Equivariant principal bundles over toric Deligne--Mumford stacks
Abstract
Let 𝒳(Σ) be the toric Deligne--Mumford stack with stacky torus 𝒯 associated to the stacky fan Σ = (N,Σ, β). A toric principal H-bundle over 𝒳(Σ) is a principal H-bundle 𝒫 equipped with a 𝒯-action lifting the 𝒯-action on 𝒳(Σ), such that the 𝒯-action and the H-action on 𝒫 commute. For a connected reductive algebraic group H over ℂ, we classify the isomorphism classes of framed toric principal H-bundles over 𝒳(Σ) in terms of piecewise linear maps from |Σ| to the cone over the Tits building of H. Using this classification, we describe the equivariant automorphism group of a toric principal H-bundle, and give a necessary and sufficient condition for an equivariant reduction of the structure group. We also show that every toric principal GL(r)-bundle over the weighted projective stack ℙ(w0,…,wn) splits equivariantly whenever r < n, and that every toric principal H-bundle over the weighted stacky projective line ℙ(a,b) splits equivariantly when H is a connected reductive algebraic group.
Related
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- A Quadratically Enriched Pushforward for Deligne--Mumford Stacks
- Compactification Independence of the Irregular Hodge Filtration on Deligne-Mumford Stacks
- On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks
- Irregular Hodge Bundles for Deligne--Mumford Landau--Ginzburg Families with Fixed Pole Orders
BibTeX
@misc{arora2026equivariant,
title = {{Equivariant principal bundles over toric Deligne--Mumford stacks}},
author = {Ramandeep Singh Arora and Chandranandan Gangopadhyay and Mainak Poddar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33488},
eprint = {2609.33488},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33488},
}
RIS
TY - UNPB
TI - Equivariant principal bundles over toric Deligne--Mumford stacks
AU - Ramandeep Singh Arora
AU - Chandranandan Gangopadhyay
AU - Mainak Poddar
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33488
UR - https://arxiv.org/abs/2609.33488
AB - Let 𝒳(Σ) be the toric Deligne--Mumford stack with stacky torus 𝒯 associated to the stacky fan Σ = (N,Σ, β). A toric principal H-bundle over 𝒳(Σ) is a principal H-bundle 𝒫 equipped with a 𝒯-action lifting the 𝒯-action on 𝒳(Σ), such that the 𝒯-action and the H-action on 𝒫 commute. For a connected reductive algebraic group H over ℂ, we classify the isomorphism classes of framed toric principal H-bundles over 𝒳(Σ) in terms of piecewise linear maps from |Σ| to the cone over the Tits building of H. Using this classification, we describe the equivariant automorphism group of a toric principal H-bundle, and give a necessary and sufficient condition for an equivariant reduction of the structure group. We also show that every toric principal GL(r)-bundle over the weighted projective stack ℙ(w_0,…,w_n) splits equivariantly whenever r < n, and that every toric principal H-bundle over the weighted stacky projective line ℙ(a,b) splits equivariantly when H is a connected reductive algebraic group.
ER -