## 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

### Citations:

Zbl 1089.53503; Zbl 1235.32020; Zbl 1201.53085
Full Text:

### References:

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