An algebraic proof for the symplectic structure of moduli space

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.