Hurwitz Spaces and Moduli Spaces as Ball Quotients via Pull-back

TL;DR

This paper explores the construction of ball quotient structures on certain moduli spaces derived from Hurwitz spaces and Deligne-Mostow spaces, using hypergeometric functions and intersection homology, to understand their geometric and topological properties.

Contribution

It introduces a pull-back method to identify subball quotients within Deligne-Mostow moduli spaces and analyzes specific examples related to binary forms and elliptic surfaces.

Findings

Identification of conditions for subball quotient structures

Detailed analysis of Gaussian and Eisenstein examples

Recovery of known moduli space structures for elliptic surfaces

Abstract

We define hypergeometric functions using intersection homology valued in a local system. Topology is emphasized; analysis enters only once, via the Hodge decomposition. By a pull-back procedure we construct special subsets S_{pi}, derived from Hurwitz spaces, of Deligne-Mostow moduli spaces DM(n,mu). Certain DM(n,mu) are known to be ball quotients, uniformized by hypergeometric functions valued in a complex ball (i.e., complex hyperbolic space). We give sufficient conditions for S_{pi} to be a subball quotient. Analyzing the simplest examples in detail, we describe ball quotient structures attached to some moduli spaces of inhomogeneous binary forms. This recovers in particular the structure on the moduli space of rational elliptic surfaces given by Heckman and Looijenga. We make use of a natural partial ordering on the Deligne-Mostow examples (which gives an easy way to see that the original list of Mostow, eventually corrected by Thurston, is in error), and so highlight two key examples, which we call the Gaussian and Eisenstein ancestral examples.