I have been getting confused a lot lately with the relation between local and global data, mainly due to a heavy focus on the local. One thing in particular is to do with the Fubini-Study form and Chern curvature.
These two 2-forms are related by $\omega_{FS}=\frac{i}{2 \pi} F$, where locally $\omega_{FS} = \frac{i}{2 \pi} \partial \overline{\partial} log(1 + |w_i|^2)$, and $F$ is the Chern connection of the bundle $O(1) \to \mathbb{C}P^n$ with (local) Hermitian metric $h_i=(1+|w_i|^2)^{-1}$.
I think what is confusing me is that the Curvature form is related to the bundle $E := O(1)$ and we have $F \in A^{1,1}(E)=(\mathbb{C}P^n, \text{End}(E))$ whilst $\omega_{FS}$ was defined without any relation to $E$ and can be shown to be globally defined over $\mathbb{C}P^n$, and thus $\omega_{FS} \in A^{1,1}(\mathbb{C}P^n)$. Given that $\omega_{FS}$ and $F$ are equal up to constants they should live in the same space?
That is a general result that if $E$ is a complex line bundle, then $\operatorname{End}(E)$ is trivial (see here). Thus
$$ A^{1,1} (M, \operatorname{End}(E)) = A^{1,1} (M)$$