Chern Classes of Tropical Manifolds

TL;DR

This paper extends the concept of Chern-Schwartz-MacPherson cycles to tropical manifolds, establishing invariance, correspondence theorems, an adjunction formula, and a Noether's formula for tropical surfaces.

Contribution

It introduces an invariant description of CSM cycles for matroids, proves correspondence theorems for tropicalizations, and extends classical formulas to tropical geometry.

Findings

Invariance of CSM cycles under integer affine transformations

Correspondence theorems for tropicalizations of subvarieties

Noether's formula for compact tropical surfaces

Abstract

We extend the definitions of Chern-Schwartz-MacPherson (CSM) cycles of matroids to tropical manifolds. To do this, we provide an alternate description of CSM cycles of matroids which is invariant under integer affine transformations. Utilising results of Esterov and Katz-Stapledon, we prove correspondence theorems for the CSM classes of tropicalisations of subvarieties of toric varieties. We also provide an adjunction formula relating the CSM cycles of a tropical manifold and a codimension-one tropical submanifold. Lastly, we establish Noether's Formula for compact tropical surfaces with a Delzant face structure. This extends the class of surfaces for which the formula had been previously proved by the third author.