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

我学习了辛商和里奇平面流形。在许多情况下，辛商与代数几何中的商是一样的，尽管它们的构造看起来非常不同。因此，这些商具有辛几何和代数几何中非常丰富的性质。另一方面，一般情况下很难明确地构造里奇平坦度规。虽然超kahler流形是里奇平面流形的例子，但它们被构造为超kahler商，它是辛商的类似物。所以我研究了超kahler商的几何。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。这项工作及其后续工作的部分工作得到了该基金的支持。在第二篇和第三篇文章中，我们证明了超kahler商虽然是非紧的，但它们作为紧超kahler流形几何的局部模型是重要的，并讨论了研究超kahler商的许多可能性。在前几篇文章中，我们只处理了光滑超kahler商，并用辛技术研究了它们。然而，如果我们试图将这些结果推广到非环超kahler商，就必须研究奇异超kahler商。要做到这一点，仅仅使用微分几何或辛方法是不够的。因此，我们建立了基于代数几何的方法框架来研究奇异环面超卡勒变异。这样，我们不仅简化了第一篇论文结果的证明，而且给出了更精确的描述。这些结果在《当代数学》上发表的一篇题为“环面超kahler变种的几何”的论文中得到了总结。

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.