Finiteness problem for hyper-Kahler varieties

The classical Shafarevich conjecture is about the finiteness of isomorphism classes of curves of given genus defined over a number field with good reduction outside a finite collection of places. It plays an important role in Falting's proof of the Mordell conjecture. Similar finiteness problems arise for higher dimensional varieties. In this talk, I will talk about finiteness problems for hyper-Kahler varieties in arithmetic geometry. This includes the unpolarized Shafarevich conjecture for hyper-Kahler varieties the cohomological generalization of the Shafarevich conjecture by replacing the good reduction condition with the unramifiedness of the cohomology. I will also explain how to generalize Orr and Skorobogatov's finiteness result on K3 surfaces to hyper-Kahler varieties, i.e. the finiteness of geometric isomorphism classes of hyper-Kahler varieties of CM type in a given deformation type defined over a number field with bounded degree. This is a joint work with Lie Fu, Teppei Takamatsu and Haitao Zou.