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Links and analytic invariants of superisolated singularities. (English) Zbl 1084.32022

The goal of this paper is to present examples and counterexamples to some of the most important conjectures regarding invariants of normal surface singularities. More precisely, the authors prove that the Seiberg-Witten invariant conjecture (of Nicolaescu and the third author), the Universal abelian cover conjecture (of Neumann and Wahl) and the geometric genus conjecture fail, at least at that generality in which they were formulated. The class of superisolated singularites play a key role in the construction of the corresponding counterexamples. The above-mentioned conjectures concerns the problem of determining what kind of invariants of an analytic complex normal surface singularity can be determined from the topology of the singularity. The article reflects the state-of-the-art of this subject and certainly sets up a guidance for next steps.

MSC:

32S55 Milnor fibration; relations with knot theory
14B05 Singularities in algebraic geometry
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)

Software:

SINGULAR
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References:

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