Polyhedral Kähler metrics on $\mathbb{CP}^n$
Martin de Borbon, Dmitri Panov
TL;DR
We classify polyhedral Kähler (PK) metrics on CP^n with cone singularities along a hyperplane arrangement by translating geometric data into weighted hyperplane configurations. The core strategy combines logarithmic connections with parabolic bundle theory: PK metrics correspond to flat, torsion-free, unitary logarithmic connections adapted to a weighted divisor, and the Kobayashi–Hitchin correspondence for parabolic bundles (Mochizuki) yields a PK metric when the parabolic second Chern character vanishes and the klt condition holds. The work introduces and utilizes the De Concini–Procesi log resolution X, Yuzvinsky’s basis of H^{2k}(X,Z), and Ohtsuki’s residue formula to relate residues to Chern classes, enabling explicit linear and quadratic constraints Q_L(a)=0 that depend only on the intersection poset. The results generalize known 2D cases (e.g., Panov) to higher dimensions and connect to complex reflection arrangements, showing how combinatorics control metric existence. Overall, the paper provides a precise, combinatorial criterion for PK metric existence on CP^n with prescribed cone angles along hyperplanes, blending hyperplane arrangement theory, parabolic geometry, and Kähler analysis to extend the landscape of conical Kähler metrics.
Abstract
We give necessary and sufficient conditions for the existence of polyhedral Kähler metrics on $\mathbb{CP}^n$ 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.