Invariants of Normal Surface Singularities

该提案有两个部分。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领域的工作，阿廷和劳弗理性，分别对极小椭圆奇点可以继续。乐观的部分是基于作者的工作Gorenstein椭圆奇点。对于具有有理同调球面链接的超曲面奇点，该提议解决了一个更强的猜想：链接决定了嵌入的拓扑类型（特别是，从Milnor纤维化导出的所有同调包），所有等变的Hodge数（特别是，几何亏格）和多重性（特别是，它预测了一个甚至比Zenski猜想更尖锐的性质）。第二部分主要介绍了L. I. Nicolaescu和作者提出，一个有理同调球面环的某个Seiberg-Witten不变量可以作为几何亏格的“最优拓扑上界”;在Q-Gorenstein情形中，它确定了它。事实上，对于Gorenstein奇点的光滑化，它预言Milnor纤维的签名等于这个Seiberg-Witten不变量的$-8$倍。这一猜想推广了Neumann和Wahl的一个较早的猜想，该猜想在光滑化的情况下，将链路的Casson不变量和Milnor纤维的签名联系起来，只要链路是整数同调球面。（解析）奇点理论是对光滑解析流形的拓扑不变量和解析不变量的推广，一个局部奇点，实际上，从拓扑学的观点看，局部曲面奇点是由它的一个3-流形（即它是其上的一个锥）的环来描述的，另一方面，它又有许多解析不变量，这些不变量把定义函数的一些有趣的性质编码起来。该建议的目的是连接这些不变量。最后的目标是一个“提升性质”，它要求许多解析不变量在拓扑水平上已经可以被读取。主要的拓扑成分是最近的Seiberg-Witten不变量。该建议将代数几何技术与拓扑和一些图的复杂组合性质结合起来。

The proposal has two parts. The main message of the first part is that forthose Gorenstein singularities whose link is a rational homology sphere thework of Artin and Laufer on rational, respectively on minimally ellipticsingularities can be continued. The optimism is partly based on the author'swork on Gorenstein elliptic singularities. For hypersurface singularitieswith rational homology sphere links the proposal addresses an even strongerconjecture: the link determines the embedded topological type (in particular,all homological package derived from the Milnor fibration), all theequivariant Hodge numbers (in particular, the geometric genus), and themultiplicity (in particular, it predicts an even sharper property thanZariski's Conjecture). The second part is concentrated on a recent conjectureof L.I. Nicolaescu and the author which states that a certain Seiberg-Witteninvariant of a rational homology sphere link can serve as an ``optimaltopological upper bound'' for the geometric genus; and in the Q-Gorensteincase, it determines it. In fact, for a smoothing of a Gorenstein singularityit predicts that the signature of the Milnor fiber equals $-8$ times thisSeiberg-Witten invariant. This conjecture generalizes an earlier conjectureof Neumann and Wahl which connects, in the presence of a smoothing,the Casson invariant of the link and the signature of the Milnor fiber,provided that the link is an integer homology sphere.One of the goals of the (analytic) singularity theory is to generalizethe topological and analytical invariants of smooth analytic manifolds.A local singularity, in fact, is the zero set of some analytic functionsFrom topological point of view, a local surface singularity is describedby its link which is a 3--manifold (i.e. it is a cone over its link).On the other hand, it has many analytic invariants codifying interestingproperties of the defining functions. The aim of the proposal is to connectthese invariants. The final goal is a ``lifting property'' which guaranteesthat many analytic invariants can be read already at the topological level.The main topological ingredient is the recent Seiberg-Witten invariant.The proposal combines techniques of algebraic geometry with topologyand sophisticated combinatorial properties of some graphs.