Hodge theory of the Goldman bracket

Abstract

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.