# Tropical Lagrangian Hypersurfaces are Unobstructed

## TL;DR

This paper constructs specific Lagrangian submanifolds in complex tori associated with tropical hypersurfaces and proves their unobstructedness in the Fukaya category, establishing a mirror symmetry correspondence.

## Contribution

It introduces a method to produce unobstructed Lagrangians from tropical hypersurfaces and demonstrates their role as mirror objects to sheaves on complex hypersurfaces.

## Key findings

- Constructed Lagrangians with tropical amoebas as moment map images.
- Proved unobstructedness of these Lagrangians in the Fukaya category.
- Established their mirror symmetry correspondence to sheaves on complex hypersurfaces.

## Abstract

We produce for each tropical hypersurface $V(\phi)\subset Q=\mathbb{R}^n$ a Lagrangian $L(\phi)\subset (\mathbb{C}^*)^n$ whose moment map projection is a tropical amoeba of $V(\phi)$. When these Lagrangians are admissible in the Fukaya-Seidel category, we show that they are unobstructed objects of the Fukaya category, and mirror to sheaves supported on complex hypersurfaces in a toric mirror.