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Study of Moduli and its Applications

模量研究及其应用

基本信息

批准号：
10304002
负责人：
MARUYAMA Masaki
金额：
$ 15.04万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A).
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10304002/
关键词：
stable vector bundle stable sheaf moduli Bogomolov's inequality reflexive sheaf simple complex boundedness exceptional line 代数的空間ベクトル束 Bogomolov不等式変形 Bogomolovの不等式

项目摘要

The results we have obtained are the following.1) The curve of the jumping lines of second kind of a stable vector bundle of rank 2 on the projective plane deeply reflects the structure of the vector bundle. We found a way to determine the multiplicity and the tangent cone at a singular pint of the curve.2) We showed a close relationship between an ordinary double point on a surface and the space of deformations of a reflexive sheaf on it.3) We generalized Bogomolov's inequality and found an interesting relationship with the moduli space of stable curves.4) Stable sheaves on a degenerating family of varieties were studied and gave a way to find the structure of the moduli space of stable sheaves on a nonsingular variety through a combination of simpler varieties.5) We constructed families of surfaces on which there is a polarization such that the dimension of the moduli space of stable vector bundles is bigger than expected even for very big second Chern classes.6) One of the subjects expected further development is the moduli space of complexes of sheaves. To get good moduli we have to give a proper definition of stable complex. Instead of this we introduced the notion of simple complex and constructed their moduli space in the category of algebraic spaces.7) One of the most important problem is the boundedness of semistable sheaves in positive characteristic cases. In fact, this has been the main target to be solved in this project. The boundedness for a surface was proved in 1973 and for the rank 2, 3 cases in 1980 by M.Maruyama. After 20 years of no progress, we could take a step forward, that is, succeeded in proving the rank 4 case. Unfortunately the method we took cannot be generalized to higher rank cases directly and we nay have to find another way to complete the proof for general cases.

得到了以下结果：1）秩为2的稳定向量丛的第二类跳线在射影平面上的曲线深刻地反映了向量丛的结构。2）证明了曲面上的普通双点与曲面上的自反层的变形空间之间的密切关系; 3）推广了Bogomolov不等式，发现了它与稳定曲线的模空间之间的一种有趣的关系; 4）证明了曲面上的普通双点与曲面上的自反层的变形空间之间的关系研究了退化簇族上的稳定层，给出了通过简单簇的组合来寻找非奇异簇上稳定层的模空间结构的方法。我们构造了曲面族，在这些曲面上存在极化，使得稳定向量丛的模空间的维数大于预期，即使对于非常大的第二陈类。层复形的模空间是有待进一步发展的课题之一。为了得到好的模，我们必须给出稳定复形的适当定义。我们引入了单复形的概念，并在代数空间的范畴内构造了它们的模空间。7）其中最重要的问题之一是半稳定层在正特征情形下的有界性。实际上，这一直是本项目要解决的主要目标。M. Maruyama在1973年证明了曲面的有界性，在1980年证明了秩为2和秩为3的曲面的有界性。在20年毫无进展的情况下，我们可以向前迈出一步，那就是成功地证明了四级案件。不幸的是，我们采取的方法不能直接推广到更高秩的情况下，我们可能不得不找到另一种方法来完成一般情况下的证明。