Automorphisms of surfaces of general type with q=1 acting trivially in cohomology

TL;DR

This paper investigates the automorphism group of complex minimal surfaces of general type with irregularity one, establishing an upper bound on automorphisms acting trivially on cohomology and characterizing cases of maximal size.

Contribution

It proves that the subgroup of automorphisms acting trivially on cohomology has size at most four and characterizes the surfaces achieving this maximum.

Findings

|Aut_0(S)| <= 4

Surfaces with |Aut_0(S)|=4 are isogenous to a product of unmixed type

Examples cover all possible geometric genera for these surfaces

Abstract

Let S be a complex minimal surface of general type with irregularity q(S)=1 and Aut_0(S) the subgroup of automorphisms acting trivially on the cohomology ring with rational coefficients. In this paper we show that |Aut_0(S)|<=4, and if the equality holds then $S$ is a surface isogenous to a product of unmixed type. Moreover, examples of surfaces with |Aut_0(S)|=4 and all possible values of the geometric genus are provided.