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 $H_i$ 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 $H_i$. 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 hyperkähler manifolds with Picard group generated by a negative class $z$, we prove that $\pm z$ is ${\BBQ}$-effective if and only if it is an MBM class. We also prove some results toward the Morrison–Kawamata cone conjecture for hyperkähler manifolds.