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Thanks, Andreas,
I am longing to understand all this at some deeper level. Maybe we may jointly make some progress here.
Last time that Zoran and David C. kept saying to me (here) something like “you sound like what you need is Arakelov geometry!” I went an tried to look at some introductions. I am afraid I ran out of steam before I was able to get a clear picture though.I should try again.
One thing that struck me, though, is that what I read Arakelov did very much sounded like… geometric quantization. One picks some 2-form, lifts it to a line bundle etc. Right? (I have to remind myself again. We should write out a definition-section on the nLab!)
I’ll go and look at the articles by Morin that you are recommending…
A question:
I have read claims (in various places, but I keep forgetting where, maybe also on your site somewhere?) that modern (Berkovich-style(?)) rigid analytic geometry is to be regarded as superior to traditional algebraic geometry, in that on the one hand it subsumes it, while on the other it generalizes it to situations where methods of algebraic geometry fail. Apparently much of the proofs in local Langlands don’t actually work in algebraic geometry, but crucially go into the analytic regime. (!?)
Could you remind me of where I might have read this? :-) And more generally, could you (or anyone reading ehre) give me some further pointers to material that would expand on this a bit?
[edit: stupid me, of course I read that on the nLab! at global analytic geometry – Motivation]
Did you ever see this grant proposal – Homotopy theory and algebraic geometry?
Yes, Duisburg-Essen seems to (have) become quite an accumulation point. (I have studied in Essen, actually (before it was fused to Essen-Duisburg)).
re #7: but is the global analytic geometry that Frederic is referring to in that entry not more general than rigid analytic geometry? I thought that was the point, but I don’t really know.
There seem to be so many to ways to formulate the non-Archimedean world, and so many ways to compare the ways, such as Ben-Bassat and Kremnizer’s Non-Archimedean analytic geometry as relative algebraic geometry
In this article, we consider Berkovich analytic spaces from the perspective of algebraic geometry relative to the closed symmetric monoidal categories of Banach spaces. This language is very universal and provides a place to compare diﬀerent geometries (Huber spaces, rigid analytic spaces, Berkovich spaces and others).
As they point out, global analytic geometry is supposed to go beyond the non-Archimedean:
In future work, we intend to embed rigid analytic spaces and Huber spaces into the categories of schemes over the opposite category to Banach algebras. We also intend develop a variant which will work with complete, convex bornological algebras, instead of Banach algebras or Frechet algebras or other types of topological vector spaces. Similarly to Paugam [33] we would like to handle the Archimedean and non-Archimedean cases with a single language.
Paugam himself writes in Global analytic geometry:
Many interesting results on polynomial equations can be proved using the mysterious interactions between algebraic, complex analytic and p-adic analytic geometry. The aim of global analytic geometry is to construct a category of spaces which contains these three geometries.
David, thanks for recalling all this.
Here is the vague picture that I am after, possibly naive:
Say we start being interested in worldsheets, hence in complex tori. We find the Witten genus by doing differential geometry with a little bit of complex geometry. Then we (or somebody at least…) refine this to the string orientation of tmf which in effect is constructed via fracture squares. By some magic, the original construction in differential geometry transmutes into one in $p$-adic geometry.
While this is understood in a step-by-step fashion, it seems to me that it is generally unclear what the conceptual picture underlying this black magic is. (Or at least so suggests the ratio of votes over answers of this MO question)
What I would like to see is a unified concept of geometry that makes this black magic become concepually clear. Some unified form of geometry where we start out wanting to compute partition functions of $(4k+2)$-branes on complex analytic worldvolumes and naturally end up decomposing this problem in a tower of problems in p-adic geometry that also knows about the underlying arithmetic geometry over the corresponding prime fields.
Can you even imagine what’s needed? Would the long sought for $p$-adic/adelic cohesion help?
Rigid analytic spaces locally modeled on Tate algebras/polydiscs will be cohesive, I am pretty sure now. One should check with the experts in suitable words, such as “are rigid spaces locally etale contractible?” (I once sent this question spelled out in some simplicial detail to one of the central experts, but just got a “thanks, but I don’t understand the question” Just using the right words might help.).
So analytic geometry, archimedean and non-archimedean should be fine. What remains to be understood is how analytification sits in the picture. We want to be talking about arithmetic spaces equipped with differential geometric structure on their analytification. Of course that’s just what Bunke-Tamme do in the archimedean case, but I don’t really understand the organizing principle yet. Something seems to be missing, but likely just on my end.
When things work out, the following ought to happen: start with a 4k+3 connection on the moduli stack of some differential cohomology theory. Transgress to a complex 2k+1 dimensional space, such that the transgressed connection becomes Kaehler. Now this should automatically know all the positive characteristic, hence arithmetic, geometries underlying this analytic one. Hence the deformation theory will be an Artin-Mazur formal group which in good cases should define a CY-cohomology theory. The corresponding equivariant theory should be the modular functor quantizing the 4k+1 brane.
For k= 0 we know this is just what happens piecewise, but it would be good to have a more conceptual way to put it all together. As far as the case k=1 and k=2 is understood at all, it seems to naturally want to sit in this picture, too. That’s why I feel this should be the right story, roughly.
(This here from my phone, bear with me)
Okay, I have tried again to ask now here on MO.
Your ratio of #votes/(# non-self answers) has to be near the highest on MO.
True, recently I wasn’t very lucky with getting replies. On the other hand, I had recently asked several questions of the kind: “this ought to be studied, but has it been studied?” and the number of votes together with the lack of answers is probably the answer itself: “yes, that is indeed an interesting question to ask and, no, it hasn’t been answered yet”.
Compiling the latest question made me remind myself about Berkovich’s local contractibility statement and I remember now that actually Berkovich spaces are not covered by polydiscs, but by inductive systems of “analytic domains”.
Did you see, Urs, Joe Berner’s edit to his answer to your MO question in #15, announcing a new paper
Thanks for the pointer! Interesting.
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