This is a very vague question. I was just reading the introduction to M. Kim's article on motivic fundamental groups and the theorem of Siegel and noticed that there are essentially three fundamental groups appearing in his work: the De Rham fundamental group, the cristalline fundamental group and the etale fundamental group.

Now, does Nori's fundamental group scheme also appear somewhere in his work? If no, why not?


It depends on what you call Nori's fundamental group scheme, of course. Nori himself has given several versions of his fundamental group scheme, and it has been vastly generalized.

If you think of the classical definition (the tannaka group of the category of essentially finite vector bundles) I don't think it does. It is of a somewhat different nature of the tannaka groups you list above : it is pro-finite, whereas the ones appearing in your list are all pro-unipotent. They are not designed for the same purposes: Nori's fundamental group was built to take into account positive characteristic phenomenons, especially torsors under finite group schemes that are not necessarily étale, whereas in Kim's fundamental groups both the de Rham version and the étale one are defined for a variety over a field of characteristic zero, and my vague understanding of the subject is that all of them are algebraic incarnations of the unipotent envelope of the non existing topological fundamental group.

Of course, this is not so simple. Nori has defined in his Phd an unipotent version of his fundamental group scheme which is related to Kim's de Rham. Also, a version of Nori's fundamental group scheme (rather, groupoid) was recently used in characteristic zero by Esnault and Hai to study the section conjecture, which is deeply interconnected with Kim's work.


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