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Extendability of group actions on K3 or Enriques surfaces

Tianchen Zhao

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.

Extendability of group actions on K3 or Enriques surfaces

Abstract

Let be a K3 or Enriques surface with good reduction. Let be a finite group acting (not necessarily linearly) on . We give a criterion for this group action to extend to a smooth model of in terms of the action of on the second -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.

Paper Structure

This paper contains 19 sections, 29 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Good reduction and the extendability problem
  3. Main results
  4. Applications of the main results
  5. Preliminaries
  6. Specialization homomorphisms
  7. RDP models and Weyl groups
  8. Group cohomology with coefficients in a non-abelian group
  9. Extendability and non-abelian group cohomology
  10. Extendability of birational maps
  11. Extendability of group actions
  12. Extendability and $\ell$-adic cohomology
  13. $G$-action and $\ell$-adic cohomology
  14. $\ell$-adic realization of $\alpha_{\mathcal{L}}$
  15. Proof of Theorem \ref{['main theorem']}
  16. ...and 4 more sections

Key Result

Theorem 1.5

Let $X$ be a K3 or Enriques surface over $K$ with good reduction. Let $G$ be a finite group acting on $X$ with base field $K_0$. Assume $K/K_0$ is finite and unramified. Then the $G$-action on $X$ is extendable if and only if there is a $G$-isomorphism

Theorems & Definitions (72)

  • Definition 1.2
  • Example 1.3
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • proof
  • Theorem 1.7
  • Definition 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 62 more