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

Helge Ruddat

Geometry & Topology 25 (2021) 3079–3132

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  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 Poincaré–Lefschetz by simplicial methods for constructible sheaves might be of independent interest.

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tropical homology, torus fibration, SYZ, integrable system, toric degeneration, mirror symmetry, versality, affine structure, Picard-Lefschetz
Mathematical Subject Classification 2010
Primary: 14J32
Secondary: 05E45, 14D06, 14T05, 32S60, 55U10
Received: 27 February 2020
Revised: 25 July 2020
Accepted: 8 September 2020
Published: 30 November 2021
Proposed: Mark Gross
Seconded: Richard P Thomas, Dan Abramovich
Helge Ruddat
Mathematisches Institut
Johannes Gutenberg-Universität Mainz
Universität Hamburg