[Paper Review] The Hodge de Rham theory of relative Malcev completion

Richard Hain|ArXiv.org|Jul 18, 1996

Homotopy and Cohomology in Algebraic Topology8 references4 citations

TL;DR

This paper develops the Hodge-de Rham theory for relative Malcev completion, generalizing Chen's de Rham theory and Morgan-Hain's Hodge theory. It establishes a foundational framework for studying fundamental group representations in algebraic geometry, with key applications to mapping class groups via mixed Hodge structures on unipotent fundamental groups of algebraic curves.

ABSTRACT

The Hodge de Rham theory of relative Malcev completion is developed in this paper. In the special case where one takes the corresponding reductive group to be trivial, one recovers Chen's de Rham theory of the fundamental group and the corresponding Hodge theory due to Morgan and the author. This work is a principal technical tool in the author's work on the mapping class groups.

Motivation & Objective

• To extend the Hodge-de Rham theory to the setting of relative Malcev completion of fundamental groups.
• To generalize Chen's de Rham theory of the fundamental group to a relative, pro-unipotent setting.
• To provide a cohomological framework for studying representations of fundamental groups in algebraic geometry.
• To establish a theoretical foundation for applications in mapping class group theory and arithmetic geometry.
• To unify and extend existing Hodge-theoretic methods in the context of unipotent fundamental groups and mixed Hodge structures.

Proposed method

• Constructs the relative Malcev completion of the fundamental group of a pointed algebraic curve over a field of characteristic zero.
• Applies the Hodge-de Rham theory to the associated pro-algebraic group via the de Rham realization of the fundamental group.
• Uses the theory of mixed Hodge structures on unipotent fundamental groups to define a Hodge-de Rham structure on the relative completion.
• Employs the de Rham cohomology of the fundamental group with coefficients in a unipotent local system.
• Applies the theory of Hodge structures on pro-algebraic groups to analyze the relative completion as a mixed Hodge structure.
• Reconstructs the Hodge-de Rham theory in the relative setting by combining Malcev completion with Hodge-theoretic techniques from algebraic geometry.

Experimental results

Research questions

• RQ1 How can the Hodge-de Rham theory be extended to the relative Malcev completion of the fundamental group?
• RQ2 What is the relationship between the Hodge-de Rham structure on the relative completion and Chen's de Rham theory?
• RQ3 How does the Hodge structure on the relative Malcev completion relate to the mixed Hodge structure on the unipotent fundamental group?
• RQ4 What is the role of the reductive group in the relative completion, and how does its triviality recover classical results?
• RQ5 In what way does this theory serve as a technical tool for studying mapping class groups?

Key findings

• The Hodge-de Rham theory of relative Malcev completion generalizes Chen's de Rham theory and Morgan-Hain's Hodge theory in the case of trivial reductive group.
• The construction yields a mixed Hodge structure on the relative Malcev completion of the fundamental group of a curve.
• The theory provides a cohomological framework for studying unipotent representations of fundamental groups in algebraic geometry.
• The relative Malcev completion admits a Hodge-de Rham structure compatible with the mixed Hodge structure on the fundamental group.
• The framework enables the study of monodromy and holonomy in the context of mapping class groups via Hodge-theoretic methods.
• The paper establishes a foundational tool for further research in arithmetic geometry and the topology of moduli spaces of curves.