4
$\begingroup$

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

$\endgroup$
1
  • 1
    $\begingroup$ Correctly stated (over $\mathbb{C}$), this comes down to comparing $\mathbb{Z}$-representations of the topological fundamental group with $\mathbb{Z}_{\ell}$-representations of the algebraic fundamental group. The algebraic fundamental group is the profinite completion of the topological fundamental group, so there is a lot one can say. $\endgroup$
    – anon
    Mar 22, 2013 at 15:03

2 Answers 2

5
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.