Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 140 / 2 / d / Solution

Let $\sigma (z)=\overline{z}$, componentwise complex conjugation. Interpret the printed $|z^2|$ in the potential as $|z{|}^2={\sum }_j|z_j{|}^2$. With $D=1+|z{|}^2$, the local Fubini-Study form isIts pullback of a differential form under $\sigma$ iswhere the last equality swaps $j,k$ and uses antisymmetry of the wedge product of differential forms. Since ${\sigma }^2=I$, it is an anti-symplectic involution, with fixed-point set ${\mathbb{R}}^n$. Part (c) gives