# Journal of the Ramanujan Mathematical Society

Volume 28, Issue 4, December 2013 pp. 423–442.

Triviality criteria for bundles over rationally connected varieties**Authors**: Indranil Biswas and João Pedro Dos Santos

**Author institution:**School of Mathematics, Tata Institute of Fundamental Research, Homi BhabhaRoad, Bombay 400 005, India

**Summary:**Let X be a separably rationally connected smooth projective variety defined over an algebraically closed field K. If E → X is a vector bundle satisfying the condition that for every morphism γ : PK1 → X the pull-back γ*E is trivial, we prove that E is trivial. If E → X is a strongly semistable vector bundle such that c1(E) and c2(E) are numerically equivalent to zero, we prove that E is trivial. We also show that X does not admit any nontrivial stratified sheaf. These results are also generalized to principal bundles over X.

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