Topological and analytical invariants of singularities

建议：DMS-0088950 PI：Andas N\methi摘要：法向曲面奇点的不变量是拓扑的，如果它是它的连接的3-流形不变量，或者如果它可以从它的最小分辨率图的组合学中确定，则它是拓扑的.该建议的驱动问题是：几何亏格和Hilbert-Samuel函数拓扑的时候？该建议有三个部分。第一部分的主要信息是，对于那些Gorenstein奇点的链接是一个合理的同调球，工作的Artin和Laufer的合理，分别对极小椭圆奇点可以continuous.The乐观的部分是基于作者最近的工作Gorenstein椭圆奇点与合理的同调球链接。对于这些奇点，作者证明了它们的几何亏格是拓扑的：它正好是椭圆序列的长度（在S. S.- T. Yau）。任何一般调查的第一步是在一些非平凡的特殊情况下测试这些假设。在第二部分的建议的情况下，悬挂奇点进行了更详细的讨论。第三部分的建议涉及更高的维推广的Artin-Laufer计划。局部解析空间是由局部定义的解析函数定义的，特别是它们携带许多解析不变量。一个非常重要的问题，在他们的分类如下：这些解析不变量可以确定从这些空间的拓扑描述？在复杂曲面奇点的情况下，拓扑结构由奇点的一个有向3-流形的连接来确定。该项目的目标是描述这个3-流形的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).