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A Barth-Lefschetz type theorem for branched coverings of Grassmannians. (English) Zbl 0835.14003

Using the “spannedness” and “\(k\)-ampleness” (in the sense of Sommese) of the vector bundle associated with a branched covering, \(f : X \to G\), of the Grassmannian \(G=\text{Gr}(r,n)\), we prove that \(f\) induces \(\mathbb{C}\)-cohomology isomorphisms between \(X\) and \(G\) in a certain range of cohomology dimensions, provided the degree of the covering is small enough. This generalizes Lazarsfeld’s theorem on branched coverings of projective space. An analogous statement is proved true for the homotopy case.
Reviewer: M.Kim (Notre Dame)

MSC:

14E20 Coverings in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14M15 Grassmannians, Schubert varieties, flag manifolds
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