Principal bundles over a smooth real projective curve of genus zero

Biswas, Indranil ; Huisman, Johannes (2008) Principal bundles over a smooth real projective curve of genus zero Advances in Geometry, 8 (3). pp. 451-472. ISSN 1615-715X

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Official URL: http://www.reference-global.com/doi/abs/10.1515/AD...

Related URL: http://dx.doi.org/10.1515/ADVGEOM.2008.029

Abstract

Let H0 denote the kernel of the endomorphism, defined by z → (z/z-)2, of the real algebraic group given by the Weil restriction of C*. Let X be a nondegenerate anisotropic conic in P2R. The principal C*-bundle over the complexification XC, defined by the ample generator of Pic(XC), gives a principal H0-bundle FH0 over X through a reduction of structure group. Given any principal G-bundle EG over X, where G is any connected reductive linear algebraic group defined over R, we prove that there is a homomorphism ρ : H0 → G such that EG is isomorphic to the principal G-bundle obtained by extending the structure group of FH0 using ρ. The tautological line bundle over the real projective line P1 R, and the principal Z/2Z- bundle P1 C → P1 R, together give a principal Gm × (Z/2Z)-bundle F on P1 R. Given any principal G-bundle EG over P1R, where G is any connected reductive linear algebraic group defined over R, we prove that there is a homomorphism ρ : Gm × (Z/2Z) → G such that EG is isomorphic to the principal G-bundle obtained by extending the structure group of F using ρ.

Item Type:Article
Source:Copyright of this article belongs to Walter de Gruyter GmbH & Co. KG.
Keywords:Principal Bundle; Real Conic; Reductive Group
ID Code:3525
Deposited On:12 Oct 2010 04:20
Last Modified:16 May 2016 14:18

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