THE HODGE DE-RHAM THEORY OF RELATIVE MALCEV COMPLETION

Suppose that X is a smooth manifold and rho : pi(1)(X,x) --> S is a re presentation of the fundamental group of X into a real reductive group with Zariski dense image. To such data one can associate the Malcev c ompletion G of pi(1)(X,x) relative to rho. In this paper we generalize Chen's iterated integrals and show that the H-0 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 rho is the mono dromy representation of a variation of Hedge structure over X, then th e coordinate ring of G has a canonical mixed Hedge structure. (C) Else vier, Paris.