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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An algebraic proof for the symplectic structure of moduli space
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by Yael Karshon PDF
Proc. Amer. Math. Soc. 116 (1992), 591-605 Request permission


Goldman has constructed a symplectic form on the moduli space $\operatorname {Hom} (\pi ,G)/G$, of flat $G$-bundles over a Riemann surface $S$ whose fundamental group is $\pi$. The construction is in terms of the group cohomology of $\pi$. The proof that the form is closed, though, uses de Rham cohomology of the surface $S$, with local coefficients. This symplectic form is shown here to be the restriction of a tensor, that is defined on the infinite product space ${G^\pi }$. This point of view leads to a direct proof of the closedness of the form, within the language of group cohomology. The result applies to all finitely generated groups $\pi$ whose cohomology satisfies certain conditions. Among these are the fundamental groups of compact Kähler manifolds.
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 116 (1992), 591-605
  • MSC: Primary 14D22; Secondary 32G13, 55N99, 57R15, 58F05
  • DOI:
  • MathSciNet review: 1112494