On exact sequences of Hodge theoretic fundamental groups

The goal of this paper is to first define a Hodge theoretic fundamental group for smooth connected complex algebraic varieties and then prove and study a right exact sequence of Hodge theoretic fundamental groups associated to a smooth projective family of algebraic varieties $f\colon X\to B$. In particular, we study when this right exact sequence is exact, relate this question to some prior results in non-abelian Hodge theory, and give an obstruction to splitting in terms of étale fundamental groups. The main examples we consider in this note is the universal curve $f\colon \mathcal{C}_g\to \mathcal{M}_g$ and the moduli space of degree $1$ line bundles on the universal curves $p\colon \mathrm{Pic}^1_{\mathcal{C}_g/\mathcal{M}_g}\to \mathcal{M}_g$.