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A non-abelian version of Deligne's Fixed Part Theorem

Hélène Esnault, Moritz Kerz

TL;DR

The work proves a non-abelian analogue of Deligne's Fixed Part theorem for polarized variations of Hodge structure by connecting finite monodromy orbits under base fundamental groups with algebraically isomonodromic extensions. It develops a robust group-theoretic framework using profinite completions and algebraic monodromy groups, then interleaves this with the non-abelian Hodge correspondence and Simpson's non-abelian Hodge locus to establish equivalences between finite orbit, extendability, and isomonodromic extendability for good morphisms. The results apply in particular to mapping class group representations and yield integrality and uniqueness statements under suitable hypotheses, while clarifying the behavior of the Mumford-Tate group in these non-abelian settings. The paper also extends classical integral-delicate fixed-part ideas to the non-abelian realm, providing tools for exploring arithmetic aspects of local systems and their Hodge-theoretic structures in families.

Abstract

We formulate and prove a non-abelian analog of Deligne's Fixed Part theorem on Hodge classes, revisiting previous work of Jost--Zuo, Katzarkov--Pantev and Landesman--Litt. To this aim we study algebraically isomonodromic extensions of local systems and we relate them to variations of Hodge structures, for example we show that the Mumford-Tate group at a generic point stays constant in an algebraically isomonodromic extension of a variation of Hodge structure. v2: a few typos ironed and Thm 1.1 5) completed. v3: there was a Schlamassel leading to a mix-up of files. Apologies. Else identical version (one minor change). v5 final version. Appears in Alg. Geom.

A non-abelian version of Deligne's Fixed Part Theorem

TL;DR

The work proves a non-abelian analogue of Deligne's Fixed Part theorem for polarized variations of Hodge structure by connecting finite monodromy orbits under base fundamental groups with algebraically isomonodromic extensions. It develops a robust group-theoretic framework using profinite completions and algebraic monodromy groups, then interleaves this with the non-abelian Hodge correspondence and Simpson's non-abelian Hodge locus to establish equivalences between finite orbit, extendability, and isomonodromic extendability for good morphisms. The results apply in particular to mapping class group representations and yield integrality and uniqueness statements under suitable hypotheses, while clarifying the behavior of the Mumford-Tate group in these non-abelian settings. The paper also extends classical integral-delicate fixed-part ideas to the non-abelian realm, providing tools for exploring arithmetic aspects of local systems and their Hodge-theoretic structures in families.

Abstract

We formulate and prove a non-abelian analog of Deligne's Fixed Part theorem on Hodge classes, revisiting previous work of Jost--Zuo, Katzarkov--Pantev and Landesman--Litt. To this aim we study algebraically isomonodromic extensions of local systems and we relate them to variations of Hodge structures, for example we show that the Mumford-Tate group at a generic point stays constant in an algebraically isomonodromic extension of a variation of Hodge structure. v2: a few typos ironed and Thm 1.1 5) completed. v3: there was a Schlamassel leading to a mix-up of files. Apologies. Else identical version (one minor change). v5 final version. Appears in Alg. Geom.
Paper Structure (22 sections, 33 theorems, 37 equations)

This paper contains 22 sections, 33 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Profinite completion of groups and Artin $K\pi1$ spaces
  3. Group theory
  4. Geometric examples
  5. An integral version of Deligne's Fixed Part Theorem
  6. Integral Fixed Part Theorem
  7. Extensions of representations of discrete groups
  8. Main group theoretic result
  9. Admissible representations
  10. Algebraically isomonodromic representations
  11. Reminder on variations of Hodge structure
  12. Polarized variations of ${\mathbb C}$-Hodge structure
  13. Characterization of a polarized variation of ${\mathbb C}$-Hodge structure via the non-abelian Hodge correspondence
  14. The Mumford-Tate group of a polarizable variation of ${\mathbb Q}$-Hodge structure
  15. Reminder on Simpson's non-abelian Hodge locus
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $f: X\to S$ be a good algebraic morphism fulfilling Ass, $s$ be a point in $S$. Let $(\mathbb L_s, {\mathsf F}_s, Q_s)$ be a $({\mathbb Z}) K \text{-}{\sf PVHS}$ on the fiber $X_s$, where $K\in \{{\mathbb Q},{\mathbb R},{\mathbb C}\}$. Then the following conditions are equivalent:

Theorems & Definitions (67)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 57 more