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 INTERSECTION COHOMOLOGY OF REPRESENTATION SPACES OF SURFACE GROUPS

2006 ◽  
Vol 17 (02) ◽  
pp. 169-182
Author(s):  
YOUNG-HOON KIEM
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Moduli Space ◽  
Local Structure ◽  
Equivariant Cohomology ◽  
Intersection Cohomology ◽  
Symplectic Reduction ◽  
Surface Groups ◽  
Representation Space ◽  
Splitting Theorem

The representation space X(G) = Hom (π, G)/G of the fundamental group π of a Riemann surface Σ of genus g ≥ 2 is the symplectic reduction of the extended moduli space defined in [6]. Using this description, we study the local structure of X(G) and show that the assumptions of the splitting theorem [11, Theorem 7.7] are satisfied. Hence the middle perversity intersection cohomology is canonically isomorphic to a subspace of the equivariant cohomology [Formula: see text] which can be computed quite explicitly. The case when G = SU(2) is discussed in detail.

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 THE VERLINDE FORMULA FOR PARABOLIC BUNDLES

2001 ◽  
Vol 63 (3) ◽  
pp. 754-768 ◽  
Author(s):  
LISA C. JEFFREY
Keyword(s):  
Riemann Surface ◽  
Line Bundle ◽  
Conjugacy Class ◽  
Fundamental Group ◽  
Moduli Space ◽  
Boundary Component ◽  
Compact Riemann Surface ◽  
Central Element ◽  
Parabolic Bundles ◽  
Verlinde Formula

Let Σg be a compact Riemann surface of genus g, and G = SU(n). The central element c = diag(e2πid/n, …, e2πid/n) for d coprime to n is introduced. The Verlinde formula is proved for the Riemann–Roch number of a line bundle over the moduli space [Mscr ]g, 1(c, Λ) of representations of the fundamental group of a Riemann surface of genus g with one boundary component, for which the loop around the boundary is constrained to lie in the conjugacy class of cexp(Λ) (for Λ ∈ t+), and also for the moduli space [Mscr ]g, b(c, Λ) of representations of the fundamental group of a Riemann surface of genus g with s + 1 boundary components for which the loop around the 0th boundary component is sent to the central element c and the loop around the jth boundary component is constrained to lie in the conjugacy class of exp(Λ(j)) for Λ(j) ∈ t+. The proof is valid for Λ(j) in suitable neighbourhoods of 0.

TORELLI THEOREM FOR MODULI SPACES OF SL(r,ℂ)-CONNECTIONS ON A COMPACT RIEMANN SURFACE

2009 ◽  
Vol 11 (01) ◽  
pp. 1-26
Author(s):  
INDRANIL BISWAS ◽  
VICENTE MUÑOZ
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Complex Structure ◽  
Real Numbers ◽  
Stable Vector Bundles ◽  
Complex Variety ◽  
Projective Curves ◽  
Torelli Theorem

Let X be any compact connected Riemann surface of genus g, with g ≥ 3. For any r ≥ 2, let [Formula: see text] denote the moduli space of holomorphic SL (r,ℂ)-connections over X. It is known that the biholomorphism class of the complex variety [Formula: see text] is independent of the complex structure of X. If g = 3, then we assume that r ≥ 3. We prove that the isomorphism class of the variety [Formula: see text] determines the Riemann surface X uniquely up to an isomorphism. A similar result is proved for the moduli space of holomorphic GL (r,ℂ)-connections on X. We also show that the Torelli theorem remains valid for the moduli spaces of connections, as well as those of stable vector bundles, on geometrically irreducible smooth projective curves defined over the field of real numbers.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

Parabolic Higgs Bundles for Real Reductive Lie Groups

2018 ◽  
pp. 653-680 ◽  
Author(s):  
Ignasi Mundet i Riera
Keyword(s):  
Riemann Surface ◽  
Lie Groups ◽  
Lie Group ◽  
Vector Bundles ◽  
Structure Group ◽  
Higgs Bundles ◽  
Local Systems ◽  
Parabolic Structure ◽  
Reductive Lie Groups ◽  
Reductive Lie Group

This chapter explains the correspondence between local systems on a punctured Riemann surface with the structure group being a real reductive Lie group G, and parabolic G-Higgs bundles. The chapter describes the objects involved in this correspondence, taking some time to motivate them by recalling the definitions of G-Higgs bundles without parabolic structure and of parabolic vector bundles. Finally, it explains the relevant polystability condition and the correspondence between local systems and Higgs bundles.

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