Automorphisms of surfaces of general type with q>=2 acting trivially in cohomology

TL;DR

This paper proves that most surfaces of general type with irregularity q≥2 have automorphism groups acting faithfully on their cohomology, providing classifications and answering a question by Catanese.

Contribution

It establishes the rational cohomological rigidity of surfaces of general type with q≥2 and classifies certain surfaces isogenous to a product regarding this property.

Findings

Surfaces of general type with q>2 are rationally cohomologically rigidified.

Minimal surfaces with q=2 are rigidified unless K^2=8X.

Classified surfaces isogenous to a product with q=2 that are not rationally cohomologically rigidified.

Abstract

A compact complex manifold X is said to be rationally cohomologically rigidified if its automorphism group Aut(X) acts faithfully on the cohomology ring H*(X,Q). In this note, we prove that, surfaces of general type with irregularity q>2 are rationally cohomologically rigidified, and so are minimal surfaces S with q=2 unless K^2=8X. This answers a question of Fabrizio Catanese in part. As examples we give a complete classification of surfaces isogenous to a product with q=2 that are not rationally cohomologically rigidified. These surfaces turn out however to be rigidified.