Surfaces of general type with q=2 are rigidified

TL;DR

This paper proves that minimal surfaces of general type with irregularity q=2 are rigidified, showing that nontrivial automorphisms acting trivially on rational cohomology imply the surface is isogenous to a product, thus establishing their rigidity.

Contribution

It characterizes surfaces of general type with q=2 that have automorphisms acting trivially on cohomology, linking them to surfaces isogenous to a product and proving their rigidity.

Findings

Surfaces with q=2 and certain automorphisms are isogenous to a product.

No nontrivial automorphism can be homotopic to the identity on these surfaces.

Such surfaces are proven to be rigidified in the sense of Catanese.

Abstract

Let $S$ be a minimal smooth projective surface of general type with irregularity $q=2$. We show that, if $S$ has a nontrivial holomorphic automorphism acting trivially on the cohomology with rational coefficients, then it is a surface isogenous to a product. As a consequence of this geometric characterization, one infers that no nontrivial automorphism of surfaces of general type with $q=2$ (which are not necessarily minimal) can be homotopic to the identity. In particular, such surfaces are rigidified in the sense of Fabrizio Catanese.