Geometry of twistor spaces

1.我们已经证明了在许多VII类复杂表面上存在反自对偶埃尔米特度量。通过对Kim-Pontecorvo方法的改进（它本身是Donaldson-Friedman方法的推广），并将此方法应用于Joyce扭量空间及其上的一些特殊的初等因子，我们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而光滑化，并且在所得流形中我们得到了VII类曲面所需的扭量空间.为了识别所得到的曲面，我们在上述基本因子上取一个有理曲线的自然循环，并考虑所得到的三元组的光滑性，结果我们可以证明Inoue-Hirzebruch曲面和抛物Inoue曲面（具有真实的参数）.在特殊线性群G=SL（2，C）下构造了一类几乎同态紧的非Kahler复流形.给定G的一个自然等变紧化X和一个无尖点的几何有限Klein群Γ，得到了这样一个流形作为X上极大不连续域的商空间。从它们的构造来看，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.