A general study on topology, analysis and geometry of singular spaces

奇异空间拓扑、分析和几何的一般研究

• 批准号: 15540086
• 负责人: YOKURA Shoji
• 金额: $ 2.3万
• 依托单位: Kagoshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540086/
• 关键词: singular varieties; characteristic classes; bivariant theories; motivic integrations; relative Grothendieck group; Atiyah-Singer's index theorem; mixed Hodge modules; ミルナー数; 指数定理; ミルナー類

(1)We developed a general theory of characteristic classes of proalgebiiac varieties, using the bivariant theory introduced by Fulton and MacPherson. This research seems to lead one to what Michael Gromov calles "Symbolic Algebraic Geometry".(2)In a joint research with the foreign collaborators, J.-P.Brasselet and J.Schurmann, I could succeed in "constructing a characteristic class version of Hirzebruch characteristics using Saito's mixed Hodge modules", which was proposed by the head investigator earlier.(3)In a theory obtained in the above (2), the relative Grothendieck group is a fundamental tool and it turned out that it is heavily related to the theory of motivic integrations. And as a biproduct of this work, we could obtain a kind of theory of unifying the three typical theories of characteristic classes of singular varieties.(4)Our construction of characteristic classes of proalgebraic varieties and the theory of vertex algebra in mathematical physics are in a sense related to each other. We want to make a further investigation on this point.(5)We could not construct a "Atiyah-Singer index theorem" for singular varieties, but to accomplish this final goal, in a oint research with J.Schuramnn and Markus Pflaum I will continue doing more research on the theory of characteristic classes using the relative Grothendieck groups.

(1)We利用富尔顿和麦克弗森提出的双变理论，发展了原代数簇的特征类的一般理论。这项研究似乎导致一个什么迈克尔格罗莫夫所谓的“符号代数几何”。(2)In与国外合作者的联合研究，J。P. Schurmann和J.Schurmann，我可以成功地“使用Saito的混合Hodge模块构建Hirzebruch特征的特征类版本”，这是由首席研究员早些时候提出的。(3)In相对Grothendieck群是一个基本工具，它与motivic积分理论密切相关。作为这一工作的一个双积，我们可以得到一种统一奇异簇的特征类的三个典型理论的理论。（4）前代数簇的特征类的构造与数学物理中的顶点代数理论在某种意义上是相互联系的。我们想对这一点作进一步的调查。(5)We我不能构造奇异簇的“Atiyah-Singer指标定理”，但为了实现这一最终目标，在与J.Schuramnn和Markus Pflaum的一项研究中，我将继续利用相对Grothendieck群对特征类理论做更多的研究。

项目成果

Tadashi Aikou: "A note of Riemannian manifolds with semi-parallel vector fields"Rep.Fac.Sci., Kagoshima Univ.. 36. 1-10 (2003)

Tadashi Aikou：“半平行向量场黎曼流形的注释”Rep.Fac.Sci.，鹿儿岛大学. 36. 1-10 (2003)

Generalized Ginzburg-Chern classes

广义Ginzburg-Chern 类

• 发表时间: 2005
• 期刊: Semiinaires et Congres, Soc.Math.France 10(in press)
• 作者: K.Miyajima;S.Yokura;S.Tsuboi;S.Tsuboi;S.Tsuboi;S.Tsuboi;S.Tsuboi;S.Tsuboi;S.Tsuboi;S.Tsuboi;S.Tsuboi;S.Yokura;S.Tsuboi;S.Yokura
• 通讯作者: S.Yokura

小柴洋一: "連続関数のRiemann積分可能性と一様連続の概念について"2003年度日本数学会秋季総合分科会歴史分科会アブストラクト. 16-16 (2003)

小芝洋一：《论连续函数的黎曼可积性和一致连续性的概念》2003年日本数学会秋季综合分会历史分会摘要16-16（2003）。

A note on Riemannian manifolds with semi-parallel vector fields

关于具有半平行向量场的黎曼流形的注解

• 发表时间: 2003
• 期刊: Rep.Fac.Sci.Kagoshima Univ. 36
• 作者: T.Aikou;T.Aikou;T.Aikou;T.Aikou;T.Aikou;T.Aikou;T.Aikou;T.Aikou
• 通讯作者: T.Aikou

Kimio Miyajima: "Analytic approach to deformation of resolution of normal isolated singularities : Formal deformations"J.Korean Math.Soc.. 40・4. 709-725 (2003)

Kimio Miyajima：“正常孤立奇点分辨率变形的分析方法：形式变形”J.Korean Math.Soc.. 709-725 (2003)。