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

Abstract

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