Tropical Vector Bundles

TL;DR

This paper introduces a construction of vector bundles on tropical varieties, especially tropical toric varieties, using valuated matroids, and explores their sections, stability, and filtrations.

Contribution

It provides a new framework for tropical vector bundles, linking them to valuated matroids and tropical linear spaces, with detailed stability and filtration theory.

Findings

Tropical vector bundles are described via valuated matroids.

Global sections and stability notions are defined for these bundles.

Filtrations like Jordan-H"older and Harder-Narasimhan are established.

Abstract

The goal of this paper is to introduce a construction of a vector bundle on a tropical variety. When the base is a tropical toric variety these tropicalize toric vector bundles, and are described by the data of a valuated matroid and some flats in the lattice of flats of the underlying matroid. The fibers are tropical linear spaces. We define global sections for tropical toric vector bundles, stability, and Jordan-H\"older and Harder-Narasimhan filtrations. Many of these require additional modularity assumptions on the defining matroid.