Kempf–Ness theorem

Kempf–Ness theorem 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, alors {displaystyle X/!/g} (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", Géométrie algébrique (Proc. Summer Meeting, Université. Copenhagen, Copenhagen, 1978), Notes de cours en mathématiques, volume. 732, Berlin, New York: Springer Verlag, pp. 233–243, est ce que je:10.1007/BFb0066647, ISBN 978-3-540-09527-9, M 0555701 This algebraic geometry–related article is a stub. Vous pouvez aider Wikipédia en l'agrandissant.

Catégories: Invariant theoryTheorems in algebraic geometryAlgebraic geometry stubs