Cohomologically or numerically trivial automorphisms of surfaces of general type

TL;DR

This paper determines the groups of cohomologically and numerically trivial automorphisms for certain surfaces of general type, producing new examples with large automorphism groups and nontrivial torsion in cohomology.

Contribution

It explicitly computes these automorphism groups for reducible fake quadrics, providing new record examples with large automorphism groups and torsion in cohomology.

Findings

A surface with |Aut_Q(S)| = 192 was constructed.

A surface with nontrivial cohomological torsion and Aut_Z(S) ≅ Z/2 was identified.

Explicit automorphism groups for reducible fake quadrics were determined.

Abstract

Our main result is the determination of the respective groups $ Aut_\mathbb{Z}(S) $ of cohomologically trivial automorphisms and $ Aut_\mathbb{Q}(S) $ of numerically trivial automorphisms for the reducible fake quadrics, that is, the surfaces $S$ isogenous to a product with $q=p_g=0$. In this way we produce new record winning examples: a surface $S$ with $|Aut_\mathbb{Q}(S)| =192$, and a surface whose cohomology has torsion with nontrivial $ Aut_\mathbb{Z}(S) \cong \mathbb{Z}/2.$