Symmetric spaces uniformizing Shimura varieties in the Torelli locus

TL;DR

This paper identifies the symmetric spaces that uniformize certain low genus counterexamples to the Coleman-Oort conjecture, linking geometric structures of Galois covers of curves to Shimura varieties.

Contribution

It determines the symmetric space uniformizing each low genus counterexample to the Coleman-Oort conjecture derived from Galois covers of curves.

Findings

Classifies symmetric spaces for low genus counterexamples.

Shows the relationship between fibrations and uniformizing symmetric spaces.

Provides explicit descriptions of the uniformizing spaces.

Abstract

An algebraic subvariety Z of A_g is totally geodesic if it is the image via the natural projection map of some totally geodesic submanifold X of the Siegel space. We say that X is the symmetric space uniformizing Z. In this paper we determine which symmetric space uniformizes each of the low genus counterexamples to the Coleman-Oort conjecture obtained studying Galois covers of curves. It is known that the counterexamples obtained via Galois covers of elliptic curves admit two fibrations in totally geodesic subvarieties. The second result of the paper studies the relationship between these fibrations and the uniformizing symmetric space of the examples.