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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 Subtwistor metric \\[4mm] on the moduli of hyperk\"ahler manifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf International conference\\[2mm] ``Geometry and analysis on metric structrures''\\[2mm] Geometric control theory laboratory \\[2mm] December 4-7, 2013. } \end{center} \newpage {\bf \blue Sub-Riemannian structures} \definition Let $M$ be a Riemannian manifold and $B\subset TM$ a sub-bundle. A {\bf\blue horizontal path} is a piecewise smooth path $\gamma:\; [b,a] \arrow M$ tangent to $B$ everywhere. A {\bf \blue sub-Riemannian}, or {\bf\blue Carno-Carath\'eodory} metric $M$ is \[ d_B(x,y):=\inf_{\gamma \text{\ horizontal}}L(\gamma):\] the infimum of the length $L(\gamma)$ for all horizontal paths connecting $x$ to $y$. \theorem {\bf \blue (Chow-Rashevskii theorem; 1938, 1939)}\\ Consider {\bf \blue the Frobenius form} $\Phi:\; \Lambda^2 B \arrow TM/B$ mapping vector fields $X, Y \in B$ to an image of $[X,Y]$ modulo $B$. Suppose that $\Phi$ is surjective. {\bf \red Then any two points can be connected by a horizontal path,} and the sub-Riemannian metric $d_B$ is finite. \newpage {\bf \blue Properties of sub-Riemannian metrics} Let $(M,B,g)$ be a sub-Riemannian manifold. \claim {\bf \red Every two points $x, y\in M$ are connected by a smooth, horizontal path $\gamma$.} Moreover, $d_B(x,y)=\inf_{\gamma\text{\ horizontal, smooth}}L(\gamma)$: the sub-Riemannian distance can be taken as infimum of the length for smooth horizontal paths connecting $x$ to $y$. \theorem {\bf \blue (ball-box theorem)} An $\epsilon$-ball in $d_B$ is {\bf \red asymptotically equivalent to a product of $\epsilon$-ball in direction of $B$ and $\epsilon^2$-ball in orthogonal direction.} \corollary The sub-Riemannian metric {\bf \red induces the standard topology on $M$.} \corollary The Hausdorff dimension of a sub-Riemannian manifold is integer, {\bf \purple and strictly bigger than $\dim M$.} \newpage \newcommand{\Gr}{\operatorname{Gr}} {\bf \blue Subtwistor metric} Throughout this talk, $H$ is a real vector space with non-degenerate scalar product of signature $(3,b-3)$, and $\Gr_{++}(H)$ -- Grassmannian of 2-dimensional positive oriented planes in $H$. The space $\Gr_{++}(H)$ is in fact a complex manifold, and it is called {\bf \blue the period space of weight 2 Hodge structures on $H$.} \definition Let $W\subset V$ be a positive 3-dimensional subspace, and $S_W=\Gr_{++}(W)\subset \Gr_{++}(H)$ a 2-dimensional sphere consisting all 2-dimensional oriented planes in $W$. Then $S_w$ is called {\bf \blue a twistor line}. \claim {\bf \purple Each pair $x, y \in \Gr_{++}(H)$ can be connected by an intersecting chain $S_{W_1}, S_{W_2}, ..., S_{W_n}$ of twistor lines;} moreover, $n\leq 3$. \definition {\bf\blue A twistor path} on $\Gr_{++}(H)$ is a piecewise smooth path $\gamma:\; [a,b]\arrow \Gr_{++}(H)$ with each smooth component sitting on a twistor line. \definition Fix a Euclidean structure on $H$, and let $g$ be the corresponding Riemannian metric on $\Gr_{++}(H)$. {\bf \blue Subtwistor metric} $d_{tw}(x,y)$ on $\Gr_{++}(H)$ is defined as $d_{tw}(x,y):=\inf_\gamma L(\gamma)$ where $L(\gamma)$ is a length of the path $\gamma$ taken with respect to $g$, and infimum is taken over all subtwistor paths connecting $x$ to $y$. \newpage {\bf \blue Properties of subtwistor metric} \question Can we connect any pair $x, y \in \Gr_{++}(H)$ with a smooth path tangent to twistor line at each point? Would the infimum of its length give the same metric? \question What about the ball-box theorem? What is a shape of a small $\epsilon$-ball in $d_{tw}$? \question What us the Hausdorff dimension $(\Gr_{++}(H), d_{tw})$? \question The definition I gave obviously can be generalized. What is an appropriate generality? \theorem {\bf \red The subtwistor metric $d_{tw}$ induces the standard topology on $\Gr_{++}(H)$.} \remark Its proof is highly non-trivial; uses a solution of Hilbert's fifth problem on continuous groups. \newpage {\bf \blue Hilbert's 5 problem} \question {\bf \blue (Hilbert, 1900)}\\ {\bf \purple ``How is Lie's concept of continuous groups of transformations of manifolds approachable in our investigation without the assumption of differentiability?''} {\bf \red Answered affirmative} by von Neumann, Gleason, Montgomery-Zippin. \theorem Let $M$ be a topological manifold equipped with a continuous group structure. {\bf \red Then $M$ admits a smooth structure compatible with the group action.} I will state the Gleason-Palais refinement of this theorem. \newpage {\bf \blue Gleason-Palais theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $M$ be a topological space. We say that $M$ {\bf\blue has Lebesgue covering dimension $\leq n$} if every open covering of $M$ has a refinement $\{U_i\}$ such that each point of $M$ belongs to at most $n+1$ element of $\{U_i\}$. A {\bf\blue Lebesgue covering dimension} of $M$ (denoted by $\dim M$) is an infimum of all such $n$. \example If $M$ is an $n$-manifold, $\dim M=n$. \claim If $X\subset M$ is a subset of a topological space, with induced topology, one has $\dim X \leq \dim M$. \theorem {\bf \blue (Gleason-Palais)}\\ Let $G$ be a topological group, which is locally path connected, and has $\dim K < \infty$ for each compact, metrizable subset $K\subset G$. {\bf \red Then $G$ is homeomorphic to a Lie group. } \newpage {\bf \blue Subtwistor norm on a Lie group} \remark We define a norm on the group $SO(H)$ compatible with the subtwistor metric on $\Gr_{++}(H)$. \definition Let $G$ be a connected component of $SO(H)$ acting on $\Gr_{++}(H)$ in a susual way. We define {\bf \blue subtwistor norm} on $G$ in such a way that {\bf \purple the bijective map $(G/G_0, \|\cdot\|_{tw})\arrow (\Gr_{++}(H), d_{tw})$ is continuous}, where $G_0\subset G$ is a stabilizer of a point $V\in \Gr_{++}(H)$. \definition An {\bf \blue elementary transform} is an element $h\in G$ fixing a codimension 2 subspace $V_1\subset V$ of signature $(1,n-3)$. {\bf\blue An elementary decomposition} of $h\in G$ is a decomposition $h=h_1h_2...h_n$, where $h_i$ are elementary transforms. Define the {\bf\blue subtwistor norm} on $G$ as $\|h\|_{tw}:= \inf(\|h_1\|+\|h_2\|+...+\|h_n\|)$, where the infinum is taken over all elementary decompositions $h=h_1 h_2 ... h_n$. \claim {\bf \purple The action of $(G, \|\cdot\|_{tw})$ on $(\Gr_{++}(H), d_{tw})$ is continuous}, and induces a homeomorphism $(G/G_0, \|\cdot\|_{tw})\arrow (\Gr_{++}(H), d_{tw})$. \newpage {\bf \blue Transformation groups and subtwistor metrics} \theorem {\bf \red The subtwistor metric $d_{tw}$ induces the standard topology on $\Gr_{++}(H)$.} {\bf \green Step 1:} Since $\Gr_{++}(H)\cong (G/G_0, \|\cdot\|_{tw})$, it suffices to show that the subtwistor norm defines the usual topology on $G$. {\bf \green Step 2:} Let $\|\cdot\|$ be the usual norm on $G$. Since $\|\cdot\|_{tw}\geq \|\cdot\|$, {\bf \purple the identity map $(G,\|\cdot\|_{tw}) \arrow (G,\|\cdot\|)$ is continuous.} {\bf \green Step 3:} {\bf \blue (Brouwer's invariance of domain theorem):}\\ {\bf \red Let $X\stackrel f\arrow Y$ be a continuous, bijective map of Hausdorff manifolds.} Then $f$ is a homeomorphism. Apply this to the identity map $(G, \|\cdot\|_{tw})\arrow G$. To prove that it is a homeomorphism, {\bf \purple it remains to show that $(G, \|\cdot\|_{tw})$ is a manifold.} {\bf \green Step 4:} Since a bijective continuous map from a compact is a homeomorphism, the identity map $(G,\|\cdot\|_{tw}) \arrow (G,\|\cdot\|)$ is a homeomorphism on compacts. Therefore, the Lebesgue covering dimension of any compact is the same in $(G,\|\cdot\|_{tw})$ and in $(G,\|\cdot\|)$, hence finite. Path connectedness of $(G,\|\cdot\|_{tw})$ is clear from its construction. Then {\bf \purple Gleason-Palais implies that $G,\|\cdot\|_{tw}$ is a manifold.} \endproof \newpage \newcommand{\Teich}{\operatorname{Teich}} \newcommand{\Comp}{\operatorname{Comp}} \newcommand{\Per}{\operatorname{\sf Per}} \newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}} {\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.} \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 {\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 \blue 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 Holomorphically symplectic manifolds} {\bf\green DEFINITION:} {\bf \blue A holomorphically symplectic manifold } is a complex manifold equipped with non-degenerate, holomorphic 2-form. {\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.} \remark Usually, one says ``hyperk\"ahler manifold'' meaning ``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 {\bf \red admits a finite covering which is a product of a torus and several simple hyperk\"ahler manifolds.} {\bf \green THEOREM:} (Fujiki). Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is simple and hyperk\"ahler. Then $C\int_M \eta^{2n}=q(\eta,\eta)^n$, for some primitive integer quadratic form $q$ on $H^2(M,\Z)$ and $C>0$. {\bf \green Definition:} This form is called {\bf \blue Bogomolov-Beauville-Fujiki form}. {\bf \purple It is defined by this relation uniquely, up to a sign.} \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 \bf $P$ 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 ${\Bbb Per}=\Gr_{++}(H^2(M,\R), q)$ {\bf \green THEOREM:} (Bogomolov) Let $M$ be a simple hyperk\"ahler manifold, and $\Teich$ its Teichm\"uller space. {\bf \red Then the period map $P:\; \Teich \arrow {\Bbb Per}$ is locally a diffeomorphism.} \newpage {\bf \blue Global Torelli theorem} \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 $V\ni x, U\ni y$, $U \cap V\neq \emptyset$. \theorem Let $M$ be a hyperk\"ahler manifold, $\Teich$ its Teichm\"uller space, and $\Teich_b$ the quotient of $\Teich$ by $\sim$. {\bf \red Then the period map $P:\; \Teich_b \arrow {\Bbb Per}$ induces a diffeomorphism on each connected component.} \remark The period space \[ {\Bbb Per}:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \ q(l,l)=0, \ \ q(l, \bar l) >0.\} \] {\bf \purple is identified with $Gr_{++}(H^2(M,\R))=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$,} which is a Grassmannian of positive oriented 2-planes in $H^2(M,\R)$. \newpage {\bf \blue Proof of Global Torelli theorem} \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 lines}. \definition We call a subspace $R\subset H^2(M,\R)$ {\bf \blue irrational} if $R^\bot \cap H^2(M,\Q)$ is empty. \theorem Let $S\subset \Perspace$ be a twistor line corresponding to an irrational plane $\Gr_{+++}(H^2(M,\R)$. {\bf \red Then it can be lifted to $\Teich$ with each of the irrational point in its preimage. } \corollary The period map $\Teich_b\arrow \Perspace$ {\bf \red is an isometry with respect to the subtwistor metrics.} \remark Now the global Torelli follows, because (being an isometry) it is also a covering. \newpage {\bf \blue Period space as a Grassmannian of positive 2-planes} \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 $SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$,} 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 \end{document}