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Comprehensive Study of Stable Bundles on Calabi-Yau Manifolds

卡拉比-丘流形上稳定丛的综合研究

基本信息

批准号：
13640035
负责人：
NAKASHIMA Tohru
金额：
$ 0.51万
依托单位：
Tokyo Metropolitan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640035/
关键词：
Calabi-Yau manifold stable vector bundle moduli space 代数幾何学

项目摘要

In this research project we planned to solve the existence problem of stable bundles on Calabi-Yau manifolds and to clarify the geometric structure of their moduli spaces. We obtained the following results concerning these subjects.As to the existence problem, we proved that the extension sheaf of two stable sheaves is again stable, under certain minimality assumption on their first Chern class. As a result, one may construct stable sheaves inductively from sheaves of lower rank. In particular, the method yields stable bundles by means of elementary transformation from globally generated line bundles on a divisor. Until recently, methods of explicit construction of stable bundles has been known only for elliptic Calabi-Yau manifolds, while our work enables us to find stable bundles on arbitrary Calabi-Yau manifolds in principle.Concerning the geometry of moduli space, we determined their birational structure in many cases. More explicity, we proved that the reflection functor defines an isomorphism between the Brill-Noether loci of the moduli spaces with different Mukai vectors (the Brill-Noether duality), which is a higher dimensional generalization of the result due to Markman-Yoshioka in the case of K3 surface. Exploiting the Brill-Noether duality, one deduces that moduli spaces have a component which is birational to the Grassmann bundle over moduli space of sheaves of lower rank. Our method applies to other varieties which are not necessarily Calabi-Yau. For example, we determined the birational structure of the moduli space of stable sheaves on certain threefolds with Del-Pezzo fibrations.

在这个研究项目中，我们计划解决Calabi-Yau流形上稳定丛的存在性问题，并阐明它们的模空间的几何结构。在存在性问题上，我们证明了两个稳定层的延伸层在其第一个陈类的极小假设下是稳定的。因此，人们可以从较低等级的叶轮感应地构造稳定的叶轮。特别地，该方法通过从全局生成的除数上的线丛进行初等变换来产生稳定的线丛。直到最近，稳定丛的显式构造方法还仅限于椭圆Calabi-Yau流形，而我们的工作原则上使我们能够在任意的Calabi-Yau流形上找到稳定丛。结合模空间的几何，我们在许多情况下确定了它们的双胞结构。更具体地，我们证明了反射函子定义了具有不同Mukai向量的模空间的Brill-Noether轨迹之间的同构(Brill-Noether对偶)，这是Markman-Yoshioka的结果在K3曲面情况下的高维推广。利用Brill-Noether对偶性，我们推导出模空间的一个分支是低阶层的模空间上的Grassmann丛的二次分支。我们的方法适用于不一定是Calabi-Yau的其他品种。例如，我们用Del-Pezzo纤维确定了某些三重结构上稳定层的模空间的双态结构。