38
$\begingroup$

In a recent talk at the University of Geneve, Dustin Clausen presented a "modified Hodge Conjecture". I found the abstract intriguing but couldn't find videos or notes available online.

If I'm reading the abstract right, his conjecture (which, as I understand from it, is part of joint work with Peter Scholze) is perhaps only similar in spirit to the classical Hodge Conjecture, but doesn't imply it/is not implied by it.

I couldn't deduce anything more precise from the abstract alone and I'd be curious to see a reference, or perhaps some seminar notes describing the statement and its relation to the classical Hodge Conjecture. Can anyone point me to any?

$\endgroup$
3
  • 9
    $\begingroup$ wouldn't it be more effective to email the speaker directly? $\endgroup$ May 17, 2023 at 13:39
  • 12
    $\begingroup$ @CarloBeenakker I didn't really want to bother him with it. Perhaps someone took notes during the talk! Or official notes already exist somewhere but I couldn't find them. That's why I asked the question as a "reference request". Also others could be just as curious as I am $\endgroup$
    – user497064
    May 17, 2023 at 13:44
  • 32
    $\begingroup$ This approach is better than emailing the speaker in one respect: we all get to see the answer. $\endgroup$
    – Ben McKay
    May 17, 2023 at 20:07

1 Answer 1

63
$\begingroup$

It's a bit of a long story, but I can at least give the idea. Let $X$ be a smooth projective variety over $\mathbb{C}$. The Hodge conjecture says that for all $p\geq 0$, the cycle class map

$$Ch^p(X)_{\mathbb{Q}} \to Hdg^p(X)_{\mathbb{Q}}$$

from codimension p algebraic cycles to Hodge classes is surjective, with rational coefficients. First, a small modification: Grothendieck gave a Chern character isomorphism

$$K_0(X)_{\mathbb{Q}} \simeq \oplus_p Ch^p(X)_{\mathbb{Q}}$$

identifying rational Chow theory with rational algebraic K-theory, so the Hodge conjecture can be equivalently reformulated as saying that the map

$$K_0(X)_{\mathbb{Q}} \to \oplus_p Hdg^p(X)_{\mathbb{Q}}$$

is surjective. Now, the proposed modification involves modifying $K_0(X)$ to a different kind of K-theory $K^{an}_0(X)$, which takes into account the analytic nature of $\mathbb{C}$. I will describe more precisely what this analytic K-theory is below, but for now let me take it as a black box. The usual K-theory maps to the analytic K-theory, and the above map extends to a map

$$K^{an}_0(X)_{\mathbb{Q}} \to \oplus_p Hdg^p(X)_{\mathbb{Q}}.$$

Then the weak form of the modified Hodge conjecture is that this map is surjective. This modified form is obviously strictly weaker than the usual Hodge conjecture. But it admits a strengthening, in a different direction. Recall that the usual cycle class map from $Ch^p(X)$ to $Hdg^p(X)$ actually factors through a more refined target, the Deligne cohomology group $H^{2p}(X;\mathbb{Z}(p))$, which is an extension of $Hdg^p(X)$ by the Griffiths intermediate Jacobian. (This Deligne cohomology is essentially the "derived" version of the definition of Hodge classes.) Now, it is known that the Hodge conjecture fails on the level of Deligne cohomology, and in fact this obstructs some inductive methods for trying to prove the Hodge conjecture by induction on dimension and using hyperplane sections (see section 3 of http://publications.ias.edu/sites/default/files/hodge.pdf). However, the situation conjecturally improves when passing to analytic K-theory. Just as with the usual cycle class map to Hodge cycles, the refined cycle class map to Deligne cohomology also extends to a map from analytic K-theory

$$K^{an}_0(X)_{\mathbb{Q}} \to \oplus_p H^{2p}(X;\mathbb{Q}(p)),$$

and we conjecture this map to be not just surjective, but an isomorphism. In fact, one can also make an analogous map out of $K^{an}_i(X)_{\mathbb{Q}}$ for any $i\in\mathbb{Z}$, with values in $\oplus_p H^{2p-i}(X;\mathbb{Q}(p))$, and we even conjecture this to be an isomorphism for all $i$.

To finish I want to say something about what this analytic K-theory is, but first, a sobering remark: one of the reasons people really care about the Hodge conjecture is that it produces strong algebraic data (algebraic cycles) out of rather weak topological/analytic data (Hodge classes). The modified Hodge conjecture does not do this: it produces analytic data instead of algebraic data. But it maybe gives a new perspective on the Hodge conjecture: if the modified Hodge conjecture is true, then the usual Hodge conjecture would essentially amount to the statement that the comparison map $K_0(X) \to K^{an}_0(X)$ is surjective on connected components, i.e. any analytic K-theory class can be continuously deformed into an algebraic K-theory class. Moreover, the modified conjecture fits the spirit of the original Hodge conjecture, in the following sense: cycle classes, or algebraic $K_0$ classes, can be viewed as a kind of universal source for cohomology classes, and the Hodge conjecture is saying that Hodge classes give universal classes. Similarly, the modified Hodge conjecture is saying is that, in some sense, Deligne cohomology is a universal cohomology theory for analytic spaces over the complex numbers. So we're not "missing" any cohomology theories; Hodge theory really has it all. This is actually why I'm interested in the statement, not because it's connected to the traditional Hodge conjecture. I just want to see if we're missing something. If we are missing something, then the modified Hodge conjecture will be wrong, and analytic K-theory will hopefully hint to us exactly what it is that we're missing.

Now, about the definition of analytic K-theory. It fits into the framework of Alexander Efimov's remarkable extension of the scope of algebraic K-theory. Usually, algebraic K-theory is defined in terms of small categories, like the category of vector bundles. Efimov says that this is a mistake, and we should think of it as an invariant of large categories, like the category of quasicoherent sheaves. (Actually, one needs to work with the derived analogs of these, namely perfect complexes and derived quasi-coherent sheaves, but let me slough over this point for the purposes of this already too lengthy explanation...) Now, this seems like an academic distinction, because the small and large categories determine each other, via passing to Ind-objects in one direction and compact objects in the other. But the point is that there are large categories which are not Ind-categories of small categories, but for which algebraic K-theory can still, magically, be defined, and with basically all the same formal properties as in the more restrictive classical context. These are the so-called dualizable categories studied by Lurie. It turns out these things are everywhere once you know to look for them, and I think it's important to get very comfortable with these non-compactly generated beasts...

Anyway, there is such a dualizable, non-compactly generated category of topological $\mathbb{C}$-vector spaces, let's call it $Mod^{an}(\mathbb{C})$ (formally it is a certain full subcategory of derived condensed $\mathbb{C}$-vector spaces), and we can build a theory of (derived) quasicoherent sheaves on $X$ where all the underlying vector spaces are objects of $Mod^{an}(\mathbb{C})$ instead of usual (smaller) category of abstract $\mathbb{C}$-vector spaces. We take Efimov's K-theory of this quasicoherent sheaf category and call it $K^{an}(X)$. The last thing to explain is what this category $Mod^{an}(\mathbb{C})$ is. It turns out that it can be described from a classical functional analysis perspective: its basic objects (which generate everything under colimits inside derived condensed $\mathbb{C}$-vector spaces) are the sequential colimits of (say) Hilbert spaces along injective transition maps which are compact operators, with singular values decaying rapidly. This is a somewhat obscure class of dual Frechet spaces, but it really is completely forced on us by the abstract considerations of Efimov's theory, and I don't believe there's any other choice than this precise one which should work.

To sum up, we replace usual vector bundles by some more nebulous analytic beasts described using functional analysis, and the hope is that this change makes K-theory match the most refined cohomology theory coming from Hodge theory.

$\endgroup$
15
  • 1
    $\begingroup$ Got it. Very nice, and thanks for taking the time! $\endgroup$
    – user497064
    May 17, 2023 at 18:51
  • 1
    $\begingroup$ "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. $\endgroup$
    – AG learner
    May 18, 2023 at 5:22
  • 1
    $\begingroup$ 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? $\endgroup$ May 18, 2023 at 6:10
  • 1
    $\begingroup$ 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. $\endgroup$
    – Will Sawin
    May 18, 2023 at 6:53
  • 2
    $\begingroup$ 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. $\endgroup$
    – David Roberts
    May 19, 2023 at 3:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.