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Suppose I have a vector bundle (or local system, or something else given by transition functions) on a Riemann surface (or generally a (complex) manifold), and I want to compute its cohomology. The trouble is, I only know the transition functions, not anything about the sections.

All I can think of is Cech cohomology, i.e. giving a cover and doing lots of calculation. To make things work, it seems to me that the cover needs to have very many opens (effectively a triangulation), which complicates calculation.

Does anyone know if there is a better way? Alternately, if anyone knows of ways to make the above approach more efficient, I'd be glad to know.

(Perhaps this is a stupid (trivial) question, but I can't find any references about this.)

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A complex vector bundle over a punctured surface is trivial. First everything together except one patch. The last patch is hopefully a disk and the overlap with the other patches is an annulus. The transition function is then a map from the circle into Gl(n,C). You can get the rest from doing things to the map so that you can compute winding numbers. :) –  Charlie Frohman May 14 '10 at 16:43
    
Does this work for holomorphic vector bundles too? –  Ketil Tveiten May 14 '10 at 17:12
    
May be I'm being dense but the total space $E\to B$ of a vector bundle is homotopy equivalent to the base, whence the cohomology of $E$ is the same as that of $B$. Or, are you meaning something else (like projectivization of $E$) when you say "cohomology of $E$"? –  Somnath Basu May 14 '10 at 17:55
    
@Somnath, when the question mentions "cohomology of a vector bundle" it means a holomorphic vector bundle and the cohomology of the sheaf of holomorphic sections. –  Joel Fine May 14 '10 at 19:09
    
It works for Holomorphic bundles too. For instance if you read papers by David Giesecker on moduli space of stable bundles that's how he does it. –  Charlie Frohman May 14 '10 at 21:18
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1 Answer 1

The simplest case of this question is if the bundle is trivial. So I think you have to accept as a building block the cohomology with trivial coefficients of your space and subspaces. But if you accept that, general cohomology can be computed in terms of these building blocks. For an open cover $X=U\cup V$, there is a Mayer-Vietoris sequence $$H^i(X;L)\to H^i(U;L)\oplus H^i(V;L)\to H^i(U\cap V;L)\to H^{i+1}(X;L)$$ I let $L$ stand for "local system," but it works as well for a coherent sheaf. If it is trivial on $U$ and $V$, then those terms are your basic building blocks (more precisely, they are only isomorphic and you have to pay attention to the isomorphisms). The transition functions show up in the restriction maps from $U$ and $V$ to the intersection. You might think of the trivialization of $L$ on the intersection as coming from the trivialization on $U$, in which case the restriction map from $U$ is the usual map on cohomology with trivial coefficients. But the restriction map from $V$ is the composite of that map with the transition function.

If you have more than two open sets in your cover, then you can iterate this procedure, but the restriction maps are no longer between cohomology with trivial coefficients and cannot be identified with transition maps. But you can compute what they are at the previous stage.

This is probably the way to go for local systems. For vector bundles, it is not so nice, since you're probably trying to compute a finite dimensional vector space on a complete variety, but the open sets are not complete and thus have infinitely dimensional spaces of sections. But if the bundle is given by transition functions, this may be your only choice.

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