Tropical Lagrangian multi-sections and toric vector bundles

TL;DR

This paper introduces tropical Lagrangian multi-sections over fans and explores their connection to toric vector bundles, providing a combinatorial criterion in dimension 2 for when such multi-sections correspond to toric vector bundles.

Contribution

It develops a new tropical geometric framework for understanding toric vector bundles and offers a mirror-symmetric construction that reinterprets existing algebraic data.

Findings

A new notion of tropical Lagrangian multi-sections over fans.

A mirror-symmetric construction for toric vector bundles.

A combinatorial condition in dimension 2 for multi-sections arising from vector bundles.

Abstract

We introduce the notion of tropical Lagrangian multi-sections over a fan and study its relation with toric vector bundles. We also introduce a "SYZ-type" construction for toric vector bundles which gives a reinterpretation of Kaneyama's linear algebra data. In dimension 2, such "mirror-symmetric" approach provides us a pure combinatorial condition for checking which rank 2 tropical Lagrangian multi-section arises from toric vector bundles.