How are motives related to anabelian geometry and Galois-Teichmuller theory?

Question

In Recoltes et Semailles, Grothendieck remarks that the theory of motives is related to anabelian geometry and Galois-Teichmuller theory. My understanding of these subjects is not very solid at this moment, but this is what I understand:

Anabelian geometry tries to ask how much information about a variety is contained in its etale fundamental group. In particular, there exist "anabelian varieties" which should be completely determined by the etale fundamental group (up to isomorphism). The determination of these anabelian varieties is currently ongoing.

Galois-Teichmuller theory tries to understand the absolute Galois group $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ in terms of the automorphisms of the "Teichmuller tower", which is constructed as follows. We begin with the moduli stacks of curves with genus $g$ and $\nu$ marked points. These moduli stacks $\mathcal{M}_{g,\nu}$ have homomorphisms to each other, which correspond to "erasing" marked points and "gluing". The Teichmuller tower $\hat{T}_{g,\nu}$ comes from the profinite fundamental groupoids of these moduli stacks.

The theory of motives is some sort of "universal cohomology theory" in the sense that any Weil cohomology theory (which is a functor from smooth projective varieties to graded algebras over a field) factors through it. This is obtained from some process of "linearization" of algebraic varieties (considering correspondences as morphisms, followed by the process of "passing to the pseudo-abelian envelope", and formally inverting the Lefschetz motive).

Related to the theory of motives is the concept of a Tannakian category, which provides a kind of higher-dimensional analogue of Galois theory. I think the category of motives is conjectured to be a Tannakian category, via Grothendieck's standard conjectures on algebraic cycles (please correct me if I am wrong about this).

So I'm guessing Tannakian categories might provide the link between the theory of motives and anabelian geometry and Galois-Teichmuller theory (which are both related to Galois theory) that Grothendieck was talking about in Recoltes et Semailles, but I'm not really sure. Either way, the ideas are still not very clear to me, and I'd like to understand the connections more explicitly.

Answer 1

Two clarifications:

For anabelian geometry, you should ask how much information about a variety is contained in the Galois action on its etale fundamental group.

While it's true that the motivic Galois group is a higher-dimensional analogue of the Galois group, it also should be true that motives are "just" a special kind of Galois representation, i.e. under the Tate conjecture the $\ell$-adic realization functor should give a faithful functor from motives to $\ell$-adic Galois representations, so the category of motives is the category of Galois representations with some restrictions placed on the objects and morphisms.

Of course these restrictions are highly nontrivial. Only for the irreducible Galois representations do we have a good conjectural description of which ones come from motives, via the Fontaine-Mazur conjecture.

So we can see that all 3 of these relate to Galois actions - the first two to Galois actions on fundamental groups, and the last to Galois actions on $\ell$-adic vector spaces. However, Galois actions are used in different ways in the three. Thinking about the three concepts, you might be led to questions like these:

• Can we construct Galois representations from the Galois action on the fundamental group of a curve? (this would be the first step in relating motives to anabelian geometry)

• Do these Galois representations arise from motives? (this would be the second step)

• Are these motives related to the geometry of a curve? (seeking a deeper connection to anabelian geometry)

• Can the class of motives arising this way be used to construct or describe the motivic Galois group? (now we bring in Grothendieck-Teichmuller theory)

I think these questions at least touch on the beginning of what Grothendieck was thinking of.

Since Grothendieck, people have heavily studied these questions, primarily in the case of unipotent quotients of the fundamental group, starting with the paper of Deligne on the fundamental group of the projective line minus three points. I think it's fair to say that the answer to all these questions is yes, with the largest caveat for the last question - I believe we can understand certain very special quotients of the motivic Galois group this way, but I don't think anyone has a strategy to construct the whole thing.

The story goes something like this:

Deligne looked at the maximal pro-$\ell$ quotient of the geometric fundamental group of the projective line minus three points. This is naturally an $\ell$-adic analytic group, and has a Lie algebra, which is an $\ell$-adic representation, and admits an action of the Galois group. This is supposed to be the $\ell$-adic realization of a motive, and Deligne worked to find the other realizations, including Hodge theory. This is a mixed motive, not a pure motive, so isn't constructed directly from linearizing algebraic varieties.

All motives generated this way are mixed Tate motives, i.e extensions of powers of the Tate motive (the inverse of the Lefschetz motive), and are everywhere unramified. One can define the Tannakian category of everywhere unramified mixed Tate motives, and the Tannakian fundamental group is known to ask faithfully on the limit of these unipotent completions.

Answer 2

There's a very deep connection between motives and Grothendieck-Teichmüller theory but it isn't well-understood yet. I can't even frame it precisely in higher genus, but at least I can frame a precise conjecture in genus zero. It has to do with motives that are connected to periods of moduli spaces on the one hand, and Grothendieck-Teichmüller being connected to automorphisms of fundamental groups of moduli spaces on the other.

So on the one hand, we have the Tannakian category $MTM$ of mixed Tate motives over Z. Goncharov and Manin gave a construction showing how to construct motivic multiple zeta values from the cohomology of the moduli spaces $M_{0,n}$ of genus zero curves with $n$ marked points, giving rise to a Tannakian subcategory of MTM, and F. Brown subsequently proved that the motivic multiple zeta category actually fills all of $MTM$. Take the Tannakian fundamental group of this category $MTM$, and take its (pro)-unipotent radical. The associated graded Lie algebra is known to be freely generated by one generator in each odd weight.

Now on the other hand, take the Grothendieck-Teichmüller group, which can be identified with the outer automorphism group of the tower of all the fundamental groups of the moduli spaces $M_{0,n}$ that commute (up to inner automorphisms) with certain standard maps between these fundamental groups coming from erasing points or subsurface inclusion (or doubling the braid strands, if you like to think of the fundamental group of $M_{0,n}$ as a braid group). This is a priori a profinite group, but it has a pro-unipotent version, which has an associated Lie algebra that is isomorphic to the graded Grothendieck-Teichmüller Lie algebra $grt$ defined by three relations that are the exact additive analogues of the three defining relations in the profinite group.

Thanks to the result that "the motivic multizeta values satisfy the associator relations", together with Brown's result, we can deduce that the Tannakian fundamental Lie algebra of $MTM$ injects into the Grothendieck-Teichmüller Lie algebra $grt$. The important conjecture is that these two Lie algebras are equal, even though one of them arises from a motivic construction which on the one hand "lifts" up actual real numbers (the real multizeta values, which are periods of the moduli spaces $M_{0,n}$ to mixed Tate motives, which then turn out to generate the full category of mixed Tate motives, and the other arises from the automorphism group of the fundamental groups of the moduli spaces $M_{0,n}$.

This answer is not disjoint from Will Sawin's answer above, but expressed differently.

Answer 3

You should look at Motivic aspects of Anabelian geometry. A, Schmidt. There, there is a reformulation of the question of Anabelian geometry using Voevodsky's Motivic Homotopy theory.