Timeline for Clausen's modified Hodge Conjecture
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| May 29, 2023 at 12:32 | comment | added | Dustin Clausen | Hi Jochen, no, that would not be so obscure! I think it corresponds to the duals of what are called "strongly nuclear" Frechet spaces. The difference is that the transition maps themselves are required to be rapid decay. So, for example, the usual space of distributions on the circle is dual nuclear but not dual strongly nuclear. | |
| May 29, 2023 at 11:53 | comment | added | Jochen Wengenroth | Does the somewhat obscure class of dual Fréchet spaces coincide with the class of duals of nuclear Fréchet spaces? | |
| May 19, 2023 at 20:39 | comment | added | AG learner | @DustinClausen I see what you are saying in the previous comment. Thanks for the clarification! | |
| May 19, 2023 at 3:44 | comment | added | David Roberts♦ | any analytic K-theory class can be continuously deformed into an algebraic K-theory class <--- nice. This reminds me of Oka theory which, in the form I'm vaguely familiar with, relates complex analytic and complex algebraic geometry, giving for instance in the best possible cases homotopy equivalence of mapping spaces on either side under suitable assumptions on domains and codomains, but at the least surjective or bijective maps on $\pi_0$ of the same. | |
| May 18, 2023 at 16:55 | comment | added | Dustin Clausen | AG learner, we completely agree on this. That's why I said that (what I was calling) "the Deligne cohomology analog of the Hodge conjecture" is a false statement, not a true statement. | |
| May 18, 2023 at 15:15 | comment | added | AG learner | @DustinClausen No, it is not. For a general quintic 3-fold, the Abel-Jacobi image of homologously trivial algebraic 1-cycles consists of only $\textit{countably}$ many points. So it is never subjective to the intermediate Jacobian. In general, the Abel-Jacobi image is a countable union translates of a polarized subtorus (which is the AJ image of cycles that are algebraically trivial) of intermediate Jacobian. As long as the Hodge structure associated to the intermediate Jacobian is not concentrated on the minimal level $H^{p,p-1}\oplus H^{p-1,p}$, the Abel-Jacobi image is not surjective. | |
| May 18, 2023 at 13:52 | comment | added | Dustin Clausen | Of course, it would be very interesting to find yet another variant of K-theory, which matches absolute Hodge cohomology... but it won't be this thing I'm calling analytic K-theory. | |
| May 18, 2023 at 13:48 | comment | added | Dustin Clausen | Hi Jan, one can prove a GAGA theorem showing that the category of quasicoherent sheaves on $X$ (algebraic $X$, but over "analytic" $\mathbb{C}$) agrees with quasicoherent sheaves on $X^{an}$. In particular, whatever this analytic K-theory is, it has to make sense for a completely general complex analytic space (not necessarily compact, let alone Kahler), which precludes anything based on a weight filtration (like absolute Hodge cohomology). More concretely, in the base case $X=\ast$ one can calculate the negative analytic K-groups and see that they match Deligne cohomology. | |
| May 18, 2023 at 11:35 | comment | added | user497064 | I don't know the details of the definition of $K_i^{\rm an}$, so I cannot tell by myself why one should really favor Deligne cohomology as a target for the conjectured isomorphism, over Beilinson's absolute Hodge cohomology. | |
| May 18, 2023 at 11:32 | comment | added | user497064 | @DustinClausen What is roughly a specific reason why you are expecting the target of your conjectured isomorphism from analytic $K$-theory to be Deligne cohomology, and not Beilinson's absolute Hodge cohomology? Deligne cohomology $H^p_{\mathcal{D}}(X^{\rm an},\mathbf{Q}(q))$ does not vanish for $p>2q$ and is considered "pathological" in this range, whereas absolute Hodge cohomology vanishes (and agrees with Deligne cohomology for $p\le 2q$). It is also an "absolute cohomology theory", like motivic, continuous étale, syntomic cohomology, etc., contrary to Deligne cohomology. | |
| May 18, 2023 at 7:12 | comment | added | Dustin Clausen | Hi Will --- actually, no, I don't know how to check even that. I believe dimension 1 should be within reach, but already dimension 2 seems quite tricky. The problem is that, unfortunately, this Efimov K-theory is not so easy to get a direct handle on. Generally speaking, the evidence for the conjecture is a bit indirect. One thing is that the statement for $K^{an}_i$ is true for $i \leq - dim(X)$ (the range in which Deligne cohomology is purely topological). Another thing is that the two sides (analytic K-theory and Deligne cohomology) have the exact same functorial/formal properties. | |
| May 18, 2023 at 6:53 | comment | added | Will Sawin | The surjectivity part of your isomorphism statement is clear for varieties of dimension $\leq 2$ since the map from the usual Chow group to Deligne cohomology is surjective in that range (since the Hodge conjecture is known and the intermediate Jacobian is just a Jacobian with an obvious connection to cycles). Is the injectivity part also proven in low dimension? Just from what you say I don't think it is obvious that the map is injective in dimension 0, though I would imagine you checked this already. | |
| May 18, 2023 at 6:10 | comment | added | Dustin Clausen | The intermediate Jacobian is the kernel of the surjective map from Deligne cohomology to Hodge classes. By "the Hodge conjecture on the level of Deligne cohomology" I meant the statement that $Ch^p(X)_{\mathbb{Q}}\to H^{2p}(X;\mathbb{Q}(p))$ is surjective. Assuming the usual Hodge conjecture, this is equivalent to saying that the Abel-Jacobi map from homologously trivial cycles to the intermediate Jacobian is surjective. Do you agree with this? | |
| May 18, 2023 at 5:22 | comment | added | AG learner | "Now, it is known that the Hodge conjecture fails on the level of Deligne cohomology." I don't think this is what the reference that is linked really means. The obstruction of induction comes from the fact that Griffiths Abel-Jacobi map (for homologously trivial cycles) is not subjective on the intermediate Jacobian, which has nothing to do with Deligne cohomology. | |
| May 17, 2023 at 18:51 | comment | added | user497064 | Got it. Very nice, and thanks for taking the time! | |
| May 17, 2023 at 18:50 | vote | accept | CommunityBot | ||
| May 17, 2023 at 18:29 | history | answered | Dustin Clausen | CC BY-SA 4.0 |