On the vector bundles over rationally connected varieties

Biswas, Indranil; dos Santos, João Pedro P.

doi:10.1016/j.crma.2009.09.006

Algebraic Geometry

On the vector bundles over rationally connected varieties
[Des fibrés vectoriels sur les variétés rationnellement connexes]

Présenté par : Jean-Pierre Demailly

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

Comptes Rendus. Mathématique, Tome 347 (2009) no. 19-20, pp. 1173-1176.

Résumé
Abstract

Soit X une variété rationnellement connexe sur $C$ et soit $E \to X$ un fibré vectoriel tel que, pour tout morphisme $γ : {CP}^{1} \to X$ , le fibré $γ^{*} E$ est trivial. Nous montrons que E est trivial. Nous en déduisons que si, pour tout γ comme avant, $γ^{*} E$ est isomorphe à $L {(γ)}^{\oplus r}$ , où $L (γ) \to {CP}^{1}$ est un fibré en droites, alors il existe un fibré en droites ζ sur X et un isomorphisme $E ≅ ζ^{\oplus r}$ .

Let X be a rationally connected smooth projective variety defined over $C$ and $E \to X$ a vector bundle such that for every morphism $γ : {CP}^{1} \to X$ , the pullback $γ^{*} E$ is trivial. We prove that E is trivial. Using this we show that if $γ^{*} E$ is isomorphic to $L {(γ)}^{\oplus r}$ for all γ of the above type, where $L (γ) \to {CP}^{1}$ is some line bundle, then there is a line bundle ζ over X such that $E = ζ^{\oplus r}$ .

Reçu le : 2009-06-28
Accepté le : 2009-09-02
Publié le : 2009-09-19

DOI : 10.1016/j.crma.2009.09.006

Affiliations des auteurs :

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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     title = {On the vector bundles over rationally connected varieties},
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     pages = {1173--1176},
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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://archive.numdam.org/articles/10.1016/j.crma.2009.09.006/

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