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.

44 pages

References in corpus (7)

• Higher Topos Theory
• Covariantly functorial wrapped Floer theory on Liouville sectors
• Microlocal Morse theory of wrapped Fukaya categories
• Arboreal Singularities in Weinstein Skeleta
• Remarks on the nonequivariant coherent-constructible correspondence for toric varieties
• Lagrangian Skeleta of Hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$
• Sectorial descent for wrapped Fukaya categories

Cited by in corpus (15)

• Microlocal Morse theory of wrapped 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 Truncated Cluster Varieties
• Mirror symmetry for perverse schobers from birational geometry
• Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties
• 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
• Local mirror symmetry via SYZ
• 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