Yoon-Joo Kim, Isotrivial Fibrations of Compact Hyper-Kähler Manifolds

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 the notion of an isotrivial elliptic fibration. In this talk, I will propose a possible classification scheme of HK manifolds admitting isotrivial Lagrangian fibrations. This method in particular rediscovers the two well-known constructions of HK manifolds, the K3^[n] and generalized Kummer constructions. I will then specialize the discussion to HK fourfolds. This is joint work with Radu Laza and Olivier Martin.