Comprehensive Study of Stable Bundles on Calabi-Yau Manifolds

在本研究计划中，我们计划解决Calabi-Yau流形上稳定丛的存在性问题，并阐明其模空间的几何结构。关于存在性问题，我们证明了两个稳定层的扩张层在其第一个Chern类的极小性假设下仍然是稳定的。因此，人们可以从较低秩的层归纳地构造稳定的层。特别是，该方法产生稳定的束通过初等变换从整体生成的线丛上的一个因子。直到最近，稳定丛的显式构造方法仅限于椭圆型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，关于模空间的几何，我们在许多情况下确定了它们的双有理结构.更明确地，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在K3曲面情形下的结果的高维推广.利用Brill-Noether对偶性，我们推导出模空间有一个分支是低阶层模空间上的Grassmann丛的双有理分支。我们的方法适用于其他品种，不一定是卡-丘。例如，我们确定了具有Del-Pezzo纤维化的稳定层在某些三重上的模空间的双有理结构。

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.