A STUDY ON THE GEOMETRY OF MODULI SPACES

模空间几何的研究

• 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

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SL（2，C）和SL（3，C）的有限子群G的交换簇的模空间的紧化以及G-轨道的模空间。主要内容有：（a）商C^3/G作为模空间的奇异性分解研究（B）模空间的Kempf稳定性和紧化研究（c）模A_<g，N>在Z[1/N]上的交换簇及相关模的规范紧化SQ_<g，N>.主要结果如下：第一，G-轨道的Hilbert格式的研究取得了显著的进展。本文对二十多年来发现的McKay对应现象作了新的解释，并将其推广到三维情形，得到了许多新的结果。首席研究员（中村）提出了一个推广的麦凯对应的三维或更高的维度，这是遵循许多相关的结果。从这个意义上说，这个项目支付了一个实质性的作用，在历史上研究麦凯的信件。除其他事项外，中村表明，G轨道的希尔伯特方案是商C^3/G的奇点的正则分解。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讶地接受了。本项目的另一个重要贡献是构造了阿贝尔簇模空间A_<g，N>的一个新的标准紧化。从不变量理论的稳定点出发，这种紧化只是稳定的紧化。在这个意义上说，它是正统的，是独特的特点，这一财产

项目成果

Kaoru Ono: "Space of geodesics on Zoll three spheres"Advanced Studies in Pure Math.. 34. 237-243 (2002)

小野薰：“佐尔三球体上的测地线空间”纯数学高级研究.. 34. 237-243 (2002)

Iku Nakamura: "The moduli space of elliptic curves with Heisenberg structure"Proceedings of Texel conference 1999, Progress in Math., Birkh\" auser. 195. 299-324 (2001)

Iku Nakamura：“具有海森堡结构的椭圆曲线的模空间”Proceedings of Texel Conference 1999, Progress in Math., Birkh" auser. 195. 299-324 (2001)

Kaoru Ono: "Space of geodesics of Zoll 3-spheres, submitted to the proceedings of JAMI conference at Johns Hopkins University March 1999"In press in Advanced Studies in Pure Mathematics.

Kaoru Ono：“Zoll 3 球体的测地线空间，提交给 1999 年 3 月在约翰·霍普金斯大学举办的 JAMI 会议论文集”，发表在《纯数学高级研究》上。

T.Katsura: "Formal Brauer groups and a stratification of the moduli of abelian surfaces"Progress in Math.. 195. 185-202 (2001)

T.Katsura：“形式布劳尔群和阿贝尔曲面模的分层”数学进展.. 195. 185-202 (2001)

I.Nakamura: "Coinvariant algebras of finite subgroups of SL(3,C)"Canadian Jour.Mathematics. (印刷中).

I. Nakamura：“SL(3,C) 有限子群的协变代数”加拿大数学杂志（正在出版）。