Geometric structures on the complement of a toric mirror arrangement

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.