Numerically trivial automorphisms of Enriques surfaces in characteristic $2$

TL;DR

This paper classifies automorphisms acting trivially on cohomology for Enriques surfaces in characteristic 2, extending known results to positive characteristic and describing special cases for supersingular surfaces.

Contribution

It extends the classification of cohomologically trivial automorphisms of Enriques surfaces to arbitrary characteristic, especially characteristic 2, and explicitly describes exceptional cases.

Findings

Automorphism group order ≤ 2 for non-supersingular surfaces

Supersingular case: automorphism group cyclic of order 1, 2, 3, 5, 7, 11 or quaternion group Q8

Automorphisms with non-trivial canonical bundle are constrained to cyclic or 2-elementary groups

Abstract

An automorphism of an algebraic surface $S$ is called cohomologically (numerically) trivial if it acts identically on the second $l$-adic cohomology group (this group modulo torsion subgroup). Extending the results of S. Mukai and Y. Namikawa to arbitrary characteristic $p > 0$, we prove that the group of cohomologically trivial automorphisms $\rm{Aut}_{\rm{ct}}(S)$ of an Enriques surface $S$ is of order $\leq 2$ if $S$ is not supersingular. If $p = 2$ and $S$ is supersingular, we show that $\rm{Aut}_{\rm{ct}}(S)$ is a cyclic group of odd order $n\in \{1,2,3,5,7,11\}$ or the quaternion group $Q_8$ of order $8$ and we describe explicitly all the exceptional cases. If $K_S \neq 0$, we also prove that the group $\rm{Aut}_{\rm{nt}}(S)$ of numerically trivial automorphisms is a subgroup of a cyclic group of order $\leq 4$ unless $p = 2$, where $\rm{Aut}_{\rm{nt}}(S)$ is a subgroup of a $2$-elementary group of rank $\leq 2$.