# Stable bundles on hypercomplex surfaces

Ruxandra Moraru; Misha Verbitsky

Open Mathematics (2010)

- Volume: 8, Issue: 2, page 327-337
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topRuxandra Moraru, and Misha Verbitsky. "Stable bundles on hypercomplex surfaces." Open Mathematics 8.2 (2010): 327-337. <http://eudml.org/doc/269722>.

@article{RuxandraMoraru2010,

abstract = {A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin’s and Gualtieri’s generalized complex geometry, (4,4)-manifolds are called “generalized hyperkähler manifolds”. We show that the moduli space of anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a (4,4)-structure.},

author = {Ruxandra Moraru, Misha Verbitsky},

journal = {Open Mathematics},

keywords = {Hopf surface; Stable bundles; Instanton bundles; Hypercomplex; Generalized hyperkaehler; stable bundles; hypercomplex manifolds; instantons; generalized hyper-Kähler structures},

language = {eng},

number = {2},

pages = {327-337},

title = {Stable bundles on hypercomplex surfaces},

url = {http://eudml.org/doc/269722},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Ruxandra Moraru

AU - Misha Verbitsky

TI - Stable bundles on hypercomplex surfaces

JO - Open Mathematics

PY - 2010

VL - 8

IS - 2

SP - 327

EP - 337

AB - A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin’s and Gualtieri’s generalized complex geometry, (4,4)-manifolds are called “generalized hyperkähler manifolds”. We show that the moduli space of anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a (4,4)-structure.

LA - eng

KW - Hopf surface; Stable bundles; Instanton bundles; Hypercomplex; Generalized hyperkaehler; stable bundles; hypercomplex manifolds; instantons; generalized hyper-Kähler structures

UR - http://eudml.org/doc/269722

ER -

## References

top- [1] Apostolov V, Cauduchon P., Grantcharov G., Bi-Hermitian structures on complex surfaces, Proc. London Math. Soc. (3), 1999, 79(2), 414–428 http://dx.doi.org/10.1112/S0024611599012058 Zbl1035.53061
- [2] Besse A., Einstein Manifolds, Springer-Verlag, New York, 1987
- [3] Bismut J.M., A local index theorem for non-Kählerian manifolds, Math. Ann., 1989, 284, 681–699 http://dx.doi.org/10.1007/BF01443359 Zbl0666.58042
- [4] Boyer C.P., A note on hyperhermitian four-manifolds, Proc. Amer. Math. Soc., 1988, 102(1), 157–164 http://dx.doi.org/10.2307/2046051 Zbl0642.53073
- [5] Braam P.J., Hurtubise J., Instantons on Hopf surfaces and monopoles on solid tori, J. Reine Angew. Math., 1989, 400, 146–172 Zbl0669.32012
- [6] Buchdahl N.P., Hermitian-Einstein connections and stable vector bundles over compact complex surfaces, Math. Ann., 1988, 280, 625–648 http://dx.doi.org/10.1007/BF01450081 Zbl0617.32044
- [7] Bredthauer A., Generalized Hyperkähler Geometry and Supersymmetry, preprint available at http://arxiv.org/abs/hep-th/0608114
- [8] Cavalcanti G.R., Reduction of metric structures on Courant algebroids, preprint Zbl1157.53324
- [9] Thomas F., Ivanov S., Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math., 2002, 6(2), 303–335 Zbl1127.53304
- [10] Gates S.J.,Jr., Hull C.M., Roček M., Twisted multiplets and new supersymmetric nonlinear σ-models, Nuclear Phys. B, 1984,248(1), 157–186 http://dx.doi.org/10.1016/0550-3213(84)90592-3
- [11] Gauduchon P., Le théorème de l’excentricité nulle, C. R. Acad. Sci. Paris Sér. A-B, 1977, 285, 387–390 (in French)
- [12] Gauduchon P., La 1-forme de torsion d’une variete hermitienne compacte, Math. Ann., 1984, 267, 495–518 (in French) http://dx.doi.org/10.1007/BF01455968 Zbl0523.53059
- [13] Gauduchon P., Hermitian connections and Dirac operators, Boll. Un. Mat. Ital. B (7), 1997, 11, 257–288 Zbl0876.53015
- [14] Goto R., On deformations of generalized Calabi-Yau, hyperKähler, G2 and Spin(7) structures I, preprint available at http://arxiv.org/abs/math/0512211
- [15] Grantcharov G., Poon Y.S., Geometry of hyper-Kähler connections with torsion, Comm. Math. Phys., 2000, 213(1), 19–37 http://dx.doi.org/10.1007/s002200000231 Zbl0993.53016
- [16] Gualtieri M., Generalized complex geometry, Ph.D. thesis, Oxford University, available at http://arxiv.org/abs/math/0401221
- [17] Hitchin N., Generalized Calabi-Yau manifolds, Q. J. Math., 2003, 54(3), 281–308 http://dx.doi.org/10.1093/qmath/hag025
- [18] Hitchin N., Instantons, Poisson structures and generalized Kähler geometry, Comm. Math. Phys., 2006, 265(1), 131–164 http://dx.doi.org/10.1007/s00220-006-1530-y Zbl1110.53056
- [19] Howe P.S., Papadopoulos G., Twistor spaces for hyper-Kähler manifolds with torsion, Phys. Lett. B, 1996, 379(1–4), 80–86
- [20] Huybrechts D., Generalized Calabi-Yau structures, K3 surfaces, and B-fields, Int. J. Math., 2005, 16 Zbl1120.14027
- [21] Ivanov S., Papadopoulos G., Vanishing theorems and string backgrounds, Classical Quantum Gravity, 2001, 18(6), 1089–1110 http://dx.doi.org/10.1088/0264-9381/18/6/309 Zbl0990.53078
- [22] Joyce D., Compact hypercomplex and quaternionic manifolds, J. Differential Geom., 1992, 35(3), 743–761 Zbl0735.53050
- [23] Kaledin D., Integrability of the twistor space for a hypercomplex manifold, Selecta Math. (N.S.), 1998, 4, 271–278 http://dx.doi.org/10.1007/s000290050032 Zbl0906.53048
- [24] Kato Ma., Compact Differentiable 4-folds with quaternionic structures, Math. Ann., 1980, 248, 79–86 http://dx.doi.org/10.1007/BF01349256 Zbl0411.57024
- [25] Li J., Yau S.-T., Hermitian Yang-Mills connections on non-Kähler manifolds, In: Mathematical aspects of string theory, World Scientific, 1987
- [26] Lübke M., Teleman A., The Kobayashi-Hitchin correspondence, World Scientific Publishing Co., Inc., River Edge, NJ, 1995 Zbl0849.32020
- [27] Nash O., Differential geometry of monopole moduli spaces, Ph.D. Thesis, University of Oxford, 2006
- [28] Obata M., Affine connections on manifolds with almost complex, quaternionic, or Hermitian structures, Jap. J. Math., 1956, 26,43–77
- [29] Pedersen H., Poon Y.S., Deformations of hypercomplex structures, J. Reine Angew. Math., 1998, 499, 81–99 Zbl0908.58073
- [30] Tyurin A.N., The Weil-Petersson metric in the moduli space of stable vector bundles and sheaves over an algebraic surface, Math. USSR-Izv., 1992, 38(3), 599–620 http://dx.doi.org/10.1070/IM1992v038n03ABEH002216 Zbl0788.14007
- [31] Verbitsky M., Hyperholomorphic vector bundles over hyperkähler manifolds, J. Algebraic Geom., 1996, 5(4), 633–669 Zbl0865.32006

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.