Hodge Theory of the Turaev Cobracket and the Kashiwara--Vergne Problem
In this paper we show that, after completing in the $I$-adic topology, the
Turaev cobracket on the vector space freely generated by the closed geodesics
on a smooth, complex algebraic curve $X$ with an algebraic framing is a
morphism of mixed Hodge structure. We combine this with results of a previous
paper (arXiv:1710.06053) on the Goldman bracket to construct torsors of
solutions of the Kashiwara--Vergne problem in all genera. The solutions so
constructed form a torsor under a prounipotent group that depends only on the
topology of the framed surface. We give a partial presentation of these groups.
Along the way, we give a homological description of the Turaev cobracket.