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Comptes Rendus Mathématique
Volume 347, n° 19-20
pages 1173-1176 (octobre 2009)
Doi : 10.1016/j.crma.2009.09.006
Received : 28 June 2009 ;  accepted : 2 September 2009
On the vector bundles over rationally connected varieties
Des fibrés vectoriels sur les variétés rationnellement connexes
 

Indranil Biswas a , João Pedro P. dos Santos b
a School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India 
b Institut de mathématiques de Jussieu, 175, rue du Chevaleret, 75013 Paris, France 

Abstract

Let X be a rationally connected smooth projective variety defined over   and   a vector bundle such that for every morphism  , the pullback   is trivial. We prove that E is trivial. Using this we show that if   is isomorphic to   for all γ of the above type, where   is some line bundle, then there is a line bundle ζ over X such that  . To cite this article: I. Biswas, J.P.P. dos Santos, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.
Résumé

Soit X une variété rationnellement connexe sur   et soit   un fibré vectoriel tel que, pour tout morphisme  , le fibré   est trivial. Nous montrons que E est trivial. Nous en déduisons que si, pour tout γ comme avant,   est isomorphe à  , où   est un fibré en droites, alors il existe un fibré en droites ζ sur X et un isomorphisme  . Pour citer cet article : I. Biswas, J.P.P. dos Santos, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.


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