  On a smooth algebraic curve X with genus greater than 1 we consider a flat
principal bundle with a reductive structure group S and a vector bundle
associated with it. To this set of information we put in correspondence a
pro-algebraic group on whose functional algebra we introduce a mixed Hodge
structure. This construction, in fact, works for any smooth algebraic variety X
which, considered as an analytic space, has a nonabelian first homotopy group,
and the rest are trivial. The Hodge structure defined in this way can be
expressed in terms of iterated integrals.
  Furthermore, considered in the context of previous work by C. Simpson, this
MHS is the local mixed Hodge structure on a nonabelian cohomological space on X
with coefficients into a Brill-Noether stack, i.e., a stack with two
non-trivial homotopy groups: a fundamental group isomorphic to the group S and
an n-th homotopy group represented by a vector space, the fiber of the vector
bundle discussed above above.
  My construction is compatible and generalizes the work of R. Hain on Hodge
structure on relative Malcev completion of the fundamental group of X.
