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Is there anything that can be said in general about the Cech cohomology of the sheaf $\mathcal{F}$ of sections of a holomorphic bundle over a contractible space $X$?

I know that such a bundle must be trivial, but I don't know much about Kunneth formulae, so haven't made much progress on that front. I've also thought about embedding it in an exact sequence involving the sheaf $\mathbb{Z}$ over $X$, but this doesn't seem to lead to any general results.

Could someone give me a hint as to whether there's anything obvious I'm missing? Otherwise just let me know that I'm looking for something non-existent! This is motivated by a desire to use the Mayer-Vietoris sequence to show that some cohomology groups are trivial by splitting the base space into contractible pieces - perhaps this isn't a method which works in general?

Many thanks in advance!

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    $\begingroup$ Contractible may be the wrong kind of space. The role of contractible spaces in topology is played by Stein spaces in complex geometry. $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 5, 2013 at 14:03
  • $\begingroup$ Thanks for alerting me to that Liviu, I'd just come across them myself. Is there a good text that would introduce me to the key properties of Stein manifolds? I come from a theoretical physics background, but have first courses in algebraic geometry, differential geometry and Riemann surfaces. Cheers! $\endgroup$
    – Edward Hughes
    Commented Apr 5, 2013 at 15:46
  • $\begingroup$ A good source is the book by Grauert and Remmert "Theory of Stein Spaces". $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 5, 2013 at 21:11
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    $\begingroup$ are there examples of contractible but non-affine algebraic varieties? (or contractible but non-Stein analytic spaces?) $\endgroup$
    – Jacob Bell
    Commented Apr 23, 2013 at 22:01

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