Chern Numbers of Matroids

TL;DR

This paper introduces Chern numbers for matroids, linking them to geometric properties of hyperplane arrangements and establishing bounds similar to classical inequalities, thus bridging combinatorics and algebraic geometry.

Contribution

It defines Chern numbers for matroids, relates them to hyperplane arrangements, and proves bounds analogous to classical geometric inequalities, extending previous results to orientable matroids.

Findings

Chern numbers of matroids are positive for rank 3.

The ratio of Chern numbers is bounded by 3 for general matroids.

For orientable matroids, the ratio is bounded by 5/2.

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.