Let $(X,\mathcal{O}\_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}\_X$-modules. Can we invoke the fact that $X$ is contractible to conclude, in some cases, that $\mathcal{F}$ is isomorphic to $\mathcal{O}\_X^{\oplus n}$ for some $n$? If you like, you may take $X$ to be the analytic space associated to a complex affine variety.


I ask because contractibility is often a useful condition when attempting to prove a fibre bundle is trivial.



Sheaves on Contractible Analytic Spaces
The so-called Oka-Grauert principle states that for any Stein space $X$ the holomorphic and the topological classification of complex vector bundles on $X$ coincide. 
The original reference is 


*Hans Grauert*, [**Analytische Faserungen über holomorph-vollständigen Räumen**](http://www.ams.org/mathscinet-getitem?mr=98199), *Math. Ann.* **135** (1958), 263-273.


As a consequence, every locally free, coherent sheaf $\mathscr{F}$ defined on a contractible subvariety $X$ of $\mathbb{C}^n$ is free.


Of course, if $\mathscr{F}$ is not locally free this is no longer true. For instance, take a closed analytic subvariety $Z \subset X$; then the ideal sheaf $\mathscr{I}\_Z \subset \mathscr{O}\_X$ is coherent but not free.


