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I am trying to study the moduli of holomorphic vector bundles fast and I'm primarily interested to understand:

1) Why and were the stability is important. 2) How are the construction methods. 3) some examples. (especially the case of vector bundles on curves)

I was looking for some references but I could not find any reasonable source online. I have the base for that and I don't want some thing very long and full of details.

I just want to see all the ideas very clearly without to much details.

Please tell me if you know any lecture note or book which contains this stuff.

Thanks in advance

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Did you look at the question mathoverflow.net/questions/44125/… ? –  mathphysicist Nov 14 '10 at 2:46
    
Yes, that question is some how general, I know many moduli theories. I'm looking for a source for this specific moduli theory. –  Mohammad F. Tehrani Nov 14 '10 at 3:01
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I don't know about online refs, but there are plenty of books by Huybrechts-Lehn, Le Potier, Mukai, Seshadri... In a nutshell, stability is what is needed to get GIT methods to work. –  Donu Arapura Nov 14 '10 at 3:49
    
Mumford's GIT. He explains the construction of many moduli spaces including vector bundles from a single point of view. If you know the theory of moduli of Riemann surfaces he explains vector bundles in the same language. –  Daniel Pomerleano Nov 14 '10 at 21:15
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2 Answers

Dear Mohammad, there is a rather elementary book Introduction to Moduli Problems and Orbit spaces by P.E. Newstead which will explain to you why stability is important, give you lots of examples (Chapter 4 is devoted to them) and which ends with a whole chapter (Chapter 5) called Vector bundles over a curve. It was written by an extremely competent expert and deliberately maintained at a quite elementary level. The author explains in the preface that his notes are an introduction to Mumford's Geometric Invariant Theory in the language of classical algebraic geometry, deliberately eschewing schemes.

On the subject of holomorphic bundles over $\mathbb P^n(\mathbb C) $ you may check Okonek, Schneider and Spindler's monograph Vector Bundles on Complex Projective Spaces, written in the language of holomorphic manofolds (the results are the same as in algebraic geometry thanks to Serre's GAGA principle).

I'd also like to mention Atiyah's classic Vector bundles over an elliptic curve published in 1957, which I still find quite instructive despite its venerable age.

And finally I should also mention the articles on moduli of vector bundles over curves written by the brilliant Indian school around the Tata Institute: M.S.Narasimhan, Seshadri, Ramanan, Nori, ...

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Twelve years after Atiyah's article classifying vector bundles on curves of genus one, Narasimhan and Ramanan published a lovely paper in the Annals, (89) no.2, 1969, p.14, where they solved the case of semi stable rank 2 vector bundles on genus two curves. This case is perhaps more typical of the higher genus situation. Basically, a rank two vector bundle is analyzed by producing a sub line bundle, whose quotient is also a line bundle, and then studying how the vector bundle is reconstructed as a twisted sum of those two line bundles. I cannot improve on the wonderful references given by Georges Elencwajg above, but I have a short 4 or 5 page set of notes from a lecture given by Daniele Arcara, in a graduate class of mine, summarizing the status of moduli of rank 2 bundles on curves in 2001, if there is some way to send it to you or attach it here as a pdf file....

I sent them to your email address at Princeton.

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thanks for email. –  Mohammad F. Tehrani Nov 15 '10 at 1:32
    
Just googling "vector bundles on curves" already yields some interesting results. The first hit for me was a set of notes by Teixidor (tufts.edu/~mteixido/files/vectbund.pdf) classifying bundles on curves of genera zero and one, and the 4th hit was a nice article by Beauville, Vector bundles on curves and theta functions, surveying results on the natural theta maps from moduli of vector bundles to projective space. google.com/… –  roy smith Nov 16 '10 at 6:24
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