Volume 21, issue 2 (2017)

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Analytic nonabelian Hodge theory

Jonathan Pridham

Geometry & Topology 21 (2017) 841–902

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–algebras, from which we can recover analytic and topological representation spaces, respectively. For a compact Kähler manifold, the C–completion also gives the natural setting for nonabelian 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.

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