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

点击翻译按钮获取中文摘要

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.