On the vector bundles over rationally connected varieties

Abstract

Let X be a rationally connected smooth projective variety defined over C and E →X a vector bundle such that for every morphism γ :CP¹ → 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) ^⊕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=ζ^⊕r.