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Moduli of framed parabolic sheaves. (English) Zbl 1095.14034
Let \(\Sigma \) be a compact Riemann surface with marked points \(p_1,\cdots, p_m\) and \(\Sigma_0\) the corresponding open Riemann surface with punctures \(p_1,\cdots,p_m\). Let \(\Delta\) be the fundamental alcove of a compact simply connected real group \(K\) with maximal torus \(T\). In [J. Hurtubise, L. Jeffrey, R. Sjamaar, Am. J. Math. 128, No. 1, 167–214 (2006; Zbl 1096.53045)], a symplectic moduli space \(M\) of “framed parabolic representations” of the fundamental group of \(X\) in \(K=\text{SU}(N)\) with a Hamiltonian torus action, is constructed. The data attached to the representation is a volume form on each of the successive quotient of the natural flag attached to the parabolic structure. Of course, \(T^m\) acts transitively on these volume forms. The reduction of this master space \(M\) at a point \((\gamma_1,\dots,\gamma_m)\) of \(\Delta^m\) can be seen as the moduli space of parabolic representations with holonomy \(\gamma_i\) at the punctures.
The authors construct a complex moduli space \(\mathcal M\) of \(\text{SL}(N)\)-sheaves with a framed parabolic structure at the marked points. To be more precise, this is the moduli space of holomorphic vector bundles together with weighted flags on fibres at finitely many points and a volume form on each successive quotient in the flags. Their construction relies on geometric invariant theory of D. Mumford by introducing a certain Gieseker space (see also the related work of A. Schmitt). One has a torus action on that space and the associated quotient is essentially the moduli space of parabolic bundles constructed by U. Bhosle [Ark. Mat. 27, No. 1, 15–22 (1989; Zbl 0692.14007)]. The main result is the existence of an homeomorphism \(M \to {\mathcal M}\) equivariant under the torus action, extending a previous work of the two first authors. Finally, an approach is suggested for dealing with the case of arbitrary reductive groups, involving the associated loop group.

MSC:
14H60 Vector bundles on curves and their moduli
53D30 Symplectic structures of moduli spaces
14D20 Algebraic moduli problems, moduli of vector bundles
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[1] Alekseev, A., Malkin, A. and Meinrenken, E.: Lie group valued moment maps, J. Differential Geom. 48 (1988), 445–495. · Zbl 0948.53045
[2] Atiyah, M. F. and Bott, R.: The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. Lond. Ser. A 308 (1982), 523–615. · Zbl 0509.14014
[3] Bhosle, U.: Parabolic vector bundles on curves, Ark. Mat. 27 (1989), 15–22. · Zbl 0692.14007
[4] Bhosle, U. and Ramanathan, A.: Moduli of parabolic G-bundles on curves, Math. Z. 202(2) (1989), 161–180. · Zbl 0686.14012
[5] Guillemin, V., Jeffrey, L. and Sjamaar, R.: Symplectic implosion, Transform Groups 7(2) (2002), 155–184. · Zbl 1015.53054
[6] Hurtubise, J. and Jeffrey, L.: Representations with weighted frames and framed parabolic bundles, Canad. J. Math. 52 (2000), 1235–1268. · Zbl 1086.53103
[7] Hurtubise, J., Jeffrey, L. and Sjamaar, R.: Group valued implosion and parabolic structures, Amer. J. Math., in press. · Zbl 1096.53045
[8] Kirwan, F.: Cohomology of Quotients in Symplectic and Algebraic Geometry, Math. Notes 31, Princeton Unversity Press, Princeton, 1984. · Zbl 0553.14020
[9] Mehta, V. and Seshadri, C. S.: Moduli of vector bundles on curves with parabolic structure, Math. Ann. 248 (1980), 205–239. · Zbl 0454.14006
[10] Mumford, D., Fogarty, J. and Kirwan, F.: Geometric Invariant Theory, Springer-Verlag, Berlin, 1994. · Zbl 0797.14004
[11] Narasimhan, M. S. and Seshadri, C. S.: Stable and unitary vector bandles on a compact Riemann surface, Ann. Math. 82, 540–567. · Zbl 0171.04803
[12] Ramanathan, A.: Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129–152. · Zbl 0289.32020
[13] Thaddeus, M.: Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), 691–723. · Zbl 0874.14042
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