Kähler quotient

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

[math]\displaystyle{ \mu^{-1}(0)/G. }[/math]

If [math]\displaystyle{ G }[/math] acts freely and properly, then [math]\displaystyle{ \mu^{-1}(0)/G }[/math] 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 [math]\displaystyle{ G }[/math] is closely related to a geometric invariant theory quotient by the complexification of [math]\displaystyle{ G }[/math].

See also

• Hyperkähler quotient