Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings

TL;DR

This paper classifies torus-equivariant vector bundles over toric schemes defined over a discrete valuation ring using piecewise linear maps into Bruhat-Tits buildings, extending existing classifications over fields and rings.

Contribution

It extends Klyachko's and Mumford's classifications to the setting of toric schemes over discrete valuation rings, providing a new framework for arithmetic geometry.

Findings

Classification of vector bundles via piecewise linear maps

Criterion for splitting into line bundles

Foundation for arithmetic studies of toric vector bundles

Abstract

We give a classification of rank $r$ torus equivariant vector bundles $\mathcal{E}$ on a toric scheme $\mathfrak{X}$ over a discrete valuation ring $\mathcal{O}$, in terms of graded piecewise linear maps $\Phi$ from the fan of $\mathfrak{X}$ to the (extended) building of $GL(r)$. This is an extension of Klyachko's classification of torus equivariant vector bundles on toric varieties over a field on one hand, and Mumford's classification of equivariant line bundles on toric schemes over $\mathcal{O}$ on the other hand. We also give a simple criterion for equivariant splitting of $\mathcal{E}$ into a sum of toric line bundles in terms of its piecewise linear map. Among other things, this work lays the foundations for study of arithmetic geometry of toric vector bundles.