Invariant connections and invariant holomorphic bundles on homogeneous manifolds

Indranil Biswas; Andrei Teleman

Open Mathematics (2014)

  • Volume: 12, Issue: 1, page 1-13
  • ISSN: 2391-5455

Abstract

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Let X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects: equivalence classes of α-invariant K-connections on X α-invariant gauge classes of K-connections on X, andα-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic Kℂ-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).

How to cite

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Indranil Biswas, and Andrei Teleman. "Invariant connections and invariant holomorphic bundles on homogeneous manifolds." Open Mathematics 12.1 (2014): 1-13. <http://eudml.org/doc/269764>.

@article{IndranilBiswas2014,
abstract = {Let X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects: equivalence classes of α-invariant K-connections on X α-invariant gauge classes of K-connections on X, andα-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic Kℂ-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).},
author = {Indranil Biswas, Andrei Teleman},
journal = {Open Mathematics},
keywords = {Invariant connection; Gauge group; Principal bundle; invariant connection; gauge group; principal bundle},
language = {eng},
number = {1},
pages = {1-13},
title = {Invariant connections and invariant holomorphic bundles on homogeneous manifolds},
url = {http://eudml.org/doc/269764},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Indranil Biswas
AU - Andrei Teleman
TI - Invariant connections and invariant holomorphic bundles on homogeneous manifolds
JO - Open Mathematics
PY - 2014
VL - 12
IS - 1
SP - 1
EP - 13
AB - Let X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects: equivalence classes of α-invariant K-connections on X α-invariant gauge classes of K-connections on X, andα-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic Kℂ-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).
LA - eng
KW - Invariant connection; Gauge group; Principal bundle; invariant connection; gauge group; principal bundle
UR - http://eudml.org/doc/269764
ER -

References

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  1. [1] Biswas I., Homogeneous principal bundles over the upper half-plane, Kyoto J. Math., 2010, 50(2), 325–363 http://dx.doi.org/10.1215/0023608X-2009-016 Zbl1204.53015
  2. [2] Biswas I., Classification of homogeneous holomorphic hermitian principal bundles over G/K, Forum Math. (in press), DOI: 10.1515/forum-2012-0131 
  3. [3] Chevalley C., Eilenberg S., Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 1948, 63(1), 85–124 http://dx.doi.org/10.1090/S0002-9947-1948-0024908-8 Zbl0031.24803
  4. [4] Donaldson S.K., Kronheimer P.B., The Geometry of Four-Manifolds, Oxford Math. Monogr., Oxford University Press, New York, 1990 Zbl0820.57002
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  6. [6] Hofmann K.H., Morris S.A., The Structure of Compact Groups, 2nd ed., de Gruyter Stud. Math., 25, Walter de Gruyter, Berlin, 2006 Zbl1139.22001
  7. [7] Husemoller D., Fibre Bundles, 3rd ed., Grad. Texts in Math., 20, Springer, New York, 1994 http://dx.doi.org/10.1007/978-1-4757-2261-1 
  8. [8] Kobayashi S., Nomizu K., Foundations of Differential Geometry, I, Interscience Tracts in Pure and Applied Mathematics, 15(I), Interscience, New York-London, 1963 
  9. [9] Kobayashi S., Nomizu K., Foundations of Differential Geometry, II, Interscience Tracts in Pure and Applied Mathematics, 15(II), Interscience, New York-London, 1969 Zbl0175.48504
  10. [10] Lübke M., Teleman A., The Universal Kobayashi-Hitchin Correspondence on Hermitian Manifolds, Mem. Amer. Math. Soc., 183(863), American Mathematical Society, Providence, 2006 Zbl1103.53014
  11. [11] Wang H., On invariant connections over a principal fibre bundle, Nagoya Math. J., 1958, 13, 1–19 Zbl0086.36502
  12. [12] Yang K., Almost Complex Homogeneous Spaces and their Submanifolds, World Scientific, Singapore, 1987 Zbl0645.53001

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