A STUDY ON THE GEOMETRY OF MODULI SPACES

研究了SL（2，C）和SL（3，C）的有限子群G的阿贝尔变体模空间的紧化性以及G轨道的模空间。我们所考虑的主要问题是：(a)商C^3/G作为模空间的奇异解的研究；(b)模空间的Kempf稳定性和紧化的研究；(C)阿贝尔变体和相关模的正则紧化SQ_< G，N>，N> / Z[1/N]。在这个项目中，每个学科都取得了显著的进步。主要结果如下：第一，g轨道希尔伯特格式的研究取得了显著进展。我们可以对二十多年来发现的麦凯对应现象给出新的解释，并将其推广到三维情况下，得到许多新的结果。首席研究员（Nakamura）提出了McKay对应到三维或更高维度的概括，随后有许多相关的结果。从这个意义上说，这个项目在研究麦凯函授的历史上发挥了重要作用。在其他事情中，Nakamura证明了G轨道的希尔伯特格式是商C^3/G奇点的标准解析。这是一个从未被观察到的新发现，违背了最小模型理论的常识。因此，这一发现被专家们惊奇地接受了。本课题的另一个重要贡献是构造了一个新的模空间A_<g，N>的正则紧化，该紧化是射影的，具有理想的紧化性质。从不变量理论的立足点来看，这个紧化就是稳定性紧化。从这个意义上说，它是正统的，具有这种性质的独特特征

Certain compactification of moduli space of abelian varieties was studied as well as moduli spaces of G-orbits for a finite subgroup G of SL(2,C) and SL(3,C). The main issues we have in mind are as follows (a) Study of a resolution of singularity of the quotient C^3/G as a moduli space (b) study of Kempf stability and compactification of moduli spaces (c) A canonical ompactification SQ_<g,N> of the moduli A_<g,N> over Z[1/N] of abelian varieties and related moduli.There were remarkable progresses on each subject during this project. The main results are as follows : first there was a remarkable progress in the study on Hilbert schemes of G-orbits. We copuld give a new explanation to the phenomenon of McKay correspondence which was discovered over twenty years, and extending it to the three dimensional case, we obtained a lot of new resluts. The head investigator (Nakamura) proposed a generalization of McKay correspondence to the three or higher dimension, which was follows by many related results. In this sense this project payed a substantial role in the history of studying McKay correspondence. Among other things Nakamura showed that the Hilbert scheme of G-orbits is the canonical resolution of singularities of the quotient C^3/G. This is a new discovery which has never been observed, against the common sense in minimal model theory. Therefore this discovery has been accepted by specialists with surprise. Another substantial contribution of this project is that we constructed a new canonical compactification of moduli space A_<g,N> of abelian varieties This compactification is projective, it enjoys a desirable property as a compactification. From the stabdpoint of invariant theory, this compactification is ust that by stability. In this sense it is orthodox and is uniquely characterized by this property