SOME MEETING POINTS OF SINGULARITY THEORY AND LOW DIMENSIONAL TOPOLOGY

ANDRÁS NÉMETHI

Abstract. We review some basic facts which connect the deformation theory of normal surface singularities with the topology of their links. The presentation contains some explicit descriptions for certain families of singularities (cyclic quotients, sandwiched singularities).

1. Introduction

The aim of the present paper is to serve as a first introductory guide for those researchers who wish to study the subtle connections between the analytic and deformation invariants of normal surface singularities and topological aspects. Since this subject is rather large, we had to make some selection: here we will say nothing about the connection of singularity theory with the Seiberg–Witten or Heegaard Floer theory, or even with lattice cohomology. Instead, we focus on some aspects regarding deformations, the Milnor fibers of the smoothings, and their connection with the Stein/symplectic fillings in topology. The manuscript can be thought as a complement of the manuscript of A. Stipsicz of this volume, where he concentrates mostly on the topological methods and aspects. Here we wish to collect some facts about the analytic part too.

2. The link, from singularity point of view

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) [60]. 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). 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 smooth 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 [42]). We write e_{v} := (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 as a plumbing graph, and M can be considered as an plumbed manifold whose plumbing graph is Γ(π) (see e.g. [24, 61] or [62, 65, 89]).

1991 Mathematics Subject Classification. Primary. 14B05, 14J17, 32S25, 57M27, 57R57. Secondary. 14E15, 32S45, 32S55, 53D10, 57M25, 57R17.

Key words and phrases. normal surface singularities, links, geometric genus, rational, elliptic, cyclic quotient, sandwiched singularities, lens spaces, deformation, smoothings, Milnor fibers, Stein fillings, symplectic fillings, open book decompositions, contact structures.

In order to have the plumbing representation, it is necessary to consider resolutions with normal crossing exceptional divisor, that is good resolutions. Among the good resolutions there is a minimal one, called the minimal good resolution. On the other hand, in some other considerations, it is preferable to work with the minimal (not necessarily good) resolution. In such a resolution, there is no rational smooth irreducible exceptional divisor E_{v} with e_{v} = −1.

The link M is called rational homology sphere (RHS) if H_{1}(M, ℚ) = 0. This is happening if and only if Γ(π) is a tree and g_{v} = 0 for all v ∈ V.

2.2.1. Γ(π) is connected and I is negative definite [61]. The converse of this was proved by Grauert [31]. More precisely, a connected 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.

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 [75], the oriented diffeomorphism type of M determines completely the equivalence class of Γ(π).

In fact, Neumann’s theorem was preceded by the following two key results. First, Mumford proved that the minimal resolution graph of (X, 0) is empty if and only if M = S^{3} [61]. Later, Orlik and Wagreich generalized this for singularities with a good ℂ^{∗}-action as follows [83]. In this case, M is a Seifert manifold which can be codified by its minimal (star-shaped) plumbing graph, and 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.

3. Some analytic invariants of normal surface singularities

3.1. The analytic type of (X, 0) is characterized by its local analytic ring O_{X,0} whose maximal ideal will be denoted by m_{0} ⊂ O_{X,0}. It determines the analytic invariants of (X, 0). We investigate the topological nature of some of the a priori analytical invariants, hence 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.

Then f_{HS}(1) = 1 and f_{HS}(2) − 1 = dim m_{0}/m_{0}^{2} equals the minimal N for which some embedding (X, 0) ⊂ (ℂ^{N}, 0) can be realized, hence 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. The geometric genus p_{g} is defined as dim_{ℂ} H^{1}(X̃, OX̃), where X̃ → X is any resolution as above (see also 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 θ := dim_{ℂ} H^{1}(X̃, θX̃). The point is that these invariants can be recovered from the sheaf-cohomology of some one-dimensional 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 h^{i}(Z) := dim H^{i}(Z, O_{Z}) (for i = 1, 2). By the theorem of formal functions (see [34, page 277]) if one takes Z with m_{v} ≫ 0 for all v, then, e.g., 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 base-points. If π has no base-points, then mult(X, 0) = −Z_{max}^{2} (cf. [111]); 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 pullback 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. By a result of Laufer [43]:

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

This can be read ‘from (X, 0)’ as well: If H_{L^{2}}^{0} denotes 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. Topological candidates for some analytic invariants

4.1. One can ask if it is possible to recover a certain analytic invariant from the link M. For a more complete discussion see e.g. [12, 65, 66, 67, 68, 69, 70, 71, 77, 78, 79, 87]. Similarly, we can ask the same question for certain analytic properties which characterize important families of singularities.

4.1.1. The Hirzebruch-Jung singularities, by definition, are characterized by the existence of a finite projection (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(p, q) with 0 < q < p and gcd(p, q) = 1; or equivalently, if the minimal resolution graph is a straight line graph with all genera zero (here we will use the traditional notation −b_{v} instead of e_{v}):

[figure: −b_{1}; −b_{2}; −b_{s}; ⋯]

where −b_{1}, …, −b_{s} are given by the continued fraction [b_{1}, b_{2}, …, b_{s}]:

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

Notice also that a Hirzebruch-Jung singularity can also be realized as a cyclic quotient singularity X_{p,q} := (ℂ^{2}, 0)/ℤ_{p}. Here the action is ξ ∗ (u, v) = (ξu, ξ^{q}v), where ξ is a primitive p-th root of unity.

For general reference on Hirzebruch–Jung singularities, see e.g. [7] or [89].

4.1.2. Artin defined the rational singularities by the vanishing p_{g} = 0. Artin’s topological characterization, cf. [3, 4, 21, 91], is reviewed in section 7.

4.1.3. The rational double points, or RDP–singularities (i.e. rational singularities with multiplicity two) are exactly the simple hypersurface singularities, hence those 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).

Additionally, the RDP’s are exactly the (Kleinian) quotient singularities ℂ^{2}/G for finite subgroups G ⊂ SL(2, ℂ). Fore more details, see e.g. [21, 23].

4.2. (Pseudo)Taut singularities. One can ask even for the possibility to characterize completely the analytic type from the topological type. The list of singularities when this is happening was established by Laufer in [44] (completing the list started by Grauert, Brieskorn, Tjurina and Wagreich). These singularities are called taut, their topological types carry unique analytical structure. They include all the Hirzebruch-Jung singularities and the rational double and triple points.

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

4.3. Topological candidates. Unfortunately, the class of (pseudo)taut singularities is rather restrictive.

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 invariant only, which guides a certain geometric phenomenon. For example, by [92, 112], 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 during the deformation.

Therefore, one tries to characterize topologically some of the discrete invariants of (X, 0) only. In general, the analytic invariants listed is section 3 are not topological, but if we restrict our study to some special families, then they might be determined from the graph. In fact, 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 them.

4.3.1. 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 [3, 4].

Z_{min} is the topological candidate for Z_{max}. Clearly Z_{min} ≤ Z_{max}. 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).

4.3.2. Another ‘pair’ is the following. The analytic invariant is the canonical divisor KX̃ of X̃. Numerically, it is codified by the (anti)canonical cycle 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. The number −(KX̃, E_{v}) can be determined topologically by the adjunction formula:

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

(Above δ(E_{v}) is the sum of delta–invariants of the singularities of E_{v}.)

One shows 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. [49]). Moreover, Z_{K} = 0 if and only if (X, 0) is a RDP-singularity and π is the minimal resolution.

Note also that 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. 4.1.3). 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.

4.3.3. The main importance of the (anti)canonical cycle Z_{K} comes from 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.

4.3.4. 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.

Note that the Gorenstein property does not impose any further topological restriction. Indeed, in [90] Popescu-Pampu proved that any numerically Gorenstein topological type can support a Gorenstein analytic structure.

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/formal) deformation theory.

Deformation theory analyses flat maps λ : (X , 0) → (T, 0) whose special fibre is identified with (X, 0), up to isomorphisms of maps which respect these identifications. Such maps are called deformations of (X, 0) over (T, 0). In this article, (T, 0) is always supposed to be 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 deformations λ are induced by a map (T, 0) → (B, 0) which is unique at the level of tangent spaces, see [32, 95]. 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 [107], or if emb dim(X, 0) = 4 [94]. 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 [104, 106] for different notions (like Whitney conditions, equisingularity, simultaneous resolutions, or μ-constant, resp. μ^{∗}-constant deformations in the case of hypersurfaces).

Here we will emphasize mainly the notion of simultaneous resolution. It also can be defined at different levels of complexity.

5.2. Simultaneous resolutions. Definitions. [106, 47, 40] 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 non-reduced 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.

5.3. Our plan is to highlight some topological criteria which assure the existence of some kind of simultaneous resolution, see [106, 40, 47, 48]. 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}.

Theorem 5.3.1. Let λ : (X , 0) → (D, 0) be a flat deformation of (X, 0). Then (i) [110] (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) [26] (p_{g})_{t} is upper 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 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, cf. [15, 14], see also the survey of Pinkham: Résolution simultanée de points doubles rationnels in [21, pages 179-203].

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. [40]); but the very weak simultaneous resolution, in general, is not.

5.4.1. Topological characterization. [47]

(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. [47], page 12), that if (X, 0) is normal Gorenstein then 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.

Above, the Gorenstein assumption cannot be dropped, cf. [40](2.8), or 6.3 here. Nevertheless, 5.4.1 has a global version without the Gorenstein assumption [40](2.1); see also [40](2.25) for a possible local analog.

5.4.2. Example. Let λ : (ℂ^{3}, 0) → (ℂ, 0) be a simple hypersurface singularity. 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 the miniversal deformation 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 exceptional divisor E of the minimal resolution of (X, 0) has two irreducible components 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, in this deformation the singular point 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.

5.5. Topological characterization of weak simultaneous resolutions. [47, 40] 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 along s. Obviously, the topological condition is equivalent with the stability of the links: M_{t} ≃ M for any t (where ≃ means ‘orientation preserving diffeomorphism’).

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

In [106] Teissier proved that the existence of a strong simultaneous resolution implies the Whitney condition along s(D). Moreover, by another result of Teissier [104], and respectively of Briançon and Speder [13], the Whitney condition is equivalent to μ_{t}^{∗} constant. Finally, the circle is completed by Laufer in [48], 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, 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.

For a deformation which admits a weak simultaneous resolution the existence of a strong simultaneous resolution is not guaranteed, even if the stable link is a rational homology sphere.

5.7.1. Example. Consider the deformation of hypersurface (even Newton non– degenerate) isolated singularities f_{t} = z_{3}^{3} + z_{2}^{4}z_{1} + z_{1}^{10} + tz_{2}^{3}z_{3} [12]. Then the link is stable under the deformation, nevertheless μ^{(2)}(f_{1}) = 7 and μ^{(2)}(f_{0}) = 8. Note that in this case the stable link is a rational homology sphere (similar examples without this additional property were constructed by Briançon and Speder [13]).

6. Smoothings and their invariants

6.1. A (flat) deformation λ : (X , 0) → (D, 0) 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 an oriented real connected 4-manifold with boundary M. As the fibers of λ are Stein, F has the homotopy type of a finite CW-complex of dimension ≤ 2. (For the choice of the ‘good’ representatives of the local Milnor balls, Milnor fibers and fibrations, see e.g. the book of Looijenga [56].)

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 [115], and proved by Greuel and Steenbrink [33]), 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 [46, 115, 58, 55, 88] (and the references therein).

For example, Dolgachev’s triangle singularity D_{pqr} (with Dolgachev numbers p, q, r, cf. 8.3.2) admits a smoothing if p + q + r < 22 [55], cannot be smoothed at all if p + q + r > 22 [115], 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 them is distinguished. It can be constructed as follows [5]: let X̃ → X be the minimal resolution of (X, 0). Then all the deformations of X̃ come from deformations of (X, 0) (since p_{g} is constantly zero). These deformations provide a component, called the Artin component. It is smooth [113].

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 and 10).

The Milnor number corresponding to the Artin component is 1, while the other Milnor number is 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. [115], (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 [113], for any rational singularity, the semi-universal deformation restricted above the Artin component admits a very weak simultaneous resolution. This is not 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, hence the Gorenstein assumption in 5.4.1 is needed.

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}.

6.5. Theorem. [22, 46, 99, 115, 58] Consider 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 a smoothing of a Gorenstein singularity, (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. Moreover, in the Gorenstein case, μ and σ are independent of the smoothing.

6.5.1. Wahl in [115] added to this list some other relations. He also explained a recipe by which one can create even more identities, modulo a conjectural globalization assumption. This assumption was proved by Looijenga in [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 an a priori analytic invariant of the smoothing λ. Recall that θ denotes h^{1}(X̃, θX̃).

6.6. Theorem. [115] Consider the minimal resolution of (X, 0) with #V_{min} irreducible exceptional components and (anti)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. [8]) 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. [8], cf. also with 7.7 here). But Wahl have found an example when the inequality is strict (cf. [8]).

For the ultimate formula for the codimension of the Artin component of a rational singularity see [19].

6.7. Remark. In general it is extremely difficult to determine the number of irreducible components of B and their dimensions. Hence, 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; see the next section.

7. Rational singularities

7.1. In this section we exemplify for rational singularities that 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. [3, 4] (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. Note also that any resolution of a 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 a singularity with such a resolution graph Γ is rational. Moreover, Z_{min}^{2} = −#{v : δ_{v} = 1}.

(c) Rational surface singularities with reduced fundamental cycle (i.e. with Z_{min} = ∑ E_{v}) are also called minimal singularities (cf. [40], section 7). E.g., the cyclic quotients are minimal.

The class of rational graphs is closed while taking subgraphs and decreasing self-intersections. The graphs of minimal singularities are obtained from those considered in (b) by decreasing the decorations.

(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. [98, 39]. A sandwiched singularity is, by definition, a normal surface singularity which is analytically isomorphic to the germ of an algebraic surface that admits a birational morphism 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 [98]. Consider a plane curve singularity (C, 0) ⊂ (ℂ^{2}, 0), and let ϕ : Y → ℂ^{2} be a (in general, non-minimal) embedded resolution 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. [3, 4] 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). Although there is a very general conjecture of Kollár about B, valid for any rational singularity [41], we present the case of cyclic quotient singularities only.

Notice that the cyclic quotient singularities are taut singularities. Hence, the next discussion shows that even in those cases when a priori we know that some invariant is topological, its representation from the graph can be really complicated.

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 [2]. Here we reproduce the combintorial part of some results obtained by J. Christophersen, J. Stevens and J. Kollár and N.I. Shepherd-Barron (after [101]).

7.6. Combinatorial characterization of B. Assume as in subsection 4.1.1 that p/q = [b_{1}, …, b_{s}]. We write p/(p − q) = [a_{2}, …, a_{e−1}]. (Here we will follow the traditional index notation.) 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_{p,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 of tuples of positive integers [k] = (k_{2}, …, k_{e−1}), (k_{i} ≥ 1), such that the continued fraction [k_{2}, …, k_{e−1}] is zero and [k] is admissible in the sense of (10.2). 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 using only diagonals which intersect at most at vertices. 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 [101]):

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 follow from a very 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) [40]. This gives possibilities for remarkable generalizations (see [41]) 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), [114], cf. also with [40]. For a cyclic quotient singularity X_{p,q}, the following facts are equivalent:

(i) Z_{K}^{2} ∈ ℤ;

(ii) (q + 1)^{2}/p ∈ ℤ;

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

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

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

Notice that r = 1 implies d = 1, p = 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_{p,q}) to be l if r > 1, respectively to be l − 1 if r = 1.

In [40], the key object which provides valuable information about 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 (anti)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 curves 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. [40, (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 [40] for the particular case of cyclic quotients.

7.10. Theorem. [Kollár–Shepherd-Barron] For any cyclic quotient singularity (X, 0), there is a one-to-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 [101].

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: the minimal resolution and the identity of (X, 0).

7.10.2. Example.[40] 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. If (X, 0) is rational with mult(X, 0) ≤ 3 then the base space B is smooth. See [37] and [102] for the description of the deformation of rational quadruple points, [38] for minimal rational singularities, and [39] 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 [41]. 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 [101] and minimal rational singularities [39].

8. 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. Similarly, χ(Z_{min}) = 0 if and only if min_{Z>0} χ(Z) = 0. Laufer in [45, page 1281] proved this via the theory of analytic deformations. For a combinatorial proof see [62, (4.3)].

8.2. Definition. (Wagreich [111], Laufer [45]) (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. The definition is independent of the choice of the resolution.

Laufer in [45] 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. 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. Laufer 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}.

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}.

8.3. Theorem. Minimally elliptic singularities. [45] 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.

8.3.1. Example. Assume that Γ(π) consists of only one vertex with g_{v} = 1, and arbitrary self-intersection e_{v}. Then it is elliptic, and is called simple elliptic of degree −e_{v}. If e_{v} ∈ {−1, −2, −3} then they are realized by the hypersurface singularities ({x^{a} + y^{b} + z^{c} + λxyz = 0}, 0) with (a, b, c) = (2, 3, 6) (Ẽ_{8}), (2, 4, 4) (Ẽ_{7}) and (3, 3, 3) (Ẽ_{6}). The case e_{v} = −4 is realized by the complete intersection (D̃_{5}) {x^{2} + y^{2} + λzw = xy + z^{2} + w^{2} = 0}.

8.3.2. Example. Dolgachev’s triangle singularity D_{−e_{1},−e_{2},−e_{3}}. Assume that the resolution graph Γ has the following form with g_{v} = 0 for all v:

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

The intersection matrix is negative definite if and only if 1+∑_{v>0} 1/e_{v} > 0. Note that 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}.

The next theorem was obtained by Laufer and Reid; M. Tomari also proved similar results for elliptic singularities in the presence of some additional assumptions [108].

8.4. Theorem. [45, 91] Assume that (X, 0) is a minimally elliptic singularity. 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).

For more about elliptic singularities, see [45, 63]. For deformation invariants of minimally elliptic singularities, see Wahl’s paper [114] and the references therein.

9. Links and fillings form contact geometry point of view.

9.1. Contact structures on the link. One of the objects which connects local complex analytic singularities with low-dimensional topology is the link of isolated singularities. One defines a contact structure on a singularity link as follows. Fix an embedding (X, 0) ⊂ (ℂ^{N}, 0). Then the distance function ρ := ∑_{k=1}^{N} |z_{k}|^{2} is strict pseudosubharmonic, and ξ := T M ∩ J(T M), the J-invariant subspace of T M, defines a contact structure on M. Its isotopy class is called the canonical contact structure of M induced by the analytic structure of (X, 0). (Here J is the almost complex structure of T X.)

This definition imposes several questions/problems for any fixed M:

(1) Classify all the possible contact structures induces by different analytic structures supported by the topological type determined by M.

(2) How is the subclass from (1) related with the class of all contact structures of M?

A partial answer to (1) was given in [16, 17]: all the possible (canonical) contact structures induces by different analytic structures are contactomorphic. Notice again that this fact is not valid in higher dimensions, see Ustilovsky’s work [109].

In fact, we conjecture that all the possible (canonical) contact structures induces by different analytic structures are even isotopic. But, definitely, in order to prove this (or even to state this) one needs first to provide a canonical construction which identifies the link M up to an isotopy (and independently of the supported analytic structure); the existing plumbing construction identifies M only up to an orientation preserving diffeomorphism.

Although there is an intense activity in classification of contact structures, and a considerably impressive list of positive results finishing the classification for lens spaces, torus bundles over circles, circle bundles over surfaces, some Seifert manifolds (thanks to the work of Giroux, Etnyre, Honda, Lisca, Stipsicz and others, see e.g. [28, 29, 35, 36, 54] and the references listed therein), Part (2) is mainly open. Recall that a contact structure is either overtwisted or tight, and all overtwisted structure are characterized by the homotopy of their underlying oriented plane field (by a result of Eliashberg). Hence, the difficulty appears in the classification of tight structures. The canonical structure is one of them. Surprisingly, even the finiteness of the possible tight structures on M is not guaranteed in general (as was shown by Colin, Giroux and Honda [20]): although the number of corresponding homotopy classes of plane fields is finite, the number of isotopy classes of tight contact structures is finite if and only if M is atoroidal (i.e. the minimal resolution graph is star-shaped with at most three ‘legs’, and all genus decorations are zero).

It is also not clear at all how one can identify in this multitude of structures the canonical one.

In the sequel the canonical contact structure will be denoted by (M, ξ_{can}).

9.2. Fillings of (M, ξ_{can}). Although in the literature there are many different versions of fillability (holomorphic, Stein, strong/weak symplectic), here we will deal only with Stein one: any Stein manifold whose contact boundary is contactomorphic to (M, ξ_{can}) is a Stein filling of (M, ξ_{can}). For a singularity link, the following questions/problems are natural:

(1) Is any (M, ξ_{can}) Stein fillable?

(2) Classify all the Stein fillings of (M, ξ_{can}).

(3) Determine all the Stein fillings ‘coming from singularity theory’.

The answer to (1) is yes: Consider the minimal resolution of (X, 0). It is a holomorphic, non-Stein filling of (M, ξ_{can}), but by a theorem of Bogomolov and de Oliveira [11] this holomorphic structure can be deformed into a Stein one. This construction already provides an example for (3); another one is given by the Milnor fibers of the smoothings of different analytic realizations (X, 0) of the topological type fixed by M (if there are any). More precisely, the existence (and uniqness) of the miniversal deformation of isolated singularities is guaranteed by results of Schlessinger [95] and Grauert [32]. In general, its base space has many irreducible components. A component is called smoothing component if the generic fiber over it (the so-called Milnor fiber) is smooth. In general, different analytic structures might have different smoothings; or for a fixed (X, 0), different smoothing components might produce diffeomorphic Milnor fibers. It might also happen that smoothings do not exist at all (for the last two situation see e.g. the case of some simple elliptic singularities [58, 80]).

In fact, in the literature basically only these two constructions are present regarding (3); it is a high desire to find some other general constructions too. We notice that rational singularities are always smoothable, moreover, the Milnor fiber of one of the smoothing component (the Artin component) is diffeomorphic with the space of the minimal resolution.

The list regarding part (2) starts with a result of Eliashberg [25] showing that (𝕊^{3}, ξ_{can}) has only one Stein filling (up to diffeomorphism), namely the ball. For links of simple and simple elliptic singularities the classification was finished by Ohta and Ono [80, 81]. Moreover, in all these cases all possible Stein fillings are provided by the minimal resolution or Milnor fibers. This parallelism sometimes is really striking. E.g. for simple elliptic singularities with degree k > 0, Ohta and Ono proved that the existence of a Stein filling with vanishing first Chern class imposes k ≤ 9. This can be compared with the fact that in the case of a Milnor fiber the Chern class is vanishing (by [96]) and the smoothability condition is the same k ≤ 9 (cf. [86]).

Fillings of links of lens spaces L(p, q) were classified by Lisca [52, 53] (as a generalization a result of McDuff [59] valid for the spaces L(p, 1), for all p ≥ 2), and of quotient surface singularities by Bhupal and Ono [10]. We will return to Lisca’s result in the next section showing that Lisca’s list agrees perfectly with the list of Milnor fibers (or, with the smoothing/all deformation components).

We would like to notice that this phenomenon, namely that all the Stein fillings are obtained either by minimal resolution or Milnor fibers – at some point of the singularity complexity – might stop. This emphasizes the importance of the research in direction (3) even more.

Moreover, in general, the finiteness of the Stein fillings might fail too: Ohta and Ono produced on some singularity links infinitely many symplectic fillings [82], also Ozbagci and Stipsicz in [85], and independently Smith in [97], have shown that certain contact structures have infinitely many Stein fillings (although they are not singularity links, one expects that at some moment similar fact will be established for some singularity links as well). For similar result see also the recent article [1] too.

9.3. (M, ξ_{can}) and open books. By a result of Giroux [30], there is a one-to-one correspondence between open book decompositions of M (up to isotopy and stabilization) and positive (that is, which induces the ambient orientation) contact structures on M (up to isotopy).

The link M of a normal surface singularity (X, 0) admits a natural family of open book decompositions, the so-called Milnor open books. They are cut out by analytic germs f : (X, 0) → (ℂ, 0) which define isolated singularities. If L_{f} ⊂ M denotes the (transversal) intersection of f^{−1}(0) with M, then the Milnor fibration of f defines an open book decomposition of M with binding L_{f}. By [17], all the Milnor open book decompositions support the same contact structure on M, namely the canonical contact structure (M, ξ_{can}). Section 12 contains a more detailed discussion about Milnor open books and their invariants.

10. Fillings. The case of cyclic quotient singularities

10.1. In this section we assume that (X, 0) = (X_{p,q}, 0) is a cyclic quotient (or Hirzebruch-Jung) singularity. Its oriented link is the oriented lens space L(p, q). Recall that (X_{p,q}, 0) is taut and rational.

For cyclic quotient singularities the classification of the Stein fillings of (M, ξ_{can}) and their connection with Milnor fibers and Milnor open books is completely understood. In this section we review this problem; the presentation follows [72].

10.2. Notations. If \underline{x} = (x_{1}, …, x_{n}) are variables, the Hirzebruch-Jung continued fraction [x_{1}, …, x_{n}] can be defined by induction on n through the formulae: [x_{1}] = x_{1} and [x_{1}, …, x_{n}] = x_{1} − 1/[x_{2}, …, x_{n}] for n ≥ 2. One shows that:

[x_{1}, …, x_{n}] = \frac{Z_{n}(x_{1}, …, x_{n})}{Z_{n−1}(x_{2}, …, x_{n})},

where the polynomials Z_{n} ∈ ℤ[x_{1}, …, x_{n}] satisfy the inductive formulae:

(10.2.1) Z_{n}(x_{1}, …, x_{n}) = x_{1} · Z_{n−1}(x_{2}, …, x_{n}) − Z_{n−2}(x_{3}, …, x_{n}) for all n ≥ 1,

with Z_{−1} ≡ 0, Z_{0} ≡ 1 and Z_{1}(x) = x. In fact, Z_{n}(\underline{x}) equals the determinant of the n × n-matrix M(\underline{x}), whose entries are M_{i,i} = x_{i}, M_{i,j} = −1 if |i − j| = 1, M_{ij} = 0 otherwise. Hence, besides (10.2.1), they satisfy many ‘determinantal relations’ too; e.g. Z_{n}(x_{1}, …, x_{n}) = Z_{n}(x_{n}, …, x_{1}). Following [84], \underline{x} ∈ ℕ^{n} is admissible if the matrix M(\underline{x}) is positive semi-definite of rank ≥ n − 1. Denote by adm(ℕ^{n}) the set of admissible n-tuples.

If \underline{x} is admissible and n > 1, then each x_{i} > 0. Moreover, if [x_{1}, …, x_{n}] is admissible then [x_{n}, …, x_{1}] is admissible too. For any r ≥ 1, denote:

K_{r} := \{\underline{k} = (k_{1}, …, k_{r}) ∈ adm(ℕ^{r}) | [k_{1}, …, k_{r}] = 0\}.

For \underline{k} = (k_{1}, …, k_{r}) ∈ K_{r} set \underline{k}′ := (k_{r}, …, k_{1}) ∈ K_{r}.

For p, q as above and HJ-expansion \frac{p}{p−q} = [a_{1}, …, a_{r}], set:

K_{r}(\frac{p}{p−q}) = K_{r}(\underline{a}) := \{\underline{k} ∈ K_{r} | \underline{k} ≤ \underline{a}\} ⊂ K_{r}.

Here, \underline{k} ≤ \underline{a} means that k_{i} ≤ a_{i} for all i.

10.3. Lisca’s Conjecture. We recall briefly the classification of fillings of lens spaces (endowed with their canonical contact structure) established by Lisca. He provides by surgery diagrams a list of compact oriented 4-manifolds W_{p,q}(\underline{k}) with boundary L(p, q). They are parametrized by the set K_{r}(\frac{p}{p−q}) of sequences of integers \underline{k} ∈ ℕ^{r} (cf. (10.2)). He showed that each manifold W_{p,q}(\underline{k}) admits a structure of Stein surface, filling (L(p, q), ξ_{can}), and that any symplectic filling of (L(p, q), ξ_{can}) is orientation-preserving diffeomorphic to a manifold obtained from one of the W_{p,q}(\underline{k}) by a composition of blow-ups. In general, the oriented diffeomorphism type of the boundary and the parameter \underline{k} does not determine uniquely the (orientation-preserving) diffeomorphism type of the fillings: for some pairs the corresponding types might coincide (see subsection (10.8) for this ‘ambiguity’).

Lisca also noted that, following the works of Christophersen [18] and Stevens [101], K_{r}(\frac{p}{p−q}) parametrizes also the irreducible components of the reduced miniversal base space of deformations of the cyclic quotient singularity X_{p,q}. Since each component of the miniversal space is in this case a smoothing component, Lisca conjectured in [53, page 768] that the Milnor fiber of the irreducible component of the reduced miniversal base space of the cyclic quotient singularity X_{p,q}, parametrized in [101] by \underline{k} ∈ K_{r}(\frac{p}{p−q}) is diffeomorphic to W_{p,q}(\underline{k}).

On the other hand, in [39], de Jong and van Straten studied by an approach completely different from Christophersen and Stevens the deformation theory of cyclic quotient singularities (as a particular case of sandwiched singularities). They also parametrized the Milnor fibers of X_{p,q} using the elements of the set K_{r}(\frac{p}{p−q}). Hence, one can formulate the previous conjecture for their parametrization as well.

10.4. The answers to the (improved) conjecture. [72] answers positively these questions. Its main results are the following:

(1) One defines an additional structure associated with any (non-necessarily oriented) lens space: the ‘order’. Its meaning is the following: geometrically it is a (total) order of the two solid tori separated by the (unique) splitting torus of the lens space; in plumbing language, it is an order of the two ends of the plumbing graph (provided that this graph has at least two vertices). Then one shows that the oriented diffeomorphism type and the order of the boundary, together with the parameter \underline{k} determines uniquely the filling.

(2) One endows in a natural way all the boundaries of the spaces involved (Lisca’s fillings W_{p,q}(\underline{k}), Christophersen-Stevens’ Milnor fibers F_{p,q}(\underline{k}), and de Jong-van Straten’s Milnor fibers F′_{p,q}(\underline{k})) with orders (this extra structure is denoted by ∗). Then one proves that all these spaces are connected by orientation-preserving diffeomorphisms which preserve the order of their boundaries: W_{p,q}(\underline{k})^{∗} ≃ F_{p,q}(\underline{k})^{∗} ≃ F′_{p,q}(\underline{k})^{∗}. This is an even stronger statement than the result expected by Lisca’s conjecture since it eliminates the ambiguities present in Lisca’s classification.

(3) In fact, [72] even provides a fourth description of the Milnor fibers constructed by a minimal sequence of blow ups of the projective plane which eliminates the indeterminacies of a rational function which depends on \underline{k}. This is in the spirit of Balke’s work [6].

(4) As a byproduct it follows that both Christophersen-Stevens and de Jongvan Straten parametrized the components of the miniversal base space in the same way.

(5) Moreover, one obtains that the Milnor fibers corresponding to the various irreducible components of the miniversal space of deformations of X_{p,q} are pairwise non-diffeomorphic by orientation-preserving diffeomorphisms whose restrictions to the boundaries preserve the order.

In the sequel we will not say more about Lisca’s construction, instead, we will describe briefly the Milnor fibers associated with the different smoothing components, with a special emphasis on the construction (10.4)(3). This will be compatible with the description of Christophersen and Stevens on the structure of the reduced miniversal base space of cyclic quotients [9, 18, 103] (cf. also with [2]). The de Jong-van Straten construction will be reviewed in the next section.

10.5. First we concentrate on X_{p,q}. It can be embedded into ℂ^{r+2} by some regular functions z_{0}, …, z_{r+1}. Some of the equations of the embedding are

(10.5.1)  z_{i−1}z_{i+1} − z_{i}^{a_{i}} = 0 for all i ∈ {1, …, r}.

Using equations (10.5.1) and induction, one shows that the restriction of each z_{i} to X_{p,q} is a rational function in (z_{0}, z_{1}) of the form

z_{i} = z_{1}^{Z_{i−1}(a_{1},...,a_{i−1})} · z_{0}^{−Z_{i−2}(a_{2},...,a_{i−1})} for i ∈ {1, …, r + 1}.

In particular, the restriction pr_{01} of the projection (z_{0}, …, z_{r+1}) ↦ (z_{0}, z_{1}) to X_{p,q} is birational, i.e. it is a ‘sandwiched representation’ of X_{p,q} (cf. next section for the terminology). (Here a comment is in order. Recall that X_{p,q} is minimal, hence, by a general construction, it admits a canonical sandwiched representation pr : X_{p,q} → ℂ^{2}, see e.g. (11.1). We wish to emphasize that the two birational maps pr_{01} and pr are different capturing two different geometrical aspects about X_{p,q}.)

The equations of X_{p,q} are weighted homogeneous, however the weights w_{i} := w(z_{i}) are not unique. With the choice w_{0} = w_{1} = 1 one has w_{i} = Z_{i−1}(a_{1}, …, a_{i−1})− Z_{i−2}(a_{2}, …, a_{i−1}) for all i ≥ 1, and 1 = w_{0} = w_{1} ≤ w_{2} ≤ ⋯ ≤ w_{r+1} = q.

10.6. The deformations. Next, we fix \underline{k} ∈ K_{r}(\underline{a}), and we denote by SCS ; \underline{k} the corresponding deformation component (as it is described by Christophersen and Stevens). Then, we consider a special 1-parameter deformation with equations Et ; \underline{k} of X_{p,q}. This deformation is determined by the deformed equations of (10.5.1) (cf. [2], [101, (2.2)]). These are:

(10.6.1)  z_{i−1}z_{i+1} = z_{i}^{a_{i}} + t · z_{i}^{k_{i}} for all i ∈ {1, …, r},

where t ∈ ℂ. Let X t ; \underline{k} be the affine space determined by the equations Et ; \underline{k} in ℂ^{r+2}. One proves that the deformation t ↦ X t ; \underline{k} has negative weight and is a smoothing belonging to the component SCS ; \underline{k}. Hence, X t ; \underline{k} is a smooth affine space for t ≠ 0 (and by [115, (2.2)]) it is diffeomorphic to the Milnor fiber of SCS ; \underline{k}.

10.7. X t ; \underline{k} as a rational surface. Similarly as for X_{p,q}, using (10.6.1), on X t ; \underline{k} all the coordinates z_{i} can be expressed as rational functions in (z_{0}, z_{1}). Indeed, for each i ∈ \{1, …, r + 1\}, on X t ; \underline{k} one has:

z_{i} = z_{0}^{−Z_{i−2}(a_{2},...,a_{i−1})}P_{i}

for some P_{i} ∈ ℤ[t, z_{0}, z_{1}]. The polynomials P_{i} satisfy the inductive relations:

(10.7.1)  P_{i−1} · P_{i+1} = P_{i}^{a_{i}} + tP_{i}^{k_{i}} · z_{0}^{(a_{i}−k_{i})·Z_{i−2}(a_{2},...,a_{i−1})}

with P_{1} = z_{1} and with the convention P_{0} = 1.

Consider the application π : ℂ^{2} \ \{z_{0} = 0\} ⟶ X t ; \underline{k} given by

(10.7.2) (z_{0}, z_{1}) ↦ (z_{0}, z_{1}, P_{2}, …, z_{0}^{−Z_{i−2}(a_{2},...,a_{i−1})}P_{i}, …, z_{0}^{−(p−q)}P_{r+1}) ∈ ℂ^{r+2}.

We are interested in the birational map ℂ^{2} −−→ X t ; \underline{k}, still denoted by π, and its extension \overline{π} : ℙ^{2} −−→ \overline{X t ; \underline{k}}, where \overline{X t ; \underline{k}} is the closure of X t ; \underline{k} in ℙ^{r+2}.

Let ρ′ ; \underline{k} : B′ℙ^{2} → ℙ^{2} be the minimal sequence of blow ups such that \overline{π} ∘ρ′ ; \underline{k} extends to a regular map B′ℙ^{2} → \overline{X t ; \underline{k}}. Let L_{∞} ⊂ ℙ^{2} be the line at infinity and by L_{0} the closure in ℙ^{2} of {z_{0} = 0}. We use the same notations for their strict transforms via blow ups of ℙ^{2}. Since the projection pr : \overline{X t ; \underline{k}} → ℂ^{2} is regular and pr ∘ π is the identity, one gets that \overline{π} ∘ ρ′ ; \underline{k} sends L_{0} and the total transform of L_{∞} in the curve at infinity \overline{X t ; \underline{k}} \ X t ; \underline{k}.

Therefore, let us modify ρ′ ; \underline{k} into ρ\underline{k} : Bℙ^{2} → ℙ^{2}, the minimal sequence of blow ups which resolve the indeterminacies of \overline{π} sitting in ℂ^{2} (hence ρ′ ; \underline{k} and ρ\underline{k} over ℂ^{2} coincide). Denote by E_{π} its exceptional curve and by C_{π} the union of those irreducible components of E_{π} which are sent to C∞ ; \underline{k}. Set Bℂ^{2} := Bℙ^{2} \ L_{∞}. Summing up all the above discussions, one obtains:

Theorem 10.7.1. The restriction of \overline{π} ∘ ρ\underline{k} induces an isomorphism Bℂ^{2} \ (L_{0} ∪ C_{π}) → X t ; \underline{k}. In particular, the Milnor fiber can be realized as the complement of the projective curve L_{∞} ∪ L_{0} ∪ C_{π} in Bℙ^{2}.

The point is that the indeterminacies of \overline{π} above ℂ^{2}, hence the modification ρ\underline{k} too, can be described precisely. This leads to the following description of the Milnor fiber.

Corollary 10.7.2. Consider the lines L_{∞} and L_{0} on ℙ^{2} as above. Blow up r − 1 + ∑_{i=1}^{r}(a_{i} − k_{i}) infinitely closed points of L_{0} in order to get the dual graph in Figure 1 of the configuration of the total transform of L_{∞} ∪L_{0} (this procedure topologically is unique, and its existence is guaranteed by the fact that \underline{k} ∈ K_{r}(\underline{a})). Denote the space obtained by this modification by Bℙ^{2}. Then the Milnor fiber X t ; \underline{k} of SCS ; \underline{k} is diffeomorphic to Bℙ^{2} \ (∪_{j=0}^{r}V_{j}).

Moreover, let T be a small open tubular neighbourhood of ∪_{j=0}^{r}V_{j}, and set F_{p,q}(\underline{k}) = Bℙ^{2} \ T. Then F_{p,q}(\underline{k}) is a representative of the Milnor fiber of SCS ; \underline{k} as a manifold with boundary whose boundary is L(p, q).

Furthermore, the marking {V_{i}}_{i} as in the Figure 1, defines on the boundary of F_{p,q}(\underline{k}) an order; denote this supplemented space by F_{p,q}(\underline{k})^{∗}. Then its ordered boundary is L(p, q)^{∗} endowed with the preferred order.

[figure: …; 1; 1 − a_{1}; −a_{2}; −a_{r}; L_{∞} = V_{0}; L_{0} = V_{1}; V_{2}; V_{r}; −1; −1; −1; −1; −1; −1; …; …; …; a_{1} − k_{1}; a_{2} − k_{2}; a_{r} − k_{r}; ; ; ; Figure 1]

Remark 10.7.3. In fact, ρ\underline{k} serves also as the minimal modification which eliminates the indeterminacy of the last component of π from (10.7.2), namely of the rational function z_{r+1} = P_{r+1}/z_{0}^{p−q}. In particular, we find the following alternative description of the Milnor fiber F_{p,q}(\underline{k}):

For each \underline{k} ∈ K_{r}(\underline{a}), define the polynomial P_{r+1} via the inductive system (10.7.1). Let ρ\underline{k} : Bℙ^{2} → ℙ^{2} be the minimal modification of ℙ^{2} which eliminates the indeterminacy points of P_{r+1}/z_{0}^{p−q} sitting in ℂ^{2}. Then the dual graph of the total transform of L_{∞} ∪L_{0} has the form indicated in Figure 1, and F_{p,q}(\underline{k}) is orientation-preserving diffeomorphic to Bℙ^{2} \ (∪_{j=0}^{r}V_{j}).

10.8. An identification criterion of the Milnor fibers. The next criterion generalizes Lisca’s criterion [53, §7] to recognize the fillings of L(p, q), it is valid for the spaces with ordered boundaries. It also implies immediately part (2) of (10.3), namely that W_{p,q}(\underline{k})^{∗} ≃ F_{p,q}(\underline{k})^{∗}.

Associate to the sequence \underline{a} the string G(\underline{a}) decorated with the entries a_{1}, …, a_{r} (this might also serve as the minimal resolution graph of X_{p,p−q}). Regarded G(\underline{a}) as a plumbed graph, it determines the plumbed 4-manifold Π(\underline{a}) (which is diffeomorphic to the minimal resolution space of X_{p,p−q}), whose oriented boundary is −L(p, q) (L(p, q) with opposite orientation).

Let F be a Stein filling of (L(p, q), ξ_{can}), e.g. one of the Milnor fibers considered above. Set V for the closed 4-manifold obtained by gluing F and Π(\underline{a}) via an orientation preserving diffeomorphism ϕ : ∂F → ∂(−Π(\underline{a})) of their boundaries. Denote by \{s_{i}\}_{1≤i≤r} the classes of 2-spheres \{S_{i}\}_{1≤i≤r} in H_{2}(Π(\underline{a})) (listed in the same order as \{a_{i}\}_{1≤i≤r}), and also their images via the monomorphism H_{2}(Π(\underline{a})) → H_{2}(V) induced by the inclusion.

Lisca’s criterion (implied also by the results of (10.7)) reads as follows:

Proposition 10.8.1. For all i ∈ {1, …, r} one has

#{e ∈ H_{2}(V) | e^{2} = −1, s_{i} · e ≠ 0, s_{j} · e = 0 for all j ≠ i} = 2(a_{i} − k_{i})

for some \underline{k} ∈ K_{r}(\underline{a}). In this way one gets the pair (\underline{a}, \underline{k}) and F is orientationpreserving diffeomorphic to F_{p,q}(\underline{k}) (≃ F_{p,q′}(\underline{k}′)).

One verifies that the above criterion is independent of the choice of the diffeomorphism ϕ. In fact, even the diffeomorphism type of the manifold V is independent of the choice of ϕ.

Notice that {S_{i}}_{1≤i≤r} and {S_{r−i}}_{1≤i≤r} cannot be distinguished, hence the above algorithm does not differentiate (\underline{a}, \underline{k}) from (\underline{a}′, \underline{k}′), or F_{p,q}(\underline{k}) from F_{p,q′}(\underline{k}′). On the other hand, these are the only ambiguities. (In fact, if r = 1, or even of r > 1 but \underline{a} and \underline{k} are symmetric, then there is no ambiguity, since (p, q, \underline{k}) = (p, q′, \underline{k}′).)

Using the notion of order of the boundaries, one can eliminate the above ambiguity. Notice that any diffeomorphism F_{p,q}(\underline{k}) → F_{p,q′}(\underline{k′}) (whenever (p, q, \underline{k}) ≠ (p, q′, \underline{k}′)) does not preserve any fixed order of the boundary. One has:

Theorem 10.8.2. All the spaces F_{p,q}(\underline{k})^{∗} are different, hence their boundaries L(p, q)^{∗} and \underline{k} ∈ K_{r}(\underline{a}) determine uniquely all the Milnor fibers up to orientationpreserving diffeomorphisms which preserve the order of the boundary.

In order to prove this, the criterion (10.8.1) is modified as follows. Let F^{∗} be a Stein filling of (L(p, q), ξ_{can}) with an order on its boundary. Consider Π(\underline{a})^{∗} with its preferred order (provided a well-determined order of the s_{i}’s, cf. [72]). Construct V as in (10.8.1), and consider the two pairs (q, \underline{k}) and (q′, \underline{k}′) provided (but undecided) by (10.8.1).

Proposition 10.8.3. If ϕ preserves (resp. reverses) the orders of the boundary then F^{∗} is orientation and order preserving diffeomorphic to F_{p,q}(\underline{k})^{∗} (resp. to F_{p,q′}(\underline{k}′)^{∗}).

11. Fillings. The case of sandwiched singularities

11.1. For the definition of minimal rational and sandwiched singularities see 7.3.

In [39] de Jong and van Straten related the theory of sandwiched surface singularities with decorated plane curve singularities.

Consider a reduced germ of plane curve (C, 0) ⊂ (ℂ^{2}, 0) with numbered branches (irreducible components) {C_{i}}_{1≤i≤r}. The multiplicity sequence associated with C_{i} is the sequence of multiplicities on the successive strict transforms of C_{i}, starting from C_{i} itself and not counting the last strict transform. The total multiplicity m(i) of C_{i} with respect to C is the sum of the sequence of multiplicities of C_{i}.

A decorated germ of plane curve is a weighted germ (C, l) such that l = (l_{i})_{1≤i≤r} ∈ (ℕ^{∗})^{r} and l_{i} ≥ m(i) for all i ∈ {1, …, r}.

Starting from a decorated germ, one can blow up iteratively points infinitely near 0 on the strict transforms of C, such that the sum of multiplicities of the strict transform of C_{i} at such points is exactly l_{i}. Such a composition of blow-ups is determined canonically by (C, l). If l_{i} is sufficiently large (in general, larger than m(i)) then the union of the exceptional components which do not meet the strict transform of C form a connected configuration of (compact) curves E(C, l). After the contraction of E(C, l), one gets a sandwiched singularity X(C, l), determined uniquely by (C, l) (for details see [39]). This follows from the fact that the collection of the irreducible components of the exceptional curve E not contained in E(C, l) are exactly the (−1) curves involved in the definition of sandwiched singularities. Conversely, for any sandwiched singularity X one can find (C, l) such that X can be represented as X(C, l).

11.2. Markings of the links [73]. By a result of Neumann [75], the information codified in the link of a surface singularity up to an orientation-preserving homeomorphism and in the weighted dual graph of the minimal good resolution are equivalent. In [89, Theorems 9.1 and 9.7] this is generalized as follows. From the resolution, the abstract link inherits a plumbing structure, that is, a family of pairwise disjoint embedded tori whose complement is fibered by circles, and such that on each torus the intersection number of the fibers from each side is ±1. (The tori correspond to the edges of the dual graph and the connected components of their complement – the “pieces of M” – correspond to the ’un-numbered vertices’.) Then, by [89, Theorem 9.7], the plumbing structure corresponding to the minimal normal crossings resolution is determined up to an isotopy by the oriented link.

Consider again X(C, l) associated with some (C, l), and let M be its link. Using the notations of the previous subsection, write E = E(C, l) + ∑_{i=1}^{r} E_{i}, where (E_{i})_{1≤i≤r} is numbered such that E_{i} is the unique irreducible component which intersects the strict transform of C_{i}. Denote by F_{i} the unique irreducible component of E(C, l) which intersects E_{i}. To F_{i} corresponds a well-defined “piece” of M. In this way, one gets a map from the set {1, ..., r} to the set of pieces of M.

Definition 11.2.1. A map from {1, ..., r} (the index set of the numbered branches of C) to the set of “pieces” of M obtained as above is called a marking of M.

Hence, each realization of M as the link of some X(C, l), where (C, l) is a decorated curve with numbered branches, provides a well-defined marking of M.

11.3. Deformation of sandwiched singularities after de Jong and van Straten. The point is that the above correspondence between sandwiched singularities and decorated plane curves extends to their deformation theory as well.

The total multiplicity of C_{i} with respect to C may be encoded also as the unique subscheme of length m(i) supported on the preimage of 0 on the normalization of C_{i}. This allows to define the total multiplicity scheme m(C) of any reduced curve contained in a smooth complex surface, as the union of the total multiplicity schemes of all its germs. Given a smooth complex analytic surface Σ, a pair (C, l) consisting of a reduced curve C ↪ Σ and a subscheme l of the normalization C̃ of C is called a decorated curve if m(C) is a subscheme of l ([39, (4.1)]). The deformations of (C, l) considered by de Jong and van Straten are:

Definition 11.3.1. (i) [39, p. 476] A 1-parameter deformation of a decorated curve (C, l) over a germ of smooth curve (S, 0) consists of:

(1) a δ-constant deformation C_{S} → S of C;

(2) a flat deformation l_{S} ⊂ C̃_{S} = C̃ × S of the scheme l, such that:

(3) m_{S} ⊂ l_{S}, where the relative total multiplicity scheme m_{S} of C̃_{S} → C_{S} is defined as the closure \overline{⋃_{s∈S\0} m(C_{s})}.

(ii) A 1-parameter deformation (C_{S}, l_{S}) is called a picture deformation if for generic s ≠ 0 the divisor l_{s} is reduced.

The singularities of C_{s≠0} are only ordinary multiple points. Hence, it is easy to draw a real picture of a deformed curve, which motivates the terminology.

Theorem 11.3.2. [39, (4.4)] All the 1-parameter deformations of X(C, l) are obtained by 1-parameter deformations of the decorated germ (C, l). Moreover, picture deformations provide all the smoothings of X(C, l).

Next, let us fix a decorated germ (C, l) and one of its picture deformations (C_{S}, l_{S}). Fix a closed Milnor ball B for the germ (C, 0). For s ≠ 0 sufficiently small, C_{s} will have a representative in B, denoted by D, which meets ∂B transversally. It is a union of immersed discs {D_{i}}_{1≤i≤r} canonically oriented by their complex structures (and whose set of indices correspond canonically to those of {C_{i}}_{1≤i≤r}). The singularities of D consist of ordinary multiple points.

Denote by {P_{j}}_{1≤j≤n} the set of images in B of the points in the support of l_{s}. It is a finite set of points which contains the singular set of D (because m_{s} ⊂ l_{s} for s ≠ 0), but it might contain some other ‘free’ points as well. There is a priori no preferred choice of their ordering. (Hence, the matrix introduced next is welldefined only up to permutations of columns.)

The Milnor fiber of the smoothing associated with the fixed picture deformation has the following description. Let β : (B̃, D̃) → (B, D) be the simultaneous blowup of the points P_{j} of D. Here D̃ := ∪_{1≤i≤r}D̃_{i}, where D̃_{i} is the strict transform of the disc D_{i} by the modification β. Let T_{i} be a sufficiently small open tubular neighbourhood of D̃_{i} in B̃ (with pairwise disjoint closures).

Proposition 11.3.3. [39, (5.1)] Let (C, l) be a standard decorated germ. Then the Milnor fiber of the smoothing of X(C, l) corresponding to the picture deformation (C_{S}, l_{S}) is orientation-preserving diffeomorphic to the compact oriented manifold with boundary F′ := B̃ \ (⋃_{1≤i≤r} T_{i}) (whose corners are smoothed).

Moreover, by this presentation (for a fixed (C, l)) one can also canonically identify the boundaries of all the Milnor fibers with the link.

Finally, one reads from the above deformation a combinatorial object too, which will be crucial in the sequel:

Definition 11.3.4. [39, page 483] The incidence matrix of a picture deformation (C_{S}, l_{S}) is the matrix I(C_{S}, l_{S}) with r rows and n columns whose entry at the i-th row and the j-th column is the multiplicity of P_{j} as a point of D_{i}.

11.4. The characterization of the Milnor fibers by the incidence matrix. First, let us start with some general remarks. Obviously, the main goal would be to extend for sandwiched singularities the statements of (10.4), valid for cyclic quotients. This is obstructed seriously in both sides of the correspondence. First, at this moment there is no classification of the Stein fillings of the canonical contact structures of sandwiched singularity links. Second, there is no classification of the smoothing components either, or of the possible Milnor fibers. (Recall that in principle it might happen that several different smoothing components have the same Milnor fiber. Also, it is also still open the characterization of those ‘combinatorial’ picture deformations which can be analytically realized.)

Nevertheless, we have the following characterization result, which provides a homological/combinatorial method to ‘separate’ some Milnor fibers associated with different smoothing components. In the next paragraphs we follow [73].

Theorem 11.4.1. (a) The incidence matrix I(C_{S}, l_{S}) associated to a picture deformation of a decorated germ (C, l) is determined (up to a permutation of its columns) by the associated Milnor fiber and the marking of the link.

(b) Consider two topologically equivalent decorated germs of plane curves, and for each one of them a picture deformation. If their incidence matrices are different up to permutation of columns, then their associated Milnor fibers are not diffeomorphic by an orientation preserving diffeomorphism which preserves the markings of the boundaries.

The proof is based on a construction which glues a special universal “cap” to each Milnor fiber. This is explained in the next subsection.

11.5. Closing the boundary of the Milnor fiber. Let (C_{S}, l_{S}) be a picture deformation of the decorated germ (C, l).

As the disc-configuration D is obtained by deforming C, its boundary ∂D := ∪_{1≤i≤r}D_{i} ↪ ∂B is isotopic as an oriented link to ∂C ↪ ∂B. Therefore, we can isotope D outside a compact ball containing all the points P_{j} till its boundary coincides with the boundary of C. From now on, D will denote the result of this isotopy. Let (B′, C′) be a second copy of (B, C), and define:

(V, Σ) := (B, D) ∪_{id} (−B′, −C′).

Here V is the oriented 4-sphere obtained by gluing the boundaries of B and −B′ by the tautological identification, and Σ := ∪_{i=1}^{r}Σ_{i}, where Σ_{i} is obtained by gluing D_{i} (perturbed by the above isotopy) and −C′_{i} along their common boundaries. Moreover, one also glues (−B′, −C′) with (B̃, D̃) in such a way that the blow-up morphism β may be extended by the identity on −B′, yielding β : (Ṽ , Σ̃) ⟶ (V, Σ). Here Σ̃ := ∪_{i=1}^{r}Σ̃_{i}, where Σ̃_{i} denotes the strict transform of Σ_{i}, i.e. Σ̃_{i} = D̃_{i} ∪ −C′_{i}. Since C′_{i} is a topological disc, and D_{i} is an immersed disc, Σ̃_{i} is a topologically embedded 2-sphere in Ṽ. Write T := ⋃_{1≤i≤r} T_{i} and set

U := −B′ ∪ T.

Since F′ = B̃ \ T (cf. 11.3.3), the closed oriented 4-manifold Ṽ is obtained by closing the boundary of F′ by the cap U. The point is that U is independent of the chosen picture deformation and it is always glued in the same way (up to an isotopy) to the boundaries of all the involved Milnor fibers.

In particular, the homology of Ṽ serves as an invariant of the Milnor fiber F′. More precisely one has the next statement (which generalizes the ‘identification criterion’ (10.8.1), and implies (11.4.1) too):

Proposition 11.5.1. (a) Up to permutations of columns, there exists a unique basis (e_{j})_{1≤j≤n} of H_{2}(Ṽ) such that e_{j}^{2} = −1 for all j, and the matrix

N(C_{S}, l_{S}) := ([Σ̃_{i}] · e_{j})_{1≤i≤r,1≤j≤n}

has only non-negative entries.

(b) For any picture deformation (C_{S}, l_{S}), the incidence matrix I(C_{S}, l_{S}) is equal to N(C_{S}, l_{S}), up to a permutations of the columns.

Corollary 11.5.2. Let (M, ξ_{can}) be a link of a sandwiched singularity endowed with its canonical contact structure. Fix the topological type of a defining decorated germ. Then there are at least as many Stein fillings (up to diffeomorphisms fixed on the boundary) of (M, ξ_{can}) as there are incidence matrices (up to permutation of columns) realised by the picture deformations of all the decorated germs with the given topology.

The above corollary captures all the Milnor fibers associated with all the analytic structures supported by (the cone over) M. Notice that it might happen that some Milnor fibers of one of the analytic structure cannot be realized by another analytic structure.

12. Milnor open books

12.1. Notations. As we already mentioned in (9.3), (M, ξ_{can}) can be studied via Milnor open books. On the other hand, Milnor open books can also be studied via the combinatorics of the resolutions. In this section we assume that M is a RHS.

For any fixed resolution (graph) we consider the lattice L := (ℤ^{|V|}, (·, ·)), and in order to emphasize the graph Γ, we write Z_{an}^{Γ} and Z_{top}^{Γ} instead of Z_{an} and Z_{top}.

Recall that for rational singularities Z_{an}^{Γ} = Z_{top}^{Γ}, hence Z_{max} = Z_{min} too. But, in general, these equalities do not hold: the structure of Z_{an}^{Γ} (hence the identification of Z_{max} too) can be very difficult, it depends essentially on the analytic structure of (X, 0). (Maybe the most general result in this direction is the combinatorial description of Z_{an}^{Γ} for any splice-quotient singularity, a family which includes all the weighted homogeneous singularities as well, cf. [66].)

12.2. (Milnor) open books. One has the following facts for the Milnor open book decompositions of M, cf. 9.3.

(1) For any such f ∈ O_{X,0}, consider an embedded good resolution π of the pair (X, f^{−1}(0)). Then the strict transform S(f) := div(π^{∗}f) − (f)_{E} intersects E transversally, and the number of intersection points (S(f), E_{v}) is exactly −((f)_{E}, E_{v}). Since the intersection form is negative definite, the collection of binding components {(S(f), E_{v})}_{v} and (f)_{E} ∈ Z_{an}^{Γ} determine each other.

Moreover, by classical results of Stallings and Waldhausen, the (topological type of the) binding L_{f} ⊂ M determines completely the open book up to an isotopy, provided that M is a rational homology sphere. (For counterexamples for these statement in the general case, see e.g. [64].)

In particular, all the Milnor open book decompositions associated with a fixed analytic type (X, 0) are completely described by the semi-groups {Z_{an}^{Γ}}_{Γ} (associated with all the possible resolutions and natural identifications of elements of them). Hence, the classification of all the Milnor open books associated with a fixed analytic type (X, 0) is, in fact, as difficult as the determination of the semi-groups Z_{an}^{Γ} of Z_{top}^{Γ}.

(2) Therefore, from topological points of view, it is more natural to consider the open books of all the analytic germs associated with all the analytic structures supported by the topological type of some (X, 0).

As we already mentioned, for a fixed topological type of (X, 0), in any (negative definite) plumbing graph Γ of M one can also define the cone Z_{top}^{Γ}. The point is that for any D ∈ Z_{top}^{Γ} there is a convenient analytic structure on (X, 0) and an analytic germ f, such that the plumbing graph can be identified with a dual embedded resolution graph (for the pair (X, f^{−1}(0))), and D is the compact part (f)_{E} [76]. Hence, modifying the analytic structure of (X, 0), we fill by the collections Z_{an}^{Γ} all the semi-group Z_{top}^{Γ}.

In particular, the collection of all the Milnor open book decompositions associated with a fixed topological type M is described by the collection of semi-groups {Z_{top}^{Γ}}_{Γ} (considered in all the possible resolution graphs with a natural identifications of elements of them).

(3) For any fixed analytic type (X, 0), the open book associated with Z_{max} is the Milnor fibration of the generic hyperplane section, in particular this open book is (resolution) graph-independent. Similarly, for a fixed topological type of (X, 0), the open book associated with Z_{min} is also graphindependent, it depends only on the topology of the link.

12.3. Invariants of Milnor open books. From the above correspondence, any property defined for elements of Z_{top}^{Γ} (or Z_{an}^{Γ}) can be translated in the language of open books. This is true also in the opposite direction, invariants of open books can be studied via the lattice L. The aim of this section is to emphasise exactly this second direction applied for some key invariants of open books.

Let us fix M, a plumbing (or, a dual resolution) graph Γ. Let us consider a Milnor open book associated with an element Z ∈ Z_{top}^{Γ}, cf. 12.2(2). In the sequel we will consider the following numerical invariants of the open book and also their description in terms of L:

(1) The number of binding components bn(Z) is given by −(Z, E).

(2) Let F_{f} be the fiber of the open book. It is an oriented connected surface with −(Z, E) boundary components. Let g(Z) be its genus (the so-called page-genus of the open book) and μ(Z) be the first Betti-number of F_{f} (the so-called Milnor number). Clearly μ(Z) = 2g(Z) − 1 − (Z, E) ≥ 2g(Z).

What is less obvious is the following identity, which connects open books with the Riemann-Roch formula: for any Z ∈ Z_{top}^{Γ} one has

g(Z) = 1 + (Z, E) + χ(−Z), and
μ(Z) = 1 + (Z, E) + 2 · χ(−Z) = g(Z) + χ(−Z).

12.4. ‘Monotone increasing’ invariants. This description allows to prove the next property of these invariants.

Definition 12.4.1. Assume that for any resolution π with graph Γ one has a map I_{Γ} : Z_{top}^{Γ} → ℤ_{≥0}. We say that I = {I_{Γ}}_{Γ} is ‘monotone increasing’ if for any two cycles Z_{i} ∈ Z_{top}^{Γ} (i = 1, 2) with Z_{1} ≤ Z_{2} one has I_{Γ}(Z_{1}) ≤ I_{Γ}(Z_{2}) for any Γ.

Remark 12.4.2. Assume that the collection of invariants {I_{Γ}}_{Γ} can be transformed into (or comes from) an invariant I which associates with any Milnor open book 𝔪 of the link an integer. For any fixed analytic type, let 𝔪_{max} be the Milnor open book associated with Z_{max} (considered in any resolution). Similarly, for any topological type, let 𝔪_{min} be the Milnor open book associated with Z_{min} (in any resolution of an analytic structure conveniently chosen); cf. 12.2(3). Then, whenever {I_{Γ}}_{Γ} is monotone increasing, one has automatically the next consequences:

(1) Fix an analytic singularity (X, 0) and consider all the Milnor open books associated with all isolated holomorphic germs f ∈ O_{X,0}. Then the minimum of integers I(𝔪) of all these Milnor open books 𝔪 is realized by the generic hyperplane section, i.e. by I(𝔪_{max}).

(2) Fix a topological type of a normal surface singularity, and consider the open books associated with all the isolated holomorphic germs of all the possible analytic structures supported by the fixed topological type. Then the minimum of all integers I(𝔪) of all these Milnor open books 𝔪 is realized by the open book associated with the Artin cycle, i.e. by I(𝔪_{min}).

The above definition is motivated by the following result:

Theorem 12.4.3. [74] Both invariants Z ↦ g(Z) and Z ↦ μ(Z) are monotone increasing.

12.5. Invariants of the canonical contact structure. In [27] Etnyre and Ozbagci consider three invariants associated with fixed contact structure defined in terms of all open book decomposition supporting it:

• the support genus sg(ξ) is the minimal possible genus for a page of an open book that supports ξ;

Meeting points of singularity theory and low dimensional topology 31

• the binding number bn(ξ) is the minimal number of of binding components for an open book supporting ξ and that has pages of genus sg(ξ);

• the norm 𝔫(ξ) of ξ is the negative of the maximal (topological) Euler characteristic of a page of an open book that supports ξ.

Now, we consider the above invariants restricted on the realm of Milnor open books (i.e. for all those open books which might appear in the analytic context). In particular, ξ will be the canonical contact structure ξ_{can}. Let us denote the corresponding invariants by sg_{an}(ξ_{can}), bn_{an}(ξ_{can}) and 𝔫_{an}(ξ_{can}). Then Theorem 12.4.3 has the following consequence:

sg_{an}(ξ_{can}) = g(Z_{min});
bn_{an}(ξ_{can}) = bn(Z_{min});
𝔫_{an}(ξ_{can}) = μ(Z_{min}) − 1.

In particular,

𝔫_{an}(ξ_{can}) − bn_{an}(ξ_{can}) = 2 · sg_{an}(ξ_{can}) − 2.

These facts answer some of the questions of [27], section 8, at least in the realm of Milnor open books. Since χ(−Z) + χ(Z) + Z^{2} = 0, one also has

g(Z_{min}) = 1 + (Z_{min}, E − Z_{min}) − χ(Z_{min}).

Therefore, if (X, 0) is rational (i.e. χ(Z_{min}) = 1) then g(Z_{min}) = (Z_{min}, E − Z_{min}) (this can be strict positive, e.g. for the E_{8}-singularity it is 1); if (X, 0) is elliptic (i.e. χ(Z_{min}) = 0) then g(Z_{min}) = 1 + (Z_{min}, E − Z_{min}) ≥ 1. In general,

g(Z_{min}) ≥ 1 − χ(Z_{min}),

hence it can be arbitrarily large.

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Rényi Institute of Mathematics, Budapest, Hungary

   E-mail address: nemethi@renyi.hu