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

Ruddat, Helge

doi:10.2140/gt.2021.25.3079

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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

DOI: 10.2140/gt.2021.25.3079

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

Keywords

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

References

Publication

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

Authors

Helge Ruddat
Mathematisches Institut Johannes Gutenberg-Universität Mainz Mainz Germany	Universität Hamburg Hamburg Germany

Volume 25, issue 6 (2021)