Polyhedral K\"ahler metrics on $\mathbb{CP}^n$

TL;DR

This paper establishes precise conditions for the existence of polyhedral K"ahler metrics with specified cone angles on complex projective space, linking geometric structures to combinatorial data of hyperplane arrangements.

Contribution

It provides necessary and sufficient linear and quadratic conditions for such metrics, using a parabolic Kobayashi-Hitchin correspondence approach.

Findings

Conditions depend on the intersection poset of the hyperplane arrangement

Existence characterized by linear and quadratic constraints on cone angles

Method relies on a parabolic Kobayashi-Hitchin correspondence

Abstract

We give necessary and sufficient conditions for the existence of polyhedral K\"ahler metrics on $\mathbb{CP}^n$ whose singular set is a hyperplane arrangement and whose cone angles are in $(0, 2\pi)$. 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.