Aspects of functoriality in homological mirror symmetry for toric varieties

Abstract

We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. In particular, we realize tropical Lagrangian sections as objects of a partially wrapped category and construct a Lagrangian correspondence mirror to the inclusion of a toric divisor. As a corollary, we prove that tropical sections generate the Fukaya-Seidel category, completing a Floer-theoretic proof of homological mirror symmetry for projective toric varieties. In the course of the proof, we develop techniques for constructing Lagrangian cobordisms and Lagrangian correspondences in Liouville domains, which may be of independent interest.

arXiv comment78 pages, 22 Figures. Updates: minor expositional changes following referee report. Version accepted to Advances in Mathematics

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Cited by in corpus (5)

• Resolutions of toric subvarieties by line bundles and applications
• Homological mirror symmetry for hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$
• Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces
• Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
• Symplectomorphisms of some Weinstein 4-manifolds