TITLE: Reference request: moduli spaces of vector bundles
QUESTION [9 upvotes]: 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

REPLY [2 votes]: 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.