Mirror symmetry and Fukaya categories of singular hypersurfaces

TL;DR

This paper defines a Fukaya category for singular hypersurfaces, proving its desirable properties, including an analog of Orlov's derived Kn"orrer periodicity and implications for homological mirror symmetry in certain degenerations.

Contribution

It introduces a new Fukaya category for singular hypersurfaces and establishes its derived equivalence to Fukaya-Seidel categories, advancing mirror symmetry understanding.

Findings

Proves an A-side analog of Orlov's derived Kn"orrer periodicity.

Shows the Fukaya category is derived equivalent to Fukaya-Seidel category of a Landau-Ginzburg model.

Demonstrates homological mirror symmetry for specific degenerations of abelian varieties.

Abstract

We consider a definition of the Fukaya category of a singular hypersurface proposed by Auroux, given by localizing the Fukaya category of a nearby fiber at Seidel's natural transformation, and show that this possesses several desirable properties. Firstly, we prove an A-side analog of Orlov's derived Kn\"orrer periodicity theorem by showing that Auroux's category is derived equivalent to the Fukaya-Seidel category of a higher-dimensional Landau-Ginzburg model. Secondly, we describe how this definition implies homological mirror symmetry for some large complex structure limit degenerations of abelian varieties.