Given a Hermitian metric on a holomorphic vector bundle we can easily define its Chern connection. But if we are given a connection $\mathcal{A}$,
$$[De=\mathcal{A}e,]$$
where $e$ is a holomorphic frame field, how do we know if we can find a Hermitian metric on this vector bundle, so that the connection $\mathcal{A}$ is the Chern connection of the metric, or other types of connections compatible with the metric?
In general vector bundles, I searched and find it's related to if the holonomy group is a subgroup of some unitary groups, and so related to curvatures by the theorem of Ambroise-Singer.
I'm think is there any simpler criterions in the case of holomorphic bundle?
Thanks for your attention!