Equivariant principal bundles on nonsingular toric varieties

TL;DR

This paper classifies equivariant principal G-bundles on nonsingular toric varieties with Abelian G, showing they always split and providing explicit parametrizations for certain cases.

Contribution

It introduces a classification framework for equivariant principal G-bundles on nonsingular toric varieties and demonstrates their splitting property, with explicit parametrizations for specific G and varieties.

Findings

Any equivariant principal G-bundle splits into a reduction to a torus intersection.

Provides explicit parametrization of isomorphism classes for complete varieties.

Establishes a classification for bundles with Abelian structure groups.

Abstract

We give a classification of the equivariant principal $G$-bundles on a nonsingular toric variety when $G$ is a closed Abelian subgroup of $GL_k(\mathbb{C})$. We prove that any such bundle splits, that is, admits a reduction of structure group to the intersection of $G$ with a torus. We give an explicit parametrization of the isomorphism classes of such bundles for a large family of $G$ when $X$ is complete.