Graded roots and singularities

András Némethi

Contents

1 Introduction 2

2 Normal surface singularities. 2 2.1 The link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The combinatorics of the link. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 The topology of the link. The Heegaard Floer homology HF^{+}(−M). . . . . 5 2.4 Some analytic invariants of the singularity. . . . . . . . . . . . . . . . . . . . 6 2.5 Rational singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.6 Weakly elliptic singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.7 Almost rational singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Graded roots [34] 10 3.1 Graded roots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 The homology of a graded root. . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Graded roots associated with plumbing graphs. . . . . . . . . . . . . . . . . 12 3.4 Characterization of rational and elliptic graphs via roots; classification. … 13 3.5 Graded roots and Heegaard–Floer homology of almost rational graphs. … 15 3.6 Example. Lens spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Line bundles associated with surface singularities [35]. 17 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Line bundles on X̃. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Some cohomological computations . . . . . . . . . . . . . . . . . . . . . . . 21 4.4 Main (conjectured) properties. . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.5 Example. The case of rational singularities. . . . . . . . . . . . . . . . . . . 24

5 The graded roots of S_{−p/q}^{3}(K). 25 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.2 The manifold S_{−p/q}^{3}(K_{f}). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.3 The main invariants of S_{−p/q}^{3}(K_{f}). . . . . . . . . . . . . . . . . . . . . . . . 28 5.4 The first part of the proof of 5.3.2: k_{r}^{2} + #J . . . . . . . . . . . . . . . . . 29 5.5 The second part of the proof of 5.3.2: (R_{τ_{[k]}}, χ_{τ_{[k]}}) . . . . . . . . . . . . . . . 32 5.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.7 S_{−p/q}^{3}(K) as Kulikov graph–manifold. . . . . . . . . . . . . . . . . . . . . . 37 6 Unicuspidal rational plane curves and S_{−d}^{3}(K). 38

6.1 The semigroup distribution property. . . . . . . . . . . . . . . . . . . . . . . 38 6.2 The semigroup distribution property and surface singularities. . . . . . . . . 39 6.3 The semigroup distribution property and graded roots/Heegaard Floer ho-

mology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1 Introduction

The present article has a two-fold goal. Firstly, it aims to advertise the graded roots introduced by the author in his study of the topology of normal surface singularities. In the body of the paper we emphasize two aspects of them: their potential role in the classification of normal surface singularities, and also their connections with the Seiberg–Witten (and Heegaard–Floer) theory of rational homology sphere 3–manifolds.

In order to make the presentation more complete, we organized some of the sections in the spirit of a review article, listing some of the important constructions and results relevant to the theory. In these sections there are less proofs, but we always provide the original sources.

As a second goal of the article, we provide also a series of new results. For example, sections 4 and 5 consist of unpublished results (although they circulated in preprint form, cf. [35, 36]). These sections contain all the proofs with all the necessary details.

The main motivation of the author in the connection of singularity theory with the Seiberg-Witten invariant of 3–manifolds was born when the article [37] was written: this article formulates a conjecture connecting topological and analytical invariants (for its generalization, see section 4). For the ‘history’ and ‘realizations’ of this conjecture the reader is invited to read [33]. In the first verifications of the conjecture for different particular families, we used the realization of the Seiberg-Witten invariants based on the Turaev’s torsion. Later, the article [51] provided a different model how one can understand these invariants via Heegaard–Floer homology. It was a big surprise for the author that some parts of the computational technique described in [51] for the Heegaard–Floer homology resonated incredibly with the technique of computational sequences initiated by Laufer and S. S.-T. Yau (and used by author too) in the computation of different singularity invariants (like the geometric genus and Hilbert–Samuel function). This alloy lead to the definition of graded roots, and to the algorithms of its computations (for almost rational plumbing graphs, a family which is also a novelty of the theory).

We definitely believe that the theory of graded roots will have many deep applications. In fact, we believe that it is the guiding structure in many phenomena. This is exemplified in 3.4, but also in the last section by its appearance in a different problem, which a priori sits rather far from the above circle of ideas, the classification of rational unicuspidal projective plane curves.

For the organization of the article, see contents above.

2 Normal surface singularities.

2.1 The link.

2.1.1. Definition. The link. Let (X, 0) be a complex analytic normal surface singularity embedded in (ℂ^{N}, 0), and let B_{ϵ} be the ϵ-ball in ℂ^{N} centered at the origin. 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) [28]. Moreover, 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.

2.1.2. The link as a plumbed manifold. Not any oriented 3-manifold can be realized as the link of a singularity. In order to see this, consider the following resolution procedure. 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 will assume that the exceptional divisor E := π^{−1}(0) is a normal crossing divisor with irreducible components {E_{j}}_{j∈J}. 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_{j}, E_{j})}_{j} and genera {g_{j}}_{j∈J} (see [19]). Sometimes we write e_{j} for (E_{j}, E_{j}). Notice that H_{2}(X̃, ℤ) is freely generated by the fundamental classes {[E_{j}]}_{j}. Let I be the intersection matrix {(E_{j}, E_{i})}_{j,i}. Since π identifies ∂X̃ with M, the graph Γ(π) can be regarded a plumbing graph, and M can be considered as an S^{1}-plumbed manifold whose plumbing graph is Γ(π).

The crucial point is that Γ(π) is connected and I is negative definite [29]. The converse is also true, it was proved by Grauert [12]: 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 gives a complete classification of the possible topological types of (analytic) normal surface singularities.

2.1.3. Assumption. We note that M is a rational homology sphere (ℚHS), i.e. H_{1}(M, ℚ) = 0, if and only if Γ(π) is a tree and g_{j} = 0 for all j ∈ J. In this article we will assume that M, or equivalently the corresponding plumbing graph, satisfies this additional property.

We recall also that M is an integral homology sphere, i.e. H_{1}(M, ℤ) = 0, if and only if additionally I is unimodular, i.e. det(I) = ±1.

2.1.4. As we already said, by the plumbing construction, any resolution graph Γ(π) determines the oriented 3-manifold M completely. The converse is also true in the following sense. We say that two 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 [41], the oriented diffeomorphism type of M determines completely the equivalence class of Γ(π).

2.1.5. In the sequel Γ will denote either a good resolution graph, or a plumbing graph of M. Moreover, X̃ denotes either the space of a good resolution, or the oriented 4-manifold obtained by plumbing disc-bundles corresponding to a plumbing graph.

2.2 The combinatorics of the link.

By the ‘combinatorics of the link’ we understand the combinatorial machinery related with a fixed resolution (plumbing) graph, or with different lattices associated with it.

2.2.1. Definition. The lattices L and L′. The exact sequence of ℤ-modules

0 → L →^{i} L′ → H → 0  (1)

will stand for the homological exact sequence

0 → H_{2}(X̃, ℤ) → H_{2}(X̃, M, ℤ) ⟶^{∂} H_{1}(M, ℤ) → 0,  (2)

or, via Poincaré duality, for

0 → H_{c}^{2}(X̃, ℤ) → H^{2}(X̃, ℤ) → H^{2}(M, ℤ) → 0.  (3)

Hence, L, considered as in (2), is freely generated by the homology classes {E_{j}}_{j∈J} (we prefer to use the same notation for the exceptional curves, their homology classes, and the bases of the lattice). For each j, consider a small transversal disc D_{j} in X̃ with ∂D_{j} ⊂ ∂X̃. Then L′ is freely generated by the (relative homology) classes {D_{j}}_{j∈J}. Notice that the morphism i : L → L′ can be identified with L → Hom(L, ℤ) given by l ↦ (l, ·). The intersection form has a natural extension to L_{ℚ} = L ⊗ ℚ, and it is convenient to identify L′ with a sub-lattice of L_{ℚ}: α ∈ Hom(L, ℤ) corresponds with the unique l_{α} ∈ L_{ℚ} which satisfies α(l) = (l_{α}, l) for any l ∈ L. By this identification, D_{j}, considered in L_{ℚ} (and written in the base {E_{j}}_{j}), is the j^{th} column of I^{−1}, and (D_{j}, E_{i}) = δ_{ji}.

This shows that the exact sequence (1) can be deduced from Γ(π) – or, in view of 2.1.4, from M – i.e. L is the free ℤ-module generated by the vertices of Γ(π), the bilinear form is I, while L′ is the dual of (L, ( , )).

2.2.2. Elements x = ∑_{j} r_{j}E_{j} ∈ L_{ℚ} will be called (rational) cycles. If x_{i} = ∑_{j} r_{j,i}E_{j} for i = 1, 2, then min{x_{1}, x_{2}} := ∑_{j} min{r_{j,1}, r_{j,2}}E_{j}. x^{2} means (x, x). We define the support |x| of x by ∪E_{j}, where the union runs over {j : r_{j} ≠ 0}.

2.2.3. ‘Positive’ cones. One can consider two types of ‘positivity conditions’ for rational cycles. The first one is considered in L. A cycle x = ∑_{j} r_{j}E_{j} ∈ L_{ℚ} is called effective, denoted by x ≥ 0, if r_{j} ≥ 0 for all j. Their collection is denoted by L_{ℚ,e}, while L′_{e} := L_{ℚ,e} ∩ L′ and L_{e} := L_{ℚ,e} ∩ L. We write x ≥ y if x − y ≥ 0. x > 0 means x ≥ 0 but x ≠ 0. ≥ provides a partial order of L_{ℚ}.

The second is the numerical effectiveness of the rational cycles, i.e. positivity considered in L′. We define L_{ℚ,ne} := {x ∈ L_{ℚ} : (x, E_{j}) ≥ 0 for all j}. In fact, L_{ℚ,ne} is the positive cone in L_{ℚ} generated by {D_{j}}_{j}, i.e. it is exactly {∑_{j} r_{j}D_{j}, r_{j} ≥ 0 for all j}. Since I is negative definite, all the entries of D_{j} are strictly negative. In particular, −L_{ℚ,ne} ⊂ L_{ℚ,e}. Similarly as above, write L_{ne} := L ∩ L_{ℚ,ne}.

Later, in 2.2.7, we will discuss a series of crucial elements of these cones. The cycle which stands as a model for all of them is the classical Artin’s cycle, introduced in [3, 4]. Artin proved that, if x_{1}, x_{2} are elements of −L_{ne} \ {0}, then min{x_{1}, x_{2}} is also an element of −L_{ne} \ {0}. In particular, −L_{ne} \ {0} has a unique minimal element, Z_{min}, the Artin’s fundamental (or minimal) cycle.

2.2.4. The canonical cycle. The canonical divisor KX̃ of X̃ numerically is codified by the canonical cycle K ∈ L′, defined by the system (of adjunction relations) (K, E_{j}) = −(E_{j}, E_{j}) − 2 for all j. Since I is non-degenerate, the system has a unique solution in L′.

Although the self–intersection K^{2} depends on the choice of the resolution π, the rational number K^{2} + #J is independent of the choice of π, and it is an invariant of the link M.

Another crucial importance of the canonical cycle K consists of its role in the Riemann-Roch formula. For this, we fix an integral cycle x ∈ L_{e} \ {0}. Although the individual cohomology dimensions h^{i}(x) := h^{i}(X̃, O_{x}) (i = 1, 2), in general, are not topological, the Euler characteristic χ(x) := h^{0}(x) − h^{1}(x) depends only on Γ(π) by the Riemann-Roch theorem: χ(x) = −(x, x + K)/2.

2.2.5. Definitions. We say that (X, 0) is numerically Gorenstein if K has integral coefficients, i.e., if K ∈ L. (This is the ‘topological counterpart’ of Gorenstein property. Indeed, (X, 0) is said to be Gorenstein if the line bundle Ω_{X\{0}}^{2} is holomorphically trivial. The topological analogue of this fact — which happens if and only if K ∈ L — is when this line bundle is topologically trivial.)

2.2.6. Characteristic elements. Spin^{c}-structures. The set of characteristic elements are defined by

Char = Char(L) := {k ∈ L′ : (k, x) + (x, x) ∈ 2ℤ for any x ∈ L}.

Notice that the canonical cycle (2.2.4) is in Char and Char = K + 2L′. There is a natural action of L on Char by x ∗ k := k + 2x whose orbits are of type k + 2L. Obviously, H acts freely and transitively on the set of orbits by [l′] ∗ (k + 2L) := k + 2l′ + 2L (in particular, they have the same cardinality).

If X̃ is a resolution as above, then the first Chern class (of the associated determinant line bundle) realizes an identification between the spin^{c}-structures Spin^{c}(X̃) on X̃ and Char ⊂ L′ = H^{2}(X̃, ℤ) (see e.g. [11], 2.4.16). The restrictions to M defines an identification of the spin^{c}-structures Spin^{c}(M) of M with the set of orbits of Char modulo 2L; and this identification is compatible with the action of H on both sets. In the sequel, we think about Spin^{c}(M) by this identification, hence any spin^{c}-structure of M will be represented by an orbit [k] := k + 2L ⊂ Char. The canonical spin^{c} structure corresponds to [K].

2.2.7. Liftings. If H is not trivial, then the exact sequence (2.2.1)(1) does not split. Nevertheless, we will consider some ‘liftings’ (set theoretical sections) of the element of H into L′. They correspond to the positive cones in L_{ℚ} considered in 2.2.3.

More precisely, for any l′ +L = h ∈ H, let l′_{e}(h) ∈ L′ be the unique minimal effective rational cycle in L_{ℚ,e} whose class is h. Clearly, the set {l′_{e}(h)}_{h∈H} is exactly Q := {∑_{j} r_{j}E_{j} ∈ L′; 0 ≤ r_{j} < 1} (the intersection of L′ with the closed/open unit rational ‘L-cube’).

Similarly, for any h = l′ +L, the intersection (l′ +L)∩L_{ℚ,ne} has a unique maximal element l′_{ne}(h), and the intersection (l′ +L)∩(−L_{ℚ,ne})) has a unique minimal element l̄′_{ne}(h) (cf. [34], 5.4). By their definitions l̄′_{ne}(h) = −l′_{ne}(−h).

The elements l̄′_{ne}(h) were introduced in [34] and were denoted there by l′_{[k]}, where [k] = K +2(l′ + L). Using these elements, one defines the distinguished representative k_{r} of [k] by k_{r} := K + 2l̄′_{ne}(h) (h = l′ + L). Sometimes, in the body of the paper, we will use these notations as well.

For some h, l̄′_{ne}(h) might be situated in Q, but, in general, this is not the case (cf. 4.5.3 and 4.5.4). In general, the characterization of all the elements l̄′_{ne}(h) is not simple (see 3.6.2 when M is a lens space, or [34] for Seifert manifold).

2.2.8. The χ-functions (Riemann-Roch formula). For any characteristic element k ∈ Char one defines

χ_{k} : L′ → ℚ by χ_{k}(l′) := −(l′, l′ + k)/2.

Clearly, χ_{k}(L) ⊂ ℤ. For k = K we recover the classical Riemann-Roch function (cf. 2.2.4). For the interpretation of χ_{k} in terms of (twisted) Riemann-Roch, consider the following.

Fix a line bundle L ∈ P ic(X̃), and set c_{1}(L) = l′ ∈ L′ (cf. 4.2). Set k := K − 2l′ ∈ Char. For any l ∈ L with l > 0 one defines the sheaf O_{l} := OX̃/OX̃(−l) supported by E. Consider the sheaf L ⊗ O_{l} and let χ(L ⊗ O_{l}) = h^{0}(L ⊗ O_{l}) − h^{1}(L ⊗ O_{l}) be its (holomorphic) Euler-characteristic. The Riemann-Roch theorem states that this can be computed combinatorially, namely

χ(L ⊗ O_{l}) = −(l, l + k)/2 = χ_{k}(l).

2.3 The topology of the link. The Heegaard Floer homology HF^{+}(−M).

2.3.1. The list of invariants of connected (oriented) 3–manifolds is huge. Any of them can be considered as a topological invariant of the singularity. The invariant becomes really interesting (for algebraic geometers) if it can be related with the analytic structure of the germ.

This list is separated into two, rather distinguishable parts. Some invariants are motivated by singularity theory, are described e.g. in terms of the lattice L, or their positive cones. The other part is produced by the classical 3–manifold invariants developed by topologists. Sometimes it is difficult to find a dictionary connecting them.

For example, our (negative definite plumbed) three manifolds are completely characterized by their fundamental groups (here, in the ℚHS case, the lens spaces are exceptions, but they are also well understood). Then, one might ask, how can one understand the integer Z_{min}^{2} (associated, say, with the minimal good resolution) in terms of π_{1}(M)?

In this article we will concentrate mainly on a rather new invariant, the Seiberg-Witten invariant of M. There are many ways to introduce it (here we will mention briefly three of them). One of them it realizes as the ‘euler-characteristic’ of Heegaard–Floer homology.

2.3.2. The Ozsváth–Szabó invariant. For any oriented rational homology 3-sphere M the Heegaard Floer homology HF^{+}(M) was introduced by Ozsváth and Szabó in [50] (see also their long list of articles). In fact, HF^{+}(M) is a ℤ[U]-module with a ℚ-grading compatible with the ℤ[U]- action, where deg(U) = −2. Additionally, HF^{+}(M) also has an (absolute) ℤ_{2}-grading; HF_{even}^{+}(M), respectively HF_{odd}^{+}(M), denote the part of HF^{+}(M) with the corresponding parity.

Moreover, HF^{+}(M) has a natural direct sum decomposition of ℤ[U]-modules (compatible with all the gradings) corresponding to the spin^{c}-structures of M:

HF^{+}(M) = ⊕_{σ∈Spin^{c}(M)} HF^{+}(M, σ).

For any spin^{c}-structure σ, one has a graded ℤ[U]-module isomorphism

HF^{+}(M, σ) = T_{d(M,σ)}^{+} ⊕ HF_{red}^{+}(M, σ),

where HF_{red}^{+}(M, σ) has a finite ℤ-rank and an induced (absolute) ℤ_{2}-grading, T_{d}^{+} is an irreducible ℤ[U]–module of infinite ℤ–rank such that ker(U|T_{d}^{+}) has rank one of degree d (for a more precise definition, see 3.2.1); in fact, d(M, σ) can also be defined via this isomorphism. One also considers

χ(HF^{+}(M, σ)) := rank_{ℤ} HF_{red,even}^{+}(M, σ) − rank_{ℤ} HF_{red,odd}^{+}(M, σ).

Then one recovers the Seiberg-Witten topological invariant of (M, σ) (see [56]) via

sw^{OSz}(M, σ) := χ(HF^{+}(M, σ)) − \frac{d(M, σ)}{2}.

With respect to the change of orientation the above invariants behave as follows: The spin^{c}- structures Spin^{c}(M) and Spin^{c}(−M) are canonically identified (where −M denotes M with the opposite orientation). Moreover,

d(M, σ) = −d(−M, σ) and χ(HF^{+}(M, σ)) = −χ(HF^{+}(−M, σ)).

Notice also that one can recover HF^{+}(M, σ) from HF^{+}(−M, σ) via (7.3) [50] and (1.1) [52].

2.3.3. We will use the notation λ(M) for the Casson-Walker invariant of M (normalized as in [23] (4.7)). If M is an integral homology sphere (i.e., if H is trivial), then there is only one spin^{c} structure, the canonical one, σ_{can}, and by [49] (1.3):

λ(M) = sw^{OSz}(M, σ_{can}).

2.3.4. The ‘original’ definition of the ‘topological’ Seiberg-Witten invariant is the following.

For any fixed spin^{c}-structure σ ∈ Spin^{c}(M), one defines the modified Seiberg-Witten invariant sw_{M}^{0}(σ) as the sum of the number of Seiberg-Witten monopoles and the Kreck-Stolz invariant, see [6, 24, 27] (for this notation, more discussions and references, and relevance with singularities, see [37]). In the present article we prefer to change its sign: we will write sw^{0}(M, σ) := −sw_{M}^{0}(σ). In general it is very difficult to compute sw^{0}(M, σ) using its analytic definition, therefore there is an intense activity to replace this definition with a different one. Besides the Ozsváth–Szabó theory, this invariant can also be recovered by the sign refined Reidemeister-Turaev torsion T_{M,σ} = ∑_{h∈H} T_{M,σ}(h)h ∈ ℚ[H] (determined by the Euler structure) associated with σ [62] via

sw^{T CW} (M, σ) := −T_{M,σ}(1) + λ(M)/|H|.

Here 1 denotes the neutral element of the group H (with the multiplicative notation).

For negative definite plumbed 3–manifolds one has ([56, 47]): sw^{0}(M, σ) = sw^{T CW} (M, σ) = sw^{OSz}(M, σ). Nevertheless, different realizations might illuminate essentially different aspects of the theory. E.g., since ∑_{σ} T_{M,σ}(1) = 0, for any rational homology sphere one has

λ(M) = ∑_{σ} sw^{T CW} (M, σ).  (1)

If we do not want to specify the source of the invariant, we just write sw(M, σ).

2.4 Some analytic invariants of the singularity.

2.4.1. Definitions. The analytic type of (X, 0) is characterized completely by its local analytic ring O_{X,0} whose maximal ideal will be denoted by m_{0} ⊂ O_{X,0}. Here are some of its discrete invariants.

• 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.

• The geometric genus is defined by p_{g} := dim_{ℂ} H^{1}(X̃, OX̃), where X̃ → X is any resolution as above. It can be recovered from the sheaf-cohomology of some one-dimensional spaces as well: by the theorem of formal functions, for x ∈ L, x = ∑_{j∈J} m_{j}E_{j} with m_{j} ≫ 0, one has p_{g} = h^{1}(x).

For a generalization of p_{g}, where the structure sheaf OX̃ will be replaced by line bundles on X̃, see § 4.

2.4.2. Question: Are the analytic invariants topological? In the theory of surface singularities, one of the guiding questions is the following: is it possible to recover (some of) the analytic invariants from the link M, or equivalently, from a resolution graph Γ(π) of (X, 0)? Even more, is it possible to characterize some important families of singularities (defined a priori via analytic terms) by their topology. For a rather detailed discussion of this problem see the review article [33]. But, in some sense, the whole philosophy of the present article, and some of its main results, are also motivated by these questions.

The ‘classical’ models for such invariant and family–characterizations are the cyclic quotient, rational and (weakly) elliptic singularities. For a detailed discussion of their properties, see also [30, 33], and references therein.

2.4.3. Example. 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:

[figure: −k_{1}; −k_{2}; −k_{s}; ⋯]

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

p/q = [k_{1}, k_{2}, …, k_{s}] = k_{1} − \frac{1}{k_{2} − \frac{1}{... − \frac{1}{k_{s}}}} , k_{1}, …, k_{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.

2.4.4. Example. Rational singularities were defined by Artin by the vanishing of the geometric genus p_{g} = 0. The next subsection (2.5) is devoted to their topological characterization. Subsection (2.6) recalls the relevant facts about the elliptic singularities.

2.5 Rational singularities.

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

The main point is that Artin succeeded in replacing the vanishing of h^{1}(x)’s by a criterion formulated in terms of χ. In fact, 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.

2.5.2. Topological characterizations of rational singularities [3, 4].

p_{g} = 0 ⇔ χ(x) ≥ 1 for all cycles x > 0 ⇔ χ(Z_{min}) = 1.

Notice that these characterizations are independent of the choice of the resolution π. If a resolution graph satisfies the second (or equivalently, the third) property of the above equivalence, we say that it is a rational graph. Any rational graph is automatically good.

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

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

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

I is automatically negative definite, hence Γ is the minimal resolution graph of some singularity. One can show that Z_{min} = ∑_{j} E_{j}, χ(Z_{min}) = 1, hence any singularity with minimal resolution graph Γ is rational. Moreover, Z_{min}^{2} = −#{v : δ_{j} = 1}. (This also shows that the multiplicity and the embedded dimension of a rational singularity can be arbitrarily large, cf. 2.5.4.)

(c) The class of rational graphs is closed while taking subgraphs and decreasing self-intersections. In particular, in the above example (b), one can decrease any of the decorations e_{j}, and still obtain a rational graph. For these modified graphs one still has Z_{min} = ∑_{j} E_{j}. Rational surface singularities with reduced fundamental cycle (i.e. with Z_{min} = ∑ E_{j}) are also called minimal singularities. E.g., the cyclic quotients are minimal.

(d) Another subclass of rational singularities has the name of sandwiched singularities. A sandwiched singularity is, by definition, a normal surface singularity that admits a birational map to (ℂ^{2}, 0). If we consider a resolution X̃ → X, then we get a diagram (X̃, E) → (X, 0) → (ℂ^{2}, 0), hence X is sandwiched between two smooth spaces via birational maps.

They also can be characterized by their (minimal) resolution graphs as follows [57]. Consider a plane curve singularity (C, 0) ⊂ (ℂ^{2}, 0), and let Y → ℂ^{2} be a (in general, non-minimal) embedded resolution ϕ of it. Consider the collection E of those irreducible exceptional divisors which are not (−1)-curves (and assume that they form a connected curve). If one contracts E then one gets a sandwiched singularity, and any of them 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 result targets the analytic invariants introduced above.

2.5.4. Theorem. [3, 4] Assume that (X, 0) is rational and k ≥ 1. Then f_{HS}(k) = −k(k − 1)/2 · Z_{min}^{2} + k. In particular, mult(X, 0) = −Z_{min}^{2} and emb dim(X, 0) = −Z_{min}^{2} + 1.

2.6 Weakly elliptic singularities.

2.6.1. Definition. A normal surface singularity (X, 0) is called weakly elliptic, in short elliptic, if its graph Γ(π) is elliptic. A graph Γ(π) is elliptic if min_{x>0} χ(x) = 0, or equivalently, χ(Z_{min}) = 0. (Wagreich used the first vanishing [63], Laufer the second one [22]; see also [30] for their equivalence.) The definition is independent of the choice of the resolution graph.

2.6.2. Remark. The set of elliptic singularities includes all the singularities with p_{g} = 1, and all the Gorenstein singularities with p_{g} = 2. But an elliptic singularity might have arbitrary large p_{g}.

In general, the geometric genus and the other analytic invariants listed in 2.4.1 of an elliptic singularity are not topological (for discussion, see e.g. [30, 33]). But Laufer, in [22], identified topologically a subclass of elliptic singularities for which p_{g} is topological (in fact, p_{g} = 1). For this characterization, it is convenient to consider the minimal resolution of (X, 0) (i.e. a resolution which has no rational (−1) curve).

2.6.3. Theorem. Minimally elliptic singularities. [22] 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 −K = Z_{min}. (ii) χ(Z_{min}) = 0 and any proper subgraph of Γ(π) supports a rational singularity. (iii) p_{g} = 1 and (X, 0) is Gorenstein.

Notice that (i-ii) give topological characterizations of Gorenstein singularities with p_{g} = 1. For additional characterizations, see [22], or [30]. Here is an example when the link is ℚHS:

2.6.4. Example. ‘Polygonal’ singularities. Assume that the resolution graph Γ has the following form with s + 1 vertices (with g_{j} = 0):

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

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

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

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

2.6.5. Remark. Laufer in [22] computed the Hilbert-Samuel function of minimally elliptic singularities from their graphs. This is generalized by the author in [32] for Gorenstein singularities with b_{1}(M) = 0. In this description, the crucial topological ingredient is the elliptic sequence of an elliptic singularity, introduced and intensively studied by S.S.-T. Yau, see [65, 66]. We prefer to recall it only in the numerically Gorenstein case. We consider a minimal resolution π with E = π^{−1}(0), and we write Z_{K} for −K. In such a case Z_{K} ≥ Z_{min}.

2.6.6. Definition. The elliptic sequence consists of a sequence {Z_{B_{j}}}_{j=0}^{m}, where Z_{B_{j}} is the fundamental cycle of B_{j} ⊂ E. We define {B_{j}}_{j} inductively as follows. For j = 0 take B_{0} = E, hence Z_{B_{0}} = Z_{min}. Then Z_{K} ≥ Z_{B_{0}}. If Z_{K} > Z_{B_{0}} then we set B_{1} := |Z_{K} − Z_{B_{0}}|. Similarly, if B_{i} is already defined for any i ≤ j, then it turns out that Z_{K} ≥ Z_{B_{0}} + ⋯ + Z_{B_{j}}. If the inequality is strict then we define B_{j+1} := |Z_{K} − Z_{B_{0}} − ⋯ − Z_{B_{j}}|, otherwise we stop. In particular, Z_{K} = ∑_{j=0}^{m} Z_{B_{j}}. The length of the elliptic sequence {Z_{B_{j}}}_{j=0}^{m} is m + 1. The case m = 0 (i.e. when the identity Z_{K} = Z_{min} holds) corresponds exactly to the minimally elliptic singularities.

It is worth mentioning that by Yau’s inequality (cf. [66, (3.9)]), m + 1 is an optimal topological upper bound for p_{g}: for a numerically Gorenstein elliptic singularity p_{g} ≤ m + 1. This (and other partial results of Yau and Tomari [60]) were the starting points of the following result:

2.6.7. Theorem.[32] Assume that the link of the elliptic Gorenstein singularity (X, 0) is a rational homology sphere. Then p_{g} is a topological invariant: it equals the length of the elliptic sequence in the minimal resolution of (X, 0). Moreover, the following holds:

(a) mult(X, 0) = max(2, −Z_{min}^{2});

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

(c) If Z_{min}^{2} ≤ −3 then dim m_{0}^{k}/m_{0}^{k+1} = −kZ_{min}^{2} (k ≥ 1).

2.7 Almost rational singularities.

Recall that rational singularities can be characterized by their graphs; those graphs which satisfy the corresponding combinatorial properties are called rational graphs, cf. 2.5.2. The next definition is in the same spirit: using a combinatorial definition we enlarge the set of rational (plumbing) graphs:

2.7.1. Definition. Assume that the plumbing graph Γ is a negative definite connected tree. We say that Γ is almost-rational (in short AR) if there exists a vertex j_{0} ∈ J of Γ such that replacing e_{j_{0}} by some e′_{j_{0}} ≤ e_{j_{0}} we get a rational graph Γ′. In general, the choice of j_{0} is not unique. Once the distinguished vertex j_{0} is fixed, we write J = {0} ∪ J^{∗} such that the index 0 ∈ J corresponds to this vertex.

2.7.2. Examples. The set of AR graphs is surprisingly large.

1) Obviously, all rational graphs are AR.

2) Any elliptic graph is AR (for a proof, see [34], (8.2)).

3) Any star-shaped graph is AR. Indeed, first blow down all the (−1)-vertices different from the central vertex (this transformation preserves the AR graphs); let this new graph be Γ̄. Then take for j_{0} the central vertex of Γ̄, and take for −e′_{j_{0}} an integer larger than the number of adjacent vertices of the central vertex of Γ̄. Then the modified graph will become minimal rational, cf. 2.5.3. (In other words: all the Seifert 3–manifolds are plumbed manifolds associated with AR graphs.)

4) The class of AR graphs is closed while taking subgraphs and decreasing the Euler numbers e_{j} (since the class of rational graphs is so).

5) The rational surgery 3–manifolds S_{−p/q}^{3}(K) considered in §5 are AR (see 5.2.4).

6) Not any graph is AR. For example, if Γ has two (or more) vertices, both with decoration −e_{j} ≤ δ_{j} − 2, then Γ is not AR. E.g.:

[figure: −4; −2; −2; −4; −4; −4; −4; −4]

This graph has the following property too: if we delete one of the (−2)–vertices, then all the components of the remaining graph are rational. Still the graph itself is not AR.

2.7.3. In fact, all the main results of the present article are related with AR graphs. Their generalization for the general case looks rather difficult (and it can be a beautiful goal for interested reader).

3 Graded roots [34]

3.1 Graded roots.

3.1.1. Preliminary remarks. The Ozsváth-Szabó ℤ[U]-module HF^{+}(M, [k]) can be computed (for any plumbed 3–manifolds M associated with an AR graph, see below) in a combinatorial way from the corresponding plumbing graph Γ. Our goal is to define an intermediate object, a graded root R_{k} associated with any negative definite plumbed graph Γ and a characteristic element k. This will contain all the needed information to determine the homological object HF^{+}, but it preserves also some additional, more subtle topological information about Γ (or, about M).

In this and next subsection we give the definition and first properties of abstract graded roots. Subsection 3.3 contains the construction of the graded roots R_{k} from the plumbing graphs Γ. [Although both Γ (the plumbing graph) and the constructed graded root R_{k} are “connected trees”, they serve rather different roles. E.g., the edges of Γ codify the corresponding gluings in the plumbing, while the edges of R_{k} codify the ℤ[U]-action. We hope the terminology will not create any confusion.]

3.1.2. Definitions.

(1) Let R be an infinite tree with vertices V and edges E. We denote by [u, v] the edge with end-points u and v. We say that R is a graded root with grading χ : V → ℤ if

(a) χ(u) − χ(v) = ±1 for any [u, v] ∈ E;

(b) χ(u) > min{χ(v), χ(w)} for any [u, v], [u, w] ∈ E, v ≠ w;

(c) χ is bounded below, χ^{−1}(k) is finite for any k ∈ ℤ, and #χ^{−1}(k) = 1 if k is sufficiently large.

(2) v ∈ V is a local minimum point of the graded root (R, χ) if χ(v) < χ(w) for any edge [v, w]. Their set is denoted by V_{lm}. In fact, V_{lm} coincides with the set of vertices with adjacent degree one.

(3) A geodesic path connecting two vertices is monotone if χ restricted to the set of vertices on the path is strict monotone. If a vertex v can be connected by another vertex w by a monotone geodesic and χ(v) > χ(w), then we write v ≻ w. ≻ is an ordering of V. For any pair v, w ∈ V there is a unique ≻-minimal vertex sup(v, w) which dominates both.

(4) If (R, χ) is a graded root, and r ∈ ℤ, then we denote by (R, χ)[r] the same R with the new grading χ[r](v) := χ(v) + r. (This can be generalized for any r ∈ ℚ as well.)

3.1.3. Examples. (1) For any integer n ∈ ℤ, let R_{n} be the tree with V = {v^{k}}_{k≥n} and E = {[v^{k}, v^{k+1}]}_{k≥n}. The grading is χ(v^{k}) = k.

(2) Let I be a finite index set. For each i ∈ I fix an integer n_{i} ∈ ℤ; and for each pair i, j ∈ I fix n_{ij} = n_{ji} ∈ ℤ with the next properties: (i) n_{ii} = n_{i}; (ii) n_{ij} ≥ max{n_{i}, n_{j}}; and (iii) n_{jk} ≤ max{n_{ij}, n_{ik}} for any i, j, k ∈ I.

For any i ∈ I consider R_{i} := R_{n_{i}} with vertices {v_{i}^{k}} and edges {[v_{i}^{k}, v_{i}^{k+1}]}, (k ≥ n_{i}). In the disjoint union ∐_{i} R_{i}, for any pair (i, j), identify v_{i}^{k} and v_{j}^{k}, resp. [v_{i}^{k}, v_{i}^{k+1}] and [v_{j}^{k}, v_{j}^{k+1}], whenever k ≥ n_{ij}. Write v̄_{i}^{k} for the class of v_{i}^{k}. Then ∐_{i} R_{i}/_{∼} is a graded root with χ(v̄_{i}^{k}) = k. It will be denoted by R = R({n_{i}}, {n_{ij}}).

Clearly V_{lm}(R) is a subset of {v̄_{i}^{n_{i}}}_{i∈I}, and this last set can be identified with I. V_{lm}(R) = I if in (ii) all the inequalities are strict. Otherwise all the indices I \ V_{lm}(R) are superfluous, i.e. the corresponding R_{i}’s produce no additional vertices.

In fact, any graded root (R′, χ′) is isomorphic (in a natural sense) with some R({n_{i}}, {n_{ij}}). Indeed, set I := V_{lm}(R′), n_{v} := χ′(v) and n_{uv} := χ′(sup(u, v)) for u, v ∈ I.

(3) Any map τ : {0, 1, …, T_{0}} → ℤ produces a starting data for construction (2). Indeed, set I = {0, …, T_{0}}, n_{i} := τ(i) (i ∈ I), and n_{ij} := max{n_{k} : i ≤ k ≤ j} for i ≤ j. Then ∐_{i} R_{i}/_{∼} constructed in (2) using this data will be denoted by (R_{τ} , χ_{τ}).

For example, for T_{0} = 4, take for the values of τ: −3, −1, −2, 0 and −2 (respectively −3, 0, −2, −1 and −2). Then the two graded roots are:

[figure: ⋮; χ = 0; R^{1} :; ⋮; χ = 0; R^{2} :]

3.2 The homology of a graded root.

3.2.1. ℤ[U]-modules. We will use the following notations. Consider the graded ℤ[U]-module ℤ[U, U^{−1}], and (following [51]) denote by T_{0}^{+} its quotient by the submodule U · ℤ[U]. This has a grading in such a way that deg(U^{−d}) = 2d (d ≥ 0). Similarly, for any n ≥ 1, define the graded module T_{0}(n) by the quotient of ℤ⟨U^{−(n−1)}, U^{−(n−2)}, …, 1, U, …⟩ by U · ℤ[U] (with the same grading). Hence, T_{0}(n), as a ℤ-module, is the free ℤ-module ℤ⟨1, U^{−1}, …, U^{−(n−1)}⟩ (generated by 1, U^{−1}, …, U^{−(n−1)}), and has finite ℤ-rank n.

More generally, for any graded ℤ[U]-module P with d-homogeneous elements P_{d}, and for any r ∈ ℚ, we denote by P [r] the same module graded in such a way that P [r]_{d+r} = P_{d}. Then set T_{r}^{+} := T_{0}^{+}[r] and T_{r}(n) := T_{0}(n)[r].

3.2.2. Definition. The ℤ[U]-modules associated with graded roots. For any graded root (R, χ), let ℍ(R, χ) (briefly ℍ(R)) be the set of functions ϕ : V → T_{0}^{+} with the following property: whenever [v, w] ∈ E with χ(v) < χ(w), then

U · ϕ(v) = ϕ(w).

Or, equivalently, for any w ≻ v one requires

U^{χ(w)−χ(v)} · ϕ(v) = ϕ(w).  (∗)

Clearly ℍ(R) is a ℤ[U]-module via (Uϕ)(v) = U · ϕ(v). Moreover, ℍ(R) has a grading: ϕ ∈ ℍ(R) is homogeneous of degree d ∈ ℤ if for each v ∈ V with ϕ(v) ≠ 0, ϕ(v) ∈ T_{0}^{+} is homogeneous of degree d − 2χ(v). Notice that in (∗) one has 2χ(v) + deg ϕ(v) = 2χ(w) + deg ϕ(w), hence d is well-defined.

Notice that any ϕ as above is automatically finitely supported.

3.2.3. From the definitions, it is clear that ℍ((R, χ)[r]) = ℍ(R, χ)[2r] for any r ∈ ℤ.

3.2.4. Proposition. Let (R, χ) be a graded root. We order V_{lm} as follows. The first element v_{1} is an arbitrary vertex with χ(v_{1}) = min_{v} χ(v). If v_{1}, …, v_{k} is already determined, and J_{k} := {v_{1}, …, v_{k}} ⊊ V_{lm}, then let v_{k+1} be an arbitrary vertex in V_{lm}\J_{k} with χ(v_{k+1}) = min_{v∈V_{lm}\J_{k}} χ(v). Let w_{k+1} ∈ V be the unique ≻-minimal vertex of R which dominates both v_{k+1}, and at least one vertex from J_{k}. Then one has the following isomorphism of ℤ[U]-modules

ℍ(R, χ) = T_{2χ(v_{1})}^{+} ⊕ ⊕_{k≥2}T_{2χ(v_{k})}(χ(w_{k}) − χ(v_{k})).

In particular, with the notations m := min_{v} χ(v) and

ℍ_{red}(R, χ) := ⊕_{k≥2}T_{2χ(v_{k})}(χ(w_{k}) − χ(v_{k})),

one has a canonical direct sum decomposition of graded ℤ[U]-modules: ℍ(R, χ) = T_{2m}^{+} ⊕ℍ_{red}(R, χ).

3.2.5. Examples.(a) ℍ(R_{n}) = T_{2n}.

(b) The graded roots R^{1} and R^{2} constructed in 3.1.3(3) are not isomorphic but their ℤ[U]-modules are isomorphic: ℍ(R^{1}) = ℍ(R^{2}) = T_{−6}^{+} ⊕ T_{−4}(1) ⊕ T_{−4}(2). Hence, in general, a graded root carries more information than its ℤ[U]-module.

3.2.6. Corollary. Let (R_{τ} , χ_{τ}) be a graded root associated with some function τ : ℕ → ℤ, cf. 3.1.3(3), which satisfies τ(1) > τ(0). Then the ℤ-rank of ℍ_{red}(R_{τ} , χ_{τ}) is:

rank_{ℤ} ℍ_{red}(R_{τ}) = −τ(0) + min_{i≥0} τ(i) + ∑_{i≥0} max{τ(i) − τ(i + 1), 0}.

The summand T_{2m}^{+} of ℍ(T_{τ} , χ_{τ}) has index m = min_{i≥0} τ(i) = min_{v} χ_{τ} (v).

3.3 Graded roots associated with plumbing graphs.

Fix a connected plumbing graph Γ whose bilinear form is negative definite. In this section we will construct a graded root (R_{k}, χ_{k}) associated with any characteristic element k.

3.3.1. The construction of (R_{k}, χ_{k}). Fix any k ∈ Char and define χ_{k} : L → ℤ by χ_{k}(x) := −( k(x) + (x, x) )/2 as in (2.2.8). For any n ∈ ℤ, we define a finite 1-dimensional simplicial complex L̄_{k,≤n} as follows. Its 0-skeleton is L_{k,≤n} := {x ∈ L : χ_{k}(x) ≤ n}. For each x and j ∈ J with x, x + E_{j} ∈ L_{k,≤n}, we consider a unique 1-simplex with endpoints at x and x + E_{j} (e.g., the segment [x, x + E_{j}] in L ⊗ ℝ). We denote the set of connected components of L̄_{k,≤n} by π_{0}(L̄_{k,≤n}). For any v ∈ π_{0}(L̄_{k,≤n}), let C_{v} be the corresponding connected component of L̄_{k,≤n}.

Next, we define the graded root (R_{k}, χ_{k}) as follows. The vertex set V(R_{k}) is ∪_{n∈ℤ}π_{0}(L̄_{k,≤n}). The grading V(R_{k}) → ℤ, still denoted by χ_{k}, is χ_{k}|π_{0}(L̄_{k,≤n}) = n.

If v_{n} ∈ π_{0}(L̄_{k,≤n}), and v_{n+1} ∈ π_{0}(L̄_{k,≤n+1}), and C_{v_{n}} ⊂ C_{v_{n+1}}, then [v_{n}, v_{n+1}] is an edge of R_{k}. All the edges E(R_{k}) of R_{k} are obtained in this way.

3.3.2. Proposition. For any k ∈ Char, (R_{k}, χ_{k}) is a graded root.

3.3.3. Some of these graded roots are not very different. Indeed, assume that k and k′ determine the same spin^{c} structure, hence k′ = k + 2l for some l ∈ L. Then χ_{k′}(x − l) = χ_{k}(x) − χ_{k}(l) for any x ∈ L. This means that the transformation x ↦ x′ := x − l realizes an identification of L̄_{k,≤n} and L̄_{k′,≤n−χ_{k}(l)}. Hence, we get:

(R_{k′}, χ_{k′}) = (R_{k}, χ_{k})[−χ_{k}(l)] whenever k′ = k + 2l for some l ∈ L.

In fact, there is an easy way to choose a graded root from the multitude {(R_{k}, χ_{k})}_{k∈[k]}. For any k ∈ Char we define

m_{k} := \frac{k^{2} − max_{k′∈[k]}(k′)^{2}}{8} ≤ 0.

Since (k + 2l, k + 2l) = (k, k) − 8χ_{k}(l), m_{k} is an integer. Set M_{[k]} := {k ∈ [k] : m_{k} = 0}.

3.3.4. Lemma. Fix a spin^{c} structure [k]. Then k_{0} ∈ M_{[k]} if and only if −χ_{k_{0}}(l) ≤ 0 for any l ∈ L. Moreover, if k_{0} and k_{0} +2l ∈ M_{[k]}, then −χ_{k_{0}}(l) = 0. In particular, any choice of k_{0} ∈ M_{[k]} provides the same graded root (R_{k_{0}}, χ_{k_{0}}), which will be denoted by (R_{[k]}, χ_{[k]}). Moreover, for any k ∈ [k]

(R_{k}, χ_{k}) = (R_{[k]}, χ_{[k]})[m_{k}].

The notation is compatible with 3.2.4: m_{k} = min χ_{k}. In fact,

k^{2} − 8 min χ_{k} = max_{k′∈[k]}(k′)^{2}.

3.3.5. The graded root associated with the canonical spin^{c} structure sometimes is denoted by (R_{can}, χ_{can}). It has the following property: #χ_{can}^{−1}(n) = 1, provided that n ≥ 1.

3.3.6. Clearly, many different plumbing graphs can provide the same 3-manifold M. But all these plumbing graphs can be connected by each other by a finite sequence of blowups/downs (−1)-vertices of degree ≤ 2. One can show that {(R_{[k]}, χ_{[k]})}_{[k]} depends only on M, i.e. it is independent of the choice of the (negative definite) plumbing graph Γ which provides M.

3.4 Characterization of rational and elliptic graphs via roots; classification.

We believe that the right object which guides the classification of normal surface singularities is the (canonical) graded root associated with the connected negative definite plumbing graphs. In order to exemplify and support this statement, we start with the following results.

Recall that (R_{can}, χ_{can}) denotes the canonical graded root. We invite the reader to recall the definitions of the roots R_{m} (m ∈ ℤ) in 3.1.3 as well.

3.4.1. Theorem. Characterization of rational graphs [34]. Let Γ be a connected, negative definite plumbing graph whose plumbed three-manifold is a rational homology sphere. Then the following facts are equivalent:

(a) Γ is rational;

(a′) #χ_{can}^{−1}(0) = 1;

(b) R_{can} = R_{0};

(c) R_{can} = R_{m} for some m ∈ ℤ;

(d) For all characteristic elements k ∈ Char, R_{k} = R_{m_{k}} for some m_{k} ∈ ℤ;

(Hb) ℍ(R_{can}) = T_{0}^{+};

(Hc) ℍ(R_{can}) = T_{m}^{+} for some m ∈ ℤ; or equivalently, ℍ_{red}(R_{can}) = 0;

(Hd) For all k ∈ Char, ℍ(R_{k}) = T_{m_{k}}^{+} for some m_{k} ∈ ℤ; or equivalently, ⊕_{[k]}ℍ_{red}(R_{[k]}) = 0. Moreover, if Γ is rational and k = K + 2l′, then

m_{k} = min χ_{k} = min_{x∈L} χ(l′ + x) − χ(l′) = χ(l′_{[k]}) − χ(l′) = χ_{k}(l′_{[k]} − l′) ≤ 0.

In particular, if Γ is rational and k_{r} = K + 2l′_{[k]}, then min χ_{k_{r}} = 0.

It is instructive to compare (a′) with the property #χ_{can}^{−1}(n) = 1, valid for any Γ and n ≥ 1; cf. 3.3.5.

The next result involves only the canonical graded root (the interested reader may clarify the other cases as well):

3.4.2. Theorem. Characterization of elliptic graphs [34]. Let Γ be a connected, negative definite plumbing graph whose plumbed three-manifold is a rational homology sphere. Then the following facts are equivalent:

(a) Γ is elliptic;

(b) R_{can} = R({n_{i}}, {n_{ij}}) for some index set I, #I = l + 1 ≥ 2, and n_{i} = 0 for any i ∈ I, and n_{ij} = 1 for any i ≠ j;

(Hb) ℍ(R_{can}) = T_{0}^{+} ⊕ ( T_{0}(1) )^{⊕l} for some l ≥ 1. Moreover, l above can be identified with the length of the elliptic sequence. In particular, if the graph Γ is minimally elliptic then l = 1.

3.4.3. Remarks. (a) The results 3.4.1 and 3.4.2 can also be interpreted as follows: The grading χ_{can} of R_{can} satisfies min χ_{can} ≥ 0 (or, equivalently min χ_{can} = 0) if and only if Γ is either rational or weakly elliptic. In this situation, if ℍ_{red}(R_{can}) = 0 then Γ is rational, otherwise it is weakly elliptic. The rank of ℍ_{red}(R_{can}) is the optimal topological upper bound for p_{g}.

(b) In both rational and weakly elliptic case, for any spin^{c}-structure [k], one has min χ_{k_{r}} = 0.

(c) One can find elliptic graphs with l = 1 which are not numerically Gorenstein (i.e. K ∉ L), hence which are not minimally elliptic. E.g., the following graph Γ has these properties:

[figure: −1; −4; −2; −3; −4; Γ :; R_{can} :; ⋮; 0]

Here the two minimal points of R_{can} correspond to the zero cycle and to Artin’s fundamental cycle.

3.4.4. Classification. Continuation of the Artin–Laufer program. By the ‘Artin-Laufer program’ we understand results in the spirit of subsections 2.5 and 2.6 (i.e., results which provide, say, the Hilbert–Samuel function and the geometric genus for a topologically identified family). Although we know some obstructions to continue this type of results (see e.g. [33]), we still hope in its future.

In order to continue this program, we first have to identify subfamilies for which one can show that they share ‘common properties’. We propose to identify these subfamilies by graded roots: For each fixed possible graded root, we consider all the singularities which have their canonical graded root identical with the fixed one. We believe that this invariant is exactly the right object which (conjecturally) guides the topological classification of singularities. (As for starting point, see 3.4.1 and 3.4.2.)

Let us formulate here a very precise, new situation/family. Let us say that a singularity is of ‘general type’ if it is not rational or elliptic. This happens if and only if χ(Z_{min}) < 0. Now, the subfamily which collects the ‘simplest’ general type singularities is characterized by the fact that they share the same canonical graded root. This is the following:

[figure: 0; ⋮]

It is not difficult to see that for this class 2 ≤ p_{g} ≤ 3, and p_{g} = 3 if and only if the singularity is Gorenstein. We expect that all the results of Laufer for minimally elliptic singularities (or, those of [32]) will have their analogs in minimally general case as well.

Notice that the class of ‘minimally general’ singularities is not empty, e.g. it contains x^{5} +y^{5} +z^{3}. But in general, it is not clear at all what abstract graded roots can be realized as canonical graded roots associated with singularities (even when we consider only numerical Gorenstein singularities, when the canonical root has a ℤ_{2} symmetry).

3.4.5. Problem. Determine all the possible canonical graded roots.

Notice that the possible resolution graphs are characterized by Grauert criterion, namely that they are negative definite. For each negative definite graph (tree) we construct the graded root. The problem is to find a combinatorial characterization of all of them.

3.5 Graded roots and Heegaard–Floer homology of almost rational graphs.

3.5.1. This subsection contains two important results. The first one says that for a plumbed 3– manifolds obtained from an AR plumbing graph, the associated Heegaard–Floer homologies can be determined purely combinatorially from the plumbing graph via the homology of the corresponding graded roots. More precisely, theorems (4.8) and (8.3) of [34] read as follows (we invite the reader to recall the definition of the distinguished representatives k_{r} in 2.2.7):

3.5.2. Theorem. Assume that Γ is an AR-graph. Then, for any [k] ∈ Spin^{c}(M)

HF_{odd}^{+}(−M, [k]) = 0,

and

HF_{even}^{+}(−M, [k]) = ℍ(R_{k_{r}}, χ_{k_{r}})[ − \frac{k_{r}^{2} + #J}{4} ].

In particular,

d(−M, [k]) = − max_{k′∈[k]} \frac{(k′)^{2} + #J}{4} = −\frac{k_{r}^{2} + #J}{4} + 2 min χ_{k_{r}}.

This is generalization of the main result of [51] (where the statement was proved for ‘almost minimal rational’ graphs).

3.5.3. Corollary. If Γ is an AR-graph, and [k] ∈ Spin^{c}(M), then:

sw^{OSZ}(−M, [k]) = \frac{k_{r}^{2} + #J}{8} + rank_{ℤ} ℍ_{red}(R_{k_{r}}, χ_{k_{r}}) − min χ_{k_{r}}.

3.5.4. We wish to emphasize that in most of the applications, the root (R_{k_{r}}, χ_{k_{r}}) is not determined by its definition 3.3.1, but by an algorithm which will be described in the sequel. This algorithm, the second main result of the subsection, is motivated by singularity theory; its idea was suggested by different computational sequences (as generalizations of Laufer’s computation sequence whose output is the Artin’s minimal cycle Z_{min}), used by Laufer, Stephen Yau and others in the combinatorics of elliptic (or other) singularities.

In order to present the algorithm, we distinguish one of vertices of our AR-graph Γ, which satisfies the definition of AR-graphs. By convention, it corresponds to the index 0 ∈ J (fixed for ever), cf. 2.7.1.

For a rational cycle x = ∑_{j} r_{j}E_{j} ∈ L′ we write pr_{j}(x) for the coefficient r_{j}.

3.5.5. The definition of the sequence {x_{[k]}(i)}_{i≥0}. We fix a class [k]. Recall the definition of l′_{[k]} from 2.2.7. Then for any integer i ≥ 0 there exists a unique cycle x_{[k]}(i) ∈ L with the following properties:

(a) pr_{0}(x_{[k]}(i)) = i;

(b) (x_{[k]}(i) + l′_{[k]}, E_{j}) ≤ 0 for any j ∈ J \ {0};

(c) x_{[k]}(i) is minimal (with respect the partial ordering ≤) with the properties (a) and (b).

(Part (b) for the canonical class [k] = [K] reads as (x_{can}(i), E_{j}) ≤ 0 for any j ≠ 0; notice the similarity with the definition of Artin’s minimal cycle.)

3.5.6. Theorem. ((9.3) of [34]) Fix a spin^{c}-structure [k]. There exists an integer T_{0} (which depend on [k]) such that χ_{k_{r}}(x_{[k]}(i + 1)) ≥ χ_{k_{r}}(x_{[k]}(i)) for any i ≥ T_{0}. Moreover, the graded root associated with τ_{[k]} : {0, …, T_{0}} → ℕ given by τ_{[k]}(i) := χ_{k_{r}}(x_{[k]}(i)) satisfies

(R_{k_{r}}, χ_{k_{r}}) = (R_{τ_{[k]}}, χ_{τ_{[k]}}).

3.5.7. Remarks. (a) In general, it is not easy to find the cycles x_{[k]}(i). Fortunately, one does not need all the coefficients of these cycles, only the values τ_{[k]}(i) = χ_{k_{r}}(x_{[k]}(i)). In most of the cases they are computed inductively using the following equality (cf. (9.1)(c) [34]):

χ_{k_{r}}(x_{[k]}(i + 1)) = χ_{k_{r}}(x_{[k]}(i) + E_{0}).

Since the right hand side is χ_{k_{r}}(x_{[k]}(i)) + 1 − (E_{0}, l′_{[k]} + x_{[k]}(i)), basically one needs only the intersections (E_{0}, x_{[k]}(i)) for any i.

(b) Clearly τ_{[k]}(0) = 0. One also shows that τ_{[k]}(1) > 0, hence 3.2.6 can be applied:

rank_{ℤ} ℍ_{red}(R_{k_{r}}, χ_{k_{r}}) − min χ_{k_{r}} = ∑_{i≥0} max ( 0 , τ_{[k]}(i) − τ_{[k]}(i + 1) )
= ∑_{i≥0} max ( 0 , −1 + (E_{0}, l′_{[k]} + x_{[k]}(i)) ).

3.6 Example. Lens spaces.

3.6.1. Notations. In this subsection M is the lens space L(p, q), for its plumbing graph see 2.4.3, whose notations we will preserve. M can also be obtained as a −p/q surgery on the unknot in S^{3}.

We invite the reader to refresh the notations of 2.2.1; in particular, we recall that {D_{j}}_{j∈J} denotes the dual base in L′. For any 1 ≤ i ≤ s we write the continued fraction [k_{i}, …, k_{s}] as a rational number n_{is}/d_{is} with n_{is} > 0 and gcd(n_{is}, d_{is}) = 1. E.g., n_{1s} = p and n_{2s} = q. Consider also q′ defined by qq′ ≡ 1 modulo p and 0 < q′ < p. ({x} := x − [x] denotes the fractional part of the real number x.)

3.6.2. The group H, the spin^{c} structures, and the elements l′_{[k]}. We write [D_{j}] for the class D_{j} + L of D_{j} in L′/L = H. In the present case, H = ℤ_{p}, and [D_{s}] is one of its generators. In fact, [D_{j}] = [n_{j+1,s}D_{s}] in H (1 ≤ j ≤ s).

Similarly, the set of spin^{c}-structures on M is the set of orbits {[−aD_{s}]}_{0≤a<p} = {−aD_{s} +L}_{0≤a<p} (we prefer to use this sign, since −D_{s} is effective). More precisely, this correspondence is [k] = K + 2(−aD_{s} + L), where a runs from 0 to p − 1. In order to emphasize the role of a, we also use the notation l′_{[−aD_{s}]} for l′_{[k]}. For any 0 ≤ a < p write

l′_{[−aD_{s}]} = −(a_{1}D_{1} + a_{2}D_{2} + ⋯ + a_{s}D_{s}).

The next discussion clarifies the relation between the integer 0 ≤ a < p (which codifies Spin^{c}(L(p, q))) and the system S(a) := (a_{1}, …, a_{s}) (the coefficients of the corresponding minimal vectors l′_{[−aD_{s}]}).

Using the definition of l′_{[−aD_{s}]} and the combinatorics of the continued fractions, one shows (for details, see [34], § 10) that the entries of (a_{1}, …, a_{s}) satisfy the system of inequalities:

(SI) \{ \begin{matrix}a_{1} ≥ 0, …, a_{s} ≥ 0 \\ n_{i+1,s}a_{i} + n_{i+2,s}a_{i+1} + ⋯ + n_{ss}a_{s−1} + a_{s} < n_{is} for any 1 ≤ i ≤ s. \end{matrix}

By this system one can identify the integers 0 ≤ a < p with the possible combinations (a_{1}, …, a_{s}) satisfying (SI). Indeed, the integer a can be recovered from (a_{1}, …, a_{s}) by

a = n_{2s}a_{1} + n_{3s}a_{2} + ⋯ + n_{ss}a_{s−1} + a_{s}.  (1)

And, any 0 ≤ a < p determines inductively the entries a_{1}, ⋯, a_{s} by the formula

a_{i} = [\frac{a − ∑_{t=1}^{i−1} n_{t+1,s}a_{t}}{n_{i+1,s}} ] (1 ≤ i ≤ s).

3.6.3. As a curiosity, we mention that the above system (SI) can also be interpreted in language of ‘generalized’ continued fractions. For any 1 ≤ i ≤ s write r_{i} := n_{is}/n_{i+1,s}. Then

\frac{a}{p} = \frac{a_{1} + \frac{a_{2} + \frac{⋯ + \frac{a_{s−1} + \frac{a_{s}}{r_{s}}}{⋯}}{r_{3}}}{r_{2}}}{r_{1}}.

The inequalities (SI) imply that all the possible fractions in the above expression are < 1; and this property guarantees the uniqness of the entries (a_{1}, …, a_{s}) in this continued fraction (for any given 0 ≤ a < p).

3.6.4. In concrete examples, the fastest way to determine S(a) = (a_{1}, …, a_{s}), 0 ≤ a < p, in the order S(p − 1), …, S(0), is the following. Start with S(p − 1) = (k_{1} − 1, k_{2} − 2, …, k_{s} − 2). Assume that S(a) = (a_{1}, …, a_{s}) is already determined. Then, if a_{s} > 0, then S(a−1) = (a_{1}, …, a_{s−1}, a_{s}−1). If a_{i} = ⋯ = a_{s} = 0, but a_{i−1} ≠ 0, then

S(a − 1) = (a_{1}, …, a_{i−2}, a_{i−1} − 1, k_{i} − 1, k_{i+1} − 2, …, k_{s} − 2).

3.6.5. The Ozsváth-Szabó Heegaard–Floer homology HF^{+}(±M). Since Γ is rational, by 3.4.1 and 3.5.2 (see also 2.3.2), HF_{red}^{+}(±M) = 0, and HF^{+}(±M, [k]) = T_{±d}^{+}, and sw^{OSZ}(M, [k]) = −d/2, where

d := d(M, [k]) = \frac{k_{r}^{2} + s}{4} = \frac{K^{2} + s}{4} − 2χ(l′_{[k]}).

Notice that (see e.g. (7.1) of [37]):

\frac{K^{2} + s}{4} = \frac{p − 1}{2p} − 3 · s(q, p),

where s(q, p) denotes the Dedekind sum

s(q, p) = ∑_{l=0}^{p−1} ((\frac{l}{p}))((\frac{ql}{p} )), where ((x)) = \{ \begin{matrix}\{x\} − 1/2 & if & x ∈ ℝ \ ℤ \\ 0 & if & x ∈ ℤ. \end{matrix}

(In fact, this formula for K^{2} + s for cyclic quotients goes back to the work of Hirzebruch.)

Finally, we determine χ(l′_{[−aD_{s}]}).

3.6.6. Proposition. For any 0 ≤ a < p one has:

χ(l′_{[−aD_{s}]}) = \frac{a(1 − p)}{2p} + ∑_{j=1}^{a} \{\frac{jq′}{p} \}.

3.6.7. Clearly, with the choice D_{1} as a generator, one has

χ(l′_{[−aD_{1}]}) = \frac{a(1 − p)}{2p} + ∑_{j=1}^{a} \{\frac{jq}{p} \}.

3.6.8. Remark. The Casson-Walker invariant is λ(L(p, q)) = p · s(q, p)/2 (see e.g. [37](5.3)). It satisfies

−λ(M) = ∑_{[k]∈Spin^{c}(M)} \frac{k_{r}^{2} + s}{8},

or, equivalently:

∑_{[k]} χ(l′_{[k]}) = \frac{p − 1}{4} − p · s(q, p).

4 Line bundles associated with surface singularities [35].

4.1 Introduction

In [37] L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (X,0) (whose link M is a rational homology sphere) with the Seiberg-Witten invariant of M associated with the ‘canonical’ spin^{c} structure of M. This is a generalization of a conjecture of Neumann and Wahl [43] formulated for complete intersection singularities with integral homology sphere links. The interested reader can find in the articles [37, 38, 39, 34, 33] the verification of the conjecture in different cases; in [26] some counterexamples, showing that the original assumptions of [37] (namely, that (X, 0) is ℚ–Gorenstein with ℚHS link) are too weak (hence one needs to consider some additional restrictions in order to give a chance to the conjecture). For related constructions and results, see also the articles of Neumann and Wahl [44, 45, 46].

Since the Seiberg-Witten theory of the link M provides a rational number for any spin^{c} structure (which are classified by H_{1}(M, ℤ)), it was a natural challenge to search for a complete set of conjecturally valid identities, which involve all the Seiberg-Witten invariants (giving an analytic – i.e. singularity theoretical – interpretation of them).

The formulation of this set of identities/properties — conjecturally valid for some ‘nice’ families of normal surface singularities — is one of the goals of the present section. In fact (similarly as in [37]), we formulate conjecturally valid inequalities which hopefully become equalities in special rigid situations. In this way, the Seiberg-Witten invariants provide optimal topological upper bounds for the dimensions of the first sheaf-cohomology of some line bundles living on a resolution of (X, 0).

The first part of the section is devoted to the construction and study of these ‘natural’ holomorphic line bundles on the resolution X̃. This construction automatically provides a natural splitting of the exact sequence

0 → P ic^{0}(X̃) → P ic(X̃) →^{c_{1}} H^{2}(X̃, ℤ) → 0.

The line-bundle construction is compatible with abelian covers. This allows us to reformulate the ‘conjecture’ in its second version, which relates the echivariant geometric genus of the universal (unbranched) abelian cover of (X, 0) with the Seiberg-Witten invariants of the link.

In the last subsection we verify the validity of the conjectured properties for rational singularities. The presentation is based on the author’s unpublished preprint [35]. Some h^{1}–computations for

the case of rational singularities were also found independently by T. Okuma [48].

4.2 Line bundles on X̃.

4.2.1. Let π : (X̃, E) → (X, 0) be a fixed good resolution of (X, 0). Similarly as above, we assume that the link M is a ℚHS, i.e. H^{1}(X̃, ℤ) = 0. Therefore, one has the exact sequence

0 → P ic^{0}(X̃) → P ic(X̃) →^{c_{1}} L′ → 0,  (1)

where P ic^{0}(X̃) = H^{1}(X̃, OX̃), P ic(X̃) = H^{1}(X̃, O∗ ; X̃) (= isomorphism classes of holomorphic line bundles on X̃), and c_{1}(L) = ∑_{j} deg(L|E_{j}) D_{j} is the set of Chern numbers (multidegree) of L.

In the sequel, we will use the same notation for l = ∑ n_{j}E_{j} ∈ L and the algebraic cycle ∑ n_{j}E_{j} of X̃ supported by E. E.g., for any l = ∑ n_{j}E_{j} ∈ L one can take the line bundle OX̃(l) := OX̃(∑ n_{j}E_{j}). Notice that c_{1}(OX̃(l)) = l, hence c_{1} admits a group-section s_{L} : L → P ic(X̃) above the subgroup L of L′. The main goal of this subsection is to construct a (natural) group section s : L′ → P ic(X̃) of (1) which extends s_{L} (and is compatible with abelian coverings).

Clearly, if P ic^{0}(X̃) = 0 (i.e. if (X, 0) is rational, see [3], or §2.5 here) then there is nothing to construct: c_{1} is an isomorphism, and s := c_{1}^{−1} identifies the lines bundles with their multidegree. Nevertheless, for non-rational singularities, even the existence of any kind of splitting of (1) is not so obvious.

4.2.2. The first construction. Notice that X̃ \ E ≈ X \ {0} has the homotopy type of M, hence the abelianization map π_{1}(X̃ \ E) = π_{1}(M) → H defines a regular Galois covering of X̃ \ E. This has a unique extension c : Z → X̃ with Z normal and c finite [13]. The (reduced) branch locus of c is included in E, and the Galois action of H extends to Z as well. Since E is a normal crossing divisor, the only singularities what Z might have are cyclic quotient singularities (situated above Sing(E)). Let r : Z̃ → Z be a resolution of these singular points of Z, such that (c ∘ r)^{−1}(E) is a normal crossing divisor.

Let (X_{a}, 0) denote the the universal (unbranched) abelian cover of (X, 0), i.e. the unique normal singular germ corresponding to the regular covering of X \ {0} associated with π_{1}(X \ {0}) → H. Notice that Z̃ is a good resolution of (X_{a}, 0). Hence, one can consider the following commutative diagram:

0 → L → L′ → H → 0
↓ ↓^{p′} ↓^{p_{H}}
0 → L_{a} → L′_{a} → H_{a} → 0

Here, the first, resp. second, horizontal line is the exact sequence 2.2.1(3) applied for the resolution X̃ → X, resp. for Z̃ → X_{a}. The vertical arrows (pull-back of cohomology classes) are induced by p = c ∘ r.

4.2.3. Lemma. p_{H} = 0. In particular, p′(L′) ⊂ L_{a} (i.e. any element p′(l′), l′ ∈ L′, can be represented by a divisor supported by the exceptional divisor in Z̃).

Proof. Denote by M_{a} the link of (X_{a}, 0). The morphism p_{H} : H^{2}(M, ℤ) → H^{2}(M_{a}, ℤ) is dual to p_{∗} : H_{1}(M_{a}, ℤ) → H_{1}(M, ℤ), which is zero since H_{1}(M_{a}, ℤ) is the abelianization of the commutator subgroup of π_{1}(M).

This shows that for any l′ ∈ L′, one can take OZ̃(p′(l′)) ∈ P ic(Z̃).

4.2.4. Theorem. The line bundle OZ̃(p′(l′)) is a pull-back of a unique element of P ic(X̃).

Proof. We break the proof into several steps. Let f : S → T be one of the maps r : Z̃ → Z, c : Z → X̃ or p : Z̃ → X̃. In each case one has a commutative diagram of type

H^{1}(T, ℤ) → P ic^{0}(T) → P ic(T) → H^{2}(T, ℤ) → 0
↓^{f′′} ↓^{f^{0,∗}} ↓^{f^{∗}} ↓^{f′}
H^{1}(S, ℤ) → P ic^{0}(S) → P ic(S) → H^{2}(S, ℤ) → 0

(I). Assume that f = c. Then:

• c′ is injective since c′ ⊗ ℚ : H^{2}(X̃, ℚ) ≈ H^{2}(Z, ℚ)^{H} ↪ H^{2}(Z, ℚ) is so and H^{2}(X̃, ℤ) is free.

• c^{0,∗} is injective with image P ic^{0}(Z)^{H}. Indeed (see also the proof of 4.2.9), c_{∗}O_{Z} has a direct sum decomposition ⊕_{χ}L_{χ} into line bundles L_{χ}, where the sum runs over all the characters of H, and for the trivial character L_{1} = OX̃. Therefore, since c is finite, H^{1}(Z, O_{Z}) = H^{1}(X̃, c_{∗}O_{Z}) = ⊕_{χ}H^{1}(X̃, L_{χ}), whose H-invariant part is H^{1}(X̃, OX̃).

• Im(H^{1}(Z, ℤ) → P ic^{0}(Z)) ∩ Imp^{0,∗} = 0. Indeed, since any element of Imp^{0,∗} is H-invariant, the above intersection is in Im(H^{1}(Z, ℤ)^{H} → P ic^{0}(Z)^{H}). But H^{1}(Z, ℤ)^{H} embeds into H^{1}(Z, ℚ)^{H} = H^{1}(X̃, ℚ) = 0, hence it is trivial.

• c^{∗} is injective, a fact which follows from the previous three statements. (II). Assume that f = r. Then r′′ is an isomorphism (since such a resolution does not modify H_{1}(·, ℤ) of the exceptional divisors), r^{0,∗} is an isomorphism (since a quotient singularity has geometric genus zero), and r′ is injective. Hence (by a diagram check) r^{∗} is also injective. (III). Assume that f = p = c ∘ r. Then:

• For any l′ ∈ L′, OZ̃(p′(l′)) is in the image of p^{∗}. Indeed, take a line bundle L ∈ P ic(X̃) with c_{1}L = l′. Then L′ := OZ̃(p′(l′)) ⊗ p^{∗}L^{−1} has trivial multidegree, and it is in P ic^{0}(Z̃)^{H}. But using (I-II) p^{0,∗} is onto on P ic^{0}(Z̃)^{H}, hence L′ is a pull-back. Hence OZ̃(p′(l′)) itself is a pull-back.

• Again, from (I-II), p^{∗} is injective; a fact which ends the proof of 4.2.4.

4.2.5. Notation. Write OX̃(l′) for the unique line bundle L ∈ P ic(X̃) with p^{∗}(L) = OZ̃(p′(l′)).

The proof of 4.2.4 also implies the following fact.

4.2.6. Corollary. s : L′ → P ic(X̃) defined by l′ ↦ OX̃(l′) is a group section of the exact sequence 4.2.1(1) which extends s_{L} (cf. 4.2.1).

4.2.7. Example. Consider an A_{2}-singularity. Then the exceptional divisor E of the minimal resolution X̃ contains two components E_{1} and E_{2} which intersect each other transversely, both with self-intersection −2. H = ℤ_{3}, and Z also contains two divisors, but they intersect each other in a singular point whose resolution consists of a −3 curve. Hence Z̃ contains a string of three rational curves F_{1}, F_{12} and F_{2} with self-intersections −1, −3 and −1. Then p′(E_{1}) = 3F_{1} + F_{12} and p′(E_{2}) = F_{12} + 3F_{2}. Since −D_{1} = (2E_{1} + E_{2})/3, one gets that p′(−D_{1}) = 2F_{1} + F_{12} + F_{2}, hence it is an integral cycle. In particular, c_{1}^{−1}(−D_{1}) is that line bundle whose p^{∗}-image is OZ̃(2F_{1} +F_{12} +F_{2}).

4.2.8. The second construction. The following result will illuminate a different aspect of the line bundles OX̃(l′). In fact, the next proposition could also serve as the starting point of a different construction of these line bundles.

Below we will write Ĥ for the Pontrjagin dual Hom(H, S^{1}) of H. Recall that the natural map

θ : H → Ĥ, induced by [l′] ↦ e^{2πi(l′,·)}

is an isomorphism.

4.2.9. Theorem. Consider the finite covering c : Z → X̃, and set Q := {∑ r_{j}E_{j} ∈ L′ : 0 ≤ r_{j} < 1} ⊂ L′ (cf. 2.2.7) as above. Then the H-eigenspace decomposition of c_{∗}O_{Z} has the form:

c_{∗}O_{Z} = ⊕_{χ∈Ĥ}L_{χ},

where L_{θ(h)} = OX̃(−l′_{e}(h)) for any h ∈ H. In particular, c_{∗}O_{Z} = ⊕_{l′∈Q}OX̃(−l′).

Proof. The proof is based on a similar statement of Kollár valid for cyclic coverings, see e.g. [18], §9.

First notice that c_{∗}L_{Z} is free. Indeed, since all the singularities of Z are cyclic quotient singularities, this fact follows from the corresponding statement for cyclic Galois coverings, which was verified in [18]. Moreover, above X̃ \ E the covering is regular (unbranched) corresponding to the regular representation of H. Therefore, rank c_{∗}L_{Z} = |H|, and it has an eigenspace decomposition ⊕_{χ}L_{χ}, where all the characters χ ∈ Ĥ appear, and L_{χ}|X̃ \E is a line bundle. By a similar reduction as above to the cyclic case, one gets that L_{χ} itself is a line bundle. Moreover, L_{1} (corresponding to the trivial character 1) equals OX̃.

Next we identify L_{χ} for any character. Fix χ_{0} ∈ Ĥ. It generates the subgroup ⟨χ_{0}⟩ in Ĥ. Write K̂ := Ĥ/⟨χ_{0}⟩. The morphism π_{1}(M) → H → ⟨χ_{0}⟩ˆ defines a well-defined normal space Y as a cyclic Galois ⟨χ_{0}⟩ˆ -covering of X̃. Clearly, one has the natural maps Z ⟶^{f} Y ⟶^{e} X̃ with e∘f = c. By similar argument as above f_{∗}O_{Z} = ⊕_{ξ∈K̂} L̄_{ξ} for some L̄_{ξ} ∈ P ic(Y) and L̄_{1} = O_{Y}. Since e_{∗}L̄_{ξ} = ⊕_{[χ]=ξ}L_{χ}, and [χ_{0}] = 0 in K̂, one gets that L_{χ_{0}} is one of the summands of e_{∗}L̄_{1} = e_{∗}O_{Y}. In particular, L_{χ_{0}} can be recovered from the cyclic covering e as the χ_{0}-eigenspace of e_{∗}O_{Y}.

Assume that the order of χ_{0} is n, i.e. ⟨χ_{0}⟩ = ℤ_{n}, and we regard χ_{0} as the distinguished generator of ℤ_{n}. Write g_{j} := [∂D_{j}] ∈ H. Then for any j ∈ J, χ_{0}(g_{j}) has the form e^{2πim_{j}/n} for some (unique) 0 ≤ m_{j} < n. Using these integers, define the divisor B := ∑_{j} m_{j}E_{j}. Since for any fixed i ∈ J one has

1 = χ_{0}([E_{i}]) = χ_{0}(∑_{j}I_{ij}g_{j}) = e^{2πi(∑_{j} m_{j}E_{j},E_{i})/n},

one gets that B/n ∈ L′. Let h = [B/n] be its class in H. Clearly, θ(h) = χ_{0} since

θ(h)(g_{j}) = e^{2πi(∑ m_{j}E_{j}/n,D_{j})} = e^{2πim_{j}/n} = χ_{0}(g_{j}).

On the other hand, by [18], 9.8 (and from the fact that all the coefficients of B/n are in the interval [0, 1)), the χ_{0}-eigenspace of e_{∗}O_{Y} is some line bundle L^{−1} with the properties (i) L^{⊗n} = OX̃(B) and (ii) e^{∗}L = O_{Y} (e′(B/n)). From (i) follows that c_{1}(L) = B/n, hence (ii), via our definition 4.2.5, reads as L = OX̃(B/n). Therefore, L^{−1} = OX̃(−B/n). Since B/n is in the unit cube, B/n = l′_{e}(h).

4.3 Some cohomological computations

4.3.1. Let (X, 0) and π : X̃ → X be as above. In this section we analyse h^{1}(X̃, L) for any L ∈ P ic(X̃).

First, recall the following general (Kodaira, or Grauert-Riemenschneider type) vanishing theorem (cf. [55], page 119, Ex. 15):

If c_{1}(L) ∈ K + L_{ℚ,ne}, then h^{1}(l, L|_{l}) = 0 for any l ∈ L, l > 0, hence h^{1}(X̃, L) = 0. The next statement is an improvement of it, valid for rational singularities:

4.3.2. Assume that (X,0) is a rational singularity. If c_{1}(L) ∈ L_{ℚ,ne}, then h^{1}(l, L|_{l}) = 0 for any l > 0, l ∈ L, hence h^{1}(X̃, L) = 0 too.

Proof. For any l > 0 there exists E_{j} ⊂ |l| such that (E_{j}, l + K) < 0. Indeed, (E_{j}, l + K) ≥ 0 for any j would imply χ(l) = −(l, l + K)/2 ≤ 0, which would contradict the rationality of (X,0) [3]. Then, from the cohomology exact sequence of

0 → L ⊗ O_{E_{j}}(−l + E_{j}) → L|_{l} → L|_{l−E_{j}} → 0

one gets h^{1}(L|_{l}) = h^{1}(L|_{l−E_{j}}), hence by induction h^{1}(L|_{l}) = 0.

We will generalize this proof as follows.

4.3.3. Proposition. Let X̃ → X be a good resolution of a normal singularity (X,0) as above. (a) For any l′ ∈ L′ there exists a unique minimal element l_{l′} ∈ L_{e} with e(l′) := l′ − l_{l′} ∈ L_{ℚ,ne}.

(b) l_{l′} can be found by the following (generalized Laufer’s) algorithm. One constructs a “computation sequence” x_{0}, x_{1}, …, x_{t} ∈ L_{e} with x_{0} = 0 and x_{i+1} = x_{i} + E_{j(i)}, where the index j(i) is determined by the following principle. Assume that x_{i} is already constructed. Then, if l′−x_{i} ∈ L_{ℚ,ne}, then one stops, and t = i. Otherwise, there exists at least one j with (l′ − x_{i}, E_{j}) < 0. Take for j(i) one of these j’s. Then this algorithm stops after a finitely many steps, and x_{t} = l_{l′}.

(c) For any L ∈ P ic(X̃) with c_{1}(L) = l′ one has:

h^{1}(L) = h^{1}(L ⊗ OX̃(−l_{l′})) − (l′, l_{l′}) − χ(l_{l′}).

In particular (since c_{1}(L⊗OX̃(−l_{l′})) ∈ L_{ℚ,ne}), the computation of any h^{1}(L) can be reduced (modulo the combinatorics of (L, (·, ·))) to the computation of some h^{1}(L′) with c_{1}(L′) ∈ L_{ℚ,ne}.

Proof. (a) First notice that since B is negative definite, there exists at least one effective cycle l with l′ − l ∈ L_{ℚ,ne} (take e.g. a large multiple of some x with (x, E_{j}) < 0 for any j). Next, we prove that if l′ − l_{i} ∈ L_{ℚ,ne} for l_{i} ∈ L_{e}, i = 1, 2, and l := min{l_{1}, l_{2}}, then l′ − l ∈ L_{ℚ,ne} as well. For this, write x_{i} := l_{i} − l ∈ L_{e}. Then |x_{1}| ∩ |x_{2}| = ∅, hence for any fixed j, E_{j} ⊄ |x_{i}| for at least one of the i’s. Therefore, (l′ − l, E_{j}) = (l′ − l_{i}, E_{j}) + (x_{i}, E_{j}) ≥ 0.

(b) First we prove that x_{i} ≤ l_{l′} for any i. For i = 0 this is clear. Assume that it is true for some i but not for i + 1, i.e. E_{j(i)} ⊄ |l_{l′} − x_{i}|. But this would imply (l′ − x_{i}, E_{j(i)}) = (l′ − l_{l′}, E_{j(i)}) + (l_{l′} − x_{i}, E_{j(i)}) ≥ 0, a contradiction. The fact that x_{i} ≤ l_{l′} for any i implies that the algorithm must stop, and x_{t} ≤ l_{l′}. But then by the minimality of l_{l′} (part a) x_{t} = l_{l′}. (cf. [20].)

(c) For any 0 ≤ i < t, consider the exact sequence

0 → L ⊗ OX̃(−x_{i+1}) → L ⊗ OX̃(−x_{i}) → L ⊗ O_{E_{j(i)}}(−x_{i}) → 0.

Since deg(L ⊗ O_{E_{j(i)}}(−x_{i})) = (l′ − x_{i}, E_{j(i)}) < 0, one gets h^{0}(L ⊗ O_{E_{j(i)}}(−x_{i})) = 0. Therefore

h^{1}(L ⊗ OX̃(−x_{i})) − h^{1}(L ⊗ OX̃(−x_{i+1})) = −χ(L ⊗ O_{E_{j(i)}}(−x_{i}))

which equals −(l′, x_{i+1} − x_{i}) + χ(x_{i}) − χ(x_{i+1}). Hence the result follows by induction.

4.3.4. Examples. If L = OX̃(l′) for some l′ ∈ L′ (cf. 4.2.5) then 4.3.3(c) reads as

h^{1}(OX̃(l′))) = h^{1}(OX̃(e(l′))) − (l′, l_{l′}) − χ(l_{l′}).

Additionally, if (X, 0) is rational then h^{1}(OX̃(e(l′))) = 0 by 4.3.2, hence h^{1}(OX̃(l′)) = −(l′, l_{l′}) − χ(l_{l′}). In particular, for (X,0) rational, h^{1}(L) depends only on topological data and it is independent of the analytic structure of (X,0) .

4.3.5. Definition. We will distinguish the following set of rational cycles:

𝕃′ := {l′ ∈ L′ : e(l′) = l′_{ne}(h) for some h ∈ H} = ⋃_{h∈H} l′_{ne}(h) + L_{e}.

It is easy to verify the following inclusions: L′_{e} ⊂ ⋃_{l′∈Q} −l′ + L_{e} ⊂ 𝕃′.

4.4 Main (conjectured) properties.

4.4.1. Preliminary words. The next ‘conjecture’/expected property is a generalization of the conjecture of [37], where only the case of canonical spin^{c} structure was considered. The conjectured property provides an optimal upper bound for h^{1}(L) (L ∈ P ic(X̃)) in terms of the topology of (X, 0) and c_{1}(L). The topological ingredient is provided by the Seiberg-Witten theory of the link.

Now, the author knows that the conjecture of [37] is not true in the high generality in which it was formulated, see [26] and [33] for details. Nevertheless, we expect that it is true for a large class of normal surface singularities (subclass of ℚ-Gorenstein singularities with ℚHS links). In this article we will not enter in the guess of the (largest) possible class for which the property is valid; we just formulate these conjectures as expected properties. But the reader might consider them as conjectures valid for, say (jokingly), NY I (‘not-yet-identified’) singularities.

We will present two versions (‘strong’ and ‘weak’), both of them having two parts. The first ones, (SIn) and (WIn), are inequalities: we expect their validity for a class of singularities (mainly) topologically identified. The second parts, (SId) and (WId), are identities: we expect their validity under some additional analytic assumption. (In the last subsection we will verify all of them for rational singularities, — without any additional assumption.)

4.4.2. Property (strong version). Let (X, 0) be a normal surface singularity whose link M is a rational homology sphere. Let π : X̃ → X be a fixed good resolution and s := #J the number of irreducible exceptional divisors of π. Consider an arbitrary l′ ∈ 𝕃′ and define the characteristic element k := K − 2l′ ∈ Char. Then, we say that (X, 0) satisfies (SIn) or (SId) if:

(SIn) for any line bundle L ∈ P ic(X̃) with c_{1}(L) = l′ one has

h^{1}(L) ≤ −sw(M, [k]) − \frac{k^{2} + s}{8}.  (1)

(SId) for L = OX̃(l′) in (1) one has equality.

(For a generalization of (SIn) see 4.4.3(d), where the restriction l′ ∈ 𝕃′ is dropped.)

4.4.3. Remarks.

(a) If L = OX̃ then h^{1}(OX̃) is the geometric genus p_{g} of (X, 0). For detailed historical remarks and list of cases for which (SIn) and (SId) are valid, see [37, 38, 39, 34, 33, 26].

(b) Notice that sw(M, [k]) depends only on the class [k] of k. In particular, the right hand side of (1) consists of the ‘periodical’ term −sw(M, [k]) and the ‘quadratic’ term −(k^{2} + s)/8.

(c) In order to prove the property, it is enough to verify it for line bundles L with c_{1}(L) = l′ of type l′ = l′_{ne}(h) (for some h ∈ H).

Indeed, write l′ in the form l′ = l′_{1} +l where l′_{1} = e(l′) = l′_{ne}(l′ +L) and l ∈ L_{e}. Let RHS(l′), resp. RHS(l′_{1}), be the right hand side of (1) for l′, resp. l′_{1}. Since [K −2l′] = [K −2l′_{1}], the Seiberg-Witten invariants are the same, hence

RHS(l′) − RHS(l′_{1}) = \frac{−(K − 2l′)^{2} + (K − 2l′_{1})^{2}}{8} = −(l, l′) − χ(l).

This combined with 4.3.3(c) shows that the statements of 4.4.2 for L and L⊗OX̃(−l) are equivalent.

(c’) Consider any set of representatives {l′}_{l′∈R} (R ⊂ 𝕃′) of the classes H, i.e. {l′ + L}_{l′∈R} = H. Then the comparison (c) can be also be done for any L with l′ := c_{1}(L) ∈ R and for L ⊗ OX̃(−l). Therefore, the validity of the property 4.4.2 follows from the verification of (1) for line bundles L with c_{1}(L) ∈ R.

E.g., one can take R = Q, or R = −Q. The importance of R = −Q is emphasized by 4.2.9. This fact is exploited in the second version of the Property.

(d) Similarly, if one verifies the inequality (1) for any l′ ∈ 𝕃′, then one gets automatically the inequality (1) for any l′ ∈ L′, hence for any L ∈ P ic(X̃). This statement follows by induction: if the inequality (1) is valid for some L, then it is valid for L ⊗ OX̃(−E_{j}) (for any j ∈ J).

Indeed, using the exact sequence 0 → L ⊗ OX̃(−E_{j}) → L → L|_{E_{j}} → 0, one gets that h^{1}(L ⊗ OX̃(−E_{j})) ≤ h^{1}(L) + 1 + (c_{1}(L), E_{j}). The proof ends with similar comparison as in (c).

The point is that if l′ ∉ 𝕃′, then the inequality (1), in general, is not sharp (optimal).

Theorem 9.6 of [34] implies the following fact, emphasizing once again the importance of almost rational singularities (cf. §2.7):

4.4.4. Theorem. Part (SIn) of the strong version of the Property 4.4.2 (hence 4.4.3(d) too) is true for any almost rational singularity.

4.4.5. Discussion. (a) Let (X_{a}, 0) be the universal abelian cover of (X, 0) with its natural Haction. Obviously, if p : Z̃ → X_{a} is a resolution of (X_{a}, 0), then Z̃ inherits a natural H-action. Recall that the geometric genus p_{g}(X_{a}, 0) of (X_{a}, 0) is h^{1}(Z̃, OZ̃). But one can define much finer invariants: consider the eigenspace decomposition ⊕_{χ∈Ĥ}H^{1}(Z̃, OZ̃)_{χ} of H^{1}(Z̃, OZ̃), and take

p_{g}(X_{a}, 0)_{χ} := dim_{ℂ} H^{1}(Z̃, OZ̃)_{χ} (for any χ ∈ Ĥ).

(b) Obviously, we can repeat the above definition for any (unbranched) abelian cover of (X,0) . More precisely, for any epimorphism H → K one can take the composed map π_{1}(M) → H → K which defines a Galois K-covering (X_{K}, 0) →(X,0) of (X,0) (with (X_{K}, 0) normal). Similarly as above, one can define p_{g}(X_{K}, 0)_{χ} for any χ ∈ K̂. But these invariants are not essentially new: all of them can be recovered from the corresponding invariants associated with the universal abelian cover. Indeed, consider χ ∈ Ĥ via K̂ ↪ Ĥ. Then p_{g}(X_{K}, 0)_{χ} = p_{g}(X_{a}, 0)_{χ}. In particular,

p_{g}(X_{K}, 0) = ∑_{χ∈K̂} p_{g}(X_{a}, 0)_{χ}.

(c) In the above definition (part (a)), p_{g}(X_{a}, 0)_{χ} is independent of the choice of Z̃, in particular one can take Z̃ considered in the proof of 4.2.4. Those facts, together with 4.2.9 show that

p_{g}(X_{a}, 0)_{θ(h)} = h^{1}(OX̃(−l′_{e}(h))) (for any h ∈ H).

Since the set {−l′_{e}(h)}_{h∈H} is a set of representatives (cf. 4.4.3(c’)), the previous version 4.4.2, restricted to the set of line bundles of type OX̃(l′), is completely equivalent with the following.

4.4.6. Property (weak version). Let (X,0) be a normal surface singularity whose link M is a rational homology sphere. Let π : X̃ → X be a fixed good resolution and s := #J the number of irreducible exceptional divisors of π. For any h ∈ H = H_{1}(M, ℤ) consider the character χ := θ(h) and the characteristic element k := K + 2l′_{e}(h) ∈ Char.

(WIn) for any h ∈ H

p_{g}(X_{a}, 0)_{θ(h)} ≤ −sw(M, [k]) − \frac{k^{2} + s}{8}.  (2)

(WId) in (2) one has equality.

4.4.7. Remark. (I). Notice that (WIn) (for all h) and 2.3.4(1) imply

p_{g}(X_{a}, 0) ≤ −λ(M) − ∑_{l′∈Q} \frac{(K + 2l′)^{2} + s}{8},  (3)

or

p_{g}(X_{a}, 0) ≤ −λ(M) − |H| · \frac{K^{2} + s}{8} + ∑_{l′∈Q} χ(l′).

(II). Assume that for some singularity one can verify the inequalities (WIn) (see e.g. 4.4.4). Then equality in (3) implies 4.4.6 with equalities for all h, i.e. (WId).

4.4.8. Example. Assume that (X,0) = {x^{2} + y^{3} + z^{12t+2} = 0}. Here we assume that t ≥ 1 (if t = 0 then (X,0) is rational, a case which will be clarified in the next section.) The following invariants of (X, 0) can be computed using [37], section 6.

The minimal good resolution graph of M is star-shaped with three arms and (normalized) Seifert invariants (α, ω) equal to (3, 1), (3, 1), (6t + 1, 2t), the self intersection of the central curve is −1, the orbifold euler number equals −1/(18t + 3), K^{2} + s = 2, λ(M) = −(24t + 1)/12, and H = ℤ_{3}.

Consider the two arms corresponding to Seifert invariants (3, 1). Both of them contain only one vertex with self intersection −3. We denote them by E_{1}, respectively E_{2}. Then Q contains exactly three element. They are 0, l′_{1} := (E_{1} + 2E_{2})/3 and l′_{2} := (2E_{1} + E_{2})/3. One can rewrite

∑_{l′∈Q} \frac{(K + 2l′)^{2} + s}{8} = |H| · \frac{K^{2} + s}{8} − ∑_{l′∈Q} χ(l′).

In our case, by an easy verification χ(l′_{1}) = χ(l′_{2}) = 1/3. Therefore, the right hand side of (Wa) is (24t + 1)/12 − 3 · 2/8 + 2/3 = 2t.

On the other hand, the universal abelian cover of (X,0) is isomorphic to a Brieskorn singularity of type (X_{a}, 0) = {u^{3} + v^{3} + w^{6t+1} = 0} (with the action ξ ∗ (u, v, w) = (ξu, ξ̄v, w), ξ^{3} = 1). And one can verify easily that p_{g}(X_{a}, 0) = 2t. Hence, for (X,0) , (2) is valid with equalities.

In fact, in this case p_{g}(X, 0) = 2t as well, hence p_{g}(X_{a}, 0)_{χ} = 0 for any χ ≠ 1.

4.5 Example. The case of rational singularities.

4.5.1. Assume that (X, 0) is rational. Then by 3.4.1 and 3.5.2 one has

−sw(M, [k]) = (k_{r}^{2} + s)/8.  (1)

4.5.2. Theorem. Property 4.4.2 (hence 4.4.6 too) is true (with equality) for any rational singularity.

Proof. By 4.4.3(c), we can assume that l′ = l′_{ne}(h) where h = l′ + L. Using 4.3.2, and the fact that l′_{ne}(h) ∈ L_{ℚ,ne}, one gets that h^{1}(OX̃(l′)) = 0. On the other hand, by (1), −sw(M, [k]) = (k_{r}^{2} + s)/8. Here (see also 2.2.7), k = K−2l′, hence k_{r} = K+2l̄′_{ne}(−l′+L) with l̄′_{ne}(−l′+L) = −l′_{ne}(−(−l′+L)) = −l′_{ne}(l′ + L) = −l′. Hence k_{r} = k and the right hand side of 4.4.2(1) is also vanishing.

Theorem 4.5.2 (and its proof) and the identity 4.5.1(1) have the following consequence (whose statement is independent of the Seiberg-Witten theory):

4.5.3. Corollary. Assume that (X,0) is rational, and fix some h ∈ H. Then

p_{g}(X_{a}, 0)_{θ(h)} = \frac{(K + 2l̄′_{ne}(h))^{2} − (K + 2l′_{e}(h))^{2}}{8} = −χ(l̄′_{ne}(h)) + χ(l′_{e}(h)).

In particular, (X_{a}, 0) is rational if and only if χ(l̄′_{ne}(h)) = χ(l′_{e}(h)) for any h ∈ H.

This emphasizes in an impressive way the differences between the two ‘liftings’ l′_{e}(h) and l̄′_{ne}(h). Recall: for a class h = l′ + L, both l′_{e}(h) and l̄′_{ne}(h) are elements of l′ + L, but the first is minimal in L_{ℚ,e} (i.e. it is the representative in Q), while the second is minimal in −L_{ℚ,ne}. Since −L_{ℚ,ne} ⊂ L_{ℚ,e}, one has l′_{e}(h) ≤ l̄′_{ne}(h). In some cases they are not equal.

For example, take the A_{4} singularity, where E has three components E_{1}, E_{2}, E_{3}, all with selfintersection −2, and E_{2} intersecting the others transversely. Then −D_{2} = (1/2, 1, 1/2) = l̄′_{ne}(h) for some h, but it is not in Q: l′_{e}(h) = −D_{2} − E_{2} = (1/2, 0, 1/2). Nevertheless, in this case, their Euler characteristics are the same (corresponding to the fact that — A_{4} being a cyclic quotient singularity — the universal abelian cover is smooth).

4.5.4. Example. Assume that (X, 0) is a rational singularity with the following dual resolution graph.

[figure: −2; −3; −3; −3]

Let X̃ be its minimal resolution, we write E_{0} for the central exceptional component, the others are denoted by E_{i}, i = 1, 2, 3. Then by a computation l′ := −(D_{1} + D_{2} + D_{3}) = (1, 2/3, 2/3, 2/3). This element is minimal in −L_{ℚ,ne} but not in L_{ℚ,e}. Its representative in Q is l′ − E_{0}. The character χ = θ([l′]) is χ(g_{0}) = 1 and χ(g_{j}) = e^{4πi/3} for j = 1, 2, 3. By a computation K = −l′, (K+2l′)^{2} = −2 and (K +2(l′ −E_{0}))^{2} = −10. Therefore, p_{g}(X_{a}, 0)_{χ} = (−2+10)/8 = 1. In particular, (X_{a}, 0) cannot be rational.

Indeed, one can verify (using e.g. [40]), that the minimal resolution of (X_{a}, 0) contains exactly one irreducible exceptional curve of genus 1 and self-intersection −3. In particular, (X_{a}, 0) is minimally elliptic with p_{g}(X_{a}, 0) = 1. This also shows that the above eigenspace is the only non-trivial one.

We reverify this last fact for the conjugate χ̄ of χ. In this case h = θ^{−1}(χ̄) is the class of −D_{0} = (1, 1/3, 1/3, 1/3) = l̄′_{ne}(h); and l′_{e}(h) = −D_{0} − E_{0}. By a calculation K + 2l̄′_{ne}(h) = E_{0} and K + 2l′_{e}(h) = −E_{0}, hence their squares are the same. In particular, p_{g}(X_{a}, 0)χ̄ = 0.

5 The graded roots of S_{−p/q}^{3}(K).

5.1 Introduction.

In this section we determine the graded roots (in particular, the Heegaard Floer homology HF^{+}(−M) and the Seiberg-Witten invariants sw^{OSZ}(M, σ)) for the oriented 3-manifold M = S_{−p/q}^{3}(K) obtained by a negative rational surgery (with coefficient −p/q) along an algebraic knot K ⊂ S^{3}. In this case, since H_{1}(M, ℤ) = ℤ_{p}, the spin^{c}-structures {σ_{a}}_{a} of M can be parametrized by integers a = 0, 1, … p − 1. The main result establishes the graded roots in terms of the integers p, q, a, and the Alexander polynomial Δ of K ⊂ S^{3}. Notice that the Alexander polynomial of an algebraic (any) knot is well-understood, it can be easily computed from most of the other invariants of the knot (e.g., in the present algebraic case, from Puiseux or Newton pairs, or from the semigroup associated with the corresponding local analytic germ). In particular, the input of the theorem is the simplest that one can hope for. Since (in some sense) all the coefficients of Δ are effectively involved, in fact, the result is optimal.

In the very recent manuscript [53], Ozsváth and Szabó computed HF^{+}(S_{p}^{3}(K)) — for any knot K and any integer surgery coefficient p — in terms of the filtered chain homotopy type of the Heegaard Floer complex associated with the pair (S^{3}, K). Compared with this, our starting data, and also the description of HF^{+}(−M), are simpler, and totally elementary; as a price for this we have to impose the ‘negativity restrictions’ for the surgery coefficient and for K.

The proof is based on the results and constructions of [34] valid for AR graphs. The needed facts are listed here in §3.5. Although [34], or §3.5 here, presents a precise algorithm how one should compute HF^{+}, its implementations in different situations sometimes is not straightforward. In the present case too, the proof and additional constructions run over two subsections. In fact, with this presentation, we also wish to advertise the efficiency, novelty and the power of [34]. (For another example, see [34], § 11, where the case of Seifert manifolds is treated.)

Subsection 2 also recalls the classical invariants of algebraic knots and connects them with the plumbing of M. The last subsection contains some concrete examples as well.

5.2 The manifold S_{−p/q}^{3}(K_{f}).

5.2.1. Review of algebraic knots. [5, 7, 14] Let K_{f} ⊂ S^{3} be the link of an irreducible complex plane curve singularity f : (ℂ^{2}, 0) → (ℂ, 0); i.e. for ϵ > 0 sufficiently small, write S^{3} = {z ∈ ℂ^{2} : ||z|| = ϵ} and take the transversal intersection K_{f} := {f = 0} ∩ S^{3} ↪ S^{3}. The natural orientations of S^{3} and of the regular part of {f = 0} induces a natural orientation on K_{f}. Since f is irreducible, K_{f} ≈ S^{1}. We will assume that {f = 0} is not smooth at the origin, i.e. K_{f} ⊂ S^{3} is not the unknot. The isotopy type of K_{f} ⊂ S^{3} is completely characterized by any of the following invariants listed below.

• The Newton pairs of f consist of g ≥ 1 pairs of integers {(p_{i}, q_{i})}_{i=1}^{g}, where p_{i} ≥ 2, q_{i} ≥ 1, q_{1} > p_{1} and gcd(p_{i}, q_{i}) = 1.

• In some topological constructions, it is preferable to replace the Newton pairs by the ‘linking pairs’ (or, the decorations of the splice diagram, cf. [7]) (p_{i}, a_{i})_{i=1}^{g}, where

a_{1} = q_{1} and a_{i+1} = q_{i+1} + p_{i+1}p_{i}a_{i} for i ≥ 1.

• Let Δ(t) be the Alexander polynomial of K_{f} ⊂ S^{3}, or equivalently, the characteristic polynomial of the algebraic monodromy acting on the first homology of the Milnor fiber of f. It is normalized by Δ(1) = 1. In terms of the (p_{i}, a_{i})_{i} pairs it is

Δ(t) = \frac{(t^{a_{1}p_{1}p_{2}···p_{g}} − 1)(t^{a_{2}p_{2}···p_{g}} − 1) ⋯ (t^{a_{g}p_{g}} − 1)(t − 1)}{(t^{a_{1}p_{2}···p_{g}} − 1)(t^{a_{2}p_{3}···p_{g}} − 1) ⋯ (t^{a_{g}} − 1)(t^{p_{1}···p_{g}} − 1)}.  (1)

The degree of Δ, or equivalently, the first Betti number of the Milnor fiber of f, is the Milnor number μ of f.

• The semigroup S of the germ f is a sub-semigroup of ℕ defined by S := {i_{0}(f, h) : h ∈ O_{(ℂ^{2},0)}}, where i_{0}(f, h) denotes the local intersection multiplicity of f and h (which equals the codimension of the ideal (f, h) in O_{(ℂ^{2},0)}). For h invertible i_{0}(f, h) is zero, hence 0 ∈ S.

It is known that S is generated by the integers β̄_{0} = p_{1}p_{2} ⋯ p_{g}, β̄_{k} = a_{k}p_{k+1} ⋯ p_{g} for 1 ≤ k ≤ g−1, and β̄_{g} = a_{g}. Moreover, #(ℕ \ S) is finite. Its cardinality is the delta-invariant δ of f, which in this case also equals μ/2, cf. [28]. It is also known that δ = μ/2 equals the minimal Seifert genus of K_{f} ⊂ S^{3}.

The largest element of ℕ \ S is μ − 1. In fact, for 0 ≤ k ≤ μ − 1 one has: k ∈ S if and only if μ − 1 − k ∉ S.

• The embedded minimal good dual resolution graph of the germ f has the following shape

[figure: K_{f}; ⋯; −1; j_{0}; Γ(f) :]

Above we emphasize only the vertices of degree one and three. The dash-line between two such vertices replaces a string [figure: ⋯] . The number of vertices of degree three is exactly g. In general, any vertex j is decorated by the self-intersection of the corresponding irreducible exceptional divisor E_{j}. In the above diagram we put only the decoration of the vertex j_{0}, which corresponds to the unique (−1)-curve, and which also supports the strict transform of {f = 0}. The other decorations are not really essential in our next discussions; for the description of the complete graph see e.g. [5], or [31], section 4.I.

The above diagram can also be identified with the plumbing graph of (S^{3}, K_{f}). In this case the self-intersections are the corresponding Euler numbers of the S^{1}-bundles, all the surfaces used in the plumbing construction are S^{2}’s, and K_{f} is a generic fiber of the unique (−1)-bundle.

The above resolution graph Γ(f) will be denoted by the following schematic diagram:

[figure: K_{f}; −1; j_{0}; Γ̄; Γ(f) :]

The polynomial Δ(t) can also be deduced from Γ(f) by A’Campo’s formula [1]. Notice that (if we disregard the arrowhead) the graph Γ̄ can be blown down completely.

We emphasize again, the information codified in the following objects – isotopy class of K_{f} ⊂ S^{3}, set of Newton pairs, set of linking pairs, Alexander polynomial, semigroup or the embedded resolution graph – are completely equivalent. This means that the polynomial Q introduced below can be deduced from any of them.

5.2.2. Definition of the polynomial Q. One has the following identity connecting the Alexander polynomial Δ and the semigroup S (probably know already by Zariski and Teissier, see [14]):

\frac{Δ(t)}{1 − t} = ∑_{k∈S} t^{k}.  (2)

Since Δ(1) = 1 and Δ′(1) = δ (use e.g. the formula (1) of 5.2.1), one gets

Δ(t) = 1 + δ(t − 1) + (t − 1)^{2} · Q(t)

for some polynomial Q(t) = ∑_{i=0}^{μ−2} α_{i}t^{i} of degree μ − 2 with integral coefficients. In fact, all the coefficients {α_{i}}_{i=0}^{μ−2} are strict positive, and:

δ = α_{0} ≥ α_{1} ≥ ⋯ ≥ α_{μ−2} = 1.

Indeed, by the above identity (2), one has δ+(t−1)Q(t) = ∑_{k∉S} t^{k}, or Q(t) = ∑_{k∉S}(t^{k−1}+⋯+t+1). This shows that

α_{i} = #{k ∉ S : k > i}.

5.2.3. The surgery 3-manifold M = S_{−p/q}^{3}(K). In the sequel we fix an oriented algebraic knot K_{f} ⊂ S^{3} and we consider the oriented manifold M := S_{−p/q}^{3}(K_{f}) obtained by −p/q-surgery along K_{f} ⊂ S^{3}, where p/q > 0 is a positive rational number (p > 0, gcd(p, q) = 1).

It is easy to see (e.g. by Kirby calculus, see e.g. [11]) that M can be represented by the following plumbing diagram (the symbols j_{0}, …, j_{s} are the ‘names’ of the corresponding vertices, they are not really parts of the decoration of the diagram):

[figure: j_{1}; j_{2}; j_{s−1}; j_{s}; −1; j_{0}; Γ̄; Γ(M) :; ⋯; −k_{1} − m_{f}; −k_{2}; −k_{s}; −k_{s−1}]

where k_{1} ≥ 1 and k_{j} ≥ 2 (2 ≤ j ≤ s) are integers determined by the continued fraction p/q = [k_{1}, k_{2}, …, k_{s}] and m_{f} = a_{g}p_{g}. In terms of the embedded resolution graph of f (cf. 5.2.1), m_{f} is the multiplicity of the pull back of the germ f along the (−1)-curve. Topologically, e.g. if one uses Kirby calculus and blows down all the vertices of Γ̄, m_{f} is a sum ∑_{i} m_{i}^{2} of squares of a sequence of linking numbers m_{i}. In fact, in the language of plane curve singularities, this sequence of m_{i}’s is exactly the ‘multiplicity sequence’ of f.

Notice that if we would start with the unknot K_{f} ⊂ S^{3}, then M = S_{−p/q}^{3}(K_{f}) would be the lens space L(p, q) (in general, normalized with 0 < q < p, hence k_{1} ≥ 2 as well). This case is completely solved in § 3.6. Therefore, in the sequel we will assume that K_{f} is not the unknot; and we will allow the case 0 < p/q ≤ 1 (i.e. k_{1} = 1) as well.

Take a point ∗ ∈ S^{3} \ K_{f} and identify S^{3} \ ∗ with ℝ^{3}. Then a Kirby diagram of M is given by the knot K_{f} ⊂ ℝ^{3} with decoration (surgery coefficient) −p/q. In particular, the Kirby diagram of the manifold −M (M with reversed orientation) — whose Heegaard Floer homology will be computed in the sequel — is given by the mirror image K_{f}^{m} of K_{f} (with respect to any plane in ℝ^{3}) with surgery coefficient p/q; i.e. −M = S_{p/q}^{3}(K_{f}^{m}).

5.2.4. Lemma. Γ(M) is an AR-graph.

Proof. We will show that if we modify the −1 decoration of j_{0} in Γ(M) into −2 (let us call this new graph by Γ(M)_{−2}), we get a sandwiched graph (cf. 2.5.3(d)). Indeed, consider the graph Γ̄. Blow up the −1 vertex j_{0}. The new decoration of j_{0} will be −2, while a new −1 vertex is created. Then blow up this new vertex (m_{f} + k_{1} − 1) times. Then its new decoration will be −m_{f} − k_{1}, while it has (m_{f} + k_{1} − 1) neighbours which are all −1 curves. Fix one of them, and blow up k_{2} − 1 times. If one continues this procedure, one gets a graph which contains Γ(M)_{−2} as a subgraph.

5.2.5. Remark. Using 3.4.1, the above proof also shows that the plumbed 3-manifold associated with Γ(M)_{−2} has trivial reduced Heegaard–Floer homology (i.e., it is an L-space in the sense of Ozsváth-Szabó).

5.3 The main invariants of S_{−p/q}^{3}(K_{f}).

5.3.1. Consider M = S_{−p/q}^{3}(K_{f}) as above. Then H := H_{1}(M, ℤ) = ℤ_{p}. There is natural identification of the spin^{c}-structures of M with the classes of ℤ_{p} – or, with the integers a = 0, 1, …, p − 1, a = 0 corresponding to the canonical spin^{c}-structure. For the precise identification, see 2.2.6 and 5.4.7. Let σ_{a} be the spin^{c}-structure associated with a.

The next theorem provides a purely combinatorial description of HF^{+}(−M) from Γ(M). More precisely, HF^{+}(−M, σ_{a}) will be expressed in terms of the delta-invariant δ, the coefficients {α_{i}}_{i} of the polynomial Q (cf. 5.2.2), and the integers p, q and a. The symbol s(q, p) denotes the Dedekind sum (see e.g. 3.6). The integer q′ is uniquely determined by 1 ≤ q′ ≤ p and qq′ ≡ 1 (mod p).

5.3.2. Theorem. For each fixed a = 0, 1, …, p − 1, — corresponding to the p different spin^{c}- structures of M — one defines the following objects :

• t_{a} := [\frac{(2δ−1)q−a−1}{p}];

• r_{a} := 3s(q, p) + 2 ∑_{j=1}^{a} \{\frac{jq′}{p} \} − \frac{(1+2a)(p−1)}{2p} + δ(1 − \frac{q+1}{p}) + \frac{δ^{2}q}{p} − \frac{2δa}{p};

• a function τ_{a} : {0, 1, …, 2t_{a} + 2} → ℤ by

\{ \begin{matrix}τ_{a}(2t) = t(1 − δ) + ∑_{i=0}^{t−1} [\frac{ip+a}{q}], (t = 0, …, t_{a} + 1); \\ τ_{a}(2t + 1) = τ_{a}(2t + 2) + α_{[(tp+a)/q]}, (t = 0, …, t_{a}). \end{matrix}

• and the graded root (R_{τ_{a}}, χ_{τ_{a}}) associated with τ_{a}. Then (R_{τ_{a}}, χ_{τ_{a}}) is the graded root of M associated with σ_{a}. Hence the following facts also hold:

HF_{odd}^{+}(−M, σ_{a}) = 0, HF_{even}^{+}(−M, σ_{a}) = ℍ(R_{τ_{a}}, χ_{τ_{a}})[r_{a}], and −(k_{r}^{2} + #J)/4 = r_{a}; d(−M, σ_{a}) = 2 · min τ_{a} + r_{a}, where min τ_{a} = τ_{a}( 2⌈t_{a}/2⌉).

The proof is based on the theorems 3.5.2 and 3.5.6. The details will be provided in the subsections 5.4 and 5.5. We continue with some remarks and corollaries.

5.3.3. Remark. Notice that for any t ∈ {0, …, t_{a}}, α_{[(tp+a)/q]} is strict positive, hence τ_{a}(2t + 1) > τ_{a}(2t + 2). On the other hand, using properties of S (see e.g. 5.2.1, or 5.5.9(2)), one can verify that the following identity also holds:

τ_{a}(2t + 1) = τ_{a}(2t) + #{γ ∈ S : γ ≤ (tp + a)/q }.

Therefore, since 0 ∈ S, one has τ_{a}(2t + 1) > τ_{a}(2t). In particular, the above representation of the graded root is the most ‘economical’: all the values are essential, see also 3.1.3(2).

This also shows that (R_{τ_{a}}, χ_{τ_{a}}) has exactly t_{a} + 2 local minimum points, and they correspond to the values τ_{a}(2t), t = 0, 1, …, t_{a} + 1.

5.3.2, 5.3.3 and 3.2.6 imply the next two corollaries.

5.3.4. Corollary.

sw^{OSz}(M, σ_{a}) = −sw^{OSz}(−M, σ_{a}) = \frac{r_{a}}{2} − ∑_{t≥0} α_{[(tp+a)/q]}.

Recall that HF^{+}(−M, σ_{a}) is a ℤ[U]-module. Denote by ker U(σ_{a}) (resp. by coker U(σ_{a})) the kernel (resp. the cokernel) of the U-action. They are finitely generated graded free ℤ-module. Let ℤ_{(r)} denote a rank one free ℤ-module whose grading is r.

5.3.5. Corollary.

ker U(σ_{a}) = ⨁_{0≤t≤t_{a}+1} ℤ_{(2τ_{a}(2t)+r_{a})}.

In particular, ker U(σ_{a}) depends only on the integers p, q a, and δ, but not on the coefficients {α_{ℓ}}_{ℓ}. On the other hand,

coker U(σ_{a}) = ⨁_{0≤t≤t_{a}} ℤ_{(2τ_{a}(2t+1)+r_{a}−2)},

which depends essentially on the coefficients {α_{ℓ}}_{ℓ} of Q. Moreover:

rank_{ℤ} ker U = ∑_{a=0}^{p−1}(t_{a} + 2) = p + (2δ − 1)q = p + rank_{ℤ} cokerU.

5.3.6. Remark. If a ≥ (2δ − 1)q, then t_{a} = −1, hence R_{τ_{a}} = R_{0} (cf. 3.1.3(1)). In particular, for such a, ℍ_{red}(R_{τ_{a}}) = 0. For all the other a’s, ℍ_{red}(R_{τ_{a}}) is not trivial. E.g., for the canonical spin^{c}-structure corresponding to a = 0, ℍ_{red}(R_{τ_{0}}) is never trivial. In particular, −M is never an L-space.

5.4 The first part of the proof of 5.3.2: k_{r}^{2} + #J

5.4.1. The index set J. According to the shape of the plumbing graph Γ(M), the index set of its vertices is J = J̄∪{1, …, s}, where J̄ is the index set of the vertices of Γ̄, and the indices {1, …, s} correspond to j_{1}, …, j_{s}. The distinguish vertex of Γ(M) is j_{0} corresponding to 0 ∈ J̄ ⊂ J. The base elements will also be denoted accordingly: E_{0}, E_{1}, …, E_{s}; or E_{j} for j ∈ J̄.

5.4.2. The graph _{δ}Γ. As we already mentioned, the sub-graph Γ̄ can be blown down. After this blow down, we get a graph which will be denoted as follows:

[figure: ⋯; −k_{1}; −k_{2}; −k_{s}; −k_{s−1}; [δ]; _{δ}Γ:]

We can think about this graph as the dual graph of a minimal resolution (of a normal surface singularity) obtained from a resolution with dual graph Γ(M). In this sense, the new vertex j_{1} is a rational curve with a singular point which has delta-invariant δ; and the decorations −k_{i} are the corresponding self-intersections. On the other hand, one can think about this graph also in the language of Kirby diagrams: j_{1} represents the knot K_{f} ⊂ S^{3}, the other vertices represent unknots, they are linked as usual with linking number one, and −k_{i} are the corresponding surgery coefficients.

The point is that a large number of numerical invariants of the graph Γ(M) can already be determined from _{δ}Γ in terms of δ and {k_{i}}_{i} (for this comparison, see the second part 5.4.6 of this subsection). The advantage of this is that the above graph _{δ}Γ is the same as the graph of a lens space L(p, q), provided that we disregard the decoration [δ]. In particular, its invariants can be computed by similar methods as those of lens spaces – in fact, in their computations we will even use the corresponding relations valid for lens spaces. This will be the subject of the first part of this subsection. Our model is subsection 3.6 where we run the algorithm for lens spaces. The reader is invited to consult this subsection and also the original source [34] and to verify that our present claims, verified in 3.6 for case q < p, are valid for q ≥ p as well.

In fact, any invariant of the lattice (which does not involve δ) equals the corresponding invariant of L(p, q). But, formulas which involve δ (like the ‘adjunction formula’ determining the canonical characteristic element, or the ‘Riemann-Roch’ formula for χ_{k}) depend essentially on δ.

5.4.3. Notations. In order to make a distinction between invariants of the graphs Γ(M) and _{δ}Γ, the invariants of the second one will have an extra ˜ -decoration; e.g. L̃ denotes the lattice of rank s with base elements {Ẽ_{i}}_{i=1}^{s}, while {D̃_{i}}_{i=1}^{s} ∈ L̃′ are the dual bases, etc.

The integers {n_{is}}_{i=1}^{s} and have the same meaning as in 3.6.1. Similarly, q′ is the unique integer with p ≥ q′ ≥ 1 and qq′ ≡ 1 (mod p) (notice the small difference with 3.6.1 where q < p, but in the present case p < q is allowed too).

5.4.4. The group H̃ and the elements l̃′_{[k]}. Clearly, cf. 3.6, H̃ = L̃′/L̃ = ℤ_{p}, and [D̃_{s}] = D̃_{s} + L̃ is one of its generators. Then [D̃_{j}] = [n_{j+1,s}D̃_{s}] in H̃ (1 ≤ j ≤ s). The set of spin^{c}-structures is the set of orbits {−aD̃_{s} + L̃}_{0≤a<p}. For any 0 ≤ a < p write

l̃′_{[−aD̃_{s}]} = −(a_{1}D̃_{1} + a_{2}D̃_{2} + ⋯ + a_{s}D̃_{s}).

By 3.6.2, the system (a_{1}, …, a_{s}) can be realized as the set of coefficients of a minimal vector l̃′_{[−aD̃_{s}]} for some 0 ≤ a < p if and only if the entries satisfy the system of inequalities (SI) of 3.6.2.

5.4.5. The canonical characteristic element. Let _{δ}K̃ denote the canonical characteristic element associated with the graph _{δ}Γ (see below). It is convenient to consider the canonical characteristic element K̃ of the graph of L(p, q) (recall that this graph is obtained from _{δ}Γ by omitting δ). The adjunction relations defining K̃ are the usual ones, but those which identify _{δ}K̃ are the following (see also 5.4.9): (_{δ}K̃, Ẽ_{j}) − k_{j} + 2 should equal twice the delta-invariant of the vertex j, namely = 2δ for j = 1, and = 0 otherwise. Hence:

δK̃ = K̃ + 2δD̃_{1}.

In particular,

δK̃^{2} + s = K̃^{2} + s + 4δ(K̃, D̃_{1}) + 4δ^{2}(D̃_{1}, D̃_{1}).

(K̃, D̃_{1}) is the Ẽ_{1}-coefficient of K̃, which equals −1 + (q + 1)/p, see e.g. [37], (5.2). Similarly, D̃_{1}^{2} = Ĩ_{11}^{−1} = −n_{2s}/p = −q/p. Also, by 3.6.5:

K̃^{2} + s = 2(p − 1)/p − 12 · s(q, p).

Therefore,

δK̃^{2} + s = \frac{2(p − 1)}{p} − 12s(q, p) − 4δ(1 − \frac{q + 1}{p}) − 4δ^{2} \frac{q}{p}.

The distinguished characteristic element k̃_{r} of the orbit −aD̃_{s} + L̃ is

k̃_{r} = _{δ}K̃ + 2l̃′_{[−aD̃_{s}]}.

Therefore,

k̃_{r}^{2} + s = _{δ}K̃^{2} + s + 4 · ( K̃ + l̃′_{[−aD̃_{s}]}, l̃′_{[−aD̃_{s}]} ) + 8δ · ( D̃_{1}, l̃′_{[−aD̃_{s}]} ).

By 3.6.6

( K̃ + l̃′_{[−aD̃_{s}]}, l̃′_{[−aD̃_{s}]} ) = \frac{a(p − 1)}{p} − 2 · ∑_{j=1}^{a} \{\frac{jq′}{p} \}.

On the other hand, by formula (1) of 3.6.2:

( D̃_{1}, l̃′_{[−aD̃_{s}]} ) = − ∑_{i=1}^{s} a_{i}(D̃_{1}, D̃_{i}) = ∑_{i=1}^{s} a_{i} \frac{n_{i+1,s}}{p} = \frac{a}{p}.

Summing all, one gets

−\frac{k̃_{r}^{2} + s}{4} = 3s(q, p) + 2 ∑_{j=1}^{a} \{\frac{jq′}{p} \} − \frac{(1 + 2a)(p − 1)}{2p} + δ(1 − \frac{q + 1}{p}) + \frac{δ^{2}q}{p} − \frac{2δa}{p}.

5.4.6. Back to the graph Γ(M). Finally, we compare the invariants of the graphs Γ(M) and _{δ}Γ.

There are two natural morphisms connecting the corresponding lattices L and L̃. The first one, π_{∗} : L → L̃ is defined by π_{∗}(E_{j}) = 0 for any j ∈ J̄, while π_{∗}(E_{i}) = Ẽ_{i} for i = 1, …, s.

In order to define the second morphism, we need an additional construction.

Let Z_{f} := ∑_{j∈J̄} m_{j}E_{j} be the cycle supported by Γ̄ which satisfies

(Z_{f} + E_{1}, E_{j}) = 0 for j ∈ J̄.

Since det Γ̄ = ±1, this system has a unique integral solution {m_{j}}_{j}. In fact, in terms of the diagram Γ(f) (cf. 5.2.1), m_{j} is the vanishing order of the pull back of f along the corresponding irreducible exceptional divisor. E.g., m_{0} = m_{f}.

Then one defines π^{∗} : L̃ → L by π^{∗}(Ẽ_{j}) = E_{j} for j ≥ 2 and π^{∗}(Ẽ_{1}) = Z_{f} + E_{1}. By the very definition follows the ‘projection formula’:

(π^{∗}(l̃), l) = (l̃, π_{∗}(l)) for l̃∈ L̃, l ∈ L.

5.4.7. The group H. π_{∗} has a natural extension to L_{ℚ} → L̃_{ℚ}, and π_{∗}(L′) ⊂ L̃′. Therefore, π_{∗} induces a group morphism H → H̃. Its surjectivity follows from π_{∗}π^{∗} = 1. In order to prove injectivity, consider an l′ ∈ L′ with π_{∗}(l′) ∈ L̃. Then π^{∗}π_{∗}(l′) is an element of L. Since π^{∗}π_{∗}(l′) − l′ is supported by Γ̄, and det Γ̄ = ±1, one gets that π^{∗}π_{∗}(l′) − l′ ∈ L as well. Hence l′ ∈ L, and π_{∗} induces an isomorphism H → H̃ = ℤ_{p}, and H is generated by D_{s}.

In the sequel, the set of spin^{c}-structures of M will be identified with the set of orbits {−aD_{s} + L}_{0≤a<p}. We denote by σ_{a} that spin^{c}-structure which correspond to the orbit −aD_{s} + L.

5.4.8. The element l′_{[−aD_{s}]}. We claim that l′_{[−aD_{s}]} = π^{∗}(l̃′_{[−aD̃_{s}]}). Indeed, l′ := π^{∗}(l̃′_{[−aD̃_{s}]}) ∈ (−L_{ℚ,ne}) by the projection formula. Next, we verify the minimality of l′. Notice that any l ∈ L with l ≥ 0 can be written in the form π^{∗}(x̃) + x̄, where x̃ ≥ 0 and x̄ is supported by Γ̄. Assume that for such π^{∗}(x̃) + x̄ ≥ 0 one has:

(l′ − π^{∗}(x̃) − x̄, E_{j}) ≤ 0 for any j ∈ J .  (∗)

Since (π^{∗}(ỹ), E_{j}) = 0 for any j ∈ J̄, one gets that (−x̄, E_{j}) ≤ 0 for any j ∈ J̄. Since Γ̄ is negative definite, it follows that x̄ ≤ 0. Using this, (∗) for j = 1, …, s gives

0 ≥ (l′ − π^{∗}(x̃), E_{j}) + (−x̄, E_{j}) ≥ (l′ − π^{∗}(x̃), E_{j}) = (l̃′_{[−aD̃_{s}]} − x̃, Ẽ_{j}).

hence, by the minimality of l̃′_{[−aD̃_{s}]}, one has x̃ = 0. But then x̄ ≥ 0 and x̄ ≤ 0 implies x̄ = 0 as well.

5.4.9. Claim: π_{∗}(K) = _{δ}K̃. First notice that by the projection formula:

π_{∗}(D_{j}) = \{ \begin{matrix}m_{j}D̃_{1} & if j ∈ J̄ \\ D̃_{j} & if j = 1, …, s. \end{matrix}

(where Z_{f} = ∑_{j∈J̄} m_{j}E_{j} as above). The adjunction relations for Γ(M) can be rewritten as

K = − ∑_{j∈J} E_{j} − ∑_{j∈J}(2 − s_{j})D_{j},

where s_{j} is the adjacent degree of the vertex j in Γ(M). They also satisfy (cf. [1]):

∑_{j∈J̄}(2 − s_{j})m_{j} = 1 − μ = 1 − 2δ.

Then, taking π_{∗} one gets π_{∗}(K) = K̃ + 2δD̃_{1} = _{δ}K̃.

5.4.10. Claim: K^{2} + #J = _{δ}K̃^{2} + s. Indeed, set KΓ̄ := K − π^{∗}(_{δ}K̃). Then KΓ̄ is supported by Γ̄ and satisfies the adjunction relations (KΓ̄, E_{j}) = −e_{j} − 2 for any j ∈ J̄, hence it is the canonical cycle of Γ̄. Moreover, since Γ̄ can be blown down (and since by an elementary blow up of a smooth point K^{2} decreases by 1), one has K2 Γ̄ = −(#J − s). By projection formula π^{∗}(_{δ}K̃) is orthogonal to KΓ̄, hence

K^{2} = (π^{∗}(_{δ}K̃) + KΓ̄)^{2} = _{δ}K̃^{2} + K2 Γ̄ = _{δ}K̃^{2} − (#J − s).

Finally, this claim, via the projection formula, shows for any spin^{c}-structure that

k_{r}^{2} + #J = k̃_{r}^{2} + s.

5.4.11. Example. Integer surgery. Assume that q = 1, i.e. M = S_{−p}^{3}(K_{f}). Then q′ = 1 as well, and

−\frac{k_{r}^{2} + #J}{4} = −\frac{k̃_{r}^{2} + s}{4} = \frac{(p + 2δ − 2 − 2a)^{2} − p}{4p}.

5.4.12. Example. (1/q)-surgery. Assume that p = 1, i.e. M = S_{−1/q}^{3}(K_{f}). Then again q′ = 1. There is only one spin^{c}-structure corresponding to a = 0, and

−\frac{K^{2} + #J}{4} = −\frac{K̃^{2} + s}{4} = qδ(δ − 1).

5.5 The second part of the proof of 5.3.2: (R_{τ_{[k]}}, χ_{τ_{[k]}})

The construction of the cycles x_{[k]}(i) is given in two steps. The first step provides similar cycles associated with the graph Γ̄, and it is based on the combinatorial properties of the graph Γ(f) (involving also some techniques of plane curve singularities). In particular, we prefer to think about Γ̄ as the dual graph of irreducible exceptional curves obtained by repeated blow ups of (ℂ^{2}, 0). The lattice of Γ̄ is denoted by L̄, and we write L̄_{ne} := {y ∈ L̄ : (y, E_{j}) ≥ 0, j ∈ J̄}.

5.5.1. The cycles {y(i)}_{i≥0}. Since Γ̄ is negative definite, (7.6) of [34] guarantees, for any i ≥ 0, the existence of a positive cycle y(i) ≥ 0 (supported by Γ̄) with the following properties:

(a) pr_{0}(y(i)) = i;

(b) (y(i), E_{j}) ≤ 0 for any j ∈ J̄ \ {0};

(c) y(i) is minimal (with respect the partial ordering ≤) with the properties (a) and (b).

E.g., y(0) = 0. Although there is very precise algorithm which determines all the cycles y(i) (see e.g. the proof of 5.5.3, or [34]), we are not interested in all the coefficients of y(i). Instead, what we really have to know is the set of intersection numbers (y(i), E_{0}) (cf. 3.5.7). Let Z_{f} be the divisorial part (supported by Γ̄) of the germ f, cf. 5.4.6, which satisfies

(Z_{f}, E_{j}) = \{ \begin{matrix}0 & if j ∈ J̄ \ \{0\}; \\ −1 & if j = 0. \end{matrix}

Recall also that S ⊂ ℕ denotes the semigroup of f, see 5.2.1.

5.5.2. Proposition.

(a) If i = tm_{f} + i_{0} with t ≥ 0 and 0 ≤ i_{0} < m_{f}, then y(i) = tZ_{f} + y(i_{0});

(b) For any i < m_{f} one has

(y(i), E_{0}) = \{ \begin{matrix}1 & if i ∉ S; \\ 0 & if i ∈ S. \end{matrix}

Proof. (a) Clearly, y′(i) := tZ_{f} + y(i_{0}) satisfies (5.5.1)(a)-(b), we need to verify (c). But if for some y ≥ 0, y′(i) − y satisfies (a)-(b), then y(i_{0}) − y would also satisfy (a)-(b) for i_{0}, hence y = 0 by minimality of y(i_{0}). Part (b) of (5.5.2) will follow from the next sequence of lemmas.

The first lemma was proved and used intensively in [34] as a general principle of rational graphs. For the convenience of the reader, we sketch its proof, for more details, see [loc. cit.].

5.5.3. Lemma. (y(i), E_{0}) ≤ 1 for any i ≥ 0.

Proof. We denote by Z_{min} Artin’s minimal cycle associated with Γ̄, i.e. the minimal nonzero cycle in −L̄_{ne}. The cycle Z_{min} can be determined by the following (Laufer’s) algorithm ([20]): Construct a ‘computation’ sequence of cycles {Z_{l}}_{l} as follows. Set Z_{0} := 0, Z_{1} := b_{0}; if Z_{l} is already constructed, but for some j ∈ J̄ one has (Z_{l}, E_{j}) > 0, then take Z_{l+1} := Z_{l} + b_{j}. If Z_{l} ∈ (−L̄_{ne}) then stop and Z_{l} = Z_{min}.

Now, by a geometric genus computation, one gets for all rational graphs that whenever (Z_{l}, E_{j}) > 0 in the above algorithm, in fact one has the equality (†): (Z_{l}, E_{j}) = 1 (cf. [20]). Since Γ̄ can be blown down, it is rational, hence (†) works.

The point is that y(i) can also be computed by a similar algorithm: Assume that y(i) is already determined. Then construct the sequence of cycles {Z_{l}}_{l} as follows. Z_{0} := y(i), Z_{1} := y(i) + E_{0}, and if (for l ≥ 1) (Z_{l}, E_{j}) > 0 for some j ∈ J̄ \ {0}, then take Z_{l+1} := Z_{l} + E_{j}, otherwise stop and write Z_{l} = y(i + 1).

Then one can verify that any sequence connecting y(i) with y(i + 1) can be considered as part of a computation sequence associated with Z_{min}, hence lemma follows by the above property (†).

5.5.4. Lemma. Fix an arbitrary i ≥ 0. Then (y(i), E_{0}) ≤ 0 if and only if i ∈ S.

Proof. ‘⇒’ If (y(i), E_{0}) ≤ 0 then y(i) ∈ (−L̄_{ne}). Since Γ̄ is the dual graph of a modification of (ℂ^{2}, 0), the cycle y(i) is the divisorial part Z_{g} of a holomorphic germ g ∈ O_{(ℂ^{2},0)}. Since the intersection multiplicity i_{0}(f, g) of the germs f and g at 0 ∈ ℂ^{2} is the multiplicity of Z_{g} along the exceptional divisor supporting the arrow of f, one gets i_{0}(f, g) = pr_{0}(y(i)) = i. But, by definition, i_{0}(f, g) ∈ S.

‘⇐’ If i ∈ S, then there exists g ∈ O_{(ℂ^{2},0)} with i_{0}(f, g) = i; i.e. with pr_{0}(Z_{g}) = i (see above) and Z_{g} ∈ (−L̄_{ne}) (a general property of divisorial parts of holomorphic germs). Then the minimality of y(i) implies y(i) ≤ Z_{g}. This, and pr_{0}(y(i)) = pr_{0}(Z_{g}), and the fact that I off the diagonal has positive entries imply that (E_{0}, y(i)) ≤ (E_{0}, Z_{g}). But, since Z_{g} ∈ (−L̄_{ne}), (E_{0}, Z_{g}) ≤ 0.

Lemma 5.5.4 can be improved for i < m_{f}:

5.5.5. Lemma. Assume that i < m_{f}. Then (y(i), E_{0}) = 0 if and only if i ∈ S.

Proof. Via 5.5.4, it is enough to prove that (y(i), E_{0}) < 0 is not possible for i < m_{f}. Indeed, assume that (y(i), E_{0}) < 0 for some i. Then, by a verification y(i) − Z_{f} ∈ (−L̄_{ne}), hence (since Γ̄ is negative definite) y(i) − Z_{f} ≥ 0. In particular, i − m_{f} = pr_{0}(y(i) − Z_{f}) ≥ 0.

5.5.6. Back to Γ(M). We consider again the graph Γ(M). We fix an orbit [k] = −aD_{s} + L for some 0 ≤ a < p (corresponding to the spin^{c}-structure σ_{a} of M). In the sequel we will use freely the notations of the previous subsection. Additionally, we write for any j = 1, …, s:

a′_{j} := n_{j+1,s}a_{j} + n_{j+2,s}a_{j+1} + … + a_{s}.

E.g., by 3.6.2, a′_{1} = a.

Consider now the sub-graph of Γ(M) with vertices j_{0}, …, j_{s} and the s edges connecting them. We wish to identify the cycle z(i) = iE_{0} + ∑_{i=1}^{s} u_{j}E_{j} which has the properties of x(i) ‘restricted on this sub-graph’. More precisely:

5.5.7. Proposition. Fix i ≥ 0 and 0 ≤ a < p.

(a) For j = 1, …, s define the integers u_{j} inductively by:

u_{1} := ⌈ \frac{iq − a}{p + qm_{f}} ⌉, u_{j} := ⌈ \frac{u_{j−1}n_{j+1,s} − a′_{j}}{n_{j,s}} ⌉ (1 < j ≤ s).

Then the cycle z(i) := iE_{0} + ∑_{j=1}^{s} u_{j}E_{j} is ≥ 0 and satisfies

(z(i) + l′_{[−aD_{s}]}, E_{j}) ≤ 0 for any j = 1, …, s.

(b) Assume that the cycle z̄(i) := iE_{0} + ∑_{j=1}^{s} ū_{j}E_{j} is ≥ 0 and satisfies

(z̄(i) + l′_{[−aD_{s}]}, E_{j}) ≤ 0 for any j = 1, …, s.

Then

ū_{1} ≥ ⌈ \frac{iq − a}{p + qm_{f}} ⌉ , and ū_{j} ≥ ⌈ \frac{ū_{j−1}n_{j+1,s} − a′_{j}}{n_{j,s}} ⌉ for 1 < j ≤ s.

Proof. This property was used in a similar situation for the plumbing graph of a Seifert manifolds, applied for its ‘legs’, cf. 11.11 of [34]. For the convenience of the reader we sketch the proof.

(a) By the definition of u_{j} one has u_{j}n_{js} ≥ u_{j−1}n_{j+1,s} − a′_{j}. This is equivalent, via the identities n_{js} = k_{j}n_{j+1,s} − n_{j+2,s}, resp. a′_{j} = n_{j+1,s}a_{j} + a′_{j+1}, with

u_{j}k_{j} − u_{j−1} + a_{j} ≥ \frac{u_{j}n_{j+2,s} − a′_{j+1}}{n_{j+1,s}} .

Using the definition of u_{j+1}, one gets u_{j}k_{j} −u_{j−1}+a_{j} ≥ u_{j+1}, which is exactly (z(i)+l′_{[−ag_{s}]}, E_{j}) ≤ 0.

(b) can be proved by a descending induction, using the above sequence of arguments in reversed order.

5.5.8. Corollary. Fix 0 ≤ a < p and write i = tm_{f} + i_{0} for some t ≥ 0 and 0 ≤ i_{0} < m_{f}. Then

x_{[k]}(i) = t · Z_{f} + y(i_{0}) + ⌈ \frac{iq − a}{qm_{f} + p} ⌉ E_{1} + ∑_{j≥2} u_{j}E_{j}.

In particular,

(x_{[k]}(i), E_{0}) = −t + ⌈ \frac{iq − a}{qm_{f} + p} ⌉ + (y(i_{0}), E_{0}).

Moreover:

χ_{k_{r}}( x_{[k]}(i + 1) ) − χ_{k_{r}}( x_{[k]}(i) ) = t + 1 − ⌈ \frac{iq − a}{qm_{f} + p} ⌉ − \{ \begin{matrix}1 & if i_{0} ∉ S \\ 0 & if i_{0} ∈ S. \end{matrix}

Proof. Since the subgraphs Γ̄ and its complement are connected in Γ(M) only through j_{0}, the two parts of x_{[k]}(i) — one supported by Γ̄, the other by its complement — have independent lives. But, for any j ∈ J̄, (x_{[k]}(i) + l′_{[k]}, E_{j}) = (x_{[k]} + π^{∗}(l̃′_{[k]}), E_{j}) = (x_{[k]}, E_{j}). Hence x_{[k]}(i) restricted on Γ̄ should be exactly y(i). Clearly, the restriction on the complement of Γ̄ should be z_{[k]}(i). Hence the first two statements follow. The last is the consequence of 3.5.7 together with the identity χ_{k_{r}}(E_{0}) = 1, for (l′_{[k]}, E_{0}) = (π^{∗}(l̃′_{[k]}), E_{0}) = 0.

5.5.9. τ_{a}. Corresponding to this and 3.5.6, we write τ_{a}(i) := χ_{k_{r}}(x_{[k]}(i)). With the notations of 5.5.8, notice that

\frac{iq − a}{qm_{f} + p} ≤ t + 1,

hence τ_{a}(i + 1) − τ_{a}(i) ≥ −1 for any i, and = −1 only in very special situations, namely if

\frac{(tm_{f} + i_{0})q − a}{qm_{f} + p} > t and i_{0} ∉ S.  (1)

In order to analyze when is this possible, we will consider sequences Seq(t) := {tm_{f} + i_{0} : 0 ≤ i_{0} < m_{f}} for fixed t ≥ 0. In such a sequence, notice that the very last element of ℕ \ S, namely μ − 1 = 2δ − 1 is strict smaller than m_{f} − 1 (a fact, which can be proved by induction over the number of Newton pairs), hence the complete set ℕ \ S sits in {0, …, m_{f} − 1}. Therefore, there exists in Seq(t) an i_{0} satisfying (1) if and only if

\frac{(tm_{f} + 2δ − 1)q − a}{qm_{f} + p} > t.

This is equivalent to t ≤ t_{a}, for t_{a} defined in 5.3.2. In other words, if i ≥ T_{0} := (t_{0} + 1)m_{f}, then τ_{a}(i + 1) ≥ τ_{a}(i), hence those values of τ_{a} provide no contribution in the graded root (cf. 3.1.3). Moreover, for t ∈ {0, …, t_{a}}, in Seq(t) one has:

Δ(i_{0}) := τ_{a}(tm_{f} + i_{0} + 1) − τ_{a}(tm_{f} + i_{0}) = \{ \begin{matrix}0 & if i_{0} ≤ (tp + a)/q, & and i_{0} ∉ S; \\ +1 & if i_{0} ≤ (tp + a)/q, & and i_{0} ∈ S; \\ −1 & if i_{0} > (tp + a)/q, & and i_{0} ∉ S; \\ 0 & if i_{0} > (tp + a)/q, & and i_{0} ∈ S. \end{matrix}

In particular, Δ(i_{0}) ≥ 0 for 0 ≤ i_{0} ≤ (tp + a)/q with exactly

A_{t} := #{γ ∈ S : γ ≤ (tp + a)/q}

times taking the value +1, otherwise zero; and Δ(i_{0}) ≤ 0 for i_{0} > (tp + a)/q with exactly

B_{t} := #{γ ∉ S : γ > (tp + a)/q}

times taking the value −1, otherwise zero. Notice that both A_{t} and B_{t} are strict positive (since 0 ∈ S, respectively 2δ − 1 ∉ S and 2δ − 1 > (tp + a)/q). This shows that

M_{t} := max_{0≤i_{0}<m_{f}} τ_{a}(tm_{f} + i_{0}) = τ_{a}(tm_{f}) + A_{t} = τ_{a}((t + 1)m_{f}) + B_{t},  (2)

and

M_{t} > max{ τ_{a}(tm_{f}), τ_{a}(tm_{f} + m_{f}) }.

Therefore, the graded root associated with the values {τ_{a}(i)}_{0≤i≤(t_{a}+1)m_{f}} is the same as the graded root associated with the values

τ_{a}(0), M_{0}, τ_{a}(m_{f}), M_{1}, τ_{a}(2m_{f}), M_{2}, …, τ_{a}(t_{a}m_{f}), M_{t_{a}}, τ_{a}(t_{a}m_{f} + m_{f}).

Finally notice, since #{γ ∉ S} = δ, one has δ −B_{t} = #{γ ∉ S : γ ≤ (tp+a)/q}, hence δ −B_{t} +A_{t} = [(tp + a)/q] + 1. Hence, by (2),

τ_{a}((t + 1)m_{f}) − τ_{a}(tm_{f}) = [\frac{tp + a}{q}] + 1 − δ.

Since τ_{a}(0) = 0, this gives τ_{a}(tm_{f}) inductively. Notice also that B_{t} = α_{[(tp+a)/q]}.

Clearly, the graded root associated with τ_{a} is the same as the graded root associated with τ̃_{a} : {0, 1, 2, …, 2t_{a} + 2} → ℤ, where τ̃_{a}(2t) := τ_{a}(tm_{f}) and τ̃_{a}(2t + 1) := M_{t}. This is the tau-function of 5.3.2 if we delete ˜ .

5.6 Examples

5.6.1. Example. Assume p = q = 1. In this case M is integral homology sphere; a = 0 and t_{0} = 2δ − 2 = μ − 2. In particular, the rank of ker U is μ. Moreover, r_{0} = δ(δ − 1) and τ_{0}(2t) = t(t − 2δ + 1)/2. The reader is invited to draw the graded root and verify that

HF^{+}(−M) = T_{0}^{+} ⊕ T_{0}(α_{δ−1}) ⊕ ⨁_{i=1}^{δ−1} T_{i(i+1)}( α_{i−1+δ})^{⊕2};

ker U = ⨁_{i=0}^{δ−1} ℤ_{(i^{2}+i)}^{2}.

5.6.2. Example. More generally, assume only that p = 1. The Heegaard Floer homology of the unique spin^{c}-structure is:

HF^{+}(−M) = T_{0}^{+} ⊕ T_{0}(α_{δ−1})^{⊕q} ⊕ ⨁_{i=1}^{(δ−1)q} T_{([l/q]+1)({l/q}q+l)}( α_{δ−1+⌈l/q⌉})^{⊕2}.

5.6.3. Remark. Apparently in 5.6.2, HF^{+}(−M) contains less information than the polynomial P, the above formula involves only the coefficients α_{ℓ} for ℓ ≥ δ − 1. But this is not the case. Indeed, since the Alexander polynomial Δ(t) is symmetric, one gets that

t^{μ−2}P (1/t) − P (t) = \frac{δ(1 + t^{μ−1}) − (1 + t + ⋯ + t^{μ−1})}{t − 1},

hence α_{μ−2−i} − α_{i} is a universal number. In particular, from {α_{ℓ}}_{ℓ≥δ−1} one can recover P.

This shows that from HF^{+}(−S_{−1/q}^{3}(K_{f})) one can recover both the integer q and the isotopy type of K_{f} ⊂ S^{3}. It looks that similar result is valid for general surgery coefficients as well.

5.6.4. Remark. In the above example it is striking a ℤ_{2}-symmetry of the graded root and of HF^{+}(−M). We will explain this for a = 0.

The point is that if the canonical characteristic element has only integral coefficients (i.e., if the graph is numerical Gorenstein), then all the theory associated with the integral cycles has a duality. This happens for example if p = 1; or if q = 1 and 2δ − 2 is divisible by p. In our case, in these situations, the function τ_{0} is stable with respect to the ℤ_{2} action i ↦ 2t_{0} + 2 − i, i.e. τ_{0}(i) = τ_{0}(2t_{0} + 2 − i). This induces a symmetry of the root and of the Heegaard Floer homology.

But, for general p/q, the graphs are not ‘numerical Gorenstein’. Even if they are, for general a, the symmetry may fail.

5.6.5. Example. Assume that K_{f} ⊂ S^{3} is the torus knot (4, 5) (i.e. g = 1 and (p_{1}, q_{1}) = (4, 5)). In this case δ = 6, μ = 12, S is generated by 4 and 5, hence

P (t) = 6 + 5t + 4t^{2} + 3t^{3} + 3t^{4} + 3t^{5} + 2t^{6} + t^{7} + t^{8} + t^{9} + t^{10}.

Then for some (p/q)-surgeries the corresponding graded roots are presented in the next Figure. [Here we did not draw all the vertices, only those ones which are either local minimums or supremums of pairs of local minimums; in fact, exactly these ones are given by the function τ_{a}. Also, the roots should be continued upward with one vertex in χ^{−1}(n) for each n ≥ 1.]

Recall that when we compute the grading of HF^{+}, the value of τ_{a} is doubled, and then shifted by r_{a}. The corresponding values of r_{a} in these five cases are:

30, 71/4, 49/4, \frac{(p + 10)^{2} − p}{4p}, 60.

In particular, in the first and fifth case (i.e. when p = 1) one has d(−M, σ_{0}) = 0, as we expected: the Heegaard Floer homology are written in 5.6.1, resp. 5.6.2. In the second and third cases (i.e. when p/q = 2 and a = 0, 1) the ℤ[U]-modules are:

( T_{−18}^{+} ⊕ T_{−16}(2)^{⊕2} ⊕ T_{−10}(1)^{⊕2} ⊕ T_{0}(1)^{⊕2} ) [71/4];
( T_{−12}^{+} ⊕ T_{−12}(3) ⊕ T_{−8}(1)^{⊕2} ⊕ T_{0}(1)^{⊕2} ) [49/4].

In the forth case it is

T_{−10}^{+} ⊕ T_{0}(1) [ \frac{(p + 1)^{2} − p}{4p} ].

[figure: 0; 0; 0; 0; 0; 0; 0; -5; 0; 0; 0; 0; -10; 0; 0; 0; 0; -15; 0; 0; 0; 0; -20; 0; 0; 0; 0; -25; 0; 0; 0; 0; -30; 0; p ≥ 11, q = 1; a = 0; p/q = 1; a = 0; p/q = 2; a = 0; p/q = 2; a = 1; p/q = 1/2; a = 0; τ_{a}(i)]

5.7 S_{−p/q}^{3}(K) as Kulikov graph–manifold.

5.7.1. Notice that 5.2.4 and 4.4.4 imply that the geometric genus p_{g} of any normal surface singularity (X, 0), whose link is diffeomorphic to M = S_{−p/q}^{3}(K), satisfies the inequality (SIn) of 4.4.2, namely

p_{g} ≤ −sw(M, σ_{can}) − (K^{2} + #J)/8.

This, via 5.3.2 and 5.3.4 reads as follows:

5.7.2. Theorem. Assume that the link M of (X, 0) is S_{−p/q}^{3}(K). Then

p_{g} ≤ ∑_{t≥0} α_{[tp/q]}.

In the spirit of the second part (SId) of 4.4.2, (see also [37]), we expect that the above upper bound for p_{g} is an optimal topological upper bound (when we run all the possible analytic structures on (X, 0)). In order to be more cautious (for some warnings see [26]), we formulate it as a question:

5.7.3. Question. Is it possible to find a special analytic singularity (X, 0) with M as its link, such that

p_{g} = ∑_{t≥0} α_{[tp/q]} ?

We invite the reader to formulate the similar inequality/identity for twisted line bundles, respectively for the other spin^{c} structures.

5.7.4. The above Question 5.7.3 can be made even more precise; i.e., we even propose a candidate for an analytic structure (X, 0). The point is that there is a natural family of normal surface singularities, the so-called Kulikov singularities, with link M. In general, the fact, that on the topological type of a singularity one can put the analytic structure of a Kulikov singularity, can be determined topologically from the combinatorics of the plumbing graphs. The corresponding graphs are called ‘Kulikov graphs’ (Karras called them ‘Kodaira graphs’). These particular graphs were studied intensively by Kulikov [16], Karras [15], Stevens [58] and others.

In the sequel, we exemplify the construction via M (in fact, proving that Γ(M) is a Kulikov graph). The construction also provides a possible analytic germ (X, 0) with special features.

Consider the plumbing graph:

[figure: j_{+}; −m_{f}; −1; j_{0}; Γ̄; Γ_{0}(M) :]

Clearly, the corresponding plumbed 3-manifold is S_{0}^{3}(K) (the manifold obtained by 0 surgery along K). The point is that this graph is negative semi-definite, hence by a theorem of Winters [67], there exists a holomorphic proper map h : Y → D (where D is a small disc, and Y is a smooth surface) such that h^{−1}(0) is a normal crossing divisor whose dual graph is Γ_{0}(M), and the generic fiber of h is a smooth curve. Then one blows up k_{1} generic points of the irreducible component of h^{−1}(0) corresponding to the vertex j_{+}. In this way one creates k_{1} (−1)–curves. Blow up k_{2} − 1 generic points of one of these new curves, and keep the other k_{1} − 1 unmodified; and continue this procedure. Then, the strict transform of h^{−1}(0) together with all the exceptional divisors form a curve. Notice that the dual graph of all the irreducible components which are not (−1)–curves is exactly Γ(M). These curves can be contracted providing a normal surface singularity (X, 0) with dual resolution graph Γ(M). The analytic structure of (X, 0) has some nice properties (e.g. h induces a nice function on it, for details see e.g. [58]). A singularity constructed in this way is called Kulikov singularity.

5.7.5. Question. Is it true that p_{g} = ∑_{t≥0} α_{[tp/q]} for a Kulikov singularity (X, 0) with minimal good resolution graph G(M) ?

6 Unicuspidal rational plane curves and S_{−d}^{3}(K).

6.1 The semigroup distribution property.

6.1.1. The main objects of this section are irreducible rational projective plane curves in the complex projective space ℙ^{2}. It is a very difficult open problem to characterize the local embedded topological types of local singular germs which can be realized as the singularities of such a projective curve C of degree d. This problem has a long and rich history providing many interesting compatibility properties connecting local invariants of the singular germs {(C, p_{i})}_{i} with some global invariants of C — like its degree, or the log-Kodaira dimension of ℙ^{2} \ C, etc. For a (non-complete) list of some of these restrictions, see e.g. [8, 9, 10] and the references therein. The simplest one is the genus formula: the sum of the Milnor numbers of the local plane curve singularities {(C, p_{i})}_{i} should be (d − 1)(d − 2). (But, this is far to be enough for the characterization.)

The article [8] proposes a new compatibility property — valid for rational cuspidal curves C. (A ‘cusp’ means a locally irreducible singularity, C is ‘cuspidal’ if it has only cusps.) Its formulation is the following. Consider a collection (C, p_{i})_{i=1}^{ν} of cusps, let Δ_{i}(t) be their characteristic polynomials, and set Δ(t) := ∏_{i} Δ_{i}(t). Its degree is 2δ, where δ is the sum of the delta-invariants of the singular points. (By the above mentioned genus formula: 2δ = (d − 1)(d − 2).) Write Δ(t) as 1 + (t − 1)δ + (t − 1)^{2}Q(t) for some polynomial Q(t), cf. 5.2.2. Let c_{l} be the coefficient of t^{(d−3−l)d} in Q(t) for any l = 0, …, d − 3. (In the notations of 5.2.2 one has c_{l} = α_{d(d−3−l)}.)

6.1.2. Conjecture A. [8] Let (C, p_{i})_{i=1}^{ν} be a collection of cusps, such that 2δ = (d − 1)(d − 2) for some integer d. If (C, p_{i})_{i=1}^{ν} can be realized as the local singularities of a degree d (automatically rational and cuspidal) projective plane curve, then

c_{l} ≤ (l + 1)(l + 2)/2 for all l = 0, …, d − 3.

In fact, the integers n_{l} := c_{l} − (l + 1)(l + 2)/2 are symmetric: n_{l} = n_{d−3−l}; and n_{0} = n_{d−3} = 0 automatically. We also mention that strict inequality (with ν > 1) may occur, cf. [8].

Let κ̄(ℙ^{2} \ C) denote the logarithmic Kodaira dimension of ℙ^{2} \ C. The main result of [8] is:

6.1.3. Theorem. [8] If κ̄(ℙ^{2} \ C) ≤ 1, then the above conjecture A is true (with n_{l} = 0).

On the other hand, rather surprisingly, in the unicuspidal case one can show the following.

6.1.4. Proposition. [8] If ν = 1 then c_{l} ≥ (l + 1)(l + 2)/2 for 0 ≤ l ≤ d − 3.

Therefore, conjecture 6.1.2 in this case can be reformulated as follows:

6.1.5. Conjecture B1. With the notations of 6.1.2, if ν = 1, then

c_{l} = (l + 1)(l + 2)/2 for all l = 0, …, d − 3.

If ν = 1, the characteristic polynomial Δ of (C, p) ⊂ (ℙ^{2}, p) is a complete embedded topological invariant of this germ, similarly as the semigroup S ⊂ ℕ. Hence, one can reformulate conjecture B1 in terms of S and d. It turns out that the collection of vanishings of all the coefficients n_{l} is replaced by a very precise and mysterious distribution of the elements of the semigroup with respect to the (half–open) intervals I_{l} := ( (l − 1)d, ld]:

6.1.6. Conjecture B2. Assume that ν = 1. Then for any l > 0, the interval I_{l} contains exactly min{l + 1, d} elements from the semigroup.

In other words, for every rational unicuspidal plane curve C of degree d, the above conjecture is equivalent with the conjectural identity D(t) ≡ 0, where:

D(t) := ∑_{k∈S} t^{⌈k/d⌉} − (1 + 2t + ⋯ + (d − 1)t^{d−2} + d(t^{d−1} + t^{d} + t^{d+1} + ⋯)).  (DP)

For an explanation of the equivalences of conjectures B1 and B2, see 6.2.4. Here we only recall the key connection between the coefficients c_{l} and the semigroup S:

c_{l} = #{k ∈ S; k ≤ ld}.

This follows from 5.2.2 and from the fact that for 0 ≤ k < μ one has: k ∈ S if and only if μ−k−1 ∉ S.

6.2 The semigroup distribution property and surface singularities.

6.2.1. Superisolated singularities. The theory of isolated hypersurface surface singularities ‘contains’ in a canonical way the theory of complex projective plane curves via the family of superisolated singularities. These singularities were introduced by I. Luengo in [25].

A hypersurface singularity f : (ℂ^{3}, 0) → (ℂ, 0), f = f_{d} + l^{d+1} (where f_{d} is homogeneous of degree d and l is linear) is superisolated if the projective plane curve C := {f_{d} = 0} ⊂ ℙ^{2} is reduced, and its singularities {p_{i}}_{i=1}^{ν} are not situated on {l = 0}. The equisingular type of f depends only on f_{d}, i.e. only on the projective curve C ⊂ ℙ^{2}. In particular, all the invariants (of the equisingular type) of f can be determined from the invariants of the pair (ℙ^{2}, C).

6.2.2. Assume in the sequel that C is irreducible, rational with exactly one singular point (C, p).

In the next discussion we follow [26, 8]. Let μ = 2δ, S and Δ be the local invariants of (C, p). Set Δ̄(t) := t^{−δ}Δ(t). Let K ⊂ S^{3} be the local embedded link of (C, p). Then, one shows that M = S_{−d}^{3}(K); in particular, H_{1}(M, ℤ) = ℤ_{d}.

In fact, if we blow up the maximal ideal of the singular point of {f = 0} we get its minimal resolution, which contains one irreducible exceptional divisor, which is isomorphic to C and has self–intersection −d. From this picture the above identification of M also follows. But from this fact one also has that the invariant K^{2} + #J (of M) equals 1 − d(d − 2)^{2}, hence it depends only on d. One also computes p_{g} = d(d − 1)(d − 2)/6.

Before we continue our discussion, we recall that the Property (SId) (conjectured for ‘nice’ singularities), applied for the germ f and the canonical spin^{c} structure reads as follows:

6.2.3. Property SWC. [37] −sw(M, σ_{can}) − (K^{2} + #J)/8 = p_{g}.

In our case, since p_{g} and K^{2} + #J depend only on d, the validity of SW C (6.2.3) would impose serious restriction on the link M = S_{−d}^{3}(K), or equivalently on the local knot K ⊂ S^{3}, which can be measured by the local invariants Δ or S. In this subsection we will explain this fact via the interpretation of sw(M, σ_{can}) by the Reidemeister-Turaev torsion, cf. 2.3.4. One shows, cf. [26], that

T_{M,σ_{can}}(1) = \frac{1}{d} ∑_{ξ^{d}=1≠ξ} \frac{Δ(ξ)}{(ξ − 1)^{2}} and λ(M) = −\frac{Δ̄(t)′′(1)}{2} + \frac{(d − 1)(d − 2)}{24}.

Consider also (motivated partly by the above formula of the torsion):

R(t) := \frac{1}{d} ∑_{ξ^{d}=1} \frac{Δ(ξt)}{(1 − ξt)^{2}} − \frac{1 − t^{d^{2}}}{(1 − t^{d})^{3}} .

Similarly,

N(t) := ∑_{l=0}^{d−3} (c_{l} − \frac{(l + 1)(l + 2)}{2})t^{d−3−l}; and D(t) := ∑_{k∈S} t^{⌊k/d⌋} − \frac{1 − t^{d}}{(1 − t)^{2}} .

Notice that this D(t) agrees with the one defined in (DP) in the previous subsection. In [8] the following facts are verified:

• R(t) = D(t^{d})/(1 − t^{d}) = N(t^{d});

• N(t) (hence R(t) too) has non-negative coefficients;

• R(1) = −p_{g} − sw(M, σ_{can}) − (K^{2} + #J)/8.

In particular, we have the equivalence of the ‘Seiberg-Witten invariant conjecture’ with the ‘semigroup distribution property’:

6.2.4. Theorem. Under the assumptions (6.2.2) the following facts are equivalent:

(a) R(1) = 0, i.e. (the conjectured) Property SW C (6.2.3) is true (for the above germ f);

(b) R(t) ≡ 0;

(c) N(t) ≡ 0, i.e. Conjecture B1 (6.1.5) is true;

(d) D(t) ≡ 0, i.e. Conjecture B2 (6.1.6) is true.

6.3 The semigroup distribution property and graded roots/Heegaard Floer homology.

6.3.1. Assume that we are in the situation of 6.2.2. In this subsection we will compare the invariants of the link M with the corresponding invariants of the Seifert 3–manifold Σ(d, d, d + 1) — this is the link of the hypersurface Brieskorn singularity x^{d} + y^{d} + z^{d+1} = 0. This connection is very surprising, the possible conceptual relationship between these two spaces (together, hopefully, with the proof of conjectures listed in 6.2.4) will be explained in a forthcoming manuscript.

6.3.2. Theorem.[10] The following facts are equivalent:

(a) Conjecture B1 (6.1.5) is true,

(b) The canonical graded roots of S_{−d}^{3}(K) and Σ(d, d, d + 1) are the same.

(c) The canonical Heegaard-Floer homologies of −S_{−d}^{3}(K) and −Σ(d, d, d+1) are the same modulo a grading shift, namely:

HF^{+}(−S_{−d}^{3}(K), σ_{can})[1 − d(d − 2)^{2}] = HF^{+}(−Σ(d, d, d + 1), σ_{can})[−d(d − 1)(d − 3)].

(d)

(sw(M, σ_{can}) + \frac{K^{2} + #J}{8} ) |_{M=S_{−d}^{3}(K)} = (sw(M, σ_{can}) + \frac{K^{2} + #J}{8} ) |_{M=Σ(d,d,d+1)}.

The proof is given in several steps. The main point is that both 3–manifolds S_{−d}^{3}(K) and Σ(d, d, d+ 1) are almost rational, cf. (2.7). In particular, in both cases, the canonical graded root can be determined via the ‘τ-construction’ 3.5.6. In the first case this is done explicitly in 5.3.2, while for the second case, see [34], last section.

6.3.3. Example. Let us rewrite 5.3.2 for the case M = S_{−d}^{3}(K) and for the canonical spin^{c} structure: a = 0, q = q′ = 1, p = d. Set c_{l} := α_{(d−3−l)d} and define τ : {0, 1, …, 2d − 4} → ℤ by

τ(2l) = \frac{l(l−1)}{2}d − l(δ − 1), τ(2l + 1) = τ(2l + 2) + c_{d−3−l}.

Then (R_{can}, χ_{can}) = (R_{τ} , χ_{τ}).

6.3.4. Example. Consider the Seifert manifold Σ(d, d, d + 1). Its canonical graded root is the following (cf. [34]). For any 0 ≤ l ≤ d − 3 write c_{l}^{u} := (l + 1)(l + 2)/2, and 2δ := (d − 1)(d − 2). Then define τ^{u} : {0, 1, …, 2d − 4} → ℤ by

τ^{u}(2l) = \frac{l(l−1)}{2}d − l(δ − 1), τ^{u}(2l + 1) = τ^{u}(2l + 2) + c_{d−3−l}^{u}.

Then (R_{can}, χ_{can}) = (R_{τ^{u}}, χ_{τ^{u}}).

Notice the shocking similarities of 6.3.3 and 6.3.4: the graded roots associated with S_{−d}^{3}(K) and Σ(d, d, d + 1) coincide exactly when c_{l} = c_{l}^{u} for all l, a system of equalities subject of Conjecture B1. This proves (a) ⇔ (b). (b) ⇒ (c) follows from 3.5.2, since in both cases

HF^{+}(−M, σ_{can})[(K^{2} + #J)/4] = ℍ(R_{can}, χ_{can}),

where the shift in the case of M = S_{−d}^{3}(K) is K^{2} + #J = 1 − d(d − 2)^{2}, while for M = Σ(d, d, d + 1) is K^{2} + #J = −d(d − 1)(d − 3). This, and the corresponding definitions show (c) ⇒ (d).

Since the Property SW C (6.2.3) is true for the Brieskorn singularity f_{BR} := x^{d} + y^{d} + z^{d+1} (cf. [38]), and the geometric genus of the superisolated singularity f equals the geometric genus of f_{BR} (both equal d(d − 1)(d − 2)/6), this last identity (d) is also equivalent with the validity of the Property SW C (6.2.3) for f — a fact, which is equivalent with (a) by 6.2.4.

6.3.5. Remark. For (d) ⇒ (b) one can also argue as follows. Regarding the Seiberg-Witten invariant of M = S_{−d}^{3}(K), by 5.3.2 and 5.3.4 one has

−sw(M, σ_{can}) − \frac{K^{2} + #J}{8} = ∑_{l≥0} τ(2l + 1) − τ(2l + 2) = ∑_{l≥0} c_{l},

and there is a similar formula for M = Σ(d, d, d + 1) with the obvious replacements.

6.3.6. Example. Assume that d = 5 and C is unicuspidal whose singular point has only one Puiseux pair (a, b) with a < b. Then by the genus formula the possible values of (a, b) are (4, 5), (3, 7) and (2, 13). It turns out that the first and the third cases can be realized, while the second not. The corresponding canonical graded roots (together with the root of Σ(5, 5, 6)) are drawn in the next picture.

[figure: 0; −1; −2; −3; −4; −5; Σ(5, 5, 6); S_{−5}^{3}(T_{3,7}); S_{−5}^{3}(T_{2,13}), S_{−5}^{3}(T_{4,5}); χ]

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