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Working paper

Morrison-Kawamata cone conjecture for hyperkahler manifolds

Let $M$ be a simple holomorphically symplectic manifold, that is, a simply connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$. We prove that the group of holomorphic automorphisms of $M$ acts on the set of faces of its Kahler cone with finitely many orbits. This is a version of the Morrison-Kawamata cone conjecture for hyperkahler manifolds. The proof is based on the following observation, proven with ergodic theory. Let $M$ be a complete Riemannian orbifold of dimension at least three, constant negative curvature and finite volume, and $\{S_i\}$ an infinite set of locally geodesic hypersurfaces. Then the union of $S_i$ is dense in $M$.