Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 16 / 5 / b / Solution

Use the normalization in which a projective line has area $2\pi$. On the affine chart of Complex projective space where $Z_0\ne 0$, set $w_j=Z_j/Z_0$ and $S=1+{\sum }_j|w_j{|}^2$. Define the Fubini-Study form byOn 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 $v\ne 0$ the Cauchy-Schwarz inequality givesHence 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.