Polyhedral K\"ahler cone metrics on $\mathbb{C}^n$ singular at hyperplane arrangements

Abstract

Let $X$ be a complex manifold and let $g$ be a polyhedral metric on it inducing its topology. We say that $g$ is a polyhedral K\"ahler (PK) metric on $X$ if it is K\"ahler outside its singular set. The local geometry of PK metrics is modelled on PK cones, and in this article we focus on an interesting class of examples of these. Following work of Couwenberg-Heckman-Looijenga, we consider a special kind of flat torsion free meromorphic connections on $\mathbb{C}^n$ with simple poles at the hyperplanes of a linear arrangement. In the case of unitary holonomy we show that, under suitable numerical conditions, the metric completion is a PK cone metric on $\mathbb{C}^n$. We apply our results to the essential braid arrangement, extending to higher dimensions the classical story of spherical metrics on $\mathbb{CP}^1$ with three cone points.