Hodge theory of the Turaev cobracket and the Kashiwara–Vergne problem

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