Yoon-Joo Kim (Universität Bonn) : Isotrivial fibrations of hyper-Kähler fourfolds

A compact hyper-Kähler (HK) manifold and its Lagrangian fibration are higher-dimensional generalizations of a K3 surface and its elliptic fibration. A Lagrangian fibration f : X -> B of a HK manifold is called isotrivial if its smooth fibers are all isomorphic to each other. This is the most special type of Lagrangian fibrations, generalizing isotrivial elliptic fibrations of K3 surfaces. In this talk, I will propose a classification strategy of HK manifolds admitting isotrivial Lagrangian fibrations. The method in particular recovers the two well-known constructions of HK manifolds from scratch, the Hilbert schemes of n points on K3 surfaces and generalized Kummer varieties. I will then specialize the discussion to HK fourfolds. This is joint work with Radu Laza and Olivier Martin.