Malcev completions, Hodge theory, and motives

Abstract

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.