Comptes Rendus
Algebraic Geometry
On the vector bundles over rationally connected varieties
[Des fibrés vectoriels sur les variétés rationnellement connexes]
Comptes Rendus. Mathématique, Volume 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.09.006

Indranil Biswas 1 ; João Pedro P. dos Santos 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
@article{CRMATH_2009__347_19-20_1173_0,
     author = {Indranil Biswas and Jo\~ao Pedro P. dos Santos},
     title = {On the vector bundles over rationally connected varieties},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1173--1176},
     publisher = {Elsevier},
     volume = {347},
     number = {19-20},
     year = {2009},
     doi = {10.1016/j.crma.2009.09.006},
     language = {en},
}
TY  - JOUR
AU  - Indranil Biswas
AU  - João Pedro P. dos Santos
TI  - On the vector bundles over rationally connected varieties
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 1173
EP  - 1176
VL  - 347
IS  - 19-20
PB  - Elsevier
DO  - 10.1016/j.crma.2009.09.006
LA  - en
ID  - CRMATH_2009__347_19-20_1173_0
ER  - 
%0 Journal Article
%A Indranil Biswas
%A João Pedro P. dos Santos
%T On the vector bundles over rationally connected varieties
%J Comptes Rendus. Mathématique
%D 2009
%P 1173-1176
%V 347
%N 19-20
%I Elsevier
%R 10.1016/j.crma.2009.09.006
%G en
%F CRMATH_2009__347_19-20_1173_0
Indranil Biswas; João Pedro P. dos Santos. On the vector bundles over rationally connected varieties. Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1173-1176. doi : 10.1016/j.crma.2009.09.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.006/

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :

Commentaires - Politique