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Malcev Completions, Hodge Theory, and Motives

Emil Jacobsen

TL;DR

The paper proves that local systems of geometric origin on smooth, connected varieties in characteristic zero with a rational point are closed under extensions, framing this as a Malcev-completeness statement in a motivic, Hodge-theoretic, and De Rham context. It introduces motivic local systems and the motivic fundamental group, showing that Malcev completions of monodromy representations carry motivic structures (Nori motives) and that the natural map from the classical fundamental group to the motivic one is Malcev complete. The authors develop a general Malcev-completeness criterion, reduce to unipotent cases via the construction of universal objects, and apply these results to Hodge theory and motives, yielding stability results for Hodge-origin and motivic local systems and giving a unified proof of Lazda’s De Rham comparison in the unipotent setting. Collectively, the work provides motivic refinements of Hain’s unipotent Malcev theorem, a motivic fundamental group framework, and a versatile toolkit for proving Malcev completeness in enriched local-system contexts. These insights unify several strands of fundamental group theory, Tannakian duality, and motivic/Hodge-theoretic structures, with implications for the structure of unipotent completions and their 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 generalisation 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 fibre.

Malcev Completions, Hodge Theory, and Motives

TL;DR

The paper proves that local systems of geometric origin on smooth, connected varieties in characteristic zero with a rational point are closed under extensions, framing this as a Malcev-completeness statement in a motivic, Hodge-theoretic, and De Rham context. It introduces motivic local systems and the motivic fundamental group, showing that Malcev completions of monodromy representations carry motivic structures (Nori motives) and that the natural map from the classical fundamental group to the motivic one is Malcev complete. The authors develop a general Malcev-completeness criterion, reduce to unipotent cases via the construction of universal objects, and apply these results to Hodge theory and motives, yielding stability results for Hodge-origin and motivic local systems and giving a unified proof of Lazda’s De Rham comparison in the unipotent setting. Collectively, the work provides motivic refinements of Hain’s unipotent Malcev theorem, a motivic fundamental group framework, and a versatile toolkit for proving Malcev completeness in enriched local-system contexts. These insights unify several strands of fundamental group theory, Tannakian duality, and motivic/Hodge-theoretic structures, with implications for the structure of unipotent completions and their 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 generalisation 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 fibre.
Paper Structure (43 sections, 61 theorems, 73 equations)

This paper contains 43 sections, 61 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Motivic Malcev completions
  3. The motivic fundamental group
  4. Other results
  5. Hodge theory
  6. De Rham fundamental groups
  7. Background
  8. The fundamental short exact sequence
  9. Motivic structures on fundamental groups
  10. The Malcev completeness criterion
  11. A sketch of the proofs
  12. Acknowledgements
  13. Affine group schemes over a field
  14. Preliminaries
  15. Unipotency
  16. ...and 28 more sections

Key Result

Theorem 1.3

Let $X$ be a smooth and connected $k$-variety with a rational point. Then local systems of geometric origin are stable under extension in $\mathop{\mathrm{LS}}\nolimits(X)$. In other words, if is a short exact sequence in $\mathop{\mathrm{LS}}\nolimits(X)$ such that $L'$ and $L"$ are of geometric origin, then so is $L$.

Theorems & Definitions (154)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3: Thm. \ref{['thm:motivic_hain']}.1
  • Remark 1.4
  • Theorem 1.5: Thm. \ref{['thm:motivic_hain']}.2
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8: Thm. \ref{['thm:motivic_hain']}.1
  • Theorem 1.9: Thm. \ref{['thm:generalized_hain']}
  • Remark 1.10
  • ...and 144 more