A homology theory for tropical cycles on integral affine manifolds and a perfect pairing

Abstract

We introduce a cap product pairing for homology and cohomology of tropical cycles on integral affine manifolds with singularities. We show the pairing is perfect over $\mathbb{Q}$ in degree one when the manifold has at worst symple singularities. By joint work with Siebert, the pairing computes period integrals and its perfectness implies the versality of canonical Calabi-Yau degenerations. We also give an intersection theoretic application for Strominger-Yau-Zaslow fibrations. The treatment of the cap product and Poincaré-Lefschetz by simplicial methods for constructible sheaves might be of independent interest.

arXiv comment49 pages, 12 illustrations, corrected version, Construction 11 got added

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Cited by in corpus (7)

• Tropically constructed Lagrangians in mirror quintic threefolds
• A tropical view on Landau-Ginzburg models
• A Note on $h^{2,1}$ of Divisors of CY Fourfolds I
• Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology class
• Tropical contractions to integral affine manifolds with singularities
• Differential forms and cohomology in tropical and complex geometry
• Cohomologically tropical varieties