
\documentclass{slides}

\usepackage{amssymb, amsmath, amscd, color, epsfig}
%\usepackage[matrix,arrow]{xy}

\newcommand{\green}{\color[rgb]{0,0.4,0}}
\newcommand{\purple}{\color[rgb]{0.4,0,0.4}}
\newcommand{\red}{\color[rgb]{0.7,0,0}}
\newcommand{\blue}{\color{blue}}


\def\eqref#1{(\ref{#1})}
\newcommand{\goth}{\mathfrak}
\newcommand{\g}{{\frak g}}
\newcommand{\arrow}{{\:\longrightarrow\:}}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\C}{{\Bbb C}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\Q}{{\Bbb Q}}
\renewcommand{\H}{{\Bbb H}}
\newcommand{\6}{\partial}
\def\1{\sqrt{-1}\:}
\newcommand{\restrict}[1]{{\left|_{{#1}}\right.}}
\newcommand{\cntrct}                % contraction with a vector field
{\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}}


\def\Bbb#1{\mathbb #1}


\newcommand{\calo}{{\cal O}}
\newcommand{\cac}{{\cal C}}

% Correcting TeX...
%\let\oldtilde=\tilde
%\renewcommand{\tilde}{\widetilde}
\renewcommand{\bar}{\overline}
\renewcommand{\phi}{\varphi}
\renewcommand{\epsilon}{\varepsilon}
\renewcommand{\geq}{\geqslant}
\renewcommand{\leq}{\leqslant}

% Operatornames
\newcommand{\even}{{\rm even}}
\newcommand{\ev}{{\rm even}}
\newcommand{\odd}{{\rm odd}}
\newcommand{\const}{{\it const}}
\newcommand{\fl}{{\rm fl}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\Hol}{\operatorname{{\cal H}ol}}
\newcommand{\Tot}{\operatorname{Tot}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\id}{\operatorname{\text{\sf id}}}
\newcommand{\Vol}{\operatorname{Vol}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Mat}{\operatorname{Mat}}
\newcommand{\Alt}{\operatorname{Alt}}
\newcommand{\Iso}{\operatorname{Iso}}
\newcommand{\Sec}{\operatorname{Sec}}
\newcommand{\Can}{\operatorname{Can}}
\newcommand{\Sing}{\operatorname{Sing}}
\newcommand{\Spin}{\operatorname{Spin}}
\newcommand{\codim}{\operatorname{codim}}
\newcommand{\coim}{\operatorname{coim}}

\newcommand{\coker}{\operatorname{coker}}
\newcommand{\slope}{\operatorname{slope}}
\newcommand{\rk}{\operatorname{rk}}
\newcommand{\Def}{\operatorname{Def}}
\newcommand{\Lie}{\operatorname{Lie}}
\newcommand{\Tw}{\operatorname{Tw}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\Diff}{\operatorname{Diff}}

\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}



\newcommand{\inbfpare}[1]{{%
  \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % 
  \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}%
}}
\newcommand{\comment}[1]{{}}

\def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}}
\def\endproof{\blacksquare}
\def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}}


\makeatletter

%\@ifundefined{Bbb}
%     {\newcommand{\Bbb}[1]{{\mathbb #1}}}%
%{}%     {\edef\Bbb#1{{\Bbb #1}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       Pagestyle                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 

   \setlength\paperheight {10in}%
    \setlength\paperwidth  {13.5in}
\setlength{\textwidth}{0.8\paperwidth}
\setlength{\textheight}{0.8\paperheight}

 \setlength{\pdfpageheight}{\paperheight}
 \setlength{\pdfpagewidth}{\paperwidth}
\addtolength{\topmargin}{-20mm}
\addtolength{\leftmargin}{-25mm}
\addtolength{\rightmargin}{-25mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Lemma, sublemma, corollary, proposition, theorem,             %
% definition,example defined there:                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcounter{section}
\newcounter{Mycounter}[section]
\newcounter{lemma}[section]
\setcounter{lemma}{0}
\renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}}
\newcommand{\lemma}{%
     \setcounter{lemma}{\value{Mycounter}}
     \refstepcounter{lemma}
     \stepcounter{Mycounter}
     {\bf \green LEMMA:\ }}

\newcounter{claim}[section]
\setcounter{claim}{0}
\renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}}
\newcommand{\claim}{%
     \setcounter{claim}{\value{Mycounter}}
     \refstepcounter{claim}
     \stepcounter{Mycounter}
     {\bf \green CLAIM:\ }}

\newcounter{corollary}[section]
\setcounter{corollary}{0}
\renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}}
\newcommand{\corollary}{%
     \setcounter{corollary}{\value{Mycounter}}
     \refstepcounter{corollary}
     \stepcounter{Mycounter}
     {\bf \green COROLLARY:\ }}

\newcounter{theorem}[section]
\setcounter{theorem}{0}
\renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}}
\newcommand{\theorem}{%
     \setcounter{theorem}{\value{Mycounter}}
     \refstepcounter{theorem}
     \stepcounter{Mycounter}
     {\bf \green THEOREM:\ }}

\newcounter{conjecture}[section]
\setcounter{conjecture}{0}
\renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}}
\newcommand{\conjecture}{%
     \setcounter{conjecture}{\value{Mycounter}}
     \refstepcounter{conjecture}
     \stepcounter{Mycounter}
     {\bf \green CONJECTURE:\ }}

\newcounter{proposition}[section]
\setcounter{proposition}{0}
\renewcommand{\theproposition}
       {\noindent{Proposition \thesection.\arabic{proposition}}}
\newcommand{\proposition}{%
     \setcounter{proposition}{\value{Mycounter}}
     \refstepcounter{proposition}
     \stepcounter{Mycounter}
     {\bf \green PROPOSITION:\ }}

\newcounter{definition}[section]
\setcounter{definition}{0}
\renewcommand{\thedefinition}
       {\noindent{Definition~\thesection.\arabic{definition}}}
\newcommand{\definition}{%
     \setcounter{definition}{\value{Mycounter}}
     \refstepcounter{definition}
     \stepcounter{Mycounter}
     {\bf \green DEFINITION:\ }}


\newcounter{example}[section]
\setcounter{example}{0}
\renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}}
\newcommand{\example}{%
     \setcounter{example}{\value{Mycounter}}
     \refstepcounter{example}
     \stepcounter{Mycounter}
     {\bf \green EXAMPLE:\ }}

\newcounter{remark}[section]
\setcounter{remark}{0}
\renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}}
\newcommand{\remark}{%
     \setcounter{remark}{\value{Mycounter}}
     \refstepcounter{remark}
     \stepcounter{Mycounter}
     {\bf \green REMARK:\ }}


\newcounter{observation}[section]
\setcounter{observation}{0}
\renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}}
\newcommand{\observation}{%
     \setcounter{observation}{\value{Mycounter}}
     \refstepcounter{observation}
     \stepcounter{Mycounter}
     {\bf \green OBSERVATION:\ }}

\newcommand{\question}{%
     {\bf \green QUESTION:\ }}

\newcommand{\exercise}{%
     {\bf \green EXERCISE:\ }}

\newcommand{\proof}{{\bf \green Proof:\ }}


\newcommand{\ps@verbit}{%
  \renewcommand{\@oddhead}{%
          \scriptsize {\it \small Ergodic complex structures \hfil
  \tiny M. Verbitsky }}
  \renewcommand{\@evenhead}{\@oddhead}
  \renewcommand{\@oddfoot}{\hfil\thepage\hfil}
  \renewcommand{\@evenfoot}{\@oddfoot}}
 
\pagestyle{verbit}


\begin{document}
\setcounter{page}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Moduli of complex structures and Ratner theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf Hyperbolicity 2015\\[2mm]
Holomorphic dynamics school\\[2mm]
Hyperbolicity in algebraic geometry conference 
\\[2mm] 
Ilhabela, 06.01.2015
}
\end{center}


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


\newpage


{\bf \blue Complex manifolds}
\newcommand{\Teich}{\operatorname{Teich}}
\newcommand{\Comp}{\operatorname{Comp}}



{\bf\green DEFINITION:} Let $M$ be a smooth manifold. 
An {\bf \blue almost complex structure} is an operator
$I:\; TM \arrow TM$ which satisfies $I^2 = - \Id_{TM}$.

The eigenvalues of this operator are $\pm \1$.
The corresponding eigenvalue 
decomposition is denoted $TM\otimes \C=T^{0,1}M\oplus T^{1,0}(M)$.

{\bf\green DEFINITION:}
An almost complex structure is {\bf \blue integrable}
if $\forall X,Y \in T^{1,0}M$, one has $[X,Y]\in T^{1,0}M$.
In this case $I$ is called {\bf \blue a complex structure operator}.
A manifold with an integrable almost complex structure
is called {\bf \blue a complex manifold}. 

{\bf\green THEOREM:} (Newlander-Nirenberg)\\
{\bf \purple This definition is equivalent to the usual one.}

\definition
{\bf \blue The space of almost complex structures}
is an infinite-dimensional Fr\'echet manifold $X_M$  of all tensors
$I^2 = - \Id_{TM}$, equipped with the natural Fr\'echet topology.

\claim
The space $\Comp$ of integrable almost complex structures
{\bf \red is a submanifold in $X_M$} (also infinite-dimensional).



\newpage

{\bf\blue Teichm\"uller space}


{\bf \green Definition:} Let $M$ be a compact complex manifold, and 
$\Diff_0(M)$ a connected component of its diffeomorphism group
({\bf\blue the group of isotopies}). Denote by $\Comp$
the space of complex structures on $M$, and let
$\Teich:=\Comp/\Diff_0(M)$. We call 
it {\bf \blue the Teichm\"uller space.}

\remark $\Teich$ is {\bf \blue a finite-dimensional
complex space} (Kodaira-Spencer-Kuranishi-Douady),
but often {\bf \red non-Hausdorff}.

\definition Let $\Diff_+(M)$ be the group of oriented
diffeomorphisms of $M$. We call $\Gamma:=\Diff_+(M)/\Diff_0(M)$ {\bf \blue the
mapping class group}. The {\bf \blue moduli space of complex
structures on $M$} is a connected component of $\Teich/\Gamma$.

\remark This terminology is {\bf \purple standard for curves.}


\remark
The topology of the moduli space $\Teich/\Gamma$ is often
bizzarre. However, {\bf \purple its points are in bijective correspondence
with equivalence classes of complex structures.}


\newpage

{\bf\blue K\"ahler manifolds}


{\bf\green DEFINITION:} A Riemannian metric $g$ on
a complex manifold $(M,I)$ is called 
{\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$.
In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$,
hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric.


{\bf\green DEFINITION:} The differential 
form $\omega\in \Lambda^{1,1}(M)$ is called
{\bf \blue the Hermitian form} of $(M,I,g)$.

{\bf\green DEFINITION:} A complex Hermitian manifold $(M,I,\omega)$
is called {\bf \blue K\"ahler} if $d\omega=0$. 
The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ 
is called {\bf \blue the K\"ahler class} of $M$, and
$\omega$ {\bf \blue the K\"ahler form}. 

{\bf \green Definition:} Let $M=\C P^n$ be a complex projective
space, and $g$ a $U(n+1)$-invariant Riemannian form. It is called
{\bf \blue Fubini-Study form on $\C P^n$}. The Fubini-Study
form is obtained by taking arbitrary Riemannian form
and averaging with $U(n+1)$.

{\bf \green Remark:} For any $x\in \C P^n$, the stabilizer
$St(x)$ is isomorphic to $U(n)$. {\bf \purple Fubini-Study form on
$T_x\C P^n= \C^n$ is $U(n)$-invariant, hence unique up to a constant.}


\newpage

{\bf\blue K\"ahler manifolds II.}


{\bf \green Claim:} {\bf \red Fubini-Study form is K\"ahler.}
Indeed, $d\omega\restrict x$ is a $U(n)$-invariant 3-form
on $\C^n$, but such a form must vanish, because $-\Id\in U(n)$

\remark 
{\bf \purple The same argument works for all symmetric spaces.}

{\bf \green Corollary:} {\bf \red Every projective manifold 
(complex submanifold of $\C P^n$) is K\"ahler.}
Indeed, a restriction of a closed form is again closed.

\definition 
The cohomology class of the K\"ahler form is called
{\bf\blue the K\"ahler class} of a manifold.

{\bf \green Hodge theory for K\"ahler manifolds (first cohomology):} \\
Let $M$ be a compact K\"ahler manifold, and $\theta\in \Omega^1(M)$
a holomorphic differential. Then $\theta$ is closed, and its
cohomology class is non-zero. This gives an injective map
$\Psi:\; H^0(\Omega^1 M)\hookrightarrow H^1(M,\C)$.
Moreover, {\bf \red any $\alpha \in H^1(M,\C)$ can be decomposed
as $\alpha= \alpha^{1,0}+\alpha^{0,1}$, with
$\alpha^{1,0}\in \im \Psi$ and $\overline{\alpha^{0,1}}\in \im \Psi$ 
represented by holomorphic differentials holomorphic.}

\definition {\bf \blue The space $\im\Psi$ is denoted  $H^{1,0}(M)$.}

\newpage

{\bf\blue Teichm\"uller space for a compact torus}

\definition
Let $\Z^{2n} \subset \C^n$ be a cocompact lattice.
Then $\C^n/\Z^{2n}$ is a complex manifold, called {\bf\blue a (compact) 
complex torus}.

\remark The space of complex structures on $R^{2n}$
is naturally identified with $GL(2n, \R)/GL(n,\C)$.

\theorem
{\bf \red Any connected component of the 
Teichm\"uller space for a compact torus is 
identified with $GL(2n, \R)/GL(n,\C)$.}

{\bf \green Proof:} Let the {\bf \blue period map} put
$(M,I)$ to $H^{1,0}(M)\subset H^1(M,\C)$, considered
as a point on $GL(2n, \R)/GL(n,\C)$.
{\bf \purple Since $M= H^{1,0}(M)/H^1(M,\Z)$, this map is 
invertible.} \endproof

\corollary 
{\bf \purple Complex structures on a torus are in (1,1)-correspondence
with $GL(2n,\Z)\backslash GL(2n, \R)/GL(n,\C)$.}

\remark Now I will prove that {\bf \red the action of
$GL(2n,\Z)$ on $GL(2n, \R)/GL(n,\C)$ is ergodic}.

\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 $(M,\mu)$ ergodically. {\bf \red Then the 
set of non-dense orbits has measure 0.}

\proof
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 of such orbits has measure 0.
\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.

\claim
Let $(M,I)$ be a manifold with ergodic complex structure,
and $I'$ another complex structure.
{\bf \purple Then there exists a sequence of diffeomorphisms
$\nu_i$ such that $\nu_i^*(I)$ converges to $I'$.}

\newpage

{\bf \blue Ergodicity of the mapping class group action}


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

\corollary 
{\bf \purple
The action of $GL(2n,\Z)$ on $GL(2n, \R)/GL(n,\C)$ is ergodic.}

\proof
Indeed, $SL(2n, \Z)$ acts on $SL(2n,\R)/SL(n,\C)$ ergodically
by Moore's theorem.
\endproof

\theorem 
Let $M=\C^n/\Lambda$ be a compact torus. {\bf \red Then $M$ is non-ergodic
if and only if the lattice $\Lambda\cong\Z^{2n}$ is rational.}

{\bf \blue Its proof uses Ratner theory.}

\remark {\bf \purple The set of such tori is countable.}


\newpage

{\bf \blue Ratner's theorem: preparatory definitions}

\definition
A matrix is called {\bf \blue unipotent} if it 
is an exponent of a nilpotent, and {\bf \blue semisimple}
if it is conjugate to a diagonal matrix.
An element $g$ in an algebraic group
is called {\bf \blue unipotent (semisimple)} if it 
is represented by a unipotent (algebraic) 
matrix for some algebraic representation.

\theorem {\bf \blue (Chevalley-Jordan decomposition)}\\
Any element $g$ of a Lie algebra
{\bf \red can be represented as $g=s+u$, where $s$ is semisimple,
$u$ nilpotent, and $s, u$ commute.} Moreover, {\bf \red such a
decomposition is unique.}

\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).

\theorem {\bf \blue (Borel and Harish-Chandra)} \\
An arithmetic subgroup of a reductive algebraic group $G$, defined over $\Q$,
{\bf \red is a lattice whenever $G$ has no non-trivial characters
over $\Q$.}


\newpage

{\bf \blue Ratner's theorem}

\theorem {\bf \blue (Ratner's theorem)}
Let $H\subset G$ be a Lie subroup generated by 
unipotents, and $\Gamma\subset G$ a lattice.
Then {\bf \red a closure of any $H$-orbit $Hx$ in $G/\Gamma$
is an orbit of a closed, connected subgroup $S\subset G$,
such that $S^x\cap \Gamma\subset S$ is a lattice.}
Here $S^x=xSx^{-1}$.

\example
Let $\Lambda \in \C^n$ be a cocompact lattice.
The corresponding complex torus is non-ergodic if and only if 
there exists an intermediate Lie group 
$H=SL(n,\C)\subset S\subsetneq SL(2n,\R)$
such that $S\cap SL(\Lambda)$ is a lattice.
{\bf \red This is equivalent to $S$ being a rational Lie
  group},
with respect to the rational structure induced by $\Lambda$.

\claim
Let $n\geq 2$, $G=SL(2n, \R)$, and 
$H\cong SL(n, \C) \subset G$.
{\bf \purple Then any closed connected
Lie subgroup $S\subset G$ containing $H$ coincides
with $G$ or with $H$.}

{\bf \green Proof:} See the next slide. \endproof

\corollary 
For any non-ergodic torus $\C^n/\Lambda$,
the intersection $SL(n,\C)\cap SL(\Lambda)$ is a lattice.
{\bf \red This is equivalent to $\Lambda$ being rational}.


\newpage

{\bf \blue Intermediate subgroups $SL(n,\C) \subset S \subset SL(2n, \R)$}


\claim
Let $n\geq 2$.
{\bf \red Then any closed, connected
Lie subgroup $S\subset GL(2n, \R) $ containing $GL(n,\C)$ coincides
with $GL(2n,\R)$ or with $GL(n,\C)$.}

{\bf \green Proof. Step 0:} $\dim_\R GL(n,\C)= 2n^2$, and 
$\dim_\R SL(2n,\R)= 4n^2$. 

{\bf \green Step 1:} 
It suffices to prove the same result for
Lie algebras. Let $\goth h= \goth{gl}(n,\C) \subset \goth{s} \subset \goth{gl}(2n, \R)=\goth g$.
As a representation of $\goth h$, the space $\goth g$ is a direct sum of
$\goth h$ (matrices commuting with $I$) and
and the space $\goth g_-$ of matrices anticommuting
with $I$. The later representation is isomorphic to $\goth h$; the isomorphism
is provided by $a \arrow a u$, for any non-degenerate matrix 
$u$ anticommuting with $I$.

{\bf \green Step 2:} The subalgebra $\goth s$ must be $\goth h$-invariant.
Therefore, it is either $\goth h$ or $\goth h \oplus \goth g_-=\goth g$.
\endproof


\newpage

{\bf \blue Further developments: hyperk\"ahler manifolds}

{\bf \red Ergodicity theorem is true for hyperk\"ahler
  manifolds:} A complex structure on a hyperk\"ahler
manifold is non-ergodic if and only if its Picard rank is maximal.

\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 $I,J,K$.

\remark {\bf \purple A hyperk\"ahler manifold is holomorphically
symplectic:} $\omega_J+\1 \omega_K$ is a holomorphic
symplectic form on $(M,I)$. 

{\bf\green THEOREM:} (Calabi-Yau) 
A compact, K\"ahler, holomorphically symplectic manifold
{\bf \red admits a unique hyperk\"ahler metric in any K\"ahler class.}

\example Take a 2-dimensional complex torus $T$,
then the singular locus of $T/{\pm1}$ is of form 
$(\C^2/{\pm1}) \times T$.
Its resolution $\widetilde {T/{\pm1}}$ is called 
{\bf \green a Kummer surface}. {\bf \red
It is holomorphically symplectic}.

\definition A complex surface is called {\bf \blue a K3 surface}
if it a deformation of a Kummer surface. K3 surface is
also hyperk\"ahler.

\newpage

{\bf \blue Further developments: Kobayashi non-hyperbolicity}


\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 $\C \arrow M$.


Using ergodicity, the following longstanding conjecture
was proven.

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

\remark This is equivalent to having an entire curve
$\C \arrow M$ (Brody).

\newpage

{\bf \blue Exercises}

\exercise
Let $H\subset G$ be a subgroup. Prove that there are no closed, connected subgroups $S$
satisfying $H\subsetneq S \subsetneq G$ when

a. $H=SO(p,k)\times SO(q-k)$, $G=SO(p,q)$, $p, q>0$

b. $H=Sp(2n,\R)$, $G=SL(2n, \R)$

c. $H=SL(n,\H)$, $G=SL(2n, \C)$

\exercise
Find all dense $G_\Z$-orbits in $G/H$ for all these cases.

%\exercise
%Prove that $H\backslash G/G_\Z$ is Hausdorff when
%$H=SO(p,q)$, $G=SO(p+q)$, $p, q>0$.

\exercise
Find a K3 surface with vanishing Kobayashi pseudodistance.



\end{document}

