Algebraic Geometry
On the vector bundles over rationally connected varieties
[Des fibrés vectoriels sur les variétés rationnellement connexes]
Comptes Rendus. Mathématique, Tome 347 (2009) no. 19-20, pp. 1173-1176.

Soit X une variété rationnellement connexe sur C et soit EX un fibré vectoriel tel que, pour tout morphisme γ:CP1X, le fibré γE est trivial. Nous montrons que E est trivial. Nous en déduisons que si, pour tout γ comme avant, γE est isomorphe à L(γ)r, où L(γ)CP1 est un fibré en droites, alors il existe un fibré en droites ζ sur X et un isomorphisme Eζr.

Let X be a rationally connected smooth projective variety defined over C and EX a vector bundle such that for every morphism γ:CP1X, the pullback γE is trivial. We prove that E is trivial. Using this we show that if γE is isomorphic to L(γ)r for all γ of the above type, where L(γ)CP1 is some line bundle, then there is a line bundle ζ over X such that E=ζr.

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DOI : 10.1016/j.crma.2009.09.006
Biswas, Indranil 1 ; dos Santos, João Pedro P. 2

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
2 Institut de mathématiques de Jussieu, 175, rue du Chevaleret, 75013 Paris, France
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Biswas, Indranil; dos Santos, João Pedro P. On the vector bundles over rationally connected varieties. Comptes Rendus. Mathématique, Tome 347 (2009) no. 19-20, pp. 1173-1176. doi : 10.1016/j.crma.2009.09.006. http://www.numdam.org/articles/10.1016/j.crma.2009.09.006/

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