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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 Eigenvalues of automorphisms of hyperk\"ahler manifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, LAG seminar, 22.10.2021 } \end{center} \newpage {\bf \blue Eigenvalues of an automorphism of a hyperk\"ahler manifold} \theorem {\bf \blue (Bogomolov, Kamenova, Lu, V.)}\\ Let $(M,I)$ be a hyperk\"ahler manifold, and $f$ an automorphism of $M$. Assume that $f$ acts on $H^{2}(M)$ with an eigenvalue $\alpha>0$. {\bf \red Then all eigenvalues of $\gamma$ have absolute value which is a power of $u:=\alpha^{1/2}$.} Moreover, {\bf \red the maximal of these eigenvalues on even cohomology $H^{2d}(M)$ is equal to $\alpha^{d}$ (with eigenspace of dimension 1), and on odd cohomology $H^{2d+1}(M)$ it is strictly less than $\alpha^{\frac{2d+1}{2}}$.} Finally, let $H_k^p(M)\subset H^p(M)$ be the direct sum of all eigenspaces associated with eigenvalues $\alpha$ satisfying $|\alpha|=u^k$. {\bf \red Then $dim H_k^p(M)= h^{k, p-k}(M)$.} \corollary \[ \lim_{n\arrow \infty} \frac {\log \Tr (f^n)\restrict{H^*(M)}}n=\alpha. \] In particular, {\bf \red the number of $k$-periodic points grows as $\alpha^{nk}$.} \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 $I,J,K$). \remark {\bf \purple Hyperk\"ahler manifolds are holomorphically symplectic.} Indeed, $\Omega:=\omega_J+\1\omega_K$ is a holomorphic symplectic form on $(M,I)$. \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. {\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.} {\bf\green DEFINITION:} For the rest of this talk, {\bf \blue 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}, or {\bf\blue maximal holonomy}, or {\bf \blue IHS} 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}. \theorem {\bf \blue (``Bochner's vanishing'')}\\ Let $M$ be a maximal holonomy hyperk\"ahler manifold. {\bf \red Then $H^{p,0}=0$ for $p$ odd, and $H^{p,0}=\C$ for $p$ even.} \newpage {\bf \blue The Bogomolov-Beauville-Fujiki form} {\bf \green THEOREM:} {\bf \blue (Fujiki).} Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is hyperk\"ahler. {\bf \red Then $\int_M \eta^{2n}=cq(\eta,\eta)^n$, for some primitive integer quadratic form $q$ on $H^2(M,\Z)$, and $c>0$ a rational 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}{2n}\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 $(3,b_2-3)$.} 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 K. Oguiso: Dynamical degree of an automorphism of a hyperk\"ahler manifold} \theorem {\bf \blue (K. Oguiso)} Let $f:\; M \arrow M$ be an automorphism of a hyperk\"ahler manifold with a real eigenvalue $\alpha>1 $ on $H^2(M)$. {\bf \red Then $h_{2d}(f)\geq \alpha^d$ for all $d\leq \dim_{\Bbb H}(M)$.} \proof {\bf \purple $H^{2d}(M)$ contains the symmetric tensor product $\Sym^d(H^2(M))$.} \endproof {\bf \green Problem:} Not precise enough: {\bf \purple we don't get estimations of number of periodic points,} because we have no control over other eigenvalues. \newpage {\bf \blue Classification of automorphisms of hyperbolic space} \remark The group $O(m,n), m, n>0$ has 4 connected components. We denote the connected component of 1 by $SO^+(m,n)$. We call a vector $v$ {\bf \blue positive} if its square is positive. \definition Let $V$ be a vector space with quadratic form $q$ of signature $(1,n)$, $\Pos(V)=\{x\in V\ \ |\ \ q(x,x)>0\}$ its {\bf \blue positive cone}, and ${\Bbb P}^+ V$ projectivization of $\Pos(V)$. Denote by $g$ any $SO(V)$-invariant Riemannian structure on ${\Bbb P}^+ V$. Then $({\Bbb P}^+ V, g)$ is called {\bf \blue hyperbolic space}, and the group $SO^+(V)$ {\bf \blue the group of oriented hyperbolic isometries}. {\bf \green Theorem-definition:} Let $n>0$, and $\alpha \in SO^+(1,n)$ is an isometry acting on $V$. Then one and only one of these three cases occurs\\ \phantom{XXl} {\bf \red (i)} $\alpha$ has an eigenvector $x$ with $q(x,x)>0$ ($\alpha$ is {\bf \blue ``elliptic isometry'')} \\ \phantom{XX} {\bf \red (ii)} $\alpha$ has an eigenvector $x$ with $q(x,x)=0$ and a real eigenvalue $\lambda_x$ satisfying $|\lambda_x|>1$ ($\alpha$ is {\bf \blue ``hyperbolic isometry'')} \\ \phantom{XX} {\bf \red (iii)} $\alpha$ has a unique eigenvector $x$ with $q(x,x)=0$ ($\alpha$ is {\bf \blue ``parabolic isometry'')}. \remark {\bf \purple All eigenvalues of elliptic and parabolic isometries have absolute value 1. Hyperbolic and elliptic isometries are semisimple} (that is, diagonalizable). \newpage {\bf \blue Automorphisms of hyperkahler manifolds} \remark Serge Cantat argues for a change of terminology to use {\bf \blue ``loxodromic''} instead of ``hyperbolic'', and using ``hyperbolic'' for automorphisms which act trivially on a codimension 2 hyperspace. \definition An automorphism of a hyperk\"ahler manifold $(M,I)$ is called {\bf \blue elliptic (parabolic, hyperbolic)} if it is elliptic (parabolic, hyperbolic) on $H^{1,1}_I(M,\R)$. \theorem {\bf \blue (E. Amerik, V.)}\\ {\bf \red Let $M$ be a hyperk\"ahler manifold, with $b_2(M)\geq 5$. Then $M$ has a deformation admitting a hyperbolic automorphism.} {\bf \green THEOREM 1:} {\bf \blue (Bogomolov, Kamenova, Lu, V.)}\\ Let $M$ be a hyperk\"ahler manifold, and $\gamma\in \Aut(H^*(M))$ an automorphism preserving the Hodge decomposition and acting on $H^{1,1}(M)$ hyperbolically. Denote by $\alpha$ the eigenvalue of $\gamma$ on $H^2(M,\R)$ with $|\alpha|>1$. Replacing $\gamma$ by $\gamma^2$ if necessary, we may assume that $\alpha>1$. {\bf \red Then all eigenvalues of $\gamma$ have absolute value which is a power of $\alpha^{1/2}$.} Moreover, {\bf \red the eigenspace of eigenvalue $\alpha^{k/2}$ on $H^d(M)$ is isomorphic to $H^{\frac{(d+k)}2, \frac{(d-k)}2}(M)$.} The proof of this result follows. \newpage {\bf \blue Hodge structures and automorphisms} \remark The Hodge decomposition {\bf \purple defines multiplicative action} of $U(1)$ on cohomology $H^*(M)$, with $t\in U(1)\subset \C$ acting on $H^{p,q}(M)$ as $t^{p-q}$. \theorem (V., 1995) Let $G$ be the group generated by $U(1)$-action for all complex structures on a hyperk\"ahler manifold. {\bf \red Then $G$ is isomorphic to $\Spin^+(H^2(M,\R), q)$} (with center acting trivially on even-dimensional forms and as -1 on odd-dimensional forms). Here $\Spin^+$ denotes the connected component, and $q$ is BBF form. \endproof {\bf \green Theorem 2:} {\bf \red The connected component of the group of automorphisms of $H^*(M)$ is mapped to $G$ surjectively and with compact kernel.} \proof $\Aut(H^*(M))$ is mapped to $SO(H^2(M,\R), q)$ by the restriction map; indeed, $\Aut(H^*(M))$ is compatible with the BBF form, as follows from the Fujiki theorem. It is surjective because $\Aut(H^*(M))$ contains the Hodge $U(1)$-action. Finally, the kernel $K$ of the map $\Aut(H^*(M))\arrow G$ acts trivially on $H^2(M)$, hence commutes with the Lefschetz $SL(2)$-triples. However, the Hodge decomposition is expressed through the Lefschetz $SL(2)$-action by $\goth{so}(1,4)$-theorem. Therefore, $K$ also preserves the Hodge type. Therefore, {\bf \purple $K$ preserves the Riemann-Hodge form, which is positive definite.} \endproof \newpage {\bf \blue $\Aut(H^*(M))$ is a direct product} {\bf \green Theorem 2:} {\bf \red The connected component of the group of automorphisms of $H^*(M)$ is mapped to $G$ surjectively and with compact kernel.} \remark By Theorem 2, the group $\Aut(H^*(M))$ is a semidirect product, $\Aut(H^*(M))= G \ltimes K$. However, elements of $K$ commute with elements of $G$, because they commute with the Hodge decomposition. {\bf \purple This gives $\Aut(H^*(M))=K\times G$.} \corollary For each $f\in \Aut(H^*(M))$, {\bf \red there exists an element $f'\in G=\Spin^+(H^2(M,\R), q)$ acting on $H^*(M)$ with eigenvalues of the same absolute value.} \proof Let $f=f' k$, where $f'\in G$, $k\in K$. Since $k$ belongs to a compact group, all its eigenvalues have absolute value 1; since $f'$ and $k$ commute, eigenvalues of $f'=fk^{-1}$ are products of eigenvalues of $f$ and eigenvalues of $k$. \endproof \corollary We obtain that {\bf \purple it suffices to prove Theorem 1 assuming that $\gamma\in G$.} \newpage {\bf \blue Eigenvalues of hyperbolic automorphisms} \lemma Let $\gamma \in SO(V^{1,n})$ be a hyperbolic automorphism of a vector space of signature $(1,n)$. {\bf \red Then there exists $\gamma'\in SO(V^{1,n})$ with all eigenvalues equal 2 except 2 of them, commuting with $\gamma$ and with $\gamma'\gamma^{-1}$ elliptic.} \proof Let $\alpha, \alpha^{-1}$ be the eigenvalues of $\gamma$ with absolute value $\neq 1$, and $X\subset V^{1,n}$ the corresponding 2-dimensional subspace. Then $X^\bot\subset V^{1,n}$ is a negative definite subspace preserved by $\gamma$. Let $\gamma'$ act as $\gamma$ on $X$ and as identity on $X^\bot$. Then $\gamma'\gamma^{-1}$ acts as isometry on $X^\bot$ and trivially on $X$, hence it has a positive eigenvector, and all its eigenvalues have absolute value 1. \endproof \remark Since eigenvalues of $\gamma$ and $\gamma'$ on $H^*(M)$ have the same absolute values, {\bf \purple it suffices to prove Theorem 1 for $\gamma$ equal to identity on a codimension 2 subspace.} \newpage {\bf \blue Eigenvalues of hyperbolic automorphisms} \claim Let $G$ be a group, and $V$ its representation. {\bf \purple Then the eigenvalues of $g$ and $xgx^{-1}$ are equal for all $x, g\in G$.} \endproof \proposition Let $(M,I)$ be a hyperk\"ahler manifold, and $\gamma\in G=\Spin^+(H^2(M,\R), q)$ a hyperbolic isometry which acts as identity on a codimension 2 subspace in $H^2(M)$. Consider the one-parametric subgroup $H_\gamma:=e^{\C \log \gamma}$ in the complexification $G_\C$ of $G$. Let $W$ act on $H^{p,q}(M)$ as a multiplication by a scalar $\1(p-q)$, and let $H= e^{\C W}$ be the corresponding one-parametric subgroups in $G_\C$. {\bf \red Then $H_\gamma$ and $H$ are conjugate by some $h\in G_\C$.} \proof Both $H$ and $H_\gamma$ act on $H^2(M, C)$ with 2-dimensional eigenspaces $X$ with eigenvalues $\lambda, \lambda^{-1}$ and as identity on $X^\bot$. However, all such $X$ are conjugate by some $h\in G_\C$. \endproof \corollary {\bf \red The eigenvalue decomposition for $\gamma$ acting on $H^*(M)$ is conjugate to the Hodge decomposition,} and the eigenspaces with absolute value $\alpha^{k/2}$ under this conjugation correspond to $H^{p,q}(M)$ with $p-q=k$. This finishes the proof of Theorem 1. \newpage {\bf \blue Topological entropy} \definition Let $K$ be a metric space. A subset $S\subset K$ is called {\bf \blue $\epsilon$-separated} if for all $x\neq y$ in $S$, $d(x,y)\geq \epsilon$. Denote my $N(K, \epsilon)$ the cardinality of a maximal $\epsilon$-separated subset of $S\subset K$. \definition Let $(M,d)$ be a metric space, and $f:\; M \arrow M$ a self-map. Denote by $d_n$ the metric $d_n(x,y)= \max_{k=0}^{n-1} d(f^k(x), f^k(y))$. The {\bf \blue topological entropy} of $f$ is the number \[ h(f):= \lim_{\epsilon\arrow 0} \limsup_{n \arrow \infty} \frac{\log N(M, d_n,\epsilon)}n. \] \remark Topological entropy counts the exponential growth of the number of $\epsilon$-separated orbits. {\bf \green Exercise:} Assume that $M$ is a compact metric space. {\bf \blue Prove that this number is independent from the choice of $d$.} \newpage {\bf \blue Gromov's theorem} \definition Let $T$ be an automorphism of a manifold $M$, and consider the corresponding action on $H^d(M,\R)$. {\bf \blue The $d$-th dynamical degree} is logarithm of the maximal absolute value of its eigenvalues. \theorem {\bf \blue (Gromov)}\\ Let $M$ be compact, K\"ahler, $f:\; M \arrow M$ its automorphism, $h_d(f)$ the $d$-th dynamical degree, and $h(f)$ topological entropy. {\bf \red Then $f(h)= \max h_d(f)$.} \proof M. Gromov, On the entropy of holomorphic maps, \url{http://www.ihes.fr/~gromov/PDF/10%5B24%5D.pdf}, 1977. S. Friedland, Entropy of algebraic maps, Proceedings of the Conference in Honor of Jean-Pierre Kahane, J. Fourier Anal. Appl. (1995), Special Issue, 215-228. \url{http://homepages.math.uic.edu/~friedlan/Dynalg.pdf} \endproof \end{document}