Chern-Schwartz-MacPherson cycles of matroids

TL;DR

This paper introduces Chern-Schwartz-MacPherson cycles for matroids, linking combinatorial structures to geometric classes and revealing their valuation properties and relation to characteristic polynomials.

Contribution

It defines CSM cycles for arbitrary matroids and establishes their geometric and combinatorial properties, including their relation to CSM classes and characteristic polynomials.

Findings

CSM cycles are balanced weighted fans supported on Bergman fan skeleta.

Degrees of CSM cycles correspond to coefficients of a shifted reduced characteristic polynomial.

CSM cycles are valuations under matroid polytope subdivisions.

Abstract

We define Chern-Schwartz-MacPherson (CSM) cycles of an arbitrary matroid. These are balanced weighted fans supported on the skeleta of the corresponding Bergman fan. In the case that the matroid arises from a complex hyperplane arrangement A, we show that these cycles represent the CSM class of the complement of A. We also prove that for any matroid, the degrees of its CSM cycles are given by the coefficients of (a shift of) the reduced characteristic polynomial, and that CSM cycles are valuations under matroid polytope subdivisions.