Galois action on universal covers of Kodaira fibrations

Abstract

F. Catanese has recently asked if there exists an element of the absolute Galois group σ ∈ Gal(Q) for which there is a Kodaira fibration f : S → B defined over a number field such that the universal covers of S and its Galois conjugate surface S are not isomorphic. The main result of this article is that every element σ 6= Id. has this property.