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

Abstract:

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.