The Hodge De Rham theory of relative Malcev completion

Suppose that X is a smooth manifold and ρ : π1 (X,N) → S is a representation of the fundamental group of X into a real reductive group with Zariski dense image. To such data one can associate the Malcev completion G of π1(X,x) relative to ρ. In this paper we generalize Chen's iterated integrals and show that the H0 of a suitable complex of these iterated integrals is the coordinate ring of G. This is used to show that if X is a complex algebraic manifold and ρ is the monodromy representation of a variation of Hodge structure over X, then the coordinate ring of G has a canonical mixed Hodge structure. © Elsevier, Paris.

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Author Richard Hain Mathematics