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

Question

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 $X$?

Answer 1

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 $H^2(X, \, \mathcal{O}_X)=H^3(X, \, \mathbb{Z})=0$.

However, it fails for a general smooth basis. You can see the discussion at p. 190 of

Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 4. Berlin: Springer (ISBN 3-540-00832-2/hbk). xii, 436 p. (2004). ZBL1036.14016.