turkmath.org

In this talk I shall attempt to show that whether or not a complex surface $S$ can be defined by polynomials with coefficients in a number field depends only on the holomorphic universal covers of certain Zariski open subsets $U\subset S$. In the particular case of Kodaira fibrations $U$ can be taken to be the whole surface $S$. This characterization of arithmeticity in terms of universal covers is very much in contrast with the corresponding situation in complex dimension one, where the universal cover is always the unit disk. This is joint work with Sebastin Reyes-Carocca.