Morrison-Kawamata cone conjecture for hyperkähler manifolds

Abstract

Let M be a simple hyperkahler manifold, that is, a simply connected compact holomorphically symplectic manifold of Kahler type with h 2,0 = 1. Assuming b2(M) 6 5, 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 statement is known as Morrison-Kawamata cone conjecture for hyperkahler manifolds. As an implication, we show that a hyperkahler manifold has only finitely many non-equivalent birational models. 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 {Si} an infinite set of complete, locally geodesic hypersurfaces. Then the union of Si is dense in M.