INVARIANTS OF NORMAL SURFACE SINGULARITIES

ANDRÁS NÉMETHI

1. Introduction

In the last period, several authors tried to understand the following problem: what kind of analytic invariants of an analytic complex normal surface singularity can be determined from the topology of the singularity? Or, in what circumstances one can hope for such a result?

The present note reflects some of the ideas and results behind these questions. It is far from being a complete summary about the invariants of such singularities. In fact, all the sections serve only one purpose: to understand the subtle relationship between analytic and topological invariants; or in the presence of a smoothing, their relation with the smoothing invariants. In the last section, when the hypersurface case is discussed, the embedded topological type also appears in the picture.

About some invarints we say less, about others we say more. The “most wanted” analytic invariant in this note is the geometric genus.

The note contains a considerable number of open questions and (old and new) conjectures. We hope that they will serve as an invitation for some more research and activity in this area, and they will contribute to the renaissance of complex surface singularities.

2. The link

2.1. Definition. Let (X, 0) be a complex analytic normal surface singularity embedded in (ℂ^{N}, 0), and let B_{ϵ} be the corresponding ϵ-ball. Then, for ϵ sufficiently small, the intersection M := X ∩ ∂B_{ϵ} is a connected compact oriented 3-manifold, whose oriented C^{∞} type does not depend on the choice of the embedding and ϵ. It is called the link of (X, 0) [61].

Moreover, one can verify that X ∩ B_{ϵ} is homeomorphic to the cone over M. In particular, M characterizes completely the local topological type of (X, 0). Therefore, if an invariant of (X, 0) can be deduced from M, we say that it is a topological invariant.

2.2. M as a plumbed manifold. The bridge between the analytical and topological type of (X, 0) is realized by a resolution, respectively by a resolution graph of (X, 0).

2 András Némethi

Fix a sufficiently small Stein representative X of (X, 0) (e.g. X ∩ B_{ϵ} as above) and let π : X̃ → X be a resolution of the singular point 0 ∈ X. In particular, X̃ is smooth, and π is a biholomorphic isomorphism above X \ {0}. We can assume that the exceptional divisor E := π^{−1}(0) is a normal crossing divisor with irreducible components {E_{v}}_{v∈V}. Such a resolution is called good. For a good resolution π, let Γ(π) be the dual resolution graph associated with π decorated with the self intersection numbers {E_{v} · E_{v}}_{v} and genera {g_{v}}_{v∈V} (see [44]). Sometimes we write e_{v} for E_{v} · E_{v}. Notice that H_{2}(X̃, ℤ) is freely generated by the fundamental classes {[E_{v}]}_{v}. Let I be the intersection matrix {E_{v} · E_{w}}_{v,w}. Since π identifies ∂X̃ with M, the graph Γ(π) can be regarded a plumbing graph, and M can be considered as an S^{1}-plumbed manifold whose plumbing graph is Γ(π). Here one should add two very important facts.

2000 Mathematics Subject Classification. Primary. 14B05, 14J17, 32S25, 57M27, 57R57. Secondary.

14E15, 32S45, 57M25.

Key words and phrases. normal surface singularities, (ℚ-)Gorenstein singularities, singularities with ℂ^{∗}- action, geometric genus, deformations, rational, elliptic and hypersurface singularities, links, Zariski’s conjecture, Neumann-Wahl conjecture, Seiberg-Witten invariants of ℚ-homology spheres, Reidemeister-Turaev torsion, Casson-Walker invariant.

The author is partially supported by NSF grant DMS-0088950.

2.2.1. First, by a result of Mumford [62], I is negative definite. Its converse was proved by Grauert [27]. More precisely, a plumbing graph can be realized as a resolution graph of a (complex analytic) normal surface singularity if and only if the associated intersection form I is negative definite. This, by the next statement, gives a complete classification of the possible topological types of (analytic) normal surface singularities.

2.2.2. By the plumbing construction, any resolution graph Γ(π) determines the oriented 3-manifold M completely. Hence, one can ask the very natural question: Does Γ(π) contain any additional information which cannot be recovered from M? The answer is negative in the following sense. We say that two plumbing graphs (with negative definite intersection forms) are equivalent if one of them can be obtained from the other by a finite sequence of blow-ups and/or blow-downs along rational (−1)-curves. Obviously, for a given (X, 0), the resolution π, hence the graph Γ(π) too, is not unique. But different resolutions provide equivalent graphs. By a result of W. Neumann [76], the oriented diffeomorphism type of M determines completely the equivalence class of Γ(π). In particular, the information carried in any Γ(π) is the same as the information carried by the link M; or, equivalently, any invariant defined from the resolution graph Γ(π) (which is constant in its equivalence class) is, in fact, an invariant of the oriented 3-manifold M.

2.2.3. We wish to mention here two key results which preceded Neumann’s result. First, Mumford proved that the minimal resolution graph of (X, 0) is empty if and only if M = S^{3} [62]. Later, Orlik and Wagreich generalized this for singularities with a good ℂ^{∗}-action as follows [83] (cf. also with 11.4). In this case, M is a Seifert manifold which can be codified by its minimal (star-shaped) plumbing graph. By [loc. cit.], this minimal plumbing graph coincides with the minimal good resolution graph of (X, 0). In particular, the minimal resolution graph can be recovered from the Seifert invariants of the link.

2.2.4. In fact, Mumford verified that for links of normal surface singularities, the isomorphism M = S^{3} can be characterized homotopically by the vanishing of the fundamental group π_{1}(M). Neumann in [76] proved that this fact can be extended in general: the link of a normal surface singularity can be recovered from its fundamental group with two exceptions, which are also completely understood.

The first exception consists of the cyclic quotient (or Hirzebruch-Jung) singularities X_{n,q}, where gcd(n, q) = 1, 0 < q < n. The corresponding links are the lens-spaces L(n, q) with π_{1}(L(n, q)) = ℤ_{n} (cf. 4.2.1). The second exceptional family is given by the cusp singularities, whose minimal resolution graph is cyclic and g_{v} = 0 for all v. In this case, M is a torus bundle over the circle (cf. 8.7.1).

2.2.5. In the above identification of a resolution graph with a plumbing graph it is convenient (and, in some sense, necessary) to consider resolutions with normal crossing exceptional divisor (i.e. good resolutions). Among the good resolutions there is a minimal one, called the minimal good resolution.

On the other hand, in some other constructions it is preferable not to assume that E is a normal crossing divisor, and work with the minimal (not necessarily good) resolution. In such a resolution, there is no rational irreducible exceptional divisor E_{v} with e_{v} = −1.

2.3. Definition. We say that M is a rational homology sphere (RHS) if H_{1}(M, ℚ) = 0. M is called an integral homology sphere (IHS) if H_{1}(M, ℤ) = 0.

M is an RHS if and only if Γ(π) is a tree and g_{v} = 0 for all v ∈ V. M is an IHS if additionally I is unimodular (i.e. det(I) = ±1).

In general, the first Betti number b_{1}(M) of M is b_{1}(Γ) + 2 ∑_{v} g_{v}, where b_{1}(Γ) is the number of independent cycles of the graph Γ.

3. Some analytic invariants

3.1. Definitions/First properties. The analytic type of (X, 0) is characterized completely by its local analytic ring O_{X,0} whose maximal ideal will be denoted by m_{0} ⊂ O_{X,0}. An invariant is called analytic if it is defined from (O_{X,0}, m_{0}). Since our main interest is to investigate the topological nature of the (a priori) analytical invariants, we deal mainly with discrete invariants.

3.1.1. The Hilbert-Samuel function is defined by

f_{HS}(k) = dim_{ℂ} O_{X,0}/m_{0}^{k} for any k ≥ 1.

Obviously, f_{HS}(1) = 1 and f_{HS}(2) = 1 + dim m_{0}/m_{0}^{2}. It is not difficult to prove that dim m_{0}/m_{0}^{2} equals the minimal N for which some embedding (X, 0) ⊂ (ℂ^{N}, 0) can be realized. Therefore,

emb dim(X, 0) := dim m_{0}/m_{0}^{2}

is called the embedding dimension of (X, 0) .

For k ≫ 1, f_{HS}(k) = P_{HS}(k) for some polynomial P_{HS} (called the Hilbert-Samuel polynomial)

P_{HS}(k) = mk^{2}/2 + a_{1}k + a_{2}.

The integer m above is called the multiplicity of (X, 0), and it is denoted by mult(X, 0). It is not difficult to verify that if (X, 0) ⊂ (ℂ^{N}, 0) is an arbitrary embedding and L a generic affine space of codimension 2 (close to the origin), then mult(X, 0) = #X ∩ L.

3.1.2. Another crucial invariant discussed in this article is the geometric genus p_{g} := dim_{ℂ} H^{1}(X̃, OX̃), where X̃ → X is a resolution as above (cf. also with 3.1.4).

The geometric genus is just one example of the many possible analytic invariants obtained from a fixed resolution π : X̃ → X via sheaf-cohomology. For example, one can replace the structure sheaf by the tangent sheaf θX̃ = (Ω1 ; X̃)^{∗}, and obtain another invariant θ := dim_{ℂ} H^{1}(X̃, θX̃).

The point is that these invariants can be recovered from the sheaf-cohomology of some onedimensional spaces as well. Indeed, let Z (resp. Z_{ℚ}) be the set of integral (resp. rational) cycles, i.e. divisors of type Z = ∑_{v∈V} m_{v}E_{v}, m_{v} ∈ ℤ (resp. m_{v} ∈ ℚ) supported on the exceptional divisor E of π. Then, for any Z ∈ Z, one can consider the ℂ–dimension h^{i}(Z) of H^{i}(Z, O_{Z}) (for i = 1, 2). By the theorem of formal functions, if one takes Z with m_{v} ≫ 0 for all v, then h^{1}(Z) = p_{g}.

3.1.3. Z has a natural partial ordering: ∑_{v} m_{v}E_{v} ≥ ∑_{v} n_{v}E_{v} if m_{v} ≥ n_{v} for all v. If Z_{1} ≥ Z_{2} but Z_{1} ≠ Z_{2} then we write Z_{1} > Z_{2}.

Some cycles Z > 0 are of special interest: they describe the possible supports of analytic functions defined on (X, 0) . More precisely, for any f ∈ m_{0} let m_{v} be the order of vanishing of f ∘ π along E_{v}, and set (f)_{E} = ∑_{v} m_{v}E_{v}. Define Z_{an} by {(f)_{E} | f ∈ m_{0}}. This yields an ordered lattice (in the sense that if Z_{1}, Z_{2} ∈ Z_{an}, then Z_{1} + Z_{2} and min{Z_{1}, Z_{2}} are elements of Z_{an}). In particular, Z_{an} has a unique minimal element (corresponding to the “generic linear hyperplane section” in (ℂ^{N}, 0)). It is called the maximal ideal cycle Z_{max} (of S.S.-T. Yau), or the “fiber cycle”. Obviously Z_{max} ≥ ∑_{v} E_{v}.

By the above definitions, the divisorial part of the sheaf π^{∗}(m_{0}) is given by Z_{max}. More precisely, there exists an ideal I, supported by a finite set V (I), such that π^{∗}(m_{0}) = I · OX̃(−Z_{max}). We say that π^{∗}(m_{0}) (or π) has no basepoint, if V (I) = ∅. If π does have some base-points, by some (iterated) blow-ups at the base-points, one can replace π by another resolution which has no basepoints. If π has no base-points, then mult(X, 0) = −Z_{max}^{2} (cf. [113]); a fact which illustrates the importance of the maximal cycle and the basepoint freeness of π.

In fact, more generally, one expects an intimate relationship between the pull-back sheaves π^{∗}(m_{0}^{k}) and the divisorial sheaves OX̃(−kZ_{max}) (hence between the Hilbert-Samuel function and the maximal ideal cycle).

3.1.4. Consider now the holomorphic line bundles Ω_{X\{0}}^{2} of holomorphic 2-forms on X \{0}. If it is holomorphically trivial then we say that (X, 0) is Gorenstein. If one of its power is holomorphically trivial then (X, 0) is ℚ–Gorenstein. E.g., complete intersections are Gorenstein.

Similarly, one can also consider the line bundle Ω2 ; X̃. Then, by a result of Laufer [45]:

p_{g} = dim_{ℂ} H^{0}(X̃ \ E, Ω2 ; X̃) / H^{0}(X̃, Ω2 ; X̃).

This can be read at the “level of (X, 0)” as well. Let H_{L^{2}}^{0} denote the global L^{2}-forms. Then

p_{g} = dim_{ℂ} H^{0}(X \ {0}, Ω_{X}^{2}) / H_{L^{2}}^{0}(X \ {0}, Ω_{X}^{2}).

4. Are the analytic invariants topological ?

4.1. The Main Question we propose to address is the following: is it possible to recover (some of) the analytic invariants from the link M, or equivalently, from a resolution graph Γ(π) of (X, 0)?

In fact, we can extend this question to some analytic properties which characterize some important families of singularities.

4.2. Examples. Here we mention only three such family-characterizations; but later, in the body of the article, the reader will find some more.

4.2.1. The Hirzebruch-Jung singularities, by definition, are characterized by the existence of a finite projection q : (X, 0) → (ℂ^{2}, 0) whose reduced discriminant space is included in the union of the coordinate axes of (ℂ^{2}, 0). On the other hand, one can show that (X, 0) is Hirzebruch-Jung if and only if its link is a lens space L(n, q) with 0 < q < n and gcd(n, q) = 1; or equivalently, if the minimal resolution graph is a straight line graph with all genera zero.

In fact, the minimal resolution graph has the following form (here we will use the traditional notation b_{v} instead of e_{v}):

[figure: −b_{1}; −b_{2}; −b_{s}; ⋯]

where all vertices have genus zero, and the self–intersections −b_{1}, …, −b_{s} are given by the continued fraction [b_{1}, b_{2}, …, b_{s}]:

n/q = b_{1} − \frac{1}{b_{2} − \frac{1}{... − \frac{1}{b_{s}}}} , b_{1}, …, b_{s} ≥ 2.

Notice also that the Hirzebruch-Jung singularities can also be realized as cyclic quotient singularities ℂ^{2}/ℤ_{n}. Indeed, the singularity X_{n,q} with link L(n, q) is the quotient of (ℂ^{2}, 0) with respect to the action ξ ∗ (u, v) = (ξu, ξ^{q}v), where ξ is a primitive n-th root of unity.

4.2.2. Artin defined the rational singularities by the vanishing p_{g} = 0. For their topological characterization, see section 7.

4.2.3. The rational double points, or RDP–singularities (i.e. rational singularities with multiplicity two) are exactly the (simple) hypersurface singularities of type A-D-E. Topologically they are characterized by the fact that their minimal resolution graphs are the well–known A-D-E (negative definite) graphs. Equivalently, any connected, negative definite graph with g_{v} = e_{v} + 2 = 0 for all v is the minimal resolution graph of some RDP (and it is of type A-D-E).

We mention also that the RDP’s are exactly the (Kleinian) quotient singularities ℂ^{2}/G for finite subgroups G ⊂ SL(2, ℂ).

4.3. Questions. On the other hand, in the last decades similar questions were asked for which the answers are unknown yet. E.g., Laufer formulated the following problems ([33], page 122): (1) give necessary conditions on Γ(π) for (X, 0) to be a hypersurface singularity; (2) give sufficient conditions on a graph Γ for a hypersurface singularity (X, 0) to exist with Γ(π) = Γ.

Neumann and Wahl formulated the same questions for complete intersections.

4.4. (Pseudo)Taut singularities. Obviously, one can ask the above Main Question for the complete analytic invariant O_{X,0}. In this case, the answer is given by Laufer in [46] (completing the list started by Grauert, Brieskorn, Tjurina and Wagreich). Namely, he classified all those topological typesfollowing example. of normal surface singularities, the so-called taut singularities, which carry unique analytical structure. They include all the Hirzebruch-Jung singularities, the rational double and triple points and cusp (or cyclic) singularities.

In fact, Laufer classified even those resolution graphs which support only a countably (equivalently, finitely) many analytic structures. They are called pseudo-taut singularities. (They can be distinguished by their topological types and θ = h^{1}(X̃, θX̃).)

4.5. Hunting for topological characterizations of some special invariants. Unfortunately, the class of (pseudo)taut singularities is rather restrictive (almost all of them are rational). On the other hand, in most of our geometrical problems we do not really need this extremely strong, complete topological characterization. In general, what we need is the behavior of some specific invariants only.

We exemplify this phenomenon by the geometric genus. It is known ([92] and [116]) that a deformation of the minimal resolution X̃ blows down to a deformation of (X, 0) if and only if p_{g} = h^{1}(OX̃) remains constant (this condition defines the sub-scheme Res of the deformation space of (X, 0), studied intensively by Artin, Brieskorn, Laufer, Schlessinger, Wahl, Lipman, and others).

Therefore, the main goal is to try to characterize topologically some of the discrete invariants of (X, 0) only. In general, the analytic invariants listed is section 3 are not topological (cf. e.g. 4.6 and 4.7), but if we restrict our study to some special families, then (hopefully) they can be determined from the graph.

Actually, in general, we have “topological candidates” (or bounds), which in nice cases coincide with their analytical counterparts. In the next paragraphs we will list some of these “candidates” for the invariants listed in section 3.

4.5.1. The topological candidate for Z_{an} is the ordered lattice Z_{top} :=the set of cycles Z ∈ Z with Z > 0 and Z · E_{v} ≤ 0 for any E_{v}. The point behind this definition is that any Z ∈ Z_{an} satisfies this property, hence Z_{an} ⊂ Z_{top}. More precisely, the definition of Z_{top} contains the topological obstruction which should be satisfied by a cycle Z if it equals (f)_{E} for some f ∈ m_{0}. Artin proved that, if Z_{1}, Z_{2} ∈ Z_{top}, then min{Z_{1}, Z_{2}} ∈ Z_{top}. In particular, Z_{top} has a unique minimal element Z_{min}, called Artin’s fundamental cycle, or the minimal cycle [4, 5].

Z_{min} is the topological candidate for Z_{max}. Clearly Z_{min} ≤ Z_{max}. Although Z_{min}, being an element Z, depends on the choice of the resolution π, its self-intersection Z_{min}^{2} is independent of π and an invariant of M.

If Z_{min} = Z_{max}, and π^{∗}(m_{0}) has no base-points, then mult(X, 0) = −Z_{min}^{2}, in particular, the multiplicity is a topological invariant (a posteriori).

4.5.2. Another “pair” is the following. The analytic invariant is the canonical divisor KX̃ of X̃. Numerically, it is codified by its “topological version” Z_{K}, which is the unique (rational) cycle in Z_{ℚ}, supported by E, satisfying Z_{K} · E_{v} = −KX̃ · E_{v} for all v ∈ V. The number −KX̃ · E_{v} can be determined topologically by the adjunction formula.

For this, first we define the delta-invariant of a curve C. If ψ : C̃ → C is the normalization of C, then ψ_{∗}OC̃/O_{C} is supported by the singular locus of C, and it is finite dimensional. Its dimension is the delta-invariant δ(C) of C. In particular, in our case, for any irreducible exceptional divisor E_{v} one has δ(E_{v}) ≥ 0, which is zero if and only if E_{v} is smooth (see e.g. [109] or [47]).

Then, by the adjunction formula

Z_{K} · E_{v} = E_{v} · E_{v} + 2 − 2g_{v} − 2δ(E_{v}) for all v.  (∗)

Notice that δ(E_{v}) = 0 for any good resolution. Since I is non-degenerate, (∗) has a unique solution in Z_{ℚ}, which is an invariant of Γ(π).

Using (∗), it is not difficult to show that in the case of the minimal resolution one has Z_{K} ·E_{v} ≤ 0 for any v. Since the intersection matrix is negative definite, this implies that either Z_{K} = 0 or all the coefficients of Z_{K} are strictly positive (see e.g. [52]).

Moreover, Z_{K} = 0 if and only if (X, 0) is a RDP-singularity and π is the minimal resolution. If π is not minimal, then some of the coefficients of Z_{K} might be negative. Although the self–intersection Z_{K}^{2} depends on the choice of the resolution π, the rational number

Z_{K}^{2} + #V is independent of the choice of π and is an invariant of the link M.

On the other hand, Z_{K}^{2} also plays an important role: if π is the minimal resolution, Z_{K}^{2} associated with π will be denoted by Z_{K,m}^{2}. Obviously, this number is also an invariant of M.

Finally notice that in the case of the minimal resolution, Z_{K} · E_{v} = 0 for some v if and only if E_{v} is a smooth, rational (−2)–curve. Sometimes we prefer to avoid such curves (and prefer to have strict inequalities K · E_{v} > 0). Therefore, in the minimal resolution we contract all the RDP– configurations of curves (cf. 4.2.3). In this way we get a modification X̃ → X with X̃ in general singular, but only with RDP’s. It is called the minimal resolution with RDP’s.

4.5.3. The main importance of the canonical cycle Z_{K} consists of its role in the Riemann-Roch formula. For this, we fix an integral cycle Z > 0. Although the individual cohomology dimensions h^{i}(Z) (i = 1, 2) in general are not topological, the Euler characteristic χ(Z) := h^{0}(Z) − h^{1}(Z) depends only on Γ(π) by the Riemann-Roch theorem. Indeed, χ(Z) = −Z · (Z − Z_{K})/2, where “·” means the natural extension of the intersection form to Z_{ℚ}.

For another important application of the topological nature of Z_{K} see 10.3.

4.5.4. Z_{K} also characterizes the “topological counterpart” of the set of Gorenstein singularities. The perfect analogue of their definition is the following: we say that (X, 0) is numerically Gorenstein if the line bundle Ω_{X\{0}}^{2} is topologically trivial. This happens if and only if Ω_{X\{0}}^{1} is topologically trivial. Moreover, in terms of a fixed resolution, (X, 0) is numerically Gorenstein if and only if Z_{K} has integral coefficients. Obviously, Gorenstein singularities are numerically Gorenstein.

4.5.5. Finally, we say a few words about the geometric genus. For p_{g}, we do not have a unique topological candidate because p_{g} can vary very much in a fixed topological type. Wagreich defined the topological invariant p_{a} := 1 − min_{Z>0} χ(Z), the so-called arithmetical genus of (X, 0), which is a lower bound for p_{g} [113]. Moreover, p_{g} = 0 if and only if p_{a} = 0. But already for p_{a} = 1, the geometric genus p_{g} can be arbitrary large (cf. 8.4). One of the goals of the present article is to discuss an “optimal topological upper bound” p_{max} of p_{g}, cf. 9.5 and 10.8.

In our conjectures and discussions we try to follow this principle: if we consider all the singularities with the same resolution graph, then p_{g} = p_{a} for the generic analytic structure, and p_{g} = p_{max} for some very specific ones. For example, in the case of numerically Gorenstein singularities, we expect that p_{g} will equal the lower bound p_{a} for generic singularities, and the upper bound p_{max} in the Gorenstein case.

The next examples show that, in general, one cannot recover the analytic invariants from M.

4.6. Example. Consider the hypersurface singularities (X_{1}, 0) = {x^{2} + y^{3} + z^{18} = 0} ⊂ (ℂ^{3}, 0) and (X_{2}, 0) = {z^{2} = y(x^{4} + y^{6})} ⊂ (ℂ^{3}, 0). One can verify that they have the same minimal resolution graph (where [1] denotes the genus of the irreducible component E_{0}, and all the other genera are zero):

[figure: −1; −2; −2; [1]]

On the other hand, in the first case p_{g} = 3, and in the second case p_{g} = 2.

Moreover, in the first case Z_{min} = Z_{max}, but in the second case Z_{min} < Z_{max} (hence Z_{an} ≠ Z_{top}). Indeed, it is easy to verify that Z_{min} = ∑_{v} E_{v}. In the first case Z_{min} = (z)_{E}, but in the second case there is no function f whose multiplicity along E_{0} is 1.

In fact, by a result of Laufer (cf. [47]), for the generic analytic structure supported by the same topological type (i.e. with the above minimal resolution graph) one gets p_{g} = 1. This generic analytical structure (X_{gen}, 0) fails to be Gorenstein (see e.g. 8.10.1), in particular emb dim(X_{gen}, 0) > 3, hence neither can we recover emb dim from M.

Notice also that mult(X_{1}, 0) = mult(X_{2}, 0) = 2, but mult(X_{gen}, 0) > 2. This follows from the fact that any singularity with multiplicity 2 has the form of a suspension hypersurface singularity {f(x, y) + z^{2} = 0} for some isolated plane curve singularity f.

4.7. Example. Examples with different multiplicities can be found easily even among hypersurfaces. For example, consider two Brieskorn singularity {x^{a} + y^{b} + z^{c} = 0} with exponents (3, 7, 21), respectively (4, 5, 20). Then they have the same minimal resolution graph: Γ(π) consist of only one vertex with genus 6 and self intersection −1. On the other hand, clearly, their multiplicities are not the same. (Moreover, their geometric genus and Milnor numbers are also different.)

For some more examples see 5.7.1 and 5.7.2.

4.8. Remark. Before the reader will lose his or her enthusiasm for our program, we point out that in all the above (negative) examples 4.6-4.7 the link is not a rational homology sphere! This distinction will be crucial later.

4.9. Why the surface singularities ? We wish to emphasize that this is the only dimension when the Main Question 4.1 really make sense.

For example, if (C, 0) is a complex analytic curve singularity, then its link consists of r disjoint copies of the circle S^{1}, where r is the number of irreducible components of (C, 0). In particular, it contains absolutely no other information about the analytic type of (C, 0).

Similarly, in higher dimensions too, the topological information codified by the link is rather weak (from the point of view of the question 4.1).

Consider for example the Brieskorn singularities (Y, 0) = {x_{1}^{2} +x_{2}^{2} +x_{3}^{2} +x_{4}^{3} = 0} ⊂ (ℂ^{4}, 0). Then, by a result of Brieskorn [11], M is diffeomorphic to S^{5}, but Y is evidently not smooth.

Or, consider the series of Brieskorn singularities (Y_{k}, 0) = {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{3}+x_{5}^{6k−1} = 0} ⊂ (ℂ^{5}, 0) (k ≥ 1). Then the link M_{k} is a topological S^{7} for any k. Write k = 28q + r with 0 ≤ r < 28. Then by another result of Brieskorn [12], the diffeomorphism type of M_{k} depends only on r, and for r = 0, 1, …, 27 one obtains all 28 possible differentiable structures on the topological 7–sphere. In particular, for any k = 28q, the link is diffeomorphic to S^{7}, but (almost) all the other invariants “increase” with k. Similar examples can be constructed in higher dimensions as well.

Then, one can ask, why is the surface case so special ? The answer lies (probably) in the rich fundamental group of the 3–dimensional link.

5. Deformations of normal surface singularities

5.1. Deformations. If one tries to clarify how the analytical invariants change in a given topological type, one can start first to understand how they can vary in an infinitesimal deformation. This is the subject of the (local) deformation theory.

Obviously, the deformation invariants of singularities are analytic. But since some of the smoothing invariants will play an important role in this article, we will treat them separately. In fact, we will even break the bipolar discussion (topological versus analytical) introduced previously, and replace it by a “triangular picture” adding the deformation invariants as a new vertex.

Deformation theory analyses flat deformations λ : (X , 0) → (T, 0) of (X, 0) (which means that (X, 0) is isomorphic to (λ^{−1}(0), 0)). In this article T is always reduced.

Since (X, 0) has an isolated singularity, one has a semi-universal deformation (X , 0) → (B, 0). It has the property that all the flat deformations λ are induced by a map (T, 0) → (B, 0) which is unique at the level of tangent spaces, see [28, 97]. The space (B, 0) is called the base space of the semi-universal deformation of (X, 0). It is smooth if (X, 0) is a complete intersection [110], or if emb dim(X, 0) = 4 [96]. But in general, B can be singular with several irreducible components (with different dimensions), and these components can intersect each other in complicated ways.

The space B, in general, is hardly computable. Usually, one first determines its Zariski tangent space (T_{X,0}^{1}, 0), the so-called first order deformations of (X, 0), and then the “obstruction space” (T_{X,0}^{2}, 0), with a map (T_{X,0}^{1}, 0) → (T_{X,0}^{2}, 0), whose fiber over the origin provides B.

5.1.1. “Trivial” deformations. Even if (in some cases) we can say very little about B (or X → B), we would still like to be able to decide when a flat deformation λ is “trivial”. Here we can think about “analytic triviality”, or “topological triviality”. Analytic triviality means that the map (T, 0) → (B, 0) is the constant map onto the origin. But, unfortunately, there is no unanimously accepted definition which would answer to all the expectations of the topological triviality. On the other hand, the multitude of the different possibilities explore an extremely rich structure of the deformations; see e.g. the papers of Teissier for different notions (like Whitney conditions, equisingularity, simultaneous resolutions, or μ-constant, resp. μ^{∗}-constant deformations in the case of hypersurfaces) [107, 109].

In this paper we will emphasize mainly the notion of simultaneous resolution (resp. its connection with the μ^{∗}-constant deformation). It also can be defined at different levels of complexity.

5.2. Simultaneous resolutions. Definitions. [109, 50, 42] Let λ : (X , 0) → (T, 0) be a flat deformation of (X, 0) as above. For any t ∈ T we write X_{t} for λ^{−1}(t), and we use similar notations for other spaces and invariants as well. Sometimes we fix a section s : T → X of λ (which identifies for any t a base-point s(t) ∈ X_{t}, or a space-germ (X_{t}, s(t))).

Let Π : X̃ → X be a proper modification, such that λ ∘ Π is flat. We say that

(i) Π is a simultaneous RDP resolution if each X̃_{t} → X_{t} is a minimal RDP resolution.

(ii) Π is a very weak simultaneous resolution if each X̃_{t} → X_{t} is a minimal resolution.

(iii) Π is a weak simultaneous resolution along s if it is a very weak resolution near s(T), and the restriction map Π : Π^{−1}(s(T))_{red} → s(T) is simple (i.e. a locally trivial deformation in the Euclidean topology).

(iv) Π is a strong simultaneous resolution along s if it is a very weak resolution near s(T) and Π : Π^{−1}(s(T)) → s(T) is simple.

We say that λ admits a simultaneous RDP (resp. very weak, weak, or strong) resolution, if there exits Π as above with (i) (resp. (ii), (iii) or (iv)).

Notice that (ii) means that λ ∘ Π is an analytic submersion. Also, from (ii) it follows that if T is smooth then Π is a resolution of X.

We emphasize that in (iii) Π^{−1}(s(T))_{red} is the topological space with the Euclidean topology, which forgets the analytic space structure of the inverse image; while in (iv) Π^{−1}(s(T)) is the nonreduced analytic space. In particular, in (iv) we require that the inverse images of the maximal ideals of the germs (X_{t}, s(t)) behave trivially.

In the next paragraphs, for simplicity, we will take the unit disc D for T.

The literature about simultaneous resolutions and deformations of normal singularities is extremely rich (the interested reader is invited to navigate the articles of Hironaka, Lipman, Schlessinger, Teissier, Lê, Brieskorn, Tjurina, Riemenschneider, Wahl, Looijenga, Arndt, Cristophersen, Behnke, Knörrer, Theo de Jong, van Straten, Pellikaan, Stevens, Vaquié, Kollár, Shepherd-Barron, etc); and we do not plan to present a comprehensive review of these results.

5.3. Our plan is to highlight some topological criteria which assure the existence of some kind of simultaneous resolution, see [109, 42, 50, 51]. Some of these topological descriptions are codified by some numerical invariants which, in general, are semi-continuous. If we do not fix a section s, then we also accept multiple singularities in the fiber X_{t}. If i is an invariant associated with germs of normal surface singularities, then ∑_{X_{t}} i will denote the sum ∑ i(X_{t}, p) over all the singular points p of the fiber X_{t}.

5.3.1. Semi-continuity. Let λ : (X , 0) → (D, 0) be a flat deformation of (X, 0). Then (i) [112] (Z_{K,m}^{2})_{t} (t ∈ D) is lower semi-continuous; i.e. (Z_{K,m}^{2})_{0} ≤ ∑_{X_{t}} Z_{K,m}^{2}. (ii) [24] (p_{g})_{t} is lower semi-continuous; i.e. p_{g}(X, 0) ≥ ∑_{X_{t}} p_{g}.

5.4. Very weak simultaneous resolutions. First we mention a result of Brieskorn [13] which says that if a deformation λ admits a simultaneous RDP resolution, then by a finite surjective base-change, λ can be transformed onto a deformation which admits a very weak simultaneous resolution. In particular, if we accept base-changes, then the obstructions to have a RDP, or a very weak simultaneous resolution are the same. Notice also that the RDP resolution is unique (if it exists, cf. [42]); but the very weak simultaneous resolution, in general, is not.

5.4.1. Topological characterization. [50]

(i) Let λ : (X , 0) → (D, 0) be a flat deformation of a normal Gorenstein surface singularity (X, 0). Then t ↦ ∑_{X_{t}} Z_{K,m}^{2} is constant if and only if λ admits a simultaneous RDP resolution.

Here one can verify (see e.g. [50], page 12), that the fact that (X, 0) is normal Gorenstein automatically assures that X and all the fibers X_{t} are normal Gorenstein.

(ii) Let λ be a family of normal Gorenstein singularities (X_{t}, s(t)). If Z_{K,m}^{2}(X_{t}, s(t)) is constant, then p_{g}(X_{t}, s(t)) is also constant.

5.4.2. Example. Let λ : (ℂ^{3}, 0) → (ℂ, 0) be a simple hypersurface singularity (with minimal resolution graph of type A-D-E). Then in the minimal resolution graph of (X, 0) = (λ^{−1}(0), 0) one has Z_{K} = 0. On the other hand all the other fibers are smooth. Hence, for any t one has ∑_{X_{t}} Z_{K,min}^{2} = 0. The existence of the very weak simultaneous resolution (after a base change) shows that the Milnor fiber of λ is diffeomorphic to X̃, the minimal resolution of (X, 0).

This deformation also exemplifies the necessity of the base change in Brieskorn result 5.4. Indeed, for a smoothing as above one needs a Galois base change with the corresponding Weyl group W. E.g., in the case of A_{1} singularity, W = ℤ_{2}, and {x^{2} +y^{2} +z^{2} = t^{2}} admits a very weak simultaneous resolution.

5.4.3. Example. Let (X, 0) be the hypersurface singularity ({z^{2} = x^{3} + y^{12}}, 0) ⊂ (ℂ^{3}, 0). Let X be {z^{2} = x^{3} + (y^{2} + t)^{6}}, a deformation over t ∈ D. The minimal resolution of (X, 0) has two irreducible exceptional divisors E = E_{1} ∪ E_{2} with g_{1} = 1, g_{2} = 0, E_{1}^{2} = −1, E_{2}^{2} = −2, E_{1} · E_{2} = 1, Z_{K} = 2E_{1} + E_{2}. Hence Z_{K}^{2} = −2. On the other hand, the singular point in this deformation splits into two singular points: for t ≠ 0, X_{t} has two singular points, each having Z_{K}^{2} = −1. Therefore, this deformation admits a very weak (or RDP) simultaneous resolution.

5.4.4. Remark. The Gorenstein assumption in 5.4.1 cannot be dropped, see e.g. [42](2.8). Nevertheless, 5.4.1 has a global version without the Gorenstein assumption [42](2.1); see also [42](2.25) for a possible local analog (suggested by Wahl). For an example when 5.4.1 fails without the Gorenstein assumption see also 6.3 here.

5.5. Topological characterization of weak simultaneous resolutions. [50, 42] Let λ : X → D be a flat family of normal surface singularities, and s : D → X a section of λ. Then the germs of {X_{t}}_{t∈D} along s are pairwise homeomorphic (i.e. each X_{t} has a singularity at s(t) such that (X_{t}, s(t)) is homeomorphic to (X, 0)) if and only if λ admits a weak simultaneous resolution. (Obviously, the topological condition is equivalent with the stability of the links: M_{t} = M for any t.)

5.5.1. Example. Assume that we are in the situation of 5.4.1 (ii); i.e. λ is a family of normal Gorenstein singularities (X_{t}, s(t)). Then if the link is constant, then evidently Z_{K,m}^{2}(X_{t}, s(t)) is constant too, hence p_{g}(X_{t}, s(t)) is also constant.

If, additionally, each (X_{t}, s(t)) has a smoothing, with Milnor number μ_{t}, then using the above statement and 6.5(2), one gets that if the link is constant then μ_{t} is also constant.

5.6. Strong simultaneous resolutions. We assume that λ is a flat deformation of isolated hypersurface singularities (X_{t}, s(t)) (t ∈ D).

We recall (see [107]), that for any isolated hypersurface singularity (X, 0), Teissier defined an invariant μ^{∗}. In the two-dimensional case, μ^{∗} = (μ^{(3)}, μ^{(2)}, μ^{(1)}, μ^{(0)}) has the following significance: μ^{(3)} is the Milnor number μ, μ^{(1)} = mult(X, 0) − 1, μ^{(0)} = 1, and finally, μ^{(2)} is the Milnor number of the plane curve singularity obtained by cutting (X, 0) by a generic hyperplane section.

In [109] Teissier proved that the existence of a strong simultaneous resolution implies the Whitney condition along s(D). Moreover, by another result of Teissier [107], and respectively of Briançon and Speder [10], the Whitney condition is equivalent to μ_{t}^{∗} constant. Finally, the circle is completed by Laufer in [51], where he proved that μ_{t}^{∗} constant implies strong simultaneous resolution for λ.

In particular, λ admits a strong simultaneous resolution if and only if (μ_{t}, μ_{t}^{(2)}, mult(X_{t})) is constant.

5.7. If one tries to compare the strong, respectively the weak simultaneous resolutions of deformations of isolated hypersurface singularities (cf. 5.5 and 5.6), then one realizes that the difference which really separates them is codified in the invariants μ_{t}^{(2)} and mult(X_{t}). Both depend essentially on the properties of the generic hyperplane sections. Let m_{t} be the maximal ideal of O_{X_{t},s(t)}. Then we recall (cf. 3.1.3) that Π_{t}^{∗}(m_{t}) = I_{t} · OX̃_{t}(Z_{max,t}). If μ_{t} and Π_{t}^{∗}(m_{t}) is constant then λ admits a strong simultaneous resolution. If for a deformation λ (with M_{t} and μ_{t} constant) a strong simultaneous resolution does not exist, then we can expect that either one of the coefficients of Z_{max,t} jumps at t = 0, or at the embedded points V (I_{t}) the structure of I_{t} is not constant.

We recall that our interest is to recover some of the analytic invariants from the link. In particular, one can consider π^{∗}(m_{0}). The above discussion suggests, that a topological characterization of π^{∗}(m_{0}) would imply that a weak simultaneous resolutions exists if and only if a strong one exists.

To stress even more: the Main Question 4.1 in this situation can be translated as follows: in what situation does the existence of a weak simultaneous resolution of a deformation λ imply the existence of a strong one?

5.7.1. Example.[51] Consider the family of weighted homogeneous isolated hypersurface singularities (X_{t}, 0) := {z^{7} + x(y^{9} + x^{27}) + tzy^{8} = 0}. Since the weights are independent of t, by [83], there exists a weak simultaneous resolution (along the singular locus). μ_{t} is constant, and equals 1350. The minimal resolution graph is independent of t, and it is

[figure: −3; −2; −2; E_{1}; E_{2}; E_{3}; [24]]

Then evidently Z_{min} = ∑ E_{v}. On the other hand, Z_{max,t} is not constant: for t = 0 one has Z_{max,0} = E_{1} + 2E_{2} + E_{3}, but for t ≠ 0 one gets Z_{max,t} = Z_{min}. (In fact, even the embedded components of the pullback of the corresponding maximal ideals jump: for each t let Γ_{t} be the minimal resolution graph of (X_{t}, 0) in which the pullback of the maximal ideal has no basepoint. Then Γ_{t} is not constant.)

This should be reflected in the invariant μ_{t}^{∗} as well. It is easy to see that μ_{t}^{(1)} is constant: mult(X_{t}, 0) = 7 for any t. But μ_{0}^{(2)} = 54 and μ_{t}^{(2)} = 50 for t ≠ 0.

5.7.2. Example.[9] By similar argument as above, the topological type (link) of the family {z^{5} + tzy^{6} + xy^{7} + x^{15} = 0} is constant. In fact, μ_{t} and mult(X_{t}, 0) are also constant. On the other hand, I_{t} and μ_{t}^{(2)} are not constant.

5.7.3. Remark. Example 5.5.1 shows that there is no local flat deformation of isolated hypersurface singularities with constant topological type but with Milnor number or geometric genus non-constant (cf. also with 4.6 and 4.7).

6. Smoothings and their invariants

6.1. Definitions/First properties. A flat deformation λ : (X , 0) → (T, 0) (dim T = 1) of (X, 0) is called a (one parameter) smoothing of (X, 0) if X_{t} = λ^{−1}(t) is smooth for t ≠ 0. We will write F for (the C^{∞} type of) X_{t} (t ≠ 0), and we will call it the Milnor fiber of λ. F is a real connected 4-manifold with boundary M. It has the homotopy type of a finite CW-complex of dimension ≤ 2. We write μ := rank H_{2}(F, ℤ), and we call it the Milnor number of the smoothing λ. In fact, H_{1}(F, ℚ) = 0 (fact conjectured by Wahl [119], and proved by Greuel and Steenbrink [31]), hence the topological Euler characteristic χ_{top}(F) of F is 1 + μ. The intersection form in H_{2}(F, ℤ) is symmetric. Let (μ_{+}, μ_{−}, μ_{0}) be its Sylvester invariants and σ := μ_{+} − μ_{−} the signature of the Milnor fiber F.

Notice that the Milnor fiber F, hence all its invariants, depend on the choice of the smoothing λ. Consider the semi-universal deformation (X , 0) → (B, 0) of (X, 0). An irreducible component B_{i}

of B is called a smoothing component of (X, 0) if the general fiber over B_{i} is smooth. Any one parameter smoothing λ of (X, 0) lies on a unique smoothing component.

6.1.1. Not every singularity (X, 0) admits a smoothing, and, in general, it is rather difficult to decide if a specific singularity has any smoothing at all. For different obstructions the reader is invited to search in the papers of Laufer, Wahl, Looijenga, Pinkham [48, 119, 58, 56, 88] (and the references listed in them).

For example, Dolgachev’s triangle singularity D_{pqr} (with Dolgachev numbers p, q, r, cf. 8.7.2) admits a smoothing if p + q + r < 22 [56], cannot be smoothed at all if p + q + r > 22 [119], and the remaining cases p + q + r = 22 are decided in [88]: all but D_{2,10,10} can be smoothed.

6.2. Example. For rational singularities all the components of B are smoothing components. One of these components is distinguished. It can be constructed as follows [6]: let X̃ → X be the minimal resolution of (X, 0). Then all the deformations of X̃ (which preserve the topological type) come from deformations of (X, 0) (since p_{g} is constantly zero). These deformations provide a component, called the Artin component. It is smooth [117].

6.3. Example. In general, even in the case of rational singularities, there are more (smoothing) components. The first example was constructed by Pinkham [86]: the cone over the rational normal curve of degree 4 has two smoothing components. In this case, the minimal resolution graph has only one vertex with e = −4 and g = 0 (hence evidently (X, 0) is rational, cf. 7.3). B has two irreducible components; one of them is the Artin component B_{1}, which has dimension 3. The other component B_{2} has dimension 1. In this case both components are smooth and intersect each other transversally (cf. also with the corresponding results of section 7).

The Milnor number corresponding to the Artin component is μ = 1, while the other Milnor number is μ = 0. This means that the Milnor fiber F above B_{2} is a rational homology ball (with boundary the lens space M = L(4, 1)).

The smoothing above B_{2} can be realized as follows, see e.g. [119], (5.9.1). Consider the hypersurface singularity {f = 0} = {xy − z^{2} = 0} in (ℂ^{3}, 0) with the Z_{2}-diagonal action (±1) ∗ (x, y, z) = (±x, ±y, ±z). The quotient singularity {f = 0}/ℤ_{2} is exactly (X, 0), and the smoothing {f = t} of f induces a one parameter smoothing of (X, 0). The Milnor fiber F_{f} of f has the homotopy type of S^{2}, and ℤ_{2} acts on F_{f} freely; hence χ_{top}(F) = 1, which shows that μ = 0.

By [117], for any rational singularity, the semi-universal deformation restricted above the Artin component admits a very weak simultaneous resolution. This is not generally true for the other components. In the example of Pinkham, the deformation above B_{2} does not admit a very weak simultaneous resolution.

Notice also that in this example if one takes the semi-universal deformation restricted above the Artin component, then the invariant ∑_{X_{t}} Z_{K,m}^{2} is not constant. Indeed, for t = 0 it is −1, otherwise it is zero. Nevertheless, a very weak simultaneous resolution exists. This shows the necessity of the Gorenstein assumption in 5.4.1.

6.4. There are interesting relations which connect the topological invariants Z_{K}^{2} + #V, b_{1}(M) (the first Betti number of the link M), the smoothing invarints μ and σ, and the analytic invariant p_{g}. For different levels of generality, see [21, 48, 102, 119, 58].

6.5. Theorem. Let λ be a smoothing of the normal surface singularity (X, 0). Then

(1) 4p_{g} = μ + σ + b_{1}(M). In addition, if (X, 0) is Gorenstein, then

(2) μ = 12p_{g} + Z_{K}^{2} + #V − b_{1}(M). In particular, for smoothing of Gorenstein singularities, (1) and (2) give

(3) σ + 8p_{g} + Z_{K}^{2} + #V = 0.

This shows that modulo the link-invariants Z_{K}^{2} + #V and b_{1}(M), there are two (independent) relations connecting p_{g}, μ and σ, provided that (X, 0) is Gorenstein. So, if by some other argument one can recover one of them from the topology of M, then all of them can be determined from M.

(2) and (3) also show that μ and σ are independent of the smoothing (if (X, 0) is Gorenstein).

6.5.1. Wahl in [119] added to this list some other relations. (He also explained clearly a recipe by which one can create even more identities. Notice also that Looijenga in [57] proved the validity of the globalization assumption used in [119]; see also the last remark of [57].) We will recall in the next paragraphs only one such identity which is strongly related to deformations.

Let λ : (X , 0) → (D, 0) be a smoothing, and consider the relative derivations θ_{X /D}. Then coker(θ_{X /D} ⊗ O_{X} → θ_{X}) has finite dimension, which will be denoted by β. Note that β is a priori analytic invariant of the smoothing λ. Recall that θ denotes h^{1}(X̃, θX̃).

6.6. Theorem. [Wahl][119] Consider the minimal resolution of (X, 0) with #V_{min} irreducible exceptional components and canonical cycle Z_{K,m}. Then, with the above notations, one has:

(1) β equals the dimension of the irreducible component B_{i} of B on which the smoothing λ occurs.

(2) β − θ + 14p_{g} = 2(μ + b_{1}(M) − #V_{min}).

If (X, 0) is Gorenstein, then this (via 6.5(2)) transforms into

(3) β = θ + 10p_{g} + 2Z_{K,m}^{2}.

Notice that (by (2)) β depends only on (X, 0) and the Milnor number of the smoothing λ; in fact, β − 2μ is independent of the smoothing. If (X, 0) is Gorenstein, then β itself is independent of the smoothing.

6.6.1. Example. If (X, 0) is rational then p_{g} = b_{1}(M) = 0. Then, for any smoothing σ = −μ (i.e. the intersection form on H_{2}(F) is negative definite, cf. 6.5(1)), 0 ≤ μ ≤ #V_{min}, and

β = θ − 2(#V_{min} − μ).

For smoothings supported by the Artin component B_{1} one has μ = #V_{min} and dim B_{1} = β = θ. Assume that (X, 0) is rational, but not RDP. Then Wahl proved (cf. [7], Appendix) that

dim T_{X,0}^{1} ≥ θ + emb dim(X, 0) − 4,

which gives a lower bound for the codimension of the Artin component in T_{X,0}^{1}. For many cases the equality holds (see e.g. [7], cf. also with 7.7 here). But Wahl have found an example when the inequality is strict (cf. [7]).

For the ultimate formula for the codimension of the Artin component of a rational singularity see 6.7. Remark. In general it is extremely difficult to determine the number of irreducible components of B, their dimensions, and so on. Then it is really remarkable if in some cases this can be done, and it is even more remarkable if this can be done only from the topology of the link. Such examples will be discussed in the next section.

7. Rational singularities

7.1. In the previous sections we listed a large number of examples showing that some specific analytic or smoothing invariants cannot always be determined from the topology of the link.

In this section, we start our list of “positive results”. As we will see, in the case of rational singularities, the topology indeed carries a lot of analytic and smoothing information.

Recall that, by definition, (X, 0) is rational if p_{g} = 0. In the sequel we fix a resolution π. It is easy to see that p_{g} = 0 if and only if h^{1}(Z) = 0 for any Z > 0. In particular, all the genera g_{v} should vanish, and Γ(π) should be a tree.

The main point is that Artin succeeded to replace the vanishings of h^{1}(Z)’s by a criterion formulated in terms of χ(Z). In fact, the beauty of the second part of the next characterization is that it is enough to consider only one cycle, namely the fundamental cycle Z_{min}. It is instructive to recall that for any normal surface singularity h^{0}(Z_{min}) = 1, hence χ(Z_{min}) ≤ 1.

7.2. Topological characterizations of rational singularities. [Artin][4, 5] (a) p_{g} = 0 if and only if χ(Z) ≥ 1 for all cycles Z > 0.

(b) p_{g} = 0 if and only if χ(Z_{min}) = 1.

Notice that the characterizations (a) and (b) are independent of the choice of the resolution π. If a resolution graph satisfies (a) or (b), we say that it is a rational graph. Any rational graph is automatically good.

7.3. Examples. (a) If a rational singularity (X, 0) is numerically Gorenstein, then in the minimal resolution Z_{K} = 0. In particular, (X, 0) is a RDP.

(b) Let Γ be an arbitrary tree. For any vertex v set δ_{v} to be the number of edges with endpoint

v. Consider the decorations g_{v} = 0 and e_{v} = \{ \begin{matrix}−δ_{v} & if δ_{v} ≠ 1 \\ −2 & if δ_{v} = 1 \end{matrix} for any v. Then the intersection

matrix I is automatically negative definite, hence Γ is the minimal resolution graph of some singularity. It is not difficult to show that Z_{min} = ∑_{v} E_{v}, χ(Z_{min}) = 1, hence any singularity with minimal resolution graph Γ is rational. Moreover, Z_{min}^{2} = −#{v : δ_{v} = 1}. [This also shows that the multiplicity and the embedded dimension of a rational singularity can be arbitrarily large (cf. 7.4).]

(c) The class of rational graphs is closed while taking subgraphs and decreasing self-intersections. In particular, in the above example (a), one can decrease any of the decorations e_{v}, and still obtain a rational graph. For these modified graphs one still has Z_{min} = ∑_{v} E_{v}.

Rational surface singularities with reduced fundamental cycle (i.e. with Z_{min} = ∑ E_{v}) are also called minimal singularities (cf. [42], section 7). E.g., the cyclic quotients are minimal.

(d) There is another important subclass of rational singularities, the so-called sandwiched singularities, studied by Zariski, Lipman, Hironaka, Spivakovsky (who invented the name of the family), Theo de Jong, van Straten and others, see e.g. [100, 38]. A sandwiched singularity is, by definition, a normal surface singularity that admits a birational map to (ℂ^{2}, 0). If we consider a resolution X̃ → X of such a singularity, then we get a diagram (X̃, E) → (X, 0) → (ℂ^{2}, 0). In particular, X is sandwiched between two smooth spaces via birational maps.

They are also characterized by their (minimal) resolution graphs [100]. Indeed, any sandwiched singularity can be obtained as follows. Consider a plane curve singularity (C, 0) ⊂ (ℂ^{2}, 0), and let Y → ℂ^{2} be a (in general, non-minimal) embedded resolution ϕ of it. Consider the collection E of those irreducible exceptional divisors which are not (−1)-curves (and assume that they form a connected curve). If one contracts E then one gets a sandwiched singularity, and any sandwiched singularity can be obtained in this way (although the choice of (C, 0) and ϕ is not unique). Notice that this can be reformulated in terms of the combinatorics of the graph as well.

The next theorem targets some of the analytic invariants introduced in section 3.

7.4. Theorem. [Artin][4, 5] Assume that (X, 0) is rational and k ≥ 1. Then:

(a) Z_{an} = Z_{top}, in particular Z_{min} = Z_{max};

(b) π^{∗}m_{0}^{k} = O(−kZ_{min}), in particular mult(X, 0) = −Z_{min}^{2};

(c) dim_{ℂ} m_{0}^{k} / m_{0}^{k+1} = −kZ_{min}^{2} + 1, in particular emb dim(X, 0) = −Z_{min}^{2} + 1;

(d) f_{HS}(k) = −k(k − 1)/2 · Z_{min}^{2} + k.

7.5. Next we discuss the base space B of the semi-universal deformation of (X, 0). For some results already mentioned in the previous section, see 6.2, 6.3, 6.6.1.

Although there is a very general conjecture of Kollár about B, valid for any rational singularity [43], we prefer to present the case of cyclic quotient singularities only.

Notice that the cyclic quotient singularities are taut singularities, in particular any analytic invariant can be deduced from the resolution graph, cf. 4.4. Hence, from the point of view of the Main Question 4.1 this case is not really exceptional. Nevertheless, we present this case in order to show, that even in those cases when a priori we know that some invarint is topological, its representation from the graph can be really complicated (and sometimes provided by a combinatorial construction with an extraordinary intrinsic beauty).

For cyclic quotient singularities, Riemenschneider in [93] determined the infinitesimal deformations T_{X,0}^{1}, and later Arndt gave the equations of the base space B [1]. Here we will reproduce the combintorial part of some results obtained by J. Christophersen, J. Stevens and J. Kollár and N.I. Shepherd-Barron. We will follow [105].

7.6. Combinatorial characterization of B. Assume as above that n/q = [b_{1}, …, b_{s}]. We write n/(n − q) = [a_{2}, …, a_{e−1}]. (Here we will follow the traditional index notation, though its reason will not be explained in this article.) These two expansions are dual; one finds the first one from the second one with Riemenschneider’s point diagram: place in the i-th row a_{i} − 1 dots, the first one under the last dot of the (i − 1)-st row; then column j contains b_{j} − 1 dots. From this it is easy to see that the length of the sequence a_{∗} (i.e. e − 2) is 1 + ∑_{j}(b_{j} − 2). Hence, the integer e is exactly emb dim(X_{n,q}). Indeed, since Z_{min} = ∑ E_{v}, 7.4 gives emb dim = 3 + ∑_{j}(b_{j} − 2).

If e = 2, then (X, 0) is smooth. If e = 3 then (X, 0) is a hypersurface singularity A_{s}; in particular B is smooth and irreducible of dimension μ = s.

In the sequel we will assume that e ≥ 4.

Let K_{e−2} be the set tuples of positive integers [k] = (k_{2}, …, k_{e−2}), (k_{i} ≥ 1), such that the continued fraction [k_{2}, …, k_{e−1}] is zero. The cardinality of this set is \frac{1}{e−2} \binom{2(e−3)}{e−3}, the Catalan number C_{e−2}. This number also describes the number of ways to subdivide an (e − 1)-gon in triangles. The correspondence between the two methods of counting is the following. Mark one of the vertices of the (e − 1)-gon by ∗ (this is called the distinguished vertex), and the others by v_{2}, …, v_{e−1}. Given a subdivision, we define k_{i} as the number of triangles having v_{i} as vertex (2 ≤ i ≤ e − 1). Then the integers [k] = (k_{2}, …, k_{e−1}) determine an element in K_{e−2}.

In a similar way we also define the integer k_{∗} associated with the distinguished vertex ∗. It satisfies

k_{∗} + ∑ k_{i} = 3(e − 3).

Then one has the following numerical identities (for details, see [105]):

7.7. Theorem. [Riemenschneider, Christophersen, Stevens]

• dim T_{X,0}^{1} = ∑_{j}(b_{j} − 1) + e − 4 = 2e − 7 + s.

• θ = ∑_{j}(b_{j} − 1) = e − 3 + s.

• There is a one-to-one correspondence between the irreducible components of B and continued fractions [k] ∈ K_{e−2} with k_{i} ≤ a_{i} for all i. For each [k] we denote the corresponding component by B_{[k]}, its dimension by β_{[k]}, and the corresponding Milnor number by μ_{[k]}.

• Each B_{[k]} is smooth.

• β_{[k]} = ∑_{j}(b_{j} − 1) − 2(e − 3 − k_{∗}) = s + k_{∗} − (e − 3 − k_{∗}).

• μ_{[k]} = s − (e − 3 − k_{∗}) = β_{[k]} − k_{∗} ≥ 0.

• For any [k] clearly k_{∗} ≤ e − 3, hence β_{[k]} ≤ θ and μ_{[k]} ≤ s.

• The Artin component corresponds to [k] = [1, 2, …, 2, 1]. This is the unique [k] with k_{∗} = e − 3. In particular, the Artin component is the unique component with β_{[k]} = θ and μ_{[k]} = s.

7.7.1. The above data follows from a very precise and detailed analysis of the first order deformation space T_{X,0}^{1}, and the corresponding equations of B in it. Of course, from this one can obtain even more information about B (e.g. about the intersection properties of the components). On the other hand, there is another, completely different approach of Kollár and Shepherd-Barron, which finds all the components of B, and all their dimensions, without computing T_{X,0}^{1} → T_{X,0}^{2} (but provides no information e.g. about their intersections) [42]. This gives possibilities for remarkable generalizations (see [43]) to situations when T_{X,0}^{1} → T_{X,0}^{2} is out of reach.

This construction depends essentially on a very special subclass of cyclic quotient singularities.

7.8. The class T of cyclic quotient singularities. [58](5.9), [118], cf. also with [42]. For a cyclic quotient singularity X_{n,q}, the following facts are equivalent:

(i) Z_{K}^{2} ∈ ℤ;

(ii) (q + 1)^{2}/n ∈ ℤ;

(iii) For some positive integers r, l and d with d ≤ r and gcd(d, r) = 1 one has n = r^{2}l and q = drl − 1.

In fact, the integers r, l, d can be recovered from n and q by

l := gcd(q + 1, n, (q + 1)^{2}/n); rl := gcd(q + 1, n); drl := q + 1.

Notice that r = 1 implies d = 1, n = l and q = l − 1; hence in this case (X, 0) is the RDP A_{l−1}. (iv) The minimal resolution graph can be recognized as follows: the graph is either an A_{k} graph,

or can be obtained by starting with one of the graphs

[figure: −4; or; −3; −2; −2; −2; −3; ⋯]

and iterating a few times (in an arbitrary order) the next step. By one step, change [b_{1}, …, b_{k}] to either [2, b_{1}, …, b_{k−1}, b_{k} + 1] or [b_{1} + 1, b_{2}, …, b_{k}, 2] and take the corresponding graph.

7.8.1. Definitions. A cyclic quotient singularity satisfying one of the above properties is called of class T. For any such singularity one defines def′(X_{n,q}) by l if r > 1, respectively by l − 1 if r = 1. [def′(X) is the dimension of Def′(X) ⊂Def(X) which corresponds to the ℚ-Gorenstein deformations, cf. [42]; r is called the index of X_{n,q}.]

In [42], the key object which guides the irreducible components of B is the set of P -resolutions.

7.9. Definition. A P -resolution of a cyclic quotient singularity (X, 0) is a partial resolution f : Y → X (i.e. a modification which is an isomorphism above X \ {0}) such that K_{Y} is ample relative to f (cf. next paragraph), and Y has only singularities of class T.

7.9.1. Fix a modification f : Y → X. Let F_{j} (1 ≤ j ≤ k) be the set of irreducible exceptional divisors of f, and set Sing(Y) = {P_{1}, …, P_{l}}. Let ρ : X̃ → Y be the minimal resolution of Y, and set f ∘ ρ = π. Denote the strict transform (via ρ) of F_{j} by F′_{j}. Let Γ_{i} (1 ≤ i ≤ l) be the subgraph of the resolution graph Γ(π) of π corresponding to the singular points P_{i}. Let Z_{K}(Γ(π)), respectively Z_{K}(Γ_{i}) be the canonical cycle of Γ(π), respectively of Γ_{i}.

Then K_{Y} is ample relative to f (i.e. K_{Y} · F_{j} > 0 for all j) if and only if

( Z_{K}(Γ(π)) −_{∑i} Z_{K}(Γ_{i}) ) · F′_{j} < 0 for any 1 ≤ j ≤ k.

In particular, the set of P -resolutions can be determined combinatorially from the plumbing graph of the link of (X, 0). The set of P -resolutions is not empty. For example, the minimal RDP resolution of (X, 0) is a P -resolution.

If f : Y → X is a P -resolution, set def′(f) := ∑_{i=1}^{l} def′(Y, P_{i}) (cf. 7.8.1).

The set of exceptional divisors F′_{j} (1 ≤ j ≤ k) form h connected, pairwise disjoint exceptional curve C_{1}, …, C_{h}, each determining a cyclic quotient singularity (X_{j}, Q_{j}) (by contracting C_{j} into the point Q_{j}). The dimension of the Artin component of (X_{j}, Q_{j}) is denoted by θ(X_{j}, Q_{j}) (cf. 7.7). Set θ(f) := ∑_{j=1}^{h} θ(X_{j}, Q_{j}).

7.9.2. [42](3.14). When one searches for all the P -resolutions, the next fact is very helpful: Any P -resolution is dominated by the maximal resolution of (X, 0).

Here, a resolution π : X̃ → X is maximal if all the coefficients of Z_{K}(π) are strictly positive, and π is not dominated by any other resolution with this property. Any cyclic quotient singularity has a unique maximal resolution.

Now, we can state (3.9) of [42].

7.10. Theorem. [Kollár–Shepherd-Barron] For any cyclic quotient singularity (X, 0), there is a oneto-one correspondence between the set of P -resolutions of (X, 0) and the set of irreducible components of the base space B of the semi-universal deformation of (X, 0). If the component B(f) corresponds to the P -resolution f, then dim B(f) =def′(f) + θ(f).

Under this correspondence, the Artin component corresponds to the minimal RDP resolution.

The correspondence between the pictures 7.7 and 7.10 is realized in [105].

7.10.1. Example. (Cf. with 6.3.) Assume that the minimal resolution graph of (X, 0) has only one vertex with g_{v} = 0 and e_{v} = −4. Then (X, 0) has two P -resolution. One of then is the minimal resolution, the other is the identity of (X, 0).

7.10.2. Example.[42] Assume that the minimal resolution X̃ of (X, 0) has the dual graph:

[figure: −3; −4; −2]

Consider also the resolution X̃′ → X with graph:

[figure: −4; −1; −5; −2]

Then (X, 0) has three P -resolutions Y_{1}, Y_{2} and Y_{3}. Y_{1} is the minimal RDP resolution obtained from X̃ by contracting the (−2)-curve. Y_{2} is obtained from X̃ by contracting the (−4)-curve. Finally, Y_{3} is obtained from X̃′ by contracting all the curves except the (−1)-curve.

The corresponding dimensions of the irreducible components of B are 6, 4 and 2.

7.11. Remark. The base space B is smooth if (X, 0) is rational with mult(X, 0) ≤ 3. See [36] and [106] for the description of the deformation of rational quadruple points, [37] for minimal rational singularities, and [38] for sandwiched singularities. All these descriptions are topological.

7.12. Kollár’s Conjectures. As we already mentioned, Kollár formulated a conjecture about the irreducible components of B, in terms of P -resolutions, for any rational singularity [43]. He also stated a simplified version of this rather complex conjecture as follows. Let (X, 0) be a rational singularity. Suppose that any irreducible exceptional divisor of the minimal resolution of (X, 0) has self-intersection at most −5. Then B has only one component, namely the Artin component. This second conjecture was verified for cyclic quotients [105] and minimal rational singularities [38].

8. “Weakly” Elliptic singularities

8.1. Recall that for any fixed resolution graph Γ(π) of a normal surface singularity (X, 0) one has χ(Z_{min}) ≤ 1. Similarly, min_{Z>0} χ(Z) ≤ 1. By 7.2, (X, 0) is rational if and only if χ(Z_{min}) = 1, or equivalently, min_{Z>0} χ(Z) = 1.

Now, assume that min_{Z>0} χ(Z) = 0. Then χ(Z_{min}) ≥ 0; but χ(Z_{min}) = 1 is impossible by the previous statement. Hence χ(Z_{min}) = 0. Conversely, if χ(Z_{min}) = 0, then min_{Z>0} χ(Z) = 0 by a result of Laufer [47], page 1281. Laufer’s argument uses the theory of analytic deformations. For a different, completely combinatorial proof see [66] (4.3).

8.2. Definition. [Wagreich, Laufer] (X, 0) is called elliptic if Γ(π) is elliptic. Γ(π) is elliptic if min_{Z>0} χ(Z) = 0, or equivalently, χ(Z_{min}) = 0. (Wagreich used the first vanishing [113], Laufer the second one [47].) The definition is independent of the choice of the resolution (graph).

8.3. Remark. Since p_{g} ≥ 1 − min_{Z>0} χ(Z), for any elliptic singularity p_{g} ≥ 1. In fact, the set of elliptic singularities includes all the singularities with p_{g} = 1, and all the Gorenstein singularities with p_{g} = 2. But an elliptic singularity might have arbitrary large geometric genus. Perhaps the following terminology removes the possible confusion: elliptic=“weakly” elliptic; {p_{g} = 1}=“strongly” elliptic. By this one emphasizes the difference between elliptic singularities and those with p_{g} = 1.

8.4. Example. Consider the hypersurface singularity (X_{m}, 0) = {x^{2}+y^{3}+z^{6+6m)} = 0} with m ≥ 0. The minimal resolution graph has m + 1 vertices

[figure: −2; −1; −2; −2; [1]; E_{1}; E_{0}; E_{2}; E_{m}; ⋯]

Then Z_{min} = ∑_{v=0}^{m} E_{v}, Z_{K} = ∑_{v=0}^{m}(m + 1 − v)E_{v}, and χ(Z_{min}) = 0. The geometric genus can be computed easily by 6.5. One gets b_{1}(M) = 2, μ = 12m + 10, and p_{g} = m + 1.

8.4.1. Notice that the graph discussed in 4.6 is elliptic (corresponds to m = 2 above). 4.6 shows that the geometric genus of an elliptic singularity, in general, is not topological.

On the other hand, Laufer in [47] identified topologically a subclass of elliptic singularities for which p_{g} is topological (in fact, p_{g} = 1). It is the set of minimally elliptic singularities. They are defined via their minimally elliptic cycle.

8.5. Definition. The minimally elliptic cycle. [Laufer] Let π be the minimal resolution. A cycle Z > 0 is called minimally elliptic if χ(Z) = 0 and for any 0 < D < Z one has χ(D) > 0.

8.5.1. Laufer in [47] proved that if χ(Z_{min}) = 0 then there is a unique minimally elliptic cycle. It will be denoted by C. Clearly, by its minimality, C ≤ Z_{min}.

8.5.2. Notice also that if (X, 0) is numerically Gorenstein, then the Z_{K} associated with the minimal resolution is in Z_{top}, hence Z_{min} ≤ Z_{K}.

8.6. Theorem. Minimally elliptic singularities. [Laufer] [47] Consider the minimal resolution π of (X, 0). Then the following statements are equivalent. If a singularity satisfies them, it is called minimally elliptic.

(i) (X, 0) is numerically Gorenstein and Z_{K} = Z_{min}.

(ii) χ(Z_{min}) = 0 and any proper subgraph of Γ(π) supports a rational singularity.

(iii) (X, 0) is numerically Gorenstein and Z_{K} is a minimally elliptic cycle.

(iv) Z_{min} is a minimally elliptic cycle.

(v) p_{g} = 1 and (X, 0) is Gorenstein.

In particular, for minimally elliptic singularities Z_{min} = Z_{K} = C.

Notice that this theorem gives a topological characterization of Gorenstein singularities with p_{g} = 1. (The first four conditions are topological. (ii) provides the name of the “family”.)

8.7. Example. Assume that Γ(π) consists of only one vertex with g_{v} = 1, and arbitrary selfintersection e_{v}. Then it is elliptic. E.g., consider the Brieskorn hypersurface singularities ({x^{a} + y^{b} + z^{c} = 0}, 0) with (a, b, c) = (2, 3, 6), (2, 4, 4) and (3, 3, 3). Then e_{v} is −1, −2, −3 respectively.

8.7.1. Example. Cusp Singularities. In this case the minimal very good resolution graph has s ≥ 3 vertices, forming a unique cycle. (Very good means that no irreducible exceptional divisor has a self-intersection point, and no pair of irreducible exceptional divisors has two intersection points). The decorations are g_{v} = 0 for all v, and the self-intersections e_{v} ≥ 2 for all v. For such a graph the intersection matrix is negative definite if and only if ∑_{v}(e_{v} − 2) > 0.

For cusps Z_{K} = Z_{min} = ∑ E_{v}. The link M is a torus bundle over the circle S^{1} with monodromy

(written multiplicatively) ( \begin{matrix}e_{s} & 1 \\ −1 & 0 \end{matrix} ) ⋯ ( \begin{matrix}e_{1} & 1 \\ −1 & 0 \end{matrix} ) (cf. [76]).

8.7.2. Example. “Polygonal” singularities. Assume that the resolution graph Γ has the following form with s + 1 vertices, all of them with g_{v} = 0:

[figure: E_{0}; 2 − s; E_{1}; e_{1}; E_{2}; e_{2}; e_{s}; E_{s}]

The intersection matrix is negative definite if and only if 2 − s + ∑_{v>0} 1/e_{v} < 0.

If s > 3 then Γ is minimal, and Z_{K} = Z_{min} = 2E_{0} + ∑_{v>0} E_{v}.

If s = 3, then Z_{min} ≠ Z_{K}. But in this case Γ is not minimal: E_{0} is a rational (−1)-curve, so it should be contracted. Then in the new minimal graph, Z_{K} = Z_{min} = ∑ E_{v} again. If s = 3 then (X, 0) is the Dolgachev’s triangle singularity D_{e_{1}−1,e_{2}−1,e_{3}−1}.

8.8. Remark. Laufer in [47] computed the Hilbert-Samuel function of minimally elliptic singularities from their graphs. This will be generalized in 8.11. Example 4.6 (see also 9.1) shows that the topological characterization of p_{g}, or of the Hilbert-Samuel function, cannot be continued for more general elliptic singularities. Nevertheless, the main message of the author’s paper [70] is that for Gorenstein singularities with b_{1}(M) = 0 the Artin–Laufer program can be continued.

In the proof, the crucial topological ingredient is the elliptic sequence of an elliptic singularity, introduced and intensively studied by S.S.-T. Yau, see [123]–[125].

8.9. Definition. (a) [S.S.-T. Yau] In the numerically Gorenstein case the definition of the elliptic sequence is the following. Let π : X̃ → X be the minimal resolution of (X, 0), and E = π^{−1}(0) as above. The elliptic sequence consists of a sequence {Z_{B_{j}}}_{j=0}^{m}, where Z_{B_{j}} is the fundamental cycle of B_{j} ⊂ E. We define {B_{j}}_{j} inductively as follows. For j = 0 take B_{0} = E, hence Z_{B_{0}} = Z_{min}. Then Z_{K} ≥ Z_{B_{0}}. If Z_{K} > Z_{B_{0}} then we set B_{1} := |Z_{K} −Z_{B_{0}}|. Similarly, if B_{i} is already defined for any i ≤ j, then Z_{K} ≥ Z_{B_{0}} +⋯+Z_{B_{j}}. If the inequality is strict then we define B_{j+1} := |Z_{K} −Z_{B_{0}} −⋯−Z_{B_{j}}|, otherwise we stop. In particular, Z_{K} = ∑_{j=0}^{m} Z_{B_{j}}. It turns out that Z_{B_{m}} is the minimally elliptic cycle C. The length of the elliptic sequence {Z_{B_{j}}}_{j=0}^{m} is m + 1. The case m = 0 (i.e. when the identity Z_{K} = Z_{min} holds) corresponds exactly to the minimally elliptic singularities.

(b) If we contract the connected exceptional curve B_{j} ⊂ X̃, we obtain a unique singular point (X̃/B_{j}, p_{j}), 0 ≤ j ≤ m. We also consider C′_{t} = ∑_{i=t}^{m} Z_{B_{i}}, (0 ≤ t ≤ m).

8.9.1. Yau’s inequality. The point of the definition 8.9(a) is that m + 1 is an optimal topological upper bound for p_{g}. Indeed, S. S.-T. Yau proved that for a numerically Gorenstein elliptic singularity p_{g} ≤ m + 1 [125, (3.9)].

8.9.2. Example. The elliptic singularity in 8.4 has length m + 1. Example 4.6 (which corresponds to m = 2) shows that strict inequality p_{g} < m + 1 can occur.

In [70], the author proved the following characterizations of the “extremal property” p_{g} = m + 1. [S. S.-T. Yau called a singularity satisfying p_{g} = m + 1 maximally elliptic.]

8.10. Theorem. Characterization of p_{g} = m + 1. [70] Assume that (X, 0) is a numerically Gorenstein elliptic singularity. Then the following facts are equivalent:

(a) p_{g} = m + 1;

(b) The line bundles O_{C′_{j+1}}(−Z_{B_{j}}) are trivial in P ic^{0}(C′_{j+1}) for all 0 ≤ j ≤ m − 1.

Moreover, in the Gorenstein case, the triviality of the line bundles O_{C′_{j+1}}(−Z_{B_{j}}) (1 ≤ j ≤ m − 1) implies the triviality of O_{C′_{1}}(−Z_{B_{0}}).

(c) The singularities (X̃/B_{j}, p_{j}) are Gorenstein for all 0 ≤ j ≤ m − 1. (Notice that (X̃/B_{m}, p_{m}) is automatically Gorenstein because it is minimally elliptic.)

The fact that a numerically Gorenstein singularity (X, 0) with p_{g} = m + 1 is Gorenstein was first proved by S. S.-T. Yau in [125, (3.11)], cf. also with Reid’s paper [91]. But the Gorenstein property of (X, 0) does not imply the extremal property p_{g} = m + 1. By the above result, all the singularity tower (X̃/B_{j}, p_{j})_{j} must be Gorenstein.

8.10.1. Examples. Assume that (X, 0) is a numerically Gorenstein elliptic singularity. (a) If m = 1 then p_{g} ≤ 2. For the generic analytic structure p_{g} = 1, and:

p_{g} = 2 ⇔ O_{C}(−Z_{min}) is trivial ⇔ (X, 0) is Gorenstein.

(b) Assume that m = 2. Then in the numerically Gorenstein case we have two obstruction line bundles in 8.10(b). For the generic analytic structure, p_{g} = 1 as above. If (X, 0) is Gorenstein, then 3 ≥ p_{g} ≥ 2. From the above characterization, p_{g} = 3 if and only if O_{C}(−Z_{B_{1}}) is trivial. But for Gorenstein singularities O_{C}(−Z_{min} − Z_{B_{1}}) is trivial. Hence, if (X, 0) is Gorenstein, then:

p_{g} = 3 ⇔ O_{C}(−Z_{min}) is trivial ⇔ (X̃/B_{1}, p_{1}) is Gorenstein.

Example 4.6 shows that for a Gorenstein (X, 0) both p_{g} = 2 and p_{g} = 3 are possible.

The next theorem is a generalization of the results of Laufer [47] and Reid ([91], chapter 4) about the minimally elliptic singularities. In their case m = 0, and the identity p_{g} = m+1 is automatically satisfied. The result completes the program of S. S.-T. Yau about general elliptic singularities. M. Tomari also obtained some similar results for elliptic singularities in the presence of some additional assumptions.

8.11. Theorem. [70] Assume that (X, 0) is an elliptic singularity with p_{g} = m + 1. Let Z_{min} be the fundamental cycle in the minimal resolution π. Then:

(a) Z_{min} = Z_{max};

(b) If Z_{min}^{2} ≤ −2, then π^{∗}(m_{0}) = O(−Z_{min}), hence mult(X, 0) = −Z_{min}^{2};

(c) If Z_{min}^{2} = −1, then π^{∗}(m_{0}) = m_{Q}O(−Z_{min}) for a smooth point Q of E, and mult(X, 0) = 2;

(d) emb dim(X, 0) = max(3, −Z_{min}^{2});

(e) If Z_{min}^{2} ≤ −3 then dim O_{X,0}/m_{0}^{k} = χ(O_{kZ_{min}}) + 1 and dim m_{0}^{k}/m_{0}^{k+1} = −kZ_{min}^{2} (k ≥ 1);

(f) H^{0}(M, O(−kZ_{min})) = m_{0}^{k} for any k ≥ 0 provided that Z_{min}^{2} ≤ −3. Hence, the O_{X,0}– algebra R := ⊕_{k≥0}H^{0}(O(−kZ_{min})) is generated by m_{0}. One can write a generator set of R in the case d := −Z_{min}^{2} ≤ 2 as well. If d = 2, then R is generated by m_{0} in degree 1 and an element y ∈ m_{0}\m_{0}^{2} in degree two. If d = 1, then R is generated by m_{0} in degree 1, by an element y ∈ m_{0}\m_{0}^{2} in degree 2, and z ∈ m_{0} \ (m_{0}^{2}, y) in degree 3.

8.10 and 8.11 show that the vanishing of the obstruction line bundles implies the expected topological behaviour of the Hilbert–Samuel function. The next theorem analyzes these line bundles:

8.12. Theorem. [70] Let (X, 0) be an elliptic Gorenstein singularity. Fix an integer 1 ≤ k ≤ m and assume that for k + 1 ≤ j ≤ m the obstruction line bundles O_{C′_{j}}(−Z_{B_{j−1}}) are trivial. (If k = m then this assumption is vacuous.) Then O_{C′_{k}}(−Z_{B_{k−1}}) has finite order, not greater than k!.

8.12.1. Example. Cf. with 4.6 and 8.10.1(b). Assume that (X, 0) is elliptic Gorenstein, and m = 2. Then O_{C}(−Z_{B_{1}} − Z_{min}) ≈ O_{C}. The above theorem (for m = k = 2) says that the order of O_{C}(−Z_{min}) divides 2, hence:

O_{C}(−Z_{min}) is trivial ⇔ p_{g} = 3; O_{C}(−Z_{min}) has order 2 ⇔ p_{g} = 2.

Using the fact that if b_{1}(M) = 0 then P ic^{0}(Z) is torsion free for any cycle Z > 0, one gets:

8.13. Theorem. [70] Assume that the link of the elliptic Gorenstein singularity (X, 0) is a rational homology sphere. Then p_{g} is a topological invariant: it equals the length of the elliptic sequence in the minimal resolution of (X, 0). In particular, all the consequences of 8.11 hold as well.

8.13.1. Example. In fact, one of the starting points of the author’s investigation was S. S.-T. Yau’s result in [124], which says that hypersurface singularities with b_{1}(M) = 0 and p_{g} = 2 satisfy m = 1. Yau’s argument was based on his classification of all possible dual resolution graphs of hypersurface singularities with p_{g} = 2 (250 cases).

8.14. Remark. For deformation invariants of minimally elliptic singularities, see Wahl’s paper [118] and the references cited there.

9. Arbitrary singularities

9.1. Discussion. Recall that our wish is to compute (some of) the analytic invariants from the topology of the link. In this section we will concentrate mainly on the geometric genus p_{g}. The previous sections teach us that p_{g} can be recovered from the (minimal) resolution graph Γ if χ(Z_{min}) = 1, or if Γ is minimally elliptic. But for more general (even elliptic) singularities, we run into obstructions.

Let us analyze some of the previous counterexamples, and try to understand these obstructions. Example 4.6 presents two hypersurface singularities with the same graph but different p_{g}. Here

the corresponding analytic structures are “nice” (and we do not want to eliminate the hypersurfaces from any kind of general result). Hence, we believe that the “guilty” object responsible for the “bad behaviour” of p_{g} is the elliptic irreducible exceptional divisor in the resolution of (X, 0), i.e. b_{1}(M) > 0.

Now, let us consider any elliptic Gorenstein singularity with p_{g} > 1. Then, by a result of Laufer [47], if one deforms the analytic structure (keeping the topological type), then for the generic analytic structure p_{g} drops to one. This is true even if the link of (X, 0) is “nice”, e.g. if b_{1}(M) = 0. This shows that if we wish to recover the p_{g} from the link, we cannot work with arbitrary (i.e. very general) analytic structures.

The conclusion is: if we wish to continue the Artin-Laufer program, we have to introduce two types of restrictions for our singularities, a topological one and an analytical one. For example, in 8.13 we considered only Gorenstein singularities with b_{1}(M) = 0.

This, and the results of the previous section for elliptic singularities, leds to the following general conjecture. Part (a) formulated for Gorenstein singularities with integral homology sphere links appears as a question in [78](3.2) motivated by the examples and results of that article. Part (a) and (b) for Gorenstein singularities with b_{1}(M) = 0 appears in [70] (page 146) as the main “message” of the article.

9.2. Conjecture. Assume that (X, 0) is ℚ-Gorenstein with b_{1}(M) = 0. Then (a) p_{g} is topological;

(b) Z_{min} = Z_{max}, and the Hilbert-Samuel function is topological.

I am tempted to add to this list even more analytic invariants, and the reader is also invited to add his or her favorite invariant. Nevertheless, we have to be careful with these generalizations. For example, we cannot expect that the Tjurina number of a hypersurface isolated singularity is a topological invariant. Even for rational singularities one can find non-topological discrete invariants, e.g. the number of irreducible components of B and their dimensions. But definitely, we expect that the set of analytic invariants computable from M (under the above restrictions) is rather considerable.

9.3. Remarks. (a) Notice that the family of ℚ-Gorenstein singularities is significantly larger than that of Gorenstein ones. Rational singularities, for example, are ℚ-Gorenstein, and any singularity with a good ℂ^{∗}-action and with b_{1}(M) = 0 is ℚ-Gorenstein [75].

(b) As a starting point, it would be very important to verify the above conjecture for some nontrivial families. It is a big desire (at least of the author of the present article) to see a proof of 9.2 for singularities whose resolution graph is star-shaped. Here there is a very precise topological candidate for p_{g}, namely the expression given by Pinkham valid for singularities with good ℂ^{∗}-action (cf. [87], and 9.4(a)). More precisely, the question is: is it true that for any ℚ-Gorenstein singularity with star-shaped graph (and all g_{v} = 0) p_{g} can be determined by Pinkham’s formula ?

9.4. Remark. In the literature one can find many results and examples of families of singularities involving topological+analytical restrictions.

(a) As we already mentioned, Pinkham in [87] proved that if 0 is the unique singular point of an affine algebraic variety with a good ℂ^{∗}-action and b_{1}(M) = 0, then p_{g} is topological. We will discuss the invariants of such singularities in section 11.

(b) In [78], Neumann and Wahl proved that in the case of a hypersurface singularity of type (X, 0) ={f(x, y) = z^{n}}, whose link is an integral homology sphere, p_{g} is topological. This was generalized for rational homology sphere links in [74]; cf. also with section 13.

(c) Laufer proved in [49] that if (X, 0) has multiplicity two and b_{1}(M) = 0, then Z_{min} = Z_{max} and p_{g} is topological (see also Dixon’s results in [18, 19]).

(d) Karras and Stevens proved in [39, 40], resp. in [103, 104], different results in this spirit for Kulikov and Kodaira singularities. Here the topological restriction is that the graph is “Kodaira” (i.e. for any E_{v} with E_{v} · Z_{min} < 0, the multiplicity of E_{v} in the fundamental cycle Z_{min} must be one). The analytic restriction is codified already in the definition of these singularities, and it implies the existence of a special function on (X, 0) (e.g. (X, 0) is the smoothing of its general hyperplane section). For details, see the above references.

9.5. Discussion. 9.2 raises immediately (at least) two questions: what is the meaning of the topological dependence of p_{g}, and what is a topological candidate for p_{g}?

Let us examine the above problem from a different point of view. Consider the parameter space of all possible analytic structures (defined in some reasonable sense) associated with a fixed topological type (or minimal resolution graph) with b_{1}(M) = 0. This has a natural stratification given by the value of p_{g}. Our goal is to identify that strata for which p_{g} is maximal, and to show that it contains (or it equals) the subset corresponding to the ℚ-Gorenstein structures. Moreover, we claim that this maximal p_{g} can be characterized topologically. This pictures emphasizes the subtlety of the problem: we do not search for the topological characterization of an analytic invariant which lives on a Zariski open set, but we wish to characterize topologically the invariant supported by a tiny stratum on which this invariant is maximal. In other words, we search for an “optimal topological upper bound” for p_{g}, which can be defined (topologically) for any normal surface singularity, and equals p_{g} for some special analytic structures.

Notice that the “length of the elliptic sequence” discussed in the previous section provides exactly this service for numerically Gorenstein elliptic singularities.

Just for comparison: the arithmetic genus p_{a} = 1 − min_{Z>0} χ(Z) is the candidate for an optimal topological lower bound for p_{g}.

9.6. Recently Neumann and Wahl initiated a general program about ℚ-Gorenstein singularities with b_{1}(M) = 0. The starting point was Neumann’s paper [75] which proves that the universal abelian cover of a singularity with a good ℂ^{∗}-action and with b_{1}(M) = 0 is a Brieskorn complete intersection whose weights can be determined from the Seifert invariants of the link. This, and other examples worked out by Neumann and Wahl (see e.g. [80] about quotient-cusps) lead to a rather complex program and package of conjectures. Briefly, it can be formulated as follows.

9.7. Conjecture. [Neumann-Wahl][79] Assume that (X, 0) is ℚ-Gorenstein with b_{1}(M) = 0. Then the universal abelian cover of (X, 0) is an isolated complete intersection singularity. Moreover, the equations of this cover, together with the action of H_{1}(M, ℤ), can be recovered from M (modulo some equisingular and equivariant deformation).

The above conjecture has a surprising aspect: being an isolated complete intersection singularity (in the world of, say, Gorenstein singularities) is extremely restrictive. The above conjecture claims (as a particular case) that any Gorenstein singularity with H_{1}(M, ℤ) = 0 is a complete intersection. [Notice also that there exists no criterion at present which would distinguish the complete intersections from the Gorenstein singularities.]

The remarkable fact in the above conjecture is that in its detailed version it provides a clear recipe for how to write down the equations of the universal abelian cover. This is done in terms of the combinatorics of the resolution graph. In order to be able to write down the equations, the graph should satisfy some interesting arithmetical properties (the so-called semigroup conditions and congruence conditions). Their validity is part of the conjecture. The construction also shows that the embedding dimension of the covering is at most the number of ends (vertices v with δ_{v} = 1) of Γ(X, 0). The reader is invited to see all the details in [79].

9.7.1. Notice that conjecture 9.7 implies the topological nature of p_{g} for the corresponding singularities. One the other hand, in order to prove 9.7, probably first one has to prove that p_{g} is topological, i.e. it is constant in a topological trivial family. Then using this, one can handle the deformation which appears in 9.7.

This reciprocal relationship is nicely exemplified by those Gorenstein singularities whose link has a star-shaped graph with b_{1}(M) = 0. If the singularity is weighted homogeneous then p_{g} is given by Pinkham’s topological formula (cf. 9.3(b)). A singularity is an equisingular deformation of a weighted homogeneous one if and only if its p_{g} equals the number predicted by Pinkham’s formula (cf. [78](3.3)).

9.8. Smoothable singularities. Assume now that (X, 0) is Gorenstein and admits a smoothing λ with Milnor number μ and signature σ. Then, by 6.5, p_{g} is topological if and only if μ, or equivalently, σ is topological. On the other hand, about σ we have really remarkable results.

Fintushel and Stern proved in [25] that for a hypersurface Brieskorn singularity whose link is an integral homology sphere, the Casson invariant λ(M) of the link M equals −#R(M)/2, where #R(M) is the number of conjugacy classes of irreducible SU(2)-representations of π_{1}(M) (a fact conjectured by Neumann and Wahl). It turns out that this number coincides with the number given by Brieskorn’s formula for σ/8. This result was generalized by Neumann and Wahl as follows.

9.9. Theorem. [Neumann-Wahl] [78] λ(M) = σ/8 for all Brieskorn-Hamm complete intersections and suspensions of irreducible plane curve singularities, provided that the link is an integral homology sphere.

Recently, they verified this property for some other particular families of singularities as well, cf. [79, 80, 81]. Moreover, they formulated the following 9.10. Conjecture. [Neumann-Wahl] [78] λ(M) = σ/8 for any isolated complete intersection singularity whose link is an integral homology sphere.

The result 9.9 was re-proved and reinterpreted by Collin and Saveliev (see [16] and [17]) using equivariant Casson invariant and cyclic covering techniques.

The reader is invited to consult [81] as well. This the recent manuscript contains the “Milnor fiber conjecture”; namely, the conjectural description from the link of the Milnor fiber of an isolated complete intresection singularity whose link is integral homology sphere.

The Neumann-Wahl conjecture 9.10 challenged the singularity community: what is the natural generalization of this identity for rational homology sphere links (and how can one interpret the deep geometric meaning of this identity).

Easy particular examples already show that the Casson-Walker invariant [115] (as an immediate natural generalization of the Casson invariant for rational homology sphere 3-manifolds) does not extend the above identity. The right generalization is the subject of the next section.

10. The Seiberg-Witten invariant of the link

10.1. Preliminaries. In this section we will assume that the link M of (X, 0) is a rational homology sphere. We will use the notation H := H_{1}(M, ℤ) (or, by Poincaré duality, H = H^{2}(M, ℤ)).

Our goal is to generalize simultaneously both conjecture 9.2 and the Neumann-Wahl conjecture 9.10. More precisely, we wish to find an optimal topological upper bound for p_{g}, which bounds p_{g} for any normal surface singularity, equals p_{g} in the ℚ-Gorenstein case, and generalizes 9.9 for smoothings of Gorenstein singularities.

This bound is provided by a certain Seiberg-Witten invariant of the link. In fact, one has to consider a certain canonical spin^{c} structure σ_{can} of M (introduced in [72]), and the Seiberg-Witten invariant associated with σ_{can} provides the above optimal bound. In the next paragraphs we recall the needed definitions, for details, see [72].

10.2. The link and its canonical spin^{c} structure. M carries a natural symmetric non-singular bilinear form b_{M} : H × H → ℚ/ℤ, called the linking form of M. Sometimes it is convenient to represent b_{M} as a discriminant form. By this we mean the following.

Assume that L is a finitely generated free Abelian group with a symmetric bilinear form ( , ) : L × L → ℤ. Set L′ := Hom_{ℤ}(L, ℤ). Then there is a natural homomorphism i_{L} : L → L′ and a natural extension of ( , ) to a rational bilinear form ( , )_{ℚ} : L′ × L′ → ℚ. If L is non-degenerate then one defines the discriminant space D(L) :=coker(i_{L}), and the discriminant bilinear form b_{D(L)} : D(L) × D(L) → ℚ/ℤ by b_{D(L)}([d_{1}], [d_{2}]) = (d_{1}, d_{2})_{ℚ} (mod ℤ).

Assume that M is the boundary of a oriented 4-manifold N with H_{1}(N, ℤ) = 0 and H_{2}(N, ℤ) torsion-free. Let L be the intersection lattice (H_{2}(N, ℤ), ( , )). Then L′ can be identified with H_{2}(N, ∂N, ℤ) and one has the exact sequence L → L′ → H → 0. If b_{1}(M) = 0 then L is nondegenerate, and (H, b_{M}) = (D(L), −b_{D(L)}).

10.2.1. Quadratic functions and forms associated with b_{M}. A map q : H → ℚ/ℤ is called a quadratic function if b(x, y) = q(x + y) − q(x) − q(y) is a bilinear form on H × H. If in addition q(nx) = n^{2}q(x) for any x ∈ H and n ∈ ℤ then q is called a quadratic form. In this case we say that the quadratic function, respectively form, is associated with b.

In the case of the link M, we denote by Q^{c}(M) (respectively by Q(M)) the set of quadratic functions (resp. forms) associated with b_{M}. Obviously Q(M) ⊂ Q^{c}(M). The set Q(M) is a G := H^{1}(M, ℤ_{2}) torsor (i.e. G acts freely and transitively on it). Similarly, Q^{c}(M) is a H torsor.

Quadratic functions appear in a natural way. For example, take an N as above, and consider the set of characteristic elements

C_{N} := {k ∈ L′ : k(x) + (x, x) ∈ 2ℤ for all x ∈ L}.

Pick one k ∈ C_{N}. Then, for any d ∈ L′ with class [d] ∈ H, define

q_{D(L),k}([d]) := \frac{1}{2} (d + k, d)_{ℚ} (mod ℤ).

Then −q_{D(L),k} is a quadratic function associated with b_{M} = −b_{D(L)}. If in addition k ∈ Im(i_{L}), then −q_{D(L),k} is a quadratic form of b_{M}.

There is an important example to consider. Assume that N is an almost-complex manifold, i.e. its tangent bundle T N carries an almost complex structure. By the Wu formula, k = −c_{1}(T N) ∈ L′ is a characteristic element.

10.2.2. The spin and spin^{c} structures of M. The set Spin(M) of the possible spin structures of M is a G-torsor. In fact, there is a natural (equivariant) identification 𝔮 : Spin(M) → Q(M). Similarly, the set Spin^{c}(M) of spin^{c} structures on M is in a natural way a H-torsor. We denote this action H × Spin^{c}(M) → Spin^{c}(M) by (h, σ) ↦ h · σ. Moreover, there is a canonical H-equivariant identification 𝔮^{c} : Spin^{c}(M) → Q^{c}(M), which is compatible with the G-equivariant identification 𝔮 : Spin(M) → Q(M).

For every σ ∈ Spin^{c}(M) we denote by 𝕊_{σ} the associated bundle of complex spinors, and we set c(σ) := c_{1}(𝕊_{σ}) ∈ H, where c_{1} denotes the first Chern class.

If N is a 4-manifold as above, then its spin^{c} structures are parametrized by the set of characteristic elements C_{N} via σ̃ ↦ c(σ̃) ∈ C_{N} (with c(σ̃) having the same meaning as above).

10.2.3. Finally, consider our singularity (X, 0), fix a resolution π : X̃ → X as above, and identify M = ∂X̃. Then L = H_{2}(X̃, ℤ) is generated by the fundamental classes {[E_{v}]}_{v}, and the canonical cycle Z_{K} can be identified with an element in L′ (acting on L via the adjunction formulas 4.5.2(∗)). Moreover, Z_{K} ∈ Im(i_{L}) if and only if (X, 0) is numerically Gorenstein. By the Riemann-Roch formula χ(Z) = −Z(Z − Z_{K})/2, hence −Z_{K} ∈ CX̃.

In particular, any fixed resolution π provides a quadratic function −q_{D(I),−Z_{K}} associated with b_{M}, which is a quadratic form if and only if (X, 0) is numerically Gorenstein. One can verify that it does not depend on the choice of the resolution π.

10.3. Definition. The canonical spin^{c} structure of M. [72] Let (X, 0), M and π be as above. Then we define the canonical spin^{c} structure of M by (𝔮^{c})^{−1}(−q_{D(I),−Z_{K}}), modified by the natural involution of Spin^{c}(M). In particular, c(σ_{can}) = [Z_{K}] ∈ H. Equivalently, σ_{can} is the restriction to M of the spin^{c} structure on X̃, given by the characteristic element Z_{K} ∈ CX̃. Clearly, if (X, 0) is numerically Gorenstein then σ_{can} is a spin structure.

Because of the topological nature of Z_{K}, σ_{can} depends only on the topology of M.

10.4. Remarks. (1) From a result of Looijenga and Wahl [58] one can also deduce the following facts. The almost complex structure on X \ {0} determines a spin^{c} structure, whose restriction to M is σ_{can}. Similarly, if π : X̃ → X is a resolution, then the almost complex structure on X̃ gives a spin^{c} structure σX̃ on X̃, whose restriction to M is σ_{can}. Here it is instructive to add the following. Assume that the intersection form ( , )X̃ is even, hence X̃ has a unique spin structure ϵX̃. The point is that, in general, σX̃ ≠ ϵX̃, and their restrictions can be different as well, even if the restriction of σX̃ is spin.

More precisely: ( , )X̃ is even if and only if Z_{K} ∈ 2L′; the restriction of σX̃ is in Spin(M) if and only if Z_{K} ∈ L; and finally, the restrictions of σX̃ and ϵX̃ are equal if and only if Z_{K} ∈ 2L.

(2) If M is a numerically Gorenstein ℤ_{2}–homology sphere, then σ_{can} is the unique spin structure of M. If M is an integral homology sphere then it is clearly numerically Gorenstein, hence the above statement applies.

(3) Assume that (X, 0) has a smoothing with Milnor fiber F whose homology group H_{1}(F, ℤ) has no torsion. Then the (almost) complex structure of F provides a spin^{c} structure on F whose restriction to M is exactly σ_{can}.

Moreover, F has a spin structure if and only if the intersection form ( , ) of F is even; and in this case the spin structure is unique. If F is spin, then its spin structure ϵ_{F} coincides with the spin^{c} structure induced by the complex structure (since the canonical bundle of F is trivial). In particular, if F is spin, σ_{can} is the restriction of ϵ_{F}, hence it is spin. This also proves that if (X, 0) has a smoothing with even intersection form and without torsion in H_{1}(F, ℤ), then it is necessarily numerically Gorenstein. [Here is worth noticing that the Milnor fiber of a smoothing of a Gorenstein singularity has even intersection form [99].]

(4) The reader is invited to see the article [58] of Looijenga and Wahl (and the references cited in it) for deep applications of the discriminant forms in singularities (especially for smoothing obstructions).

10.5. The Seiberg-Witten invariants of M. To describe the Seiberg-Witten invariants one has to consider an additional geometric data belonging to the space of parameters

P = {u = (g, η) : g = Riemann metric, η = closed two-form}.

Then for each spin^{c} structure σ on M one defines the (σ, g, η)-Seiberg-Witten monopoles. For a generic parameter u, the Seiberg-Witten invariant sw_{M}(σ, u) is the signed monopole count. This integer depends on the choice of the parameter u and thus it is not a topological invariant. To obtain an invariant of M, one needs to alter this monopole count by the Kreck-Stolz invariant KS_{M}(σ, u), cf. [55] (or see [41] for the original “spin version”). Then, by [14, 55, 59], the rational number

\frac{1}{8}KS_{M}(σ, u) + sw_{M}(σ, u)

is independent of u and thus it is a topological invariant of the pair (M, σ). We denote this modified Seiberg-Witten invariant by sw_{M}^{0}(σ).

10.6. The Reidemeister-Turaev torsion and the Casson-Walker invariant. For any spin^{c} structure σ on M, we denote by

𝒯_{M,σ} = ∑_{h∈H} 𝒯_{M,σ}(h) h ∈ ℚ[H]

the sign-refined Reidemeister-Turaev torsion associated with σ (see [111]). We think of 𝒯_{M,σ} as a function H → ℚ given by h ↦ 𝒯_{M,σ}(h). Let λ(M) be the Casson-Walker invariant of M normalized as in [54, §4.7]. Then by a result of Nicolaescu [82], one has:

sw_{M}^{0}(σ) = −\frac{1}{|H|}λ(M) + 𝒯_{M,σ}(1).  (1)

Below we will present a formula for 𝒯_{M,σ} in terms of Fourier transform. For this, consider the Pontryagin dual Ĥ := Hom(H, U(1)) of H. Then a function f : H → ℂ and its Fourier transform f̂: Ĥ → ℂ satisfy:

f̂(χ) = ∑_{h∈H} f(h)χ̄(h); f(h) = \frac{1}{|H|} ∑_{χ∈Ĥ} f̂(χ)χ(h).

It is known that 𝒯̂_{M,σ}(1) = 0.

10.7. Remark. Probably this is the right place to mention the following. Those readers who are interested more in singularities, and much less in Seiberg-Witten theory, can consider 10.6(1) as the “definition” of sw_{M}^{0}(σ). In fact, in 10.10 we will provide very precise formulas for λ(M)/|H| and 𝒯_{M,σ}(1) in terms of any good resolution graph. These formulas, together with (1) “define” sw_{M}^{0}(σ), and one can “avoid” the gauge theory and Turaev’s torsion theory of the link.

Now we are ready to formulate the next conjecture.

10.8. Conjecture. [Némethi-Nicolaescu][72] Assume that (X, 0) is a normal surface singularity whose link M is a rational homology sphere. Then, conjecturally, the following facts hold:

(1) For any (X, 0), there is a topological upper bound for p_{g} given by:

sw_{M}^{0}(σ_{can}) − (Z_{K}^{2} + #V)/8 ≥ p_{g}.

(2) If (X, 0) is ℚ-Gorenstein, then in (1) one has equality.

(3) In particular (cf. 6.5(3)), if (X, 0) is Gorenstein which admits a smoothing with signature σ, then

−sw_{M}^{0}(σ_{can}) = σ/8.

10.8.1. Remark. In the above conjecture one has automatically built in the following statements as well. They provide new topological characterization of the rational and minimally elliptic singularities.

For any normal singularity (X, 0) the topological invariant sw_{M}^{0}(σ_{can}) − (Z_{K}^{2} + #V)/8 is nonnegative. Moreover, this topological invariant is zero if and only if (X, 0) is rational. Similarly, assume that (X, p) is numerically Gorenstein. Then this topological invariant is 1 if and only if (X, 0) is minimally elliptic.

10.9. Remark. If one wants to prove conjecture 10.8, one has to clarify (at least) three steps, each of them with a different flavour and mathematical machinery. They cannot be eliminated, and none of them can substitute for any of the others. Assume e.g. that (X, 0) is ℚ-Gorenstein.

[AG] The “algebraic-geometrical” step is to verify that p_{g} is independent of the analytic structure, and should provide some kind of graph-theoretical/combinatorial interpretation of p_{g}. This step is in fact the reformulation of the Artin-Laufer program.

[T] The “topological” step is to compute sw_{L}^{0}(σ_{can}) from the resolution (plumbing) graph. For this step we provided a solution in 10.10. But even with these formulas, in general, it is extremely difficult to compute sw_{M}^{0}(σ_{can}) (since the formula for the torsion involves a Fourier inversion, and a complicated regularization procedure). [The author believes in the existence of a simpler formula, cf. 13.1(a).]

[C] The “combinatorial” step is to establish the identity 10.8(2), once both p_{g} and sw_{M}^{0}(σ_{can}) are represented in terms of the decorated graph. Notice that even in those cases when we know rather satisfactory answers for steps [AG] and [T], this last step [C] can be very difficult, involving combinatorial/number theoretical identities, generalized Dedekind sums, Dedekind-Fourier sums, Taylor-Laurent expansions, etc; see e.g. sections 11 and 12.

10.9.1. Cases when the conjecture 10.8 is proved:

(a) Cyclic quotient and RDP singularities [72],

(b) some (sporadic) families of rational, elliptic and hypersurface singularities [72],

(c) “Polygonal” singularities (cf. 8.7.2) [72],

(d) singularities with good ℂ^{∗}-action [73],

(e) suspension singularities (i.e. of type {f(x, y) + z^{n} = 0}, where f is irreducible [74].

Before we give a more detailed presentation of (some of) these examples, we present the precise formulas of the topological terms Z_{K}^{2} + #V, λ(M) and 𝒯_{M,σ}(1) involved in the conjecture in terms of a (good) resolution graph.

10.10. Formulas. (For details, see [72]). We fix a resolution graph Γ with intersection matrix I. I_{vw}^{−1} denotes the entries of the inverse matrix I^{−1}. Recall that for any vertex v, δ_{v} denotes the number of adjacent edges.

(a) Clearly, H is isomorphic to coker(I), hence |H| = | det(I)|.

(b) [72]:

Z_{K}^{2} + #V = ∑_{v} e_{v} + 3#V + 2 + ∑_{v,w}(2 − δ_{v})(2 − δ_{w})I_{vw}^{−1}.

[For cyclic quotient singularities, the above formula goes back to the work of Hirzebruch.]

(c) The Casson-Walker invariant can be computed from the general surgery formulas of Lescop [54] (see also the thesis of Raţiu [90]):

−\frac{24}{|H|} λ(M) = ∑_{v} e_{v} + 3#V + ∑_{v}(2 − δ_{v}) I_{vv}^{−1}.

(d) The next formula of the sign-refined Reidemeister-Turaev torsion is proved in [72] using Turaev’s surgery formulas.

We will use the following notations. Consider a good resolution π. For each v ∈ V, we define g_{v} ∈ H = H_{1}(∂X̃, ℤ) generated by a small loop around E_{v} (i.e. by an S^{1}-fiber above a generic point of E_{v} considered in the plumbing representation of M). Clearly, the element g_{v} ∈ H can be determined from the resolution graph as well.

For any spin^{c} structure σ of M consider the unique element h_{σ} ∈ H such that h_{σ} · σ_{can} = σ. Then, for any χ ∈ Ĥ \ {1} the following identity holds:

𝒯̂_{M,σ}(χ̄) = χ̄(h_{σ}) · ∏_{v∈V} (χ(g_{v}) − 1)^{δ_{v}−2}.

The right hand side should be understood as follows. If χ(g_{v}) ≠ 1 for all v (with δ_{v} ≠ 2), then the expression is well-defined. Otherwise, the right hand side is computed via a regularization procedure. More precisely, fix a vertex v_{∗} ∈ V so that either χ(g_{v_{∗}}) ≠ 1, or it has an adjacent vertex u with χ(g_{u}) ≠ 1. Next, let b⃗^{∗} be the column vector indexed by V, with a 1 in the place v_{∗} and zero otherwise. Then find w^{∗} ∈ ℤ^{n} with entries {w_{v}^{∗}}_{v} in such a way that Iw^{∗} = −m^{∗} · b⃗^{∗} for some integer m^{∗} > 0. Then

𝒯̂_{M,σ}(χ̄) = χ̄(h_{σ}) · lim_{t→1} ∏_{v∈V} (t^{w_{v}^{∗}} χ(g_{v}) − 1)^{δ_{v}−2}.

One of the main difficulties in the above regularization procedure is that different characters may have rather different supports {w : χ(g_{w}) ≠ 1}. In particular, for different characters, one has to take different vertices v_{∗}, making the regularization rather non-uniform. In lucky cases (e.g. when the graph is star-shaped) for any character one can choose the central vertex for v_{∗}.

10.11. Notation. In the sequel we will use the following notation for Dedekind sums (see [89]):

s(h, k) = ∑_{l=0}^{k−1} ((\frac{l}{k} ))((\frac{hl)}{k})),

where {x} denotes the fractional part of x, and ((x)) the Dedekind symbol

((x)) = \{ \begin{matrix}\{x\} − 1/2 & if & x ∈ ℝ \ ℤ \\ 0 & if & x ∈ ℤ. \end{matrix}

10.12. Example. Cyclic quotient singularities. [72] We show that 10.8(2) is true for (X_{n,q}, 0). Indeed, from 10.10 (see also [58, 5.9] or [34]):

K^{2} + #V = \frac{2(n − 1)}{n} − 12 · s(q, n).

The link is L(n, q) with resolution graph 4.2.1. From 10.10(d): 𝒯̂_{M,σ_{can}}(χ̄) = [(χ − 1)(χ^{q} − 1)]^{−1}. Using the Fourier inversion 10.6, and the identity ([89, 18a]):

\frac{1}{n} ∑_{ζ^{n}=1, ζ≠1} \frac{1}{(ζ − 1)(ζ^{q} − 1)} = −s(q, n) + \frac{n − 1}{4n}.  (10.1)

one gets

𝒯_{M,σ_{can}}(1) = \frac{n − 1}{4n} − s(q, n).

The Casson-Walker contribution is λ(L(n, q))/n = s(q, n)/2. Hence 10.8(2) follows.

X_{n,q} is numerically Gorenstein if and only if q = n − 1. [In all other cases σ_{can} is not spin.] If q = n − 1, then s(n − 1, n) = −(n − 1)(n − 2)/(12n).

11. Singularities with good ℂ^{∗}-actions

11.1. In this section we discuss the validity of the conjecture 10.8(2) for singularities with good ℂ^{∗}-action. For more details, see [73] (and the references cited there). In this section b_{1}(M) = 0.

A complex affine algebraic variety X admits a ℂ^{∗}-action if and only if the affine coordinate ring A admits a grading A = ⊕_{k}A_{k}. Following Orlik-Wagreich, we say that the action is good if A_{k} = 0 for k < 0 and A_{0} = ℂ. This means that the point 0 corresponding to the maximal ideal ⊕_{k>0}A_{k} is the only fixed point of the action. Additionally, we assume that (X, 0) is normal. Notice also that by [75], if a normal surface singularity with b_{1}(M) = 0 admits a good ℂ^{∗} action then it is ℚ-Gorenstein.

11.2. Theorem. [73] For these singularities 10.8(2) is true:

sw_{M}^{0}(σ_{can}) − \frac{K^{2} + #V}{8} = p_{g}.  (∗)

This follows from the topological identity 11.7, and it is based on the concrete computation of all the invarints involved in terms of the Seifert invariants of the link. p_{g} is computed via Pinkham’s formula [87] (cf. also with Dolgachev’s work, see e.g. [20]).

11.2 has the following corollary, which can also be applied to singularities without ℂ^{∗}-action (but with some other type of additional rigidity properties).

11.3. Corollary. Assume that the link M of a normal surface singularity (X, 0) is a Seifert 3manifold with b_{1}(M) = 0. If (X, 0) is rational, or minimally elliptic, or Gorenstein elliptic, then the identity (∗) holds.

Indeed, in the case of these singularities, all the numerical invariants involved in the conjecture are characterized by the link. Moreover, each family (with fixed topological type) contains a special representative which admits a good ℂ^{∗}-action. In fact, the above corollary can automatically be extended to any family of singularities with these two properties.

11.4. The Seifert invariants. [35, 75, 77] we denote the set of (unnormalized) Seifert invariants of M by (α_{i}, β_{i}) = 1, where α_{i} > 0 and gcd(α_{i}, β_{i}) = 1. The number e := − ∑ β_{i}/α_{i} is called the orbifold Euler number of M. M is a link of singularity if and only if e < 0. Recall that the set of β_{i}’s is not unique, but any other choice changes each β_{i} within its residue class modulo α_{i} in such a way that the sum e = − ∑_{i}(β_{i}/α_{i}) is constant.

The set of normalized Seifert invariants {(α_{i}, ω_{i})}_{i=1}^{ν} is defined as follows. Write e = b+∑ ω_{i}/α_{i} for some integer b, and 0 ≤ ω_{i} < α_{i} with ω_{i} ≡ −β_{i} (mod α_{i}). Clearly, these properties define {ω_{i}}_{i} uniquely. Notice that b ≤ e < 0. In the sequel we assume ν ≥ 3. (Recall that for cyclic quotient singularities, the identity (∗) was verified in 10.12.)

For each i, consider the continued fraction α_{i}/ω_{i} = [b_{i1}, b_{i2}, ⋯ b_{iν_{i}}] with b_{ij} ≥ 2. Then the minimal resolution graph of (X, 0) is a star-shaped graph with ν arms. The central vertex has decoration b and the arm corresponding to the index i has ν_{i} vertices decorated by −b_{i1}, …, −b_{iν_{i}} (the vertex decorated by −b_{i1} is connected to the central vertex).

We will distinguish those vertices v ∈ V of the graph which have δ_{v} ≠ 2. We will denote by v̄_{0} the central vertex, and by v̄_{i} the end-vertex of the i^{th} arm for all 1 ≤ i ≤ ν. Then gv̄_{0} is exactly the class of the generic fiber. The group H has the following presentation:

H = ab⟨gv̄_{0}, gv̄_{1}, … gv̄_{ν} | g−b ; v̄_{0} = ∏_{i=1}^{ν} gω_{i} ; v̄_{i}, gv̄_{0} = gα_{i} ; v̄_{i} for all i⟩.

Let α := lcm(α_{1}, …, α_{ν}). The order of the group H and the order o of the subgroup ⟨gv̄_{0}⟩ can be determined by (cf. [75]):

|H| = α_{1} ⋯ α_{ν}|e|, o := |⟨gv̄_{0}⟩| = α|e|.

11.5. Invariants computed from the plumbing graph. Assume that M is a Seifert manifold with e < 0 and genus zero. Then one has the following formulae for some of its invariants. For more information, see [72] and [73].

(i) The Casson-Walker invariant. [54] (6.1.1):

−\frac{24}{|H|}λ(M) = \frac{1}{e}(2 − ν + ∑_{i=1}^{ν} \frac{1}{α_{i}^{2}} ) + e + 3 + 12 ∑_{i=1}^{ν} s(β_{i}, α_{i}).

(ii) K^{2} + #V. [72] (5.4):

K^{2} + #V = \frac{1}{e}(2 − ν + ∑_{i=1}^{ν} \frac{1}{α_{i}} )^{2} + e + 5 + 12 ∑_{i=1}^{ν} s(β_{i}, α_{i}).

(iii) The coefficient r_{0} of Ev̄_{0} in Z_{K} is (see e.g. [72], 5.2 and 5.5):

r_{0} = 1 + \frac{1}{e}(2 − ν + ∑_{i=1}^{ν} \frac{1}{α_{i}} ).

R := r_{0} − 1 is called the “exponent of (X, 0)”, and −R the “log discrepancy of Ev̄_{0}”. Some authors (see e.g. [75]) prefer to use the notation χ_{M} := 2 − ν + ∑_{i=1}^{ν}(1/α_{i}) = eR instead.

(iv) The Reidemeister-Turaev sign-refined torsion. For any χ ∈ Ĥ (and t ∈ ℂ) set

P̂_{χ}(t) := \frac{(t^{α}χ(gv̄_{0}) − 1)^{ν−2}}{∏_{i=1}^{ν} (t^{α/α_{i}} χ(gv̄_{i}) − 1)}.

Then, by 10.10(d) one has 𝒯̂_{M,σ_{can}}(χ̄) = lim_{t→1} P̂_{χ}(t). By Fourier inversion (see [73]):

𝒯_{M,σ_{can}}(1) = lim_{t→1} ( ∑_{l≥0} max ( 0 , 1 − lb + ∑_{i=1}^{ν} ⌊ \frac{−lω_{i}}{α_{i}} ⌋ ) t^{ol} − \frac{1}{|H|} · P̂_{1}(t) ).

(v) The geometric genus of (X, 0). Let M be a Seifert manifold with e < 0 and Seifert invariants as above. Define the Dolgachev-Pinkham (topological) invariant of M by

DP_{M} := ∑_{l≥0} max (0 , −1 + lb − ∑_{i=1}^{ν} ⌊ \frac{−lω_{i}}{α_{i}} ⌋ ).

If (X, 0) is a normal surface singularity with a good ℂ^{∗}–action and b_{1}(M) = 0, then, by [87] (5.7): p_{g} = DP_{M}. Using max(0, x) − max(0, −x) = x, one gets

𝒯_{M,σ_{can}}(1) − DP_{M} = lim_{t→1} ( ∑_{l≥0} ( 1 − lb + ∑_{i=1}^{ν} ⌊ \frac{−lω_{i}}{α_{i}} ⌋ ) t^{ol} − \frac{1}{|H|} · P̂_{1}(t) ).  (∗∗)

11.6. Some Laurent series. The right hand side of (∗∗) is a difference of two series, both having poles of order two at t = 1. Their Laurent expansions at t = 1 are the following. In fact, we prefer to expand the series in terms of the powers of t^{o} − 1 (instead of t − 1).

∑_{l≥0} ( 1 − lb + ∑_{i=1}^{ν} ⌊ \frac{−lω_{i}}{α_{i}} ⌋ ) t^{ol} = \frac{−e}{(t^{o} − 1)^{2}} + \frac{−e − χ_{M}/2}{t^{o} − 1} + \frac{2 − χ_{M}}{4} + ∑_{i=1}^{ν} s(β_{i}, α_{i}) + R(t),

with lim_{t→1} R(t) = 0.

\frac{P̂_{1}(t)}{|H|} = \frac{−e}{(t^{o} − 1)^{2}} + \frac{−e − χ_{M}/2}{t^{o} − 1} + E + Q(t),

where lim_{t→1} Q(t) = 0 and

E := −\frac{(e + 1)(e + 5)}{12e} + \frac{1}{4} ∑_{i} (1 − \frac{1}{α_{i}} ) + \frac{1}{12e} ∑_{i} (1 − \frac{1}{α_{i}} )(4 + \frac{1}{α_{i}} ) − \frac{1}{4e} ∑_{i<j} (1 − \frac{1}{α_{i}} )(1 − \frac{1}{α_{j}} ).

These two expansions and (∗∗) provide the next identity, which proves 10.8(2).

11.7. Theorem. [73] Let M be a Seifert 3–manifold with e < 0 and b_{1}(M) = 0. Then the invariants 𝒯_{M,σ_{can}}(1), λ(M), K^{2} + #V and DP_{M} are connected by the following identity:

𝒯_{M,σ_{can}}(1) − \frac{λ(M)}{|H|} = \frac{K^{2} + #V}{8} + DP_{M}.

11.8. Corollary. Assume that (X, 0) is Gorenstein and admits a smoothing. Let χ(F) denote the topological Euler characteristic of the Milnor fiber F. Then

2 · χ(F) = 24 · 𝒯_{M,σ_{can}}(1) + \frac{1}{e}(2 − ν + ∑_{i} \frac{1}{α_{i}^{2}} ) − \frac{1}{e}(2 − ν + ∑_{i} \frac{1}{α_{i}} )^{2}.

Notice that by [31] the first Betti number of F is zero. Hence χ(F) = 1 + μ, where μ is the Milnor number of the smoothing. Assume in the above formula that |H| = 1. Then 𝒯_{M,σ_{can}}(1) = 0, and −1/e = α = ∏_{i} α_{i}. Moreover, by [75], (X, 0) is the Brieskorn-Hamm complete intersection X(α_{1}, …, α_{ν}). Hence 11.8 recovers the Milnor number formula (3.10.b) of [30].

11.9. Remark. The Poincaré series. Let (X, 0) be a normal singularity with a good ℂ^{∗}-action and affine graded coordinate ring A = ⊕_{k}A_{k}. Then its Poincaré series is defined by p_{(X,0)}(t) = ∑_{k} dim(A_{k})t^{k}. The point is that it can be computed topologically (cf. [75]):

p_{(X,0)}(t) = ∑_{l≥0} max ( 0 , 1 − lb + ∑_{i=1}^{ν} ⌊ \frac{−lω_{i}}{α_{i}} ⌋ ) t^{l}.

Similarly, if (X_{ab}, 0) denotes the universal abelian cover of (X, 0), then p_{(X_{ab},0)}(t) = P̂_{1}(t) (cf. [75]). Therefore (cf. 11.5(iv)):

𝒯_{M,σ_{can}}(1) = lim_{t→1} (p_{(X,0)}(t^{o}) − p_{(X_{ab},0)}(t)/|H| ).  (†)

Notice that for many special families, the Poincaré series p_{(X,0)}(t) is computed, see e.g. [114]. For p_{(X_{ab},0)}(t) = P̂_{1}(t) one can use 11.5. Finally notice the “mysterious” similarity between the Fourier transform of the Reidemeister-Turaev torsion, and the formula [75](4.2) of Neumann of the Poincaré series of the graded affine ring associated with (X_{ab}, 0).

11.10. Example. Complete intersections. Notice that both Poincaré series in (†) are rational functions with pole of order 2 at t = 1. Next, we will analyse the first three terms of the corresponding Laurent expansions.

The above formulae provides p_{(X,0)}(t) in terms of the Seifert invariants. Nevertheless, if (X, 0) ⊂ (ℂ^{n}, 0) is a complete intersection with weights q_{1}, …, q_{n} (where gcd(q_{1}, …, q_{n}) = 1) and degrees d_{1}, …, d_{n−2}, then one also has (cf. e.g. with [114]):

p_{(X,0)}(t) = \frac{∏_{i=1}^{n−2}(1 − t^{d_{i}})}{∏_{j=1}^{n}(1 − t^{q_{j}})}.

Its Laurent expansion at t = 1 is:

p_{(X,0)}(t) = \frac{∏_{i} d_{i}}{∏_{j} q_{j}} [\frac{1}{(t − 1)^{2}} + \frac{∑_{i} d_{i} − ∑_{j} q_{j} + 2}{2(t − 1)} + F + U(t)],

where lim_{t→1} U(t) = 0 and

F := \frac{1}{6} ∑_{i} d_{i}^{2} + \frac{1}{4} ∑_{i<j} d_{i}d_{j} + \frac{1}{4} ∑_{i} d_{i} + \frac{1}{12} ∑_{j} q_{j}^{2} + \frac{1}{4} ∑_{i<j} q_{i}q_{j} − \frac{1}{4} ∑_{j} q_{j} − \frac{1}{4} ∑_{i,j} d_{i}q_{j} + \frac{1}{12}.

By (†), the coefficients of (t^{o} − 1)^{k} (k = −2, −1) of p_{(X,0)}(t^{o}) and P̂_{1}(t)/|H| should agree, hence:

−e = ∏_{i} d_{i}/ ∏_{j} q_{j} and R = ∑_{i} d_{i} − ∑_{j} q_{j}.

The coefficients of (t^{o} − 1)^{0} provide 𝒯_{M,σ_{can}}(1) = −e · F − E.

12. Brieskorn-Hamm rational homology spheres

In the previous section we determined (some of) the invariants of singularities with good ℂ^{∗}- action. Even if the Brieskorn-Hamm complete intersections are particular cases of singularities with ℂ^{∗}-action, we will reconsider this special family. Our goal is to present a different set of identities from which conjecture 10.8 follows. In fact, there are to major differences. First, here we provide formulas in terms of the weights of the Brieskorn polynomials. Second, we will use a signature formula, instead of Pinkham’s formula for p_{g}. This section is based on [72].

12.1. Notations. (see e.g. [32]) Fix n ≥ 3 positive integers a_{i} ≥ 2 (i = 1, …, n), and for a generic set of complex numbers {c_{j,i}} (i = 1, …, n, j = 1, …, n − 2), consider the affine variety

X(a_{1}, …, a_{n}) := {z ∈ ℂ^{n} : c_{j,1}z_{1}^{a_{1}} + ⋯ + c_{j,n}z_{n}^{a_{n}} = 0 for all 1 ≤ j ≤ n − 2} .

Then (X, 0) is a Brieskorn-Hamm complete intersection singularity with link M = Σ(a_{1}, …, a_{n}). By [35, 77], M is an oriented Seifert 3-manifold with Seifert invariants

(g; \underbrace{(α_{1}, β_{1}), ⋯, (α_{1}, β_{1})}_{s_{1}}; ⋯; \underbrace{(α_{n}, β_{n}), ⋯, (α_{n}, β_{n})}_{s_{n}}),

where g is the genus of the base of the Seifert fibration, and the pairs (α_{i}, β_{i}) (each considered s_{i} times) are the unnormalized Seifert invariants (cf. 11.4). Hence

e = − ∑_{i=1}^{n} s_{i} · \frac{β_{i}}{α_{i}} < 0.  (e)

Set

a := lcm (a_{i}; 1 ≤ i ≤ n), q_{i} := \frac{a}{a_{i}} , 1 ≤ i ≤ n; A := ∏_{i=1}^{n} a_{i}.

Then the Seifert invariants are as follows (see [35, Sec. 7] or [77]):

α_{i} := \frac{a}{lcm (a_{j}; j ≠ i)}, s_{i} := \frac{∏_{j≠i} a_{j}}{lcm (a_{j}; j ≠ i)} = \frac{Aα_{i}}{aa_{i}} , (1 ≤ i ≤ n);

g := \frac{1}{2}(2 + (n − 2)\frac{A}{a} − ∑_{i} s_{i}).

By these notations −e = A/(a^{2}). Notice that the integers {α_{j}}_{j=1}^{n} are pairwise coprime. Therefore the integers β_{j} are determined from (e). In fact ∑_{j} q_{j}β_{j} = 1, hence β_{i}q_{j} ≡ 1 (mod α_{j}) for any j. Similarly as above, we set α := lcm(α_{1}, …, α_{n}). The generic orbit (in the previous section gv̄_{0}) will be denoted by h.

It is clear that M is a rational homology sphere if and only if g = 0. A different characterization is as follows. For different levels of generality, see [12, 32, 73].

12.2. Proposition. If b_{1}(M) = 0, then (a_{1}, …, a_{n}) (after a possible permutation) has (exactly) one of the following forms:

(i) (a_{1}, …, a_{n}) = (db_{1}, db_{2}, b_{3}, …, b_{n}), where the integers {b_{j}}_{j=1}^{n} are pairwise coprime, and gcd(d, b_{j}) = 1 for any j ≥ 3;

(ii) (a_{1}, …, a_{n}) = (2^{c}b_{1}, 2b_{2}, 2b_{3}, b_{4} …, b_{n}), where the integers {b_{j}}_{j=1}^{n} are odd and pairwise coprime, and c ≥ 1.

12.3. Some invariants in the case 12.2(i). (See [72].)

• α_{j} = b_{j} for j = 1, …, n;

• s_{1} = 1, s_{2} = 1, and s_{j} = d for j ≥ 3, and ν = 2 + (n − 2)d;

• α = B := ∏_{j=1}^{n} b_{j}, and −e · B = 1, hence |⟨h⟩| = 1.

• H = ⊕_{j≥3}(ℤ_{b_{j}})^{d−1}, hence |H| = ∏_{j≥3} b_{j}^{d−1}.

𝒯_{M,σ_{can}}(1) = \frac{B · d(d − 1)}{24} ∑_{j≥3}(1 − \frac{1}{b_{j}^{2}} ).

−\frac{λ(M)}{|H|} = −\frac{B}{24}(−d(n − 2) + ∑_{j=1}^{n} \frac{s_{j}}{b_{j}^{2}} ) − \frac{1}{24B} + \frac{1}{8} + \frac{1}{2} ∑_{j=1}^{n} s_{j}s(β_{j}, b_{j}).

• The signature σ of the Milnor fiber is computed by Hirzebruch in terms of cotangent sums. Here we prefer to use another version proved in [78, 1.12]. This, in the case (i), reads as follows.

σ = −1 + \frac{1}{3B} (1 − (n − 2)d^{2}B^{2} + B^{2} ∑_{j=1}^{n} \frac{s_{j}^{2}}{b_{j}^{2}} ) − 4 ∑_{j=1}^{n} s_{j}s(β_{j}, b_{j}).

12.4. The invariants in the case 12.2(ii). (Cf. [72].)

• a = 2^{c}B and A = 2^{c+2}B, where B := ∏_{j} b_{j}. Moreover, α_{1} = 2^{c−1}b_{1} and α_{j} = b_{j} for j ≥ 2.

• s_{j} = 2 for j ≤ 3 and s_{j} = 4 for j ≥ 4, and ν = 4n − 6.

• α = 2^{c−1}B, hence −e^{−1} = 2^{c−2}B. Hence |⟨h⟩| = 2 and |H| = 2^{c}B^{3}/(b_{1}b_{2}b_{3})^{2}.

• The self-intersection b is even.

• ⟨h⟩ ≈ ℤ_{2}, and there is an exact sequence 0 → ⟨h⟩ → H → Q → 0 with Q = ⊕_{j≥1}(ℤ_{α_{j}})^{s_{j}−1}.

𝒯_{M,σ_{can}}(1) = \frac{2^{c−1}B}{8} + \frac{2^{c−1}B}{24} ∑_{j≥1} \frac{s_{j}(s_{j} − 1)}{2}(1 − \frac{1}{α_{j}^{2}} ).

−\frac{λ(M)}{|H|} = −\frac{2^{c−2}B}{24} (−4(n − 2) + ∑_{j=1}^{n} \frac{s_{j}}{α_{j}^{2}} ) − \frac{1}{3 · 2^{c+1}B} + \frac{1}{8} + \frac{1}{2} ∑_{j=1}^{n} s_{j}s(β_{j}, α_{j}).

σ = −1 + \frac{1}{3 · 2^{c−2}B} (1 − (n − 2)2^{2c}B^{2} + 2^{2c−4}B^{2} ∑_{j=1}^{n} \frac{s_{j}^{2}}{α_{j}^{2}} ) − 4 ∑_{j=1}^{n} s_{j}s(β_{j}, α_{j}).

Using these data, the verification of 10.8(3) is elementary.

13. Hypersurface singularities

13.1. The conjecture 10.8 and suspension singularities. Before we start to present the problems we propose to address in this section, we wish to say a few words about the verification of conjecture 10.8 in the suspension case.

If one tries to prove conjecture 10.8(2) or (3), one can try to decompose the link by splicing into Seifert manifolds in such a way that this decomposition is reflected at the level of singularities as well. Then, one hopes for (some kind of) additivity of the analytic invariant p_{g} (or equivalently, of one of the smoothing invariants μ or σ); and then also for a similar additivity of the Seiberg-Witten invariant of the link.

This program is complete in the case of suspension hypersurfaces. These singularities have the special form (X, 0) = {f(x, y) + z^{n} = 0} ⊂ (ℂ^{3}, 0), where f is an isolated plane curve singularity.

In the case when the link M of a suspension singularity is an integral homology sphere, then the additivity property of the signature was proved in [78], while the additivity of the Casson invariant (of integral homology spheres with respect to splicing) was proved (independently) by Akbulut-McCarthy, Boyer-Nicas and Fukuhara-Maruyama (according to [8] and [78]).

In the case when the link is a rational homology sphere, and f is irreducible, the additivity of the signature was proved in [68] (using [64], see also [65] for a summary); while the additivity of the Seiberg-Witten invariant was proved in [74]. This proves conjecture 10.8(3) for this family. For details, see [74].

We want to add two more remarks about the proof of the additivity of sw_{M}^{0} in this last case.

(a) As we know, sw_{M}^{0}(σ_{can}) = −λ(M)/|H| + 𝒯_{M,σ_{can}}(1). In [74] there is a complete description of the behaviour of both invariants λ(M) and 𝒯_{M,σ_{can}}(1) with respect to the splicing. (In general, the splicing properties of λ(M) are easier, while the properties of 𝒯_{M} much harder.) The point is that these topological invariants behave very badly with respect to the splicing; in particular they are far from being additive in any sense. On the other hand, their needed linear combination behaves nicely (showing that the independent study of the two summands creates a lot of unnecessary complications). Therefore, it would be really important to find a different representation of sw_{M}^{0} which would help to analyze the splicing properties in a better way.

(b) In [74] we realized (with much surprise) that the behaviour of sw_{M}^{0} with respect to splicing and cyclic covers (constructions topological in nature) definitely prefers some special algebraic situations.

13.2. Remark. Hypersurface singularities show very convincingly how subtle the computation of the analytic invariants from the topology is. Indeed, in this case, one can use, instead of the topological type of M, a much richer topological information, namely, the embedded topological type of (X, 0) (see below); and still, some of the analytic invariants persist in resisting topological characterization. The goal of this section is to show how the embedded topological type modifies the previous picture.

13.3. Definition. The embedded topological type. Let g : (ℂ^{3}, 0) → (ℂ, 0) be an isolated hypersurface singularity. Then for ϵ sufficiently small the embedding M = M(g) = ∂B_{ϵ} ∩ X ⊂ ∂B_{ϵ} = S^{5} does not depend on the choice of ϵ. The pair M ⊂ S^{5} characterizes the embedded topological type of g. We say that two germs g and h have the same embedded topological type if there exists a homeomorphism (S^{5}, M(g)) → (S^{5}, M(h)). This is equivalent to the existence of a homeomorphism (ℂ^{3}, 0) → (ℂ^{3}, 0) which sends {g = 0} to {h = 0} [94].

13.4. Definition. The Milnor fibration. Hodge numbers. By a classical result of Milnor, g/|g| : S^{5} \ M → S^{1} is an open-book decomposition whose fiber is diffeomorphic to F [61]. This induces a monodromy action on H^{2}(F, ℤ) with characteristic polynomial Δ(t).

In fact, the vanishing cohomology H^{2}(F, ℂ) carries a mixed Hodge structure compatible with the monodromy action. This provides the corresponding equivariant Hodge numbers (see e.g. [101, 102]). It is well-known that the Milnor number, signature and the geometric genus can be computed from them [102]. The set of equivariant Hodge numbers can be codified in the set of spectral pairs (see e.g. [101]); if one forgets the weight filtration then one gets the set of spectral numbers (or spectrum of g), introduced and studied by Arnold, Steenbrink and Varchenko.

13.5. The following question is very natural: which discrete invariants of g can be determined from the embedded topological type (S^{5}, M) of g? Evidently, any invariant of the link M is an invariant of (S^{5}, M), but our goal is to detect some other ones.

For example, in [108] Teissier proved that the Milnor number μ can be recovered from (S^{5}, K). In particular, via 6.5, the geometric genus can also be deduced from the embedded topological type. Lê extended Teissier’s result for the characteristic polynomial Δ(t) [53]. In fact, F has the homotopy type of the universal cover of the link complement S^{5} \ M, hence the Galois action can be identified with the monodromy action. In particular, Δ(t) is the Reidemeister torsion of the link complement.

But the next (more than 30 year old) conjecture shows that, in some cases, we have to work hard to obtain such a characterization. Moreover, it also tells us, that if for hypersurface singularities, and (even) using the embedded topological type, the characterization of the multiplicity is so difficult, then for an arbitrary normal surface singularities the characterization of the multiplicity from the link itself cannot be very easy.

13.6. Zariski’s Conjecture. [126] The multiplicity of an isolated singularity g : (ℂ^{3}, 0) → (ℂ, 0) can be recovered from the embedded topological type of g.

This has been verified so far only for quasi-homogeneous singularities (see [29, 84, 121, 122], or by a combination of [95] and [98], cf. the paragraph after 13.11) and for suspension singularities (with f irreducible) [60] (see 13.9 below), and some sporadic cases.

13.7. We hope in 13.6, since the embedded topological type, indeed, is very rich. E.g., Brieskorn singularities (see e.g. 4.7) show clearly that, in general, the embedded topological type contains more information about g than the abstract link M (see also 13.7.1).

On the other hand, in some special cases, even the complicated embedding can be characterized by some simpler invariants. E.g., Xu and Yau in [121] proved that the embedded topological type of a weighted homogeneous singularity can be recovered from M and Δ(t). This fact is not true in general: Artal Bartolo found a pair of (superisolated) singularities with the same M and Δ(t), but different monodromies [3].

13.7.1. Examples. We consider two suspension singularities of type {f(x, y) + z^{n} = 0}. In both cases f has two Newton pairs: (3, 7) and (40, 1) in the first case; (4, 5) and (42, 1) in the second one. In the first case n = 42, in the second n = 40. In both cases the minimal good resolution graphs are the same:

[figure: [6]; −841; [6]; −841; −1; −2; −2; −2; −21; ⋯]

where the number of −2 curves is 19. On the other hand, their Milnor numbers are different. For details, see [60].

Therefore, the next result is rather surprising.

13.8. Theorem. [60] Let f : (ℂ^{2}, 0) → (ℂ, 0) be an irreducible plane curve singularity, n ≥ 2, and M the link of the hypersurface singularity (X, 0) = ({f(x, y) + z^{n} = 0}, 0). Then, except for two pathological cases S1 and S2 (which can be characterized perfectly in terms of M, and are completely understood), from the link M one can recover completely the Newton pairs of f and the integer n (provided that we disregard the “z-axis ambiguity”, cf. 13.8.1(a) below). In both exceptional cases the corresponding links have non-trivial first Betti numbers.

In particular, the above statement holds without any exception provided that the link is a rational homology sphere.

On the other hand, in the cases S1 and S2, the link together with the Milnor number of the hypersurface singularity f + z^{n} determine completely the Newton pairs of f and the integer n.

13.8.1. Here some remarks are in order.

(a) Notice that a Brieskorn singularity {x^{a} + y^{b} + z^{c} = 0} can be written in three different way as a suspension. For such a singularity, if one succeeds in recovering from the link the set of integers {a, b, c}, then one has to make a choice for the z-axis. This, in general, is not unique, and provides the “z-ambiguity”.

(b) M determines the number s of Newton pairs of f in all the cases.

The exceptional case S1 appears when s = 1, and the corresponding singularities have the equisingular types of some special Brieskorn singularities. They are classified. A possible S1-coincidence is exemplified in 4.7.

The exceptional case S2 appears for s = 2 with some other strong additional restrictions, and again it is completely understood. In this case any link M can be realized by at most two possible pairs (f, n). It is exemplified in 13.7.1. For more details, see [60].

In all other cases, e.g. when s ≥ 3, the theorem assures uniqueness. This is slightly surprising. Indeed, as s increases, one may expect more and more complicated special families providing interesting coincidences for their links. But, it turns out that this is not the case: if the resolution graph of (X, 0) has a more and more complicated structure, then it becomes more “over-determined”, and it leaves no room for any ambiguity for f and n.

(c) In general, it is very difficult to characterize those resolution graphs (or links) which can be realized by, say, hypersurface singularities, or complete intersections, or by any family of germs defined by some analytic property. The proof of 13.8 gives a complete characterization of those graphs which can be realized as resolution graphs of some {f + z^{n} = 0} for some irreducible f.

(d) For the case when f is arbitrary (i.e. reduced), but n = 2, and the link is a rational homology sphere, Laufer established the above uniqueness in [49]. In [85], Pichon verified that for any fixed n, any link can be realized in finitely many ways as the link of f + z^{n} = 0 (f reduced).

The main point in 13.8 is that from the Newton pairs of f and from the integer n one can recover the equisingular type of (X, 0), in particular, “almost” all discrete invariants of {f + z^{n} = 0}. This last statement follows from the description of the invariants of plane curve singularities and different “Sebastiani-Thom type” formulae; for more details see e.g. [2, 98, 68, 69, 65].

13.9. Corollary. [60] Assume that (X, 0) = {g(x, y, z) := f(x, y) + z^{n} = 0} is a suspension hypersurface singularity with b_{1}(M) = 0. Then the link M determines completely the following data:

(1) The embedded topological type M ⊂ S^{5}; in particular, the Milnor fibration and all the homological package derived from it.

(2) All the equivariant Hodge numbers associated with the vanishing cohomology of g; in particular, the geometric genus of (X, 0).

(3) The multiplicity of g.

[The same is true even if b_{1}(M) > 0, provided that (X, 0) is not of type S1 or S2.] More generally, if b_{1}(M) > 0, then M together with the Milnor number μ of g determines completely (1-2-3).

In particular, Zariski’s conjecture is true for suspensions of irreducible plane curve singularities.

13.9.1. More remarks about the redused case. We expect that even for reducible f, the link of f + z^{n} contains all the information about the equisingularity type of f and the integer n “generically”, i.e. excepting a set of coincidences which is “small” inside of suspension singularities. At this moment it is extremely difficult to guess how large is the set of these exceptions. Coincidences with star-shaped graphs are known, and can be classified. But definitely there are other type of coincidences as well. We expect that this case inevitably will contain a much larger list of pathological cases.

For example, the minimal resolution graphs of the suspension singularities {(x^{19} + y^{38})(x^{32} + y^{16}) + z^{3} = 0} and {(x^{5} + y^{6})(x^{10} + y^{3}) + z^{45} = 0} are the same, namely:

[figure: [18]; [15]; −2; −2]

Moreover, the above S2-coincidence warns us that we can expect rather strange coincidences as well. We also notice that if f is reducible, the graph of f + z^{n} in general is not a tree, a fact which enlarges the possible graphs considerably.

On the other hand, we conjecture that even for reducible f, if the link of f + z^{n} is a rational homology sphere, then it determines the equisingularity type of f and the integer n in a unique way.

The above results motivate the following conjecture.

13.10. Conjecture. [Mendris-Némethi][60] Let g : (ℂ^{3}, 0) → (ℂ, 0) be an isolated hypersurface singularity whose link M is a rational homology sphere. Then M characterizes completely the embedded topological type of g, the equivariant Hodge numbers and the multiplicity of g.

13.10.1. Cases when conjecture 13.10 is proved:

(a) suspension singularities with f irreducible [60] (cf. 13.9); or with n = 2 [49],

(b) weighted homogeneous singularities [60].

For part (b), it is enough to recover from M the weights of g. For this, notice that using [121] (see also [83]), one can recover the weights from M and Δ(t) (even if M is not a rational homology sphere). The point is that if b_{1}(M) = 0, then Δ(t) can be computed from the Seifert invariants of the link (using the topological description of p_{(X,0)}(t), cf. 11.9, and a recent result of Ebeling [22]).

Steenbrink and Varchenko succeeded to understand many deep properties of hypersurface singularities via their spectrum. This led Steenbrink to formulate the following questions.

13.11. Questions. [Steenbrink][33] Let g be an isolated hyperurface singularity.

(a) Is the spectrum of g an invariant of its embedded topological type?

(b) Does the spectrum of g determine its multiplicity?

Clearly, an affirmative answer to (a) and (b) would imply Zariski’s conjecture. Notice that in the case of quasi-homogeneous singularities ℂ^{3} → ℂ, the answer to both questions is yes. (a) is answered positively in [95], while [98] shows that from the spectrum one can recover the weigths, which clearly provide the multiplicity. (See also [63] for the general realtionship between the spectral pairs and embedded topological type.)

Now, we invite the reader to review section 5, especially 5.7. The question from the end of 5.7 can be reformulated in the following concrete conjecture.

13.12. Conjecture. Let λ : (X , 0) → (D, 0) be a flat deformation of hypersurface isolated singularities (X_{t}, 0) (t ∈ D) (i.e. with the section s(t) = 0). Assume that for each t the link M_{t} is a rational homology sphere. Then λ admits a strong simultaneous resolution along s if and only if it admits a weak simultaneous resolution along s.

We end this section with another conjecture about isolated hypersurface singularities, which does not really take part in the above program, but it is substantially connected with it, and continuously fascinates the author (for more comments see e.g. [68, 69, 65], and the references listed in them).

13.13. Conjecture.

(a) [Durfee] [21] The signature σ of an isolated singularity g : (ℂ^{3}, 0) → (ℂ, 0) is negative. In fact, conjecturally, even a stronger inequality holds, namely p_{g} ≤ μ/6.

(b) [Neumann] [120] Neumann generalized the first part of Durfee’s conjecture as follows: The signature is semicontinuous. More precisely, if the isolated hypersurface singularity f is adjacent to the isolated hypersurface singularity g, then σ(f) ≤ σ(g). [Notice that μ is semicontinuous.]

References

[1] Arndt, J.: Verselle Deformationen zyklischer Quotientensingularitäten, Diss. Hamburg, 1988.

[2] Arnold, V.I and Gusein-Zade, S.M. and Varchenko, A.N.: Singularities of Differentiable maps, Volume 1 and 2, Monographs Math., 82-83, Birkhäuser, Boston, 1988.

[3] Artal Bartolo, E.: Forme de Jordan de la monodrmomie des singularités superisolées de surfaces, Mem. Amer. Math. Soc., 525 (1994).

[4] Artin, M.: Some numerical criteria for contractibility of curves on algebraic surfaces. Amer. J. of Math., 84, 485-496, 1962.

[5] Artin, M.: On isolated rational singularities of surfaces. Amer. J. of Math., 88, 129-136, 1966.

[6] Artin, M.: Algebraic construction of Brieskorn’s resolutions, J. of Algebra 29 (1974), 330-348.

[7] Behnke, K. and Knörrer, H.: On infinitesimal deformations of rational surface singularities, Comp. Math., 61 1987), 103-127.

[8] Boyer, S. and Nicas, A.: Varieties of group representations and Casson’s invariant for rational homology 3–spheres, Trans. AMS 322 (2), 507-522, 1990.

[9] Briançon J. and Speder, J.: La trivialité topologique n’implique pas les conditions de Whitney, C. R. Acad. Sc., Paris, 280 (1975), 365-367.

[10] Briançon J. and Speder, J.: Les condition de Whitney impliquent “μ^{∗} constant”, Ann. Inst. Fourier (Grenoble), 26 (1976), 153-164.

[11] Brieskorn, E.: Examples of singular normal complex spaces which are topological manifolds, Proc. Nat. Acad. Sci.. 55 (1966), 1395-1397.

[12] Brieskorn, E.: Beispiele zur Differentialtoplogie von Singularitäten, Invent. Math., 2 (1966), 1-14.

[13] Brieskorn, E,: Singular elements in semi-simple algebraic groups, Proc. Int. Con. Math. Nice, 2 (1971), 279-284.

[14] Chen, W: Casson invariant and Seiberg-Witten gauge theory, Turkish J. Math., 21(1997), 61-81.

[15] Christophersen, J. A. and Gustavsen, T. S.: On infinitesimal deformations and obstructions for rational surface singularities, J. Algebraic Geometry, 10 (2001) (1), 179-198.

[16] Collin, O. and Saveliev, N.: A geometric proof of the Fintushel-Stern formula, Adv. in Math., 147(1999), 304-314 .

[17] Collin, O. and Saveliev, N.: Equivariant Casson invariants via gauge theory, J. reine Angew. Math., 541 (2001), 143-169.

[18] Dixon, D. : Dissertation, Purdue University, 1977.

[19] Dixon, D.J.: The fundamental divisor of normal double points of surfaces, Pacific J. Math., 80, no. 1, 105-115, 1979.

[20] Dolgachev, I.V.: Automorphic forms and weighted homogeneous singularities, Funkt. Anal. Jego. Prilozh., 9(1975), 67-68 . English translation in Funct. Anal. Appl., 9(1975), 149-151.

[21] Durfee, A.: The Signature of Smoothings of Complex Surface Singularities, Math. Ann., 232, 85-98, 1978.

[22] Ebeling, W.: Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity, math.AG/0109210, Manuscripta Math., to appear.

[23] Eisenbud, D. and Neumann, W.: Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. of Math. Studies 110, Princeton University Press, 1985.

[24] Elkik, R.: Singularités rationelles et Déformations, Inv. Math., 47 (1978), 139-147.

[25] Fintushel, R. and Stern, R.J.: Instanton homology of Seifert fibered homology 3–spheres, Proc. London Math. Soc., (3) 61, 109-137, 1991.

[26] Fujita, G.: A splicing formula for the Casson-Walker’s invariant, Math. Annalen, 296, 327-338 (1993).

[27] Grauert, H.: Über Modifikationen und exceptinelle analytische Mengen, Math. Annalen, 146 (1962), 331-368.

[28] Grauert, H.: Über die Deformationen Isolierten Singularitäten Analytischer Mengen, Inv. Math., 15 (1972), 171-198.

[29] Greuel, G.-M.: Constant Milnor number implies constant multiplicity for quasi-homogeneous singularities, Manuscripta Math., 56 (1986), 159-166.

[30] Greuel G.-M. and Hamm, H.A.: Invarianten quasihomogener vollständiger Durchschnitte, Invetiones math., 49 (1978), 67-86.

[31] Greuel, G.-M. and Steenbrink, J.: On the topology of smoothable singularities, Proc. of Symp. in Pure Math., 40 Part 1 (1983), 535-545.

[32] Hamm. H.A.: Exotische Sphären als Umgebungsrn̈der in speziellen komplexen Räumen, Math. Ann., 197 (1972), 44-56.

[33] Hauser, H. and Randell, R.: Report on the problem session, Contemporary Mathematics, Vol. 90, 119-134, 1989.

[34] Ishii, S: The invariant −K^{2} and continued fractions for 2-dimensional cyclic quotient singularities, preprint.

[35] Jankins, M. and Neumann, W.D.: Lectures on Seifert Manifolds, Brandeis Lecture Notes, 1983.

[36] de Jong, T. and van Straten, D.: On the base space of a semi-universal deformation of rational quadruple points, Annals of Math., 134 (2) (1991), 653-678.

[37] de Jong, T. and van Straten, D.: On the deformation theory of rational surface singularities with reduced fundamental cycle, J. Alg. Geom.. 3 (1994), 117-172.

[38] de Jong, T. and van Straten, D.: Deformation theory of sandwiched singularities, Duke Math. J., 95 (3) (1998), 451-522.

[39] Karras, U.: On Pencils of Curves and Deformations of Minimally Elliptic Singularities, Math. Ann., 247, 43-65, 1980.

[40] Karras, U.: Methoden zur Berechnung algrebraischer Invarianten und zur Konstruktion von Deformationen normaler Flächensingularitäten, Habilitationsschrift, Dortmund 1981.

[41] Kreck, M. and Stolz, S.: Nonconnected moduli spaces of positive sectional curvature metrics, J. of the AMS., 6(1993), 825-850.

[42] Kollár, J. and Shepherd-Barron, N.I.: Threefolds and deformations of surface singularities, Invent. math., 91 (1988), 299-338.

[43] Kollár, J.: Flips, flops, minimal models, etc., Surveys in Diff. Geom., 1 (1991), 113-199.

[44] Laufer, H.B.: Normal two–dimensional singularities. Annals of Math. Studies, 71, Princeton University Press, 1971.

[45] Laufer, H.B.: On rational singularities, Amer. J. of Math., 94, 597-608, 1972.

[46] Laufer, H.B.: Taut two–dimensional singularities, Math. Ann., 205, 131-164, 1973.

[47] Laufer, H.B.: On minimally elliptic singularities, Amer. J. of Math., 99, 1257-1295, 1977.

[48] Laufer, H.B.: On μ for surface singularities, Proceedings of Symposia in Pure Math., 30, 45-49, 1977.

[49] Laufer, H.B.: On normal two–dimensional double point singularities, Israel Journal of Mathematics, Vol. 31, Nos 3-4, 315-334, 1978.

[50] Laufer, H.B.: Weak simultaneous resolution for deformations of Gorenstein surface singularities, Proc. of Symp. in Pure Math., 40, Part 2 (1983), 1-29.

[51] Laufer, H.B.: Strong Simultaneous Resolution for Surface Singularities, Adv. Studies in Pure Math., 8 (1986), 207-214. Complex Analytic Singularities.

[52] Laufer, H.B.: The multiplicity of isolated two–dimensional hypersurface singularities, Transactions of the AMS, 302 Number 2, 489-496, 1987.

[53] Lê Dũng Tráng: Topologie des singularités des hypersurfaces complexes, Astérisque, 7-8 (1973), 171-182.

[54] Lescop, C.: Global Surgery Formula for the Casson-Walker Invariant, Annals of Math. Studies, vol. 140, Princeton University Press, 1996.

[55] Lim, Y: Seiberg-Witten invariants for 3-manifolds in the case b_{1} = 0 or 1, Pacific J. of Math., 195(2000), 179-204.

[56] Looijenga, E.: The smoothing components of a triangle singularity. I, Proc. of Symp. in Pure Math., 40 Part 2, (1983), 173-183.

[57] Looijenga, E.: Riemann-Roch and smoothing of singularities, Topology 25 (3) (1986), 293-302.

[58] Looijenga, E. and Wahl, J.: Quadratic functions and smoothing surface singularities, Topology, 25 (1986), 261-291.

[59] Marcolli, M. and and Wang, B.L.: Seiberg-Witten invariant and the Casson-Walker invariant for rational homology 3-spheres, math.DG/0101127, Geometriæ Dedicata, to appear.

[60] Mendris, R. and Némethi, A.: The link of {f(x, y) + z^{n} = 0} and Zariski’s Conjecture, submitted. math.AG/0207212.

[61] Milnor, J.: Singular points of complex hypersurfaces, Annals of Math. Studies 61, Princeton University Press 1968.

[62] Mumford, D.: The topology of normal singularities of an algebraic surface and criterion for simplicity, IHES Publ. Math. 9, 5-22, 1961.

[63] Némethi, András: The real Seifert form and the spectral pairs of isolated hypersurface singularities, Compos. Math., 98, 23-41, 1995.

[64] Némethi, A.: The equivariant signature of hypersurface singularities and eta–invariant, Topology Vol. 34, No. 2, 243-259, 1995.

[65] Némethi, A.: The signature of f(x, y) + z^{n}, Proceedings of Real and Complex Singularities, Liverpool, August 1996; London Math. Soc. Lecture Notes Series, 263, 131-149, 1999.

[66] Némethi, A.: Five lectures on normal surface singularities, lectures delivered at the Summer School in Low dimensional topology Budapest, Hungary, 1998; Bolyai Society Math. Studies 8 (1999), 269-351.

[67] Némethi, A.: Casson invariant of cyclic coverings via eta–invariant and Dedekind sums, Topology and its Applications, 102(2), 181-193, 2000.

[68] Némethi, A.: Dedekind sums and the signature of f(x, y) + z^{N}, Selecta Mathematica, New series, 4, 361-376, 1998.

[69] Némethi, A.: Dedekind sums and the signature of f(x, y) + z^{N},II., Selecta Mathematica, New series, 5, 161-179, 1999.

[70] Némethi, A.: “Weakly” Elliptic Gorenstein Singularities of Surfaces, Inventiones math., 137, 145-167, 1999.

[71] Némethi, A.: The resolution of some surface singularities, I., (cyclic coverings); Proceedings of the AMS Conference, San Antonio, 1999; Contemporary Mathematics 266, 89-128.

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

[73] Némethi, A. and Nicolaescu, L.I.: Seiberg-Witten invariants and surface singularities II (singularities with good C^{∗}-action), submitted. math.AG/0201120.

[74] Némethi, A. and Nicolaescu, L.I.: Seiberg-Witten invariants and surface singularities III (splicings and cyclic covers), submitted. math.AG/0207018.

[75] Neumann, W.: Abelian covers of quasihomogeneous surface singularities, Proc. of Symposia in Pure Mathematics, vol. 40, Part 2, 233-244.

[76] Neumann, W.D.: A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Transactions of the AMS, 268 Number 2, 299-344, 1981.

[77] Neumann, W.D. and Raymond, F.: Seifert manifolds, plumbing, μ-invariant and orientation reserving maps, Algebraic and Geometric Topology, Lecture Notes in Math. 664, 161-196.

[78] Neumann, W. and Wahl, J.: Casson invariant of links of singularities, Comment. Math. Helv. 65, 58-78, 1991.

[79] Neumann, W. and Wahl, J.: Universal abelian covers of surface singularities, arXiv:math.AG/0110167.

[80] Neumann, W. and Wahl, J.: Universal abelian covers of quotient-cusps, arXiv:math.AG/0101251.

[81] Neumann, W. and Wahl, J.: Complex surface singularities with integral homology sphere links, arXiv:math.AG/0301165.

[82] Nicolaescu, L.I.: Seiberg-Witten invariants of rational homology spheres, math.DG/0103020.

[83] Orlik, P. and Wagreich, Ph.: Isolated singularities of algebraic surfaces with ℂ^{∗} action, Ann. of Math. (2) 93, 205-228, 1971.

[84] O’Shea, D.: Topological trivial deformations of isolated quasi-homogeneous hypersurface are equimultiple, Proc. AMS, 101 (1987) 260-262.

[85] Pichon, A.: Singularities of Complex surfaces with Equations z^{k} = f(x, y) = 0, International Math. Research Notices, 5 (1997), 241-246.

[86] Pinkham, H.: Deformations of algebraic varieties with G_{m} action, Astérisque 20 (1974), 1-131.

[87] Pinkham, H.: Normal surface singularities with C^{∗}–action, Math. Ann., 227, 183-193, 1977.

[88] Pinkham, H.: Smoothing of the D_{pqr} singularities, p + q + r = 22, Proc. of Symp. in Pure Math., 40 Part 2, (1983), 373-377.

[89] Rademacher, H. and Grosswald, E.: Dedekind Sums, The Carus Math. Monographs, MAA, 1972.

[90] Ratiu, A.: The Jones–Witten invariants of tree manifolds, These de Doctorat, Universite Paris 7, 1996.

[91] Reid, M.: Chapters on Algebraic Surfaces. In: Complex Algebraic Geometry, IAS/Park City Mathematical Series, Volume 3 (J. Kollár editor), 3-159, 1997.

[92] Riemenschneider, O.: Bemerkungen zur Deformationstheorie Nichtrationaler Singularitäten, Manus. Math., 14, 91-99, 1974.

[93] Riemenschneider, O.: Deformationen von Quotintensingularitäten (nach zyklischen Gruppen), Math. Ann., 209 (1974), 211-248.

[94] Saeki, O.: Topological types of complex isolated hypersurface singularities, Kodai Math. J., 12 (1990), 23-29.

[95] Saeki, O.: Real Seifert form determines the spectrum for quasihomogeneous hypersurface singularities in ℂ^{3}, J. Math. Soc. Japan, 52 (2) (2000), 409-431.

[96] Schaps, M.: Deformations of Cohen-Macauley Schemes of codimension 2 and Non-Singular Deformations of Space Curves, Am. J. Math., 99 (1977), 669-685.

[97] Schlessinger, M.: Functors of Artin Rings, Trans. AMS, 130 (1968), 208-222.

[98] Schrauwen, R. and Steenbrink, J. and Stevens, J.: Spectral Pairs and Topology of Curve Singularities, Proc. Sumpos. Pure Math., 53, 305-328, 1991.

[99] Seade, J.A.: A cobordism invariant for surface singularities, Proc. of Symp. in Pure Math., 40 (2) (1983), 479-484.

[100] Spivakovsky, M.: Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Annals of Math., 131 (1990), 411-491.

[101] Steenbrink, J.H.M.: Mixed Hodge structures on the vanishing cohomology, Nordic Summer School/NAVF, Symposium in Mathematics, Oslo, 1976.

[102] Steenbrink, J.H.M.: Mixed Hodge structures associated with isolated singularities, Proc. Symp. Pure Math., 40, Part 2 (1983), 513-536.

[103] Stevens, J.: Elliptic Surface Singularities and Smoothings of Curves, Math. Ann., 267, 239-247, 1984.

[104] Stevens, J.: Kulikov singularities; Thesis, Leiden 1985 [105] Stevens, J.: On the versal deformation of cyclic quotient singularities, LNM, 1462 (1991), 302-319.

(Singularity theory and its applications, Warwick 1989)

[106] Stevens, J.: Partial resolutions of rational quadruple points, Int. J. of Math., 2 (2) (1991), 205-221.

[107] Teissier, B.: Cycles evanescents, sections planes et conditions de Whitney, Asterisque 7-8 (1973), 285-362.

[108] Teissier, B.: Deformation à type topologique constant II, Séminaire Douady-Verdier 1972.

[109] Teissier, B.: Résolution simultanée I, II, LNM 777 (1980), 71-146.

[110] Tjurina, G.-N.: Locally Flat Deformations of Isolated Singularities of Complex Spaces, Math. USSR Izvestia 3 (1969), 967-999.

[111] Turaev, V.G.: Torsion invariants of Spin^{c}-structures on 3-manifolds, Math. Res. Letters, 4(1997), 679-695.

[112] Vaquié, M.: Résolution simultanée de surfaces normales, Ann. Inst. Fourier, 35 (1985), 1-38.

[113] Wagreich, Ph.: Elliptic singularities of surfaces. Amer. J. of Math., 92, 419-454, 1970.

[114] Wagreich, Ph.: The structure of quasihomogeneous singularities, Proc. Symp. in Pure Math., 40:2, 598-611 (1983)

[115] Walker, K.: An extension of the Casson’s invariant, Ann. of Math. Studies 126, Princeton University Press, 1992.

[116] Wahl, M.J.: Equisingular deformations of normal surface singularities, I, Ann. of Math., 104, 325-356, 1976.

[117] Wahl, M. J.: Simultaneous resolution of rational singularities, Compositio Math., 38 (1) (1979), 43-54.

[118] Wahl, M. J.: Elliptic Deformations of Minimally Elliptic Singularities, Math. Ann., 253 (1980), 241-262.

[119] Wahl, J.: Smoothings of normal surface singularities, Topology, 20, 219-246, 1981.

[120] Weber, C.: Problems, Noeuds, tresses et singularités, Monographie 31 de L’Enseignement Math. (1983), 247-259.

[121] Xu, Y and Yau, S. S.-T. Yau: Classification of topological types of isolated quasi-homogeneous two dimensional hypersurface singularities, Manuscripta Math., 64 (1989), 445-469.

[122] Yau, S.S.-T.: Topological types of isolated hypersurface singularities, Contemporary Math., 101 (1989), 303-321.

[123] Yau, S.S.-T.: On almost minimally elliptic singularities, Bulletin of the AMS, 83 Number 3, 362-364, 1977.

[124] Yau, S.S.-T.: Gorenstein singularities with geometric genus equal to one, Amer. J. of Math., 101, 813-854, 1979.

[125] Yau, S.S.-T.: On maximally elliptic singularities, Transactions of the AMS, 257 Number 2, 269-329, 1980.

[126] Zariski, O.: Some open questions in the theory of singularities, Bull. A. M. S., 77, 481-491, 1971.

Department of Mathematics, Ohio State University, Columbus, OH 43210

   E-mail address: nemethi@math.ohio-state.edu