Non-displaceable toric fibers on compact toric manifolds via tropicalizations

TL;DR

This paper introduces a combinatorial method using tropicalizations to identify all non-displaceable Lagrangian toric fibers in compact toric manifolds, linking tropical geometry with Floer theory.

Contribution

It provides a new combinatorial approach to locate non-displaceable fibers, confirming all bulk-balanced fibers are strongly bulk-balanced, thus unifying previous Floer-theoretic results.

Findings

All strongly bulk-balanced fibers are located via tropicalizations.

The method detects all non-displaceable fibers accessible by Floer theory.

A combinatorial intersection of tropicalizations identifies these fibers.

Abstract

We give a combinatorial way to locate non-displaceable Lagrangian toric fibers on compact toric manifolds. By taking the intersection of certain tropicalizations coming from combinatorial data of a moment polytope, we locate all strongly bulk-balanced fibers introduced in [FOOOSurv]. As an application, we show that every bulk-balanced fiber defined in [FOOOToric2] is strongly bulk-balanced. Thus, the method indeed detects the positions of all non-displaceable fibers that can be detected by Lagrangian Floer theory developed in [FOOOToric1] and [FOOOToric2].