Towards the cone conjecture for hyperkahler manifolds

Let $M$ be a holomorphically symplectic manifold, and $K$ be its Kaehler cone. We show that all faces of the Kahler cone of $M$ are hyperplanes orthogonal to certain homology classes, called MBM classes. The MBM classes can be characterized as homology classes which can be represented by a minimal curve in some deformation of $M$. For a deformation of a Hilbert scheme on $K3$, this result gives a simple proof of the Morrison-Kawamata cone conjecture (proven by Markman and Yoshioka in a forthcoming paper). This is a joint work with Ekaterina Amerik.