# Does any projective bundle on a compact complex manifold have an associated holomorphic vector bundle?

Let $$X$$ be a compact complex manifold, and $$f: Y\to X$$ a proper surjective holomorphic map with fiber $$\mathbb{CP}^n$$. Is there always a holomorphic vector bundle $$E$$ of rank $$n+1$$ such that $$Y$$ is biholomorphic to $$\mathbb{P}(E)$$ over $$X$$?

It is classically known that this is true when $$\dim X=1$$ ("ruled surfaces" = "geometrically ruled surfaces").
It is also true when $$\dim X=2$$, provided that $$H^2(X, \, \mathcal{O}_X)=H^3(X, \, \mathbb{Z})=0$$.