#### Volume 21, issue 2 (2017)

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Analytic nonabelian Hodge theory

### Jonathan Pridham

Geometry & Topology 21 (2017) 841–902
##### 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.

##### Keywords
nonabelian Hodge theory, twistor structures, $C^*$–algebras
##### Mathematical Subject Classification 2010
Primary: 32G13, 32G20
##### Publication
Received: 7 July 2014
Revised: 4 January 2016
Accepted: 7 April 2016
Published: 17 March 2017
Proposed: Richard Thomas
Seconded: Peter Teichner, Jim Bryan
##### Authors
 Jonathan Pridham School of Mathematics and Maxwell Institute The University of Edinburgh James Clerk Maxwell Building The King’s Buildings Mayfield Road Edinburgh EH9 3FD United Kingdom