Extendability of group actions on K3 or Enriques surfaces

TL;DR

This paper establishes criteria for extending finite group actions on K3 and Enriques surfaces over good reduction models, generalizing previous Galois action results and linking surface reduction properties.

Contribution

It provides a new cohomological criterion for extending group actions on K3 and Enriques surfaces, broadening understanding of their symmetries in algebraic geometry.

Findings

Extendability criterion based on second ℓ-adic cohomology

Symplectic actions extend if residue characteristic does not divide group order

Relation between good reduction of Enriques surfaces and their K3 covers

Abstract

Let $X$ be a K3 or Enriques surface with good reduction. Let $G$ be a finite group acting (not necessarily linearly) on $X$. We give a criterion for this group action to extend to a smooth model of $X$ in terms of the action of $G$ on the second $\ell$-adic cohomology groups. In particular, we generalize the result on the extendability of Galois actions on K3 surfaces by Chiarellotto, Lazda, and Liedtke. As an application, we prove that a symplectic linear group action is extendable if the residue characteristic does not divide its order. Lastly, we relate the good reduction of Enriques surfaces with that of their K3 double covers.