Local Mixed Hodge Structure on Brill-Noether Stacks

Abstract

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.