Nonabelian Hodge Theory

Classically, Hodge theory and related constructions provided extra structure to abelian topological invariants of the usual topological spaces associated to algebraic varieties over (C. I would like to explain how, in analogy with these abelian constructions, certain nonabelian topological invariants of complex algebraic varieties have extra structures. We will be concerned with two related invariants, the space R of framed 71-dimensional representations of %\(X,x), and M, its universal categorical quotient by the action of G£(n). These are related to the nonabelian cohomology H(X,G^(n)), which is the set of isomorphism classes of representations, and which has a structure of non-Hausdorff space. The moduli space M is the set of Jordan equivalence classes of representations, the universal Hausdorff space to which the cohomology space maps. The representation space R is the cohomology of X relative to a choice of base point x. We will interpret R and M as nonabelian analogues of the abelian cohomology H(X,(C).

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