Ricci-flat manifolds and the global structure of their moduli spaces

里奇平坦流形及其模空间的全局结构

基本信息

批准号：
15540062
负责人：
KONNO Hiroshi
金额：
$ 2.3万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

I studied symplectic quotients and Ricci-flat manifolds. In many cases symplectic quotients turn out to be the same as quotients in algebraic geometry, although their construction seems very different. As a result, these quotients have very rich properties from symplectic geometry as well as algebraic geometry. On the other hand, it is difficult to construct Ricci-flat metric explicitly in general. Although hyperkahler manifolds are examples of Ricci-flat manifolds, they are constructed as hyperkahler quotients, which are analogues of symplectic quotients. So I studied the geometry of hyperkahler quotients. (89 words)In the first paper, I described the variation of hyperkahler structures of toric hyperkahler manifolds. Part of this work and its subsequent works were supported by this fund. In the second and third articles, we showed that, although hyperkahler quotients are non-compact, they are important as local models of the geometry of compact hyperkahler manifolds, and we also discussed many possibilities of the study of hyperkahler quotients. In the first articles, we treated only smooth hyperkahler quotients, and studied them by symplectic techniques. However, if we try to generalize these results to non-toric hyperkahler quotients, it is necessary to study singular hyperkahler quotients. To do that, it is not enough to use only differential geometric or symplectic methods. So we developed the framework of the method based on algebraic geometry to study singular toric hyperkahler varieties. Thus we succeeded in not only simplifying the proof of the results in the first paper, but also giving more precise descriptions. These results were summarized in a paper "The geometry of toric hyperkahler varieties", which was submitted to Contemporary Math.

我研究了辛商和里奇平坦流形。在许多情况下，辛商与代数几何中的商相同，尽管它们的结构似乎非常不同。因此，这些商具有非常丰富的辛几何和代数几何性质。另一方面，一般来说，明确地构造 Ricci 平坦度量是很困难的。尽管超卡勒流形是利玛窦平坦流形的示例，但它们被构造为超卡勒商，这是辛商的类似物。所以我研究了超卡勒商的几何。 (89字)在第一篇论文中，我描述了环面超卡勒流形的超卡勒结构的变化。本研究的部分工作及其后续工作得到了该基金的支持。在第二篇和第三篇文章中，我们表明，虽然超卡勒商是非紧的，但它们作为紧超卡勒流形几何的局部模型很重要，并且我们还讨论了研究超卡勒商的许多可能性。在第一篇文章中，我们仅处理平滑的超卡勒商，并通过辛技术对其进行研究。然而，如果我们尝试将这些结果推广到非复曲面超卡勒商，则有必要研究奇异超卡勒商。为此，仅使用微分几何或辛方法是不够的。因此我们开发了基于代数几何的方法框架来研究奇异复曲面超卡勒簇。这样我们不仅成功地简化了第一篇论文结果的证明，而且给出了更精确的描述。这些结果总结在一篇论文“The Geometry of toric hyperkahler ×”中，该论文已提交给《Contemporary Math》。