**Statement.** Let $X$ be a smooth complex projective variety that is a $\mathbb CP^k$ bundle over $\mathbb CP^n$ in *analtytic topology*. It is *well known* that there exists a rank $k+1$ complex vector bundle $V$ over $\mathbb CP^n$ such that $X$ is isomorphic as a *projective variety* to the projectivisation $\mathbb PV$. 


**Question.** I would like to find a precise reference to this statement, is there such a reference? 


(I know that one is supposed to say that the Brauer group of $\mathbb CP^n$ is trivial, and this is why the **statement** holds. 
But I can't find any place in the literature where this **statement** about projective bundles is stated and I need to find such a reference :( )


**PS.** I realised, that Section 6.2 of the beautiful paper by Arnaud Beauville 
<https://arxiv.org/abs/1507.02476>
would do as a reference to the **statement**. Yet this section speaks of something far more general. I still hope for something more direct, maybe along the lines of what Angelo suggested in his comment (or at least explicitly stated).



$\mathbb CP^k$ bundles over $\mathbb CP^n$ are projectivisations of vector bundles. Any reference?
I would like to give a final answer to this question (it took me some time to get it, thanks to Angelo and Donu) which is, in fact, the following statement.


**Statement'**. Let $Y$ be a projective variety with $H^2(Y,\mathbb {\cal O}^\*)=0$. Then any holomorphic $\mathbb P^m$-bundle over it is a projectivisation of a holomoprhic vector bundle over it. 


*Proof.* Please look into the comment of Angelo to the answer of Arun and into the comments of Donu to the earlier answer that I gave.


Coming back to the original question, we just notice (as Angelo did), that $H^2(\mathbb CP^n, {\mathcal O}^\*)=0$.


I will accept this answer, to finish the story. I still wonder if the **Statement'** was never written in any paper or book. 


