Analytic non-abelian Hodge theory

TL;DR

This paper develops a refined non-abelian Hodge theory framework using completions with Banach and C*-algebras, revealing new analytic and topological invariants and structures for compact Kähler manifolds.

Contribution

It introduces new invariants via Banach and C*-algebra completions, connecting non-abelian Hodge theory with analytic and topological representation spaces.

Findings

C*-completion yields a pro-C*-dynamical system with a pure Hodge structure.

Representations correspond to pluriharmonic local systems in Hilbert spaces.

Studies of cohomology reveal splittings of Hodge and twistor structures.

Abstract

The pro-algebraic fundamental group can be understood as a completion with respect to finite-dimensional non-commutative algebras. We introduce finer invariants by looking at completions with respect to Banach and C*-algebras, from which we can recover analytic and topological representation spaces, respectively. For a compact Kaehler manifold, the C*-completion also gives the natural setting for non-abelian Hodge theory; it has a pure Hodge structure, in the form of a pro-C*-dynamical system. Its representations are pluriharmonic local systems in Hilbert spaces, and we study their cohomology, giving a principle of two types, and splittings of the Hodge and twistor structures.