Invariants of Normal Surface Singularities

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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.