Catanese has recently asked if there exists an element of the absolute Galois group for which there is a Kodaira fibration defined over a number field such that the universal covers of and its Galois conjugate surface are not isomorphic. The main result of this article is that every element has this property.
"Galois action on universal covers of Kodaira fibrations." Duke Math. J. 169 (7) 1281 - 1303, 15 May 2020. https://doi.org/10.1215/00127094-2019-0078