A Miyaoka–Yau inequality for hyperplane arrangements in CPn

Let be a hyperplane arrangement in . We define a quadratic form on that is entirely determined by the intersection poset of . Using the Bogomolov–Gieseker inequality for parabolic bundles, we show that if is such that the weighted arrangement is stable , then . As an application, we consider the symmetric case where all the weights are equal. The inequality gives a lower bound for the total sum of multiplicities of codimension intersection subspaces of . The lower bound is attained when every intersects all the other members of along codimension subspaces; extending from to higher dimensions a condition found by Hirzebruch for line arrangements in the complex projective plane.