The mixed Hodge structure on the fundamental group of a punctured Riemann surface

Given a compact Riemann surface $\bar{X}$ of genus $g$ and distinct points $p$ and $q$ on $\bar{X}$, we consider the non-compact Riemann surface $X:=\bar{X}\setminus\{q\}$ with basepoint $p\in X$. The extension of mixed Hodge structures associated to the first two steps of $\pi_1(X,p)$ is studied. We show that it determines the element $(2g\,q-2\,p-K)$ in $\operatorname{Pic}^0(\bar{X})$, where $K$ represents the canonical divisor of $\bar{X}$ as well as the corresponding extension associated to $\pi_1(\bar{X},p)$. Finally, we deduce a pointed Torelli theorem for punctured Riemann surfaces.