Study of singularities and geometry by means of representation theory

通过表示论研究奇点和几何

• 批准号: 08404001
• 负责人: NAKAMURA Iku
• 金额: $ 10.43万
• 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: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08404001/
• 关键词: Representation; Singularity; Simple singularity; McKay correspondence; Abelian variety; moduli; Stability; McKay対応; マッカイ対応; 商特異点; ヒルベルト スキーム; スタビリィティ(安定性); ヒルベルトスキーム; マッケイ対応; トーリック多様体; ツイスター空間; ハイパーケーラー多様体; モノドロミ-; 横断性定理

We studied two-dimensional McKay correspondence and a canonical compactification of the moduli of abelian varieties.(I) For any finite subgroup of SL(2) we can construct a minimal resolution of a simple surface singularity CィイD12ィエD1/G as the Hilbert scheme of G-orbits. Via this construction we are able to provide a new explanation of two-dimensional McKay correspondence.(ii) We proved that for any finite abelian subgroup G of SL(3) the Hilbert scheme of G-orbits is a crepant resolution of the singularity CィイD13ィエD1/G. For simple finite subgroup G of SL(3) (there are only two such) we determined the structure of the Hilbert scheme of G-orbits.(iii) We constructed a canonical compactification SQィイD1g,NィエD1 of the moduli of abelian varieties over Z[ζィイD2NィエD2, 1/N]. Any point of SQィイD2g,NィエD2 is represented by an isomorphism class of a possibly singular abelian variety with certain level structure.

我们研究了二维McKay对应关系和Abelian变体模量的规范压实。（i）对于SL（2）的任何有限亚组，我们可以将简单的表面奇异性CII D12 D1/G构造为G-orbits的希尔伯特方案。通过这种结构，我们能够提供二维McKay对应关系的新解释。（ii）证明，对于SL（3）的任何有限的Abelian子组G（3）G-Orbits的Hilbert方案是G-Orbits的Hilbert方案，是奇异性CII CII CII CII D13E D1/g的奶油分辨率。对于SL（3）的简单有限亚组G（只有两个），我们确定了Hilbert g-Orbits方案的结构。（iii）我们构建了一个经典的紧凑型SQII D1G，NIE D1，NIE D1，ABELIAN变量的Moduli在Z [ζid2nie d2nie d2，1/N]上。 NIE D2的SQII D2G的任何点都由具有一定水平结构的可能单数Abelian品种的同构类别表示。

项目成果

Go-o Ishikawa: "Symplectic and Lagrange stabilities of open Whitney umbrellas" Invent Math.126. 215-234 (1996)

Go-o Ishikawa：“打开惠特尼伞的辛和拉格朗日稳定性”发明数学126。

Y.Ito: "Hilbert schemes and simple singularities" New trends in algebraic geometry,Cambridge Univ Press. 151-233 (1999)

Y.Ito：“希尔伯特方案和简单奇点”代数几何新趋势，剑桥大学出版社。

G. Ishikawa: "Transversalities for Langlange singularities" Singularities and Differential Equations. 33. 93-104 (1996)

G. Ishikawa：“Langlange 奇点的横截性”奇点和微分方程。

I.Nakamura: "Compactification of the moduli of abelian varieties over Z[ζ_<N1>υ_<N1>]" Comptes Rerdus Acad.Sci.Paris.327. 875-880 (1998)

I.Nakamura：“Z[z_<N1>υ_<N1>] 上阿贝尔簇模的紧化”Comptes Rerdus Acad.Sci.Paris.327 (1998)。

Y.Ito: "Mokay correspondence and Hilbert schemes" Proc.Japan Acad.72. 135-138 (1996)

Y.Ito：“Mokay 通信和希尔伯特方案”Proc.Japan Acad.72。