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

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 $Q(\mathbf{a})\le 0$. As an application, we consider the symmetric case where all the weights are equal. The inequality $Q(a,\dots ,a)\le 0$ 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.

120 pages. Accepted for publication in J. Lon. Math. Soc