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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Ergodic complex structures}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf 
Master class: Around Torelli's theorem for K3 surfaces\\[2mm]
Arithmetic, Geometric and Dynamical aspects\\[2mm]
Institut de Recherche 
de Math\'ematique Avanc\'ee, Strasbourg\\ October 28-November 1 2013
}
\end{center}

\newpage

{\bf \blue Ergodic complex structures}
\def\Teich{\operatorname{Teich}}
\newcommand{\Comp}{\operatorname{Comp}}
\newcommand{\St}{\operatorname{St}}


\definition
Let $M$ be a smooth manifold.
{\bf \blue A complex structure} on $M$ is an endomorphism $I\in \End TM$,
$I^2=-\Id_{TM}$ such that the eigenspace bundles of $I$
are {\bf \blue involutive}, that is, 
satisfy $[T^{1,0}M, T^{1,0}M]\subset T^{1,0}M$.

\remark
Let $\Comp$ be the space of such tensors equipped with 
a topology of convergence of all derivatives. 
{\bf \purple It is a Fr\'echet manifold}.

\definition
The diffeomorphism group $\Diff$ is a Fr\'echet Lie
group acting on $\Comp$ in a natural way.
A complex structure is called {\bf \blue ergodic}
if its $\Diff$-orbit is dense in $\Comp$.

\remark
The ``moduli space'' of complex structures (if it exists)
is identified with $\Comp/\Diff$; {\bf \purple existence of ergodic
complex structures guarantees that the moduli space 
does not exist} (all points are 
non-separable).

\theorem
Let $M$ be a compact torus, $\dim_\C M \geq 2$,
or a simple hyperk\"ahler manifold.  
{\bf \red A complex structure on $M$ is ergodic if and only if
$\Pic(M)$ is not of maximal rank.}

\newpage

{\bf \blue Period space as a Grassmannian of positive 2-planes}

\proposition
The period space 
\[
{\Bbb Per}:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \  q(l,l)=0, q(l, \bar l) >0.\}
\]
{\bf \red is identified with 
$Gr_{+,+}(H^2(M,\R))=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$,} which
is a Grassmannian of positive oriented 2-planes in $H^2(M,\R)$.

{\bf \green 
Proof. Step 1:} Given $l\in {\Bbb P}H^2(M, \C)$, {\bf \purple the space
generated by $\Im l, \Re l$ is 2-dimensional,} because 
$q(l,l)=0, q(l, \bar l)$ implies that $l \cap H^2(M,\R)=0$.

{\bf \green  Step 2:} {\bf \purple This 2-dimensional plane is 
positive,} because 
 $q(\Re l, \Re l) = q(l+ \bar l, l+ \bar l) = 2 q(l, \bar l)>0$.

{\bf\green  Step 3:} Conversely, for any 2-dimensional positive
plane  $V\in H^2(M,\R)$, {\bf \purple 
the quadric $\{l\in V \otimes_\R \C\ \ |\ \ q(l,l)=0\}$
consists of two lines;} a choice of a line is determined by orientation.
\endproof

\newpage

{\bf \blue Birational Teichm\"uller moduli space}

\definition
Let $M$ be a topological space. We say that $x, y \in M$
are {\bf \blue non-separable} (denoted by $x\sim y$)
if for any open sets $V\ni x, U\ni y$, $U \cap V\neq \emptyset$.

\theorem (Huybrechts)
Two points $I,I'\in \Teich$ {\bf \purple are non-separable if and only
if there exists a bimeromorphism $(M,I)\arrow (M,I')$
which is non-singular in codimension 2.}

\definition
The space $\Teich_b:= \Teich/\sim$ is called {\bf \blue the
birational Teichm\"uller space} of $M$.

\newcommand{\Per}{\operatorname{\sf Per}}
\newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}}

\theorem {\bf \red The period map $\Teich_b\stackrel\Per \arrow \Perspace$ 
is an isomorphism,}
for each connected component of $\Teich_b$.

\theorem 
Let $(M,I)$ be a hyperk\"ahler manifold, and $W$ 
a connected component of its birational
moduli space. {\bf \red Then 
the set of isomorphism classes of complex
holomorphically symplectic structures
is identified with ${\Perspace}/\Gamma$,
where ${\Perspace}=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$
and $\Gamma$ is 
a finite index subgroup in $O(H^2(M, \Z), q)$}, called {\bf \blue the
monodromy group}.

\newpage


{\bf \blue Ergodic complex structures}

\definition
Let $(M,\mu)$ be a space with measure,
and $G$ a group acting on $M$ preserving measure.
This action is {\bf\blue ergodic} if all
$G$-invariant measurable subsets $M'\subset M$
satisfy $\mu(M')=0$ or $\mu(M\backslash M')=0$.

\claim
Let $M$ be a manifold, $\mu$ a Lebesgue measure, and
$G$ a group acting on $M$ ergodically. {\bf \red Then the 
set of non-dense orbits has measure 0.}

{\bf\green Proof. Step 1:}
Consider a non-empty open subset $U\subset M$. 
Then $\mu(U)>0$, hence $M':=G\cdot U$ satisfies 
$\mu(M\backslash M')=0$. For any orbit $G\cdot x$
not intersecting $U$, $x\in M\backslash M'$.
Therefore the set $Z_U$ of such orbits has measure 0.

{\bf\green Proof. Step 2:} Choose a countable base
$\{U_i\}$ of topology on $M$. Then the set of 
points with dense orbits is $M \backslash \bigcup_i Z_{U_i}$.
\endproof

\definition
Let $M$ be a complex manifold, $\Teich$ its Techm\"uller
space, and $\Gamma$ the mapping group acting on $\Teich$
{\bf\blue An ergodic complex structure} is a complex
structure with dense $\Gamma$-orbit.



\newpage

{\bf \blue Ergodicity of the monodromy group action}

\definition
{\bf\blue A lattice} in a Lie group is a discrete
subgroup $\Gamma\subset G$ such that $G/\Gamma$ has finite
volume with respect to Haar measure.

\theorem (Calvin C. Moore, 1966)
Let $\Gamma$ be a lattice in a non-compact 
simple Lie group $G$ with finite center, and $H\subset G$ a 
non-compact subgroup. {\bf \red Then the left action of $\Gamma$
on $G/H$ is ergodic.}


\theorem Let ${\Perspace}$ be a component of 
a birational Teichm\"uller space, and
$\Gamma$ its monodromy group. Let $\Perspace_e$ be
a set of all points with dense orbits. {\bf \red Then 
$Z:=\Perspace\backslash \Perspace_e$ has measure 0.}


{\bf \green Proof:}
 Let $G=SO(b_2-3,3)$, $H=SO(2) \times SO(b_2-3,1)$.
{\bf \purple Then $\Gamma_I$-action on $G/H=\Perspace$ is ergodic,}
by Moore's theorem. Therefore, all orbits outside of a measure
0 set are dense.
\endproof


\remark 
Generic deformation of $M$ has no rational curves,
and no non-trivial birational models. Therefore,
{\bf \purple outside of a measure zero subset,
$\Teich=\Teich_b$.} This implies that 
{\bf \red almost all complex structures on $M$ are ergodic.}

\newpage

{\bf \blue Ratner's theorem}

\definition
Let $G$ be a connected Lie group equipped with
a Haar measure. {\bf \blue A lattice} $\Gamma\subset G$
is a discrete subgroup of finite covolume (that is,
$G/\Gamma$ has finite volume).

\remark {\bf \purple Arithmetic lattices in simple Lie groups 
have finite covolume} (Borel, Harish-Chandra).

\theorem
Let $H\subset G$ be a Lie subroup generated by 
unipotents, and $\Gamma\subset G$ an arithmetic lattice.
Then {\bf \red a closure of any $H$-orbit in $G/\Gamma$
is an orbit of a closed, connected subgroup $S\subset G$,
such that $S\cap \Gamma\subset S$ is a lattice.}

\remark
A closure of $H\cdot x$ in $G/\Gamma$
and a closure of $x\cdot \Gamma$ in $H\backslash G$ are related
as follows. Let $\pi_1:\; G \arrow H\backslash G$,
$\pi_2:\; G \arrow G/\Gamma$ be the projections.
Then the connected component of
$\pi_2^{-1}(\overline {H\cdot x}_{G/\Gamma})$ is the connected component
of $\pi_1^{-1}(\overline {x\cdot \Gamma}_{H\backslash G}).$

\example
Let $V$ be a real vector space with a non-degenerate
bilinear symmetric form of signature $(3,k)$, $k>0$, 
$G:=SO^+(V)$ a connected component of the 
isometry group, $H\subset G$ a subgroup
fixing a given positive 2-dimensional plane,
$H\cong SO^+(1,k)\times SO(2)$, and $\Gamma\subset G$ an arithmetic
lattice. Consider the quotient $\Perspace:=G/H$. 
{\bf \purple Then 
a closure of $\Gamma\cdot J$ 
in $G/H$ is $\Gamma\cdot J \cdot S$, for some Lie subgroup $S \supset H$.}
Moreover, $S=H$ if and only if the orbit $\Gamma\cdot J$ is closed.


\newpage

{\bf \blue Characterization of ergodic complex structures}


\claim Let $G=SO^+(3,k)$, and 
$H\cong SO^+(1,k)\times SO(2)\subset G$.
Then {\bf \purple any closed connected
Lie subgroup $S\subset G$ containing $H$ coincides
with $G$ or with $H$.}

\corollary
Let $J\in \Perspace=G/H$. Then {\bf \red either $J$ is ergodic, or
its $\Gamma$-orbit is closed in $\Perspace$}.

\remark By Ratner's theorem, in the latter case
the $H$-orbit of $J$ has finite volume in $G/\Gamma$.
Therefore, {\bf \purple its intersection with $\Gamma$ is a lattice in 
$H$.} This brings

\corollary
Let $J\in \Perspace$ be a point, such that its 
$\Gamma$-orbit is closed in $\Perspace$. Consider its stabilizer
$\St(J)\cong H \subset G$. {\bf \red Then $\St(J)\cap \Gamma$
is a lattice in $\St(J)$.}

\corollary 
Let $J$ be a non-ergodic complex structure on a hyperk\"ahler
manifold, and $W\subset H^2(M,\R)$ be a plane generated
by $\Re\Omega, \Im \Omega$. {\bf \red Then $W$ is rational.}

\remark 
This can be used to show that {\bf \purple any hyperk\"ahler
manifold is Kobayashi non-hyperbolic.}

\newpage

{\bf \blue Kobayashi pseudometric}

\remark The results further on are from a joint paper
arXiv:1308.5667 by Ljudmila Kamenova, Steven Lu, Misha Verbitsky.

\definition
{\bf \blue Pseudometric} on $M$ 
is a function $d:\; M \times M \arrow \R^{\geq 0}$
which is symmetric: $d(x,y)=d(y,x)$ and satisfies the
triangle inequality $d(x,y)+d(y,z) \geq d(x,z)$.

\remark 
Let ${\goth D}$ be a set of pseudometrics. {\bf \purple Then
$d_{\max}(x,y):= \sup_{d\in {\goth D}}d(x,y)$ is also a pseudometric.}

\definition
The {\bf \blue Kobayashi pseudometric} on a complex manifold $M$
is $d_{\max}$ for the set ${\goth D}$ of all pseudometrics
such that any holomorphic map from the 
Poincar\'e disk to $M$ is distance-non-increasing.

\theorem 
Let $\pi:\; {\cal M} \arrow X$ be a smooth holomorphic family,
which is trivialized as a smooth manifold: ${\cal M}=M \times X$,
and $d_x$ the Kobayashi metric on $\pi^{-1}(x)$. {\bf \red Then $d_x(m,m')$
is upper continuous on $x$.} \endproof

\corollary
Denote the diameter of the Kobayashi pseudometric by
$\diam(d_x):= \sup_{m,m'}d_x(m,m')$. {\bf \purple 
Then the Kobayashi diameter of a fiber of $\pi$ 
is an upper continuous function: 
$\diam:\; X \arrow \R^{\geq 0}$.}




\newpage

{\bf \blue Vanishing of Kobayashi pseudometric}

\theorem
Let $(M,I)$ be a complex manifold
with vanishing Kobayashi pseudometric. Then {\bf \red the Kobayashi
pseudometric vanishes for all ergodic complex structures
in the same deformation class.}

{\bf \green Proof:}
Let $\diam:\; \Comp \arrow \R^{\geq 0}$
map a complex structure $J$ to the diameter of the Kobayashi
pseudometric on $(M,J)$. Let $J$ be an 
ergodic complex structure. The set of points $J'=\nu(J)\in \Comp$
such that $(M,J')$ is biholomorphic to $(M,J)$
is dense, because $J$ is ergodic. By upper semi-continuity, 
$0=\diam(I) \geq \inf_{J'=\nu(J)} \diam (J)$.
\endproof

\example
Let $M$ be a projective K3 surface. Then the Kobayashi
metric on $M$ vanishes. {\bf \purple Since all non-projective K3 are
ergodic,} the Kobayashi metric vanishes on non-projective K3 surfaces
as well.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\theorem
Let $M$ be a compact simple hyperk\"ahler manifold. Assume that
a deformation of $M$ admits a holomorphic Lagrangian fibration
and the Picard rank of $M$ is not maximal. {\bf \red Then the Kobayashi
pseudometric on $M$ vanishes.}

\theorem
Let $M$ be a Hilbert scheme of K3. {\bf \red Then the Kobayashi
pseudometric on $M$ vanishes.}




\newpage

{\bf \blue Kobayashi hyperbolic manifolds}

\definition
{\bf \blue An entire curve} is a non-constant 
map $\C \arrow M$.

\definition
A compact complex manifold $M$ is called {\bf \blue Kobayashi
hyperbolic} if there exist no entire curves $\C \arrow M$.

\theorem {\bf \blue (Brody, 1975)}\\
Let $I_i$ be a sequence of complex structures on $M$
which are not hyperbolic, and $I$ its limit. Then
$(M,I)$ is also not hyperbolic.

\theorem
{\bf \red All hyperk\"ahler manifolds are non-hyperbolic.}


\remark
{\bf \purple This theorem would follow if we produce an
ergodic complex structure which is non-hyperbolic.}
Indeed, a closure of its orbit is the whole $\Teich$,
and a limit of non-hyperbolic complex structures
is non-hyperbolic.

\newpage


{\bf \blue Twistor spaces and hyperk\"ahler geometry}


\definition
A {\bf \blue hyperk\"ahler structure} on a manifold $M$
is a Riemannian structure $g$ and a triple of complex
structures $I,J,K$, satisfying quaternionic relations
$I\circ J = - J \circ I =K$, such that $g$ is K\"ahler
for $I,J,K$.

\definition 
{\bf \color{blue} Induced complex structures} 
on a hyperk\"ahler manifold are 
complex structures of form 
$S^2 \cong \{ L:= aI + bJ +c K, \ \ \ a^2+b^2+c^2=1.\}$

\definition
A {\bf\blue twistor space} $\Tw(M)$ of a hyperk\"ahler manifold
is {\bf \purple 
a complex manifold obtained by gluing these complex structures into
a holomorphic family over $\C P^1$.} More formally:

 Let $\Tw(M) := M \times S^2$. Consider the complex structure $I_m:T_mM \to T_mM$ 
on $M$ induced by $J \in S^2 \subset {\Bbb H}$. Let $I_J$
denote the complex structure on $S^2 = \C P^1$.

The operator $I_{\Tw} = I_m \oplus I_J:T_x\Tw(M) \to T_x\Tw(M)$ 
satisfies $I_{\Tw} ^2 = -\Id$. {\bf \purple It defines 
an almost complex structure on $\Tw(M)$.} This almost 
complex structure is known to be integrable (Obata).


\newpage

{\bf \blue Entire curves in twistor fibers}

\theorem {\bf \blue (F. Campana, 1992)}\\
Let $M$ be a hyperk\"ahler manifold, and 
$\Tw(M)\stackrel \pi \arrow \C P^1$ its twistor
projection. {\bf \red Then there exists an entire curve in some
fiber of $\pi$.}

\claim
{\bf \purple There exists a twistor family which has only ergodic
fibers. }

{\bf\green Proof:} There are only countably many complex
structures which are not ergodic. \endproof

\theorem
{\bf \red All hyperk\"ahler manifolds are non-hyperbolic.}


{\bf\green Proof:} Let $\Tw(M)\arrow \C P^1$ be a twistor family
with all fibers ergodic. {\bf \purple By Campana's theorem, one of these
fibers, denoted $(M, I)$, is non-hyperbolic.} Since any complex structure
$I'\in \Teich$ lies in the closure of $\Diff(M)\cdot I$, 
all complex structures $I'\in \Teich$ are non-hyperbolic. \endproof 



\end{document}



