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GLOBAL CONSTRUMONS OF MODULI SPACES

模空间的全局构造

基本信息

批准号：
17340018
负责人：
NAGATOMO Yasuyuki
金额：
$ 4.15万
依托单位：
Kyushu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17340018/
关键词：
ASD connections vector bundles moduli spaces quaternion manifolds Lie groups twistor spaces vanishing theorems harmonic mans ASD接続コホモロジー消滅定理

项目摘要

In 2005, we focus our attention on submanifolds which appear as singular sets of ideal instantons. Those are zero loci of the twistor sections satisfying linear equations which are linearization of the (higher dimensional instanton equations. Moreover, we construct embeddings of the Wolf spares into Grassmannian_, which turn out to be minimal embeddings. We also obtain vanishing theorems for cohomology groups.In 2006, we succeeded to find some relations between harmonic mappings into Grasmannians and the Yang-Mills connections, which are essential and important steps to our subject. We obtain a condition for a map of a Riemannian manifold into Grassmannian to be a harmonic map. We use this condition to obtain the classification of harmonic maps with constant energy density from holonomy irreducible homogeneous manifold s into Grassmannian manifolds. In addition, we can show that a vector bundles on a real Grassmannian manifold with some topological type admits a unique ASD connection u … More p to gauge equivalence.In 2007, we consider the cases that a harmonic map into Grassmannian is a totally geodesic one. As a result, we obtain the classification of totally geodesic immersions of irreducible type. In this classification, we obtain an integral formula which indicates the dimension of Grassmannian, which is the target space of the mapping. In the case of the complex projective line, we can show that an indecomposable totally geodesic immersion is an totally geodesic immersion of the irreducible type. To obtain the result, we use the above characterization of a harmonic map and construct a variant of the spherical function theory on homogeneous vector bundles. This implies that we can classify all totally geodesic immersions of complex projective line into Grassmannians. We develop an analogue of the "geometry of the twistor sections" on symmetric spaces of compact type. This gives us pairs of totally geodesic submanifolds on almost symmetric spaces of compact type. These pairs are intimately related to vector bundles and sections of them. Indeed, we can construct a function using a section, which is an isoparametric function on every Grassmann manifold. This function gives a family of submanifolds as level sets. We can find one and only minimal submanifold in this family. Less

在2005年，我们把我们的注意力集中在子流形出现的理想瞬子的奇异集。这些是满足线性方程组的扭振截面的零轨迹，线性方程组是高维瞬子方程的线性化。此外，我们构造了Wolf备件到Grassmannian_的嵌入，证明了它们是极小嵌入。在2006年，我们成功地发现了到Grasmannian的调和映射与Yang-Mills联络之间的一些关系，这是我们研究这一课题的重要步骤。本文给出了黎曼流形到格拉斯曼流形的映射是调和映射的一个条件。利用这一条件，我们得到了完整不可约齐次流形到格拉斯曼流形的具有常能量密度的调和映射的分类。此外，我们还证明了具有某种拓扑类型的真实的Grassmannian流形上的向量丛存在唯一的ASD联络u ...更多信息在2007年，我们考虑了到Grassmannian的调和映射是全测地的情形。从而得到了不可约型全测地浸入的分类。在这种分类中，我们得到了一个积分公式，它指示的格拉斯曼维数，这是映射的目标空间。在复射影直线的情形下，我们可以证明一个不可分解的全测地浸入是一个不可约型的全测地浸入。为了得到这个结果，我们利用调和映射的上述特征，构造了齐次向量丛上的球函数理论的一个变体。这意味着我们可以把复射影直线的所有全测地浸入分类为Grassmannian。我们开发了一个类似的“几何的扭量部分”对称空间的紧凑型。这就给出了紧型几乎对称空间上的全测地子流形对。这些对与向量丛及其截面密切相关。事实上，我们可以用截面构造一个函数，它是每个格拉斯曼流形上的等参函数。这个函数给出一个子流形族作为水平集。我们可以在这个族中找到一个且唯一的极小子流形。少