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 $$G$$.