Residues on Singular Varieties

单一品种的残留

• 批准号: 18340015
• 负责人: SUWA Tatsuo
• 金额: $ 3.92万
• 依托单位: Niigata University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18340015/
• 关键词: singular varieties; localization of characteristic classes; residues; Chern classes; Atiyah classes; holomorphic foliations; complex dynamical systems; intersection theory; 局所Euler障害

The head investigator and the others did research on residues on singular varieties and related subjects. More specifically:1. We developed a theory of localization of Chern classes by frames of vector bundles on singular varieties. In the previous year, we gave explicit expressions (analytic, algebraic and topological)of the residues at an isolated singularity. In this research, we gave an expression in the case the singularity is not isolated.2. As an application of 1 above, we developed an analytic intersection theory on singular varieties. This clarifies global intersections, local intersections and the relation between the two. In the global case, the localization theory of Chern classes is very effective and in the local case, Grothendieck residues on singular varieties play an essential role. The two situations are related by the residue theorem.3. As a summary of collaboration with J.-P. Brasselet and J. Seade, we almost finished writing a book on the characteristic classes of singular varieties utilizing indices and residues of vector fields. This also includes a new simple proof of the Proportionality Theorem, which describes a fundamental property of the local Euler obstruction of singular varieties.4. In the collaboration with M. Abate, F. Bracci and F. Tovena, we started to construct a localization theory of Atiyah classes of holomorphic vector bundles. This theory is expected to be very interesting and have many applications.5. Besides the above, Ohmoto obtained important results on the characteristic classes of varieties with group actions, Oka on the fundamental group of the complement of algebraic curves, Saito on Lie algebras and singularities, Tajima on Milnor and Tjurina numbers, Yokura on motivic characteristic classes, respectively.

首席调查员和其他人对奇异品种和相关主题的残留物进行了研究。更具体地说：1。我们通过奇异品种上的矢量束框架开发了Chern类定位的理论。在上一年，我们在孤立的奇异性下给出了残基的明确表达（分析，代数和拓扑）。在这项研究中，我们在这种情况下表达了一个奇异性。2。作为上述1的应用，我们开发了关于单数品种的分析交集理论。这阐明了全球交叉点，局部交集以及两者之间的关系。在全球案例中，Chern类的本地化理论非常有效，在当地情况下，关于奇异品种的Grothendieck残基起着至关重要的作用。这两种情况与残基定理相关3。作为与J.-P。合作的摘要Brasselet和J. Seade，我们几乎完成了一本关于利用矢量领域索引和残留物的奇异品种的特征类别的书。这还包括一个新的简单证据，证明了比例定理，该证明描述了当地欧拉障碍物的基本特性。4。在与Abate，F。Bracci和F. Tovena的合作中，我们开始构建一个atiyah类别类别的Holomorphic Vector Bundles的本地化理论。预计该理论将非常有趣，并且具有许多应用。5。除上述内容外，ohmoto还获得了具有小组动作的特征性类别的重要结果，OKA在代数曲线的基本群体上，对代数和奇异性的saito，Milnor和Tjurina数字上的Tajima，Yokura的tajima，Yokura的动机特征类别。

项目成果

Proportionality of indices of 1-forms on singular varieties

奇异变体上 1-形式指数的比例性

• 发表时间: 2007
• 期刊: Advanced Studies in Pure Mathematicss 46
• 作者: T.Suwa;J.-P.Brasselet;J.Seade
• 通讯作者: J.Seade

Singularities in Geometry and Topology

几何和拓扑中的奇点

• 发表时间: 2007
• 作者: Tomohiro Ogawa;Hiroshi Nagaoka;J. -P. Brasselet and T. Suwa
• 通讯作者: J. -P. Brasselet and T. Suwa

Introduction to Complex Analytic Geometry

复杂解析几何简介

• 发表时间: 2007
• 期刊: Proc. Trieste Singularity Summer School and Workshop (ICTP, Trieste, 2005) ed. J-P.Brasselet et.al., World Scientific
• 作者: A. A. Davydov;G. Ishikawa;S. Izumiya;W.-Z. Sun;G. Ishikawa;大本 亨;S. Yokura;G. Ishikawa;G. Ishikawa;T. Akita;T. Akita;T. Suwa;T. Ohmoto;T. Suwa
• 通讯作者: T. Suwa

Localization of Atiyah classes

Atiyah 类的本地化

• 发表时间: 2007
• 作者: T. Akita;T. Akita;T. Suwa
• 通讯作者: T. Suwa

Residue Theoretical Approach to Intersection Theory

相交理论的剩余理论方法

• 发表时间: 2008
• 期刊: Proceedings of IX Workshop on Real and Complex Singularities, Sao Carlos, Brazil, Contemporary Math., AMS.(55 pages). (to appear)
• 作者: A. A. Davydov;G. Ishikawa;S. Izumiya;W.-Z. Sun;G. Ishikawa;大本 亨;S. Yokura;G. Ishikawa;G. Ishikawa;T. Akita;T. Akita;T. Suwa
• 通讯作者: T. Suwa