On regular surfaces of general type with numerically trivial automorphism group of order $4$

TL;DR

This paper characterizes regular minimal surfaces of general type with ample canonical bundle that have a trivial automorphism group of order four acting on rational cohomology, showing they are isogenous to a product of two curves.

Contribution

It provides a complete characterization of such surfaces achieving the maximal automorphism group order, identifying them as isogenous to a product of two curves of unmixed type.

Findings

Surfaces with automorphism group of order 4 are isogenous to a product of two curves.

The automorphism group in these cases is isomorphic to (Z/2Z)^2.

Unbounded families of such surfaces are constructed.

Abstract

Let $S$ be a regular minimal surface of general type over the field of complex numbers, and $\mathrm{Aut}_\mathbb{Q}(S)$ the subgroup of automorphisms acting trivially on $H^*(S,\mathbb{Q})$. It has been known since twenty years that $|\mathrm{Aut}_\mathbb{Q}(S)|\leq 4$ if the invariants of $S$ are sufficiently large. Under the assumption that $K_S$ is ample, we characterize the surfaces achieving the equality, showing that they are isogenous to a product of two curves, of unmixed type, and that the group $\mathrm{Aut}_\mathbb{Q}(S)$ is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$. Moreover, unbounded families of surfaces with $\mathrm{Aut}_\mathbb{Q}(S)\cong(\mathbb{Z}/2\mathbb{Z})^2$ are provided.