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Morrison-Kawamata cone conjecture for hyperkahler manifolds
Annales Scientifiques de l'Ecole Normale Superieure. 2017. Vol. 50. No. 4. P. 973–993.
Amerik E., Verbitsky M.
Let M be a simple hyperk¨ahler manifold, that is, a simply connected compact holomorphically symplectic manifold of K¨ahler 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 K¨ahler cone with finitely many orbits. This statement is known as Morrison-Kawamata cone conjecture for hyperk¨ahler manifolds. As an implication, we show that a hyperk¨ahler 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 manifold 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.
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