Chern-Schwarz-Macpherson classes are one way to extend the notion of Chern classes to singular and non-complete varieties. In this talk, I will provide a combinatorial analogue of these classes for matroids. In this setting, the CSM class is given by Minkowski weights supported on the Bergman fan of the matroid. For matroids arising from hyperplane arrangements over \mathbb{C} these Minkowski weights encode the CSM class of the complement of the arrangement in various compactifications.

One goal in doing this is to obtain a Chow theoretic description of matroid invariants such as the characteristic polynomial, h-vector, and conjecturally Speyer’s g-polynomial and the Tutte polynomial. Secondly, these combinatorial CSM classes can be used to define Chern classes of tropical manifolds which are locally modelled on Bergman fans of matroids.

This is based on joint work with Lucia Lopez de Medrano and Felipe Rincon and also work in progress with Alex Fink and David Speyer.