Is there a classification of the holomorphic Hermitian vector bundles $\pi:E\rightarrow M$, over a given complex Hermitian manifold, which are projectively flat and the curvature is proportional to the Kähler 2-form:
$$ R^E=s\omega $$

Projectively flat means the curvature 2-form of the Chern connection on $E$ is a multiple $\alpha 1_E$ of the identity of $E$ by a $(1,1)$-form $\alpha$ (see e.g. famous book S.Kobayashi, *Differential Geometry of Complex Vector Bundles*, Math. Soc. of Japan, Iwanami Schoten and Princeton UP, 1987). But what about when $\alpha=s\omega$, a scalar multiple of the Kähler 2-form not necessarily closed? And what about if it is closed?

If $E=TM$ and $M$ is Kähler, and, moreover, $s$ is constant, then the answer is the constant holomorphic sectional curvature manifolds (99% sure).

Thank you very much for answers.