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

TL;DR

This paper proves that symplectic automorphisms of certain elliptic surfaces act trivially on the Albanese kernel of the zero-th Chow group, with specific exceptions related to the surface's geometric genus and irregularity.

Contribution

It establishes the trivial action of symplectic automorphisms on the Albanese kernel for elliptic surfaces, extending to fibration-preserving automorphisms and special cases like elliptic K3 surfaces.

Findings

Symplectic automorphisms act trivially on the Albanese kernel unless p_g=q in {1,2}.

Automorphisms preserving fibrations and inducing trivial action on H^{i,0} also act trivially on CH_0(S)_alb.

For elliptic K3 surfaces, the intersection of fibration-preserving and symplectic automorphisms acts trivially on CH_0(S)_alb.

Abstract

Let $S$ be a complex smooth projective surface of Kodaira dimension one. We show that the group $\mathrm{Aut}_s(S)$ of symplectic automorphisms acts trivially on the Albanese kernel $\mathrm{CH}_0(S)_\mathrm{alb}$ of the $0$-th Chow group $\mathrm{CH}_0(S)$, unless possibly if the geometric genus and the irregularity satisfy $p_g(S)=q(S)\in\{1,2\}$. In the exceptional cases, the image of the homomorphism $\mathrm{Aut}_s(S)\rightarrow \mathrm{Aut}(\mathrm{CH}_0(S)_\mathrm{alb})$ has order at most 3. Our arguments actually take care of the group $\mathrm{Aut}_f(S)$ of fibration-preserving automorphisms of elliptic surfaces $f\colon S\rightarrow B$. We prove that, if $\sigma\in\mathrm{Aut}_f(S)$ induces the trivial action on $H^{i,0}(S)$ for $i>0$, then it induces the trivial action on $\mathrm{CH}_0(S)_\mathrm{alb}$. As a by-product we obtain that if $S$ is an elliptic K3 surface, then $\mathrm{Aut}_f(S)\cap \mathrm{Aut}_s(S)$ acts trivially on $\mathrm{CH}_0(S)_\mathrm{alb}$.