Residues on Singular Varieties

首席调查员和其他人对单一品种和相关主题的残留进行了研究。更具体地说：1.利用奇异簇上向量丛的框架，建立了陈氏类的局部化理论。在前一年，我们给出了孤立奇点上的留数的显式表达式(解析的、代数的和拓扑的)。在本研究中，我们给出了在奇点不是孤立的情况下的表达式。作为上述1的一个应用，我们发展了关于奇异簇的解析交理论。这澄清了全局交叉口、局部交叉口以及两者之间的关系。在全局情形下，Chern类的局部化理论是非常有效的，而在局部情形下，奇异变种上的Grothendieck残差起着至关重要的作用。这两种情况通过留数定理联系在一起。作为与J.P.Brasselet和J.Seade合作的总结，我们几乎写完了一本关于利用向量场的指数和剩余来刻画奇异变种的特征类的书。这也包括了一个新的简单的比例定理的证明，该定理描述了奇异变量的局部欧拉阻塞的一个基本性质。在M.Abate，F.Bracci和F.Tovena的合作下，我们开始构造全纯向量丛的Atiyah类的局部化理论。预计这一理论将非常有趣，并有许多应用。此外，Ohmoto关于群作用簇的特征类，Oka关于代数曲线补的基本群，Saito关于李代数和奇点，Tajima关于Milnor和Tjuina数，Yokura关于动机特征类分别得到了重要的结果。

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.