Lagrangian submanifolds from tropical hypersurfaces

TL;DR

This paper demonstrates how smooth tropical hypersurfaces in three-dimensional space can be lifted to smooth embedded Lagrangian submanifolds in complex three-dimensional space, advancing the understanding of their geometric relationship.

Contribution

It provides a rigorous proof that smooth tropical hypersurfaces in $\

Findings

Tropical hypersurfaces in $\

Lagrangian submanifolds can be constructed from tropical hypersurfaces in $\

The method uses Lagrangian pairs of pants as building blocks.

Abstract

We prove that a smooth tropical hypersurface in $\mathbb{R}^3$ can be lifted to a smooth embedded Lagrangian submanifold in $(\mathbb{C}^*)^3$. This completes the proof of the result announced in the article "Lagrangian pairs pants" arXiv:1802.02993. The idea of the proof is to use Lagrangian pairs of pants as the main building blocks.