On symplectic automorphisms of a surface with genus two fibration and their action on $\mathrm{CH}_0$
Jiabin Du, Wenfei Liu
TL;DR
The article addresses how symplectic automorphisms of a surface $S$ with a genus two fibration interact with the zero-th Chow group via the Albanese kernel, proving a sharp bound $|\mathrm{Aut}_s(S)|\le 2$ when $\chi(\mathcal{O}_S)\ge 5$ and showing that these automorphisms act trivially on $\mathrm{CH}_0(S)_{\mathrm{alb}}$ in key geometric scenarios. The authors employ Xiao’s canonical map bounds for genus two fibrations, analyze quotient constructions under a Klein four group with the hyperelliptic involution, and use criteria related to finite-dimensional Chow motives and the Bloch–Beilinson philosophy to deduce trivial actions on $\mathrm{CH}_0(S)_{\mathrm{alb}}$. They demonstrate that for $\chi(\mathcal{O}_S)\ge 5$, automorphisms must preserve the fibration, and in the generic finite-canonical-map case, either the automorphism group is trivial on CH$_0$ or forces a $(\mathbb{Z}/2\mathbb{Z})^2$-symmetry with a degree-4 canonical map. Collectively, the results provide concrete structural constraints on symplectic automorphisms of genus two fibrations and advance Bloch–Beilinson-type expectations in this geometric setting.
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 $χ(\mathcal{O}_S)\geq 5$, 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 $σ\in \mathrm{Aut}(S)$ acts trivially on $H^{i,0}(S)$ for $0\leq i\leq 2$, then it also acts trivially on $\mathrm{CH}_0(S)_{\mathrm{alb}}$.