Geometry of twistor space

1.我们对Joyce的一个共轭得到了一个肯定的回答：一个容许2-环面的光滑作用的单连通完备自对偶流形与复射影平面的m个副本的连通和mCP（2）同构，并且它的自对偶结构与Joyce在1995年构造的一个例子同构.我们的方法是使用与给定的自对偶流形相关联的扭量空间。事实上，更一般地说，没有简单的连通性假设，我们已经分类紧凑的自对偶流形，承认一个行动的两个环面。作为这些研究的副产品，我们可以确定非常精确的结构作为一个复杂的流形的扭量空间与乔伊斯自对偶度量。特别地，我们已经证明了存在扭量空间的一个很好的双纯模型，它被实现为环面的纤维空间，并且后者的结构完全由环面对mCP（2）的原始光滑作用的不变量决定。扭量空间是一个Moishezon流形，构造一个自然的双有理投影模型是很有趣的.本文研究了Joyce扭量空间的变形空间，证明了它的Kuranishi空间是非奇异的。此外，利用这种变形，我们给出了对任意整数m[大于或等于]4，扭量空间代数维数为2的mCP（2）上正型自对偶结构的第一个例子。这个结果决定了与mCP（2）相关的扭量空间的代数维数分布。在4CP（2）的情形下，我们将扭量空间描述为一般的椭圆纤维空间，并通过取其分支覆盖构造了一族有趣的紧复流形，其标准丛是平凡的。

1. We have obtained an affimative answer to a conjugate of Joyce to the effect that a simply connected and complete self-dual manifold which admits a smooth action of a 2-torus is diffeomorphic to a connected sum mCP (2) of m copies of complex projective plane and its self-dual structure is isomorphic to one of the examples constructed by Joyce himself in 1995. Our method was to use the twistor space associated to the given self-dual manifold. In fact, more generally, without simple-connectivity assumption we have classified compact self-dual manifolds which admit an action of a two torus. As a byproduct of these investigation we could determine very precise structure as a complex manifold of the twistor space associated to Joyce self-dual metric. In particular we have shown that there exists a nice bimeromorphic model of the twistor space which is realized as a fiber space of torus surfaces and that this latter structure is completely determined by the invariant of the original smooth action of the torus on mCP (2). The twistor space is a Moishezon manifold and it is interesting to construct a natural birational projecitve model.2. We have studied the deformation space of Joyce twistor space and have shown that its Kuranishi space is nonsingular. Furthermore, using such deformation we have given, for every integer m【greater than or equal】4 the first examples of self-dual structure of positive type on mCP (2) whose twistor space has algebraic dimension two. This result determines distribution of algebraic dimensions of twistor spaces associated to mCP (2). Furthermore, in the case of 4CP (2) we have given a detailed description of the twistor space as a general elliptic fiber space and then by taking its branched covering constructed a interesting family of compact complex manifolds whose canonical bundle is trivial.