Rational Curves on Hyperkähler Manifolds
Let M be an irreducible holomorphically symplectic manifold. We show that all faces of the Kähler cone of M are hyperplanes Hi orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kähler cone is a connected component of a complement of the positive cone to the union of all Hi. 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 hyperkähler 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 hyperkähler manifolds.