= 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> is
$$
\widetilde\omega_{FS}=\frac{i}{2}\sum_{j,k}
\left(\frac{\delta_{jk}}D-\frac{\overline z_jz_k}{D^2}\right)dz_j\wedge d\overline z_k.
$$
Its <pullback of a differential form> under $\sigma$ is
$$
\sigma^*\widetilde\omega_{FS}
=\frac{i}{2}\sum_{j,k}
\left(\frac{\delta_{jk}}D-\frac{z_j\overline z_k}{D^2}\right)d\overline z_j\wedge dz_k
=-\widetilde\omega_{FS},
$$
where 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
$$
\boxed{\mathbb R^n\subset(\mathbb C^n,\widetilde\omega_{FS})\text{ is Lagrangian}.}
$$
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