Representations with Weighted Frames and Framed Parabolic Bundles

Hurtubise, J. C. ; Jeffrey, L. C.

Canadian journal of mathematics, Tome 52 (2000) no. 6, pp. 1235-1268

Résumé

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 ${{\mathcal{M}}_{h}}$ of holomorphic bundles on the unpunctured surface with a parabolic structure at the puncture points. For $G\,=\,\text{SU}\left( 2 \right)$ , 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 $\mathcal{P}$ of moduli of “framed parabolic bundles”, which encompass respectively all of the spaces ${{M}_{h}}$ , ${{\mathcal{M}}_{h}}$ , in the sense that one can obtain ${{M}_{h}}$ from $P$ by symplectic reduction, and ${{\mathcal{M}}_{h}}$ from $\mathcal{P}$ by a complex quotient. This allows us to explain certain features of the toric geometry of the $\text{SU(2)}$ moduli spaces discussed by Jeffrey and Weitsman, by giving the actual toric variety associated with their integrable system.

Détail

DOI : 10.4153/CJM-2000-052-4

Mots-clés : 58F05, 14D20

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