Superforms, Tropical Cohomology, and Poincar\'e Duality

TL;DR

This paper proves a fundamental isomorphism between two cohomology theories for polyhedral spaces and establishes Poincaré duality for tropical manifolds, advancing the understanding of tropical geometry and its algebraic structures.

Contribution

It introduces a canonical isomorphism between superform Dolbeault cohomology and tropical cohomology, and proves Poincaré duality for tropical manifolds.

Findings

Canonical isomorphism between two cohomology theories

Poincaré duality holds for tropical manifolds

Advances the mathematical framework of tropical geometry

Abstract

We establish a canonical isomorphism between two bigraded cohomology theories for polyhedral spaces: Dolbeault cohomology of superforms and tropical cohomology. Furthermore, we prove Poincar\'e duality for cohomology of tropical manifolds, which are polyhedral spaces locally given by Bergman fans of matroids.