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


Author: Rainer H. Kaenders
Journal: Proc. Amer. Math. Soc. 129 (2001), 1271-1281
MSC (2000): Primary 14H40, 14H30; Secondary 14F35
DOI: https://doi.org/10.1090/S0002-9939-00-05675-6
Published electronically: October 20, 2000
MathSciNet review: 1712897
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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.


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Additional Information

Rainer H. Kaenders
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitäts- straße, 40225 Düsseldorf, Germany
Email: kaenders@cs.uni-duesseldorf.de

DOI: https://doi.org/10.1090/S0002-9939-00-05675-6
Received by editor(s): March 3, 1999
Received by editor(s) in revised form: July 26, 1999
Published electronically: October 20, 2000
Additional Notes: The author was partly supported by grant ERBCHBICT930403 (HCM) from the European Community and The Netherlands Organisation for Scientific Research (NWO)
Communicated by: Leslie D. Saper
Article copyright: © Copyright 2000 American Mathematical Society

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