Projectivity of analytic $\mathbb{P}^1$ bundles Let $f: X\to Y$ be a smooth analytic $\mathbb{P}^1$-bundle from a complex manifold $X$ to a complex projective manifold $Y$. Is $X$ a projective manifold?
 A: Question: "Let $f:X→Y$ be a smooth analytic $P^1$-bundle from a complex manifold $X$ to a complex projective manifold $Y$. Is $X$ a projective manifold?"
Answer: You find an explicit proof in "Compact complex surfaces" Peters/Van de Ven/.... page 190 of the following result: If $Y$ is a smooth compact complex curve and $f$ is an analytic fiber bundle with structure group $PGL(n+1,\mathbb{C})$ where the fibers of $f$ is $\mathbb{P}^n$, then there is an algebraic vector bundle $V$ of rank $n+1$ on $C$ with $X \cong \mathbb{P}(V^*)$.
Note: If $Z$ is a complex manifold, there is an "equivalence of categories" between the category of finite rank holomorphic complex vector bundles on $Z$ and finite rank locally trivial $\mathcal{O}_Z$-modules. If $Z$ is projective it follows $Z$ is algebraic and by Hartshorne, Ex II.7.10 any $\mathbb{P}^n$-bundle (in the sense of Ex.7.10) $\pi: w \rightarrow Z$ on $Z$ is on the form $W \cong \mathbb{P}(V^*)$ for some locally free sheaf $V$ on $Z$. And since  $Z$ is projective it follows $\mathbb{P}(V^*)$ is projective. Hence the general result holds in some cases when $Z$ is projective.
