= Kähler quotient
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{wiki}
Let a compact <Lie group> $G$ act on a <Kähler manifold> $(M,\omega,J,g)$ by <holomorphic maps> and <isometries>, with an equivariant <moment map> $\mu$. If $c$ is a <regular value> fixed by the <coadjoint action> and $G$ acts freely on $\mu^{-1}(c)$, the Kähler quotient is
$$
M//_cG=\mu^{-1}(c)/G.
$$
It is a <Kähler manifold> of complex <dimension> $\dim_{\mathbb C}M-\dim_{\mathbb R}G$. Compactness supplies properness of the action. If $\iota$ includes the level and $\pi$ is the quotient projection, the reduced <Kähler form> satisfies $\pi^*\omega_c=\iota^*\omega$. The <metric> and <complex structure> descend through the $g$-orthogonal complement of the <orbit> tangent spaces inside the level tangent spaces. This horizontal space is $J$-invariant; the reduction theorem proves that the descended structure is integrable.<An Introduction to Lie Groups and Symplectic Geometry by Robert L. Bryant>[Lecture 8, Theorem 3 and central-level discussion, printed pp.135–136]{ref}
For scalar <circle group> multiplication on $\mathbb C^{n+1}$, use the <standard symplectic form> and convention $\iota_{X_\xi}\omega=d\mu^\xi$. Then $\mu(z)=-|z|^2/2$. Reducing at $c=-R^2/2$ divides the radius-$R$ <sphere> by its <Hopf fibration> and gives <Complex projective space> with the corresponding scaled <Fubini-Study metric>, as computed in <Fubini-Study form from circle reduction>.<An Introduction to Lie Groups and Symplectic Geometry by Robert L. Bryant>[Lecture 8, projective-space example, printed pp.136–138]{ref}
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