Pullback of a $1$-form on $\mathbb CP^1$

Question

Consider $X = \mathbb CP^1$, and define $h : T_{[0:1]}X \times T_{[0:1]}X \to \mathbb C, (u,v) \to \overline u v $. I want to extend $h$ to all $\mathbb CP^1$ by the action of $\rm{SU}_2(\mathbb C)$. The action is transitive so it will define a global $2$-form if the stabilizer of any point acts trivially on it. $\rm{SU}(\mathbb C)_{[0:1]}$ $ = \{g_a := \begin{pmatrix} e^{ia} & 0 \\ 0 & e^{-ia} \end{pmatrix} \mid a \in \mathbb R \}$. Here I'm stuck : for me the action would be something like $g_a \cdot h = \overline{e^{ia}u} e^{-ia}v = e^{-2ia} \overline u v$ which is not invariant. I am sure I made a stupid mistake but I can't see how. Thanks in advance !

Answer 1

I did a mistake in the stabilizer which is simply $\text{diag}(e^{ia}, e^{ia})$. Thus $\omega$ is invariant by the stabilizer.