Topological and analytical invariants of singularities

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

Proposal: DMS-0088950 PI: Andas N\methi Abstract: An invariant of a normal surface singularity is topological if it is a 3-manifold invariant of its link, or equivalently, if it can be determined from the combinatorics of its minimal resolution graph. The driving question of the proposal is: when are the geometric 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 genus is topological: it is exactly the length of the elliptic sequence (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 important question 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 structure is determined from the link of the singularity which is an oriented 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).