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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]

The Marsden-Weinstein theorem says that for a Hamiltonian group action, if $c$ is a regular value fixed by the coadjoint action and $G$ acts freely and properly on ${\mu }^{-1}(c)$, thenis a symplectic manifold with a unique form characterized by ${\pi }^{\ast }{\omega }_c={\iota }^{\ast }\omega$, where $\iota$ includes the level set and $\pi$ is the quotient projection. Its dimension is $\dim M-2\dim G$.

Equip ${\mathbb{C}}^{n+1}$ with the standard symplectic formLet the circle group act by $e^{i\theta }\cdot z=e^{i\theta }z$. Its generator is $X={\sum }_j(-y_j{\partial }_{x_j}+x_j{\partial }_{y_j})$, andConsequently the moment map in our convention is $\mu (z)=-|z{|}^2/2$. It is invariant, hence equivariant because the circle group is abelian. The level ${\mu }^{-1}(-1)$ is the sphere of radius $\sqrt{2}$; it is regular, the circle group acts freely there, and properness follows from compactness of the group. The quotient is Complex projective space, by $z\mapsto [z]$, a scaled Hopf fibration. The Marsden-Weinstein theorem therefore produces a reduced symplectic form on ${\mathbb{CP}}^n$.

To identify it rather than only assert its existence, use the primitiveOn the affine chart choose the local section $s(w)=\sqrt{2}(1,w)/\sqrt{S}$ of the quotient. Direct substitution givesDifferentiating, using $\bar{\partial }\partial =-\partial \bar{\partial }$, givesThis is the Fubini-Study form from circle reduction. The unit sphere instead produces half this form; our radius $\sqrt{2}$ is exactly what gives the $2\pi$ line-area normalization used above.