Cohomologically or numerically trivial automorphisms of surfaces of general type
Fabrizio Catanese, Davide Frapporti
TL;DR
The paper investigates cohomologically trivial and numerically trivial automorphisms of minimal surfaces of general type, focusing on surfaces isogenous to a product (fake quadrics). It develops a comprehensive group-theoretic framework using Hurwitz generating vectors, normalizers, and centralizers to compute Aut$_\mathbb{Z}(S)$ and Aut$_\mathbb{Q}(S)$, with emphasis on the unmixed case. It establishes sharp bounds and concrete maxima, notably Aut$_\mathbb{Q}(S)\leq 192$ for $q=p_g=0$ with equality at $G\cong (\mathbb{Z}/2)^3$, and constructs a surface with Aut$_\mathbb{Q}(S)=192$, plus an example with Aut$_\mathbb{Z}(S)\cong \mathbb{Z}/2$ and torsion in $H^2(S,\mathbb{Z})$. It further clarifies the irregular case, showing that for $q\ge 1$ nontrivial Aut$_\mathbb{Q}(S)$ is tightly constrained (often $\mathbb{Z}/2$ when $q=2$ and $|\mathrm{Aut}_{\mathbb{Q}}(S)|\leq 4$ when $q=1$), highlighting the special role of products of curves and the interplay between geometry and cohomology. These results deepen understanding of automorphism behavior on complex surfaces of general type and identify maximal symmetry phenomena in this class.
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.$