Ottaviani, Giorgio 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 C. R. Acad. Sci., Paris, Sér. I 305, 257-260 (1987). 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 Cited in 7 Documents 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 Keywords:splitting of vector bundle; Grassmannians; quadrics; vanishing of cohomology groups Citations:Zbl 0126.168 PDFBibTeX XMLCite \textit{G. Ottaviani}, C. R. Acad. Sci., Paris, Sér. I 305, 257--260 (1987; Zbl 0629.14012)