Mirror symmetry for very affine hypersurfaces

Abstract

We show that the category of coherent sheaves on the toric boundary divisor of a smooth quasiprojective toric DM stack is equivalent to the wrapped Fukaya category of a hypersurface in a complex torus. Hypersurfaces with every Newton polytope can be obtained. Our proof has the following ingredients. Using recent results on localization, we may trade wrapped Fukaya categories for microlocal sheaf theory along the skeleton of the hypersurface. Using Mikhalkin-Viro patchworking, we identify the skeleton of the hypersurface with the boundary of the Fang-Liu-Treumann-Zaslow skeleton. By proving a new functoriality result for Bondal's coherent-constructible correspondence, we reduce the sheaf calculation to Kuwagaki's recent theorem on mirror symmetry for toric varieties.

arXiv comment44 pages

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Cited by in corpus (16)

• Microlocal Morse theory of wrapped Fukaya categories
• Calabi-Yau structures on topological Fukaya categories
• Sheaf quantization in Weinstein symplectic manifolds
• Homological mirror symmetry at large volume
• Lagrangian Skeleta of Hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$
• Mirror symmetry for Berglund-Hübsch Milnor fibers
• Mirror symmetry for perverse schobers from birational geometry
• Mirror Symmetry for Truncated Cluster Varieties
• Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties
• Local mirror symmetry via SYZ
• Homological mirror symmetry for hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$
• Gamma II for toric varieties from integrals on T-dual branes and homological mirror symmetry
• Versality in mirror symmetry
• Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces
• Variation of GIT and Variation of Lagrangian Skeletons I: Flip and Flop
• Smoothing, scattering, and a conjecture of Fukaya