$,
$\cS_{n,k}$ being the tautological isotropic \hbox{subbundle.\kern-15pt}
\end{frame}

\begin{frame}
\frametitle{Yau's challenge and $S^6$}
In complex dimension 2, it is known that there exist compact almost
complex manifolds that cannot be given a complex structure: by
Van de Ven (1966), for $X$ a complex surface,\vskip4pt
\centerline{\alert{$p=c_1^2(X)$, $q=c_2(X)$
is in the region $\{p\leq 8q,\,p+q\equiv 0(12)\}$,}}\vskip4pt
but the only restriction for $X$
almost complex is $p+q\equiv 0(12)$.\pause

\begin{block}{Yau's challenge} For $n\geq 3$, find a
compact almost complex $n$-fold that cannot be given a complex structure .
\end{block}\pause

\vskip-2pt
The sphere $S^6$ can be realized as the set of octonions $x\in\bO$ such that
$x^2=-1$ (${}\Leftrightarrow \Re x=0$ and $|x|=1$).\pause\\
A natural non integrable almost complex structure is then given by\vskip3pt
\centerline{\alert{$J_xh=xh$,
$h\in T_{S^6,x}\Leftrightarrow \Re h=0~\hbox{and}~xh+hx=0$.}}\vskip3pt
\claim{$S^6$ is strongly suspected of not carrying a complex structure!}

\end{frame}

\begin{frame}
\frametitle{Application to complex structures on $S^6$}
\vskip-4pt
The octonion embedding $f:S^6\hookrightarrow\bO=\bR^{2k}$, $k=4$ (which has
trivial rank 2 normal bundle), yields a
universal embedding $\varphi:S^6\to Z_{3,4}$ where
$\dim Z_{3,4}=46$, rank~$\cD_{3,4}=43$ (corank 3).\vskip4pt\pause

By passing to the Grassmannian bundle we get a map $\psi:S^6\to W_{3,4}$
where $\dim  W_{3,4}<46+3\times 40=166$, $W_{3,4}$ being equipped with
bundles $\cE_{3,4}\supset\cS_{3,4}$ of respective coranks $3$ and $43$, and
at the homotopy level the question is whether  $\overline\partial\psi
\subset\cE_{3,4}$ can be retracted to a section with values in $\cS_{3,4}$
over the whole $S^6$.\vskip4pt\pause

If the answer is negative, this would prove that there are no complex 
structures on $S^6$ (it is well known that $S^6$ admits only
two almost complex structures up to homotopy, $J_0$ given by the octonions
and its conjugate $-J_0$).\vskip4pt\pause

\alert{In general, this approach could yield topological obstructions
for an almost complex structure to be homotopic to a complex structure.}

\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 $\bC^N$.\pause\\
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]{FFig5}{FFig5}
$\raise2.8cm\hbox{\bf Proof:}$
\vskip-3.2cm\strut
\kern1.5cm\pgfuseimage{FFig5}\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 Thank you for your attention}
\pgfdeclareimage[height=6.5cm]{CalabiYau}{CalabiYau}
\strut\kern2cm\pgfuseimage{CalabiYau}

\end{frame}

\end{document}