Geometric structures on the complement of a toric mirror arrangement

TL;DR

This paper investigates geometric structures on the complement of toric mirror arrangements linked to root systems, establishing flat connections, affine and projective structures, and identifying conditions for complex hyperbolic metrics and ball quotient mappings.

Contribution

It introduces a family of flat, torsion-free connections on toric arrangement complements, connecting hypergeometric functions with geometric structures and hyperbolic metrics.

Findings

Connections are torsion-free and flat.

Existence of a family of affine and projective structures.

Identification of parameter regions with hyperbolic metrics and ball quotient biholomorphisms.

Abstract

We study geometric structures on the complement of a toric mirror arrangement associated with a root system. Inspired by those special hypergeometric functions found by Heckman-Opdam, as well as the work of Couwenberg-Heckman-Looijenga on geometric structures on projective arrangement complements, we consider a family of connections on a total space, namely, a $\mathbb{C}^{\times}$-bundle on the complement of a toric mirror arrangement (=finite union of hypertori, determined by a root system). We prove that these connections are torsion free and flat, and hence define a family of affine structures on the total space, which is equivalent to a family of projective structures on the toric arrangement complement. We then determine a parameter region for which the projective structure admits a locally complex hyperbolic metric. In the end, we find a finite subset of this region for which the orbifold in question can be biholomorphically mapped onto a Heegner divisor complement of a ball quotient.