Kähler quotient

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 isIt 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 ${\pi }^{\ast }{\omega }_c={\iota }^{\ast }\omega$. 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.^[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.^[ref]