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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 2 \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 and moduli of geometric structures (2)} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize Misha Verbitsky } \\[20mm] {\tiny\bf TWENTY-THIRD G\"OKOVA GEOMETRY / TOPOLOGY CONFERENCE\\ May 30 - June 4 (2016)\\ G\"okova, Turkey} \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 Plan:} 1. Hyperk\"ahler manifolds: basic facts 2. Teichm\"uller space of hyperk\"ahler structures 3. Teichm\"uller space of symplectic structures of a hyperk\"ahler manifold. 4. Ergodic action of the mapping class group on the space of symplectic structures. \newpage {\bf \blue Teichm\"uller space for symplectic structures (reminder)} \definition Let $\Gamma(\Lambda^2 M)$ be the space of all 2-forms on a manifold $M$, and $\Symp\subset \Gamma(\Lambda^2 M)$ the space of all symplectic 2-forms. We equip $\Gamma(\Lambda^2 M)$ with $C^\infty$-topology of uniform convergence on compacts with all derivatives. Then $\Gamma(\Lambda^2 M)$ is a Frechet vector space, and $\Symp$ a Frechet manifold. \definition Consider the group of diffeomorphisms, denoted $\Diff$ or $\Diff (M)$ as a Frechet Lie group, and denote its connected component (``group of isotopies'') by $\Diff_0$. The quotient group $\Gamma:=\Diff/\Diff_0$ is called {\bf\blue the mapping class group} of $M$. \definition {\bf \blue Teichm\"uller space of symplectic structures on $M$} is defined as a quotient $\Teich_s:= \Symp/\Diff_0$. The quotient $\Teich_s/\Gamma=\Symp/\Diff$, is called {\bf\blue the moduli space of symplectic structures.} \newpage {\bf \blue Moser's theorem (reminder)} \definition Define {\bf \blue the period map} $\Per:\; \Teich_s \arrow H^2(M,\R)$ mapping a symplectic structure to its cohomology class. \theorem {\bf \blue (Moser, 1965)}\\ The {\bf \red Teichm\"uler space $\Teich_s$ is a manifold} (possibly, non-Hausdorff), and the {\bf \red period map $\Per:\; \Teich_s \arrow H^2(M,\R)$ is locally a diffeomorphism.} The proof is based on another theorem of Moser. {\bf \green Theorem 1:} {\bf \blue (Moser)}\\ Let $\omega_t$, $t\in S$ be a smooth family of symplectic structures, parametrized by a connected manifold $S$. Assume that the cohomology class $[\omega_t]\in H^2(M)$ is constant in $t$. {\bf \purple Then all $\omega_t$ are diffeomorphic.} \newpage {\bf \blue Non-Hausdorff points on symplectic Teichm\"uller space} Example of D. McDuff found in Salamon, Dietmar, {\em\green Uniqueness of symplectic structures}, Acta Math. Vietnam. 38 (2013), no. 1, 123-144. Let $M= S^1\times Z^1 \times S^2 \times S^2$ with coordinates $\theta_1, \theta_2 \in S^1\subset \C^*$ and $z_1, z_2\in S^2$. Let $\phi_{\theta, z}\; \C P^1 \arrow \C P^1$ be a rotation around the axis $z\in \C P^1$ by the angle $\theta$. {\bf \blue Consider the diffeomorphism $\Psi:\; M \arrow M$ mapping $(\theta_1, \theta_2, z_1, z_2)$ to $(\theta_1, \theta_2, z_1, \phi_{\theta_1, z_1}(z_2))$.} \theorem Let $\omega_\lambda$ be the product symplectic form on $M = T^2 \times \C P^1 \times \C P^1$ obtained as a product of symplectic forms of volume 1, 1, $\lambda$ on $T^2$, $\C P^1$, $\C P^1$. {\bf \red The form $\Psi^*(\omega_1)$ is homologous, but not diffeomorphic to $\omega_1$.} However, {\bf \red the form $\Psi^*(\omega_\lambda)$ is diffeomorphic to $\omega_\lambda$ for any $\lambda\neq 1$.} {\small (D. McDuff, {\em\green Examples of symplectic structures}, Invent. Math. 89 (1987), 13-36.)} \newpage {\bf \blue Complex manifolds} {\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.} \newpage {\bf \blue The Hodge decomposition in linear algebra} \definition {\bf \blue The Hodge decomposition} $V\otimes_\R \C:= V^{1,0}\oplus V^{0,1}$ is defined in such a way that $V^{1,0}$ is a $\1$-eigenspace of $I$, and $V^{0,1}$ a $-\1$-eigenspace. \remark Let $V_\C:= V \otimes_\R \C$. The Grassmann algebra of skew-symmetric forms $\Lambda^n V_\C:=\Lambda^n_\R V \otimes _\R C$ admits a decomposition \[ \Lambda^n V_\C= \bigoplus_{p+q=n} \Lambda^p V^{1,0} \otimes \Lambda^q V^{0,1} \] We denote $\Lambda^p V^{1,0} \otimes \Lambda^q V^{0,1}$ by $\Lambda^{p,q}V$. The resulting decomposition $\Lambda^n V_\C= \bigoplus_{p+q=n}\Lambda^{p,q}V$ is called {\bf \blue the Hodge decomposition of the Grassmann algebra}. \remark The operator $I$ induces $U(1)$-action on $V$ by the formula $\rho(t)(v) = \cos t\cdot v + \sin t \cdot I(v)$. We extend this action on the tensor spaces by muptiplicativity. \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}. \remark {\bf \purple This is equivalent to $\nabla\omega=0$,} where $\nabla$ is Levi-Civita connection. \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 $I,J,K$. {\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 $I,J,K$). \claim A compact hyperk\"ahler manifold $M$ {\bf \purple has maximal holonomy of Levi-Civita connection $Sp(n)$} if and only if {\bf \purple $\pi_1(M)=0$, $h^{2,0}(M)=1$.} \theorem {\bf \blue (Bogomolov decomposition)}\\ {\bf \red Any compact hyperk\"ahler manifold has a finite covering isometric to a product of a torus and several maximal holonomy hyperk\"ahler manifolds.} \newpage {\bf \blue Holomorphically symplectic manifolds} {\bf\green DEFINITION:} {\bf \blue A holomorphically symplectic manifold} is a complex manifold equipped with non-degenerate, holomorphic $(2,0)$-form. \remark In this talk, all holomorphically symplectic manifolds are assumed to be K\"ahler and compact. \remark A hyperk\"ahler manifold {\bf \purple has three symplectic forms\\ $\omega_I:= g(I\cdot, \cdot)$, $\omega_J:= g(J\cdot, \cdot)$, $\omega_K:= g(K\cdot, \cdot)$.} \claim In these assumptions, $\omega_J + \1\omega_K$ is holomorphic symplectic on $(M,I)$. {\bf\green THEOREM:} {\bf \blue (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 of maximal holonomy.} \newpage {\bf \blue EXAMPLES.} \example An even-dimensional complex torus. \example {\bf \purple A non-compact example:} $T^* \C P^n$ (Calabi). \remark $T^*\C P^1$ {\bf \blue is a resolution of a singularity $\C^2/{\pm1}$.} \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}. \remark Take a symmetric square $\Sym^2 T$, with a natural action of $T$, and let $T^{[2]}$ be a blow-up of a singular divisor. {\bf \purple Then $T^{[2]}$ is naturally isomorphic to the Kummer surface $\widetilde {T/{\pm1}}$.} \definition A complex surface is called {\bf \blue K3 surface} if it a deformation of the Kummer surface. \theorem {\bf \blue (a special case of Enriques-Kodaira classification)}\\ Let $M$ be a compact complex surface which is hyperk\"ahler. {\bf \red Then $M$ is either a torus or a K3 surface.} \newpage {\bf \blue Hilbert schemes} \definition A {\bf\blue Hilbert scheme} $M^{[n]}$ of a complex surface $M$ is a classifying space of all ideal sheaves $I\subset \calo_M$ for which the quotient $\calo_M/I$ has dimension $n$ over $\C$. \remark A Hilbert scheme {\bf \purple is obtained as a resolution of singularities} of the symmetric power $\Sym^n M$. \theorem (Fujiki, Beauville) {\bf \red A Hilbert scheme of a hyperk\"ahler surface is hyperk\"ahler.} \example {\bf\blue A Hilbert scheme of K3} is hyperk\"ahler. \example Let $T$ be a torus. Then it acts on its Hilbert scheme freely and properly by translations. For $n=2$, the quotient $T^{[n]}/T$ is a Kummer K3-surface. For $n>2$, a universal covering of $T^{[n]}/T$ is called {\bf \blue a generalized Kummer variety}. \remark There are 2 more ``sporadic'' examples of compact hyperk\"ahler manifolds, constructed by K. O'Grady. {\bf \purple All known compact hyperkaehler manifolds are these 2 and the two series:} Hilbert schemes of K3, and generalized Kummer. \newpage {\bf \blue Main result} \definition A symplectic structure $\omega$ on a hyperk\"ahler manifold is called {\bf \blue standard} if $\omega$ is a K\"ahler form for some hyperk\"ahler structure. \remark Any known symplectic structure on a hyperk\"ahler manifold or a torus is of this type. {\bf \purple It was conjectured that non-standard symplectic structures don't exist.} \theorem (E. Amerik, V.) Let $M$ be a maximal holonomy hyperk\"ahler manifold. Then the period map $\Per:\; \Teich_s \arrow H^2(M,\R)$ {\bf \red is an open embedding on the set of all standard symplectic structures}, and {\bf \red its image is the set of all cohomology classes $v$ such that $q(\omega, \omega) >0$,} where $q$ is a quadratic form on cohomology defined below. \newpage {\bf \blue 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 positive definite on $\langle \Omega, \bar \Omega, \omega\rangle$, where $\omega$ is a K\"ahler form. \newpage {\bf \blue MBM classes} \definition {\bf \blue K\"ahler cone} of a K\"ahler manifold is the set of all cohomology classes $\omega\in H^{1,1}(M)$ \definition {\bf \blue Face} of a K\"ahler cone $K$ is a subset $V\cap \6 K$ containing an open subset of $V$, for some hyperplane $V\subset H^{1,1}(M)$. \definition Let $M$ be a hyperk\"ahler manifold. A homology class $z\in H_2(M,\Q)$ is called {\bf \blue an MBM class} (monodromy birational minimal) if for some complex structure in the same deformation class, the annihilator $z^\bot$ contains a face of its K\"ahler cone. \definition A cohomology class $z\in H^2(M,\Q)$ is called {\bf \blue MBM class} if it becomes MBM after an identification $H^2(M,\Q)\cong H_2(M,\Q)$ provided by the Bogomolov-Beauville-Fujiki form. \newpage {\bf \blue Properties of MBM classes} \newcommand{\Pos}{\operatorname{\sf Pos}} \definition {\bf \blue Negative class} on a hyperk\"ahler manifold is $\eta\in H^2(M,\R)$ satisfying $q(\eta,\eta)<0$. \theorem (E. Amerik, V.) Let $(M,I)$ be a hyperk\"ahler manifold, $\rk H^{1,1}(M,\Z)=1$, and $z\in H_{1,1}(M,I)$ a non-zero negative class. {\bf \red Then $z$ is MBM if and only if $\pm z$ is $\Q$-effective,} that is, $\lambda z$ is represented by a complex curve. \endproof \definition {\bf \blue Positive cone} $\Pos(M)$ on a K\"ahler surface is the one of the two components of \[ \left \{v\in H^{1,1}(M,\R)\ \ |\ \ \int_M\eta\wedge\eta>0\right \} \] which contains a K\"ahler form. \theorem (E. Amerik, V.) Let $(M,I)$ be a hyperk\"ahler manifold, and $S\subset H_{1,1}(M,I)$ the set of all MBM classes in $H_{1,1}(M,I)$. Consider the corresponding set of hyperplanes $S^\bot:=\{W=z^\bot\ \ |\ \ z\in S\}$ in $H^{1,1}(M,I)$. {\bf \red Then the K\"ahler cone of $(M,I)$ is a connected component of $\Pos(M,I)\backslash \cup S^\bot$}, where $\Pos(M,I)$ is a positive cone of $(M,I)$. \endproof %\newpage % %{\bf \blue Equivalence of hyperk\"ahler structures} % %\definition %Let $(M,I,J,K,g)$ and $(M,I',J',K',g')$ be two %hyperk\"ahler structures. We say that these structures %are {\bf\blue equivalent} if the corresponding quaternionic algebras %coincide. % %\proposition\label{_hk_equiv_Proposition_} %Let $M$ be a hyperk\"ahler manifold, %and $(M,I,J,K,g)$ and $(M,I',J',K',g')$ be two %hyperk\"ahler structures of maximal holonomy. {\bf \red Then the following %conditions are equivalent.} \\ %\phantom{AAA} (i) $g$ is proportional to $g'$. \\ %\phantom{AAA} (ii) The hyperk\"ahler structure $(M,I,J,K,g)$ is equivalent to % $(M,I',J',K',g')$. % %{\bf\green Proof of (i) $\Rightarrow$ (ii):} If $g$ is proportional to $g'$, %the corresponding Levi-Civita connections coincide. %Its holonomy group is $Sp(n)$, %and {\bf \purple holonomy stabilizer in $\End(TM)$ is a quaternionic %algebra, generated by $I,J,K$ and by $I',J',K'$.} % %\newpage % %{\bf \blue Equivalence of hyperk\"ahler structures (2)} % % %{\bf \green Proof of (ii) $\Rightarrow$ (i):} Conversely, assume %that the algebras generated by $I,J,K$ and by $I',J',K'$ %coincide. Recall that a manifold is called {\bf\blue %hypercomplex} if it is equipped with a triple %complex structures $I,J,K$ satisfying quaternionic %relations. By Obata's theorem, there exists a %unique torsion-free connection preserving $I,J,K$ %on any hypercomplex manifold; this connection %is called {\bf\blue the Obata connection}. Since the %Levi-Civita connection on $(M,I,J,K,g)$ satisfies %this condition, it coincides with the Obata %connection for $(M,I,J,K)$ and for $(M,I',J',K')$ %(the latter is true because the corresponding %hypercomplex manifolds are equivalent). However, %$g$ and $g'$ are invariant with respect to the %holonomy of Levi-Civita connection, which is equal %$Sp(n)$. {\bf \purple The space of $Sp(n)$-invariant %symmetric tensors is 1-dimensional, hence %$g$ and $g'$ are proportional.} %\endproof \newpage {\bf \blue Teichm\"uller space of hyperk\"ahler structures } \definition Consider the space $\Hyp$ of all hyperk\"ahler structures $(I,J,K,g)$, let $\Teich_h:= \Hyp/\Diff_0$ be the corresponding Teichm\"uller space, called {\bf \blue Teichm\"uller space of hyperk\"ahler structures}. %\remark A quotient %$\Hyp/SU(2)$ {\bf \purple is naturally identified with the set %of equivalence classes of hyperk\"ahler structures.} %\remark %As shown above, {\bf for manifolds %with maximal holonomy the quotient $\Hyp_m:=\Hyp/SU(2)$ is %also identified with the space of all hyperk\"ahler %metrics of fixed volume, say, volume 1.} % %\definition %Define {\bf\blue the Teichm\"uller space of %hyperk\"ahler structures} as the quotient $\Hyp_m/\Diff_0$, %where $\Diff_0$ is the connected component of the group %of diffeomorphisms $\Diff$, and {\bf\blue the moduli %of hyperk\"ahler structures} as $\Hyp_m/\Diff$. % %\remark %For most geometric structures, the Teichm\"uller spaces %and especially the moduli spaces are non-Hausdorff. %However, for manifolds of maximal holonomy the moduli space of %hyperk\"ahler structures is Hausdorff. This is because %$\Hyp_m$ is identified with the space of all hyperk\"ahler %metrics of fixed volume. However, there is a metric on the moduli %space of all metrics, known as {\bf \blue Gromov-Hausdorff %metric,} and a metric space is necessarily Hausdorff. \definition Consider the space $\Perspace_h$ of all triples $x,y,z\in H^2(M,\R)$ satisfying $x^2 = y^2=z^2 >0$. Let $\Per:\; \Teich_h \arrow\Perspace_h$ the map associating to a hyperk\"ahler structure $(M,I,J,K,g)$ the triple $\omega_I, \omega_J,\omega_K$. This map called {\bf\blue the period map for the Teichm\"uller space of hyperk\"ahler structures}, and $\Perspace_h$ {\bf\blue the period space of hyperk\"ahler structures}. \theorem (E. Amerik, V.) Let $M$ be a hyperk\"ahler manifold of maximal holonomy, and $\Per:\; \Teich_h \arrow\Perspace_h$ the period map for the Teichm\"uller space of hyperk\"ahler structures. Then {\bf \red $\Per$ is an open embedding for each connected component.} Moreover, {\bf \red its image is the set of all spaces $W\in \Perspace_h$ such that the orthogonal complement $W^\bot$ contains no MBM classes.} \\ \ \ \\ {\bf \green Ingredients of its proof:} Follows from Calabi-Yau theorem, global Torelli theorem for complex structures of hyperk\"ahler type, and the description of the K\"ahler cone in terms of the MBM classes. Main idea: {\bf \purple bijective correspondence between hyperk\"ahler structures and pairs $(I, [\omega])$, where $I$ is a complex structure of hyperk\"ahler type, and $[\omega]$ a K\"ahler class on $(M,I)$.} \endproof \newpage {\bf \blue Torelli theorem for symplectic structures} \theorem Let $M$ be a maximal holonomy hyperk\"ahler manifold. Then the period map $\Per:\; \Teich_s \arrow H^2(M,\R)$ {\bf \red is an open embedding on the set of all standard symplectic structures}, and {\bf \red its image is the set of all cohomology classes $v$ such that $q(v,v) > 0$.} {\bf \green Proof. Step 1:} Let $P:\; \Teich_h \arrow \Teich_s$ be the forgetful map putting $\omega_I, \omega_J, \omega_K$ to $\omega_I$. {\bf \purple Calabi-Yau implies that $P$ is surjective.} Indeed, any K\"ahler form can be deformed to a Ricci-flat K\"ahler form in the same cohomology class. {\bf \green Step 2:} From Torelli theorem for hyperk\"ahler structures it follows that {\bf \purple the fiber $P^{-1}(\omega)$ of $P$ is the space of pairs $x,y\in H^2(M)$ satisfying $x^2=y^2=\omega^2$, such that the space $\langle \omega, x, y\rangle^\bot$ contains no MBM classes.} {\bf \green Step 3:} Since the fibers of $P$ are complements to subsets of codimension 2, they are connected. By Moser's theorem, for each $(M,\omega_I, \omega_J, \omega_K) \in P^{-1}(\omega)$ {\bf \purple the symplectic forms $\omega_I$ are diffeomorphic. } \newpage {\bf \blue Torelli theorem for symplectic structures (2)} \theorem Let $M$ be a maximal holonomy hyperk\"ahler manifold. Then the period map $\Per:\; \Teich_s \arrow H^2(M,\R)$ {\bf \red is an open embedding on the set of all standard symplectic structures}, and {\bf \red its image is the set of all cohomology classes $v$ such that $q(v,v) > 0$.} {\bf \green Step 4:} Consider the diagram \[ \begin{CD} \Teich_h @>P>> \Teich_s \\ @VV{\Per_h}V @V{\Per_s}VV\\ {\small \begin{array}{c} \{x,y,z\in H^2(M)|x^2=y^2=z^2>0, \\ \text{$\langle x, y, z\rangle^\bot$ contains no MBM classes} \} \end{array}} @>{P'}>> \{x \in H^2(M)|x^2>0\} \\ \end{CD} \] The map $\Per_h$ is an isomorphism as shown, and the fibers of $P$ are identified with fibers of $P'$ as follows from Moser's theorem and Step 3. Therefore, $\Per_s$ is injective. The rest of the arrows are surjective as shown, {\bf \purple hence $\Per_s$ is also surjective.} \endproof \newpage {\bf \blue Ergodicity of mapping class group action} \theorem (V., 2009)\\ Let $M$ be a maximal holonomy hyperk\"ahler manifold. {\bf \purple Then the image of the mapping class group $\Gamma$ in $O(H^2(M,\Z))$ has finite index.} \corollary {\bf \red $\Gamma$ acts on $\Teich_s$ with dense orbits.} {\bf \green Proof:} We use a theorem of Calvin Moore: \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 semisimple Lie subgroup. {\bf \red Then the left action of $\Gamma$ on $G/H$ is ergodic.} Applying this theorem to $\Gamma$ inside $G=SO(H^2(M,\R), q)$ and $H$ the stabilizer of $\omega\in H^2(M,\R)$, we obtain that the action of $\Gamma$ on $\Teich_s\subset H^2(M,\R)$ {\bf \purple is ergodic, hence has dense orbits.} \endproof \question The Teichm\"uller space of standard symplectic structures on K3 is Hausdorff, as shown above. {\bf \red Are there any non-Hausdorff non-standard symplectic structures in the same connected component of $\Teich_s$?} \end{document}