Research on Complex Analytic Geometry and Singularity Theory

主要研究了向量场和全纯奇异叶理的指标和留数，奇异簇的特征类，Cech-de Rham上同调理论和分层空间上的积分理论。让我们更具体地说。(1)与J.Seade合作研究开流形上叶状的Baum-Bott剩余的剩余定理及其应用。关于这一点的联合论文已发表在Mathematische Annalen。(2)在与J.Seade的另一次合作中，我们研究了具有孤立奇点的簇上的向量场的各种指数，并得到了这种簇的“附加公式”。研究结果发表在一份联合论文中。(3)作为（2）中公式的应用，得到了具有孤立奇点的局部完全交簇的Chem-Schwartz-MacPherson类的一个公式.该研究结果已发表在C.R.Acad.Sci.上，巴黎。(4)作为（2）中公式的推广，我们与D.Lehmann和J.Seade合作，引入了一个广义Milnor数，并对可能具有非孤立奇点的簇得到了一个类似的公式。研究结果发表在一份联合论文中。(5)在与B.Khanedani的合作工作中，我们研究了复曲面上奇异全纯叶理的不变量，得到了各种公式。关于这些问题的联合论文将发表在北海道数学杂志上。(6)在与T.本田的合作工作中，我们证明了复曲面上亚纯函数的一个留数公式，并给出了一些应用。结果写在一篇联合论文中。(7)在与J的合作中，P. Scholelet，我们研究了正弦全纯叶理的Nash修正，并作为应用，证明了Baum-Bott的一个猜想。研究结果发表在一份联合论文中。

The research was done mainly on the indices and residues of vector fields and holomorphic singular foliations, the charactreistic classes of singular varieties, the Cech-de Rham cohomology theory and integration theory on stratified spaces. Let us be more specific.(1) Collaboration with J.Seade on the residue theorem for the Baum-Bott residues of foliations on open manifolds and its applications. The joint paper on this has been published in Mathematische Annalen.(2) In another collaboration with J.Seade, we investigated various indices of vector fields on varieties with isolated singularities and we obtained an "adjunction formula" for such varieties. The results are written in a joint paper.(3) As an application of the formula in (2), a formula for the Chem-Schwartz-MacPherson class of a local complete intersection variety with isolated singularities is obtained. The result has been published in C.R.Acad.Sci., Paris.(4) As a generalization of the formula in (2), in a collaboration with D.Lehmann and J.Seade, we introduced a generalized Milnor number and obtained a similar formula for varieties with possibly non-isolated singularities. The results are written in a joint paper.(5) In a joint work with B.Khanedani, we studied the invariants of singular holomorphic foliations on complex surfaces and obtained various formulas. The joint paper on these will appear in Hokkaido Math.J.(6) In a joint work with T.Honda, we proved a residue formula for meromorphic functions on complex surfaces and gave some applications. The results are written in a joint paper.(7) In a collaboration with J.-P.Brasselet, we studied the Nash modification associated with a sinular holomorphic foliation and, as an application, we proved a conjecture of Baum-Bott in some cases. The results are written in a joint paper.