Hyper Kahler manifolds

设X是一个紧致黎曼流形，其里奇曲率为零。那么X的完整群列表包含了四个有趣的完整群：SU(n)、Sp(m)、G_2和Spin(7)。李群SU(n)是Calabi-Yau流形的完整群，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。这四种几何形状之间有许多有趣的共同特性。作者在研究超Kahler流形的基础上，得到了一些有可能统一这些不同几何结构的模空间结果的思想。他的主要成果如下：(1)这些几何结构的变形空间的平滑性是它们最显著的共同特性之一。作者证明这些变形可以看作是一类特殊的微分形式的变形，并构造了一类新的微分形式的变形理论。然后用一定的精确形式给出了变形的阻力。因此，他证明了障碍在上同调（拓扑）论证中消失。这个结果可以看作是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的类型定理在这些情况下成立。(3)作为应用，得到了Calabi-Yau结构和特殊lagrange子流形的光滑模空间。

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.