Zero-cycles and the Cayley-Oguiso automorphism

TL;DR

This paper investigates the action of specific automorphisms on the Chow group of zero-cycles of quartic K3 surfaces, revealing that symplectic automorphisms act trivially while anti-symplectic automorphisms act as negation.

Contribution

It establishes a precise relationship between the nature of automorphisms and their action on the Chow group of zero-cycles for certain quartic K3 surfaces.

Findings

Symplectic automorphisms act as the identity on the Chow group.

Anti-symplectic automorphisms act as minus the identity.

Automorphisms of infinite order have a predictable action on zero-cycles.

Abstract

Cayley and Oguiso have constructed certain quartic K3 surfaces $S$, with automorphisms $g$ of infinite order. We show that when $g$ is symplectic (resp. anti-symplectic), it acts as the identity (resp. minus the identity) on the degree zero part of the Chow group of zero-cycles of $S$.