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.


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
Full Text: DOI arXiv


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