Jump to content

Kempf vanishing theorem

From Wikipedia, the free encyclopedia

This is an old revision of this page, as edited by Citation bot (talk | contribs) at 00:22, 16 February 2019 (Add: jstor. Removed parameters. | You can use this bot yourself. Report bugs here. | User-activated.). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

In algebraic geometry, the Kempf vanishing theorem, introduced by Kempf (1976), states that the higher cohomology group Hi(G/B,L(λ)) (i > 0) vanishes whenever λ is a dominant weight of B. Here G is a reductive algebraic group over an algebraically closed field, B a Borel subgroup, and L(λ) a line bundle associated to λ. In characteristic 0 this is a special case of the Borel–Weil–Bott theorem, but unlike the Borel–Weil–Bott theorem, the Kempf vanishing theorem still holds in positive characteristic.

Andersen (1980) and Haboush (1980) found simpler proofs of the Kempf vanishing theorem using the Frobenius morphism.

References

  • Andersen, Henning Haahr (1980), "The Frobenius morphism on the cohomology of homogeneous vector bundles on G/B", Annals of Mathematics, Second Series, 112 (1): 113–121, doi:10.2307/1971322, ISSN 0003-486X, JSTOR 1971322, MR 0584076
  • "Kempf_vanishing_theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Haboush, William J. (1980), "A short proof of the Kempf vanishing theorem", Inventiones Mathematicae, 56 (2): 109–112, doi:10.1007/BF01392545, ISSN 0020-9910, MR 0558862
  • Kempf, George R. (1976), "Linear systems on homogeneous spaces", Annals of Mathematics, Second Series, 103 (3): 557–591, doi:10.2307/1970952, ISSN 0003-486X, JSTOR 1970952, MR 0409474