Hyperbolic geometry of the ample cone of a hyperkahler manifold

Abstract

Let $M$ be a compact hyperkahler manifold with maximal holonomy (IHS). The group $H^2(M, R)$ is equipped with a quadratic form of signature $(3, b_2-3)$, called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational Hodge lattice $H^{1,1}(M,Q)$, has signature $(1,k)$. This gives a hyperbolic Riemannian metric on the projectivisation of the positive cone in $H^{1,1}(M,Q)$, denoted by $H$. Torelli theorem implies that the Hodge monodromy group $\Gamma$ acts on $H$ with finite covolume, giving a hyperbolic orbifold $X=H/\Gamma$. We show that there are finitely many geodesic hypersurfaces which cut $X$ into finitely many polyhedral pieces in such a way that each of these pieces is isometric to a quotient $P(M')/Aut(M')$, where $P(M')$ is the projectivization of the ample cone of a birational model $M'$ of $M$, and $Aut(M')$ the group of its holomorphic automorphisms. This is used to prove the existence of nef isotropic line bundles on a hyperkahler birational model of a simple hyperkahler manifold of Picard number at least 5, and also illustrates the fact that an IHS manifold has only finitely many birational models up to isomorphism, originally deduced by Markman and Yoshioka from the Morrison-Kawamata cone conjecture.