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\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{\proof}{{\bf \green Proof:\ }} \newcommand{\pstep}{{\bf \green Proof. Step 1:\ }} \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Teichmuller space of geometric structures 4 \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 Teichm\"uller spaces for geometric structures,\\ lecture 4} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize Misha Verbitsky } \\[20mm] {\tiny\bf Conference - Teichm\"uller Theory in Higher Dimension and Mirror Symmetry \\ April 24-28, 2017} \end{center} \newcommand{\Per}{\operatorname{\sf Per}} \newcommand{\Gr}{\operatorname{\sf Gr}} \newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}} \newcommand{\Teich}{\operatorname{Teich}} \newcommand{\Comp}{\operatorname{Comp}} \newcommand{\Symp}{\operatorname{Symp}} \newcommand{\Hyp}{\operatorname{Hyp}} \newpage {\bf \blue Hyperk\"ahler manifolds (reminder)} \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 A hyperk\"ahler manifold {\bf \purple has three symplectic forms\\ $\omega_I:= g(I\cdot, \cdot)$, $\omega_J:= g(J\cdot, \cdot)$, $\omega_K:= g(K\cdot, \cdot)$.} {\bf\green REMARK:} This is equivalent to $\nabla I=\nabla J = \nabla K=0$: the parallel translation along the connection preserves $I, J,K$. {\bf\green DEFINITION:} Let $M$ be a Riemannian manifold, $x\in M$ a point. The subgroup of $GL(T_xM)$ generated by parallel translations (along all paths) is called {\bf \blue the holonomy group} of $M$. {\bf\green REMARK:} {\bf \purple A hyperk\"ahler manifold can be defined as a manifold which has holonomy in $Sp(n)$ } (the group of all endomorphisms preserving $I,J,K$). \newpage {\bf \blue Holomorphically symplectic manifolds (reminder)} {\bf\green DEFINITION:} A holomorphically symplectic manifold is a complex manifold equipped with non-degenerate, holomorphic $(2,0)$-form. \remark Hyperk\"ahler manifolds are holomorphically symplectic. Indeed, $\Omega:=\omega_J+\1\omega_K$ is a holomorphic symplectic form on $(M,I)$. {\bf\green THEOREM:} (Calabi-Yau) A compact, K\"ahler, holomorphically symplectic manifold admits a unique hyperk\"ahler metric in any K\"ahler class. {\bf\green DEFINITION:} For the rest of this talk, {\bf \red a hyperk\"ahler manifold is a compact, K\"ahler, holomorphically symplectic manifold.} {\bf\green DEFINITION:} A hyperk\"ahler manifold $M$ is called {\bf \blue simple} if $\pi_1(M)=0$, $H^{2,0}(M)=\C$. {\bf \purple Bogomolov's decomposition:} Any hyperk\"ahler manifold admits a finite covering which is a product of a torus and several simple hyperk\"ahler manifolds. {\bf \red Further on, all hyperk\"ahler manifolds are assumed to be simple}. \newpage {\bf \blue The Bogomolov-Beauville-Fujiki form (reminder)} {\bf \green THEOREM:} (Fujiki). Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is hyperk\"ahler. Then $\int_M \eta^{2n}=c q(\eta,\eta)^n$, for some primitive integer quadratic form $q$ on $H^2(M,\Z)$, and $c>0$ an integer number. {\bf \green Definition:} This form is called {\bf \blue Bogomolov-Beauville-Fujiki form}. {\bf \purple It is defined by the Fujiki's relation uniquely, up to a sign.} The sign is determined from the following formula (Bogomolov, Beauville) \begin{align*} \lambda q(\eta,\eta) &= \int_X \eta\wedge\eta \wedge \Omega^{n-1} \wedge \bar \Omega^{n-1} -\\ &-\frac {n-1}{n}\left(\int_X \eta \wedge \Omega^{n-1}\wedge \bar \Omega^{n}\right) \left(\int_X \eta \wedge \Omega^{n}\wedge \bar \Omega^{n-1}\right) \end{align*} where $\Omega$ is the holomorphic symplectic form, and $\lambda>0$. {\bf \green Remark:} {\bf \red $q$ has signature $(b_2-3,3)$.} It is negative definite on primitive forms, and positive definite on $\langle \Omega, \bar \Omega, \omega\rangle$, where $\omega$ is a K\"ahler form. \newpage {\bf \blue The Teichm\"uller space and the mapping class group (reminder)} \renewcommand{\Teich}{\operatorname{\sf Teich}} {\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 $\widetilde \Teich$ the space of complex structures on $M$, and let $\Teich:=\widetilde \Teich/\Diff_0(M)$. We call it {\bf \blue the Teichm\"uller space.} {\bf \green Remark:} $\Teich$ is {\bf \blue a finite-dimensional complex space} (Kodaira-Spencer-Kuranishi), but often {\bf \red non-Hausdorff}. {\bf \green Definition:} Let $\Diff(M)$ be the group of diffeomorphisms of $M$. We call $\Gamma:=\Diff(M)/\Diff_0(M)$ {\bf \blue the mapping class group}. \remark For hyperk\"ahler manifolds, we take for $\Teich$ {\bf \blue the space of all complex structures of hyperk\"ahler type}, that is, {\bf \red holomorphically symplectic and K\"ahler type}. It is open in the usual Teichm\" uller space. \newpage {\bf \blue The period map (reminder)} {\bf \green Remark:} For any $J\in \Teich$, $(M,J)$ is also a simple hyperk\"ahler manifold, hence $H^{2,0}(M,J)$ is one-dimensional. {\bf \green Definition:} Let $P:\; \Teich \arrow {\Bbb P}H^2(M, \C)$ map $J$ to a line $H^{2,0}(M,J)\in {\Bbb P}H^2(M, \C)$. The map $P:\; \Teich \arrow {\Bbb P}H^2(M, \C)$ is called {\bf\blue the period map}. \remark {\purple $P$ maps $\Teich$ into an open subset of a quadric,} defined by \[ {\Bbb Per}:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \ q(l,l)=0, q(l, \bar l) >0\}. \] It is called {\bf \blue the period space} of $M$. \newpage {\bf \blue Global Torelli theorem and mapping class group (reminder)} \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 $V\ni x, U\ni y$, $U \cap V\neq \emptyset$. {\bf \green THEOREM:} {\bf \blue (D. Huybrechts) }\\ If $I_1$, $I_2\in \Teich$ are non-separable points, then $P(I_1)=P(I_2)$, and $(M, I_1)$ {\bf \purple is birationally equivalent} to $(M, I_2)$ \definition Let $M$ be a topological space for which $M/\sim$ is Hausdorff. Then $M/\sim$ is called {\bf \blue a Hausdorff reduction} of $M$. \definition The space $\Teich_b:= \Teich/\sim$ is called {\bf \blue the birational Teichm\"uller space} of $M$. \theorem {\bf \blue (``Global Torelli theorem'')}\\ {\bf \red The period map $\Teich_b\stackrel \Per \arrow \Perspace$ is a diffeomorphism,} for each connected component of $\Teich_b$. {\bf \green Theorem:} {\bf \blue (``Mapping class group is arithmetic'')}\\ Let $M$ be a simple hyperk\"ahler manifold, and $\Gamma$ the mapping class group. Then \\ (i) $\Gamma\restrict{H^2(M,\Z)}$ {\bf \red is a finite index subgroup of $O(H^2(M, \Z), q)$.}\\ (ii) The map $\Gamma\arrow O(H^2(M, \Z), q)$ {\bf \red has finite kernel.} \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 {\bf \blue (ergodic actions have dense orbits)}\\ 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 Step 2:} Choose a countable base $\{U_i\}$ of topology on $M$. Then the set of points in 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. \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 $I'$.} % This is another characterization of ergodic complex structures. \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 $L\subset \Perspace$ such that the orbit $\Gamma\cdot L$ is dense (such points are called {\bf \blue ergodic}). {\bf \red Then $Z:=\Perspace\backslash \Perspace_e$ has measure 0.} {\bf \green Proof. Step 1:} Let $G=SO(b_2-3,3)$, $H=SO(2) \times SO(b_2-3,1)$. {\bf \purple Then $\Gamma$-action on $G/H$ is ergodic,} by Moore's theorem. {\bf \green Step 2:} Ergodic orbits are dense, non-ergodic orbits have measure 0. \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). \example By Borel and Harish-Chandra theorem, {\bf \purple any integer lattice in a simple Lie group has finite covolume.} \definition {\bf \blue Unipotent element} in a Lie group is an exponent of a nilpotent element. \theorem Let $H\subset G$ be a Lie subroup generated by unipotents, and $\Gamma\subset G$ an arithmetic lattice. Then {\bf \red the closure of any $\Gamma$-orbit in $G/H$ is an orbit of a Lie subgroup $S\subset G$, such that $S\cap \Gamma\subset S$ is a lattice.} \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 acting trivially on a given positive 2-dimensional plane, $H\cong SO^+(1,k)$, 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 an orbit of a closed connected Lie subgroup $S\subset G$ containing $H$.} \newpage {\bf \blue Classification of $\Gamma$-orbits on $\Perspace$} \claim Let $G=SO(3,k)$ be a group of oriented isometries of $V=\R^{3, k}$, and $H\cong SO(1,k)\subset G$. Denote by $\goth h$, $\goth g$ their ie algebras. Then {\bf \red any Lie algebra $\goth s$ such that $\goth h \subsetneq \goth s \subsetneq \goth g$ is isomorphic to $\goth{so}(2, k)$.} This is the Lie algebra of the Lie group $S=SO(2,k)$ fixing a positive vector $v\in V$. \corollary Let $J\in \Perspace=G/H$, and $\Gamma\subset G$ be am arithmetic lattice. {\bf \red Then one of three things happens.}\\ \phantom{XX} (i) ether $J$ is ergodic, \\ \phantom{XX} (ii) or the closure of $\Gamma$-orbit of $J$ is an orbit of $S$ \\ or its connected component $S^+$,\\ \phantom{XX} (iii) or the orbit $\Gamma \cdot J$ is closed. \newpage {\bf \blue Characterization of ergodic complex structures} \remark By Ratner's theorem, the $S^+$-orbit of $J$ in (ii) and the $H$-orbit of $J$ in (iii) has finite volume in $G/\Gamma$ Therefore, {\bf \purple its intersection with $\Gamma$ is a lattice in $H$.} This brings \corollary Consider the action of the mapping class group $\Gamma$ of a hyperk\"ahler manifold on its period space $\Perspace$. 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 complex structure with closed $\Gamma$-orbit on a hyperk\"ahler manifold, $\Omega$ its holomorphic symplectic form, and $W\subset H^2(M,\R)$ a plane generated by $\Re\Omega, \Im \Omega$. {\bf \red Then $W$ is rational.} Similarly, one has \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 $W$ contains a rational vector.} \end{document}