Complex projective space as a circle symplectic reduction

The zero level of $\mu$ is ${\mu }^{-1}(0)=\{z:\mathrm{\mid }z{\mathrm{\mid }}^2=2c\}$, the sphere of radius $r=\sqrt{2c}>0$.

The circle acts freely on this sphere: if $e^{i\theta }z=z$ with $z\mathrm{\ne }0$ then $e^{i\theta }=1$. The value $0$ is therefore regular by [F3], and the circle is compact, so the action is proper.

By [F2] the reduction $M_0={\mu }^{-1}(0)\mathrm{/}{S}^1$ is a symplectic manifold of dimension $2n-1-1=2n-2$, and its form is the unique form pulled back from ${\iota }^{\ast }{\omega }_0$. The quotient of the sphere by the scalar circle action is the Hopf quotient $S_r^{2n-1}\mathrm{/}{S}^1$, the standard model of ${\mathbb{C}\mathbb{P}}^{n-1}$: scalar multiplication and $z\mapsto \lambda z$ identify the same line, and the quotient is free away from the origin.

In that model the Fubini--Study form is defined exactly by the basic-form property ${\pi }^{\ast }{\omega }_{\mathrm{F}\mathrm{S}}={\iota }^{\ast }{\omega }_0$ on the sphere, so the reduced form is the Fubini--Study form in the normalization fixed by the sphere of radius $r$; replacing $c$ by ${\lambda }^{2}c$ replaces the sphere of radius $r$ by the sphere of radius $\lambda r$ and rescales the reduced form by ${\lambda }^2$. For $n=1$ the sphere is $S^1$, the circle acts transitively, and the quotient is a single point with the zero form.