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As I understand it, an open cover of a Base Space and associated holomorphic transition functions on the intersection are sufficient data to define (up to isomorphism perhaps) a holomorphic (complex) line bundle. So if we cover S1 with open sets U and V which intersect in open sets P and Q such that P and Q have empty intersection (you will note I am avoiding Latex here. Basically we cover S1 with a pair of horseshoes). We define the UV- transition function to be +1 on P and -1 on Q (locally constant thus holomorphic) we have the set up for an (infinite) Mobius band. What I am strugling to define are the corresponding trivialisations from U or V to UxC or VxC that correspond to this transition function. Even Grifiths & Harris has some "hand waving" about identifying points which does not help. I just want to see the explicit map on the fibres over U and V.

Any suggestions?

Thanks Noel R

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    $\begingroup$ A quick suggestion: work backwards. Construct the bundle using the trivialization, and convince yourself its a Mobius strip. (By the way, the bundle isn't holomorphic, and this isn't algebraic geometry, it's really basic topology.) $\endgroup$
    – Donu Arapura
    Commented Sep 21, 2010 at 14:17
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    $\begingroup$ @Donu: well, it's real algebraic geometry (and, yes, basic topology). $\endgroup$
    – Pete L. Clark
    Commented Sep 21, 2010 at 15:05
  • $\begingroup$ Thank you for your prompt response to the dual of the question asked (which I now understand from the response below was not even well formed). Btw I was seeking to understand this on my way to a proper appreciation of the Picard Group (or is that not now algebraic geometry either?) $\endgroup$
    – Noel Robinson
    Commented Sep 21, 2010 at 15:29
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    $\begingroup$ The boundaries between alg. geom and topology are pretty fluid, and I didn't mean to suggest otherwise. But as Scott suggested below, G & H is not the best place to first learn this stuff. $\endgroup$
    – Donu Arapura
    Commented Sep 21, 2010 at 16:29

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There appears to be some confusion in the question: The circle $S^1$ is not a complex manifold, so it does not admit a meaningful notion of holomorphic line bundle. If you try to construct a complex line bundle on the circle, you will find that it is automatically a trivial line bundle. If you want to construct a Möbius band as a line bundle, you should take a real line bundle, with the transition functions you specified.

I would recommend a more topological text than Griffiths and Harris for information about the construction you are trying to do. Perhaps Hatcher's book on vector bundles and K-Theory will help. It is available on his web page.

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  • $\begingroup$ Thank you for pointing out my silly mistake so promptly. (Wood for the trees and all that) and for the references. Back to Picard! $\endgroup$
    – Noel Robinson
    Commented Sep 21, 2010 at 15:30

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