Lagrangian pairs of pants

TL;DR

This paper introduces Lagrangian pairs of pants, a new class of Lagrangian submanifolds in cotangent bundles, which serve as fundamental components for lifting tropical varieties to smooth Lagrangian submanifolds in complex tori.

Contribution

It constructs Lagrangian pairs of pants as graphs of differentials of functions on blow-ups of coamoebas, enabling the lifting of tropical curves to Lagrangian submanifolds in complex tori.

Findings

Construction of Lagrangian pairs of pants in cotangent bundles.

Method to lift tropical curves to Lagrangian submanifolds in $(\mathbb{C}^*)^2$.

New examples of Lagrangian submanifolds in toric varieties, including monotone cases.

Abstract

We construct a Lagrangian submanifold, inside the cotangent bundle of a real torus, which we call a Lagrangian pair of pants. It is given as the graph of the differential of a smooth function defined on the real blow up of a Lagrangian coamoeba. Lagrangian pairs of pants are the main building blocks in a construction of smooth Lagrangian submanifolds of $(\mathbb{C}^*)^n$ which lift tropical subvarieties in $\mathbb{R}^n$. As an example we explain how to lift tropical curves in $\mathbb{R}^2$ to Lagrangian submanifolds of $(\mathbb{C}^*)^2$. We also give several new examples of Lagrangian submanifolds inside toric varieties, some of which are monotone.