Analytic nonabelian Hodge theory

Abstract

The proalgebraic fundamental group can be understood as a completion with respect to finite-dimensional noncommutative algebras. We introduce finer invariants by looking at completions with respect to Banach and $C^{\ast }$–algebras, from which we can recover analytic and topological representation spaces, respectively. For a compact Kähler manifold, the $C^{\ast }$–completion also gives the natural setting for nonabelian Hodge theory; it has a pure Hodge structure, in the form of a pro-$C^{\ast }$–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.