In this article we show that any hyperbolic Inoue surface (also called Inoue-Hirzebruch surface of even type) admits anti-self-dual bihermitian structures. The same result holds for any of its small deformations as far as its anti-canonical system is nonempty. Similar results are obtained for parabolic Inoue surfaces. Our method also yields a family of anti-self-dual hermitian metrics on any half Inoue surface. We use the twistor method of Donaldson-Friedman [13] for the proof.
Abstract.We consider a complex surface M with anti-self-dual hermitian metric h and study the holomorphic properties of its twistor space Z . We show that the naturally defined divisor line bundle [X] is isomorphic to the -j power of the canonical bundle of Z , if and only if there is a Kahler metric of zero scalar curvature in the conformai class of h . This has strong consequences on the geometry of M, which were also found by C Boyer [3] using completely different methods. We also prove the existence of a very close relation between holomorphic vector fields on M and Z in the case that M is compact and Kahler.
Abstract.We consider a complex surface M with anti-self-dual hermitian metric h and study the holomorphic properties of its twistor space Z . We show that the naturally defined divisor line bundle [X] is isomorphic to the -j power of the canonical bundle of Z , if and only if there is a Kahler metric of zero scalar curvature in the conformai class of h . This has strong consequences on the geometry of M, which were also found by C Boyer [3] using completely different methods. We also prove the existence of a very close relation between holomorphic vector fields on M and Z in the case that M is compact and Kahler.
Abstract. On an almost quaternionic manifold (M 4n , Q) we study the integrability of almost complex structures which are compatible with the almost quaternionic structure Q. If n ≥ 2, we prove that the existence of two compatible complex structuresis an oriented conformal 4-manifold, we prove a maximum principle for the angle function I 1 , I 2 of two compatible complex structures and deduce an application to anti-self-dual manifolds.By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure J on the twistor space Z of an almost quaternionic manifold (M 4n , Q) and show that J is a complex structure if and only if Q is quaternionic. This is a natural generalization of the Penrose twistor constructions.
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