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1. CJM 2000 (vol 52 pp. 1235)

Hurtubise, J. C.; Jeffrey, L. C.
Representations with Weighted Frames and Framed Parabolic Bundles
There is a well-known correspondence (due to Mehta and Seshadri in the unitary case, and extended by Bhosle and Ramanathan to other groups), between the symplectic variety $M_h$ of representations of the fundamental group of a punctured Riemann surface into a compact connected Lie group~$G$, with fixed conjugacy classes $h$ at the punctures, and a complex variety ${\cal M}_h$ of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For $G = \SU(2)$, we build a symplectic variety $P$ of pairs (representations of the fundamental group into $G$, ``weighted frame'' at the puncture points), and a corresponding complex variety ${\cal P}$ of moduli of ``framed parabolic bundles'', which encompass respectively all of the spaces $M_h$, ${\cal M}_h$, in the sense that one can obtain $M_h$ from $P$ by symplectic reduction, and ${\cal M}_h$ from ${\cal P}$ by a complex quotient. This allows us to explain certain features of the toric geometry of the $\SU(2)$ moduli spaces discussed by Jeffrey and Weitsman, by giving the actual toric variety associated with their integrable system.

Categories:58F05, 14D20

2. CJM 2000 (vol 52 pp. 582)

Jeffrey, Lisa C.; Weitsman, Jonathan
Symplectic Geometry of the Moduli Space of Flat Connections on a Riemann Surface: Inductive Decompositions and Vanishing Theorems
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 $d$.

Category:58F05

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