Galois action on universal covers of Kodaira fibrations

Abstract

Catanese has recently asked if there exists an element of the absolute Galois group $\sigma \in Gal\left(\overline{\mathbb{Q}}\right)$ for which there is a Kodaira fibration $f:S\to B$ defined over a number field such that the universal covers of $S$ and its Galois conjugate surface $S^{\sigma }$ are not isomorphic. The main result of this article is that every element $\sigma \ne \mathrm{Id}$ has this property.