Fubini-Study form from circle reduction (${\omega }_R=\frac{i{R}^2}{2}\partial \bar{\partial }\log (1+|w{|}^2)$)

The scalar circle group action on ${\mathbb{C}}^{n+1}$ with the standard symplectic form has moment map $\mu =-|z{|}^2/2$ in the convention ${\iota }_{X_H}\omega =dH$. Reduction of the radius-$R$ sphere gives Complex projective space and the form ${\omega }_R=(i{R}^2/2)\partial \bar{\partial }\log (1+|w{|}^2)$. Its area on a complex projective line is $\pi R^2$. Thus $R=\sqrt{2}$ yields the Fubini-Study form normalized to projective-line area $2\pi$, while $R=1$ yields half that form. The Hopf fibration identifies the quotient.