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 –algebras, from which we can recover analytic and topological representation spaces, respectively. For a compact Kähler manifold, the –completion also gives the natural setting for nonabelian Hodge theory; it has a pure Hodge structure, in the form of a pro-–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.
Citation
Jonathan Pridham. "Analytic nonabelian Hodge theory." Geom. Topol. 21 (2) 841 - 902, 2017. https://doi.org/10.2140/gt.2017.21.841
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