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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 Ergodic complex structures} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[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 Ergodic complex structures} \def\Teich{\operatorname{Teich}} \newcommand{\Comp}{\operatorname{Comp}} \newcommand{\St}{\operatorname{St}} \definition Let $M$ be a smooth manifold. {\bf \blue A complex structure} on $M$ is an endomorphism $I\in \End TM$, $I^2=-\Id_{TM}$ such that the eigenspace bundles of $I$ are {\bf \blue involutive}, that is, satisfy $[T^{1,0}M, T^{1,0}M]\subset T^{1,0}M$. \remark Let $\Comp$ be the space of such tensors equipped with a topology of convergence of all derivatives. {\bf \purple It is a Fr\'echet manifold}. \definition The diffeomorphism group $\Diff$ is a Fr\'echet Lie group acting on $\Comp$ in a natural way. A complex structure is called {\bf \blue ergodic} if its $\Diff$-orbit is dense in $\Comp$. \remark The ``moduli space'' of complex structures (if it exists) is identified with $\Comp/\Diff$; {\bf \purple existence of ergodic complex structures guarantees that the moduli space does not exist} (all points are non-separable). \theorem Let $M$ be a compact torus, $\dim_\C M \geq 2$, or a simple hyperk\"ahler manifold. {\bf \red A complex structure on $M$ is ergodic if and only if $\Pic(M)$ is not of maximal rank.} \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 $Gr_{+,+}(H^2(M,\R))=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$,} which is a Grassmannian of positive oriented 2-planes in $H^2(M,\R)$. {\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 Birational Teichm\"uller moduli space} \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 (Huybrechts) Two points $I,I'\in \Teich$ {\bf \purple are non-separable if and only if there exists a bimeromorphism $(M,I)\arrow (M,I')$ which is non-singular in codimension 2.} \definition The space $\Teich_b:= \Teich/\sim$ is called {\bf \blue the birational Teichm\"uller space} of $M$. \newcommand{\Per}{\operatorname{\sf Per}} \newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}} \theorem {\bf \red The period map $\Teich_b\stackrel\Per \arrow \Perspace$ is an isomorphism,} for each connected component of $\Teich_b$. \theorem Let $(M,I)$ be a hyperk\"ahler manifold, and $W$ a connected component of its birational moduli space. {\bf \red Then the set of isomorphism classes of complex holomorphically symplectic structures is identified with ${\Perspace}/\Gamma$, where ${\Perspace}=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$ and $\Gamma$ is a finite index subgroup in $O(H^2(M, \Z), q)$}, called {\bf \blue the monodromy group}. \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 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 Proof. Step 2:} Choose a countable base $\{U_i\}$ of topology on $M$. Then the set of points with 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. \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 with dense orbits. {\bf \red Then $Z:=\Perspace\backslash \Perspace_e$ has measure 0.} {\bf \green Proof:} Let $G=SO(b_2-3,3)$, $H=SO(2) \times SO(b_2-3,1)$. {\bf \purple Then $\Gamma_I$-action on $G/H=\Perspace$ is ergodic,} by Moore's theorem. Therefore, all orbits outside of a measure 0 set are dense. \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). \remark {\bf \purple Arithmetic lattices in simple Lie groups have finite covolume} (Borel, Harish-Chandra). \theorem Let $H\subset G$ be a Lie subroup generated by unipotents, and $\Gamma\subset G$ an arithmetic lattice. Then {\bf \red a closure of any $H$-orbit in $G/\Gamma$ is an orbit of a closed, connected subgroup $S\subset G$, such that $S\cap \Gamma\subset S$ is a lattice.} \remark A closure of $H\cdot x$ in $G/\Gamma$ and a closure of $x\cdot \Gamma$ in $H\backslash G$ are related as follows. Let $\pi_1:\; G \arrow H\backslash G$, $\pi_2:\; G \arrow G/\Gamma$ be the projections. Then the connected component of $\pi_2^{-1}(\overline {H\cdot x}_{G/\Gamma})$ is the connected component of $\pi_1^{-1}(\overline {x\cdot \Gamma}_{H\backslash G}).$ \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 fixing a given positive 2-dimensional plane, $H\cong SO^+(1,k)\times SO(2)$, 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 $\Gamma\cdot J \cdot S$, for some Lie subgroup $S \supset H$.} Moreover, $S=H$ if and only if the orbit $\Gamma\cdot J$ is closed. \newpage {\bf \blue Characterization of ergodic complex structures} \claim Let $G=SO^+(3,k)$, and $H\cong SO^+(1,k)\times SO(2)\subset G$. Then {\bf \purple any closed connected Lie subgroup $S\subset G$ containing $H$ coincides with $G$ or with $H$.} \corollary Let $J\in \Perspace=G/H$. Then {\bf \red either $J$ is ergodic, or its $\Gamma$-orbit is closed in $\Perspace$}. \remark By Ratner's theorem, in the latter case the $H$-orbit of $J$ has finite volume in $G/\Gamma$. Therefore, {\bf \purple its intersection with $\Gamma$ is a lattice in $H$.} This brings \corollary 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 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 $W$ is rational.} \remark This can be used to show that {\bf \purple any hyperk\"ahler manifold is Kobayashi non-hyperbolic.} \newpage {\bf \blue Kobayashi pseudometric} \remark The results further on are from a joint paper arXiv:1308.5667 by Ljudmila Kamenova, Steven Lu, Misha Verbitsky. \definition {\bf \blue Pseudometric} on $M$ is a function $d:\; M \times M \arrow \R^{\geq 0}$ which is symmetric: $d(x,y)=d(y,x)$ and satisfies the triangle inequality $d(x,y)+d(y,z) \geq d(x,z)$. \remark Let ${\goth D}$ be a set of pseudometrics. {\bf \purple Then $d_{\max}(x,y):= \sup_{d\in {\goth D}}d(x,y)$ is also a pseudometric.} \definition The {\bf \blue Kobayashi pseudometric} on a complex manifold $M$ is $d_{\max}$ for the set ${\goth D}$ of all pseudometrics such that any holomorphic map from the Poincar\'e disk to $M$ is distance-non-increasing. \theorem Let $\pi:\; {\cal M} \arrow X$ be a smooth holomorphic family, which is trivialized as a smooth manifold: ${\cal M}=M \times X$, and $d_x$ the Kobayashi metric on $\pi^{-1}(x)$. {\bf \red Then $d_x(m,m')$ is upper continuous on $x$.} \endproof \corollary Denote the diameter of the Kobayashi pseudometric by $\diam(d_x):= \sup_{m,m'}d_x(m,m')$. {\bf \purple Then the Kobayashi diameter of a fiber of $\pi$ is an upper continuous function: $\diam:\; X \arrow \R^{\geq 0}$.} \newpage {\bf \blue Vanishing of Kobayashi pseudometric} \theorem Let $(M,I)$ be a complex manifold with vanishing Kobayashi pseudometric. Then {\bf \red the Kobayashi pseudometric vanishes for all ergodic complex structures in the same deformation class.} {\bf \green Proof:} Let $\diam:\; \Comp \arrow \R^{\geq 0}$ map a complex structure $J$ to the diameter of the Kobayashi pseudometric on $(M,J)$. Let $J$ be an ergodic complex structure. The set of points $J'=\nu(J)\in \Comp$ such that $(M,J')$ is biholomorphic to $(M,J)$ is dense, because $J$ is ergodic. By upper semi-continuity, $0=\diam(I) \geq \inf_{J'=\nu(J)} \diam (J)$. \endproof \example Let $M$ be a projective K3 surface. Then the Kobayashi metric on $M$ vanishes. {\bf \purple Since all non-projective K3 are ergodic,} the Kobayashi metric vanishes on non-projective K3 surfaces as well. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \theorem Let $M$ be a compact simple hyperk\"ahler manifold. Assume that a deformation of $M$ admits a holomorphic Lagrangian fibration and the Picard rank of $M$ is not maximal. {\bf \red Then the Kobayashi pseudometric on $M$ vanishes.} \theorem Let $M$ be a Hilbert scheme of K3. {\bf \red Then the Kobayashi pseudometric on $M$ vanishes.} \newpage {\bf \blue Kobayashi hyperbolic manifolds} \definition {\bf \blue An entire curve} is a non-constant map $\C \arrow M$. \definition A compact complex manifold $M$ is called {\bf \blue Kobayashi hyperbolic} if there exist no entire curves $\C \arrow M$. \theorem {\bf \blue (Brody, 1975)}\\ Let $I_i$ be a sequence of complex structures on $M$ which are not hyperbolic, and $I$ its limit. Then $(M,I)$ is also not hyperbolic. \theorem {\bf \red All hyperk\"ahler manifolds are non-hyperbolic.} \remark {\bf \purple This theorem would follow if we produce an ergodic complex structure which is non-hyperbolic.} Indeed, a closure of its orbit is the whole $\Teich$, and a limit of non-hyperbolic complex structures is non-hyperbolic. \newpage {\bf \blue Twistor spaces and hyperk\"ahler geometry} \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 $I,J,K$. \definition {\bf \color{blue} Induced complex structures} on a hyperk\"ahler manifold are complex structures of form $S^2 \cong \{ L:= aI + bJ +c K, \ \ \ a^2+b^2+c^2=1.\}$ \definition A {\bf\blue twistor space} $\Tw(M)$ of a hyperk\"ahler manifold is {\bf \purple a complex manifold obtained by gluing these complex structures into a holomorphic family over $\C P^1$.} More formally: Let $\Tw(M) := M \times S^2$. Consider the complex structure $I_m:T_mM \to T_mM$ on $M$ induced by $J \in S^2 \subset {\Bbb H}$. Let $I_J$ denote the complex structure on $S^2 = \C P^1$. The operator $I_{\Tw} = I_m \oplus I_J:T_x\Tw(M) \to T_x\Tw(M)$ satisfies $I_{\Tw} ^2 = -\Id$. {\bf \purple It defines an almost complex structure on $\Tw(M)$.} This almost complex structure is known to be integrable (Obata). \newpage {\bf \blue Entire curves in twistor fibers} \theorem {\bf \blue (F. Campana, 1992)}\\ Let $M$ be a hyperk\"ahler manifold, and $\Tw(M)\stackrel \pi \arrow \C P^1$ its twistor projection. {\bf \red Then there exists an entire curve in some fiber of $\pi$.} \claim {\bf \purple There exists a twistor family which has only ergodic fibers. } {\bf\green Proof:} There are only countably many complex structures which are not ergodic. \endproof \theorem {\bf \red All hyperk\"ahler manifolds are non-hyperbolic.} {\bf\green Proof:} Let $\Tw(M)\arrow \C P^1$ be a twistor family with all fibers ergodic. {\bf \purple By Campana's theorem, one of these fibers, denoted $(M, I)$, is non-hyperbolic.} Since any complex structure $I'\in \Teich$ lies in the closure of $\Diff(M)\cdot I$, all complex structures $I'\in \Teich$ are non-hyperbolic. \endproof \end{document}