Geometry of twistor spaces

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1. We have shown the existence of anti-self-dual hermitian metrics on many of the complex surface of class VII. Having modified the method of Kim-Pontecorvo, which itself is the generalization of the method of Donaldson-Friedman, and applying this method to Joyce twistor spaces together with some special elementary divisors on them, we show that the resulting 3-dimensional complex space with normal crossings can be smoothed by deformations and among the resulting manifolds we get desired twistor space for surface of class VII. In order to identify the resulting surface we take a natural cycle of rational curves on the above elementary divisors and consider the smoothing of the resulting triples ; as a result we could show that the Inoue-Hirzebruch surface and parabolic Inoue surfaces (with real parameters).2. We have constructed a wide class of almost homomorphism compact non-Kahler complex manifolds under the special linear group G=SL(2,C). Given a natural equivariant compactification X of G and a geometrically finite Klein group Γ without cusps such a manifold is obtained as the quotient space by Γ of a maximal domain of discontinuity on X. From their construction they are intimately related with the 3-dimensional hyperbolic manifolds.