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Deligne, in his 1987 paper on the fundamental group of $\mathbb{P}^1 \setminus \{0,1,\infty\}$ (in "Galois Groups over $\mathbb{Q}$"), defines a system of realisations for a motivic fundamental group. Namely, Tannakian categories are defined of unipotent lisse $\ell$-adic sheaves, unipotent vector bundles with a flat connection and the crystalline equivalent given by unipotent overconvergent isocrystals, whereas the automorphism groups of their fibre functors define the $\ell$-adic, de Rham and crystalline fundamental groups respectively. Now, I do not claim to have a deep understanding of these constructions, nor of the paper as a whole (or to even have read it fully in detail, given its size!), but I was left with a question: why does the word unipotent appear in all the constructions?

Having read parts of Minhyong Kim's work it is understandable that in this context, these subcategories generated by the unipotent objects are not only enough, but seemingly exactly what is needed. In other words, one still acquires the Tannakian structure (which if I'm not mistaken exists on the full categories as well) and thus has fundamental groups which are affine group schemes and can start playing the game of using unipotency and quotients in the lower central series to construct moduli spaces of path torsors, moving gradually away from abelianness and deeper into Diophantine information. This however doesn't answer the original motivation.

I apologise if the answer is hidden in Deligne's original paper or if the context makes it obvious to everyone but me.

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2 Answers

up vote 11 down vote accepted

Essentially because the Tannakian theory gives in the unipotent case (and only in that case) a reasonably sized answer with an easy motivic interpretation.

For the size you should be aware that already in the topological situation the group scheme associated by Tannaka theory to the fundamental group of $\mathbb P^1$ minus three points (i.e., the free group on two generators) is huge. For one thing each irreducible representation gives rise to a reductive quotient and there are continuous families (i.e., positive dimensional varieties) of such representations. From this one can see that the group scheme maps onto a product of reductive groups where the index set are the points of some algebraic variety (see What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra? for the case of $1$-dimensional representations).

Added to this is the fact that most of the topological representations do not have geometric origin and hence have no motivic interpretation. If one looks at $\ell$-adic representations of the fundamental group over $\mathbb Q$ of $\mathbb P^1$ minus three points (or of suitable germs if one wants a theory over $\mathbb C$) and adds mixedness assumptions, then the Tannakian category should have a motivic interpretation which also should be independent (in some suitable sense) of $\ell$ and should be comparable to its cristalline equivalent. This however all depends on the Langlands program and hence is currently beyond our reach.

If one sticks to unipotent representations then essentially all these problems disappear. A unipotent representation (over some field $k$ of characteristic $0$) of the free group $F$ on two elements factors through a nilpotent quotient $\Gamma$ of $F$ and such a nilpotent quotient has a Malcev completion, a unipotent algebraic group $G$ over $\mathbb Q$ of dimension the rank of $\Gamma$, such that the Tannakian category of unipotent representations of $\Gamma$ over $k$ is equivalent to the category of $k$-representations of $G$. Passing to the limit gives us a pro-unipotent algebraic group $G_\infty$ over $\mathbb Q$ whose category of $k$-representations is equivalent to the category of unipotent $k$-representations of $F$. Furthermore, the Lie algebra of $G_\infty$ has a nice cohomological description; it is the free Lie algebra generated by $H_1(X,\mathbb Q)$, where $X$ is $\mathbb P^1$ minus three points.

The motivic side of things now comes along very gracefully: For one of several natural categories that has an appropriate $H_1(X)$ in it there is a corresponding relative Tannakian description of unipotent families of objects over $X$. As examples we have unipotent variations of rational Hodge structures, geometrically unipotent $\mathbb Q_{\ell}$-adic sheaves over $\mathbb Q$ and successive extensions of constant $F$-iso-crystals. In all these cases these categories are described by representations of a pro-unipotent algebraic group object in the appropriate base category (rational Hodge structure, $\mathbb Q_{\ell}$-adic sheaves over $\mathbb Q$ and $F$-isocrystals) and in all the cases its Lie algebra is the free Lie algebra on $H_1(X)$.

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I'm sure other people can, and will, give more detailed and insightful answers, so I'll be brief. From my point of view, the unipotent part of the motivic fundamental group is an important approximation to the true object. Nevertheless it is an approximation. The full motivic fundamental group should be the group associated the Tannakian category of all motivic local systems (once it's constructed!), and this would include motives associated to smooth projective families which are not unipotent.

One had a similar situation in Hodge theory, where most of the initial effort was on the unipotent side, but in the last few years people have started looking at the deeper structure. More information about this can be found in http://arxiv.org/abs/0902.4252

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While Torsten's answer is correct as well, my sympathies are with Donu's answer. The non-motivic features of algebraic completions, say failure to commute with base-change, do not detract from their general interest. Also the moduli arising in reductive completions should be a powerful tool in many contexts. In my view, 'motivic or not' is somewhat a secondary issue. On the other hand, it seems much work remains to be done before the deep theories of bigger completions (say, Simpson, Toen, etc. in addition to Donu's link) can be applied to concrete questions, as in Diophantine geometry. –  Minhyong Kim May 27 '10 at 20:19
    
Thank you for your answers. I was expecting an answer along these lines but have accepted Torten's as it gives some more details on the difference between having unipotency and not. –  Frank May 29 '10 at 13:19
    
Don't get me wrong, I would only be too happy to have someone understand properly the non-unipotent part. I think that if one is to use Tannakian techniques a generalisation of Tannakian categories will be needed where the category is allowed to grow like a scheme under base extension and not just like a zero-dimensional scheme (or rather stack) that is the case with current Tannakian categories (a case in point is a pro-torus whose character group is the additive group of the base). The unipotent theory gives a complete and satisfying answer today but it is clearly not the whole story. –  Torsten Ekedahl May 29 '10 at 15:22
    
Thanks to all for comments and answers, which are quite enlightening. I certainly agree that many deep mysteries remain. –  Donu Arapura May 29 '10 at 16:37
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