In Chern-Weil theory, we choose an arbitrary connection $\nabla$ on a complex vector bundle $E\rightarrow X$, obtain its curvature $F_\nabla$, and then we get Chern classes of $E$ from the curvature form. A priori it looks like these live in $H^*(X;\mathbb{C})$, but by an argument that I don't feel like I really understand, they're in the image of $H^*(X;\mathbb{Z})$, which is where they're usually considered to actually live. I've also recently been learning about the Atiyah-Singer index theorem, and I get the impression that whenever I see a arbitrary constants in geometry that end up having to live in $\mathbb{Z}$ I should ask myself whether the index theorem is lurking in there somewhere. Is there anything to this wild guess?
Can one use Atiyah-Singer to prove that the Chern-Weil definition of Chern classes are $\mathbb{Z}$-cohomology classes?
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differential-geometry
complex-geometry
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3I'm not sure how often anyone gets to make a comment like this, but... this question might actually be more suited for MO than math.SE. :-) – 2011-04-29
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0I think Jesse is right. An expert's answer to your question could help other non-experts in understanding the Atiyah-Singer theorem, and such an answer would better serve and interest the MO readers than the M.SE ones. Atiyah-Singer is strong voodoo, after all. – 2011-04-29
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0I don't know, I think it would also be good to leave the question here for at least a while and see if anyone has a go at answering it. I would also be quite interested in the answer. If you *do* post it on MO, please link to it here. – 2011-04-30
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0If you accept the proof that Chern classes are characterized by the naturality, sum formula, and normalization axioms, then doesn't it follow that the Chern-Weil definition must lie in $\mathbb Z$-cohomology? But anyways, I am interested in an answer to this question. – 2011-05-02
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0@Jesse: Yeah there's definitely a lot going on in the index theorem, but it seemed like this might nevertheless be a pretty low-level question, since at its heart it's really just a question of identifying the appropriate elliptic differential operator. If there are still no answers in a few days, I'll move the question to MO (and I'll certainly link it). – 2011-05-02
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1@Eric: Certainly there's the algebraic topology proof (or even definition, really) that these are $\mathbb{Z}$-classes, but I saw an analytic argument in one my of lectures the other day that had nothing to do with classifying spaces or normalization. I think it's really cool when the same story can be carried all the way through in two such distinct settings. – 2011-05-02
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0@Aaron: Do you have a link to the analytic proof? – 2011-05-03
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1@Eric: The case of line bundles is Proposition 4.4.12 in Huybrechts' book, which just chases through the Cech-de Rham complex. I think the proof in my class was different, but maybe it was essentially equivalent. I'll try to find someone who took notes that day... – 2011-05-05
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0Promoted: http://mathoverflow.net/questions/69085/can-one-use-atiyah-singer-to-prove-that-the-chern-weil-definition-of-chern-classe – 2011-06-29