We prove that, on a smooth, connected variety in characteristic zero admitting a
rational point, local systems of geometric origin are stable under extension in the
category of all local systems. As a consequence of this, we obtain a (Nori)
motivic strengthening of Hain’s theorem on Malcev completions of monodromy
representations.
Our methods are Tannakian, and rely on an abstract criterion for “Malcev
completeness”, which is proved in the first part of the paper. A couple of
secondary applications of this criterion are given: an alternative proof of
D’Addezio–Esnault’s theorem, which says that local systems of Hodge origin are
stable under extension in the category of all local systems; a generalization of
the theorem of Hain, mentioned above, which also affirms a conjecture of
Arapura; and an alternative proof of a theorem of Lazda, which under suitable
assumptions gives an isomorphism between the relative unipotent de Rham
fundamental group and the unipotent de Rham fundamental group of the special
fiber.
Keywords
motive, Tannakian category, fundamental group, Malcev
completion, unipotent completion, Hodge theory, local
system, monodromy
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