Hyperbolic structures on a toric arrangement complement

Abstract

This thesis studies the geometric structures on toric arrangement complements. Inspired by the special hypergeometric functions associated with a root system, we consider a family of connections on a total space which is the product of the complement of a toric arrangement (=finite union of hypertori) and $\mathbb{C}^{\times}$. We prove that these connections are torsion free and flat, and hence define a family of projective structures on the toric arrangement complement. We then further show that the projective structure underlies a hyperbolic structure when the multiplicity parameter lies in some certain region so that the complement admits a complex ball structure. This phenomenon could also be viewed as the geometry of the (trigonometric) Calogero-Moser system. Moreover, we prove that there also exists a family of Frobenius algebra structures on the total space, which indicates a connection with Gromov-Witten theory.