Numerically trivial automorphisms of Enriques surfaces in characteristic 2

Abstract

An automorphism of an algebraic surface $S$ is called cohomologically (numerically) trivial if it acts identically on the second cohomology group (this group modulo torsion subgroup). Extending the results of Mukai and Namikawa to arbitrary characteristic $p > 0$, we prove that the group of cohomologically trivial automorphisms $\operatorname{Aut}_{\operatorname{ct}}(S)$ of an Enriques surface $S$ is of order $\le 2$ if $S$ is not supersingular. If $p = 2$ and $S$ is supersingular, we show that $\mathrm{Aut}_{\operatorname{ct}}(S)$ 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 $\mathrm{Aut}_{\operatorname{nt}}(S)$ of numerically trivial automorphisms is a subgroup of a cyclic group of order $\le 4$ unless $p = 2$, where $\mathrm{Aut}_{\operatorname{nt}}(S)$ is a subgroup of a 2-elementary group of rank $\le 2$.