Recording: https://disk.pku.edu.cn:443/link/21483613E7EBAD10312B51853959F8A0
Valid Until: 2026-06-30 23:59
Abstract: 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-Kähler varieties in arithmetic geometry. This includes the unpolarized Shafarevich conjecture for hyper-Kähler 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-Kähler varieties, i.e. the finiteness of geometric isomorphism classes of hyper-Kähler 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.
Bio: Zhiyuan Li is an associated professor at Shanghai Center for Mathematical Sciences. His research interests are algebraic geometry and arithmetic geometry.
