In one of the problems on my differential geometry homework, I must compute the Chern class of $P := \{ z \in \mathbb{C}^{n+1}: |z| = 1 \}$, which is a principal $S^1$-bundle over $\mathbb{C} P^n$. I know that a representative of the Chern class is $-\sigma^*\omega_{FS}$, where $\omega_{FS}$ is the Fubini study form on $\mathbb{C} P^n$ and $\sigma: P \to \mathbb{C} P^n$ is projection. I want to show that $-\sigma^*\omega_{FS}$ is the curvature of a connection, i.e. that $$-\sigma^*\omega_{FS} = d \theta + \frac{1}{2}[\theta \wedge \theta] \ \ \ \ (\star)$$ for a one-form $\theta$ that is $U(1)$-equivariant and horizontal. The condition that $\theta$ is horizontal means that $\theta$ must satisfy $$\theta(d_eX_p(\xi)) = \xi \ \ \ \ (\dagger)$$ for any $\xi \in T_eS^1$. (Here, for a fixed $p \in P$, $X_p: S^1 \to P$ is the map $\lambda \mapsto \lambda p$.)
I am working in local holomorphic coordinates $z_1, \dots z_n, \bar{z}_1, \dots \bar{z}_n$. My proposed $\theta$ is $$ \theta = \frac{ \sum_{j} (\bar{z}_j dz_j - z_j d \bar{z}_j)}{2|z|^2}.$$ One can verify that $d \theta = -\sigma^*\omega_{FS}$. Since $S(1)$ is abelian, $\theta$ satisfies $(\star)$. I still have to prove that $(\dagger)$ holds. Any insights would be greatly appreciated.