Homological mirror symmetry for hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$

Abstract

We prove a homological mirror symmetry result for maximally degenerating families of hypersurfaces in $({\mathbb{C}}^{\ast }{)}^n$ (B-model) and their mirror toric Landau-Ginzburg A-models. The main technical ingredient of our construction is a "fiberwise wrapped" version of the Fukaya category of a toric Landau-Ginzburg model. With the definition in hand, we construct a fibered admissible Lagrangian submanifold whose fiberwise wrapped Floer cohomology is isomorphic to the ring of regular functions of the hypersurface. It follows that the derived category of coherent sheaves of the hypersurface quasi-embeds into the fiberwise wrapped Fukaya category of the mirror. We also discuss an extension to complete intersections.

arXiv comment91 pages; v2 minor revision: improved exposition and added examples in response to referee report

References in corpus (13)

• Covariantly functorial wrapped Floer theory on Liouville sectors
• On partially wrapped Fukaya categories
• Mirror symmetry for very affine hypersurfaces
• Auslander orders over nodal stacky curves and partially wrapped Fukaya categories
• Enumerative meaning of mirror maps for toric Calabi-Yau manifolds
• Floer theory and reduced cohomology on open manifolds
• SYZ mirror symmetry for toric Calabi-Yau manifolds
• Homological mirror symmetry for higher dimensional pairs of pants
• A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology
• Mirror symmetry for the Landau-Ginzburg A-model $M={\mathbb{C}}^n,W=z_1\cdots z_n$
• Aspects of functoriality in homological mirror symmetry for toric varieties
• Categorical mirror symmetry on cohomology for a complex genus 2 curve
• Mirror symmetry for honeycombs

Cited by in corpus (1)

• Persistence of unknottedness of clean Lagrangian intersections