Analytic non-abelian Hodge theory
arXiv:1212.3708 · doi:10.2140/gt.2017.21.841
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.
47 pp. v2 m-convex hypotheses added, final version, to appear in Geometry and Topology; v3 minor correction