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Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \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:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Global Torelli theorem for hyperk\"ahler manifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[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 Hyperk\"ahler manifolds} \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} {\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 $H^1(M)=0$, $H^{2,0}(M)=\C$. {\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\green Remark:} A simple hyperk\"ahler manifold is always simply connected (Cheeger-Gromoll theorem). {\bf \red Further on, all hyperk\"ahler manifolds are assumed to be simple}. \newpage {\bf \blue The Teichm\"uller space and the mapping class group} \newcommand{\Teich}{\operatorname{Teich}} \newcommand{\Comp}{\operatorname{Comp}} {\bf \green Definition:} Let $M$ be a compact complex manifold, and $\Diff_0\subset \Diff$ 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 $\Teich:=\Comp/\Diff_0$. We call it {\bf \blue the Teichm\"uller space.} {\bf \green Remark:} $\Teich$ is {\bf \blue a finite-dimensional complex space} (Kodaira), but often {\bf \red non-Hausdorff}. {\bf \green Definition:} We call $\Gamma:=\Diff/\Diff_0$ {\bf \blue the mapping class group}. The {\bf \blue moduli space of complex structures on $M$} is a connected component of $\Teich/\Gamma$, but this quotient is not always well-defined. {\bf \green Remark:} This terminology is {\bf \purple standard for curves.} \remark For hyperk\"ahler manifolds, it is convenient to take for $\Teich$ {\bf \blue the space of all complex structures of hyperk\"ahler type}, that is, {\bf \red holomorphically symplectic and K\"ahler}. It is open in the usual Teichm\" uller space. \remark Two ingredients of global Torelli theorem:\\ {\bf \green (a) determine the mapping class group\\ (b) determine the Teichm\"uller space.} \newpage {\bf \blue The Bogomolov-Beauville-Fujiki form} {\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 Lefschetz $\goth{sl}(2)$-action} Let $(M, I, g)$ be a Kaehler manifold, $\omega$ its Kaehler form. Consider the following operators.\\ 1. $L(\alpha):= \omega\wedge \alpha$.\\ 2. $\Lambda := L^*$ (Hermitian dual of $L$).\\ 3. The Weil operator $W_I\restrict{\Lambda^{p,q}(M)}=\1(p-q)$ \claim {\bf \red The triple $L, \Lambda, H$ satisfies the relations for the $\goth{sl}(2)$ Lie algebra:} $[L,\Lambda]=H$, $[H,L]=2L$, $[H,\Lambda]=2\Lambda$. \definition $L, \Lambda, H$ is called {\bf \blue the Lefschetz $\goth{sl}(2)$-triple}. \theorem The ${\goth sl}(2)$-action $\langle L, \Lambda, H\rangle$ and the action of Weil operator commute with Laplacian, hence {\bf \red preserve the harmonic forms on a K\"ahler manifold}. \corollary Any cohomology class can be represented as a sum of closed $(p,q)$-forms, {\bf \purple giving a decomposition $H^i(M) = \bigoplus_{p+q=i}H^{p,q}(M)$, with $\overline{H^{p,q}(M)} = H^{q,p}(M)$. } \newpage {\bf \blue Riemann-Hodge pairing} \theorem Let $(M,\omega)$ be a K\"ahler $n$-manifold. Consider the following pseudo-Hermitian form on $H^d(M)$: \[ B(\alpha, \beta):= \int_M\alpha\wedge \bar \beta\wedge \omega^{n-d} \] {\bf \red Then $B$ is sign-definite on a space $H^{p,q}_k(M)$ of $(p,q)$-forms which have weight $k$ with respect to Lefschetz $SL(2)$-action.} \corollary Let $G$ be a group of automorphisms of the algebra $H^*(M,\R)$ preserving the $(p,q)$-decomposition and fixing a K\"ahler class $\omega$. {\bf \red Then $G$ is compact.} {\bf \green Proof:} Since $G$ fixes $\omega$, $G$ commutes with the Lefschetz $SL(2)$-action, hence it fixes a sign-definite form on each space $H^{p,q}_k(M)$. \endproof \newpage {\bf \blue Automorphisms of cohomology.} \theorem Let $M$ be a simple hyperk\"ahler manifold, and $G\subset GL(H^*(M))$ a group of automorphisms of its cohomology algebra preserving the Pontryagin classes. Then $G$ acts on $H^2(M)$ {\bf \red preserving the BBF form.} Moreover, the map $G\arrow O(H^2(M, \R), q)$ {\bf \red is surjective on a connected component, and has compact kernel.} {\bf \green Proof. Step 1:} Fujiki formula $v^{2n}= q(v,v)^n$ implies that $\Gamma_0$ {\bf \purple preserves the Bogomolov-Beauville-Fujiki up to a sign. } The sign is fixed, if $n$ is odd. {\small {\bf \green Step 2:} For even $n$, the sign is also fixed. Indeed, $G$ preserves $p_1(M)$, and (as Fujiki has shown) $v^{2n-2}\wedge p_1(M)= q(v,v)^{n-1} c$, for some $c\in \R$. The constant $c$ is positive, {\bf \purple because the degree of $c_2(B)$ is positive} for any non-trivial stable bundle with $c_1(B)=0$. } {\bf \green Step 3:} ${\goth o}(H^2(M, \R), q)$ acts on $H^*(M, \R)$ by derivations preserving Pontryagin classes (V., 1995). Therefore $\Lie(G)$ surjects to ${\goth o}(H^2(M, \R), q)$. {\bf \green Step 4:} {\bf \purple The kernel $K$ of the map $G \arrow G\restrict{H^2(M,\R)}$ is compact,} because it commutes with the Hodge decomposition and Lefschetz ${\goth sl}(2)$-action, hence preserves the Riemann-Hodge form. \endproof \newpage {\bf \blue Sullivan's theorem} {\bf \green Theorem:} (Sullivan) Let $M$ be a compact, simply connected K\"ahler manifold, $\dim_\C M\geq 3$. Denote by $\Gamma_0$ the group of automorphisms of an algebra $H^*(M, \Z)$ preserving the Pontryagin classes $p_i(M)$. Then {\bf \red the natural map $\Diff(M)/\Diff_0\arrow \Gamma_0$ has finite kernel, and its image has finite index in $\Gamma_0$.} {\bf \green Theorem:} Let $M$ be a simple hyperk\"ahler manifold, and $\Gamma_0$ as above. Then \\ (i) $\Gamma_0\restrict{H^2(M,\Z)}$ {\bf \blue is a finite index subgroup of $O(H^2(M, \Z), q)$.}\\ (ii) The map $\Gamma_0\arrow O(H^2(M, \Z), q)$ {\bf \blue has finite kernel.} {\bf \green Proof:} Follows from the computation of $G=\Aut(H^*(M, \R), p_1, ..., p_n)$ done earlier. Indeed, the kernel of $\Gamma_0\restrict{H^2(M,\Z)}$ is a set of integer points of a compact Lie group, hence finite. The image of $\Gamma_0=G_\Z$ has finite index in $O(H^2(M, \Z), q)$, because the corresponding map of Lie groups is surjective. \endproof \newpage {\bf \blue Computation of the mapping class group} \corollary The mapping class group {\bf \red $\Gamma$ is mapped to $O(H^2(M, \Z), q)$ with finite kernel and finite index.} {\bf \green Proof:} By Sullivan, $\Gamma$ is mapped to $\Gamma_0$ with finite kernel and finite index, and $\Gamma_0\arrow O(H^2(M, \Z), q)$ has finite kernel and finite index, as shown above. \endproof \theorem (Kollar-Matsusaka, Huybrechts) {\bf \purple There are only finitely many connected components } of $\Teich$. \corollary Let $\Gamma_I$ be the group of elements of mapping class group preserving a connected component of Teichm\"uller space containing $I\in \Teich$. {\red \bf Then $\Gamma_I$ has finite index in $\Gamma$}. \remark $\Gamma_I$ is a group generated by monodromy of all Gauss-Manin local systems for all deformations of $(M,I)$. It is known as {\bf \blue the monodromy group} of $(M,I)$. \newpage {\bf \blue The period map} {\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}. \newcommand{\Per}{\operatorname{\sf Per}} \newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}} \remark {\purple $P$ maps $\Teich$ into an open subset of a quadric,} defined by \[ \Perspace:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \ q(l,l)=0, q(l, \bar l) >0. \] It is called {\bf \blue the period space} of $M$. \remark $\Perspace=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$ {\bf \green THEOREM:} Let $M$ be a simple hyperk\"ahler manifold, and $\Teich$ its Teichm\"uller space. Then\\ (i) (Bogomolov) {\bf \red The period map $P:\; \Teich \arrow {\Bbb Per}$ is etale.}\\ (ii) (Huybrechts) It is {\bf \red surjective.} \remark Bogomolov's theorem implies that {\bf \purple $\Teich$ is smooth.} It is {\bf \red usually non-Hausdorff}. \newpage {\bf \blue Birational equivalence and non-separable points} \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:} (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 $(M, I_2)$. \corollary {\bf \red The set of all non-separable points in $\Teich$ belongs to a union of countably many divisors.} {\bf \green Proof. Step 1:} Let $z\in H^2(M,\Z)$ be a cohomology class, and $\Teich_z$ the set of all $I\in \Teich$ such that $z\in H^{1,1}(M,I)$. The relation $z\in H^{1,1}(M,I)$ is equivalent to $q(z,\Omega)=0$, hence \[ \Per(\Teich_z)\subset \{l\in {\Bbb P}(z^\bot)\ \ | \ \ q(l,l)=0, q(l, \bar l) >0. \] {\bf \purple This shows that $\Per(\Teich_z)$ is a divisor.} {\bf \green Step 2:} If $(M,I)$ is a holomorphic symplectic manifold with non-trivial birational equivalence to $M'$, it contains a rational curve $C$. {\bf \purple Therefore, $I\in \Teich_z$, where $z=[C]$ is its homology class.} \endproof \newpage {\bf \blue Hausdorff reduction} \remark {\bf \blue A non-Hausdorff manifold} is a topological space locally diffeomorphic to $\R^n$. \definition Let $M$ be a topological space for which $M/\sim$ is Hausdorff. Then $M/\sim$ is called {\bf \blue a Hausdorff reduction} of $M$. {\bf \green Problems:} \\ 1. {\bf \red $\sim$ is not always an equivalence relation.\\ 2. Even if $\sim$ is equivalence, the $M/\sim$ is not always Hausdorff.} \remark {\purple A quotient $M/\sim$ is Hausdorff, if $M \arrow M/\sim$ is open, and the graph $\Gamma_\sim \in M\times M$ is closed.} \newpage {\bf \blue Weakly Hausdorff manifolds} \definition A point $x\in X$ is called {\bf\blue Hausdorff} if $x\not\sim y$ for any $y\neq x$. \definition Let $M$ be an $n$-dimensional real analytic manifold, not necessarily Hausdoff. Suppose that {\bf \purple the set $Z\subset M$ of non-Hausdorff points is contained in a countable union of real analytic subvarieties} of $\codim\geq 2$. Suppose, moreover, that (S) For every $x\in M$, there is a closed neighbourhood $B\subset M$ of $x$ and a continuous surjective map $\Psi:\; B \arrow \R^n$ to a closed ball in $\R^n$, {\bf \purple inducing a homeomorphism} on an open neighbourhood of $x$. Then $M$ is called {\bf\blue a weakly Hausdorff manifold}. \remark {\bf \purple The period map satisfies (S)}. Also, the non-Hausdorff points of $\Teich$ {\bf \purple are contained in a countable union of divisors.} \theorem A {\bf \red weakly Hausdorff manifold $X$ admits a Hausdorff reduction.} In other words, the quotient $X/\sim$ is a Hausdorff. Moreover, $X \arrow X/\sim$ is locally a homeomorphism. This theorem is proven using 1920-ies style point-set topology. \newpage {\bf \blue Birational Teichm\"uller moduli space} \definition The space $\Teich_b:= \Teich/\sim$ is called {\bf \blue the birational Teichm\"uller space} of $M$. \theorem {\bf \red The period map $\Teich_b\stackrel \Per \arrow \Perspace$ is an isomorphism,} for each connected component of $\Teich_b$. The proof is based on two results. \proposition {\bf \blue (The Covering Criterion)} Let $X\stackrel \phi \arrow Y$ be an etale map of smooth manifolds. Suppose that each $y\in Y$ has a neighbourhood $B\ni y$ diffeomorphic to a closed ball, such that for each connected component $B' \subset \phi^{-1}(B)$, $B'$ projects to $B$ surjectively. {\bf \red Then $\phi$ is a covering.} \proposition {\bf \red The period map satisfies the conditions of the Covering Criterion.} \newpage {\bf \blue Global Torelli theorem} \definition Let $M$ be a hyperkaehler manifold, $\Teich_b$ its birational Teichm\"uller space, and $\Gamma$ the mapping class group. The quotient $\Teich_b/\Gamma$ is called {\bf \blue the birational moduli space} of $M$. \remark The birational moduli space is obtained from the usual moduli space {\bf \purple by gluing some (but not all) non-separable points.} {\bf \red It is still non-Hausdorff.} \theorem Let $(M,I)$ be a hyperk\"ahler manifold, and $W$ a connected component of its birational moduli space. {\bf \blue Then $W$ is isomorphic to ${\Perspace}/\Gamma_I$, where ${\Perspace}=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$ and $\Gamma_I$ is an arithmetic subgroup in $O(H^2(M, \R), q)$}. {\bf \green A CAUTION:} Usually ``the global Torelli theorem'' is understood as a theorem about Hodge structures. For K3 surfaces, {\bf \purple the Hodge structure on $H^2(M,\Z)$ determines the complex structure}. For $\dim_\C M >2$, {\bf \red it is false}. \newpage {\bf \blue The marked moduli space} \definition Let $\Gamma$ be the mapping class group, and $K\subset \Gamma$ the kernel of the natural map $\Gamma \arrow GL(H^2(M, \Z))$. {\bf \red It is finite,} as we have shown. The quotient $\Teich/K$ is called {\bf\blue the marked moduli space}. \theorem The natural map $\Teich \arrow \Teich/K$ {\bf \red is a homeomorphism on each connected component.} {\bf \green Proof. Step 1:}\\ Let $I\in \Teich$ be a fixed point of a subgroup $K_I\subset K$. By Bogomolov's theorem, $T_I\Teich$ is naturally identified with $H^{1,1}_I(M)$. Since the action of $K$ on $H^2(M)$ is trivial, any $\alpha\in K_I$ acts trivially on $T_I\Teich$. Therefore, {\bf \purple $K_I$ acts as identity on a connected component of $\Teich$ containing $I$}. {\bf \green Step 2:} From Step 1, obtain that {\bf \red the quotient map $\Teich\stackrel\Psi \arrow \Teich/K$ is a finite covering,} hence {\bf \purple it induces a finite covering of the corresponding Hausdorff reductions.} However, $\Psi$ induces an isomorphism on each connected component of $\Teich_b$, because {\bf \red each component of $\Teich_b$ is isomorphic to $\Perspace$.} \endproof \newpage {\bf \blue The Hodge-theoretic Torelli theorem} \remark The group $O(p,q)$ ($p, q >0$) has {\bf \blue 4 connected components}, corresponding to the orientations of positive $p$-dimensional and negative $q$-dimensional planes. \definition Let $M$ be a hyperkaehler manifold. One says that {\bf \blue the Hodge-theoretic Torelli theorem holds for $M$} if \[ \Teich/\Gamma_I\arrow {\Perspace}/O^+(H^2(M, \Z), q), \] where $O^+(H^2(M, \Z), q)$ is a subgroup of $O(H^2(M, \Z), q)$ preserving orientation on positive 3-planes. Equivalently, {\bf \purple it is true if $M$ is uniquely determined by its Hodge structure}. \remark The Hodge-theoretic Torelli theorem {\bf\purple is true for K3 surfaces}. {\bf \red It is false} for all other known examples of hyperkaehler manifolds. {\bf \green Problems:} \\ 1. The moduli space $\Teich/\Gamma$ {\bf \blue is not Hausdorff} (Debarre, 1984). Indeed, bimeromorphically equivalent hyperk\"ahler manifolds have isomorphic Hodge structures.\\ 2. {\bf \red The covering $\Teich_b/\Gamma_I\arrow {\Perspace}/O^+(H^2(M, \Z), q)$ is non-trivial}, because the map $\Gamma_I\arrow O^+(H^2(M, \Z), q)$ is not surjective (Namikawa, 2002). \newpage {\bf \blue The birational Hodge-theoretic Torelli theorem} \definition {\bf \blue The birational Hodge-theoretic Torelli theorem is true for $M$} if $\Gamma_I$ (the stabilizer of a Torelli component in the mapping class group) {\bf \purple is isomorphic to $O^+(H^2(M, \Z), q)$.} \remark If a birational Hodge-theoretic Torelli theorem holds for $M$, then any deformation of $M$ is up to a bimeromorphic equivalence {\bf \red determined by the Hodge structure on $H^2(M)$.} \theorem (Markman) The for $M=K3^{[n]}$, the group $\Gamma_I$ {\bf \blue is a subgroup of $O^+(H^2(M, \Z), q)$ generated by oriented reflections.} \theorem Let $M=K3^{[n+1]}$ with $n$ a prime power. Then the (usual) global Torelli theorem holds birationally: {\bf \blue two deformations of a Hilbert scheme with isomorphic Hodge structures are bimeromorphic}. {\bf \red For other $n$, it is false} (Markman). \end{document}