Invariants of Normal Surface Singularities

该提案由两部分组成。第一部分的主要信息是，对于那些链接是有理同调球面的Gorenstein奇点，Artin和Laufer分别关于有理极小椭圆奇点的工作可以继续下去。这种乐观在一定程度上是基于作者对Gorenstein椭圆奇点的研究。对于具有有理同调球链的超曲面奇点，该方案解决了一个更强的猜想：该链决定嵌入的拓扑类型(特别是从Milnor纤维得到的所有同调包)、所有等变Hodge数(特别是几何亏格)和多重性(特别地，它预测了比Zariski猜想更尖锐的性质)。第二部分集中在L.I.Nicolaescu和作者最近提出的一个猜想：有理同调球环的某个Seiberg-Wittenant不变量可以作为几何亏格的“最优拓扑上界”；在q-Gorenstein型的情况下，它决定了它。事实上，对于Gorenstein奇点的平滑，它预测Milnor纤维的签名等于$-8$乘以这个Seiberg-Witten不变量。这个猜想推广了Neumann和Wahl早期的一个猜想，该猜想将链环的Casson不变量和Milnor纤维的特征联系起来，只要链环是整数同调球面。(解析)奇点理论的目的之一是推广光滑解析流形的拓扑不变量和解析不变量。从拓扑的角度来看，局部奇点实际上是一些解析函数的零集。从拓扑的角度来看，局部曲面奇点是由它的链环描述的，它是一个三维流形(即它是它的链环上的一个锥形)。另一方面，它具有许多涉及定义函数的性质的解析不变量。该提议的目的是将这些不变量联系起来。最后的目标是一个“提升性质”，它保证许多解析不变量在拓扑层上已经可以读取。主要的拓扑成分是最近的Seiberg-Witten不变量。该方案结合了代数几何和一些图的拓扑学和复杂的组合性质。

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.