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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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{\proof}{{\bf \green Proof:\ }} \newcommand{\pstep}{{\bf \green Proof. Step 1:\ }} \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Teichmuller space of geometric structures 3 \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 3} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[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 Plan:} 0. Hyperk\"ahler geometry and Teichm\"uller spaces (reminder) 1. Period space and birational Teichm\"uller space 2. Global Torelli theorem 3. Subtwistor metrics and the proof of global Torelli \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} {\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 $M$. \remark ${\Bbb Per}=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$ {\bf \green THEOREM:} {\bf \blue (Bogomolov)} Let $M$ be a simple hyperk\"ahler manifold, and $\Teich$ its Teichm\"uller space. Then {\bf \red The period map $P:\; \Teich \arrow {\Bbb Per}$ is etale.}\\ \remark Bogomolov's theorem implies that {\bf \purple $\Teich$ is smooth.} It is {\bf \red non-Hausdorff} even in the simplest examples. \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. 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 $M$. {\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.} \theorem The Teichm\"uller space of a hyperk\"ahler manifold {\bf \red admits a Hausdorff reduction}. \newpage {\bf \blue Global Torelli theorem} \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 Proven later today.} \newpage {\bf \blue Period space as $\Gr_{++}(3, b_2-3)$} \proposition The period space \[ {\Bbb Per}:= \{l\in {\Bbb P}H^2(M, \C)\ | \ q(l,l)=0, q(l, \bar l) >0\}\] {\bf \red is identified with $\frac{SO(3,b_2-3)}{SO(2) \times SO(1,b_2-3)}$,} which is a Grassmannian of positive oriented 2-planes in $H^2(M,\R)$. {\bf \green Proof. Step 1:} Given $l\in {\Bbb P}H^2(M, \C)$, {\bf \purple the space generated by $\Im l, \Re l$ is 2-dimensional,} because $q(l,l)=0, q(l, \bar l)$ implies that $l \cap H^2(M,\R)=0$. {\bf \green Step 2:} {\bf \purple This 2-dimensional plane is positive,} because $q(\Re l, \Re l) = q(l+ \bar l, l+ \bar l) = 2 q(l, \bar l)>0$. {\bf\green Step 3:} Conversely, for any 2-dimensional positive plane $V\in H^2(M,\R)$, {\bf \purple the quadric $\{l\in V \otimes_\R \C\ \ |\ \ q(l,l)=0\}$ consists of two lines;} a choice of a line is determined by orientation. \endproof \newpage {\bf \blue Period space and hyperk\"ahler lines} \definition Let $(M,I,J,K)$ be a hyperk\"ahler manifold. {\bf \blue A hyperk\"ahler 3-plane} in $H^2(M,\R)$ is a positive oriented 3-dimensional subspace $W$, generated by $\omega_I, \omega_J, \omega_K$. \remark The set of oriented 2-dimensional planes in $W$ is identified with $S^2 =\C P^1$. It is called {\bf \blue the twistor family} of a hyperk\"ahler structure. A point in the twistor family corresponds to a complex structure $aI + bJ + cK \in {\Bbb H}$, with $a^2+b^2+c^2=1$. We call the corresponding $\C P^1\subset \Teich$ {\bf \blue the twistor curves}. \remark Let $I\in \Teich$ be a complex structure, and ${\cal K}(I)$ its K\"ahler cone. The set of twistor curves passing through $I$ {\bf \red is parametrized by ${\cal K}(I)$,} by Calabi-Yau theorem. {\bf \purple The corresponding 3-dimensional subspaces are generated by $\Per(I)+\omega$, where $\omega\in {\cal K}(I)$.} \newpage {\bf \blue A K\"ahler cone for generic hyperk\"ahler manifolds} \newcommand{\NS}{\operatorname{\sf NS}} \newcommand{\Pos}{\operatorname{\sf Pos}} \definition {\bf \blue Neron-Severi lattice}, or {\bf\blue Hodge lattice} of a manifold is $\NS(M):=H^{1,1}(M)\cap H^2(M,\Z)$ \theorem Let $M$ be a hyperkaehler manifold with $\NS(M)$ of rank 0 or 1 and generated by $z$ with $z^2 > 0$. {\bf \red Then the K\"ahler cone is equal to the positive cone $\Pos(M)$,} that is, one of two components of the set \[ \{ \nu \in H^{1,1}(M) \ | \ q(\nu, \nu)> 0\}. \] {\bf \green Proof:} Nakai-Moischezon-Huybrechts-Boucksom: obstructions to K\"ahlerness of $\eta\in \Pos(M)$ are curves $C$ such that $\int_C \eta \leq 0$. For any non-zero $x, y \in \Pos(M)$, such that $[C]=x^{\dim_\C M-1},$ one has $\int_Cy= \int_M x^{\dim_\C M-1}y>0$. \endproof \newpage {\bf \blue Generic hyperk\"ahler lines} \definition Let $S\subset \Teich$ be a $\C P^1$ associated with a twistor family. It is called {\bf \blue generic} if it passes through a point $I\in \Teich$ with $\NS(M,I)=0$. \remark For a generic point $I$ in such $S$, one has $\NS(M,I)=0$. {\bf \purple This condition is equivalent to $l^\bot \cap H^2(M,\Z)=0$,} where $l\in \Perspace$ is the corresponding 2-plane. \remark A 3-plane $W\subset H^2(M,\R)$ corresponds to a generic twistor family {\bf \purple if and only if its orthogonal complement $W^\bot\subset H^2(M,\R)$ does not contain rational vectors. } \definition A hyperk\"ahler 3-plane $W\subset H^2(M,\R)$ is called {\bf \blue generic} if $W^\bot\cap H^2(M,\Z)=0$. The corresponding $\C P^1\subset \Perspace$ in the period space is called {\bf \blue a GHK line}. \newpage {\bf \blue A lifting property for GHK lines} \remark Consider a 3-plane $W= \langle \omega_I,\omega_J,\omega_K\rangle$ associated with a hyperk\"ahler structure, and let $S$ be the set of oriented 2-planes in $W$. Denote by $S_{ng}$ the set of $x\in S$ satisfying $x^\bot \cap H^2(M,\Z)\neq 0$. If $W$ is generic, then $S_{ng}$ is countable. \theorem {\bf \blue (A lifting property for GHK lines)}\\ Let $W\subset H^2(M,\R)$ be a generic 3-plane, and $S\subset \Perspace$ the corresponding GHK line. Consider the period map $P:\; \Teich \arrow \Perspace$. {\bf \red Then $P^{-1}(S)$ is a union of a countable set mapped to $S_{ng}$, and a disconnected set of rational curves bijectively mapped to $S$.} {\bf \green Proof. Step 1:} Let $x\notin S_{ng}$ We are going to prove that for all $I\in P^{-1}(x)$, {\bf \purple $y$ is contained in a connected component of $P^{-1}(S)$, bijectively mapped to $S$.} {\bf \green Step 2:} Notice that $NS(I)=x^\bot \cap H^2(M,\Z)=0$. Therefore the K\"ahler cone of $(M,I)$ {\bf \purple is one of two components of the set $\{\omega \in P(I)^\bot\ \ |\ \ q(\omega,\omega)>0\}$.} {\bf \green Step 3:} For each positive 3-plane $W\subset H^2(M,\R)$, $W= \langle \omega_I,\omega_J,\omega_K\rangle$ for some hyperk\"ahler structure $I,J,K$. {\bf \purple Then the twistor family associated with $I,J,K$ is mapped to $S$.} \endproof \newpage {\bf \blue Subtwistor metric on the Teichm\"uller space} \definition Let $g_0$ be a Riemannian metric on $\Perspace$, and $g$ its lift to $\Teich$. Define {\bf \blue the subtwistor metric} $d$ as the distance function $d(x,y)$ given by infimum of the length (in $g$) for all paths from $x$ to $y$ going through GHK curves which intersect in points $z\in \Teich$ with $\NS(z)=0$. Define the subtwistor metric on $\Perspace$ in the same way, by using paths which go through $S^2$ obtained from 3-dimensional subspaces $W\subset H^2(M,\R)$ containing no rational vectors. \theorem {\bf \red The subtwistor metric induces the standard topology} on any open subset $W\subset \Teich_b$. {\bf \green Remark:} Its proof {\bf \purple follows from Gleason-Palais-Montgomery} classification of continuous groups. \theorem {\bf \red The period map induces an isometry} from $\Teich_b$ to $\Perspace$ with respect to the subtwistor metric. \proof Immediately follows from the GHK lifting property. \endproof \corollary {\bf \red The map $\Teich_b\stackrel \Per \arrow \Perspace$ is a homeomorphism.} {\bf \green This proves the Global Torelli theorem.} \end{document}