Mirror symmetry for very affine hypersurfaces

TL;DR

This paper establishes a mirror symmetry equivalence between the category of coherent sheaves on a toric boundary divisor and the wrapped Fukaya category of a hypersurface in a complex torus, using microlocal sheaf theory and skeleton identification.

Contribution

It introduces a new functoriality result for Bondal's correspondence and connects the skeleton of hypersurfaces with known mirror symmetry frameworks.

Findings

Equivalence between coherent sheaves on toric boundary and wrapped Fukaya category

Identification of hypersurface skeleton with Fang-Liu-Treumann-Zaslow skeleton

Reduction of sheaf calculations to known mirror symmetry results

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.