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

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 ${\mathrm{Aut}}_s(S)\to \mathrm{Aut}({\mathrm{CH}}_0(S{)}_{\mathrm{alb}})$ has order at most 3. Our arguments actually take care of the group ${\mathrm{Aut}}_f(S)$ of fibration-preserving automorphisms of elliptic surfaces $f:S\to B$. 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 ${\mathrm{CH}}_0(S{)}_{\mathrm{alb}}$.

18 pages; the result in the exceptional cases slightly weakened; to appear in SCIENCE CHINA Mathematics

References in corpus (1)

• Bounded automorphism groups of compact complex surfaces

Cited by in corpus (1)

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