Existence of Lagrangian fibrations on holomorphic symplectic manifolds
Justin Sawon (Colorado State)

Abstract: A K3 surface is an elliptic fibration iff it contains a divisor with square zero. Holomorphic symplectic manifolds are higher-dimensional generalizations of K3 surfaces. Matsushita showed that a fibration on a holomorphic symplectic manifold must have (holomorphic) Lagrangian tori for smooth fibres; this is the higher-dimensional analogue of an elliptic fibration. There is a conjectural condition ensuring the existence of a Lagrangian fibration on a holomorphic symplectic manifold. In this talk I will prove this conjecture for a `generic' Hilbert scheme of points on a K3 surface. Reference: math.AG/0509224