Links and analytic invariants of superisolated singularities

By I. Luengo-Velasco, Alejandro Melle-Hernández, A. N'emethi. 2003

60 citations recorded by the source corpus.

Using superisolated singularities we present examples and counterexamples to some of the most important conjectures regarding invariants of normal surface singularities. More precisely, we show 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). Moreover, we also show that for Gorenstein singularities (even with integral homology sphere links) besides the geometric genus, the embedded dimension and the multiplicity (in particular, the Hilbert-Samuel function) also fail to be topological; and in general, the Artin cycle does not coincide with the maximal (ideal) cycle.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2197d299335395bc83ec6e33b46a0ee1b3e6b85e

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In Geometry and Topology

1. A. N'emethi
2. Walter D. Neumann
3. J. Wahl
4. A. Pichon
5. Tomohiro Okuma
6. A. Stipsicz
7. Zolt'an Szab'o
8. Stavros Garoufalidis

In Mathematical Physics

1. A. N'emethi
2. J. Wahl
3. Walter D. Neumann
4. A. Pichon
5. A. Stipsicz
6. Tomohiro Okuma
7. Zolt'an Szab'o
8. Stavros Garoufalidis

Citations in the corpus

• On the Ozsvath-Szabo invariant of negative definite plumbed 3-manifolds (2003, background): “After the first version of the present article was completed, the author realized that the above inequality – and equality for Gorenstein singularities – is not true for non-AR singularities; for details see [8].”
• Complex surface singularities with integral homology sphere links (2003, background): “If only greater weight is allowed the result always is an equisingular deformation.”
• Lattice cohomology of normal surface singularities (2007, background): “ spinc–structure was considered. The generalization to any spinc–structure appeared in [13], where it was formulated for any Q–Gorenstein singularity (with rational homology sphere link). The article [10] shows that we cannot expect the validity of the identities in this generality. Nevertheless, we”
• GRADED ROOTS AND SINGULARITIES (2007, background): “Now, the author knows that the conjecture of [37] is not true in the high generality in which it was formulated, see [26] and [33] for details.”
• The thick-thin decomposition and the bilipschitz classification of normal surface singularities (2011, background): “=0, where each fk is a homogeneous polynomial of degree k, is superisolated if the projective plane curve C :={(x, y, z):fd(x, y, z)=0}⊂P is reduced with isolated singularities {pi}i and these points pi are not situated on the projective curve {(x, y, z):fd+1(x, y, z)=0} (see [24] and [25]).”
• Complete intersection singularities of splice type as universal abelian covers (2004, background): “…57M25, 57N10 Keywords: Surface singularity, Gorenstein singularity, rational homology sphere, complete intersection singularity, abelian cover Proposed: Robion Kirby Received: 31 October 2004 Seconded: Ronald Fintushel, Ronald Stern Revised: 18 April 2005 c© Geometry & Topology Publications”