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 .
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 .
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@article{CRMATH_2009__347_19-20_1173_0, author = {Biswas, Indranil and dos Santos, Jo\~ao Pedro P.}, 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}, url = {http://www.numdam.org/articles/10.1016/j.crma.2009.09.006/} }
TY - JOUR AU - Biswas, Indranil AU - dos Santos, João Pedro P. 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 UR - http://www.numdam.org/articles/10.1016/j.crma.2009.09.006/ DO - 10.1016/j.crma.2009.09.006 LA - en ID - CRMATH_2009__347_19-20_1173_0 ER -
%0 Journal Article %A Biswas, Indranil %A dos Santos, João Pedro P. %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 %U http://www.numdam.org/articles/10.1016/j.crma.2009.09.006/ %R 10.1016/j.crma.2009.09.006 %G en %F CRMATH_2009__347_19-20_1173_0
Biswas, Indranil; dos Santos, João Pedro P. 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. http://www.numdam.org/articles/10.1016/j.crma.2009.09.006/
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