Conjectures about q-de Rham complex: examples

Peter Scholze formulated several conjectures about $ q-$ de Rham complex here. Especially intriguing to me is Conjecture $ 3.2$ : Betti-de Rham comparison isomorphism.

In this paper the author emphasized the lack of nontrivial examples and suggests to do a computation for the universal family of elliptic curves.

Question 1: Has anyone done this computation (or any other interesting computations)? I will be especially interested in those involving basic hypergeometric functions.

Question 2: What does conjecture $ 3.2$ mean in practice? Naively one might hope that it leads to some $ q-$ deformation of periods.

Leafwise de Rham cohomology(A true definition of Differential forms along leaves)

For a foliated space $ (M, \mathcal{F})$ , one associate a leafwise de Rham cohomology. This cohomology and trace class operators on this cohomology and trace interpretations for closed orbits of certain flow on $ M$ is the main object of this paper”Number theory and dynamical system of foliated manifolds.

But in the later paper, I did not find a very precise definition of “Differential forms along leaf”.

So I try to find other papers or talk to find a precise definition for this concept. Then I found a definition at page 8 of this talk “Lefschetz trace formula for flow on foliated manifolds” which give a local representation for such forms. But my problem is the following:

I think that such representation, which is quoted below, is NOT invariant under foliation charts:

$ $ \omega\sum_{\alpha_1<\alpha_2<\ldots<\alpha_k} a_{\alpha}(x,y) dx_{\alpha_1}\wedge dx_{\alpha_2}\wedge \ldots\wedge dx_{\alpha_k}$ $

Am I mistaken?

What is a precise definition and precise local representations of “Differential forms along leaves”?

Is there literature on a de Rham analogue of the Mumford-Tate group or ell-adic monodromy group?

Let $ X$ be a smooth projective variety over $ \mathbb{Q}$ . The theory of motives predicts that for each cohomology theory, there should be a distinguished Zariski closed subgroup of $ GL(H^k_{\bullet}(X))$ , the motivic Galois group. This group has conjectural descriptions for Betti and $ \ell$ -adic cohomology, and I am wondering if there is any literature on this group for de Rham cohomology.

The Betti cohomology $ H^k_B(X)$ is a rational Hodge structure, hence there is a representation $ \mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_m\to GL(H^k_B(X)\otimes\mathbb{R})$ . The Mumford-Tate group $ MT^k(X)\leq GL(H^k_B(X))$ is defined to be the smallest Zariski-closed subgroup containing the image of the representation. The Hodge conjecture would imply that $ MT^k(X)$ has finite index in the Betti motivic Galois group.

The $ \ell$ -adic cohomology $ H^k(X;\mathbb{Q}_\ell)$ is a representation of the absolute Galois group, and the $ \ell$ -adic monodromy group $ G_\ell^k(X)\leq GL(H^k(X;\mathbb{Q}_\ell))$ is defined to be the smallest Zariski-closed subgroup containing the image of the representation. The Tate conjecture would imply that $ G_\ell^k(X)$ is the full $ \ell$ -adic motivic Galois group.

Is there any literature on an analogue of the Mumford-Tate group or the $ \ell$ -adic monodromy group inside $ GL(H^k_{dR}(X))$ ?

I believe that a conjecture of Ogus would imply that the de Rham motivic Galois group is smallest Zariski closed subgroup whose $ \mathbb{Q}_p$ points contain $ F_p$ for all sufficiently large $ p$ (where $ F_p\in GL(H^k_{dR}(X))(\mathbb{Q}_p)$ is the crystalline Frobenius). However I have some doubt about this, because AndrĂ©’s book on motives states the relationship between the Hodge conjecture and the Mumford-Tate group (Proposition, and the relationship between the Tate conjecture and the $ \ell$ -adic monodromy group (Proposition, but does not give an analogous statement for the Ogus conjecture.

I also believe that the period conjecture of Grothendieck would imply that the de Rham motivic Galois group is the smallest Zariski closed subgroup containing all elements $ \varphi\in GL(H^k_{dR}(X))(\mathbb{Q})$ whose image in $ GL(H^k_{B}(X))(\mathbb{C})$ under the Betti-de Rham comparison isomorphism is contained in $ GL(H^k_{B}(X))(\mathbb{Q})$ . However, I also cannot find a statement like this in AndrĂ©’s book.

Exterior Product on de rham homology

Given a smooth manifold $ M$ and its differential graded commutstive de Rham algebra $ (\Omega(M),d,\wedge)$ , the wedge product $ \wedge$ can be projected onto the de Rham cohomology $ (H_{dR}(M),\wedge)$ .

What is an elegant way to proof that the wedge product is, indeed, well defined on $ H_{dR}(M)$ ?

Equivalence De Rham und Dolbeault Groupoids

I believe there is an error or incompleteness in Goldman’s and Xia’s Proof on the equivalence of the De Rham and Dolbeault Groupoids.

In page 33 of they want to proof that $ Mor(D_1,D_2)\to Mor(S(D_1),S(D_2))$ is bijective.

They do that by proving that $ Mor(D_1,D_2)$ and $ Mor(S(D_1),S(D_2))$ are either $ \varnothing$ or $ \mathrm{C}^*$ depending on whether $ D_1$ and $ D_2$ (or $ S(D_1)$ and $ S(D_2)$ ) are equivalent.

This however does not exclude the possibility that $ Mor(D_1,D_2)\to Mor(S(D_1),S(D_2))$ is actually the empty morphism $ \varnothing \to \mathrm{C}^*$ . For that they would need that $ S_*$ is injective, but they do not proof that, in fact that is the purpose of the Corollary 4.1.10

Am I getting something wrong? How can I proof that $ S_*$ is injective?

De Rham cohomology and extension of scalars

Let $ K$ be a field of characteristic zero and let $ X$ be a smooth variety over $ K$ . Given a field extension $ L/K$ , I denote by $ X_L$ the variety $ X \times_{Spec(k)} Spec(L)$ .

What is the easiest way to prove that algebraic de Rham cohomology behaves well with respect to this operation, in the following sense? $ $ H^\ast_{dR}(X) \otimes_K L \simeq H^\ast_{dR}(X_L) $ $

Serre spectral sequence for de Rham cohomology

Suppose we a given a fibration of manifolds $ p\colon E\to M$ with a path connected fiber $ F$ and simply connected $ M$ , then we have the Serre spectral sequence with

$ $ E_2^{p,q} = H^p(M,\underline{H^q(F)}) $ $

The standard proof of its convergence to $ H^n(E)$ is purely topological and goes for singular or cellular cohomology. Can one give the proof in terms of de Rham cohomology?

In fact, I’m even more interested in de Rham cohomology with compact support. Now we have some difficulties in defining the local system as cohomology with compact support are no longer homotopy invariant but I hope these problems can be overcome.

Notational question on Kunneth Formula for de Rham cohomology

I got to learn the Kunneth Formula for de Rham cohomology as following.

$ $ H^n(X\times Y)=\sum_{n=p+q} H^p(X)\otimes H^q(Y). $ $

And I could find same notation from

Actually, it is quite unfamiliar to use $ \sum$ for spaces. I think it should be $ \oplus$ instead of $ \sum$ . It might be stupid question and maybe I am wrong and missing something. I am looking for some clarification for it!

And clear explanation for this would be appreciated! Thanks in advance.