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I’m having an e-mail conversation with Michael Harris, the number theorist, who mentions the possibility of “a much higher version of the Grothendieck-Riemann-Roch theorem”. I see we don’t have a page even for Riemann-Roch. There must be a wonderful nPOV on Grothendieck–Hirzebruch–Riemann–Roch, Arithmetic Riemann-Roch and Riemann-Roch for algebraic stacks.
I was just thinking the other day we should have an entry on this (especially as advertisement of the nPOV), but I don’t feel particularly up to writing it, at least not in a short amount of time.
Hirzebruch-Riemann-Roch is much simpler and less powerful so it is known in many modern setup, like for dg-categories, where it is not clear at all how to get a version of Grothendieck-RR. There is as of today STILL NO KNOWN NATURAL PROOF of Grothendieck-Riemann-Roch. All proofs are dirty at least in part, e.g. one proves it for some special cases, like projective spaces first, then uses this to generalize and so on. Thus, it is far from an exercise to put it in some easy perspective. I mean the statement looks natural and simple but without getting natural proof one can not really fully understand the origin. Still also no noncommutative version of GRR (unlike HRR) is known. I know of very good people working on it, and having very slow progress.
Arithmetic RR ? Come on, the understanding is even below the algebraic case. Because the Arakelov geometry itself is a syncresy of algebraic picture and of analytic picture at infinity. Nikolai Durov attempted a cleaner approach aimed at developing certain K-theory for generalized schemes a la Durov (using nonabelian derived functors and so on), but this work is not finished (see his thesis) to my knowledge.
We have now a stub: Grothendieck-Riemann-Roch theorem.
If it’s a failure of naturality is it likely to have a good abstract description. Or can we hope the failure is systematic?
It has. As the index theorem has. There is a relation, see
This should have a natural interpretation in terms of orientation in generalized cohomology.
Unfortunately, I have no time to look deeper into this for moment.
Interpretation likely yes, but the proof, even for the ordinary varieties, does not follow from general nonsense. At least nobody, even Grothendieck, did not succeed as of now.
It also does not hold for all schemes. One needs some assumptions like smooth quasi-projective; really the study of morphisms in algebraic geometry is very subtle and here rather essential. I mean deriving etc. does not correct if you are out of the class required. Such conditions appear even more radical in some light noncommutative generalizations. This subject is, as everything in the intersection theory, closely related to various gradings on cycles. This kind of thing is still not touched seriously by -categorical community. It is very interesting.
There is a lot of interesting relations with the subject of deformation quantization. Not only Bressler-Nest-Tsygan but see also Kontsevich and also Caldararu.
It also does not hold for all schemes.
Also, it is not restricted to algebraic geometry. For instance it works for smooth submersions equipped with K-orientation.
Bunke/Schick even give a refinement to differential cohomology here.
Something wrong with your link again.
But you see, this things are special cases of algebraic geometry. Namely, topological K-theory is a special case of algebraic K-theory, and everything can be phrased in terms of algebraic geometry of spectra. People in K-theory have many GRR variants there. Eventually in the correct setup one needs to take care of types of morphisms.
However it would be interesting if the situations like in Bunke have proof which is entirely general and does not rely on special computations, even in the topological situation.
I have fixed the link. I have included the references also in the entry.
Not sure if I follow the remark about algebraic geometry. Not everything in K-theory is about algebraic geometry, just because there is algebraic K-theory. Here the statement is: there is a notion of GRR over spaces which are not schemes.
Surely, there are zillions of setups for GRR. Like for mirror symmetry one has the toy example of the topological T-duality.
In my vague memory, this topological statement was explained to me by Bressler as a special case of a statement about algebraic K-theory. To understand the relation to index theorem, one also involves topological Hochschild and topological cyclic homology (at the level of sheaves of topological spectra). Why I emphasise schemes is not the setup, as there are many, but the fact that the spectra and cycles are involved. Of course, one can form such categories that their spectra (now in the sense of spectral theory, not in the sense of topology) will be topological or differential manifolds. Cycles have various dimensions, pushforwards etc. The intersection theory is notoriously difficult and rich in the algebraic situation, while it simplifies a lot in differential and topological situations.
Now, you see, one can compare to the index theorem. The advantage of the algebraic one is that one has a formal version, which does not depend on analytic details and conditions, so it may be argued as more fundamental.
P.S. You see the study of morphisms also in Bunke-Schick. In this setup as well one needs a proper morphism again. This is essential, not technical, condition. (One can of course use usual tricks like limit to proper objects and get slight extensions).
David, can you share any information on what Michael Harris meant by “much higher version”?
All proofs are dirty at least in part, e.g. one proves it for some special cases, like projective spaces first, then uses this to generalize and so on
could we use Jim Dolan’s version of algebraic geometry, where projective spaces have a nice universal property, to get this first step? This would remove a little of the arbitrariness.
Todd #13, I’ll ask him. It was in the context of a response he’s writing to some of Voevodsky’s claims about HoTT, so should be very relevant to us.
David Roberts: I did not see so far anything essentially new in Dolan’s “version” of algebraic geometry, so I am skeptical that his insight has power to attack GRR.
15: This sounds very interesting.
So here’s the reply from Michael:
I didn’t mean anything very specific. It’s more a perspective on the drive for generality. It has definite historic roots: class field theory unified various special cases of reciprocity laws, but when extended to function fields over finite fields the reciprocity law turned out to be equivalent to the Riemann-Roch theorem for curves. In the 1950s the Riemann-Roch theorem was generalized by Hirzebruch, then by Grothendieck; meanwhile Serre’s “groupes algebriques et corps de classes” reinterpreted class field theory for curves in terms of generalized Jacobians. Then the Hirzebruch-Riemann-Roch correspondence became a special case of the Atiyah-Singer index theorem; but so did the Selberg trace formula, which Langlands saw as the key to his functoriality conjectures. And there was SGA 6, which led to higher algebraic K-theory and Arakelov theory. Generalized Jacobians reappeared in the geometric Langlands correspondence, which is now the subject of an attempted synthesis (by Langlands, in part with Frenkel and Ngo) involving both the Selberg trace formula and the Hitchin fibration. Meanwhile, Kapustin and Witten have reinterpreted the geometric Langlands correspondence as a consequence of mirror symmetry for Hitchin fibrations, using the framework of gauge theory, which in turn is an outgrowth of the index theorem. But all this is just what one might call 1-reciprocity; higher reciprocity conjectures have been formulated by Kapranov, and are being taken up again by Parshin, and in connection with motivic cohomology can be seen as a generalization of GRR to higher dimensions, undoubtedly incorporating Arakelov theory. And this can continue indefinitely.
To support the discussion on Arakelov geometry originating above, I added an extensive reference section to its entry.
Just a reply to Zoran#16: while you probably have good reason to be skeptical, it’s probably also true that you’ve only seen a small sliver of what Jim Dolan has been up to in recent years. I know from personal communication that he is interested in investigating GRR through his various lenses, although I’m pretty sure that he would agree with you that he hasn’t yet acquired a satisfying understanding of GRR. He told me he is very open to the idea of studying it more deeply.
@David: thank you!
Todd, well, I surely wish you prove to be right. To my earlier disappointment, he was rather apologetic and tending to be limiting in the scientific scope in the cafe discussion on this topic while discussing with Urs and John and David Ben-Zvi. In my feeling, if anything is soon to appear really new from what I see there so far is a possible new insight in the Tannakian theory in algebraic geometry like contexts.
As for “higher dimensional GRR” one is alluding in 17, for arithmetic RR, this “higher” modifier is partly about looking at Chern character for higher K-groups. In any case, as there are many Chern characters starting in some sort of K-theory or cyclic homology and ending in Chow groups or some simpler cohomology theory, including in noncommutative geometry, algebraic geometry, spectra in homotopy theory etc. there are many GRR-s. Having higher K-groups is known for many cases (sometimes one works at the level of spectra having all -s at the same time). I understand that it is a huge achievement to go higher in arithmetic setup, regarding the geometric depth behind the establishment of related objects like the higher class field theory, so though in other setup rather easy, in this setup “higher” is interesting, but not new for the POV.
Hi Todd,
just a side remark on your comment in #19. I have no doubts at all about Jim Dolan having awesome insights. But nevertheless, it is unfortunately true that the statement
you’ve only seen a small sliver of what Jim Dolan has been up to
implies that the answer to David Roberts’ question #14
could we use Jim Dolan’s version of algebraic geometry
(my emphasis) is: unfortunately no. Possibly he could, but since we don’t know about it, we seem to be left with looking for ideas elsewhere.
And in this special case I have to agree with Zoran’s #20: the small sliver that we did get to see made me think that Lurie’s Tannaka duality for geometric stacks was the relevant insight to turn to.
But maybe one could improve on this situation: maybe you could every now and then communicate to the outside world some of Jim Dolan’s insights? I’d certainly be interested, in as far as it is suitable for my ears.
re Zoran’s #21
in this setup “higher” is interesting, but not new for the nPOV.
Yes, this is my impression, too. The Chern character is ultimately given by a map of spectra or similar, so this is fully -categorical already.
The only way to “go higher” here seems to go to some -categorical generalization for . But “cohomology theory” in this context, whatever that should mean, is almost entirely unexplored. I’d tend to doubt that the GRR theorem would be the place to start investigating this. But maybe I am wrong.
Urs and Zoran: yes, I agree with all of this. I can try to do my best to help communicate (and I’m aware that I’ve been coming up short).
I fully agree with the impression suggested in Urs 23 (as he nicely phrased) and pledge along with Urs 22 for Todd’s messaging from Dolan to extend beyond the “sliver”.
Zoran: okay. Just to forestall a possible misunderstanding that might be in #20, let me emphasize that I am in no way hinting that Dolan does indeed have secret insights into GRR. It’s just to say that you never know with him, since he holds a great deal in reserve and is slow to publicize his thoughts, and that conversation was after all back in 2009.
To add to that, I am generally slow to catch his drift! Urs #22, #23 seem to summarize our current situation well. I’ll try to be of help, when and if I can.
While I’d be interested in hearing Jim Dolan lore, may I re-iterate my remark from #9, #11: in either case, peculiarities of algebraic geometry seem to be besides the point in the present discussion. GRR is a statement not restricted to algebraic geometry.
Well, Urs, this can not be resolved. You see many statements have easier special cases or versions. Like I mentioned mirror symmetry vs. topological T-duality. Now, only once one knows the essence, one can claim if the essence is seen at all in simpler cases. Why I insist on algebraic version is that people told me other algebraic generalizations which are in a way simpler than GRR while generalizing and told in completely different (spectral) terms which are impossible to relate to topological case.
Of course, if something is true, every true fact is implying it. So often, there is a phenomenon which is at the boundary of two different phenomena, which have common intersection. For example, GRR and Atiyah-Singer index theorem agree (and imply each other) in some situations. That does not mean that GRR and index theorem are the same mathematical entities in general. So I can not defend by direct argument, but my gut feeling is that the algebraic geometry is not a “peculiar” case only, but has some deep essential aspects lost in some simpler versions. That is, it is not only a complication.
maybe you could every now and then communicate to the outside world some of Jim Dolan’s insights?
I second this!
Sheesh, okay already!
Well, Urs, this can not be resolved.
On the contrary, this has long been resolved. There is GRR for spaces which are not schemes, not algebraic spaces, not algebraic stacks, not objects in algebraic geoemtry.
Have people ever dropped by Jim’s notebook?
I didn’t know this. So I am looking at January 2012… Hm, that leaves me rather disappointed.
I’ll better wait for Todd’s version ;-)
On the contrary, this has long been resolved. There is GRR for spaces which are not schemes
I know there are, it seems you did not read the rest of my post. I emphasised this as well. The fact that there are easy versions does not imply that the essence is inherited in those toy examples. At least one can not say until the essence is discovered. And having no natural proof yet for the classical GRR is the indication that the essence is not discovered, and I aluded to some other indications (which I am not authorized nor competent to say precise). Hence not resolved. See more carefully what I wrote above.
@yes, try <notebook360x.blogspot.com/2011/12/toposes-of-toric-quasicoherent-sheaves.html>, it’s more coherent.
EDIT: Oh, Todd already gave this link :S Just noticed
September 2010 and earlier in Jim’s notebook seem more understandable.
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