Residues on Singular Varieties

首席研究员和其他人对单一品种和相关科目的残留进行了研究。更具体地说:1。提出了奇异变异上Chern类的向量束框架局部化理论。在前一年，我们给出了孤立奇点处的残数的显式表达式（解析式、代数式和拓扑式）。在本研究中，我们给出了奇异点非孤立情况下的表达式。作为上面1的一个应用，我们发展了奇异变量的解析交理论。这阐明了全局交叉点、局部交叉点以及两者之间的关系。在全局情况下，Chern类的局部化理论是非常有效的，而在局部情况下，奇异变种上的Grothendieck残数起着至关重要的作用。这两种情况是由剩余定理联系起来的。作为与j.p合作的总结。Brasselet和J. Seade，我们几乎完成了一本关于利用向量场的指标和残数的奇异变的特征类的书。这也包括了比例定理的一个新的简单证明，它描述了奇异变异体局部欧拉阻塞的一个基本性质。在与M. Abate， F. Bracci和F. Tovena的合作中，我们开始构建全纯向量束的Atiyah类的局部化理论。这一理论预计会非常有趣，并有许多应用。此外，Ohmoto在群作用的变体的特征类上取得了重要成果，Oka在代数曲线补的基本群上取得了重要成果，Saito在Lie代数和奇点上取得了重要成果，Tajima在Milnor数和Tjurina数上取得了重要成果，Yokura在动机特征类上取得了重要成果。

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.