Chern Numbers of Matroids

Abstract

We define Chern numbers of a matroid. These numbers are obtained when intersecting appropriate matroid Chern-Schwartz-MacPherson cycles defined by L\'opez de Medrano, Rinc\'on, and Shaw. We prove that when a matroid arises from a complex hyperplane arrangement the Chern numbers of the matroid correspond to the Chern numbers of the log cotangent bundle. A matroid of rank 3 has two Chern numbers. We prove that they are positive and that their ratio is bounded by 3, which is analogous to the Bogomolov-Miyaoka-Yau inequality. If the matroid is orientable, we generalize a result of Eterovi\'c, Figuera, and Urz\'ua to prove that the ratio is bounded above by 5/2. Finally, we give a formula for the Chern numbers of the uniform matroid of any rank.