Kempf vanishing theorem

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, Bibcode:1980InMat..56..109H, doi:10.1007/BF01392545, ISSN 0020-9910, MR 0558862, S2CID 121863316
• 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