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I must preface by confessing complete ignorance in the subject. I've read introductory texts about the theory of motives, but I am certainly no expert.

In http://www.math.ias.edu/files/deligne/GaloisGroups.pdf Deligne talks about (introduces?) the motivic fundamental group. But what is the purpose of this object?

Motives come up in cohomology in order to unify the different Weil cohomologies. But I only know of one way to define the algebraic fundamental group! After all, in cohomology you have a choice of coefficients (in a sheaf), but in the definition of the fundamental group (to my knowledge) there is no equivalent to this. Is the point that just as the algebraic fundamental group classifies etale covers, we can do this for other Grothendieck topologies as well? In what sense would the motivic fundamental group unify these?

The above was really one question: i. What theories does the motivic fundamental group unify, and in what sense does it do so?

I will add two more: ii. Is the existence of the motivic fundamental group conjectural?

and

iii. What purpose does the motivic fundamental group serve other than unifying? Deligne makes some references to conjectures that arise as prediction related to the motivic fundamental group. What insight does it provide?

I'm well aware that Deligne's text probably has all the answers to these questions, but I find it to be a hard read, so the more I know coming in the more I will take out of it.

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Perhaps some way of making sense out of some sort of monodromy action. Just a shot in the dark. –  Spice the Bird Jul 29 '11 at 17:18

2 Answers 2

up vote 8 down vote accepted

As in Birdman's comment, the motivic fundamental group is unifying the notion of monodromy action on the fibers of local systems of "geometric origin."

To explain this, let us start with the case of a field $K$. We have a semisimple $\mathbb{Q}$-linear Tannakian category $\operatorname{Mot}_K$ of (pure) motives over $K$ for which fiber functors are cohomology theories, i.e., it makes sense to have an $L$-valued fiber functor for a field $L$, and this is the same as a Weil cohomology theory for smooth proper $K$-varieties with values in $L$-vector spaces. A motivic Galois group, to my understanding, is attached to a cohomology theory/fiber functor $F$ of $\operatorname{Mot}_K$.

Then the motivic Galois group is the associated group scheme/$L$ whose representations are given by the category $\operatorname{Mot}_{K}\underset{\mathbb{Q}}{\otimes}L$, i.e., it is the group scheme of automorphisms of the fiber functor $F$. So it is "the group which acts on $F$-cohomology of (smooth projective) varieties." Since this category is semi-simple, the motivic Galois group is pro-reductive. E.g., the absolute Galois group (considered as a discrete group scheme) of $K$ acts on $\ell$-adic cohomology, so there is a homomorphism from $\operatorname{Gal}(K)$ to the motivic Galois group of $K$ corresponding to the fiber functor defined by $\ell$-adic cohomology.

For, say, a smooth variety $X$ over $K$, there should be a category of "motivic sheaves" on $X$, or rather, a semi-simple category of pure motivic sheaves contained in an Artinian category of mixed motivic sheaves. You should have e.g. an $\ell$-adic" fiber functor from the mixed category to $\ell$-adic perverse sheaves on $X$ which sends pure guys to (cohomologically shifted) lisse sheaves (alias local systems). E.g., if $K=\mathbb{F}_q$, then this is the category of pure (resp. mixed) perverse sheaves on $X$. If $K=\mathbb{C}$, this should be a full subcategory of pure (resp. mixed) polarizable Hodge modules on $X$. For any smooth proper (resp. just any) map $f:Y\to X$, there should an object in the category of pure (resp. mixed) motivic sheaves on $X$ corresponding to push-forward of the structure sheaf on $Y$.

The motivic fundamental group act on the ``fibers" of pure motivic sheaves on $X$. I.e., for a $K$-point of $X$, you should get a functor to the category of $K$-motives. This is a motivic incarnation of taking the fiber of a local system. Then given our cohomology theory $F$, we obtain a functor from pure motivic sheaves on $X$ to $L$-vector spaces, and the automorphisms of this functor will be the $F$-realization of the motivic Galois group of $X$.

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A short answer: The different Weil cohomology theories do not only provide cohomology groups or rings but also come with extra structure; e.g. l-adic cohomology comes with an action from a Galois group and de Rham cohomology comes with a Hodge structure (and Hodge structures can also be expressed as being an action from some group). This extra structure varies with the cohomology theory.

The motivic fundamental group should unify these extra structures -- they all should be shadows of an action of the motivic fundamental group.

For a start see e.g. the "motivic Galois group" section on the wikipedia page here and these notes by Sujatha Ramdorai. The book by Yves Andre referenced there is maybe a good next step.

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The notes are great! –  James D. Taylor Jul 29 '11 at 22:05

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