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. This algebraic geometry–related article is a stub.
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