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Stable bundles on hypercomplex surfaces

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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.
  • [1] Apostolov V, Cauduchon P., Grantcharov G., Bi-Hermitian structures on complex surfaces, Proc. London Math. Soc. (3), 1999, 79(2), 414–428
  • [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
  • [4] Boyer C.P., A note on hyperhermitian four-manifolds, Proc. Amer. Math. Soc., 1988, 102(1), 157–164
  • [5] Braam P.J., Hurtubise J., Instantons on Hopf surfaces and monopoles on solid tori, J. Reine Angew. Math., 1989, 400, 146–172
  • [6] Buchdahl N.P., Hermitian-Einstein connections and stable vector bundles over compact complex surfaces, Math. Ann., 1988, 280, 625–648
  • [7] Bredthauer A., Generalized Hyperkähler Geometry and Supersymmetry, preprint available at
  • [8] Cavalcanti G.R., Reduction of metric structures on Courant algebroids, preprint
  • [9] Thomas F., Ivanov S., Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math., 2002, 6(2), 303–335
  • [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
  • [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)
  • [13] Gauduchon P., Hermitian connections and Dirac operators, Boll. Un. Mat. Ital. B (7), 1997, 11, 257–288
  • [14] Goto R., On deformations of generalized Calabi-Yau, hyperKähler, G2 and Spin(7) structures I, preprint available at
  • [15] Grantcharov G., Poon Y.S., Geometry of hyper-Kähler connections with torsion, Comm. Math. Phys., 2000, 213(1), 19–37
  • [16] Gualtieri M., Generalized complex geometry, Ph.D. thesis, Oxford University, available at
  • [17] Hitchin N., Generalized Calabi-Yau manifolds, Q. J. Math., 2003, 54(3), 281–308
  • [18] Hitchin N., Instantons, Poisson structures and generalized Kähler geometry, Comm. Math. Phys., 2006, 265(1), 131–164
  • [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
  • [21] Ivanov S., Papadopoulos G., Vanishing theorems and string backgrounds, Classical Quantum Gravity, 2001, 18(6), 1089–1110
  • [22] Joyce D., Compact hypercomplex and quaternionic manifolds, J. Differential Geom., 1992, 35(3), 743–761
  • [23] Kaledin D., Integrability of the twistor space for a hypercomplex manifold, Selecta Math. (N.S.), 1998, 4, 271–278
  • [24] Kato Ma., Compact Differentiable 4-folds with quaternionic structures, Math. Ann., 1980, 248, 79–86
  • [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
  • [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
  • [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
  • [31] Verbitsky M., Hyperholomorphic vector bundles over hyperkähler manifolds, J. Algebraic Geom., 1996, 5(4), 633–669
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