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 
• Hazewinkel, Michiel, ed. (2001), "Kempf_vanishing_theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Kempf_vanishing_theorem 
• 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, Bibcode: 1980InMat..56..109H 
• 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