Tropical Feichtner-Yuzvinsky and positivity criterion for fans

TL;DR

This paper establishes a deep connection between the Chow ring of simplicial fans and tropical cohomology, providing new criteria for positivity and ampleness in tropical geometry, with applications to matroids.

Contribution

It proves an isomorphism between the Chow ring and tropical cohomology for simplicial fans and introduces a tropical ampleness criterion, extending classical algebraic geometry concepts.

Findings

Chow ring of simplicial fans is isomorphic to middle degree tropical cohomology.

Tropical analogue of Kleiman's ampleness criterion for fans.

Chow ring of matroids represented as cohomology of tropical manifolds.

Abstract

We prove that the Chow ring of any simplicial fan is isomorphic to the middle degree part of the tropical cohomology ring of its canonical compactification. Using this result, we prove a tropical analogue of Kleiman's criterion of ampleness for fans. In the case of tropical fans that are homology manifolds, we obtain an isomorphism between the Chow ring of the fan and the entire tropical cohomology of the canonical compactification. When applied to matroids, this provides a new representation of the Chow ring of a matroid as the cohomology ring of a projective tropical manifold.