Reference request: moduli spaces of vector bundles I am trying to study the moduli spaces of holomorphic vector bundles quickly, and I'm primarily interested in understanding:


*

*Why and where the stability condition is used.

*How are the moduli spaces constructed.

*What are the examples, especially in 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 notes or a book which contains this stuff.
Thanks in advance
 A: 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.
A: 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, ...
