The Kahler cone of a compact hyperkahler manifold

This paper is a sequel to [11]. We study a number of questions only touched upon in [11] in more detail. In particular: What is the relation between two birational compact hyperkahler manifolds? What is the shape of the cone of all Kahler classes on such a manifold? How can the birational Kahler cone be described? Most of the results are motivated by either the well-established two-dimensional theory, i.e. the theory of K3 surfaces, or the theory of Calabi-Yau threefolds and string theory.