Tropical Lagrangian Hypersurfaces are Unobstructed

Abstract

We produce for each tropical hypersurface $V(\varphi )\subset Q={\mathbb{R}}^n$ a Lagrangian $L(\varphi )\subset ({\mathbb{C}}^{\ast }{)}^n$ whose moment map projection is a tropical amoeba of $V(\varphi )$. 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.

48 Pages. First portion of arXiv:1904.06005 which was split in two. This version has been accepted for publication by the Journal of Topology