  In this paper we show that, after completion in the I-adic topology, the
Goldman bracket on the space spanned by homotopy classes of loops on a smooth,
complex algebraic curve is a morphism of mixed Hodge structure. We prove
similar statements for the natural action (defined by Kawazumi and Kuno) of the
loops in X on paths from one "boundary component" to another. These results are
used to construct torsors of isomorphisms of the the completed Goldman Lie
algebra with the completion of its associated graded Lie algebra. Such
splittings give torsors of partial solutions to the Kashiwara--Vergne problem
(arXiv:1611.05581) in all genera. Compatibility of the cobracket with Hodge
theory is established in arXiv:1807.09209.
