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://www.numdam.org/articles/10.1016/j.crma.2009.09.006/

Bibliographie
Cité par

[1] Andreatta, M.; Wiśniewski, J.A. On manifolds whose tangent bundle contains an ample subbundle, Invent. Math., Volume 146 (2001), pp. 209-217

[2] Behrend, K.; Manin, Yu. Stacks of stable maps and Gromov–Witten invariants, Duke Math. J., Volume 85 (1996), pp. 1-60

[3] Biswas, I.; Bruzzo, U. On semistable principal bundles over a complex projective manifold, Int. Math. Res. Not. IMRN, Volume 12 (2008) (Art. ID rnn035)

[4] Campana, F. On twistor spaces of the class $C$ , J. Diff. Geom., Volume 33 (1991), pp. 541-549

[5] Fulton, W.; Pandharipande, R. Notes on stable maps and quantum cohomology | arXiv

[6] Grothendieck, A. Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math., Volume 79 (1957), pp. 121-138

[7] Kollár, J. Fundamental groups of rationally connected varieties, Michigan Math. J., Volume 48 (2000), pp. 359-368

[8] Kollár, J. Rationally connected varieties and fundamental groups, Budapest, 2001 (Bolyai Soc. Math. Stud.), Volume vol. 12, Springer, Berlin (2003), pp. 69-92 | arXiv

[9] Kollár, J.; Miyaoka, Y.; Mori, S. Rationally connected varieties, J. Algebraic Geom., Volume 1 (1992), pp. 429-448

[10] Laumon, G.; Moret-Bailly, L. Champs algébriques, Ergeb. Math. Grenzgeb., vol. 39, Springer, 2000

[11] Lütkebohmert, W. On compactification of schemes, Manuscr. Math., Volume 80 (1993), pp. 95-111

[12] Mumford, D. Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Oxford University Press, London, 1970

[13] Simpson, C.T. Higgs bundles and local systems, Publ. Math. Inst. Hautes Études Sci., Volume 75 (1992), pp. 5-95

Cité par Sources :