The geometry of the mixed Hodge structure on the fundamental group

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 ∈ X. The extension of mixed Hodge structures associated to the first two steps of π 1 (X, p) is studied. We show that it determines the element (2g q − 2 p − K) in Pic 0 (¯ X), where K represents the canonical divisor of ¯ X as well as the corresponding extension associated to π 1 (¯ X, p). Finally, we deduce a pointed Torelli theorem for punctured Riemann surfaces.

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

“…For the definition of iterated integrals and of the mixed Hodge structure (MHS) on the fundamental group we refer to the introductory article [Hai87b].…”

“…For the fundamental group π 1 (X, p) of the compact Riemann surface (X, p), Hain and Pulte ([Hai87b], [Pul88]) proved that the extension of mixed Hodge structures associated to the quotient of its group ring by W −3 determines the base point (see Theorem 2.1). From this result and from the classical Torelli theorem they derived a pointed Torelli theorem (see Theorem 2.3) as a corollary.…”

Geometrical and Topological Methods in Time Domain Antenna Synthesis

“…At this point, it is very fruitful to pause and consider the geometric consequences of what we have done. Equation (40) implies 0(6i,, onU nfUj (41) while equation (6) implies 0(6ii) = 1 on Ui n Uj (42) From this vantage we see that there are several interesting and relatively simple questions which we may ask.…”

The mixed Hodge structure on the fundamental group of the fiber type 2-arrangement

“…In general, the shuffle formula of iterated integrals (see [HI;(2.11 A pair (A,b) is called a pointed fiber type 2-arrangement. Two pointed fiber type 2-arrangements (.4,6) and (*4',6') are cross ratio equivalent if there is a one-to-one correspondence between A and A f satisfying following conditions (1), (2) …”