Complex vector bundle that does not admit holomorphic structure

Question

Is there any example of a complex vector bundle $E \to M$ whose base space $M$ is also a complex manifold, but where the total space $E$ does not admit a holomorphic structure compatible with the bundle structure?

Of course, I have already run into this question before. But I am not specifically interested in the case where the base is a Fano or even an algebraic variety.

Answer 1

Complex line bundles over $X$ are clssified by $H^2(X,\mathbb{Z})$. Hence choose some cohomology class in $H^2(X,\mathbb{Z})$ but not in $H^{1,1}(X)$ will give you an example.

Answer 2

Since I can't comment, I have to give an answer here: Take any smooth vector bundle $\mathcal{E} \to X$ over a complex manifold $X$. The complexification $\mathcal{E} \otimes \mathbf{C}$ is a complex vector bundle. In general, this is far from a holomorphic vector bundle, however.