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Vanishing theorems for tensor powers of a positive vector bundle

Part of the Lecture Notes in Mathematics book series (LNM,volume 1339)

Abstract

Let E be a holmorphic vector bundle of rank r over a compact complex manifold X of dimension n. It is shown that the Dolbeault cohomology groups H p,q (X, E ⊗k ⊗(det E)l) vanish if E is positive in the sense of Griffiths and p+qn+1, lr+C (n, p, q). The proof rests on the wellknown fact that every tensor power E ⊗k splits into irreducible representations of Gl(E), each component being canonically isomorphic to the direct image on X of a positive homogeneous line bundle over a flag manifold of E. The vanishing property is then obtained by a suitable generalization of Le Potier's isomorphism theorem, combined with a new curvature estimate for the bundle of X-relative differential forms on the flag manifold of E.

Keywords

  • Vector Bundle
  • Line Bundle
  • Irreducible Representation
  • Spectral Sequence
  • Holomorphic Vector Bundle

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© 1988 Springer-Verlag

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Demailly, JP. (1988). Vanishing theorems for tensor powers of a positive vector bundle. In: Sunada, T. (eds) Geometry and Analysis on Manifolds. Lecture Notes in Mathematics, vol 1339. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083049

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  • DOI: https://doi.org/10.1007/BFb0083049

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  • Print ISBN: 978-3-540-50113-8

  • Online ISBN: 978-3-540-45930-9

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