Analytic nonabelian Hodge theory

Pridham, Jonathan¹

¹ 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

Geometry & topology, Tome 21 (2017) no. 2, pp. 841-902.

Résumé

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.

Détail

DOI : 10.2140/gt.2017.21.841

Classification : 32G13, 32G20
Keywords: nonabelian Hodge theory, twistor structures, $C^*$–algebras

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