Superforms, Tropical Cohomology, and Poincar\'e Duality

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.