Topological and analytical invariants of singularities

摘要：法向曲面奇异形态的不变量，如果是其连杆的3流形不变量，则可以由其最小分辨率图的组合来确定。这个提议的驱动问题是：几何属和Hilbert-Samuel函数什么时候是拓扑的？该提案由三部分组成。第一部分的主要信息是，对于那些连杆是有理同调球的Gorenstein奇点，Artin和Laufer分别关于有理的最小椭圆奇点的工作可以继续下去。这种乐观部分是基于作者最近对具有有理同调球连杆的Gorenstein椭圆奇点的研究。对于这些奇点，作者在其他事实中证明了它们的几何属拓扑性：它正是椭圆序列的长度（在s.s.t意义上）。邱)。任何一般性研究的第一步都是在一些重要的特殊情况下检验猜想。在建议的第二部分，对悬性奇点的情况进行了更详细的讨论。该建议的第三部分涉及Artin-Laufer程序的高维推广。局部解析空间是由局部定义的解析函数定义的，特别是它们带有许多解析不变量。在它们的分类中有一个非常重要的问题是：这些解析不变量能否从这些空间的拓扑描述中确定？在复杂曲面奇异点的情况下，从奇异点的链路确定拓扑结构，奇异点是无取向的3流形。这个项目的目标是用这个3流形来描述奇异空间的一些分析信息（在可能的情况下）。

Proposal: DMS-0088950PI: Andas N\methiAbstract: An invariant of a normal surface singularity istopological if it is a 3-manifold invariant of its link, orequivalently, if it can be determined from the combinatorics of itsminimal resolution graph. The driving question of the proposal is: when are thegeometric genus and the Hilbert-Samuel function topological?The proposal has three parts. The main message of the first part is that for those Gorenstein singularities whose link is a rational homology sphere, the work of Artin and Laufer on rational, respectively on minimally elliptic singularities can be continued.The optimism is partly based on the author's recent work on Gorenstein elliptic singularities with rational homology sphere links. For these singularities the author proved among other facts that their geometric genusis topological: it is exactly the length of the ellipticsequence (in the sense of S. S.-T. Yau). The first step in any general investigation is the testing of the conjectures in some non-trivial particular cases. In the second part of the proposal the case of suspension singularities is discussed in more details.The third part of the proposal deals with higher dimensional generalizations of the Artin-Laufer program. Local analytic spaces are defined by locally defined analytic functions,in particular they carry a lot of analytic invariants. A very importantquestion in their classification is the following: can these analytic invariants be determined from the topological description of these spaces? In the case of complex surface singularities, the topological structureis determined from the link of the singularity which is anoriented 3-manifold. The goal of the project is to describe some of the analytic information of the singular space in terms of this 3-manifold(in those cases when it is possible).