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\begin{document}
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\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}



