= Solution
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 $\mu^{-1}(c)$, then
$$
M_c=\mu^{-1}(c)/G
$$
is a <symplectic manifold> with a unique form characterized by $\pi^*\omega_c=\iota^*\omega$, 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 form>
$$
\omega_{\mathbb C}=\sum_{j=0}^n dx_j\wedge dy_j=\frac i2\sum_{j=0}^n dz_j\wedge d\bar z_j.
$$
Let the <circle group> act by $e^{i\theta}\cdot z=e^{i\theta}z$. Its generator is $X=\sum_j(-y_j\partial_{x_j}+x_j\partial_{y_j})$, and
$$
\iota_X\omega_{\mathbb C}=-d\left(\frac12|z|^2\right).
$$
Consequently 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 $\sqrt2$; 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 primitive
$$
\lambda_0=\frac12\sum_j(x_jdy_j-y_jdx_j)
=\frac1{4i}\sum_j(\bar z_jdz_j-z_jd\bar z_j),\qquad d\lambda_0=\omega_{\mathbb C}.
$$
On the affine chart choose the local section $s(w)=\sqrt2(1,w)/\sqrt S$ of the quotient. Direct substitution gives
$$
s^*\lambda_0=\frac1{2iS}\sum_j(\bar w_jdw_j-w_jd\bar w_j)
=\frac1{2i}(\partial-\bar\partial)\log S.
$$
Differentiating, using $\bar\partial\partial=-\partial\bar\partial$, gives
$$
\boxed{\omega_c=s^*\omega_{\mathbb C}=d(s^*\lambda_0)=i\partial\bar\partial\log S=\omega_{\mathrm{FS}}.}
$$
This is the <Fubini-Study form from circle reduction>. The unit sphere instead produces half this form; our radius $\sqrt2$ is exactly what gives the $2\pi$ line-area normalization used above.
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