Title: The Matsushita alternative

Abstract: Compact hyperkahler manifolds are higher-dimensional generalizations of K3 surfaces. The only nontrivial fibration structures they admit are fibrations by Lagrangian tori, which in particular yield complete integrable systems. In this talk I'll explain how to prove a conjecture of Matsushita that such fibrations are either isotrivial or vary maximally in moduli. I will also compare this to known results about the Hitchin system and deduce some applications to the Chow theory of hyperkahler manifolds.