A Miyaoka-Yau inequality for hyperplane arrangements in $\mathbb{CP}^n$

TL;DR

This paper establishes a Miyaoka-Yau type inequality for hyperplane arrangements in complex projective space, linking stability conditions to intersection properties and providing bounds on multiplicities of intersection subspaces.

Contribution

It introduces a quadratic form based on the intersection poset and applies the Bogomolov-Gieseker inequality to derive new bounds for stable arrangements.

Findings

The quadratic form Q is non-positive for stable arrangements.

A lower bound for the sum of multiplicities of codimension 2 intersections is established.

Equality cases correspond to arrangements with maximal intersection properties.

Abstract

Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{CP}^n$. We define a quadratic form $Q$ on $\mathbb{R}^{\mathcal{H}}$ that is entirely determined by the intersection poset of $\mathcal{H}$. Using the Bogomolov-Gieseker inequality for parabolic bundles, we show that if $\mathbf{a} \in \mathbb{R}^{\mathcal{H}}$ is such that the weighted arrangement $(\mathcal{H}, \mathbf{a})$ is stable, then $Q(\mathbf{a}) \leq 0$. As an application, we consider the symmetric case where all the weights are equal. The inequality $Q(a, \ldots, a) \leq 0$ gives a lower bound for the total sum of multiplicities of codimension $2$ intersection subspaces of $\mathcal{H}$. The lower bound is attained when every $H \in \mathcal{H}$ intersects all the other members of $\mathcal{H} \setminus \{H\}$ along $(1-2/(n+1))|\mathcal{H}| + 1$ codimension $2$ subspaces; extending from $n=2$ to higher dimensions a condition found by Hirzebruch for line arrangements in the complex projective plane.