User martin - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:15:04Z http://mathoverflow.net/feeds/user/3500 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19721/gauge-theory-construction-of-vector-bundles-on-singular-varieties gauge theory construction of vector bundles on singular varieties Martin 2010-03-29T12:16:52Z 2010-03-29T22:57:09Z <p>This is sort of a follow-up to: <a href="http://mathoverflow.net/questions/19635/gauge-theory-construction-of-moduli-of-vector-bundles" rel="nofollow">http://mathoverflow.net/questions/19635/gauge-theory-construction-of-moduli-of-vector-bundles</a></p> <p>If I have a complex compact algebraic curve with at worst nodal singularities, is there an analytic description of holomorphic structures on the trivial bundle in terms of (0,1) forms satisfying some constraint? presumably one wants the forms to pick up singular behavior of some kind at the node. Also, is the description well-behaved in families as we smooth the curve? for instance, in the sense that whatever infinite-dimensional vector space is "locally constant" as we smooth the curve. </p> <p>Other mild singularity types would be interesting too!</p> http://mathoverflow.net/questions/12758/deligne-simpson-problem-in-the-symmetric-group/12825#12825 Answer by Martin for Deligne-Simpson problem in the symmetric group Martin 2010-01-24T06:44:24Z 2010-01-24T06:44:24Z <p>(wanted to make this a comment but couldn't figure how)</p> <p>As Dmitri says, you can express the number of ramified covers using the group algebra of S_n, not just whether it's nonzero; once you do this, an inclusion/exclusion argument based on splitting up the conjugacy classes into smaller partitions lets you figure out the number of these that are irreducible. this is an in-principle answer that's of course totally useless in practice</p> http://mathoverflow.net/questions/19721/gauge-theory-construction-of-vector-bundles-on-singular-varieties/19781#19781 Comment by Martin Martin 2010-03-31T03:51:03Z 2010-03-31T03:51:03Z No, not nonsingular - but where the node is disconnecting - the Picard variety is still a torus, so we're all good but I wasn't sure if the higher rank bundles could still be studied via gauge theory... thanks again for the answers! http://mathoverflow.net/questions/19721/gauge-theory-construction-of-vector-bundles-on-singular-varieties/19781#19781 Comment by Martin Martin 2010-03-30T05:07:01Z 2010-03-30T05:07:01Z Thanks Tim! Are things any easier if we're in a situation with no vanishing cycles? so for instance, the Picard variety stays compact now.