Toric vector bundles, valuations and tropical geometry

TL;DR

This paper introduces valuation and tropical geometry perspectives on toric vector bundles, providing new classifications, criteria for ampleness and global generation, and a polytopal encoding related to matroids.

Contribution

It offers three equivalent classifications of toric vector bundles using valuations and tropical geometry, extending existing matroid and polytope frameworks.

Findings

Classifies toric vector bundles via piecewise linear maps and valuations.

Interprets ampleness and global generation as convexity conditions.

Associates polytopes to vector bundles encoding global section dimensions.

Abstract

A toric vector bundle $\mathcal{E}$ is a torus equivariant vector bundle on a toric variety. We give a valuation theoretic and tropical point of view on toric vector bundles. We present three (equivalent) classifications of toric vector bundles, which should be regarded as repackagings of the Klyachko data of compatible $\mathbb{Z}$-filtrations of a toric vector bundle: (1) as piecewise linear maps to space of $\mathbb{Z}$-valued valuations, (2) as valuations with values in the semifield of piecewise linear functions, and (3) as points in tropical linear ideals over the semifield of piecewise linear functions. Moreover, we interpret the known criteria for ampleness and global generation of $\mathcal{E}$ as convexity conditions on its piecewise linear map in (1). Finally, using (2) we associate to $\mathcal{E}$ a collection of polytopes indexed by elements of a certain (representable) matroid encoding the dimensions of weight spaces of global sections of $\mathcal{E}$. This recovers and extends the Di Rocco-Jabbusch-Smith matriod and parliament of polytopes of $\mathcal{E}$. This is a follow up paper to arXiv:1806.05613.