Families of Riemann Surfaces, Uniformization and Arithmeticity

TL;DR

This paper explores the uniformization of families of Riemann surfaces, characterizing which holomorphic motions produce universal covers and linking these to the arithmeticity of the families and associated complex surfaces.

Contribution

It provides a characterization of uniformizing domains via holomorphic motions and establishes criteria for arithmeticity of families and surfaces based on their universal covers.

Findings

Identifies which holomorphic motions yield uniformizing domains.

Characterizes arithmetic families of Riemann surfaces over number fields.

Connects universal covers of complex surfaces to their arithmetic properties.

Abstract

A consequence of the results of Bers and Griffiths on the uniformization of complex algebraic varieties is that the universal cover of a family of Riemann surfaces, with base and fibers of finite hyperbolic type, is a contractible 2-dimensional domain that can be realized as the graph of a holomorphic motion of the unit disk. In this paper we determine which holomorphic motions give rise to these uniformizing domains and characterize which among them correspond to arithmetic families (i.e. families defined over number fields). Then we apply these results to characterize the arithmeticity of complex surfaces of general type in terms of the biholomorphism class of the 2-dimensional domains that arise as universal covers of their Zariski open subsets. For the important class of Kodaira fibrations this criterion implies that arithmeticity can be read off from the universal cover. All this is very much in contrast with the corresponding situation in complex dimension one, where the universal cover is always the unit disk.