2021/06/24 by Martin de Borbon, de Borbon, Martin, Dmitri Panov +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Meromorphic and Entire Functions #math.DG
paper · pdf · doi:10.48550/arxiv.2106.13224
119 pages, 4 figures
arxiv created 2021/06/24 · openalex publication_date 2021/06/24 · arxiv updated 2021/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Kähler (PK) metric on X if it is Kähler 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 ℂn 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 ℂn. We apply our results to the essential braid arrangement, extending to higher dimensions the classical story of spherical metrics on \mathbbCP1 with three cone points.