Given a complex manifold M equipped with an action of a group G, and a holomorphic principal H–bundle EH on M, we introduce the notion of a connection on EH along the action of G, which is called a G–connection. We show some relationship between the condition that EH admits a G–equivariant structure and the condition that EH admits a (flat) G–connection. The cases of bundles on homogeneous spaces and smooth toric varieties are discussed.
@article{bwmeta1.element.doi-10_1515_coma-2015-0013,
author = {Indranil Biswas and S. Senthamarai Kannan and D. S. Nagaraj},
title = {Equivariant principal bundles for G--actions and G--connections},
journal = {Complex Manifolds},
volume = {2},
year = {2015},
zbl = {06545108},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_coma-2015-0013}
}
Indranil Biswas; S. Senthamarai Kannan; D. S. Nagaraj. Equivariant principal bundles for G–actions and G–connections. Complex Manifolds, Tome 2 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_coma-2015-0013/
[1] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207. | Zbl 0078.16002
[2] I Biswas, A. Dey and M. Poddar, Equivariant principal bundles and logarithmic connections on toric varieties, Pacific Jour. Math. (to appear), arXiv:1507.02415. | Zbl 06537053
[3] N. Bourbaki, ´El´ements demath´ematique. XXVI. Groupes et alg`ebres de Lie. Chapitre 1: Alg`eebres de Lie, Actualit´es Sci. Ind. No. 1285, Hermann, Paris 1960.
[4] I. Moerdijk and J. Mrˇcun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003.