  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.
