 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-17/5/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Let

λ = \frac{1}{2} \sum_{j} (x_{j} d y_{j} - y_{j} d x_{j})

C^{3}

, so

d λ = ω_{s t}

. On a projective line, work in the affine coordinate

w = ρ e^{i θ}

and lift it to the unit three-sphere by

s (w) = \frac{( w , 1 )}{1 + ∣ w ∣ ^{2}} .

This sphere lies in a complex two-plane in

C^{3}

. The defining reduction identity for the Fubini-Study form gives

Ω = s^{*} ω_{s t}

on this chart. Differentiating the lift, or using the fact that real radial rescaling contributes no imaginary part to

\overset{z}{ˉ} d z

, gives

s^{*} λ = \frac{1}{2} \frac{ρ ^{2}}{1 + ρ ^{2}} d θ, Ω = \frac{ρ}{( 1 + ρ ^{2} ) ^{2}} d ρ \land d θ .

The missing point at infinity has zero area, so

\int_{H} Ω = 2 π \int_{0}^{\infty} \frac{ρ d ρ}{( 1 + ρ ^{2} ) ^{2}} = π .

This normalization is important: it differs from conventions in which a projective line has area

1

2 π

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