Geometry of singularities of maps

本研究的出发点是以下事实：对映射奇异性的研究主要有两个方向：（1）在对映射的一般奇异性进行分类后，研究映射的一般扰动;（2）对去奇异化的几何理解和等奇异性问题的研究.第一个方向是研究喷流空间中的E-Boardmabn层，讨论它们的Cohen-Macaulay性质;这在交叉理论中变得很重要。我们显示了一个完整的结果的问题是否莫林的理想，其中经常轨迹是科恩-博德曼地层，确定科恩-麦考利空间，或没有。本文还讨论了Ronga的去奇异化作用，并构造了支持于N-n ^<n-p+1，1>层位的复合体。这与映射芽f：（C ^n，0）→（C^2，0）的一般扰动有关。本文给出了n = 2，3，4时的n ^<n-p+1，1>轨迹方程的显式描述，并给出了在这些情况下一般形变中尖点个数的显式表达式，讨论了Newton滤子在奇点不变量计算中的应用，在第二个方向上，讨论了Blow-analytic映射。给出了一个加权形式的blow-analytic平凡性定理，澄清了福井定义的blow-analytic不变量的性质，研究了blow-analytic同构，并研究了具有双Lipschitz性质的blow-analytic映射.但通常的分层理论中的正则性条件对于拓扑等价来说太强了。根据上面的结果，我们可以期望正则性条件的一个较弱的版本。我们定义了一个加权形式的正则性条件，并构造了一个加权形式的分层理论。

Our starting point of this research is the following fact : There are two main directions in the research of singularities of maps, which are (1) Investigation of generic perturbation of maps after classification generic singularities, (2) Geometric understanding of desingularization and investigating equisingular problem.In the first direction, we investigate Thom-Boardmabn strata in the jet space and discuss their Cohen-Macaulay property, which become important in the context of intersection theory. We show a complete result for the problem whether Morin's ideal, which regular locus is the Thom-Boardman strata, determines Cohen-Macaulay spaces, or not. We also discuss Ronga's desingularization and construct complecies supported at Σ^<n-p+1,1> strata. This is related with a generic perturbation of a map germ f : (C^n,0) → (C^2,0). We obtain an explicit description of equations for Σ^<n-p+1,1> locus when n = 2,3,4 and conclude an explicit formula for the number of cusps appeared in a generic deformation in these cases.We also discuss the use of Newton filtration in the computation of invariants of singularities.In the second direction, we discuss blow-analytic maps. We showed a weighted version of blow-analytic triviality theorem, clarification of the behavior of blow-analytic invariant which was defined by Fukui, investigation of blow-analytic isomorphisms, and and work about blow-analytic map with biLipschitz property.Stratification theory is a tool to construct topological triviality. But the usual regularity condition in stratification theory is too strong for topological equivalence. By the above result, we can expect a weaker version of the regularity condition. We define a weighted version of regularity condition and construct a weighted version of stratification theory.