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 Aut_ct(𝑆) of an Enriques surface 𝑆 is of order ≤ 2 if 𝑆 is not supersingular. If 𝑝 = 2 and 𝑆 is supersingular, we show that Aut_ct(𝑆) is a cyclic group of odd order 𝑛 ∈ {1, 2, 3, 5, 7, 11} or the quaternion group 𝑄₈ of order 8 and we describe explicitly all the exceptional cases. If 𝐾_𝑆 ≠ 0, we also prove that the group Aut_nt(𝑆) of numerically trivial automorphisms is a subgroup of a cyclic group of order ≤ 4 unless 𝑝 = 2, where Aut_nt(𝑆) is a subgroup of a 2-elementary group of rank ≤ 2.