Timeline for Does there exist a GRR-like generalization of the AS Index Theorem?
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| Apr 1 at 2:08 | answer | added | Zhaoting Wei | timeline score: 8 | |
| Aug 10, 2020 at 0:07 | comment | added | David Ben-Zvi | To amplify Dustin's comment, I believe the Families Index Theorem applies to maps that are "smooth" in the sense of algebraic geometry, i.e., submersions, while GRR applies to general proper maps, such as resolutions of singularities -- i.e. the families allowed in algebraic geometry are typically much less constrained than in differential topology. Then again maybe all that's missing (given general functoriality) is to know a version of Atiyah-Singer for closed embeddings, and maybe that's well known? | |
| Aug 9, 2020 at 15:09 | comment | added | Aaron Bergman | @MathCrawler Nonetheless, from a little googling, this thesis seems to have a generalization: repository.upenn.edu/cgi/… | |
| Aug 9, 2020 at 15:06 | comment | added | Aaron Bergman | Are you ok with the case of a product? | |
| Aug 9, 2020 at 5:02 | comment | added | MathCrawler | @AaronBergman Even restricting to the projective case over the complex numbers, I do not see how the Families' Index Theorem yields (GRR). Could you be more explicit (provide some details, cite sources, etc.)? | |
| Aug 9, 2020 at 2:23 | comment | added | Aaron Bergman | Protective $\longrightarrow$ projective of course. | |
| Aug 9, 2020 at 1:59 | comment | added | Aaron Bergman | If I remember correctly, the Toledo-Tong stuff deals with the fact that you may not have a resolution by locally frees on a complex analytic manifold. I suspect that if you grant such a resolution (taking the protective case, say), things would go through in a straightforward manner. I’m not an expert, however, so I’m hoping someone will chime in, and I don’t need to try to work through the details :) | |
| Aug 9, 2020 at 1:57 | comment | added | Aaron Bergman | @MathCrawler It seems to me that multiple generalizations are going on here. GRR both generalizes HRR to the relative case and to the case of coherent sheaves. The families index theorem, I believe, gives you the relative generalization you are looking for. However, for coherent sheaves you have to be a bit more careful because they can be trickier on analytic manifolds than on algebraic varieties, even in the smooth case. Cont’d. | |
| Aug 9, 2020 at 0:04 | history | edited | MathCrawler | CC BY-SA 4.0 |
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| Aug 8, 2020 at 23:50 | comment | added | MathCrawler | @AaronBergmann Rather not, I think, for else, (GRR) would arise just by specializing some parameters in its formulation, just as (HRR) arises from (GRR) by specializing the morphism and from (ASI) by specializing the operator, and I know of no way to formulate the Index Theorem for Families in such a way that specializing some parameters ieads to (GRR). By the way, this would make the appearance of elaborate papers like [6] 14 years later look somewhat strange. | |
| Aug 8, 2020 at 23:42 | history | edited | MathCrawler | CC BY-SA 4.0 |
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| Aug 8, 2020 at 23:25 | history | edited | MathCrawler | CC BY-SA 4.0 |
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| Aug 8, 2020 at 15:47 | history | edited | MathCrawler | CC BY-SA 4.0 |
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| Aug 8, 2020 at 15:27 | history | edited | MathCrawler | CC BY-SA 4.0 |
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| Aug 7, 2020 at 8:51 | comment | added | Dustin Clausen | I guess the families index theorem corresponds to the special case of GRR where the map is smooth. | |
| Aug 7, 2020 at 4:40 | history | edited | MathCrawler | CC BY-SA 4.0 |
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| Aug 7, 2020 at 1:14 | comment | added | Aaron Bergman | Isn’t this just the families index theorem, proved in Atiyah-Singer IV? | |
| Aug 7, 2020 at 0:38 | history | edited | MathCrawler | CC BY-SA 4.0 |
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| Aug 7, 2020 at 0:18 | history | asked | MathCrawler | CC BY-SA 4.0 |