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

TL;DR

This paper investigates symplectic automorphisms of genus two fibered surfaces, establishing bounds on their size and their trivial action on certain Chow groups, supporting conjectures in algebraic geometry.

Contribution

It proves an upper bound on the size of symplectic automorphism groups for certain surfaces and verifies their trivial action on the Albanese kernel under specific conditions.

Findings

If $ ext{chi}( ext{O}_S) extgreater=5$, then $| ext{Aut}_s(S)| extless=2$.

Under some conditions, $ ext{Aut}_s(S)$ acts trivially on $ ext{CH}_0(S)_{ ext{alb}}$.

Automorphisms acting trivially on $H^{i,0}(S)$ also act trivially on $ ext{CH}_0(S)_{ ext{alb}}$.

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 $\chi(\mathcal{O}_S)\geq 5$, then $|\mathrm{Aut}_s(S)|\leq 2$. 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 $0\leq i\leq 2$, then it also acts trivially on $\mathrm{CH}_0(S)_{\mathrm{alb}}$.