The Donaldson-Hitchin-Kobayashi Correspondence for Parabolic Bundles over Orbifold Surfaces

2001 ◽  
Vol 53 (6) ◽  
pp. 1309-1339 ◽  
Author(s):  
Brian Steer ◽  
Andrew Wren
Keyword(s):  
Riemann Surfaces ◽  
Einstein Metrics ◽  
Approximation Theorem ◽  
Effective Divisor ◽  
Parabolic Bundles ◽  
Kähler Surface ◽  
Holomorphic Bundles ◽  
Suitable Approximation

AbstractA theorem of Donaldson on the existence of Hermitian-Einstein metrics on stable holomorphic bundles over a compact Kähler surface is extended to bundles which are parabolic along an effective divisor with normal crossings. Orbifold methods, together with a suitable approximation theorem, are used following an approach successful for the case of Riemann surfaces.

MODULI OF STABLE PAIRS FOR HOLOMORPHIC BUNDLES OVER RIEMANN SURFACES

1991 ◽  
Vol 02 (05) ◽  
pp. 477-513 ◽  
Author(s):  
STEVEN B. BRADLOW ◽  
GEORGIOS D. DASKALOPOULOS
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Holomorphic Section ◽  
Stable Bundles ◽  
Space Problem ◽  
Yang Mills ◽  
Closed Riemann Surface ◽  
Holomorphic Bundles ◽  
Stable Pairs ◽  
Compact Kähler Manifold

It this paper we study the space of gauge equivalence classes of pairs [Formula: see text] where [Formula: see text] represents a holomorphic structure on a complex bundle, E, over a closed Riemann Surface, and ϕ is a holomorphic section. We define a space of stable pairs and consider the moduli space problem for this space. The space of stable pairs, [Formula: see text], is related to the space of solution to the Vortex (Hermitian-Yang-Mills-Higgs) equation. Using the parameter, τ, which appears in this equation we can define subspaces [Formula: see text] within [Formula: see text]. We show that under suitable restrictions on τ and the degree of E, the space [Formula: see text] is naturally a finite dimensional, Hausdorff, compact Kähler manifold. We show further that there is a natural holomorphic map from this space onto the Seshadri compactification of the moduli space of stable bundles and that this map is generically a fibration.

K�hler-einstein metrics in holomorphic bundles

1987 ◽  
Vol 21 (2) ◽  
pp. 144-146 ◽  
Author(s):  
D. V. Alekseevskii ◽  
A. M. Perelomov
Keyword(s):  
Einstein Metrics ◽  
Holomorphic Bundles

scholarly journals Holomorphic connections on holomorphic bundles on Riemann surfaces

2021 ◽  
Author(s):  
Hassan Azad ◽  
Indranil Biswas
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Principal Bundles ◽  
Holomorphic Bundles

AbstractWe investigate aspects of holomorphic connections on holomorphic principal bundles over a Riemann surface.

scholarly journals Metric Properties of Parabolic Ample Bundles

2018 ◽  
Vol 2020 (23) ◽  
pp. 9336-9369
Author(s):  
Indranil Biswas ◽  
Vamsi Pritham Pingali
Keyword(s):  
Riemann Surfaces ◽  
Chern Classes ◽  
Type Result ◽  
Hermitian Metrics ◽  
Stable Bundles ◽  
Metric Properties ◽  
Parabolic Bundles

Abstract We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern–Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favor for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type result holds and that there are metrics on stable bundles over surfaces whose Schur forms are positive.

Stable Parabolic Bundles over Elliptic Surfaces and over Riemann Surfaces

2000 ◽  
Vol 43 (2) ◽  
pp. 174-182 ◽  
Author(s):  
Christian Gantz ◽  
Brian Steer
Keyword(s):  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Elliptic Surfaces ◽  
Parabolic Bundles

AbstractWe show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method.

MODULI OF STABLE PAIRS FOR HOLOMORPHIC BUNDLES OVER RIEMANN SURFACES II

1993 ◽  
Vol 04 (06) ◽  
pp. 903-925 ◽  
Author(s):  
STEVEN BRADLOW ◽  
GEORGIOS D. DASKALOPOULOS
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Projective Variety ◽  
Algebraic Varieties ◽  
Stable Bundles ◽  
Topological Information ◽  
Holomorphic Bundles ◽  
Stable Pairs

In this paper we continue our investigation of the moduli space of stable pairs introduced in Part I. We obtain certain topological information, and we give a proof that this moduli space admits the structure of a nonsingular projective variety. We show that the natural map from the moduli space of stable pairs onto the Seshadri compactification of stable bundles is a morphism of algebraic varieties.

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.

Sign in / Sign up

Export Citation Format

Share Document