scholarly journals Complete sets of relations in the cohomology rings of moduli spaces of holomorphic bundles and parabolic bundles over a Riemann surface

2004 ◽  
Vol 89 (03) ◽  
pp. 570-622 ◽  
Author(s):  
Richard Earl ◽  
Frances Kirwan
Keyword(s):  
Riemann Surface ◽  
Moduli Spaces ◽  
Cohomology Rings ◽  
Parabolic Bundles ◽  
Complete Sets ◽  
Holomorphic Bundles

scholarly journals Representations with Weighted Frames and Framed Parabolic Bundles

2000 ◽  
Vol 52 (6) ◽  
pp. 1235-1268 ◽  
Author(s):  
J. C. Hurtubise ◽  
L. C. Jeffrey
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Moduli Spaces ◽  
Symplectic Reduction ◽  
Toric Geometry ◽  
Parabolic Bundles ◽  
Parabolic Structure ◽  
Complex Variety ◽  
Holomorphic Bundles ◽  
Connected Lie Group

AbstractThere 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 Mh 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 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 of moduli of “framed parabolic bundles”, which encompass respectively all of the spaces Mh, , in the sense that one can obtain Mh from P by symplectic reduction, andMh from 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.

scholarly journals GRASSMANNIAN-FRAMED BUNDLES AND GENERALIZED PARABOLIC STRUCTURES

2013 ◽  
Vol 24 (12) ◽  
pp. 1350090 ◽  
Author(s):  
USHA BHOSLE ◽  
INDRANIL BISWAS ◽  
JACQUES HURTUBISE
Keyword(s):  
Riemann Surface ◽  
Moduli Spaces ◽  
Parabolic Bundles ◽  
Universal Spaces ◽  
Equivariant Compactification

We build compact moduli spaces of Grassmannian-framed bundles over a Riemann surface, essentially replacing a group by a bi-equivariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin–Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles (in the sense that all of the moduli can be obtained as quotients), and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle.

scholarly journals Intersection pairings on singular moduli spaces of bundles over a Riemann surface and their partial desingularisations

2006 ◽  
Vol 11 (3) ◽  
pp. 439-494
Author(s):  
Lisa Jeffrey ◽  
Young-Hoon Kiem ◽  
Frances C. Kirwan ◽  
Jonathan Woolf
Keyword(s):  
Riemann Surface ◽  
Moduli Spaces ◽  
Singular Moduli

scholarly journals Logarithmic Connections, WZNW Action, and Moduli of Parabolic Bundles on the Sphere

2021 ◽  
Author(s):  
Claudio Meneses ◽  
Leon A. Takhtajan
Keyword(s):  
Moduli Spaces ◽  
Vector Bundles ◽  
Riemann Sphere ◽  
Parabolic Bundle ◽  
Parabolic Bundles ◽  
Logarithmic Connection ◽  
Definition Of ◽  
Suitable Notion ◽  
Marked Points ◽  
Explicit Relation

AbstractModuli spaces of stable parabolic bundles of parabolic degree 0 over the Riemann sphere are stratified according to the Harder–Narasimhan filtration of underlying vector bundles. Over a Zariski open subset $$\mathscr {N}_{0}$$ N 0 of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $$\mathscr {S}$$ S is defined as the regularized critical value of the non-compact Wess–Zumino–Novikov–Witten action functional. The definition of $$\mathscr {S}$$ S depends on a suitable notion of parabolic bundle ‘uniformization map’ following from the Mehta–Seshadri and Birkhoff–Grothendieck theorems. It is shown that $$-\mathscr {S}$$ - S is a primitive for a (1,0)-form $$\vartheta $$ ϑ on $$\mathscr {N}_{0}$$ N 0 associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that $$-\mathscr {S}$$ - S is a Kähler potential for $$(\Omega -\Omega _{\mathrm {T}})|_{\mathscr {N}_{0}}$$ ( Ω - Ω T ) | N 0 , where $$\Omega $$ Ω is the Narasimhan–Atiyah–Bott Kähler form in $$\mathscr {N}$$ N and $$\Omega _{\mathrm {T}}$$ Ω T is a certain linear combination of tautological (1, 1)-forms associated with the marked points. These results provide an explicit relation between the cohomology class $$[\Omega ]$$ [ Ω ] and tautological classes, which holds globally over certain open chambers of parabolic weights where $$\mathscr {N}_{0} = \mathscr {N}$$ N 0 = N .

scholarly journals The Mod Two Cohomology of the Moduli Space of Rank Two Stable Bundles on a Surface and Skew Schur Polynomials

2019 ◽  
Vol 71 (03) ◽  
pp. 683-715 ◽  
Author(s):  
Christopher W. Scaduto ◽  
Matthew Stoffregen
Keyword(s):  
Riemann Surface ◽  
Group Action ◽  
Moduli Space ◽  
Class Group ◽  
Cohomology Ring ◽  
Mapping Class ◽  
Schur Polynomials ◽  
Stable Bundles ◽  
Cohomology Rings ◽  
Cup Product

AbstractWe compute cup-product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping-class group action.

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