Polyhedral Kähler metrics on \mathbbCPⁿ

2025/10/20 by Martin de Borbon, de Borbon, Martin, Dmitri Panov +1
Mathematics · #Algebraic Geometry (math.AG) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2510.17447

openalex publication_date 2025/10/20 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28

Abstract

We give necessary and sufficient conditions for the existence of polyhedral Kähler metrics on \mathbbCPⁿ whose singular set is a hyperplane arrangement and whose cone angles are in (0, 2π). These conditions take the form of linear and quadratic constraints on the cone angles and are entirely determined by the intersection poset of the arrangement. Our proof of existence relies on a parabolic version of the Kobayashi-Hitchin correspondence, due to T. Mochizuki.

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