and let $\pi:S^3\to \mathbb{CP}^1$ be the Hopf principal $U(1)$-bundle, with the standard right circle action and affine coordinate $w=z_1/z_0$ on the chart $U_0:=\{[z_0:z_1]\in \mathbb{CP}^1:z_0\ne 0\}$. Let $\alpha\in \Omega^1(S^3;i\mathbb R)$ be the standard Hopf connection form
restricted to $S^3$, and let $\Omega\in \Omega^2(\mathbb{CP}^1;i\mathbb R)$ be the unique descended curvature form satisfying $\pi^*\Omega=d\alpha$. If the Fubini-Study form $\omega_{FS}\in\Omega^2(\mathbb{CP}^1;\mathbb R)$ is normalized on $U_0$ by
Consequently, for the tautological complex line bundle $\mathcal O(-1)\to \mathbb{CP}^1$ associated to this Hopf bundle with the standard defining representation convention, one has