= Solution
Use the normalization in which a projective line has area $2\pi$. On the affine chart of <Complex projective space> where $Z_0\ne0$, set $w_j=Z_j/Z_0$ and $S=1+\sum_j|w_j|^2$. Define the <Fubini-Study form> by
$$
\boxed{\omega_{\mathrm{FS}}=i\partial\bar\partial\log S
=i\sum_{j,k}\frac{S\delta_{jk}-\bar w_jw_k}{S^2}\,dw_j\wedge d\bar w_k.}
$$
On another chart the corresponding potential differs by $\log|h|^2$ for a nowhere-zero <holomorphic function> $h$, whose $\partial\bar\partial$ is zero. Thus these local <differential forms> glue to a global form. It is real and closed. Its Hermitian coefficient matrix is positive definite, since for $v\ne0$ the <Cauchy-Schwarz inequality> gives
$$
\sum_{j,k}\frac{S\delta_{jk}-\bar w_jw_k}{S^2}v_j\bar v_k
=\frac{S|v|^2-|\sum_j\bar w_jv_j|^2}{S^2}\geq\frac{|v|^2}{S^2}>0.
$$
Hence it is a <Kähler form> and in particular a <symplectic form>.
On a <complex projective line> with $w=x+iy$ this is $2(1+|w|^2)^{-2}dx\wedge dy$, whose total area is $2\pi$. Thus $[\omega_{\mathrm{FS}}/(2\pi)]$ is the positive generator, equivalently $c_1(\mathcal O(1))$. Another common normalization uses half this form and gives line area $\pi$; the scale must be carried consistently into <symplectic reduction>.
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