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\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}