About "de-Rham" and "l-adic" local systems - comparison

Question

Hello,

Suppose that $k$ is an algebraically closed field of char. 0.

Let $X$ be a smooth connected variety over $k$.

Then I have the category $A$ of Regular Singular smooth $D$-modules on $X$ (i.e. algebraic vector bundles equipped with regular singular algebraic flat connections).

For a category $B$, I am less sure. I would like to say "local systems of $k$-vector spaces for the etale topology on $X$", but maybe this is not good as $k$ is not finite or of $l$-adic nature in general.

So:

1) Can one make sense from category $B$ and wish $A$ and $B$ to be equivalent?

2) Anyway, there seems to be a functor from the category of local systems (of finite sets) on the etale topology to $A$, by "tensoring" with the constant $D$-module (as $D$-modules are etale local). What can one say about this functor?

3) On a "decategorified" level, what can one say about the etale fundamental group, versus the group which we get by Tannakian formalism from $A$?

Thank you, Sasha

Answer 1

I cannot say much about the $\ell$-adic side. I will give "classical" answers to 1)-3):As you know, the Riemann-Hilbert correspondence says that on a smooth complex variety $X$ the category of $A$ of vector bundles with flat regular singular connection is equivalent to the category of representations of the topological fundamental group $\pi_1^{\text{top}}$ on finite dimensional complex vector spaces. Lets write this category $\operatorname{Repf}_{\mathbb{C}} \pi_1^{\text{top}}(X)$ (neglecting base points).

Since $\pi_1^{\operatorname{et}}(X)$ is the profinite completion of the abstract group $\pi_1^{\operatorname{top}}(X)$, a representation of $\pi_1^{\operatorname{top}}(X)\rightarrow GL(V)$ which factors through a finite quotient can be thought of a representation of $\pi_1^{\operatorname{et}}(X)\rightarrow GL(V)$ which is continuous with respect to the profinite topology on the left and the discrete topology on the right. Hence, given an etale covering of $f:Y\rightarrow X$, Galois theory associates with it a finite $\pi^{\operatorname{et}}(X)$-set, which we can linearize to and get a representation and then a $\mathcal{D}$-module. But what does this mean concretely? It is not difficult to check that ${f_*}\mathcal{O}_Y$ is a $\mathcal{O}_{X}$-coherent $\mathcal{D}_X$-module (hence a vector bundle), and it is a theorem that it is regular singular (Gauss-Manin).

About your third question: The pro-algebraic affine group scheme associated with the Tannaka category $\operatorname{Repf}_{\mathbb{C}} \pi_1^{\operatorname{top}}(X)$ is by definition the pro-algebraic completion of the finitely generated group $\pi_1^{\operatorname{top}}(X)$. The etale fundamental group is the profinite completion of this group. And amazingly, the profinite completion "controls" the pro-algebraic completion:

Theorem: Let $f:G\rightarrow H$ be a morphism of finitely generated (abstract) groups. Then $f$ induces an isomorphism on pro-algebraic completions if and only if it induces an isomorphism on profinite completions.

I am told that this was first discovered by Malcev, and then independently rediscovered by Grothendieck. Grothendieck precisely had the application the the Riemann-Hilbert correspondence in mind.

See: Grothendieck, Alexander Représentations linéaires et compactification profinie des groupes discrets. (French. English summary) Manuscripta Math. 2 1970 375–396.

Answer 2

One has big problems trying to make category $B$ work.

Standard Lefschetz principle arguments justify Lars' use of $\mathbb C$ even if $k$ is not, in fact $\mathbb C$.

Thus, we can write $A$ as the category of representations of $\pi_1^{top} \to GL_n (\mathbb C) $

Any $l$-adic construction is going to be about continuous representations $\pi_1^{et} \to GL_n(\mathbb C)$ for some topology on $GL_n(\mathbb C)$.

To get an equivalence of categories in any kind of nice way, we clearly need every representation $\pi_1^{top} \to GL_n(\mathbb C)$ to extend to a continuous representation of $\pi_1^{et}$. So its image must lie in a compact subgroup. But $GL_n(\mathbb Q_l)$ has elements, like $\frac{1}{l} I $, that do not lie in any compact subgroup. I don't see any way to modify this construction to fix that bug.

For 3), you may find my answer here interesting.