Residues on Singular Varieties

首席研究员和其他人对单一品种和相关主题的残留物进行了研究。更具体地说：1.我们发展了一个利用奇异簇上向量丛框架的Chern类局部化理论。在前一年，我们给出了明确的表达式（解析，代数和拓扑）的残留物在一个孤立的奇点。在本研究中，我们给出了奇异性不是孤立的情况下的一个表达式.作为上述[1]的应用，我们发展了奇异簇的解析交理论。这阐明了全局交叉点、局部交叉点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，奇异簇上的Grothendieck剩余起着至关重要的作用。这两种情况通过留数定理联系起来.作为与J. - P. Atelet和J. Seade，我们几乎完成了一本关于奇异簇的特征类的书，利用向量场的指数和剩余。这也包括一个新的简单证明的比例定理，它描述了一个基本属性的局部欧拉阻塞的奇异变量。在与M. Abate，F. Bracci和F. Tovena，我们开始构建全纯向量丛的Atiyah类的局部化理论。这一理论预计将是非常有趣的，并有许多应用。除了上述，Ohmoto获得重要成果的特征类品种与群行动，奥卡的基本群体的补充代数曲线，齐藤李代数和奇异性，田岛的Milnor和Tjurina数，横仓motivic特征类，分别。

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.