title: Kähler quotient
text: # 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 $G$.
  
  