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Critères de scindage pour les fibrés vectoriels sur les Grassmanniennes et les quadriques. (Splitting criteria for vector bundles on Grassmannians and quadrics). (French) Zbl 0629.14012

Let \({\mathbb{E}}\) be a vector bundle on a \({\mathbb{C}}\)-projective space \({\mathbb{P}}_ n\). G. Horrocks [Proc. Lond. Math. Soc., III. Ser. 14, 689-713 (1964; Zbl 0126.168)] established the following criterion: Theorem: \({\mathbb{E}}\) splits (i.e. \({\mathbb{E}}\) is a direct sum of line bundles) iff (*) \(H^ i({\mathbb{P}}_ n,{\mathbb{E}}(t))=0\) for all \(i\in]0,n[\) and all \(t\in {\mathbb{Z}}\) where \({\mathbb{E}}(t):={\mathbb{E}}\otimes {\mathcal O}_{{\mathbb{P}}_ n(t)}.\)
Here the author tries to extend this result to a vector bundle E on Grassmannians G(k,n) and quadrics \(Q_ n\). Certainly in these cases the vanishing of cohomology groups (*) is not enough to guarantee the splitting of \({\mathbb{E}}.\)
The author announces, for vector bundles \({\mathbb{E}}\) on G(k,n) (resp. \(Q_ n)\), necessary and sufficient conditions for \({\mathbb{E}}\) to split which take into account the behavior of the quotient bundle (resp. the “spinor” bundle).
In the special case where \(rank({\mathbb{E}})=2\), simplified necessary and sufficient conditions for splitting of \({\mathbb{E}}\) are also announced.
Reviewer: Vo Van Tan

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14M15 Grassmannians, Schubert varieties, flag manifolds
14F25 Classical real and complex (co)homology in algebraic geometry

Citations:

Zbl 0126.168
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