The proposal has two 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 work on Gorenstein elliptic singularities. For hypersurface singularities with rational homology sphere links the proposal addresses an even stronger conjecture: the link determines the embedded topological type (in particular, all homological package derived from the Milnor fibration), all the equivariant Hodge numbers (in particular, the geometric genus), and the multiplicity (in particular, it predicts an even sharper property than Zariski's Conjecture). The second part is concentrated on a recent conjecture of L.I. Nicolaescu and the author which states that a certain Seiberg-Witten invariant of a rational homology sphere link can serve as an ``optimal topological upper bound'' for the geometric genus; and in the Q-Gorenstein case, it determines it. In fact, for a smoothing of a Gorenstein singularity it predicts that the signature of the Milnor fiber equals $-8$ times this Seiberg-Witten invariant. This conjecture generalizes an earlier conjecture of 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 generalize the topological and analytical invariants of smooth analytic manifolds. A local singularity, in fact, is the zero set of some analytic functions From topological point of view, a local surface singularity is described by 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 interesting properties of the defining functions. The aim of the proposal is to connect these invariants. The final goal is a ``lifting property'' which guarantees that 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 topology and sophisticated combinatorial properties of some graphs.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Christopher W. Stark
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Ohio State University
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