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

TL;DR

This paper establishes a homological mirror symmetry correspondence for hypersurfaces in complex tori and their mirror Landau-Ginzburg models, introducing a fiberwise wrapped Fukaya category and demonstrating a quasi-embedding of derived categories.

Contribution

It introduces a fiberwise wrapped Fukaya category for toric Landau-Ginzburg models and constructs a Lagrangian submanifold linking the categories of hypersurfaces and their mirrors.

Findings

Proves homological mirror symmetry for hypersurfaces in $( ext{C}^*)^n$.

Constructs a fiberwise wrapped Fukaya category.

Shows a quasi-embedding of derived categories.

Abstract

We prove a homological mirror symmetry result for maximally degenerating families of hypersurfaces in $(\mathbb{C}^*)^n$ (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.