= Fubini-Study form from circle reduction
{c}
{title2=$\omega_R=\frac{iR^2}{2}\partial\bar\partial\log(1+|w|^2)$}
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 $\omega_R=(iR^2/2)\partial\bar\partial\log(1+|w|^2)$. Its area on a <complex projective line> is $\pi R^2$. Thus $R=\sqrt2$ 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.
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