Geometry of twistor spaces

1. 我们证明了在许多VII类复曲面上反自对偶厄米度量的存在性。对Kim-Pontecorvo方法进行了改进，该方法本身就是Donaldson-Friedman方法的推广，并将该方法应用于Joyce扭转空间及其上的一些特殊初等因子，证明了所得到的具有法向交叉的三维复空间可以通过变形进行光滑，并在所得到的流形中得到了VII类曲面所需的扭转空间。为了确定所得到的曲面，我们在上述初等因子上取有理曲线的自然循环，并考虑所得三元组的平滑；结果表明，Inoue- hirzebruch曲面和抛物型Inoue曲面（带实参数）在特殊线性群G=SL（2，C）下构造了一类广义的几乎同态紧致非kahler复流形。给定G的自然等变紧化X和无尖的几何有限Klein群Γ，通过Γ在X上的最大不连续域上得到了一个流形作为商空间。从它们的构造上看，它们与三维双曲流形密切相关。

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.