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.

几何形状中的模量是赋予通用几何结构的一组几何对象。众所周知，不仅模量空间本身是一个丰富的几何对象，而且通常是研究几何形状的有用工具。例如，隐居连接的模量空间强烈反映了基本歧管的差异几何特性。另一方面，由于隐居纽约连接不过是一个稳定的向量捆绑包，因此我们意识到，在不同领域独立构建的模子空间有时彼此相吻合。这一直在促进从各种观点研究模量空间理论的方向。在这个项目中，我们与我们部门的拓扑，差异几何学，数字理论，代数几何形状和交换代数的专家合作，对矢量束的分类空间，各种连接的模量和模量进行了研究。我们已经得到以下结果。1。我们完成了稳定平面上稳定滑轮的贝蒂数量的计算，并在投射平面上排名第二，此外，我们在规范的表面上得到了类似的结果。2。我们可以澄清抛物面稳定的稳定矢量在投射平面上和瞬间的抛物面稳定矢量之间的有趣关系。使用此方法，我们可以证明瞬时的模量空间是连接的。3。等级2的稳定皮带的模量空间的标准，射线平面由抛物线稳定矢量套件的Moduli空间所主导。它们是在矢量束的基本变换的概括下相关的。4。我们可以对具有理性双点及其变形的表面上的反身滑轮进行深入研究。5。我们将矢量束的结果应用于结构范围理论。