A classification of equivariant principal bundles over nonsingular toric varieties

Biswas, Indranil; Dey, Arijit; Poddar, Mainak

doi:10.1142/s0129167x16501159

Cited by 11 publications

(24 citation statements)

References 12 publications

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“…Several attempts have been made recently to study the more general category of toric principal bundles (cf. [5,7,8,22]). In this article, we prove some foundational results regarding the classification of torus equivariant principal bundles over complex toric varieties and their morphisms.…”

Section: Introductionmentioning

confidence: 99%

“…Let X be a complex toric variety with respect to a given action of an algebraic torus T , and let G be a complex linear algebraic group. A description of T -equivariant principal G-bundles when X is nonsingular was given in the article [5]. It is in terms of admissible collections of homomorphisms ρ σ : T − G and group elements P (τ, σ) ∈ G that satisfy some extension and cocycle conditions respectively (see Definition 2.5).…”

Section: Introductionmentioning

confidence: 99%

“…Two admissible collections are said to be equivalent if they satisfy certain conjugacy relations. The isomorphism classes of T -equivariant principal G-bundles over X are claimed to be in bijective correspondence with the equivalence classes of admissible collections [5,Theorem 3.2]. However, the definition of the equivalence relation for admissible collections in [5] is insufficient to prove that equivalent admissible collections produce isomorphic torus equivariant principal bundles.…”

Section: Introductionmentioning

confidence: 99%

“…The isomorphism classes of T -equivariant principal G-bundles over X are claimed to be in bijective correspondence with the equivalence classes of admissible collections [5,Theorem 3.2]. However, the definition of the equivalence relation for admissible collections in [5] is insufficient to prove that equivalent admissible collections produce isomorphic torus equivariant principal bundles. We note here that extension conditions are required not only in the definition of admissible collections, but also in defining their equivalence (see Definition 3.4).…”

Section: Introductionmentioning

confidence: 99%

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Classification, reduction and stability of toric principal bundles

Dasgupta¹,

Khan²,

Biswas³

et al. 2020

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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 Gbundles E over X and the morphisms between them. When G is connected and reductive, we characterize the equivariant automorphism group AutT (E ) of E as the intersection of certain parabolic subgroups of G that arise naturally from the T -action on E . We then give a criterion for the equivariant reduction of the structure group of E to a Levi subgroup of G in terms of AutT (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. Contents 1. Introduction 2. Preliminaries 3. Classification of equivariant principal bundles 4. Kaneyama type description 5. Morphisms of equivariant principal bundles 6. Examples 6.1. The tangent frame bundle 6.2. Equivariant automorphisms of the tangent frame bundle 7. Equivariant reduction of structure group 7.1. Equivariant reduction of structure group to a Levi subgroup 7.2. Equivariant splitting of an equivariant principal bundle 8. Stability of equivariant principal bundles Acknowledgement

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Classification, reduction and stability of toric principal bundles

Dasgupta¹,

Khan²,

Biswas³

et al. 2020

Preprint

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“…Equivariant vector bundles over a nonsingular complete toric variety up to isomorphism were first classified by Kaneyama [19], [20] by involving both combinatorial and linear algebraic data modulo an equivalence relation. Recently this work has been generalized for equivariant principal G-bundles over smooth complex toric variety, where G is a complex linear algebraic group [3]. Later in a foundational paper [25], Klyachko classified equivariant vector bundles more systematically.…”

Section: Introductionmentioning

confidence: 99%

Stability of equivariant vector bundles over toric varieties

Dasgupta¹,

Dey²,

Khan³

2019

Preprint

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We give a complete answer to the question of (semi)stability of tangent bundle of any nonsingular projective complex toric variety with Picard number 2 by using combinatorial crietrion of (semi)stability of an equivariant sheaf. We also give a complete answer to the question of (semi)stability of tangent bundle of all toric Fano 4-folds with Picard number ≤ 3 which are classified by Batyrev [1]. We have constructed a collection of equivariant indecomposable rank 2 vector bundles on Bott tower and pseudo-symmetric toric Fano varieties. Further in case of Bott tower, we have shown the existence of an equivariant stable rank 2 vector bundle with certain Chern classes with respect to a suitable polarization.

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Tannakian Classification of Equivariant Principal Bundles on Toric Varieties

Biswas

Dey

Poddar

2020

Transformation Groups

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Let X be a complete toric variety equipped with the action of a torus T , and G be a reductive algebraic group, defined over C. We introduce the notion of a compatible Σ-filtered algebra associated to X, generalizing the notion of a compatible Σ-filtered vector space due to Klyachko. 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 Σ-filtered algebras associated to X.

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A classification of equivariant principal bundles over nonsingular toric varieties

Cited by 11 publications

References 12 publications

Classification, reduction and stability of toric principal bundles

Classification, reduction and stability of toric principal bundles

Stability of equivariant vector bundles over toric varieties

Tannakian Classification of Equivariant Principal Bundles on Toric Varieties

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