On symplectic automorphisms of a surface with genus two fibration and their action on ${\mathrm{CH}}_0$

Abstract

Let $S$ be a complex smooth projective surface with a genus two fibration, and ${\mathrm{Aut}}_s(S)$ the group of symplectic automorphisms, fixing every holomorphic 2-forms (if any) on $S$. Based on the work of Jin-Xing Cai, we observe in this paper that, if $\chi ({\mathcal{O}}_S)\ge 5$, then $|{\mathrm{Aut}}_s(S)|\le 2$. Then we go on to verify, under some conditions, that ${\mathrm{Aut}}_s(S)$ acts trivially on the Albanese kernel ${\mathrm{CH}}_0(S{)}_{\mathrm{alb}}$ of the 0-th Chow group, which is predicted by a conjecture of Bloch and Beilinson. As a consequence, if an automorphism $\sigma \in \mathrm{Aut}(S)$ acts trivially on $H^{i,0}(S)$ for $0\le i\le 2$, then it also acts trivially on ${\mathrm{CH}}_0(S{)}_{\mathrm{alb}}$.

11 pages

References in corpus (1)

• On symplectic automorphisms of elliptic surfaces acting on ${\mathrm{CH}}_0$