Studies on Characteristic Classes of Singular Varieties, Motives ans Their Related Topics

奇异品种的特征类、动机及其相关话题研究

• 批准号: 12640081
• 负责人: YOKURA Shoji
• 金额: $ 2.24万
• 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640081/
• 关键词: singular variety; characteristic class; motief; bivariant theory; representation theory; Riemann-Roch; ミルナー類; 同変チャーン類

(1). In a joint work with Lars Emstrom we showed the unique existence of the bivariant Chern class with values in the bivariant Chow groups(2). A blow-up map is a local complete intersection morphism. However, a Verdier-type Riemann-Roch does not hold for this blow-up map. And motivated by this result we showed several other otherresults.(3).We obserbved that there exist various bivariant constructive functions other than those of Fulton and MacPherson. For example, any constructible function itself can be a bivariant on without imposing any geometric or topological condition on it. With this we point out that a statement made by Fulton and macPherson in their book (Categorical frameworks for the study of singular spaces) is false and furthermore we gave a modified statement of it and etc.(4). We made several remarks on the so-called Ginzburg-Chern class introduced by Victor Ginzburg in Geometric Representation Theory.(5). Moivated by the results in (4), we showed axiomatically that if … More there exist a bivariant chern class from the bivariant constructible function to the bivariant homology theory and if we restrict ourselves to the morphisms to nonsingular varieties the bivariant Chen class is unique and furthermore we showed that it is nothing but the Ginzburg-Chern class.(6). Based on the results in (5), we investigated categories of morphisms in which the Ginzburg-Chern class can be captured as a bivariant Chern class, in particular we treated smooth morphisms between nonsingular varieties.(7).Furthermore we consider morphisms with target varieties being nonsingular, and we defined another group of bivariant consructible functions which is larger than that of Fulton-MacPherson's bivariant constructible functions and we showed that the Ginzburg-Chern class can be captured as a bivariant Chern class from this bivariant constructible function to the bivariant homology theory. And in the case of arbitrary morphisms, motivated by the results obtsained in (7), we introduced formal bivariant Chern classes and we are investigating them further. Less

（1）。在与Lars Emstrom的共同作品中，我们展示了双变量Chern类的独特存在，其值在双变量的Chow群中（2）。爆炸图是局部完整的交叉态度。但是，Verdier型Riemann-Roch在此爆炸地图上不满意。通过这种结果，我们展示了其他几个其他结果。（3）。我们观察到，除了富尔顿和麦克弗森（Fulton）和麦克弗森（MacPherson）以外，还有各种双变量的建设性功能。例如，任何可构造函数本身都可以是双变量的，而不会在其上施加任何几何或拓扑条件。因此，我们指出，富尔顿和麦克弗森在他们的书中发表的陈述（用于研究单数空间的分类框架）是错误的，而且我们给出了修改的陈述，等等。（4）。我们对维克多·金茨堡（Victor Ginzburg）在几何代表理论中提出的所谓的金茨堡 - 切尔恩（Ginzburg-Chern）阶级（5）发表了几句话。（5）。在（4）中的结果中，我们在公理上表明，如果……更多的是一个双变量的樱桃，从双变量的构造函数到双变量同源理论，如果我们将自己限制在形态上到非义化的形态到非发挥品种，那么bivariant chen chen类是独特的，我们表明了它是独一无二的，除了麦格尔格 - 吉尔格格尔格（Ginzburg-Chern）。 Based on the results in (5), we investigated categories of morphisms in which the Ginzburg-Chern class can be captured as a bivariant Chern class, in particular we treated smooth morphisms between nonsingular variances.(7).Furthermore we consider morphisms with target variances being nonsingular, and we defined another group of bivariant constructible functions which is larger than that of富尔顿·麦克弗森（Fulton-Macpherson）的双变量可构造功能，我们表明，从这个双变量构造函数到双变量同源性理论，金茨堡 - 切尔恩阶级可以被捕获为双变量的chern类。如果是由（7）中获得的结果激发的任意形态，我们引入了正式的双变量Chern类，我们正在进一步研究它们。较少的

项目成果

Tsuboi,S. and Guillen,F.: "Simultaneous Cubic Hyper-resolutions of Locally Trivial Analytic Families of Complex Projective Varieties and Cohomological Descent, The Reports of the Faculty of Science"Kagoshima University. No. 33. 1-33 (2000)

坪井，S.

Shoji Yokura: "Bivariant theories of constructible functions and Grothendieck transformations"Topology and Its Applications. (印刷中). 14 (2001)

Shoji Yokura：“可构造函数和格洛腾迪克变换的双变理论”拓扑及其应用（出版中）。

J.-P.Brasselet and Shoji Yokura: "Remarks on bivariant constructible functions,"Advanced Studies in Pure Mathematics. 29. 53-77 (2000)

J.-P.Brasselet 和 Shoji Yokura：“关于双变量可构造函数的评论”，纯数学高级研究。

Shoji Yokura: "Verdier-Riemann-Roch for Chern class and Milnor class"Erwin Schrodinger Institute Preprint Series. 933. 24 (2000)

Shoji Yokura：“Verdier-Riemann-Roch for Chern class and Milnor class”埃尔温·薛定谔研究所预印本系列。

Shoji Tsuboi: "A Certain Degenerate Ordinary Singularity of Dimension Three"Proceedings of the Eighth International Conference on Finite or Infinite Dimensional Complex Analysis, Shandon University. (印刷中).

Shoji Tsuboi：“第三维的某种退化普通奇异性”第八届国际有限或无限维复分析会议论文集，山东大学。