Toric principal bundles, piecewise linear maps and Tits buildings

TL;DR

This paper introduces a new classification framework for toric principal bundles using piecewise linear maps to Tits buildings, extending Klyachko's work and applying to various algebraic groups.

Contribution

It defines piecewise linear maps to Tits buildings and shows they classify toric principal G-bundles, generalizing existing classifications for vector bundles.

Findings

Classifies toric principal G-bundles via piecewise linear maps

Recovers Klyachko's classification of toric vector bundles

Provides new classifications for orthogonal and symplectic toric bundles

Abstract

We define the notion of a piecewise linear map from a fan $\Sigma$ to $\tilde{\mathfrak{B}}(G)$, the cone over the Tits building of a linear algebraic group $G$. Let $X_\Sigma$ be a toric variety with fan $\Sigma$. We show that when $G$ is reductive the set of integral piecewise linear maps from $\Sigma$ to $\tilde{\mathfrak{B}}(G)$ classifies the isomorphism classes of (framed) toric principal $G$-bundles on $X_\Sigma$. This in particular recovers Klyachko's classification of toric vector bundles, and gives new classification results for the orthogonal and symplectic toric principal bundles.