Journal of the Ramanujan Mathematical Society

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Triviality Criteria for Bundles Over Rationally Connected Varieties


Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India
2 Universit´e de Paris 6, Institut de Mathematiques de Jussieu, 4, Place Jussieu, 75005, Paris, France
     

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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 : P1K −→ 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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  • Triviality Criteria for Bundles Over Rationally Connected Varieties

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Authors

Indranil Biswas
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India
Joao Pedro Dos Santos
Universit´e de Paris 6, Institut de Mathematiques de Jussieu, 4, Place Jussieu, 75005, Paris, France

Abstract


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 : P1K −→ 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.