A STUDY ON THE GEOMETRY OF MODULI SPACES

模空间几何的研究

基本信息

批准号：
12304001
负责人：
NAKAMURA Iku
金额：
$ 19.16万
依托单位：
HOKKAIDO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12304001/
关键词：
Abelian variety Moduli Compactification McKay correspondence Theta function Calabi-Yau Manifolds Coinvariant algebra Quiver variety テータ関数マッカイ対応余不変式環 Abel多様体 Griffiths領域 Floer homology 一般化Whittaker模型楕円曲線レベルN構造 Abel 多様体

项目摘要

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

研究了Abelian品种模量空间的某些紧凑型以及用于SL（2，C）和SL（3，C）的有限亚组G的G-Orbits的模量空间。我们想到的主要问题如下（a）研究商的奇异性C^3/g作为模量空间的分辨率（b）对Moduli空间的KEMPF稳定性和压实的研究（c）规范化的sq_ <g，n> abelian verian and abelian and abelian and abelian and abelian nibie sq_ <g，n]在这个项目中的每个主题。主要结果如下：首先，关于G-Orbits希尔伯特计划的研究取得了显着进展。我们对二十年来发现的McKay通信现象给出了新的解释，并将其扩展到三维情况，我们获得了许多新的改造。首席研究员（Nakamura）提出了McKay对应于三个或更高维度的对应关系的概括，这是许多相关结果所遵循的。从这个意义上讲，这个项目在研究麦凯信函的历史中发挥了重要作用。 Nakamura表明，G-Orbits的Hilbert方案是对商C^3/g的奇异性的规范分辨率。这是一个从未观察到的新发现，反对最小模型理论的常识。因此，这一发现已被惊喜的专家接受。该项目的另一个实质性贡献是，我们建立了一种新的典型的压实，用于模量空间a_ <g，n> abelian品种的这种压缩是射影的，它具有理想的特性作为紧凑型。从不变理论的稳定点来看，这种压实是通过稳定性的。从这个意义上说，它是东正教，是由此特性的独特特征