Equivariant vector bundles on toric schemes over semirings

TL;DR

This paper introduces a framework for equivariant vector bundles on schemes over semirings, demonstrating their splitting into line bundles on toric schemes and extending classification results to this setting.

Contribution

It develops a new notion of equivariant vector bundles over semirings, proves their splitting into line bundles on toric schemes, and extends Klyachko's classification theorem to this context.

Findings

Toric vector bundles over idempotent semifields split into sums of line bundles.

The equivariant Picard group is studied and characterized.

A version of Klyachko's classification theorem is established for this setting.

Abstract

We introduce a notion of equivariant vector bundles on schemes over semirings. We do this by considering the functor of points of a locally free sheaf. We prove that every toric vector bundle on a toric scheme $X$ over an idempotent semifield equivariantly splits as a sum of toric line bundles. We then study the equivariant Picard group $\text{Pic}_G(X)$. Finally, we prove a version of Klyachko's classification theorem for toric vector bundles over an idempotent semifield.