Prof. Zhiyuan Li，Shanghai Center for Mathematical Sciences

Bloch's conjecture for a surface X over an algebraically closed field k states that every homologically trivial correspondence acts as 0 on the Albanese kernel. When X is a K3 surface, this conjecture has been proved by Voisin and Huybrechts for symplectic automorphisms of finite order. In this talk, I will talk about the recent progress on Bloch's conjecture for symplectic autoequivalences on a K3 surface. In particular, we show that the conjecture holds for symplectic automorphisms of arbitrary order provided the Neron-Severi lattice is not a two twist. This also gives a new proof of Voisin's result on symplectic involutions. I will also mention the application to Bloch's conjecture for hyper-K?hler varieties. This is a joint work with X.Yu and R. Zhang.