K\"{a}hler cone-manifolds arising from a projective arrangement

TL;DR

This paper explores the geometric structure of complements of hyperplane arrangements in projective space, revealing they form Kähler cone-manifolds even without Schwarz conditions, generalizing Thurston's cone metrics on spheres.

Contribution

It demonstrates that the space associated with Dunkl systems forms a Kähler cone-manifold without Schwarz conditions, extending Thurston's cone metrics framework.

Findings

The space is a Kähler cone-manifold without Schwarz conditions.

Generalizes Thurston's cone metrics on spheres.

Identifies new geometric structures from hyperplane arrangements.

Abstract

Given a hyperplane arrangement of some type in a projective space, the Dunkl system, developed by Couwenberg, Heckman and Looijenga, is used to study the geometric structures on its complement, and as a consequence it leads to the discovery of new ball quotients when the Schwarz conditions are imposed. In this paper, we study how the space, investigated in this system, looks like when there is no Schwarz conditions imposed. As a result, we show that the space in question is still of a particular type of structure, namely, the structure of a cone-manifold. With this point of view, Thurston's work on moduli of cone metrics on the sphere appears as a special case in this set-up.