权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Algebraic and Geometric Study on the Structure of Moduli Spaces

模空间结构的代数和几何研究

基本信息

批准号：
05452003
负责人：
MARUYAMA Masaki
金额：
$ 4.29万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (B)
财政年份：
1993
资助国家：
日本
起止时间：
1993 至 1994
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-05452003/
关键词：
Moduli Vector Bundle Stable Sheaf Hermit-Einstein connection Instanton Parabolic Stable Sheaf Betti Number Projective Plane 位相構造代数多様体 Weil予想

项目摘要

Moduli in geometry is a set of geometric objects endowed with the universal geometric structure. It is known that not only a moduli space itself is a rich geometric object but also it is often a useful tool in studying geometry. For example the moduli space of Hermit-Nesting connections reflects strongly the differential geometric nature of the base manifold. On the other hand, as the fact that a Hermit-Nesting connection is nothing but a stable vector bundle shows us, we realize that moduli spaces constructed independently in different fields sometimes coincide with each other. This has been promoting direction of studying the theory of moduli spaces from various viewpoints. In this project we have carried out our study on classifying spaces, moduli of various connections and moduli of vector bundles, in cooperation with the specialists of topology, differential geometry, number theory, algebraic geometry and commutative algebra in our department. We have got the following results.1.We completed the computation of Betty numbers of the moduli spaces of stable sheaves of rank 2 on the projective plane and furthermore we got a similar results on ruled surfaces.2.We could clarify an interesting relationship between parabolic stable vector bundles on the projective plane and instantaneous. Using this we could prove that the moduli spaces of instantaneous are connected.3.The standard of the moduli spaces of stable sheaves of rank 2 the on projective plane are dominated by those of the moduli spaces of parabolic stable vector bundles. They are related under a generalization of the elementary transformation of vector bundles.4.We could develop deep study of reflexive sheaves on surfaces with rational double points and their deformations.5.We applied our results on vector bundles to the theory of conformal field theory.

几何中的模是一组被赋予普遍几何结构的几何对象。模空间不仅是一个丰富的几何对象，而且是研究几何的一个有用工具。例如，Hermit-Nesting联络的模空间强烈地反映了基流形的微分几何性质。另一方面，由于Hermit-Nesting联络只不过是一个稳定的向量丛，我们认识到，在不同的领域中独立构造的模空间有时是相互重合的。这为从多角度研究模空间理论提供了一个新的方向。在这个项目中，我们与本系的拓扑学、微分几何学、数论、代数几何学和交换代数学的专家合作，对空间的分类、各种联络的模和向量丛的模进行了研究。得到了如下结果：1.计算了射影平面上秩为2的稳定层的模空间的Betty数，并在直纹面上得到了类似的结果; 2.阐明了射影平面上抛物稳定向量丛与瞬时值之间的一个有趣的关系. 3.射影平面上秩为2的稳定层的模空间的标准由抛物稳定向量丛的模空间的标准支配。它们在向量丛的初等变换的推广下相互关联。4.我们可以对具有有理二重点的曲面上的自反层及其变形进行深入的研究。5.我们将向量丛的结果应用到共形场论理论中。