Superforms, tropical cohomology, and Poincaré duality

Abstract

We establish a canonical isomorphism between two bigraded cohomology theories for polyhedral spaces: Dolbeault cohomology of superforms and tropical cohomology. Furthermore, we prove Poincaré duality for cohomology of tropical manifolds, which are polyhedral spaces locally given by Bergman fans of matroids.Furthermore, the authors would like to thank the Graduierten Kolleg "GRK 1692" by the Deutsche Forschungsgemeinschaft for making possible the lecture series by the second author that inspired this collaboration. SuperformsIf we consider the sections of this diagram over a basic open subset, then the first row is exact by Proposition 3.11. By Lemma 4.32 the second row is also exact. This shows that both rows are exact sequences of sheaves on X. Thus we have a commutative diagram of acyclic resolutions of L p , thus PD induces isomorphisms on the cohomology of the complexes of global sections. This precisely means that X has PD.When X is a compact tropical manifold, the above theorem immediately implies the following.Corollary 4.34. Let X be a compact tropical manifold of dimension n. Then

Monodromy map for tropical Dolbeault cohomology

“…Assuming X compact, they define a map φ∩ : H p,q (X) ⊗ R → H p+1,q−1 (X) ⊗ R using their eigenwave φ. In fact, as shown in [JSS15], one can compute the tropical homology H p,q (X) ⊗ R, or rather the tropical cohomology, via superforms in [CLD12] as well. Then our construction in §2 would give rise to a map N X : H p,q (X) ⊗ R → H p+1,q−1 (X) ⊗ R (or rather on cohomology) for any tropical space X.…”

Cellular Sheaf Cohomology in Polymake

“…Notice that we obtain dim H q (B(M ); F p ) = dim H BM d−q (B(M ); F d−p ) for d = 1, which is the dimension of the Bergman fan. This is the homological version of Poincaré duality for matroidal fans and tropical manifolds from [16].…”

A comparison of positivity in complex and tropical toric geometry

“…On the spaces X an Σ and N Σ , there are sheaves of bigraded algebras of smooth differential forms. Both are denoted A •,• or, when we want to stress the underlying space, by A N Σ is a sheaf of R-algebras and was introduced by Smacka, Shaw and the third author [14], based on work of Lagerberg [16]. The smooth differential forms on X an Σ are called complex forms while the forms on N Σ are called Lagerberg forms.…”