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