Biswas, Indranil ; Gomez, Tomas L. ; Holla, Yogish I. (2002) Reduction of structure group of principal bundles over aprojective manifold with Picard number one International Mathematics Research Notices, 2002 (3). 889 903. ISSN 10737928

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Official URL: http://imrn.oxfordjournals.org/cgi/content/abstrac...
Related URL: http://dx.doi.org/10.1155/S1073792802110026
Abstract
Let X be a complex projective manifold with Pic(X) ≅ Z. Let G be a connected reductive algebraic group. A principal Gbundle will becalled split if it admits a reductionof structure group to a maximal torus of G. This corresponds tothe usual definition of a split vector bundle as a direct sum ofline bundles. Let E be a principal Gbundle on X, and let ρ G→ G* be an injective homomorphism.We prove that if the associated principal G*bundle E(G*) is split, then E admits a reduction to a Borel subgroupof G. Furthermore, if we assume that X is Fano or has trivialcanonical bundle, then E is also split. We apply this to obtain generalizations for principal bundles of two results about vector bundles on projective spaces. Namely, a principal Gbundle on CP is split if its restriction to a plane CP CPis split. Also, a principal Gbundle E on CP is trivial if there is a point p such that the restriction of E to every line through p is trivial.
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Deposited On:  12 Oct 2010 04:19 
Last Modified:  16 May 2016 14:19 
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