Kempf–Ness theorem: Difference between revisions
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{{Short description|Criterion for vector stability in algebraic geometry}} |
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In [[algebraic geometry]], the '''Kempf–Ness theorem''', introduced by {{harvs|txt|last1=Kempf|first1=George|author1-link=George Kempf|first2=Linda|last2=Ness|year=1979|author2-link=Linda Ness}}, gives a criterion for the [[Geometric invariant theory#Stability|stability]] of a vector in a [[group representation|representation]] of a complex [[reductive group]]. If the [[complex number|complex]] [[vector space]] is given a [[Norm (mathematics)|norm]] that is [[Invariant (mathematics)|invariant]] under a [[maximal compact subgroup]] of the reductive group, then the Kempf–Ness theorem states that a vector is stable if and only if the norm attains a minimum value on the [[Group action (mathematics)|orbit]] of the vector. |
In [[algebraic geometry]], the '''Kempf–Ness theorem''', introduced by {{harvs|txt|last1=Kempf|first1=George|author1-link=George Kempf|first2=Linda|last2=Ness|year=1979|author2-link=Linda Ness}}, gives a criterion for the [[Geometric invariant theory#Stability|stability]] of a vector in a [[group representation|representation]] of a complex [[reductive group]]. If the [[complex number|complex]] [[vector space]] is given a [[Norm (mathematics)|norm]] that is [[Invariant (mathematics)|invariant]] under a [[maximal compact subgroup]] of the reductive group, then the Kempf–Ness theorem states that a vector is stable if and only if the norm attains a minimum value on the [[Group action (mathematics)|orbit]] of the vector. |
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Latest revision as of 02:15, 20 July 2023
In algebraic geometry, the Kempf–Ness theorem, introduced by George Kempf and Linda Ness (1979), gives a criterion for the stability of a vector in a representation of a complex reductive group. If the complex vector space is given a norm that is invariant under a maximal compact subgroup of the reductive group, then the Kempf–Ness theorem states that a vector is stable if and only if the norm attains a minimum value on the orbit of the vector.
The theorem has the following consequence: If X is a complex smooth projective variety and if G is a reductive complex Lie group, then (the GIT quotient of X by G) is homeomorphic to the symplectic quotient of X by a maximal compact subgroup of G.
References
[edit]- Kempf, George; Ness, Linda (1979), "The length of vectors in representation spaces", Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Mathematics, vol. 732, Berlin, New York: Springer-Verlag, pp. 233–243, doi:10.1007/BFb0066647, ISBN 978-3-540-09527-9, MR 0555701
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