Rational curves on hyperkahler manifolds

Abstract

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 $\pm z$ is Q-effective if and only if it is an MBM class. We also prove some results towards the Morrison-Kawamata cone conjecture for hyperkahler manifolds.