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

Let H$\mathcal {H}$ be a hyperplane arrangement in CPn$\mathbb {CP}^n$ . We define a quadratic form Q$Q$ on RH$\mathbb {R}^{\mathcal {H}}$ that is entirely determined by the intersection poset of H$\mathcal {H}$ . Using the Bogomolov–Gieseker inequality for parabolic bundles, we show that if a∈RH$\mathbf {a}\in \mathbb {R}^{\mathcal {H}}$ is such that the weighted arrangement (H,a)$(\mathcal {H}, \mathbf {a})$ is stable, then Q(a)⩽0$Q(\mathbf {a}) \leqslant 0$ . As an application, we consider the symmetric case where all the weights are equal. The inequality Q(a,…,a)⩽0$Q(a, \ldots, a) \leqslant 0$ gives a lower bound for the total sum of multiplicities of codimension 2$\hskip.001pt 2$ intersection subspaces of H$\mathcal {H}$ . The lower bound is attained when every H∈H$H \in \mathcal {H}$ intersects all the other members of H∖{H}$\mathcal {H}\setminus \lbrace H\rbrace$ along (1−2/(n+1))|H|+1$(1-2/(n+1))|\mathcal {H}| + 1$ codimension 2$\hskip.001pt 2$ subspaces; extending from n=2$n=2$ to higher dimensions a condition found by Hirzebruch for line arrangements in the complex projective plane.