Rational Curves on Hyperkahler Manifolds

Let M be an irreducible holomorphically symplectic manifold. We show that all faces of the Kahler cone of M are hyperplanes H-i orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kahler cone is a connected component of a complement of the positive cone to the union of all H-i. We provide several characterizations of the MBM classes. We show the invariance of MBM property by deformations, as long as the class in question stays of type (1, 1). For hyperkahler manifolds with Picard group generated by a negative class z, we prove that +/-z is Q-effective if and only if it is an MBM class. We also prove some results toward the Morrison-Kawamata cone conjecture for hyperkahler manifolds.