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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.

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up vote 26 down vote accepted

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, 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.

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