Symplectic automorphisms of a surface with genus two fibration and their action on CH$_0$

Jiabin Du and Wenfei Liu

Vol. 342 (2026), No. 2, 259–273

DOI: 10.2140/pjm.2026.342.259

Abstract

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

Keywords

surface of general type, fibration of genus two, symplectic automorphism, Chow group, Bloch–Beilinson conjecture

Mathematical Subject Classification

Primary: 14J50, 14J29, 14C15

Milestones

Received: 20 January 2025

Revised: 25 May 2025

Accepted: 12 January 2026

Published: 23 April 2026