Moduli spaces of flat tori and elliptic hypergeometric functions

TL;DR

This paper explores the structure of moduli spaces of flat tori and elliptic hypergeometric functions, extending geometric and cohomological methods to analyze monodromy and flat surface properties.

Contribution

It generalizes Thurston's geometric results on flat spheres to flat tori using analytical and cohomological techniques, complementing a twin paper's geometric approach.

Findings

Explicit constructions of Veech on flat surfaces.

Generalization of Thurston's geometric results.

Analysis of monodromy of elliptic hypergeometric functions.

Abstract

In the genus one case, we make explicit some constructions of Veech on flat surfaces and generalize some geometric results of Thurston about moduli spaces of flat spheres as well as some equivalent ones but of an analytico-cohomological nature of Deligne-Mostow, which concern the monodromy of Appell-Lauricella hypergeometric functions. In the twin paper arXiv:1604.01812, we follow Thurston's approach and study moduli spaces of flat tori with conical singularities and prescribed holonomy by means of geometrical methods relying on surgeries for flat surfaces. In the present paper, we study the same objects making use of analytical and cohomological methods, more in the spirit of Deligne-Mostow's paper.