Equivariant vector bundles on complexity-one T-varieties and Bruhat-Tits buildings

TL;DR

This paper provides a combinatorial classification of torus-equivariant vector bundles on complexity-one T-varieties, extending previous classifications for line bundles and toric varieties using Bruhat-Tits buildings.

Contribution

It introduces a new classification framework for equivariant vector bundles on complexity-one T-varieties via piecewise affine maps to Bruhat-Tits buildings.

Findings

Classification extends Petersen-Süss's work on line bundles.

Generalizes Klyachko's classification for toric varieties.

Uses Bruhat-Tits buildings to describe vector bundles over DVRs.

Abstract

We give a combinatorial classification of torus equivariant vector bundles on a (normal) projective T-variety of complexity-one. This extends the classification of equivariant line bundles on complexity-one T-varieties by Petersen-S\"uss on one hand, and Klyachko's classification of equivariant vector bundles on toric varieties on the other hand. A main ingredient in our classification is the classification of torus equivariant vector bundles on toric schemes over a DVR in terms of piecewise affine maps to the (extended) Bruhat-Tits building of the general linear group.