Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 17 / 5 / i / Solution

Let $\lambda =\frac{1}{2}{\sum }_j(x_{j}d{y}_j-y_{j}d{x}_j)$ on ${\mathbb{C}}^3$, so $d\lambda ={\omega }_{st}$. On a projective line, work in the affine coordinate $w=\rho e^{i\theta }$ and lift it to the unit three-sphere byThis sphere lies in a complex two-plane in ${\mathbb{C}}^3$. The defining reduction identity for the Fubini-Study form gives $\Omega =s^{\ast }{\omega }_{st}$ on this chart. Differentiating the lift, or using the fact that real radial rescaling contributes no imaginary part to $\bar{z}dz$, givesThe missing point at infinity has zero area, soThis normalization is important: it differs from conventions in which a projective line has area $1$ or $2\pi$.