A Functorial Perspective on Homological Mirror Symmetry for Hypersurfaces

Abstract

In this thesis, we prove two homological mirror symmetry results for hypersurfaces of different kinds. In the first, 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. In the second part (joint work with Benjamin Gammage), we prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. We find that the complement of a very affine hypersurface has in fact two natural mirrors, one of which is a derived scheme. These two mirrors are related via a non-geometric equivalence mediated by Kn"orrer periodicity; Auroux's conjectures require some modification to take this into account. Our proof also uses new symplectic techniques for gluing Liouville sectors which may be of independent interest.