##
**Anti-self-dual bihermitian structures on Inoue surfaces.**
*(English)*
Zbl 1206.53077

The existence of anti-self-dual bi-Hermitian structures on any hyperbolic Inoue surface and also on any of its small deformations preserving the unique anti-canonical divisor on it is shown. Similar results are obtained for parabolic Inoue surfaces. The method used to obtain these results yields also a family of anti-self-dual Hermitian metrics on any half Inoue surface. The basic properties of hyperbolic, half and parabolic Inoue surfaces are presented and the Kuranishi family of deformations of associated pairs is studied. The Joyce twistor space is defined and it is explained how its singular model together with the natural anti-canonical divisor can be obtained by analytical modifications. The differential geometric implications of the obtained results are emphasized, here including the relations with generalized Kähler and locally conformally Kähler structures.

Reviewer: Gheorghe Zet (Iaşi)

### MSC:

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |

81T13 | Yang-Mills and other gauge theories in quantum field theory |