Symplectic Geometry of the Moduli Space of Flat Connections on a Riemann Surface: Inductive Decompositions and Vanishing Theorems

http://dx.doi.org/10.4153/CJM-2000-026-4
Canad. J. Math. 52(2000), 582-612

  Published:2000-06-01
 Printed: Jun 2000

This paper treats the moduli space ${\cal M}_{g,1}(\Lambda)$ of representations of the fundamental group of a Riemann surface of genus $g$ with one boundary component which send the loop around the boundary to an element conjugate to $\exp \Lambda$, where $\Lambda$ is in the fundamental alcove of a Lie algebra. We construct natural line bundles over ${\cal M}_{g,1} (\Lambda)$ and exhibit natural homology cycles representing the Poincar\'e dual of the first Chern class. We use these cycles to prove differential equations satisfied by the symplectic volumes of these spaces. Finally we give a bound on the degree of a nonvanishing element of a particular subring of the cohomology of the moduli space of stable bundles of coprime rank $k$ and degree $d$.