Classification, reduction and stability of toric principal bundles

TL;DR

This paper classifies torus-equivariant principal G-bundles over complex toric varieties, characterizes their automorphism groups, and explores conditions for reductions and stability, extending known results to a broader class of bundles.

Contribution

It provides a comprehensive classification of T-equivariant principal G-bundles, characterizes automorphism groups, and establishes criteria for reductions and stability, generalizing Kaneyama's theorem.

Findings

Classification of all T-equivariant principal G-bundles over X.

Characterization of automorphism groups as intersections of parabolic subgroups.

Equivalence of stability and equivariant stability for principal G-bundles.

Abstract

Let $X$ be a complex toric variety equipped with the action of an algebraic torus $T$, and let $G$ be a complex linear algebraic group. We classify all $T$-equivariant principal $G$-bundles $\mathcal{E}$ over $X$ and the morphisms between them. When $G$ is connected and reductive, we characterize the equivariant automorphism group $\text{Aut}_T(\mathcal{E} )$ of $\mathcal{E}$ as the intersection of certain parabolic subgroups of $G$ that arise naturally from the $T$-action on $\mathcal{E}$. We then give a criterion for the equivariant reduction of the structure group of $\mathcal{E}$ to a Levi subgroup of $G$ in terms of $\text{Aut}_T(\mathcal{E} )$. We use it to prove a principal bundle analogue of Kaneyama's theorem on equivariant splitting of torus equivariant vector bundles of small rank over a projective space. When $X$ is projective and $G$ is connected and reductive, we show that the notions of stability and equivariant stability are equivalent for any $T$-equivariant principal $G$-bundle over $X$.