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Unobstructed K-deformations of generalized complex structures and bi-Hermitian structures. (English) Zbl 1252.53095

In [V. Apostolov, P. Gauduchon and G. Grantcharov, Proc. Lond. Math. Soc., III. Ser. 92, No. 1, 200–202 (2006; Zbl 1089.53503)], the following question is adressed. Which compact complex surfaces admit non-trivial bi-Hermitian structures?
The present paper answers this question in the case of Kähler surfaces and non-trivial bi-Hermitian structures that satisfy a torsion condition. This is done using the correspondence between these bi-Hermitian structures and generalized Kähler structures [M. Gualtieri, Ann. Math. (2) 174, No. 1, 75–123 (2011; Zbl 1235.32020)], together with deformation theory.
The author introduces K-deformation of generalized complex structures. These deformations are shown to be unobstructed. With the stability result of generalized Kähler structures [the author, J. Differ. Geom. 84, No. 3, 525–560 (2010; Zbl 1201.53085)], the author shows that a compact Kähler surface admits a non-trivial bi-Hermitian structure with the torsion condition and the same orientation if and only if it has a non-zero holomorphic Poisson structure.
Examples such that del Pezzo surfaces or ruled surfaces are studied in more detail.

MSC:

53D18 Generalized geometries (à la Hitchin)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C56 Other complex differential geometry
53D17 Poisson manifolds; Poisson groupoids and algebroids
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[1] Apostolov, V.; Dloussky, G., Bihermitian metrics on Hopf surfaces, Math. res. lett., 15, 827-839, (2008) · Zbl 1162.53053
[2] Apostolov, V.; Gauduchon, P.; Grantcharov, G., Bihermitian structures on complex surfaces, Proc. lond. math. soc., 79, 414-428, (1999), (Corrigendum, 92 (2006), 200-202) · Zbl 1035.53061
[3] Bartocci, C.; Macrì, E., Classification of Poisson surfaces, Commun. contemp. math., 7, 89-95, (2005) · Zbl 1071.14514
[4] Bogomolov, F.A., Hamiltonian Kähler manifolds (English, Russian original), Sov. math. dokl., Dokl. akad. nauk SSSR, 243, 1101-1104, (1978), Translation from
[5] Demazure, M., Surfaces de del Pezzo II, III, IV, V, Lecture notes in math., 777, 23-69, (1980) · Zbl 0444.14024
[6] Goto, R., Moduli spaces of topological calibrations, calabi – yau, hyperkähler, \(\operatorname{G}_2\) and spin(7) structures, Internat. J. math., 15, 211-257, (2004) · Zbl 1046.58002
[7] Goto, R., On deformations of generalized calabi – yau, hyperkähler, \(\operatorname{G}_2\) and spin(7) structures
[8] Goto, R., Poisson structures and generalized Kähler submanifolds, J. math. soc. Japan, 61, 107-132, (2009) · Zbl 1160.53014
[9] Goto, R., Deformations of generalized complex and generalized Kähler structures, J. differential geom., 84, 525-560, (2010) · Zbl 1201.53085
[10] Goto, R., Deformations of generalized Kähler structures and bihermitain structures
[11] M. Gualtieri, Generalized complex geometry, D.Phil. Thesis, University of Oxford, 2003. · Zbl 1235.32020
[12] Gualtieri, M., Generalized complex geometry, Ann. of math., 174, 75-123, (2011) · Zbl 1235.32020
[13] Hitchin, N., Instantons, Poisson structures and generalized Kähler geometry, Comm. math. phys., 265, 131-164, (2006) · Zbl 1110.53056
[14] Hitchin, N., Bihermitian metrics on del Pezzo surfaces, J. symplectic geom., 5, 1-8, (2007) · Zbl 1187.32017
[15] Kodaira, K., Complex manifolds and deformations of complex structures, () · JFM 65.1320.02
[16] Kosmann-Schwarzbach, Y., Derived brackets, Lett. math. phys., 69, 61-87, (2004) · Zbl 1055.17016
[17] Miyajima, K., A note on the Bogomolov-type smoothness on deformations of the regular parts of isolated singularities, Proc. amer. math. soc., 125, 485-492, (1997) · Zbl 0861.32012
[18] Sakai, F., Anti-Kodaira dimension of ruled surfaces, Sci. rep. saitama univ. ser. A, 10, 2, 1-7, (1982) · Zbl 0496.14022
[19] Sakai, F., Anticanonical models of rational surfaces, Math. ann., 269, 389-410, (1984) · Zbl 0533.14016
[20] Tian, G., Smoothness of the universal deformation space of compact calabi – yau manifolds and its petersson – weil metric, (), 629-646
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