Equivariant principal bundles for G–actions and G–connections

Indranil Biswas 1 , S. Senthamarai Kannan 2 , and D. S. Nagaraj 3
  • 1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
  • 2 Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Kelambakkam 603103, India
  • 3 The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India

Abstract

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.

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  • [1] M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.

  • [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.

  • [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.

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