13
$\begingroup$

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.

$\endgroup$
30
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.