Reference request: moduli space of vector bundles - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:55:24Z http://mathoverflow.net/feeds/question/46006 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46006/reference-request-moduli-space-of-vector-bundles Reference request: moduli space of vector bundles Mohammad F.Tehrani 2010-11-14T02:25:05Z 2010-11-15T01:09:30Z <p>I am trying to study the moduli of holomorphic vector bundles fast and I'm primarily interested to understand:</p> <p>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)</p> <p>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.</p> <p>I just want to see all the ideas very clearly without to much details.</p> <p>Please tell me if you know any lecture note or book which contains this stuff.</p> <p>Thanks in advance</p> http://mathoverflow.net/questions/46006/reference-request-moduli-space-of-vector-bundles/46064#46064 Answer by Georges Elencwajg for Reference request: moduli space of vector bundles Georges Elencwajg 2010-11-14T19:38:05Z 2010-11-14T19:38:05Z <p>Dear Mohammad, there is a rather elementary book <em>Introduction to Moduli Problems and Orbit spaces</em> 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 <em>Vector bundles over a curve</em>. 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. </p> <p>On the subject of holomorphic bundles over \$\mathbb P^n(\mathbb C) \$ you may check Okonek, Schneider and Spindler's monograph <em>Vector Bundles on Complex Projective Spaces</em>, written in the language of holomorphic manofolds (the results are the same as in algebraic geometry thanks to Serre's GAGA principle).</p> <p>I'd also like to mention Atiyah's classic <em>Vector bundles over an elliptic curve</em> published in 1957, which I still find quite instructive despite its venerable age.</p> <p>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, ...</p> http://mathoverflow.net/questions/46006/reference-request-moduli-space-of-vector-bundles/46095#46095 Answer by roy smith for Reference request: moduli space of vector bundles roy smith 2010-11-15T01:03:20Z 2010-11-15T01:09:30Z <p>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....</p> <p>I sent them to your email address at Princeton.</p>