Kempf–Ness theorem

Short description: Criterion for vector stability in algebraic geometry

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 [math]\displaystyle{ X /\!/ G }[/math] (the GIT quotient of X by G) is homeomorphic to the symplectic quotient of X by a maximal compact subgroup of G.

References

• 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, 732, Berlin, New York: Springer-Verlag, pp. 233–243, doi:10.1007/BFb0066647, ISBN 978-3-540-09527-9