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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An algebraic proof for the symplectic structure of moduli space


Author: Yael Karshon
Journal: Proc. Amer. Math. Soc. 116 (1992), 591-605
MSC: Primary 14D22; Secondary 32G13, 55N99, 57R15, 58F05
MathSciNet review: 1112494
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Abstract: 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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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1112494-2
PII: S 0002-9939(1992)1112494-2
Article copyright: © Copyright 1992 American Mathematical Society