I'm a physicist and I'm trying to figure out some things about parabolic bundles. In particular I'd like to understand which is the relationship, if present, between the parabolic degree and the first chern class of the bundle. A rigorous definition of the degree of a bundle will be appreciated too. Thanks
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$\begingroup$ Is this over a curve, or a polarized variety? $\endgroup$– abxCommented Mar 15, 2014 at 16:46
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$\begingroup$ Is over a curve $\endgroup$– popoolmicaCommented Mar 15, 2014 at 16:59
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$\begingroup$ The usual degree and the first Chern class are defined for vector bundles (not parabolic). If your bundle has a parabolic structure, you get the parabolic degree by adding a correction term defined in terms of the parabolic structure. What is the problem? $\endgroup$– abxCommented Mar 15, 2014 at 17:18
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1$\begingroup$ I'm not really sure why the reaction to this question is so negative. My guess is that the parabolic degree ought to be expressible as Chern-Weil integral for a suitably chosen metric on the restriction of the bundle to the open part. Some results along these lines can be found in papers of Biquard or T. Mochizuki. $\endgroup$– Donu ArapuraCommented Mar 15, 2014 at 21:08
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1$\begingroup$ (If you're willing to work with stacks) If your parabolic bundles have rational weights, then they come from some vector bundle on a root stack over your curve, and you can take the first Chern class there. Then by taking a degree on the stack you get back the parabolic degree. For some details see for example "Fibrés paraboliques et champ des racines" by Niels Borne, math.univ-lille1.fr/~borne/Recherche/ParRac.pdf $\endgroup$– Mattia TalpoCommented Mar 16, 2014 at 15:05
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