Kähler quotient

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

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

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

By the Kempf-Ness theorem, a Kähler quotient by a compact Lie group ${\displaystyle G}$ is closely related to a geometric invariant theory quotient by the complexification of ${\displaystyle G}$.[2]