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

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 Q in degree 1 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 Poincare-Lefschetz by simplicial methods for constructible sheaves might be of independent interest.

Automated summary

The paper introduces a cap product pairing for homology and cohomology of tropical cycles on integral affine manifolds with singularities, showing its perfection over Q in degree 1 under certain conditions. Collaboration with Siebert leads to computations of period integrals and implications for the versatility of canonical Calabi-Yau degenerations. Additionally, there is an intersection-theoretic application for Strominger-Yau-Zaslow fibrations, with the simplicial methods for constructible sheaves potentially of independent interest.