Unified approach of Ricci-flat manifolds

作为Ricci平坦黎曼流形的完整群出现的可能李群的列表表明，存在四类有趣的李群：SU(N)，Sp(M)，G_2和Spin(7)。特殊的酉群SU(N)作为Calabi-Yau流形的完整群出现，Sp(M)是HyperK“Ahler流形的完整群。例外李群G_2和自旋(7)分别是7维和8维流形的完整群，称为G_2和自旋(7)流形.这四类黎曼流形之间有表面上的区别，但作者认为这四种结构是由特殊的闭微分形式定义的几何结构.他得到了这些结构变形问题的一种新方法。他证明了在一定的上同调条件下，形变空间成为有限维的光滑流形。因此，他得到了这四种结构的模空间的统一构造。这种方法是相当…的更一般，他预计应该存在许多几何结构，他的方法可以有效地应用于这些结构上。事实上，他发展了(1)全纯辛结构和(2)广义几何结构的变形问题：(继续下一页)(1)全纯辛结构作者研究了不需要K“ahler的全纯辛结构。得到了无障碍变形的一个新判据和局部Torelli型定理。(2)广义几何结构广义几何结构是Hitchinis最近提出的一种广义几何结构的概念，其思想是用流形上切丛和余切丛的直和代替切丛。然后把复结构和实辛结构看作是广义复结构的特例。作者着重于Clifford代数，证明了广义结构可以适当地理解为由共形Pin群下的作用量定义的结构。然后，他得到了广义Calabi-Yau结构、HyperK“Ahler G_2结构和自旋(7)结构的自然概念，建立了广义结构的变形理论。特别地，他得到了广义Calabi-Yau结构和自旋(7)结构的无阻变形

The list of possible Lie groups arising as holonomy groups of Ricci-flat Riemannian manfolds implies that there are four interesting classes of Lie groups : SU(n), Sp(m), G_2 and Spin(7).The special unitary group SU(n) arises as the holonomy group of Calabi-Yau manifolds and Sp(m) is the holonomy group of hyperK"ahler manifolds. The exceptional Lie group G_2 and Spin(7) are respectively holonomy groups of 7 and 8 dimensional manifolds, which are called G_2 and Spin(7) manifolds.There are superficial differences between these four classes of Riemannian manifolds, however the author shows that these four structures are regarded as geometric structures defined by special closed differential forms. He obtains a new approach of deformation problems of these structures. He shows that under certain cohomological condition, deformation space becomes a smooth manifolds of finite dimension. Hence he obtains a unified construction of moduli spaces of these four structures.This approach is quite g … More eneral and he expects that there should exist many geometric structures on which his approach can be applied effectively. In fact, he develops deformation problems of (1) holomorphci symplectic structures and (2) generalized geometric structures : (CONTINUE TO NEXT PAGE)(1)holomorphic symplectic structuresThe author studies holomorphic symplectic structures which are not necessary K"ahlerian. He obtains a new criterion of unobstructed deformations and local Torelli type theorem. He also shows that the criterion holds on complex Nilmanifolds and further constructs an example of compact holomorphic symplectic manifold which has just obstructed deformations.(2)generalized geometric structuresA notion of generlized geometric structures, which is recently introduced by HitchinIs based on an idea replacing the tangent bundle with the direct sum of the tangent and cotangent bundle on a manifold. Then complex structures and real symplectic structures are regarded as special cases of generalized complex structures.The author focuses on the Clifford algebra and shows that generalized structures can be suitably understood as structures defined by the action under the conformal pin group.Then he obtains a natural notion of generalized Calabi-Yau, hyperK"ahler G_2 and Spin(7) structures and establishes a deformation theory of generalized structures.In particular, he has unobstructed deformations of generalized Calabi-Yau and Spin(7) structures. Less