Kähler quotient

In mathematics, specifically in complex geometry, the Kähler quotient of a Kähler manifold X by a Lie group G acting on X by preserving the Kähler structure and with moment map \mu : X \to \mathfrak{g}^* (with respect to the Kähler form) is the quotient

:\mu^{-1}(0)/G.

If G acts freely and properly, then \mu^{-1}(0)/G is a new Kähler manifold whose Kähler form is given by the symplectic quotient construction.

By the Kempf-Ness theorem, a Kähler quotient by a compact Lie group G is closely related to a geometric invariant theory quotient by the complexification of G.