Tannakian classification of equivariant principal bundles on toric varieties

TL;DR

This paper extends Klyachko's classification of equivariant vector bundles on toric varieties to principal G-bundles using Tannakian formalism, establishing an equivalence of categories in characteristic zero.

Contribution

It introduces compatible Σ-filtered algebras to classify equivariant principal G-bundles on toric varieties, generalizing previous vector bundle classifications.

Findings

Establishes an equivalence between T-equivariant principal G-bundles and compatible Σ-filtered algebras.

Generalizes Klyachko's vector bundle classification to principal G-bundles.

Applicable in characteristic zero fields.

Abstract

Let $X$ be a complete toric variety equipped with the action of a torus $T$ and $G$ a reductive algebraic group, defined over an algebraically closed field $K$. We introduce the notion of a compatible $\Sigma$--filtered algebra associated to $X$, generalizing the notion of a compatible $\Sigma$--filtered vector space due to Klyachko, where $\Sigma$ denotes the fan of $X$. We combine Klyachko's classification of $T$--equivariant vector bundles on $X$ with Nori's Tannakian approach to principal $G$--bundles, to give an equivalence of categories between $T$--equivariant principal $G$--bundles on $X$ and certain compatible $\Sigma$--filtered algebras associated to $X$, when the characteristic of $K$ is $0$.