Residues on Singular Varieties

首席研究员和其他人对单一品种的残留物进行了研究。1.建立了向量丛的Chern类在奇异簇上关于截集的剩余理论。给出了孤立奇点上剩余的解析、代数和拓扑表达式. 2.引入了奇异簇上函数重数的概念，并证明了Riemann曲面上全纯映射的Iversen公式的推广. 3.在顶Chern类中，Thom类包含局部信息，并产生解析.通过Bochner-Martinelli内核的代数和拓扑不变量。我们发现了其他陈省身类的“中间Thom类”。P. Atelet和J.Seade，我们证明了奇异变元上1-形式的局部Euler阻塞的“比例定理”。5.与F.Bracci合作，我们证明了奇异复曲面的全纯自映射的不动点上抛物曲线的存在性，作为我们的留数理论的应用。6.除此之外，Ito在Poincare-Hopf型定理、Ohmoto在代数栈的特征类、Oka在代数曲线的补的基本群、Tajima在Milnor数和Tjurina数、Yokuraon motivic特征类等方面分别得到了重要结果.

The head investigator and the others did research on residues on singular varieties. More specifically :1.We developed a theory of residues of Chern classes of vector bundles on singular varieties with respect to collections of sections. We also gave explicit expressions (analytic, algebraic and topological) of the residues at an isolated singular point.2.We introduced the notion of multiplicity for functions on singular varieties and proved a generalization of a formula of Iversen for holomorphic maps onto a Riemann surface.3.In the case of the top Chern class, the Thom class contains local informations and produces analytic., algebraic and topological invariants through the Bochner-Martinelli kernel. We found the "intermediate Thom classes" for other Chern classes.4.In a collaboration with J.-P.Brasselet and J.Seade, we proved a "proportionality theorem" for the local Euler obstruction of 1-forms on singular varieties.5.In a collaboration with F.Bracci, we prove the existence of parabolic curves at a fixed point of a holomorphic self-map of a singular complex surface, as an application of our residue theory. For this, we developed the intersection theory of curves in singular surfaces, using the Grothendieck residues on singular varieties.6.Besides the above, Ito obtained important results on the Poincare-Hopf type theorems, Ohmoto on the characteristic classes of algebraic stacks, Oka on the fundamental group of the complement of algebraic curves, Tajima on Milnor and Tjurina numbers, Yokuraon motivic characteristic classes, respectively.