Chern Classes of Tropical Manifolds

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.