Aspects of functoriality in homological mirror symmetry for toric varieties

TL;DR

This paper advances homological mirror symmetry for toric varieties by demonstrating tropical Lagrangian sections generate the Fukaya-Seidel category and developing new techniques for Lagrangian cobordisms and correspondences.

Contribution

It provides a Floer-theoretic proof of homological mirror symmetry for projective toric varieties and introduces methods for constructing Lagrangian cobordisms and correspondences.

Findings

Tropical Lagrangian sections generate the Fukaya-Seidel category.

Constructed a Lagrangian correspondence mirror to the inclusion of a toric divisor.

Developed techniques for Lagrangian cobordisms in Liouville domains.

Abstract

We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. In particular, we realize tropical Lagrangian sections as objects of a partially wrapped category and construct a Lagrangian correspondence mirror to the inclusion of a toric divisor. As a corollary, we prove that tropical sections generate the Fukaya-Seidel category, completing a Floer-theoretic proof of homological mirror symmetry for projective toric varieties. In the course of the proof, we develop techniques for constructing Lagrangian cobordisms and Lagrangian correspondences in Liouville domains, which may be of independent interest.