Kempf vanishing theorem
{{short description|Theorem in algebraic geometry}}
In algebraic geometry, the Kempf vanishing theorem, introduced by {{harvs|txt|last=Kempf|year=1976|authorlink=George Kempf}}, 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.
{{harvtxt|Andersen|1980}} and {{harvtxt|Haboush|1980}} found simpler proofs of the Kempf vanishing theorem using the Frobenius morphism.
References
- {{Citation | last1=Andersen | first1=Henning Haahr | title=The Frobenius morphism on the cohomology of homogeneous vector bundles on G/B | doi=10.2307/1971322 |mr=584076 | year=1980 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=112 | issue=1 | pages=113–121| jstor=1971322 }}
- {{eom|id=Kempf_vanishing_theorem}}
- {{Citation | last1=Haboush | first1=William J. | title=A short proof of the Kempf vanishing theorem | doi=10.1007/BF01392545 |mr=558862 | year=1980 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=56 | issue=2 | pages=109–112| bibcode=1980InMat..56..109H | s2cid=121863316 }}
- {{Citation | last1=Kempf | first1=George R. | title=Linear systems on homogeneous spaces | jstor= 1970952 |mr=0409474 | year=1976 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=103 | issue=3 | pages=557–591 | doi=10.2307/1970952}}
Category:Theorems in algebraic geometry
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