Hyper Kahler manifolds

设X是Ricci曲率为零的紧致黎曼流形. X的完整群列表包括四类有趣的完整群：SU（n），Sp（m），G_2和Spin（7）.李群SU（n）是Calabi-Yau流形的完整群，而Sp（m）是超Kahler流形的完整群。G_2和Spin（7）分别作为7维和8维流形的完整群出现。这四种几何之间有许多有趣的共同性质。作者的研究是基于对超Kahler流形的研究，他从中得到了一些想法，这些想法有可能统一这些不同类型的几何结构的模空间结果。主要结果如下：（1）这些几何结构的变形空间的光滑性是它们最显著的共同性质之一。作者证明了这些变形可以看作是一类特殊微分形式的变形，并构造了一类新的微分形式的变形理论。然后通过一定的精确形式给出了变形的一个障碍。因此，他表明，障碍消失方面的上同调（拓扑）的论点。这个结果可以被认为是Kodaira-Spencer理论的自然推广。(2)他还构造了模空间并证明了局部Torelli类型定理在这些情况下成立。(3)作为应用，得到了Calabi-Yau结构的光滑模空间和特殊Lagrange子流形。

Let X be a compact Riemannian manifold with vanishing Ricci curvature. Then the list of holonomy group of X includes four interesting classes of the holonomy groups: SU(n), Sp(m), G_2 and Spin(7). The Lie group SU(n) arises as the holonomy group of Calabi-Yau manifolds and Sp(m) is the holonomy group of hyper Kahler manifolds. G_2 and Spin(7) occur as the holonomy groups of 7 and 8 dimensional manifolds respectively. There are many intriguing common properties between these four geometries. The author's research is based on the study of hyper Kahler manifolds, from which he obtains some ideas with the potential to unify moduli spaces results of these different kinds of geometric structures. His main results are the followings:(1) One of the most remarkable common property is smoothness of the deformation spaces of these geometric structures. The author show that these deformations can be regarded as deformations of special kind of differential forms and he constructs a new kind of deformations theory of these differential forms. Then an obstruction of the deformation is given by the certain exact forms. Hence he shows that the obstruction vanishes in terms of cohomological (topological) argument. This result can be considered as a natural generalization of Kodaira-Spencer theory.(2) He also constructs the moduli space and shows that the local Torelli's type theorem holds in these cases.(3) As an application he obtain the smooth moduli space of Calabi-Yau structures and Special Lagrangian submanifolds.