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% CHANGELOG
% v. 0.1, Dec 11, 2015. Started the file (LK)
% v. 0.2, Dec 17. Made the changes that we discussed (LK) 
% v. 0.2.1, Dec 30. Did not change much, added a few marginpars (MV)
% -- 2016 --
% v. 0.3, April 3. Made some corrections following our conversation (LK)
% v. 0.4, April 4. Restructured some of the proofs (LK)
% v. 1.0, April 5, Couple of corrections. Need to deal with symplectic automorphisms (MV).
% v. 1.1, April 6, Added a couple of lemmas (LK)
% v. 1.2, April 9, Removed an extra one - MV
% v. 2.0, acknowledgements and reference updated - MV, April 17, 2016
% v. 2.1, more reference - MV, April 17
% v. 2.2, April 18, corrected some typos (LK)
% v. 3.0, March 2, 2017, couple of minor corrections suggested by referees (LK)

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\renewcommand{\leftmark}{{\scriptsize Algebraic non-hyperbolicity}}
\renewcommand{\rightmark}{{\small 
L. Kamenova, M. Verbitsky}}


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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\LARGE\bf
Algebraic non-hyperbolicity of hyperk\"ahler manifolds 
with Picard rank greater than one\\[4mm]
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Ljudmila Kamenova, Misha Verbitsky\footnote{Partially supported 
by RSCF grant 14-21-00053 within AG Laboratory NRU-HSE.}

\end{center}

{\small \hspace{0.10\linewidth}
\begin{minipage}[t]{0.85\linewidth}
{\bf Abstract.} 
A projective manifold is algebraically hyperbolic if the degree of any curve
is bounded from above by its genus times a constant, which is independent 
from the curve.
This is a property which follows from Kobayashi hyperbolicity.
We prove that hyperk\"ahler manifolds are not algebraically 
hyperbolic when the Picard rank is at least 3, or if
the Picard rank is 2 and the SYZ conjecture on existence
of Lagrangian fibrations is true.
We also prove that if the automorphism group of a hyperk\"ahler 
manifold is infinite then it is algebraically non-hyperbolic. 
\end{minipage}
}


\section{Introduction} 

In \cite{_Verbitsky:ergodic_} M. Verbitsky proved that all hyperk\"ahler 
manifolds are Kobayashi non-hyperbolic. It is interesting to inquire if 
projective hyperk\"ahler manifolds are also algebraically 
non-hyperbolic (\ref{_alge_hype_Definition_}). 
For a given projective manifold algebraic non-hyperbolicity 
implies Kobayashi non-hyperbolicity. We prove algebraic non-hyperbolicity 
for projective hyperk\"ahler manifolds with infinite group of 
automorphisms. 

\hfill

%\marginpar{This is true for all projective manifolds, and probably
%known already.}
\theorem
Let $M$ be a projective hyperk\"ahler manifold with infinite automorphism 
group. Then $M$ is algebraically non-hyperbolic. 

\hfill

If a projective hyperk\"ahler manifold has Picard rank at least three, 
we show that it is algebraically non-hyperbolic. For the case when the 
Picard rank equals to two we need an extra assumption in order to prove 
algebraic non-hyperbolicity. The SYZ conjecture states that a nef 
parabolic line bundle on a hyperk\"ahler manifold gives rise to a 
Lagrangian fibration (\ref{syz}). 

\hfill

\theorem
Let $M$ be a projective hyperk\"ahler manifold with Picard rank $\rho$. 
Assume that either $\rho > 2$, or $\rho =2$ and 
the SYZ conjecture holds. 
%there is no class $\eta \in H^{1,1}(M, \Z)$ with $q(\eta)=0$. 
Then $M$ is algebraically non-hyperbolic. 


\section{Basic notions}


\definition
A {\em hyperk\"ahler manifold of maximal holonomy} 
(or {\em irreducible holomorphic symplectic}) 
manifold 
$M$ is a compact complex K\"ahler manifold with $\pi_1(M)=0$ and 
$H^{2,0}(M)=\C \sigma$, where $\sigma$ is everywhere non-degenerate. 
From now on we would tacitly assume that hyperk\"ahler manifolds are
of maximal holonomy.

\hfill

Due to results of Matsushita, holomorphic maps from hyperk\"ahler 
manifolds are quite restricted. 

\hfill

\begin{theorem} (Matsushita, \cite{_Matsushita:fibred_}) \label{fibr}
Let $M$ be a hyperk\"ahler manifold and $f\colon M\rightarrow B$ a 
proper surjective morphism with a smooth base $B$. Assume that $f$ has 
connected fibers and $0 < \dim B < \dim M$. Then $f$ is Lagrangian 
and $\dim_{\C} B = n$, where $\dim_{\C} M = 2n$. 
\end{theorem}

\hfill

Following \ref{fibr}, we call the surjective morphism 
$f\colon M\rightarrow B$ a {\em Lagrangian fibration} on the 
hyperk\"ahler manifold $M$. 
A dominant map $f\colon M\dashrightarrow B$ 
is a {\em rational Lagrangian fibration} if there exists a 
birational map $\phi \colon M \dasharrow M'$ between hyperk\"ahler 
manifolds such that the composition 
$f \circ \phi^{-1} \colon M' \rightarrow B$ is 
a Lagrangian fibration. J.-M. Hwang proved that if the base $B$ of a 
hyperk\"ahler Lagrangian fibration is smooth, then $B \cong {\mathbb P}^n$ 
(see \cite{Hwang}). 

\hfill

\definition
Given a hyperk\"ahler manifold $M$, there is a non-degenerate primitive 
form $q$ on $H^2(M,\Z)$, called the {\em Beauville-Bogomolov-Fujiki form} 
(or the {\em ``BBF form''} for short), 
of signature $(3,b_2-3)$, and satisfying the {\em Fujiki relation} 
$$\int_M \alpha^{2n}=c\cdot q(\alpha)^n\qquad\text{for }\alpha \in 
H^2(M,\Z),$$ 
with $c>0$ a constant depending on the topological type of $M$. 
This form generalizes the intersection pairing on K3 surfaces. 
A detailed description of the form can be found in 
\cite{_Beauville_}, \cite{_Bogomolov:defo_} and \cite{_Fujiki:HK_}. 

\hfill

Notice that given a Lagrangian fibration $f\colon M \rightarrow {\mathbb P}^n$, 
if $h$ is the hyperplane class on $\mathbb P^n$, and $\alpha=f^*h$, 
then $\alpha$ belongs to the birational K\"ahler cone of $M$ and 
$q(\alpha)=0$. The following SYZ conjecture states that the converse 
is also true. 

\hfill

\conjecture (Tyurin, Bogomolov, Hassett-Tschinkel,
Huybrechts, Sawon) \label{syz}
If $L$ is a line bundle on a hyperk\"ahler manifold $M$ with $q(L)=0$, 
and such that $c_1(L)$ belongs to the birational K\"ahler cone of $M$, 
then $L$ defines a rational Lagrangian fibration.

\hfill

For more reference on this conjecture, please see
\cite{_Hassett_Tschinkel:SYZ_conj_}, 
\cite{_Sawon_}, \cite{_Huybrechts:lec_} and \cite{_Verbitsky:SYZ_}.
This conjecture is known for deformations of Hilbert schemes of points on 
K3 surfaces %$\text{K3}^{[n]}$ 
(Bayer--Macr\`i \cite{_Bayer_Macri_}; Markman \cite{_Markman:SYZ_}), 
and for deformations of the generalized Kummer varieties $K_n(A)$ 
(Yoshioka \cite{_Yoshioka_}).
% combined with Theorem 1.2 in 
%Matsushita's paper \cite{_Matsushita:isotropic_}).

%\hfill

%\definition \label{mbm}
%A negative class $\alpha \in H^{1,1}(M,\Z)$ (i.e., $q(\alpha)<0$) is called 
%an {\em MBM class} if for some isometry $\gamma \in SO(H^2(M, \Z))$ in the 
%monodromy group, $\gamma (\alpha)^\perp \subset H^{1,1}(M,\Z)$ contains a 
%face of the K\"ahler cone of a birational model $M'$ of $M$. 

%\hfill

%Geometrically, the MBM classes are negative integral $(1,1)$-classes 
%that are represented by minimal rational curves on deformations of $M$ 
%after identifying $H_2(M, \Q)$ with $H^2(M, \Q)$ via the BBF form 
%(Amerik-Verbitsky, \cite{_AV:MBM_}). 

%\hfill

%\definition
%The {\em Kobayashi pseudometric} on $M$ is the maximal 
%pseudometric $d_M$ such that all holomorphic maps 
%$f\colon (D,\rho) \rightarrow (M,d_M)$ are distance 
%decreasing, where $(D,\rho)$ is the unit disk with the Poincar\'e metric. 

\hfill

%A manifold $M$ is {\em Kobayashi hyperbolic} if $d_M$ is a metric, 
%otherwise it is called {\em Kobayashi non-hyperbolic}.
In \cite{_Verbitsky:ergodic_} M. Verbitsky proved that all hyperk\"ahler 
manifolds are Kobayashi non-hyperbolic. In \cite{klv} together with 
S. Lu we proved that the Kobayashi pseudometric vanishes 
identically for K3 surfaces and for hyperk\"ahler manifolds deformation 
equivalent to Lagrangian fibrations under some mild assumptions. 
In \cite{Demailly} Demailly introduced the following notion. 

\hfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition\label{_alge_hype_Definition_}
A projective manifold $M$ is {\em algebraically hyperbolic} 
if for any Hermitian metric $h$ on $M$ there exists a constant $A>0$ 
such that for any holomorphic map $\phi \colon C \rightarrow M$ 
from a curve of genus $g$ to $M$ we have that 
$2g-2 \geq A \int_C \phi^\ast \omega_h,$ where $\omega_h$ is the K\"ahler 
form of $h$. 

\hfill

%If $M$ is a smooth quasi-projective variety, there is a weaker notion, 
%{\em weakly algebraic hyperbolic}, if there is no algebraic 
%subvariety $X \subset M$ such that its normalization $\tilde X$ is 
%either $\C^\ast$ or an elliptic curve. 
In this paper all varieties we 
consider are smooth and projective. For projective varieties, 
Kobayashi hyperbolicity implies algebraic hyperbolicity (\cite{Demailly}). 
%which in turn implies weak algebraic hyperbolicity. 
Here we explore non-hyperbolic 
properties of projective hyperk\"ahler manifolds. Algebraic non-hyperbolicity
implies Kobayashi non-hyperbolicity. 

%\hfill

%\marginpar{for Misha}
%{\bf Question}: Shall we define ``algebraic Kobayashi pseudometric'', 
%a pseudometric such that it is non-degenerate if and only if $M$ is 
%algebraically hyperbolic? 


\section{Main Results}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\proposition\label{_hk_with_Lagra_non_h_Proposition_}
Let $M$ be a hyperk\"ahler manifold admitting a (rational) Lagrangian 
fibration. Then $M$ is algebraically non-hyperbolic. 

\hfill

\begin{proof}
We use the fact that the fibers of a Lagrangian fibrations are 
abelian varieties (\cite{_Matsushita:fibred_}). The isogeny self-maps 
on an abelian variety provide curves of fixed genus and arbirary large 
degrees, and therefore they are algebraically non-hyperbolic. 

\hfill

An alternative way of proving this proposition is by using the 
following result whose proof was suggested by Prof. K. Oguiso. 

\hfill

\lemma \label{ratlc}
If a hyperk\"ahler manifold $M$ admits a Lagrangian fibration, 
then there exists a rational curve on $M$. 

\hfill

Indeed, 
in \cite{HO} J.-M. Hwang and K. Oguiso give a Kodaira-type classification 
of the general singular fibers of a holomorphic Lagrangian fibration. 
All of the general singular fibers are covered by rational curves. 
The locus of singular fibers is non-empty (e.g., Proposition 4.1 in 
\cite{Hwang}), and therefore there is a rational curve on $M$. 
%\end{proof}

\hfill

According to \ref{ratlc}, $M$ contains a rational curve, and 
therefore, $M$ is algebraically non-hyperbolic. 
This finishes the proof of \ref{_hk_with_Lagra_non_h_Proposition_}.
\end{proof}

\hfill

\hfill

\lemma \label{_infinite_orbit_Lemma_}
Let $M$ be a projective hyperk\"ahler manifold with infinite automorphism 
group $\Gamma$. Consider the natural map $f: \Gamma \arrow \Aut(H^{1,1}(M))$. 
Then the elements of the K\"ahler cone have infinite orbits with respect to 
$f(\Gamma)$. 

\hfill

\begin{proof}
See the discussion in section 2 of \cite{_O:bimer_}. 
\end{proof}

\hfill

\lemma \label{_fin_im_Lemma_}
Let $M$ be a projective hyperk\"ahler manifold, and $\Gamma$ its 
automorphism  group. Consider the natural map 
$g: \Gamma \arrow \Aut(H_{tr}^2(M)) \times \Aut(H^{1,1}(M))$. Then $g(\Gamma)$ 
is finite in the first component $\Aut(H_{tr}^2(M))$. 

\hfill

\begin{proof}
This has been proven by Oguiso, see \cite{_Oguiso:Aut_}. 
The idea is that the BBF form restricted to the transcendental part 
$H_{tr}^2(M)$ is of K3-type. Then we can apply Zarhin's theorem 
(Theorem 1.1.1 in \cite{_Zar_}) to deduce that $g(\Gamma) \subset 
\Aut(H_{tr}^2(M))$ is finite. 
\end{proof}

\hfill

\theorem \label{Aut}
Let $M$ be a projective hyperk\"ahler manifold with infinite automorphism 
group. Then $M$ is algebraically non-hyperbolic. 

\hfill

\begin{proof} 
In the notations introduced above, 
for any K\"ahler class $w$ on $M$, its $f(\Gamma)$-orbit is infinite by 
\ref{_infinite_orbit_Lemma_}.
Fix a polarization $w$ on $M$ with normalization $q(w)=1$. 
For a given constant $C>0$ consider the set 
$${\cal D}_C = \{ x\in H^{1,1}(M, \Z)\ \  |\ \  q(x) \geq 0,\ \  
q(x,w) \leq C \}.$$
Notice that ${\cal D}_C$ is compact. 
Indeed, $y= x - q(x,w)w$ is orthogonal to $w$ with respect to the BBF 
form $q$. The quadratic form $q$ is of type $(1, \rho - 1)$ on 
$H^{1,1}(M, \Z)$ and since $q(w)>0$, the restriction 
$q|_{w^\perp}$ is negative-definite. A direct computation shows that 
$q(y)=q(x) - 2q(x,w)^2 + q(x,w)^2 q(w) = q(x) -q(x,w)^2 \geq -C^2$. 
The set ${\cal D}_C$ is equivalent to the set of elements 
$\{y \in w^\perp| q(y) \geq -C^2 \}$, which is compact because 
$q|_{w^\perp}$ is negative-definite. 
Since the set ${\cal D}_C$ is compact, 
$\sup_{x \in \Gamma \cdot \eta} \deg x = \infty$, which means 
there is a class of a curve $\eta$ with $q(\eta)>0$. However, all 
curves in the orbit $\Gamma \cdot \eta$ have constant genus. 
Since their degrees could be arbitrarily high, then $M$ is algebraically 
non-hyperbolic. 
\end{proof}

\hfill

\lemma \label{rat_curve}
Let $M$ be a hyperk\"ahler manifold such that the positive cone 
does not coincide with the K\"ahler cone. Then $M$ contains a 
rational curve. 

\hfill

{\bf Proof:} 
This is a classical result that Boucksom and Huybrechts knew in the 
early 2000's 
\cite{_Boucksom-cone_,_Huybrechts:cone_}. \endproof

\hfill

\theorem
Let $M$ be a hyperk\"ahler manifold with Picard rank $\rho$. 
Assume that either $\rho > 2$ or $\rho =2$ and the SYZ conjecture holds. 
%there is no class $\eta \in H^{1,1}(M, \Z)$ with $q(\eta)=0$. 
Then $M$ is algebraically non-hyperbolic. 

\hfill

\begin{proof}
Notice that the Hodge lattice $H^{1,1}(M,\Z)$ of a hyperk\"ahler manifold
has signature $(1,k)$. Therefore, for $\rho \geq 2$, the Hodge lattice
contains a vector with positive square, and $M$ is projective
(\cite{Huy}). 
First, consider the case when $\rho > 2$. If the K\"ahler cone 
coincides with the positive cone, then the automorphism group 
$\Aut(M)$ is commensurable with the group of 
isometries $SO(H^2(M, \Z))$ (Theorem 2.17 in \cite{_AV:Aut_})
preserving the Hodge type. By \ref{_fin_im_Lemma_}, this group
is commensurable with the group of isometries of the Hodge lattice
$H^{1,1}(M,\Z)$.
By Borel and Harish-Chandra's theorem (\cite{BHC}), if $\rho>2$,
any arithmetic subgroup of $SO(1, \rho -1)$ is a lattice.
However, Borel density theorem implies that any lattice in a non-compact simple Lie group 
is Zariski dense (\cite{_Borel:density_}). 
Therefore, for $\rho>2$, $SO(H^{1,1}(M, \Z))$ is infinite. 
In this case $\Aut(M)$ is also infinite and 
we can apply \ref{Aut}. On the other hand, if the K\"ahler cone does not 
coincide with the positive cone, then by \ref{rat_curve} there 
is a rational curve on $M$. Therefore, $M$ is algebraically non-hyperbolic. 

Now let $\rho =2$. Assume the positive cone and the K\"ahler cone 
coincide. If there is no $\eta \in H^{1,1}(M, \Z)$ with $q(\eta)=0$, 
then by Theorem 87 in 
\cite{Dickson}, $SO(H^{1,1}(M, \Z))$ is isomorphic to $\Z \times \Z/2\Z$. 
Therefore, both $SO(H^{1,1}(M, \Z))$ and $\Aut(M)$ are infinite and 
we can apply \ref{Aut}. 
If there is $\eta \in H^{1,1}(M, \Z)$ with $q(\eta)=0$, then 
the SYZ conjecture implies that $\eta$ defines a rational fibration 
on $M$ and we could apply \ref{_hk_with_Lagra_non_h_Proposition_}. 
If $\rho=2$ and the positive and the K\"ahler cones are 
different (i.e., the positive cone is divided into K\"ahler chambers), 
then there is a nef class $\eta \in H^{1,1}(M, \Z)$ with 
$q(\eta)=0$. Since we assumed that the SYZ conjecture holds, the class 
$\eta$ defines a Lagrangian fibration on $M$. Applying 
\ref{_hk_with_Lagra_non_h_Proposition_} we conclude that $M$ is 
algebraically non-hyperbolic. 
\end{proof}


\hfill

\remark We conjecture that all projective hyperk\"ahler manifolds
are algebraically non-hyperbolic. However, our proof fails for manifolds
with Picard rank 1.


\hfill

{\bf Acknowledgments.} This work was inspired by a question of Erwan 
Rousseau about algebraic non-hyperbolicity of hyperk\"ahler manifolds. 
The paper was started while the first-named author was visiting 
Universit\'e libre de Bruxelles and she is grateful to Joel Fine for 
his hospitality. It was finished at the SCGP during the second-named author's 
stay there. We are grateful to the SCGP for making this possible. 



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Chern est nulle.}  J. Diff. Geom. {\bf 18} (1983) 755 - 782.

\bibitem[Bog]{_Bogomolov:defo_} Bogomolov, F., 
        {\em Hamiltonian K\"ahler manifolds}, 
        Sov. Math. Dokl. {\bf 19} (1978) 1462 - 1465.

\bibitem[Bor]{_Borel:density_} 
 A. Borel, 
{\em Density properties for certain subgroups of semi-simple groups without
    compact components,} Ann. of Math. (2) 72 (1960), 179-188.



\bibitem[Bou]{_Boucksom-cone_} 
Boucksom, S., {\em Le c\^one k\"ahl\'erien d'une vari\'et\'e 
hyperk\"ahl\'erienne}, C. R. Acad. Sci. Paris Ser. I Math. 333 (2001), no. 10, 935--938.


\bibitem[BHC]{BHC} Borel, A., Harish-Chandra, 
        {\em Arithmetic subgroups of algebraic groups,} 
        Ann. Math. (2) {bf 75} (1962) 485 - 535.

%\bibitem{cam} Campana, F., 
%        {\em On twistor spaces of class $\mathcal C$,} 
%        J. Diff. Geom. {\bf 33} (1991) 541 - 549. 

\bibitem[CR]{curt_rein} Curtis, W. C., Reiner, I., 
        {\em Representation theory of finite groups and 
         associative algebras,} 
         Interscience, New York (1962). 

\bibitem[De]{Demailly} Demailly, J.-P., 
        {\em Algebraic criteria for Kobayashi hyperbolic projective 
        varieties and jet differentials,}
        Proc. Symp. Pure Math., Vol {\bf 62.2} (1997) 285 - 360. 

\bibitem[Di]{Dickson} Dickson, L. E., 
        {\em Introduction to the theory of numbers,}
        Dover Publ. Inc., New York (1954). 

\bibitem[F]{_Fujiki:HK_} Fujiki, A., 
        {\em On the de Rham Cohomology Group of a Compact 
        K\"ahler Symplectic Manifold}, Adv. Stud.
        Pure Math. 10 (1987) 105 - 165.


\bibitem[HT]{_Hassett_Tschinkel:SYZ_conj_}
Brendan Hassett, Yuri Tschinkel, {\em 
Rational curves on holomorphic symplectic fourfolds},
arXiv:math/9910021,  
Geom. Funct. Anal. 11 (2001), no. 6, 1201--1228.


\bibitem[Hu1]{Huy} 
Huybrechts, D., 
        {\it Compact hyper-K\"ahler manifolds: basic results,}
        Invent. Math. {\bf 135} (1999) 63 - 113.


\bibitem[Hu2]{_Huybrechts:cone_}
Huybrechts, D.,
{\em The K\"ahler cone of a compact hyperk\"ahler manifold},
Math. Ann. 326 (2003), no. 3, 499--513, arXiv:math/9909109.


\bibitem[Hu3]{_Huybrechts:lec_}
 Huybrechts, Daniel, 
{\em Compact hyperk\"ahler manifolds, Calabi-Yau 
manifolds and related geometries,}
Universitext, Springer-Verlag, Berlin, 2003, 
Lectures from the Summer School held in
Nordfjordeid, June 2001, pp. 161-225.


\bibitem[Hw]{Hwang} Hwang, J.-M., 
        {\em Base manifolds for fibrations of projective irreducible 
        symplectic manifolds,} 
        Invent. Math. {\bf 174} (2008) 625 - 644.

\bibitem[HO]{HO} Hwang, J.-M., Oguiso, K., 
        {\em Characteristic foliation on the discriminant hypersurface 
        of a holomorphic Lagrangian fibration,}
        Am. J. Math. {\bf 131} (2009) 981 - 1007. 

%\bibitem{kv} Kamenova, L., Verbitsky, V., 
%        {\em Families of Lagrangian fibrations on hyperkaehler
%        manifolds}, Adv. Math. 260 (2014) 401-413. 

\bibitem[KLV]{klv} Kamenova, L., Lu, S., Verbitsky, M., 
        {\em Kobayashi pseudometric on hyperkahler manifolds,} 
        J. London Math. Soc. (2014) 90 (2): 436 - 450. 

%\bibitem[Ma1]{_Markman:survey_} Markman, E., 
%        {\em A survey of Torelli and monodromy results for 
%        holomorphic-symplectic varieties,} 
%        Proceedings of the conference ``Complex and Differential Geometry'', 
%        Springer Proceedings in Mathematics {\bf 8} (2011) 257 - 322. 

\bibitem[Mar]{_Markman:SYZ_} Markman, E., 
        {\em Lagrangian fibrations of holomorphic-symplectic varieties of 
        $K3^{[n]}$-type}, 
Springer Proceedings in Mathematics \& Statistics {\bf 71} (2014) 241 - 283. 
Algebraic and complex geometry. In honour of Klaus Hulek's 60th birthday. 

\bibitem[Mat]{_Matsushita:fibred_} Matsushita, D.,  
        {\em On fibre space structures of a projective irreducible 
        symplectic manifold,}
        Topology {\bf 38} (1999), No. 1, 79 - 83. Addendum, 
        Topology {\bf 40} (2001) No. 2, 431 - 432.

%\bibitem{_Matsushita:isotropic_} Matsushita, D., 
%        {\em On isotropic divisors on irreducible symplectic manifolds}, 
%        arXiv:1310.0896 [math.AG]. 

\bibitem[O1]{_Oguiso:Aut_} Oguiso, K., 
        {\em Automorphisms of hyper\"ahler manifolds in the view 
        of topological entropy,}
        AMS, Contemp. Math. 422 (2007) 173 - 185. 

\bibitem[O2]{_O:bimer_} Oguiso, K., 
        {\em Bimeromorphic automorphism groups of non-projective 
        hyperk\"ahler manifolds - a note inspired by C. T. McMullen, }
        J. Differ. Geom. 78 (2008) 163 - 191. 

\bibitem[Saw]{_Sawon_} 
J. Sawon, 
{\em Abelian fibred holomorphic symplectic
  manifolds}, Turkish
Jour. Math. 27 (2003), no. 1, 197-230.

\bibitem[V1]{_Verbitsky:ergodic_} Verbitsky, M., 
        {\em Ergodic complex structures on hyperk\"ahler manifolds},
        Acta Math. {\bf 215} (2015) 161 - 182.

\bibitem[V2]{_Verbitsky:SYZ_}
Verbitsky, M., 
{\em Hyperkahler SYZ conjecture and semipositive line bundles},
 arXiv:0811.0639, GAFA 19, No. 5 (2010) 1481-1493.


%\bibitem{_Verbitsky:degenerate_twistor_} Verbitsky, M., 
%        {\em Degenerate twistor spaces for hyperkahler manifolds,} 
%        J. Geom. Phys. {\bf 91} (2015) 2 - 11.

\bibitem[Y]{_Yoshioka_} Yoshioka, K., 
        {\em Bridgeland's stability and the positive cone of the moduli spaces 
        of stable objects on an abelian surface}, 
        arXiv:1206.4838 [math.AG]. 

\bibitem[Z]{_Zar_} Zarhin, Yu., 
        {\em Hodge groups of K3 surfaces,}
        J. Reine Angew. Math. 341 (1983) 193 – 220.


\end{thebibliography}

\noindent {\sc Ljudmila Kamenova\\
Department of Mathematics, 3-115 \\
Stony Brook University \\
Stony Brook, NY 11794-3651, USA,} \\
\tt kamenova@math.sunysb.edu
\\

\noindent {\sc Misha Verbitsky\\
{\sc Laboratory of Algebraic Geometry,\\
National Research University HSE,\\
Faculty of Mathematics, 7 Vavilova Str., \\Moscow, Russian Federation,}\\
\tt  verbit@mccme.ru}, also: \\
{\sc Universit\'e libre de Bruxelles, CP 218,\\
Bd du Triomphe, 1050 Brussels, Belgium
}


\end{document}





1604.02601.extracted_bib
[AV1]{_AV:MBM_ %Amerik, E., Verbitsky, M. {\em Rational curves on %hyperk\"ahler manifolds, Int. Math. Res. Not. IMRN 2015, no. 23, 13009-13045. %
1604.02601.extracted_bib
[AV2]{_AV:Teichmuller_ %Amerik, E., Verbitsky, M., %{\em Teichm\"uller space for hyperk\"ahler and symplectic structures, %J. Geom. Phys. 97 (2015) 44-50.
1604.02601.extracted_bib
[AV]{_AV:Aut_ Amerik, E., Verbitsky, M., {\em Construction of automorphisms of hyperk\"ahler manifolds, arXiv:1604.03079 [math.AG].
1604.02601.extracted_bib
[BM]{_Bayer_Macri_ Bayer, A., Macri, E., {\em MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Invent. Math. {198 (2014) 505 - 590.
1604.02601.extracted_bib
[Be]{_Beauville_ Beauville, A., {\em Varietes K\"ahleriennes dont la premi\`ere classe de Chern est nulle. J. Diff. Geom. {18 (1983) 755 - 782.
1604.02601.extracted_bib
[Bog]{_Bogomolov:defo_ Bogomolov, F., {\em Hamiltonian K\"ahler manifolds, Sov. Math. Dokl. {19 (1978) 1462 - 1465.
1604.02601.extracted_bib
[Bor]{_Borel:density_ A. Borel, {\em Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2) 72 (1960), 179-188.
1604.02601.extracted_bib
[Bou]{_Boucksom-cone_ Boucksom, S., {\em Le c\^one k\"ahl\'erien d'une vari\'et\'e hyperk\"ahl\'erienne, C. R. Acad. Sci. Paris Ser. I Math. 333 (2001), no. 10, 935--938.
1604.02601.extracted_bib
[BHC]{BHC Borel, A., Harish-Chandra, {\em Arithmetic subgroups of algebraic groups, Ann. Math. (2) {bf 75 (1962) 485 - 535. %
1604.02601.extracted_bib
{cam Campana, F., % {\em On twistor spaces of class $\mathcal C$, % J. Diff. Geom. {33 (1991) 541 - 549.
1604.02601.extracted_bib
[CR]{curt_rein Curtis, W. C., Reiner, I., {\em Representation theory of finite groups and associative algebras, Interscience, New York (1962).
1604.02601.extracted_bib
[De]{Demailly Demailly, J.-P., {\em Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Proc. Symp. Pure Math., Vol {62.2 (1997) 285 - 360.
1604.02601.extracted_bib
[Di]{Dickson Dickson, L. E., {\em Introduction to the theory of numbers, Dover Publ. Inc., New York (1954).
1604.02601.extracted_bib
[F]{_Fujiki:HK_ Fujiki, A., {\em On the de Rham Cohomology Group of a Compact K\"ahler Symplectic Manifold, Adv. Stud. Pure Math. 10 (1987) 105 - 165.
1604.02601.extracted_bib
[HT]{_Hassett_Tschinkel:SYZ_conj_ Brendan Hassett, Yuri Tschinkel, {\em Rational curves on holomorphic symplectic fourfolds, arXiv:math/9910021, Geom. Funct. Anal. 11 (2001), no. 6, 1201--1228.
1604.02601.extracted_bib
[Hu1]{Huy Huybrechts, D., {Compact hyper-K\"ahler manifolds: basic results, Invent. Math. {135 (1999) 63 - 113.
1604.02601.extracted_bib
[Hu2]{_Huybrechts:cone_ Huybrechts, D., {\em The K\"ahler cone of a compact hyperk\"ahler manifold, Math. Ann. 326 (2003), no. 3, 499--513, arXiv:math/9909109.
1604.02601.extracted_bib
[Hu3]{_Huybrechts:lec_ Huybrechts, Daniel, {\em Compact hyperk\"ahler manifolds, Calabi-Yau manifolds and related geometries, Universitext, Springer-Verlag, Berlin, 2003, Lectures from the Summer School held in Nordfjordeid, June 2001, pp. 161-225.
1604.02601.extracted_bib
[Hw]{Hwang Hwang, J.-M., {\em Base manifolds for fibrations of projective irreducible symplectic manifolds, Invent. Math. {174 (2008) 625 - 644.
1604.02601.extracted_bib
[HO]{HO Hwang, J.-M., Oguiso, K., {\em Characteristic foliation on the discriminant hypersurface of a holomorphic Lagrangian fibration, Am. J. Math. {131 (2009) 981 - 1007. %
1604.02601.extracted_bib
{kv Kamenova, L., Verbitsky, V., % {\em Families of Lagrangian fibrations on hyperkaehler % manifolds, Adv. Math. 260 (2014) 401-413.
1604.02601.extracted_bib
[KLV]{klv Kamenova, L., Lu, S., Verbitsky, M., {\em Kobayashi pseudometric on hyperkahler manifolds, J. London Math. Soc. (2014) 90 (2): 436 - 450. %
1604.02601.extracted_bib
[Ma1]{_Markman:survey_ Markman, E., % {\em A survey of Torelli and monodromy results for % holomorphic-symplectic varieties, % Proceedings of the conference ``Complex and Differential Geometry'', % Springer Proceedings in Mathematics {8 (2011) 257 - 322.
1604.02601.extracted_bib
[Mar]{_Markman:SYZ_ Markman, E., {\em Lagrangian fibrations of holomorphic-symplectic varieties of $K3^{[n]$-type, Springer Proceedings in Mathematics \& Statistics {71 (2014) 241 - 283. Algebraic and complex geometry. In honour of Klaus Hulek's 60th birthday.
1604.02601.extracted_bib
[Mat]{_Matsushita:fibred_ Matsushita, D., {\em On fibre space structures of a projective irreducible symplectic manifold, Topology {38 (1999), No. 1, 79 - 83. Addendum, Topology {40 (2001) No. 2, 431 - 432. %
1604.02601.extracted_bib
{_Matsushita:isotropic_ Matsushita, D., % {\em On isotropic divisors on irreducible symplectic manifolds, % arXiv:1310.0896 [math.AG].
1604.02601.extracted_bib
[O1]{_Oguiso:Aut_ Oguiso, K., {\em Automorphisms of hyper\"ahler manifolds in the view of topological entropy, AMS, Contemp. Math. 422 (2007) 173 - 185.
1604.02601.extracted_bib
[O2]{_O:bimer_ Oguiso, K., {\em Bimeromorphic automorphism groups of non-projective hyperk\"ahler manifolds - a note inspired by C. T. McMullen, J. Differ. Geom. 78 (2008) 163 - 191.
1604.02601.extracted_bib
[Saw]{_Sawon_ J. Sawon, {\em Abelian fibred holomorphic symplectic manifolds, Turkish Jour. Math. 27 (2003), no. 1, 197-230.
1604.02601.extracted_bib
[V1]{_Verbitsky:ergodic_ Verbitsky, M., {\em Ergodic complex structures on hyperk\"ahler manifolds, Acta Math. {215 (2015) 161 - 182.
1604.02601.extracted_bib
[V2]{_Verbitsky:SYZ_ Verbitsky, M., {\em Hyperkahler SYZ conjecture and semipositive line bundles, arXiv:0811.0639, GAFA 19, No. 5 (2010) 1481-1493. %
1604.02601.extracted_bib
{_Verbitsky:degenerate_twistor_ Verbitsky, M., % {\em Degenerate twistor spaces for hyperkahler manifolds, % J. Geom. Phys. {91 (2015) 2 - 11.
1604.02601.extracted_bib
[Y]{_Yoshioka_ Yoshioka, K., {\em Bridgeland's stability and the positive cone of the moduli spaces of stable objects on an abelian surface, arXiv:1206.4838 [math.AG].
1604.02601.extracted_bib
[Z]{_Zar_ Zarhin, Yu., {\em Hodge groups of K3 surfaces, J. Reine Angew. Math. 341 (1983) 193 – 220.