Mathematisches Forschungsinstitut Oberwolfach

Report No. 42/2006

Singularities

Organised by

Joseph H.M. Steenbrink (Nijmegen) Duco van Straten (Mainz) Victor A. Vassiliev (Moscow)

September 10th – September 16th, 2006

Abstract.

Introduction by the Organisers Workshop: Singularities Table of Contents

András Némethi (joint with Gábor Braun)

Invariants of Newton non-degenerate surface singularities . . . . . . . . . . . . . . 5

Abstracts

Invariants of Newton non-degenerate surface singularities András Némethi (joint work with Gábor Braun)

In general, it is a rather challenging task to connect the analytic and topological invariants of normal surface singularities. The program which aims to recover different discrete analytic invariants from the abstract topological type of the singularity (i.e. from the oriented homeomorphism type of the link K, or from the resolution graph) can be considered as the continuation of the work of Artin, Laufer, Tomari, S. S.-T. Yau (and the author) about rational and elliptic singularities, it includes the efforts of Neumann and Wahl to recover the possible equations of the universal abelian covers [7], and of the author and Nicolaescu about the possible connections of the geometric genus with the Seiberg–Witten invariants of the link [6]. See [5] for a review of this program.

In order to have a chance for this program, one has to consider a topological restriction (maybe, the weakest one for which we still hope for positive results is when we ask for the link to be a rational homology sphere, RHS in the sequel), and also restriction about the analytic type of the singularity. [3] shows that the Gorenstein condition, in general, is not sufficient; but even for hypersurface singularities one may expect pathologies.

For isolated hypersurface singularities a famous conjecture was formulated by Zariski [9], which predicts that the multiplicity is determined by the embedded topological type. For hypersurface germs with RHS links, Mendris and the author in [4] formulated (and verified for suspension singularities) even a stronger conjecture, namely that already the abstract link determines the embedded topological type, the multiplicity and equivariant Hodge numbers (of the vanishing cohomology).

The goal of this talk is to present the validity of this stronger conjecture for isolated singularities with non–degenerate Newton principal part (NNPP in the sequel; for its definition see [2]). In fact, we proved that from the link (provided that it is a RHS) one can recover the Newton boundary (up to a natural ambiguity — see below, and up to a permutation of coordinates); hence, in fact, the ‘equation of the germ’ (up to an equisingular deformation). This is the maximum what we can hope for.

First, we define an equivalence relation ∼ of Newton boundaries: two diagrams Γ_{1} and Γ_{2} are equivalent if both define isolated singularities and Γ_{1} ⊂ Γ_{2}. (This, at the level of germs, can be described by a linear deformation.) Although the structure of an equivalence class is not immediate from the definition, one can identify in each class an easily recognizable representative; we call it d–minimal representative.

Next, we notice that there is an algorithm, given by toric resolution, for details see Oka’s article [8], which provides a possible resolution graph G(Γ) (or, equivalently, a plumbing graph of the link) from the Newton boundary Γ. (Equivalent graphs provide plumbing graphs related by blowing ups/downs, hence the same link.) Our main result says that Oka’s algorithm essentially can be inverted:

Theorem. Assume that the Newton diagrams Γ_{1} and Γ_{2} determine isolates singularities with NNPP whose links are RHS. Assume that the good minimal resolution graphs associated with G(Γ_{1}) and G(Γ_{2}) are the same. Then (up to a permutation of coordinates) Γ_{1} ∼ Γ_{2}. In particular, from the link K, one can identify the ∼– equivalence class of the Newton boundary (up to a permutation of coordinates), or even the d–minimal representative of this class.

In fact, we proved an even stronger result: one can recover the corresponding class of Newton diagrams (or its distinguished representative) already from the orbifold diagram G^{o} associated with the good minimal resolution graph. This diagram, a priori, contains less information then the resolution graph, it codifies only its shape and some subgraph–determinants. (G^{o}, although has different decorations, it is equivalent with det Γ together with the ‘splice diagram’ considered in [7].)

Since under the deformations corresponding to ∼ most of the invariants of the germs are stable, one has the following

Corollary. Let f be an isolated germ with NNPP whose link is RHS. Then the oriented topological type of its link determines completely its Milnor number, geometric genus, spectral numbers, multiplicity, and finally, its embedded topological type.

Such a statement is highly non-trivial for any of the above invariants. For the history of the problem regarding the Milnor number and the geometric genus the reader is invited to consult [5]. Here we emphasize only the following:

• Regarding the embedded topological type, notice that the result shows that if a RHS 3-manifold can be embedded into S^{5} as the embedded link of an isolated hypersurface singularity with NNPP, then this embedding is unique. (Notice the huge difference with the case of plane curves, and also with the higher dimensional case, when already the Brieskorn singularities provide a big variety of embeddings S^{2n−1} ⊂ S^{2n+1}, n ≠ 2.)

• Moreover, such a link can be realized by a germ f with NNPP essentially in a unique way up to a sequence of linear μ-constant deformations (corresponding to ∼) and permutation of coordinates.

Regarding the main theorem, some more comments are in order.

• The assumption that the link is RHS is necessary: the germs {x^{a}+y^{b}+z^{c} = 0} with exponents (3,7,21) and (4,5,20) share the same minimal resolution graph.

• The proof of the main theorem is, in fact, a constructive algorithm which provides the d–minimal representatives of the corresponding class of diagrams from the orbifold diagram G^{o}. Hence, one may check effectively that an arbitrary

Singularities  7

resolution graph can or cannot be realized by a hypersurface singularity with NNPP. Indeed, if one runs our algorithm and it fails, then it definitely is not of this type. If the algorithm goes through and provides some candidate for a Newton diagram, then one has to compute the graph (orbifold diagram) of this candidate (by Oka’s procedure) and compare with the initial one. If they agree then the answer is yes; if they are different, the answer again is no (this may happen since our algorithm uses only a part of the information of G^{o}).

E.g., one can check that the following resolution graph cannot be realized by an isolated singularity with NNPP (although it can be realized by a suspension {z^{2} + g(x, y) = 0}, where g is an irreducible plane curve singularity with Newton pairs (2,3) and (1,3)).

[figure: −7; −3; −3; −1; −3; −2]

We mention that, in general, there is no procedure which would decide if a graph can be the resolution graph of a hypersurface isolated singularity (this is one of the open problems asked by Laufer [1], p. 122; for suspension singularities is solved in [4]).

References

[1] H. Hauser and R. Randell, Report on the problem session, Contemporary mathematics, vol. 90, 1989, pp. 119–134.

[2] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1–31.

[3] Melle-Hernández A. Luengo, I. and A. Némethi, Links and analytic invariants of superisolated singularities, J. of Algebraic Geometry 14 (2005), 543–565.

[4] R. Mendris and A. Némethi, The link of {f(x, y) + z^{n} = 0} and Zariski’s Conjecture, Compositio Math. 141 (2005), no. 2, 502–524.

[5] A. Némethi, Invariants of normal surface singularities, Contemporary Mathematics, vol. 354, 2004, pp. 161–208.

[6] A. Némethi and L. Nicolaescu, Seiberg-Witten invariants and surface singularities, Geometry and Topology 6 (2002), 269–328.

[7] W. D. Neumann and J. Wahl, Universal abelian covers of surface singularities, Trends on Singularities, A. Libgober and M. Tibăr, eds., Birkhäuser, 2002, pp. 181–190.

[8] M. Oka, On the resolution of the hypersurface singularities, Complex analytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam, 1987, pp. 405–436.

[9] O. Zariski, Some open questions in the theory of singularities, Bull. A.M.S. 77 (1071), 481–491.

Reporter: András Némethi