Moduli of framed parabolic sheaves.

*(English)*Zbl 1095.14034Let \(\Sigma \) 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.

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.

Reviewer: Julien Keller (London)

##### 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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\textit{J. Hurtubise} et al., Ann. Global Anal. Geom. 28, No. 4, 351--370 (2005; Zbl 1095.14034)

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