Analytic nonabelian Hodge theory

Pridham, Jonathan

doi:10.2140/gt.2017.21.841

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Analytic nonabelian Hodge theory

Jonathan Pridham

Geometry & Topology 21 (2017) 841–902

DOI: 10.2140/gt.2017.21.841

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 $C^{*}$ –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.

Keywords

nonabelian Hodge theory, twistor structures, $C^*$–algebras

Mathematical Subject Classification 2010

Primary: 32G13, 32G20

References

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

Volume 21, issue 2 (2017)