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: $$ R^Ee=s\,\omega\otimes e \ ,\ \ \forall e\in E\ .$$
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.
Thank you very much for answers.