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

Abstract

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