Chern classes of tropical vector bundles

TL;DR

This paper introduces tropical vector bundles, defines their Chern classes, and classifies all such bundles on elliptic curves, expanding the tropical geometry framework.

Contribution

It develops the theory of tropical vector bundles, including Chern classes, and provides a full classification on elliptic curves, a novel advancement in tropical geometry.

Findings

Defined tropical vector bundles and rational sections

Established properties of tropical Chern classes

Classified all vector bundles on elliptic curves

Abstract

We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.