The talk is based on joint work with Dmitri Panov. Given a hyperplane arrangement in complex projective space, we associate to it a quadratic form (that we call the Hirzebruch quadratic form). We show that this form is semi-negative (under certain stability assumptions) and it vanishes precisely when there is a polyhedral Kähler metric with cone singularities along the hyperplane arrangement. I will also fit this result into a broader context of Kähler-Einstein metrics and the Miyaoka-Yau inequality for log pairs, where many question remain to be answered, based on previous work with Cristiano Spotti.