LATTICE COHOMOLOGY OF NORMAL SURFACE SINGULARITIES

ANDRÁS NÉMETHI^{∗}

Dedicated to Professor Heisuke Hironaka on his 77th birthday.

Abstract. For any negative definite plumbed 3–manifold M we construct from its plumbed graph a graded ℤ[U]–module. This, for rational homology spheres, conjecturally equals the Heegaard–Floer homology of Ozsváth and Szabó, but it has even more structure. If M is a complex singularity link then the normalized Euler-characteristic can be compared with the analytic invariants. The Seiberg–Witten Invariant Conjecture of [16, 13] is discussed in the light of this new object.

1. Introduction

The article is a symbiosis of singularity theory and low-dimensional topology. Accordingly, it is preferable to separate its goals in two categories.

From the point of view of 3–dimensional topology, the article contains the following main result. For every negative definite plumbed 3–manifold it constructs a graded ℤ[U]–module from the combinatorics of the plumbing graph. This for rational homology spheres conjecturally equals the Heegaard–Floer homology of Ozsváth and Szabó. In fact, it has more structure (e.g. instead of a ℤ_{2}, odd/even grading it has a ℤ grading like a usual homology, see (5.2.6)(c)). The existence of these extra structures for arbitrary 3–manifolds might be an interesting subject for further investigation.

The motivations and aims from the point of view of singularity theory are the following.

In [16] 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. The conjecture generalized a conjecture of Neumann and Wahl [20] which formulated the relationship for complete intersection singularities with integral homology sphere links. The conjecture [16] was verified in different cases, see [3, 12, 16, 17, 18, 19].

Since the Seiberg-Witten theory provides a rational number for any spin^{c}–structure, it was a natural challenge to search for a complete set of conjecturally valid identities, which involve all spin^{c}–structures. The preprint [13] proposed such identities, connecting the sheaf–cohomology of holomorphic line bundles associated with the analytic type of the singularity with the Seiberg–Witten invariants of the link. The identities were supported by a proof valid for rational singularities.

But, a few months later, [10] appeared with a list of counterexamples. This posed a lot of questions: what kind of guiding principles were wrongly interpreted in the original conjectures? How can one ‘correct’ them?

The present manuscript aims to answer some of them.

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

14E15, 32S45, 57M25.

Key words and phrases. normal surface singularities, line bundles, geometric genus, rational singularities, elliptic singularities, almost rational singularities, Seiberg-Witten invariant, Heegaard-Floer homology, Seiberg-Witten Invariant Conjecture.

Communicated by S. Mukai. Received February 19, 2007. Revised September 6, 2007.

∗A. Rényi Institute of Mathematics, 1053 Budapest, Reáltanoda u. 13-15, Hungary; nemethi@renyi.hu; http://www.renyi.hu/˜ nemethi.

The author is partially supported by OTKA Grants.

2 András Némethi

First, let us recall in short the original conjecture (for canonical spin^{c}–structure). One fixes a topological type (identified by a rational homology sphere link) and considers the Seiberg–Witten invariant of this link (normalized with a certain invariant K^{2} + s, see below). About this the conjecture predicted two things: First, that it is an upper bound for the geometric genus of all the possible analytic structures supported by the fixed topological type. Second, that this bound is optimal, and it is realized by all ℚ–Gorenstein analytic structures.

Well, both expectations were wrong, but the nature of the two errors are completely different. Regarding the second part, the ‘Seiberg–Witten invariant identity’, the error can be localized easily. Indeed, the conjecture was over–optimistic: the identity is not valid for every ℚ–Gorenstein singularity. Nevertheless, it is proved for large classes of singularities, and we expect that the list will be continued. Hence, the form of the identity shouldn’t be modified, just we expect its validity for a subclass of ℚ–Gorenstein singularities. At this moment, it is hopeless to identify exactly this subclass, nevertheless, in [3] it has a description exclusively in terms of the analytic structure — independently of the Seiberg–Witten theory; and [23] suggests that it can be identified by some vanishing properties.

In fact, we were more concern about the inequality part: Laufer type computation sequences identified a possible topological upper bound for the geometric genus, which in all cases explicitly analysed (at the time of [16, 13]) coincided with the Seiberg–Witten invariant, and the computation sequence technique resonated perfectly with the theory of Ozsváth and Szabó from [25]. Then, which part of this line of argument fails in general? The present article gives the following answer: There exists a cohomology theory {ℍ^{q}}_{q≥0}, such that its normalized Euler–characteristic (conjecturally) equals the Seiberg–Witten invariant. On the other hand, its 0^{th} normalized ‘Betti–number’ (or invariants related with it) serves as topological upper bound for the geometric genus (and fits with computation sequence constructions). In simpler cases (e.g. for rational, elliptic or star–shaped resolution graphs) one has a vanishing ℍ^{q} = 0 for all q ≥ 1, hence the Seiberg–Witten invariant was able to serve as an upper bound. But, in general, this is not the case: the geometric genus of those analytic structures for which the ‘Seiberg–Witten invariant identity’ holds, are not extremal.

The article starts with the construction of this cohomology theory: the lattice cohomology. Here, we do not restrict ourselves to the rational homology sphere case. The construction provides from the plumbing graph of the link M (or, from the associated intersection lattice) a graded ℤ[U]–module ℍ^{q}(M, σ) for each q ≥ 0, and for all torsion spin^{c}–structures σ of M.

We emphasize and exemplify more the case ℍ^{0}. ℍ^{0}, as a combinatorial ℤ[U]–module associated with the link, is not new in the literature: it was considered by Ozsváth and Szabó in [25] in Heegaard–Floer homology computations of some special plumbed 3–manifolds (under the notation ℍ^{+}). Later, in [12], the author computed ℍ^{0} for a larger class of 3–manifolds (‘almost rational’ graph–manifolds). In the present article, in section 4, we prove similar characterization and structure results for ℍ^{∗} valid for rational, elliptic and almost rational graphs. Moreover, we analyze examples with ℍ^{1} ≠ 0 too. Section 5 connects ℍ^{∗} with the Heegaard–Floer homology.

Section 6 deals with the theory of line bundles associated with surface singularities. (It contains some parts from the unpublished [13] and from the lecture notes [15]. Some similar h^{1}–computations for the case of rational singularities were also found independently by T. Okuma [22].) In this section we determine a topological upper bound for the dimension of the sheaf–cohomologies of these line bundles in terms of their Chern classes. The description sits in ℍ^{0}.

The last section 7 presents the ‘Seiberg–Witten invariant conjecture’ (the unmodified conjectured identities), with examples and more comments.

Acknowledgement. The author was strongly influenced by the articles of Péter Ozsváth and Zoltán Szabó, and benefited enormously from private conversations with them. In fact, the idea of the existence of a cohomology theory ℍ^{∗} (for plumbed rational homology spheres) was formulated by them, and Conjecture (5.2.4) is part of their program too. The author also thanks the help of András Stipsicz in different aspects of Heegaard–Floer theory.

The first version of the manuscript was read by my colleagues Gábor Braun and Gyula Lakos, and they provided valuable comments.

Lattice cohomology 3

2. Preliminaries.

2.1. Negative definite plumbing graphs.

2.1.1. Let (X, 0) be a complex analytic normal surface singularity with link M. Fix a sufficiently small Stein representative X of the germ (X, 0) and let π : X̃ → X be a good resolution of the singular point 0 ∈ X. Let E := π^{−1}(0) be the exceptional divisor with irreducible components {E_{j}}_{j∈J} and write Γ(π) for the dual resolution graph associated with π. Recall that Γ(π) is connected and the intersection matrix I := {(E_{j}, E_{i})}_{j,i} is negative definite. We write e_{j} for E_{j}^{2}, g_{j} for the genus of E_{j} (j ∈ J), and g := ∑_{j} g_{j}. Moreover, let c be the number of independent cycles in (the topological realization of) Γ. E.g., c = 0 if and only if Γ(π) is a tree. The rank of H_{1}(M, ℤ) is c + 2g. Hence, M is a rational homology sphere (i.e. H_{1}(M, ℚ) = 0) if and only if g = c = 0.

2.1.2. Since π identifies ∂X̃ with M, the graph Γ(π) can be viewed as a plumbing graph and M as the associated S^{1}-plumbed manifold. In the sequel Γ will denote either a good resolution graph as above, or a negative definite plumbing graph of M. Similarly, X̃ denotes either the space of a good resolution, or the oriented 4-manifold obtained by plumbing disc-bundles corresponding to Γ.

2.2. The combinatorics of the plumbing.

2.2.1. Definition. The lattices L and L′. The image of the boundary operator ∂ : H_{2}(X̃, M, ℤ) → H_{1}(M, ℤ) is the torsion subgroup H of H_{1}(M, ℤ). The exact sequence of ℤ-modules

(1)  0 → L →^{i} L′ → H → 0

will stand for the homological exact sequence

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

(or for its Poincaré dual). Here L is freely generated by the homology classes {E_{j}}_{j∈J} and is equipped with the intersection form (·, ·). 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 we will regard Hom(L, ℤ) as 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. Hence, the exact sequence (1) can be recovered completely from the lattice L.

2.2.2. 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}.

The unique rational cycle K ∈ L′ which satisfies the system (of adjunction relations) (K, E_{j}) = −(E_{j}, E_{j}) − 2 + 2g_{j} for all j is called the canonical cycle. Then Char = K + 2L′. There is a natural action of L on Char by l ∗ k := k + 2l 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.

If X̃ is a 4–manifold as above, then H^{2}(X̃, ℤ) has no 2–torsion. Therefore, 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. [4], 2.4.16). On the other hand, the spin^{c}– structures on M form an H_{1}(M, ℤ) torsor. In the image of the restriction Spin^{c}(X̃) → Spin^{c}(M) are exactly those spin^{c}–structures of M whose Chern classes are restrictions L′ → H^{2}(M, ℤ) = H_{1}(M, ℤ), i.e. are torsion elements sitting in H. We call them torsion structures, and we denote them by Spin_{t}^{c}(M). One has an identification of Spin_{t}^{c}(M) with the set of L–orbits of Char, and this identification is compatible with the action of H on both sets. In the sequel, we think about Spin_{t}^{c}(M) by this identification: any torsion spin^{c}-structure of M will be represented by an orbit [k] := k + 2L ⊂ Char. The canonical spin^{c}–structure (is torsion and) corresponds to [K].

We write Ĥ for the Pontrjagin dual Hom(H, S^{1}) of H. One has a natural isomorphism

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

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.

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}.

2.2.4. Liftings. 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}.

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. [12], 5.4). By their definitions l̄′_{ne}(h) = −l′_{ne}(−h).

For some h, l̄′_{ne}(h) might be situated in Q, but, in general, this is not the case. In general, the characterization of all the elements l̄′_{ne}(h) is not simple (see e.g. [12]).

2.2.5. 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 the interpretation of χ_{k} as (twisted) Riemann-Roch formula, consider the following. Let X̃ be a resolution as in (2.1.1), and fix a holomorphic line bundle L ∈ P ic(X̃), and write c_{1}(L) = l′ ∈ L′ for its Chern class. Set k := K − 2l′ ∈ Char. For any l ∈ L with l > 0 one defines the sheaf O_{l} := OX̃/OX̃(−l) supported by E (see e.g. 6.1.1). 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}) = χ_{k}(l).

3. The lattice cohomology associated with L.

3.1. Lattice cohomology associated with ℤ^{s} and a system of weights.

3.1.1. We consider a free ℤ–module, with a fixed basis {E_{j}}_{j}, denoted by ℤ^{s}. It is also convenient to fix a total ordering of the index set J, which in the sequel will be denoted by {1, …, s}.

Our goal is to define a graded ℤ[U]–module associated with the pair (ℤ^{s}, {E_{j}}_{j}) and a system of weights, which will be introduces in (3.1.4). First we set some notations regarding ℤ[U]–modules.

3.1.2. ℤ[U]-modules. Consider the graded ℤ[U]–module T := ℤ[U, U^{−1}], and (following [25]) 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, the quotient of ℤ⟨U^{−(n−1)}, U^{−(n−2)}, …, 1, U, …⟩ by U · ℤ[U] (with the same grading) defines the graded module T_{0}(n). Hence, T_{0}(n), as a ℤ–module, is freely 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 (by ℚ) 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]. (Hence, for m ∈ ℤ, T_{2m}^{+} = ℤ⟨U^{−m}, U^{−m−1}, …⟩.)

3.1.3. The cochain complex. ℤ^{s} ⊗ ℝ has a natural cellular decomposition into cubes. The set of zero–dimensional cubes is provided by the lattice points ℤ^{s}. Any l ∈ ℤ^{s} and subset I ⊂ J of cardinality q defines a q–dimensional cube, which has its vertices in the lattice points (l+∑_{j∈I′} E_{j})_{I′}, where I′ runs over all subsets of I. On each such cube we fix an orientation. This can be determined, e.g., by the order (E_{j_{1}}, …, E_{j_{q}}), where j_{1} < ⋯ < j_{q}, of the involved base elements {E_{j}}_{j∈I}. The set of oriented q–dimensional cubes defined in this way is denoted by Q_{q} (0 ≤ q ≤ s).

Let C_{q} be the free ℤ–module generated by oriented cubes □_{q} ∈ Q_{q}. Clearly, for each □_{q} ∈ Q_{q}, the oriented boundary ∂□_{q} has the form ∑_{k} ε_{k} □_{q−1}^{k} for some ε_{k} ∈ {−1, +1}. Here, in this sum, we write only those (q − 1)–cubes which appear with non–zero coefficient. These are called faces of □_{q}.

It is clear that ∂ ∘ ∂ = 0. But, obviously, the homology of the chain complex (C_{∗}, ∂) (or, of the cochain complex (Hom_{ℤ}(C_{∗}, ℤ), δ)) is not very interesting: it is just the (co)homology of ℝ^{s}. A more interesting (co)homology can be constructed as follows. For this, we consider a set of compatible weight functions w_{q} : Q_{q} → ℤ (0 ≤ q ≤ s).

3.1.4. Definition. A set of functions w_{q} : Q_{q} → ℤ (0 ≤ q ≤ s) is called a set of compatible weight functions if the following hold:

(a) For any integer k ∈ ℤ, the set w_{0}^{−1}( (−∞, k] ) is finite;

(b) for any □_{q} ∈ Q_{q} and for any of its faces □_{q−1} ∈ Q_{q−1} one has w_{q}(□_{q}) ≥ w_{q−1}(□_{q−1}). (In the sequel sometimes we will omit the index q of w_{q}.)

Assume that we already fixed a set of compatible weight functions {w_{q}}_{q}. Then we set F^{q} := Hom_{ℤ}(C_{q}, T_{0}^{+}). Notice that F^{q} is, in fact, a ℤ[U]–module by (p ∗ ϕ)(□_{q}) := p(ϕ(□_{q})) (p ∈ ℤ[U]). Moreover, F^{q} has a ℤ–grading: ϕ ∈ F^{q} is homogeneous of degree d ∈ ℤ if for each □_{q} ∈ Q_{q} with ϕ(□_{q}) ≠ 0, ϕ(□_{q}) is a homogeneous element of T_{0}^{+} of degree d − 2 · w(□_{q}). (In fact, the grading is 2ℤ–valued; hence, the reader interested only in the present construction may divide all the degrees by two. Nevertheless, we prefer to keep the present form in our presentation because of its resonance with the Heegaard–Floer homology of the link.)

Next, we define δ_{w} : F^{q} → F^{q+1}. For this, fix ϕ ∈ F^{q} and we show how δ_{w}ϕ acts on a cube □_{q+1} ∈ Q_{q+1}. First write ∂□_{q+1} = ∑_{k} ε_{k}□_{q}^{k}, then set

(δ_{w}ϕ)(□_{q+1}) := ∑_{k} ε_{k} U^{w(□_{q+1})−w(□_{q}^{k})} ϕ(□_{q}^{k}).

3.1.5. Lemma. δ_{w} ∘ δ_{w} = 0, i.e. (F^{∗}, δ_{w}) is a cochain complex.

Proof. With the obvious notations, (δ_{w}^{2}ϕ)(□_{q+2}^{j}) equals

∑_{k} ε_{k}^{j}U^{w(□_{q+2}^{j})−w(□_{q+1}^{k})} ∑_{l} ε_{l}^{k}U^{w(□_{q+1}^{k})−w(□_{q}^{l})}ϕ(□_{q}^{l}) = ∑_{l} U^{w(□_{q+2}^{j})−w(□_{q}^{l})} ( ∑_{k} ε_{k}^{j}ε_{l}^{k} ) ϕ(□_{q}^{l}).

But, for any l, ∑_{k} ε_{k}^{j}ε_{l}^{k} = 0 since ∂^{2} = 0. □

3.1.6. In fact, (F^{∗}, δ_{w}) has a natural augmentation too. Indeed, set m_{w} := min_{l∈ℤ^{s}} w_{0}(l) and choose l_{w} ∈ ℤ^{s} such that w_{0}(l_{w}) = m_{w}. Then one defines the ℤ[U]–linear map

ϵ_{w} : T_{2m_{w}}^{+} ⟶ F^{0}

such that ϵ_{w}(U^{−m_{w}−s})(l) is the class of U^{−m_{w}+w_{0}(l)−s} in T_{0}^{+} for any integer s ≥ 0.

3.1.7. Lemma. ϵ_{w} is injective, and δ_{w} ∘ ϵ_{w} = 0.

Proof. Since ϵ_{w}(U^{−m_{w}−s})(l_{w}) = U^{−s}, the injectivity is clear. Take □ ∈ Q_{1} with ∂□ = a − b. Then

(δ_{w}ϵ_{w})(t)(□) = U^{w(□)−w(a)}ϵ_{w}(t)(a) − U^{w(□)−w(b)}ϵ_{w}(t)(b) = U^{w(□)}t − U^{w(□)}t = 0. □

We invite the reader to verify that ϵ_{w} and δ_{w} are morphisms of ℤ[U]–modules, and are homogeneous of degree zero.

3.1.8. Definitions. The homology of the cochain complex (F^{∗}, δ_{w}) is called the lattice cohomology of the pair (ℝ^{s}, w), and it is denoted by ℍ^{∗}(ℝ^{s}, w). The homology of the augmented cochain complex

0 ⟶ T_{2m_{w}}^{+} ⟶^{ϵ_{w}} F^{0} ⟶^{δ_{w}} F^{1} ⟶^{δ_{w}} …

is called the reduced lattice cohomology of the pair (ℝ^{s}, w), and it is denoted by ℍ_{red}^{∗}(ℝ^{s}, w). If the pair (ℝ^{s}, w) is clear from the context, we omit it from the notation. Clearly, for any q ≥ 0, both ℍ^{q} and ℍ_{red}^{q} admit an induced graded ℤ[U]–module structure and ℍ^{q} = ℍ_{red}^{q} for q > 0. Moreover, the ℤ–grading of F^{q} induces a ℤ–grading on ℍ^{q} and ℍ_{red}^{q}; the homogeneous part of degree d is denoted by ℍ_{d}^{q}, or ℍ_{red,d}^{q}.

It is easy to see that ℍ^{∗}(ℝ^{s}, w) depends essentially on the choice of w.

3.1.9. Lemma. One has a graded ℤ[U]–module isomorphism ℍ^{0} = T_{2m_{w}}^{+} ⊕ ℍ_{red}^{0}.

Proof. Consider the isomorphism U^{−m_{w}} : T_{0}^{+} → T_{2m_{w}}^{+}. Then define r_{w} : ℍ^{0} → T_{2m_{w}}^{+} by r_{w}(ϕ) := U^{−m_{k}}ϕ(l_{w}). Since r_{w} ∘ ϵ_{w} = 1, the exact sequence 0 → T_{2m_{w}}^{+} ⟶^{ϵ_{w}} ℍ^{0} → ℍ_{red}^{0} → 0 splits. □

3.1.10. Next, we present another realization of the modules ℍ^{∗}.

3.1.11. Definitions. For each n ∈ ℤ, define S_{n} = S_{n}(w) ⊂ ℝ^{s} as the union of all the cubes □_{q} (of any dimension) with w(□_{q}) ≤ n. Clearly, S_{n} = ∅, whenever n < m_{w}. For any q ≥ 0, set

𝕊^{q}(ℝ^{s}, w) := ⊕_{n≥m_{w}} H^{q}(S_{n}, ℤ).

Then 𝕊^{q} is ℤ (in fact, 2ℤ)–graded, the d = 2n–homogeneous elements 𝕊_{d}^{q} consist of H^{q}(S_{n}, ℤ). Also, 𝕊^{q} is a ℤ[U]–module; the U–action is given by the restriction map r_{n+1} : H^{q}(S_{n+1}, ℤ) → H^{q}(S_{n}, ℤ). Namely, U ∗ (α_{n})_{n} = (r_{n+1}α_{n+1})_{n}. Moreover, for q = 0, the fixed base–point l_{w} ∈ S_{n} provides an augmentation (splitting) H^{0}(S_{n}, ℤ) = ℤ ⊕ H̃^{0}(S_{n}, ℤ), hence an augmentation of the graded ℤ[U]–modules

𝕊^{0} = T_{2m_{w}}^{+} ⊕ 𝕊_{red}^{0} = (⊕_{n≥m_{w}}ℤ) ⊕ (⊕_{n≥m_{w}} H̃^{0}(S_{n}, ℤ)).

3.1.12. Theorem.

(a) There exists a graded ℤ[U]–module isomorphism, compatible with the augmentations:

ℍ^{∗}(ℝ^{s}, w) = 𝕊^{∗}(ℝ^{s}, w).

(b) For any degree d, there exists an integer N(d) ≥ 0, such that

ℍ_{red,d}^{∗} ∩ im(U^{N(d)}) = 0.

(c) For any n one has U^{n−m_{w}+1}ℍ_{2n}^{∗} = 0. If there exists N such that S_{n} is contractible for any n ≥ N, then U^{N−m_{w}}ℍ_{red}^{∗} = 0.

Proof. (a) Let F_{d}^{q} be the set of d = 2n–homogeneous elements ϕ ∈ F^{q}. Since δ_{w} is homogeneous of degree zero, (F_{d}^{∗}, δ_{w}) is a complex. Let (C^{∗}(S_{n}), δ) be the usual cochain complex of S_{n}. Then the two complexes can be naturally identified. Indeed, take ϕ ∈ F_{d}^{q}. Then, for any □_{q}, ϕ(□_{q}) has the form a_{ϕ}(□_{q})U^{w(□_{q})−n}. Hence a_{ϕ}(□_{q}) ∈ ℤ is well–defined for any q–cube □_{q} of S_{n}, and the correspondence ϕ ↦ a_{ϕ} realizes the bijection F_{d}^{∗} → C^{∗}(S_{n}).

Since H̃_{q}(ℝ^{s}, ℤ) = 0, for any n there exists N such that H̃_{q}(S_{n}) → H̃_{q}(S_{n+N}) is trivial. (b) is the dual statement of this. (c) follows from (a).

□

3.1.13. Remark. Although ℍ_{red}^{∗}(ℝ^{s}, w) has finite ℤ–rank in any fixed homogeneous degree, in general, it is not finitely generated over ℤ[U]. E.g., set s = 1, and define w_{0} by

w_{0}(−n) = w_{0}(n) = [n/2] + 4{n/2} for any n ∈ ℤ_{≥0},

where [ ] and { } are the integral, respectively the fractional parts; and let w_{1} on the segment [n, n+1] take the value max{w_{0}(n), w_{0}(n + 1)}. Then ℍ_{red}^{0} = ⊕_{k≥1}T_{k}(1)^{2}.

3.1.14. Restrictions. Assume that T ⊂ ℝ^{s} is a subspace of ℝ^{s} consisting of union of some cubes (from Q_{∗}). Let C_{q}(T) be the free ℤ–module generated by q–cubes of T, F^{q}(T) = Hom_{ℤ}(C_{q}(T), T_{0}^{+}). Then (F^{∗}(T), δ_{w}) is a complex, whose homology will be denoted by ℍ^{∗}(T, w). It has a natural graded ℤ[U]–module structure. The restriction map induces a natural graded ℤ[U]–module homogeneous homomorphism (of degree zero)

r^{∗} : ℍ^{∗}(ℝ^{s}, w) → ℍ^{∗}(T, w).

3.2. Lattice cohomology associated with Γ and k ∈ Char.

3.2.1. We consider a graph Γ as in §2 and we fix a characteristic element k ∈ Char. Notice that Γ automatically provides a free ℤ–module L = ℤ^{s} with a fixed bases {E_{j}}_{j}. Using Γ and k, we define a set of compatible weight functions {w_{q}}_{q}.

The definition reflects our effort to connect the topology of a singularity–link (e.g. the lattice cohomology) with analytic invariants. For more detailed motivation, see (4.2.4) and (6.2).

For any g ≥ 0 and n ≥ 0, let M^{g}(n) be the maximum of all possible dimensions of sheaf– cohomologies H^{1}(C, L), where C runs over all Riemann surfaces of genus g and L is a holomorphic line bundle on C with holomorphic Euler–characteristic χ(L) = n. (This number exists, in fact M^{g}(n) ≤ g.)

Now, we define {w_{q}}_{q} as follows. For q = 0 we set w_{0} := χ_{k} (cf. 2.2.5). Since the intersection form is negative definite, (3.1.4)(a) is satisfied.

Next, we define w_{1}. Consider a segment S ∈ Q_{1} with vertices l and l + E_{j} for some l ∈ L and j ∈ J. We set

w_{1}(S) := max { χ_{k}(l) , χ_{k}(l + E_{j}) } + M^{g_{j}}( |χ_{k}(l) − χ_{k}(l + E_{j})|).

Finally, for any □_{q} ∈ Q_{q} (q ≥ 2) set

w_{q}(□_{q}) := max { w_{1}(S) : S is a segment of □_{q} }.

3.2.2. Examples. (a) Assume that g_{j} = 0 for all j. Since M^{0}(n) = 0 for any n ≥ 0, for any q

w(□_{q}) = max{χ_{k}(v) : v is a vertex of □_{q}}.

(b) Assume that g_{j} ≤ 1 for any j. Since M^{1}(n) = 0 for n ≥ 1 and M^{1}(0) = 1, the definition of w_{1} might be modified into

w_{1}(S) = max { χ_{k}(l) , χ_{k}(l + E_{j}) , min{ χ_{k}(l) , χ_{k}(l + E_{j}) } + g_{j} }.

(c) By a vanishing theorem, in general, M^{g}(n) = 0 whenever n ≥ g.

3.2.3. Definition. The ℤ[U]–modules ℍ^{∗}(ℝ^{s}, w) and ℍ_{red}^{∗}(ℝ^{s}, w) obtained by these weight functions are called the lattice cohomologies associated with the pair (Γ, k) and are denoted by ℍ^{∗}(Γ, k), respectively ℍ_{red}^{∗}(Γ, k). We also write m_{k} := m_{w} = min_{l∈L} χ_{k}(l).

3.2.4. Theorem. ℍ_{red}^{∗}(Γ, k) is finitely generated over ℤ.

Proof. We start the proof with the following statement: Fact: There exist X ∈ L_{e} and an increasing infinite sequence of cycles {x_{i}}_{i≥0} with x_{0} = X, such that

(a) x_{i+1} = x_{i} + E_{j(i)} for some j(i), i ≥ 0,

(b) if x_{i} = ∑_{j} m_{i,j}E_{j}, then for all j, m_{i,j} tends to infinity as i tends to infinity,

(c) χ_{k}(x_{i+1}) − χ_{k}(x_{i}) ≥ g_{j(i)}.

Similarly, there exists Y ∈ L_{e} and an increasing infinite sequence of cycles {y_{i}}_{i≥0}, with y_{0} = Y and similar properties as in (a)–(b), and (c) χ_{k}(−y_{i+1}) − χ_{k}(−y_{i}) ≥ g_{j(i)}.

Indeed, take a cycle Z ∈ L such that (Z, E_{j}) < 0 for any j. Let {z_{i}}_{i=0}^{t} be an increasing sequence with z_{0} = 0, z_{t} = Z, z_{i+1} = z_{i} + E_{j(i)} (0 ≤ i < t). Then for m sufficiently large, X = mZ, and the sequence {m′Z + z_{i}} (where m′ ≥ m and 0 ≤ i < t) works. A similar statement is valid for Y = mZ (and similar type of sequence) with m ≫ 0.

Fix X, Y ∈ L_{e}, such that −Y ≤ l_{w} ≤ X. Let T (−Y, X) = {r ∈ ℝ^{s} : −Y ≤ r ≤ X}. T (−Y, X) has a natural cube–decomposition compatible with the decomposition of ℝ^{s}, hence by (3.1.14), one has a map r_{−Y,X}^{∗} : ℍ_{red}^{∗}(ℝ^{s}, w) → ℍ_{red}^{∗}(T (−Y, X), w).

Set X and Y as in Fact; clearly we may assume that −Y ≤ l_{w} ≤ X. We claim that r_{−Y,X}^{∗} is an isomorphism. Indeed, consider the restriction map r_{l,i}^{∗} : ℍ_{red}^{∗}(T (−y_{l}, x_{i+1}), w) → ℍ_{red}^{∗}(T (−y_{l}, x_{i}), w). If l ∈ T (−y_{l}, x_{i+1}) \ T (−y_{l}, x_{i}) then l = z + E_{j(i)}, z ≤ x_{i} and the coefficients of E_{j(i)} in z and x_{i} are the same. Hence, (x_{i}, E_{j(i)}) ≥ (z, E_{j(i)}). This implies that

χ_{k}(z + E_{j(i)}) − χ_{k}(z) ≥ χ_{k}(x_{i+1}) − χ_{k}(x_{i}) ≥ g_{j(i)},

which also shows (via 3.2.2(c)) that w_{1}[z, z + E_{j(i)}] = w_{0}(z + E_{j(i)}) ≥ w_{0}(z). Hence, the retract T (−y_{l}, x_{i+1}) → T (−y_{l}, x_{i}), which sends cycles of type z + E_{j(i)} (as above) to z (and preserves all cycles of different type) induces an isomorphism r_{l,i}^{∗}. Similar argument works if we move from y_{l} to y_{l+1}. Now, property (b) guarantees that r_{−Y,X}^{∗} is an isomorphism. On the other hand, ℍ_{red}^{∗}(T (−Y, X), w) is finitely generated over ℤ. □

3.2.5. Corollary. For any pair (Γ, k), the space S_{n} is contractible for n sufficiently large.

Proof. Fix X, Y as in Fact of the proof of (3.2.4). Let n be so large that T (−Y, X) ⊂ S_{n}. Then, the same argument as in the proof of (3.2.4) shows that S_{n} ∩ T (−y_{l}, x_{i}) ↪ S_{n} ∩ T (−y_{l}, x_{i+1}) admits a deformation retract. Hence, by induction, T (−Y, X) ⊂ S_{n} have the same homotopy type. □

If g = 0, one may also prove the contractibility of S_{n} for n ≫ 0 by verifying that S_{n} is a deformation retract in the real ellipsoid {x ∈ ℝ^{s} : χ_{k}(x) ≤ n}, which is obviously contractible.

3.2.6. Definitions. We will consider the following (euler–characteristic type) numerical invariants:

eu(ℍ^{0}(Γ, k)) := −m_{k} + rank_{ℤ}(ℍ_{red}^{0}(Γ, k)),
eu(ℍ^{∗}(Γ, k)) := −m_{k} +_{∑q}(−1)^{q} rank_{ℤ}(ℍ_{red}^{q}(Γ, k)).

3.2.7. Remark. There is a symmetry present in the picture. Indeed, the involution x ↦ −x (x ∈ L′) induces identities χ_{−k}(−l) = χ_{k}(l), hence isomorphisms

ℍ^{∗}(Γ, k) = ℍ^{∗}(Γ, −k) and ℍ_{red}^{∗}(Γ, k) = ℍ_{red}^{∗}(Γ, −k).

Notice that the involution [k] ↦ [−k] corresponds to the natural involution of Spin_{t}^{c}(M) ⊂ Spin^{c}(M).

Regarding the canonical structure, [K] = [−K] if and only if K ∈ L. In singularity theory, such graphs are called ‘numerical Gorenstein’ (when the tangent bundle on X \ 0 is topologically trivial). On the other hand, this happens if and only if the canonical spin^{c}–structure is spin.

3.3. Dependence of ℍ^{∗}(Γ, k) on k ∈ Char.

3.3.1. Fix Γ as above. Above we defined for any k ∈ Char a graded ℤ[U]–module ℍ^{∗}(Γ, k). Some of these graded roots are not very different. Indeed, assume that [k] = [k′] (cf. 2.2.2), hence k′ = k +2l for some l ∈ L. Then χ_{k′}(x − l) = χ_{k}(x) − χ_{k}(l) for any x ∈ L. Therefore, the transformation x ↦ x′ := x − l realizes the following identification:

3.3.2. Lemma. If k′ = k + 2l for some l ∈ L, then: ℍ^{∗}(Γ, k′) = ℍ^{∗}(Γ, k)[−2χ_{k}(l)].

In fact, there is an easy way to choose one module from the multitude {ℍ^{∗}(Γ, k)}_{k∈[k]}. Indeed, set m_{k} = min_{l∈L} χ_{k}(l) as above. Since (k + 2l)^{2} = k^{2} − 8χ_{k}(l), we get

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

Set M_{[k]} := {k ∈ [k] : m_{k} = 0}. Hence, if k_{0} and k_{0} + 2l ∈ M_{[k]}, then −χ_{k_{0}}(l) = 0. In particular, for any fixed orbit [k], any choice of k_{0} ∈ M_{[k]} provides the same module ℍ^{∗}(Γ, k_{0}). In the sequel we will denote this module by ℍ^{∗}(Γ, [k]). Notice that with this notation, for any k ∈ [k]

ℍ^{∗}(Γ, k) = ℍ^{∗}(Γ, [k])[2m_{k}].

Recall that the set of orbits [k] is the index set of the torsion spin^{c}–structures of M, cf. (2.2.2).

3.3.3. Distinguished representative. There is another more sophisticated way to choose a representative from a class [k]. Let [k] = K + 2(l′ + L). Then in the class l′ + L (corresponding to an element of H) one can chose l̄′_{ne} ∈ L′, cf. (2.2.4). The distinguished representative of [k] is, by definition, k_{r} := K + 2l̄′_{ne}. For example, the distinguished characteristic element in [K] is K itself. In [12], the elements k_{r} had a key role. The following result basically was proved there:

3.3.4. Proposition. Fix a representative k_{r} = K + 2l̄′_{ne} as above. Then in Fact (cf. proof of (3.2.4)) one may take Y = 0. This means that there exists an increasing sequence {y_{i}}_{i≥0} with y_{0} = 0, y_{i+1} = y_{i} + E_{j(i)} for some j(i) ∈ J for all i ≥ 0, all the coefficients of y_{i} tend to infinity, and finally, for any i ≥ 0 one has

χ_{k_{r}}(−y_{i+1}) − χ_{k_{r}}(−x_{i}) ≥ g_{j(i)}.

Proof. Notice that χ_{k_{r}}(−y_{i+1}) − χ_{k_{r}}(−y_{i}) = −e_{j(i)} − 1 + g_{j(i)} + (l̄′_{ne} − y_{i}, E_{j(i)}).

Therefore, if in a graph with g = 0 we can find a sequence with the wanted properties, then the same sequence will work if we decorate the vertices of the graph with some g_{j}. Hence, we may assume that g = 0. In this case the statement follows from [12] (6.1)(b), and its proof. In short, the argument is the following. Take Y > 0 (arbitrary large) provided by Fact. Then one can connect −Y to 0 with an increasing sequence along which χ_{k_{r}} is decreasing. Indeed, for any y < 0 there exists j so that E_{j} is in the support of y, and χ_{k_{r}}(y + E_{j}) ≤ χ_{k_{r}}(y). (If not, then (E_{j}, y + l̄′_{ne}) ≤ 0 for all E_{j} supported by y. But the same inequality automatically works for all other components. Hence y + l̄′_{ne} ∈ −L_{ℚ,ne} with y < 0, a contradiction.) □

3.3.5. Corollary.

(a) ℍ^{∗}(Γ, k_{r}) = ℍ^{∗}((ℝ_{≥0})^{s}, k_{r}), i.e., in the construction of ℍ^{∗}(Γ, k_{r}) one may only work with effective cycles from L_{e} instead of L (in other words, only with cubes sitting in ℝ_{≥0}^{s}).

(b) With respect to the canonical characteristic element K, S_{n}(K) is connected for all n ≥ 1.

Proof. (a) follows from a combination of (3.3.4) with the proof of (3.2.4). (b) was proved in (6.1)(d)

[12] under the assumption c = g = 0. The very same proof (based on part (a)) can be adopted. □

3.4. (In)dependence of Γ.

3.4.1. Clearly, many different negative definite 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 of (−1)-vertices with genus zero and whose number of incident edges is ≤ 2.

3.4.2. Proposition. The set ℍ^{∗}(Γ, [k]), where [k] runs over Spin_{t}^{c}(M), depends only on M and is independent of the choice of the (negative definite) plumbing graph Γ which provides M.

Proof. First we assume that Γ′ is obtained from Γ by ‘blowing up a smooth point of one of the exceptional curves’. More precisely, Γ′ denotes a graph with one more vertex and one more edge than Γ: we glue to a vertex j_{0} by the new edge the new vertex with decoration −1 and genus 0, while the decoration of E_{j_{0}} is modified from e_{j_{0}} into e_{j_{0}} − 1, and we keep all the other decorations. We will use the notations L(Γ), L(Γ′), L′(Γ), L′(Γ′). Similarly, write I, I′ for the corresponding intersection forms. Set E_{new} for the new base element in L(Γ′). The following facts can be verified:

• Consider the maps π_{∗} : L(Γ′) → L(Γ) defined by π_{∗}(∑ x_{j}E_{j} + x_{new}E_{new}) = ∑ x_{j}E_{j}, and π^{∗} : L(Γ) → L(Γ′) defined by π^{∗}(∑ x_{j}E_{j}) = ∑ x_{j}E_{j} + x_{j_{0}}E_{new}. Then I′(π^{∗}x, x′) = I(x, π_{∗}x′). This shows that I′(π^{∗}x, π^{∗}y) = I(x, y) and I′(π^{∗}x, E_{new}) = 0 for any x, y ∈ L(Γ).

• Set the (nonlinear) map: c : L′(Γ) → L′(Γ′), c(l′) := π_{ℚ}^{∗}(l′) + E_{new}. Then c(Char(Γ)) ⊂ Char(Γ′) and c induces an isomorphism between the orbit spaces Char(Γ)/2L(Γ) and Char(Γ′)/2L(Γ′).

• Consider k ∈ Char(Γ) and write k′ := c(k) ∈ Char(Γ′). Then for any x ∈ L(Γ) one has: χ_{k}(x) = χ_{k′}(π^{∗}x). Moreover, for any z ∈ L(Γ′), write z in the form π^{∗}π_{∗}z + aE_{new} for some a ∈ ℤ. Then χ_{k′}(z) = χ_{k′}(π^{∗}π_{∗}z) + χ_{k′}(aE_{new}) = χ_{k}(π_{∗}(z)) + a(a + 1)/2. Hence, the projection in the direction E_{new} provides a homotopy equivalence S_{n}(Γ′, k′) → S_{n}(Γ, k).

In fact, this can be done in two steps. Let Π^{∗} be the union of cubes of Γ′ with all vertices in π^{∗}(L(Γ)) ∪ (π^{∗}(L(Γ)) − E_{new}). Then S_{n}(Γ′, k′) has a deformation retract (via projection in E_{new} direction) into Π^{∗} ∩ S_{n}(Γ′, k′). On the other hand, the projection of the later one onto S_{n}(Γ, k) is a homotopy equivalence (by checking the liftings of the cubes).

There is a similar verification in the case when one blows up “an intersection point” corresponding to two indices i_{0} and j_{0} with (E_{i_{0}}, E_{j_{0}}) = 1. (The only difference is that π^{∗}(∑ x_{j}E_{j}) = ∑ x_{j}E_{j} + (x_{j_{0}} + x_{i_{0}})E_{new}.) The details are left to the reader. □

3.4.3. Remarks.

(a) Lattice homology. Obviously, there exists a parallel homological theory as well (already used in [25] for q = 0). Indeed, take F_{q} := C_{q} ⊗_{ℤ} T_{0}^{+}, and define ∂_{w} : F_{q} → F_{q−1} by

∂_{w}(□_{q} ⊗ t) = ∑_{k} ε_{k} · □_{q−1}^{k} ⊗ U^{w(□_{q})−w(□_{q−1}^{k})}t.

Then ℍ_{∗}(F_{∗}, ∂_{w}) is the corresponding lattice homology of the pair (ℤ^{s}, w). Similarly as above, it equals ⊕_{n}H_{∗}(S_{n}, ℤ). If w is given as in (3.2.1), then we get the lattice homology ℍ_{∗}(Γ, k) of (Γ, k).

(b) Graded root. For each Γ whose plumbed manifold is rational homology sphere, and k ∈ Char(Γ), the author in [12] constructed a graded root, from which one recovers by a natural procedure ℍ^{0}(Γ, k). Using the weight functions w_{0} and w_{1} of (3.2.3), one can define in a similar way a graded root for any Γ (whose vertices of degree n correspond to the connected components of S_{n}) with similar properties to those from [12].

(c) It might happen, that some non–empty real ellipsoids {x ∈ ℝ^{s} : (χ_{k} ⊗ ℝ)(x) ≤ n} contain no lattice points at all. In fact, min(χ_{k} ⊗ ℝ) − m_{k} can be arbitrarily large. Take for example the rational −2 curve and blow up in n different points. Then min χ_{K} ⊗ ℝ = −n/8, but m_{K} = 0.

3.5. Path cohomology.

3.5.1. Construction. Fix ℤ^{s} and compatible weight functions w_{0} and w_{1} as in (3.1.3–3.1.4).

We consider a sequence γ := {x_{i}}_{i=0}^{t} so that x_{0} = 0, x_{i} ≠ x_{j} for i ≠ j, and x_{i+1} = x_{i} ± E_{j(i)} for 0 ≤ i < t. We write T for the union of 0–cubes marked by the points {x_{i}}_{i} and segments of type [x_{i}, x_{i+1}]. Then, by (3.1.14), we get a graded ℤ[U]–module ℍ^{∗}(T, w), which is called the path–cohomology associated with the ‘path’ γ and weights {w_{q}}_{q}. It is denoted by ℍ^{∗}(γ, w). It has an augmentation with T_{2m_{γ}}^{+} , where m_{γ} := min_{i} w_{0}(x_{i}), and one gets the reduced path cohomology ℍ_{red}^{0}(γ, w) with

ℍ^{0}(γ, w) = T_{2m_{γ}}^{+} ⊕ ℍ_{red}^{0}(γ, w).

Similarly as in (3.2.6), we consider its ‘Euler–invariant’

eu(ℍ^{0}(γ, w)) := −m_{γ} + rank_{ℤ} ℍ_{red}^{0}(γ, w).

3.5.2. Lemma. ℍ^{q}(γ, w) = 0 for q ≥ 1, and

eu(ℍ^{0}(γ, w)) = −w_{0}(0) + ∑_{i=0}^{t−1} w_{1}([x_{i}, x_{i+1}]) − w_{0}(x_{i+1}).

Proof. Use induction comparing the paths {x_{i}}_{i=0}^{n−1} and {x_{i}}_{i=0}^{n} (0 < n ≤ t). □

3.5.3. Restriction map. Examples. In general, the restriction map r^{0} : ℍ^{0}(ℝ^{s}, w) → ℍ^{0}(γ, w) is not onto. Indeed, let us fix a graph Γ and weights as in (3.2.1), and we will study different paths connecting x_{0} = 0 with x_{t} = Z_{min}, the Artin’s cycle (the unique minimal element of −L_{ne} \ 0). In order to simplify the picture, we assume that c + g = 0, k = K, and χ(Z_{min}) ≤ 0 (i.e. Γ is not rational, cf. 4.1.1). We write χ := χ_{K}. There is an ‘optimal’ way to find Z_{min}, given by Laufer’s algorithm [7]: start with x_{0} = 0, take for x_{1}, say, E_{1} arbitrarily; if x_{i} (already constructed) is in −L_{ne}, then stop, set t = i, and x_{t} = Z_{min}; if (x_{i}, E_{j(i)}) > 0 for some j(i) then take x_{i+1} = x_{i} + E_{j(i)} and continue the algorithm with x_{i+1}.

If one considers any path γ_{L} connecting 0 and Z_{min} provided by Laufer’s algorithm, then χ(x_{1}) = χ(E_{1}) = 1, and after that χ will decrease to χ(Z_{min}), hence ℍ^{0}(γ_{L}, K) = T_{2χ(Z_{min})}^{+} ⊕ T_{0}(1).

Assume that the multiplicity of E_{1} in Z_{min} is ≥ 2. Then one may take the ‘non–optimal’ increasing path γ connecting 0 by Z_{min}, by taking x_{0} = 0, x_{1} = E_{1}, x_{2} = 2E_{1}, and after that we proceed according to Laufer’s algorithm. Then the maximum χ–value reached is χ(2E_{1}) = 2 − e_{1} ≥ 3 and ℍ^{0}(γ, K) = T_{2χ(Z_{min})}^{+} ⊕ T_{0}(2 − e_{1}).

One may verify that in the first case of γ_{L} the restriction r^{0} is onto, while in the second case it is not. E.g., if Γ is minimally elliptic (see 4.2), then ℍ^{0}(Γ, K) = ℍ^{0}(γ_{L}, K) = T_{0}^{+} ⊕ T_{0}(1) and r^{0} is an isomorphism, while in the second case r^{0} is not onto (by a rank argument). Moreover, in this second case, eu(ℍ^{0}(γ, K)) > eu(ℍ^{0}(Γ, K)).

3.5.4. Lemma. Fix two end–points, say 0 and l ∈ L. We consider all the paths P(l) as in (3.5.1) with x_{0} = 0 and x_{t} = l.

(a) There exists γ ∈ P(l) such that r^{0} : ℍ^{0}(Γ, w) → ℍ^{0}(γ, w) is onto.

(b) If for some γ ∈ P(l) the restriction r^{0} is onto then eu(ℍ^{0}(γ, w)) ≤ eu(ℍ^{0}(Γ, w)). In particular,

min_{γ∈P(l)} eu(ℍ^{0}(γ, w)) ≤ eu(ℍ^{0}(Γ, w)).

Proof. (a) Take any γ from P(l). If n ≫ 0 then S_{n} ∩ γ contains all the vertices and segments of Γ, hence it is contractible, H^{0}(S_{n} ∩ γ, ℤ) = ℤ and the restriction r_{n}^{0} : H^{0}(S_{n}, ℤ) → H^{0}(S_{n} ∩ γ, ℤ) is onto. If r^{0} is not onto, then let n be the largest integer for which r_{n}^{0} is not onto. This means that there exists x_{i} and x_{j} (i < j) so that the path [x_{i}, x_{j}] of γ is not in S_{n}, but there is a path γ_{ij} connecting x_{i} with x_{j} in S_{n}. Then replace [x_{i}, x_{j}] by γ_{ij}. Notice that the higher degree homologies (for n′ > n) remain unmodified. Repeating this procedure after a finite step we get the wished path.

The proof of (b) is left to the reader. □

4. Examples.

The ℤ[U]–module ℍ^{0}(Γ, k) is not new, it is the combinatorial module considered in [25, 12] (where it was denoted by ℍ^{+}). In fact, regarding ℍ^{0}(Γ, k), [12] is one of our main sources of examples.

4.1. The case of rational graphs.

4.1.1. Definition. By its very definition, a singularity is rational of its geometric genus p_{g} is vanishing. By [1, 2], this can be characterized combinatorially: a singularity is rational if and only if its graph satisfies min_{l∈L_{e}\0} χ_{K}(l) > 0 (or, equivalently, χ_{K}(Z_{min}) = 1, or χ_{K}(Z_{min}) > 0). Therefore, a (connected, negative definite) graph with this property is called rational. For them c = g = 0 automatically.

In [12] the author have given a different characterization:

4.1.2. Proposition. Assume that Γ is a negative definite connected plumbing graph with c = g = 0. Then Γ is rational if and only if ℍ_{red}^{0}(Γ, K) = 0. Moreover, in this case, ℍ_{red}^{0}(Γ, k) = 0, and also m_{k} = 0, for any k ∈ Char.

Even if one drops the assumption c = g = 0, one can prove:

4.1.3. Proposition. If ℍ^{0}(Γ, K) = T_{0}^{+} then Γ is rational.

Proof. Using (3.3.5) we get χ_{K}|L_{e} ≥ 0. Hence χ_{K}(E_{j}) = 1 − g_{j} ≥ 0. Assume that χ_{K}(Z_{min}) = 0. Then Z_{min} cannot be connected by 0 in S_{0} ∩ L_{e} since w_{1}([0, E_{j}]) = 1 by (3.2.2)(b). This contradicts the assumption, hence χ_{K}(Z_{min}) > 0, i.e. Γ is rational. □

We add to this the following vanishing result:

4.1.4. Proposition. If Γ is rational then ℍ_{red}^{∗}(Γ, k) = 0 for any k ∈ Char.

Proof. By (3.3.2) we may replace k with any characteristic element in its class. Take the distinguished representative k_{r}, cf. (3.3.3). The result follows from the proof of (3.2.4) once we show that one may take X = Y = 0 in Fact. By (3.3.4) we may take Y = 0. Hence, we have to show that for Γ rational there exists an increasing sequence {x_{i}}_{i≥0} with x_{0} = 0, x_{i+1} = x_{i}+E_{j(i)}, all the coefficients of x_{i} tend to infinity, and χ_{k_{r}}(x_{i+1}) ≥ χ_{k_{r}}(x_{i}). For this take a sequence {z_{i}}_{i=0}^{t} which connects 0 and Z_{min} provided by Laufer’s algorithm, cf. (3.5.3). Then z_{0} = 0, z_{1} = E_{1}, and (E_{j(i)}, z_{i}) = 1 for 1 ≤ i < t [7]. Hence χ_{k_{r}}(z_{1}) = 1 − (l̄′_{ne}, E_{1}) ≥ 1 and χ_{k_{r}}(z_{i+1}) − χ_{k_{r}}(z_{i}) = −(l̄′_{ne}, E_{j(i)}) ≥ 0. Hence, the sequence {mZ_{min} + z_{i}} with m ≥ 0 and 0 ≤ i < t works. □

Therefore, the above proof combined with the proof of (3.2.5) gives

4.1.5. Corollary. If Γ is rational, then S_{n} is contractible for any k ∈ Char whenever is non– empty.

4.2. The case of elliptic graphs.

4.2.1. Definition. [8, 31] A connected negative definite graph is elliptic if min_{l∈L_{e}\0} χ_{K}(l) = 0.

In this case Γ might have a cycle or a vertex with genus one, but in any case c + g ≤ 1. The next characterization result was proved in [12] for c = g = 0, here we verify the general situation.

4.2.2. Proposition. Γ is elliptic if and only if ℍ^{0}(Γ, K) = T_{0}^{+} ⊕ T_{0}(1)^{ℓ} for some ℓ ≥ 1.

Proof. Notice that, by (3.3.5), ℍ^{0}(Γ, K) = ℍ^{0}(ℝ_{≥0}^{s}, K). Assume that Γ is elliptic. Then χ_{K}|L_{e} ≥ 0. Moreover, by (3.3.5), S_{n}(K) is connected for n ≥ 1, hence ℍ^{0}(Γ, K) = T_{0}^{+} ⊕ T_{0}(1)^{ℓ} for some ℓ ≥ 0. Since Γ is not rational, ℓ ≠ 0. Conversely, if ℍ^{0} has that form, then χ_{K}|L_{e} ≥ 0 and there exists a cycle x ∈ L_{e} \ 0 with χ_{K}(x) = 0, hence Γ is elliptic. □

In fact, in the ‘classical’ theory of elliptic singularities, there is a combinatorial integer which guides the main topological and analytical properties, namely, the length of the elliptic sequence ℓ^{es}, introduced by Laufer and S. S.-T. Yau (see e.g. [32, 33]). E.g., Yau proved that ℓ^{es}+1 is a topological upper bound for the geometric genus, and [11] shows that it is realized by any Gorenstein singularity when c = g = 0. The ‘simplest’ elliptic singularities, the minimally elliptic ones, are characterized by ℓ^{es} = 0, or by the identity Z_{min} = −K [8].

The point is that the above integer ℓ provided by (4.2.2), in fact, equals ℓ^{es} + 1. In particular, for minimally elliptic singularities one has ℓ = 1.

We exemplify the above proposition (4.2.2) for three minimally elliptic singularity. The simplest case, when c = g = 0, is the hypersurface singularity {x^{2} + y^{3} + z^{7} = 0}, whose minimal good resolution graph has four vertices and three edges with E_{1}^{2} = −1, E_{2}^{2} = −2, E_{3}^{2} = −3, E_{4}^{2} = −7, and E_{1} is connected with the others. Then ker(U) has rank 2, the generating lattice points are the zero cycle and Z_{min} = 6E_{1} + 3E_{2} + 2E_{3} + E_{4}. (It general, the generators of ker(U) correspond to lattice points important in singularity theory as well, cf. [12, 14].)

4.2.3. Example. Assume that Γ consists of three vertices, each pair of vertices is connected by an edge, the self–intersections are −2, −2, −3, and g = 0. In this case −K = ∑ E_{j} = Z_{min}, hence Γ is minimally elliptic [8]. With the notation l = ∑_{j} x_{j}E_{j}, 2χ_{K}(l) = (x_{1} − x_{2})^{2} + (x_{2} − x_{3})^{2} + (x_{3} − x_{1})^{2} + x_{3}^{2} − x_{3}, which is always non–negative on L. It is zero at (0, 0, 0) and (1, 1, 1). They cannot be connected by a segment, hence S_{0} consists of two points. On the other hand, there are a lot of lattice points l in S_{1}: with the third coordinate x_{3} = −1 one has the pair (x_{1}, x_{2}) = (−1, −1), with x_{3} = 0 the pairs P = {(−1, −1), (−1, 0), (0, −1), (0, 0), (1, 0), (0, 1), (1, 1)}; with x_{3} = 1 the pairs P + (1, 1), and for x_{3} = 2 the pair (2, 2). (They are situated symmetrically with respect to −K/2.) Hence, S_{1} consists of 16 lattice points. One can verify that they can be connected by segments, S_{1} contains eight 2–cubes and one 3–cube, and S_{1} is contractible. And this is the case for all S_{n} with n ≥ 1.

4.2.4. Example. Assume that Γ has only one vertex with self–intersection −1 and g = 1. In this case again K = −E and the graph is minimally elliptic (it is the ‘simple–elliptic’ singularity Ẽ_{8}: {x^{2} + y^{3} + z^{6} = 0}). Notice that there are only two lattice points l with χ_{K}(l) = 0, namely the zero cycle and E, and they are connected by the segment [0, E] from Q_{1}. Hence, w_{1}[0, E] = 0 would imply ℍ_{red}^{0}(Γ, K) = 0. Therefore, in order to have the ‘right’ result, we are forced to put 1 for the weight of this segment, a fact compatible with (3.2.1)-(3.2.2).

Then, with this weight functions, one has: ℍ^{q} = 0 for q > 0 and ℍ^{0}(Γ, K) = T_{0}^{+} ⊕ T_{0}(1).

Notice that in the case of a minimally elliptic singularity, K is integral [8], hence [K] = [−K] (cf. 3.2.7). We may add the following vanishing result for the other spin^{c}–structures:

4.2.5. Proposition. If Γ is minimally elliptic, and the associated minimal (resolution) graph is good, then ℍ_{red}^{∗}(Γ, [k]) = 0 for any [k] ≠ [K].

Proof. By (3.4.2) we may assume that Γ is minimal. Then the proof of (4.1.4) can be adopted. Indeed, consider the representative k_{r}. Since [k_{r}] ≠ [K], k_{r} ≠ 0, hence there exists at least one j with (l̄′_{ne}, E_{j}) < 0. By [8] (p. 1261-1262), there exists a computation sequence {z_{i}}_{i=0}^{t} for Z_{min} so that the last E_{j(t−1)} is E_{j}, (z_{i}, E_{j(i)}) = 1 for 1 ≤ i ≤ t − 2, and (z_{t−1}, E_{j(t−1)}) = 2 (this fact uses the minimality of Γ). Then the proof of (4.1.4) works in this case too. □

4.3. The case of almost rational graphs.

4.3.1. Definition. [12] Assume that the graph Γ is connected and negative definite with c + g = 0. We say that Γ is almost-rational if there exists a vertex j_{0} ∈ J of Γ such that replacing its Euler number e_{j_{0}} by some e′_{j_{0}} ≤ e_{j_{0}} we get a rational graph. (In general, the choice of j_{0} is not unique.)

4.3.2. Examples. Almost rational graphs include: rational graphs, elliptic graphs (with c+g = 0), star-shaped graphs (with central vertex of genus zero). But there are more ‘exotic’ ones as well; e.g. the plumbing graph of the rational surgery 3–manifolds S_{r}^{3}(K), where r ∈ ℚ_{<0} and K is an algebraic knot in S^{3} (see e.g. [14, 15]). On the other hand, not every graph is almost rational. For example, if Γ has two (or more) vertices j with −e_{j} + 2 less than or equal to the valency of the vertex j, then Γ is not almost rational (e.g. the graph from (4.4.1)).

For almost rational graphs, ℍ^{0} might be rather complicated module (see e.g. [12] for the explicit description in the case of star–shaped graphs). On the other hand, we have:

4.3.3. Theorem. For any almost rational graph, ℍ^{q}(Γ, k) = 0 for any q > 0 and k ∈ Char.

Proof. The complete proof is rather technical and long, and we will omit it. It is based on the results of [12], §9. In fact, one can construct an infinite increasing path γ: {x_{i}}_{i≥0} with x_{0} = 0, so that the restriction r^{∗} : ℍ^{∗}(Γ, k) → ℍ^{∗}(γ, k) is an isomorphism (the fact that r^{0} is an isomorphism is the main result of [12] §9). The isomorphism is induced by a deformation retract whose existence is proved by a combination of results from [12] with the proof of (4.1.4). □

4.4. Examples with non–vanishing ℍ^{1}.

4.4.1. Example. Consider the following graph:

[figure: −2; −1; −13; −1; −2; −3; −3; E_{1}; E_{2}; E; E′_{2}; E′_{1}; E_{3}; E′_{3}]

On the right hand side we give names to the base elements. Set χ := χ_{K}. We prefer to write any l ∈ L in the form l = l_{x} + zE + l_{y}, where l_{x} = ∑ x_{i}E_{i}, l_{y} = ∑ y_{i}E′_{i}, x_{i}, y_{i}, z ∈ ℤ (i = 1, 2, 3); or in the form (x_{1}, x_{2}, x_{3}; z; y_{1}, y_{2}, y_{3}). Then −K = (7, 14, 5; 3; 7, 14, 5) and Z_{min} = (3, 6, 2; 1; 3, 6, 2), with χ(Z_{min}) = χ(2Z_{min}) = −1. In fact, m_{K} = −1 too. Then, it turns out that

ℍ^{0}(Γ, K) = T_{−2}^{+} ⊕ T_{−2}(1) ⊕ T_{0}(1) ⊕ T_{0}(1),

where the generators of ker(U) with homogeneous degree −2 are (the dual classes of) Z_{min} and 2Z_{min}, while with degree 0 are the (dual classes of) zero cycle and −K. Moreover, there exists a non–trivial class in ℍ^{1} of homogeneous degree 0. In fact,

ℍ^{1}(Γ, K) = T_{0}(1), and ℍ^{q}(Γ, K) = 0 for q ≥ 2.

In order to see (at least part of) these, we will analyse S_{−1} and S_{0}. Since χ(l) = χ(−K − l), we can use all the time the χ–symmetry of the lattice points with respect to −K/2. If z = 0 then χ(l) = χ(l_{x}) + χ(l_{y}), and since l_{x} and l_{y} are supported by rational subgraphs, χ(l_{x}) ≥ 0, χ(l_{y}) ≥ 0. Hence, the lattice points in S_{−1} have z = 1 or z = 2, and they correspond by the above symmetry.

Let us assume that z = 1. Then χ(l) = χ(l_{x}) + χ(l_{y}) − x_{2} − y_{2} + 1. Therefore, with the notation

f(x) := χ(l_{x}) − x_{2} = ( 2x_{1}^{2} + x_{2}^{2} + 3x_{3}^{2} − 2x_{1}x_{2} − 2x_{2}x_{3} − x_{2} − x_{3} )/2,

we have χ(l) = f(x) + f(y) + 1. By real calculus the minimum of f over ℝ^{3} is > −2, hence its minimum over ℤ^{3} is ≥ −1. Therefore, χ(l) = −1 if and only if f(x) = f(y) = −1. By a computation, the integral solutions of f(x) = −1 are the triplets

A := {(1, 2, 1), (1, 3, 1), (2, 3, 1), (2, 4, 1), (2, 4, 2), (2, 5, 2), (3, 5, 2), (3, 6, 2)}.

Therefore, points (x, 1, y) with x ∈ A and y ∈ A (denoted simply by (A, 1, A)) are in S_{−1}. Let B = (7, 14, 5) − A. Then, by symmetry, we get that the set of lattice points of S_{−1} is (A, 1, A) ∪ (B, 2, B). They determine two contractible connected components of S_{−1} in which Z_{min} and 2Z_{min} are ‘representatives’.

Next, we plan to solve the equation χ(l) = 0 with z = 1. Then, f(x) + f(y) = −1. f(x) = 0 has 24 integral solutions, namely union of the triplets A′ :=

{(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 2, 0), (0, 1, 1), (1, 1, 1), (0, 2, 1), (2, 2, 1), (1, 4, 1), (3, 4, 1), (2, 5, 1), (3, 5, 1)},

and the triplets of type A′′ = (4, 8, 3) − A′. Set Ã := A ∪ A′ ∪ A′′, and B̃ = (7, 14, 5) − Ã. Then the points of type

X := (A, 1, Ã) ∪ (Ã, 1, A) ∪ (B, 2, B̃) ∪ (B̃, 2, B)

are in S_{0}. Since Ã ∩ B is not empty, all the points from X can be connected by segments. In fact, S_{0} has three connected components, one of them contains the zero cycle, the other contains −K, and the third one, CS_{0}, contains all the points from X.

Finally, notice that the two intersection points P = Ã ∩ B = (4, 8, 3) and Q = A ∩ B̃ = (3, 6, 2) create a loop in CS_{0}. Indeed, half of it is the connecting path of (P; 1; Q) and (Q; 1; P) through points in X with z = 1, the other half connects (P; 2; Q) with (Q; 2; P) through points in X with z = 2. This loop can be contracted only in S_{1} (which is contractible).

4.4.2. Example. In the above example the subgraph of {E_{1}, E_{2}, E_{3}} is a ‘cusp’, Γ was obtained by gluing two cusps to the ‘central’ curve E. One may create non–trivial higher dimensional modules by gluing k cusps to a central curve E which has self–intersection −6k − 1.

5. Heegaard–Floer homology and singularity links.

5.1. Heegaard–Floer homology. In this section we will assume that M is an oriented rational homology 3–sphere.

5.1.1. Review. Heegaard–Floer homology HF^{+}(M) was introduced by Ozsváth and Szabó in [24] (and intensively studied in a series of articles). 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. 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 [28]) via

sw(M, σ) := χ(HF^{+}(M, σ)) − 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) [24] and (1.1) [26].

5.1.2. Example. If M is an integral homology sphere then for the unique (=canonical) spin^{c}– structure σ_{can}, sw(M, σ_{can}) equals the Casson invariant λ(M) (normalized as in [9] (4.7)).

5.2. Lattice homology and Heegaard–Floer homology.

5.2.1. Assume that Γ is a connected negative definite plumbing graph whose associated plumbed 3–manifold is a rational homology sphere. Our goal is to recover the Heegaard–Floer homology of M in a purely combinatorial way from Γ. We write #J = s.

5.2.2. Theorem. [25, 12] Assume that Γ is an almost rational graph. Then, for any [k] ∈ Spin^{c}(M)

HF_{odd}^{+}(−M, [k]) = 0,

and

HF_{even}^{+}(−M, [k]) = ℍ^{0}(Γ, [k])[ − max_{k′∈[k]} \frac{(k′)^{2} + s}{4}].

In particular (cf. 3.3.2), for any k ∈ [k] one has

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

5.2.3. Corollary. If Γ is an almost rational graph, then for any k ∈ Char:

−sw(M, [k]) − \frac{k^{2} + s}{8} = − min χ_{k} + rank_{ℤ} ℍ_{red}^{0}(Γ, k) = eu(ℍ^{0}(Γ, k)).

5.2.4. Conjecture. Let M be a plumbed rational homology sphere associated with a connected negative definite graph Γ. Then for any k ∈ Char the identity (∗) of (5.2.2) is valid, and

(∗∗)  −sw(M, [k]) − \frac{k^{2} + s}{8} = − min χ_{k} + ∑_{q}(−1)^{q} rank_{ℤ} ℍ_{red}^{q}(Γ, k) = eu(ℍ^{∗}(Γ, k)).

In fact, we predict that with d = d(M, [k]):

HF_{red,even}^{+}(−M, [k]) = ⨁_{q even} ℍ_{red}^{q}(Γ, [k])[−d], and HF_{red,odd}^{+}(M, [k]) = ⨁_{q odd} ℍ_{red}^{q}(Γ, [k])[−d].

5.2.5. Example. Take Γ from (4.4.1), and k = K. Then (∗) is true by [25], Corollary 1.5. Moreover, by (4.4.1), eu(ℍ^{∗}) = −(−1) + 3 − 1 = 3, (K^{2} + s)/8 = −1. On the other hand, the Casson invariant of M is −2 (using, e.g., the formula of Raţiu, see (5.3) [16]). Hence, (∗∗) is valid as well.

5.2.6. Remarks. (a) The above identities are not valid (in this form) when c + g > 0.

(b) (4.1.2–4.1.4), or (5.2.2) shows that if Γ is rational then M is an L–space (in the sense of Ozsváth and Szabó, i.e. HF_{red}^{+}(M) = 0). From the perspective of Conjecture (5.2.4), we expect that this is an ‘if and only if’ correspondence: Γ is rational if and only if M is an L–space. Notice that by (4.1.2), if ℍ_{red} = 0 then Γ is rational.

(c) Although we expect an identification of the ℍ^{∗} modules with the Heegaard–Floer modules HF^{+}, the lattice cohomology (apparently) contains more structure (at least, the author is not able to recover them in HF^{+}). For their existence the explanation is, maybe, that the involved 3–manifolds are rather special. We list here three such extra properties.

(i) The (absolute) grading of {ℍ^{q}}_{q≥0} (with respect to q) is indexed by ℤ in contrast with the ℤ_{2} (even/odd) grading of HF^{+}.

(ii) Consider from (3.1.12) the identity ℍ^{∗} = ⊕_{n}H^{∗}(S_{n}, ℤ). How can the ring structure of each H^{∗}(S_{n}, ℤ) exploited?

(iii) For a fixed graph Γ, consider any distinguished representative k_{r}. Since χ_{k_{r}}(l) ≥ χ_{K}(l) for any l ∈ L, we get S_{n}(k_{r}) ⊂ S_{n}(K), hence a natural ring homomorphism H^{∗}(S_{n}(K)) → H^{∗}(S_{n}(k_{r})), or R : ℍ^{∗}(Γ, K) → ℍ^{∗}(Γ, k_{r}). Notice that in the case of rational or elliptic graphs it happens that properties of the module associated with the canonical spin^{c}–structure ‘dominates’ all the others, cf. (4.1.2) or (4.2.5). Is it possible to say something similar in general? Is R onto?

6. Line bundles associated with surface singularities.

Starting from this section, we start to analyse the analytic aspects of the singularity (X, 0) as well. The analytic type is preserved in the complex manifold structure of the resolution X̃. Holomorphic line bundles on X̃ codify a lot of information about it.

6.1. Cohomological computations.

6.1.1. Let π : (X̃, E) → (X, 0) be a fixed good resolution of (X, 0). Let P ic(X̃) be the group of isomorphism classes of holomorphic line bundles on X̃ and c_{1} : P ic(X̃) → L′, c_{1}(L) = ∑_{j} deg(L|E_{j}) D_{j} the set of Chern classes of L. We prefer to use the same notation for l = ∑ n_{j}E_{j} ∈ L and divisors ∑ n_{j}E_{j} of X̃ supported by E. Hence, we can consider the line bundle OX̃(l) := OX̃(∑ n_{j}E_{j}). If l > 0, we write χ(l) for χ_{K}(l) = χ(O_{l}) (cf. (2.2.5)). We write |l| for the support of l.

In this subsection we analyse h^{1}(L) := dim H^{1}(X̃, L) for any L ∈ P ic(X̃). First, recall the following general (Grauert-Riemenschneider type) vanishing theorem (cf. [27], page 119, Ex. 15):

6.1.2. 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:

6.1.3. 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. From the point of view of the next discussion, it is instructive to see the 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) [1]. Then, using

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. □

This will be generalized in two different ways. First we show that the computation of any h^{1}(L) can be reduced to the computation of some h^{1}(L′) with c_{1}(L′) ∈ L_{ℚ,ne}.

6.1.4. 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 sequence x_{0}, x_{1}, …, x_{t} ∈ L_{e} with x_{0} = 0 and x_{i+1} = x_{i} + E_{j(i)}, where each 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) Since ( ,) is negative definite, there exists l ∈ L_{e} with l′ − l ∈ L_{ℚ,ne} (take e.g. a large multiple of some Z with (Z, 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. [7].)

(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. □

6.1.5. Examples. Rational singularities. If (X, 0) is rational then c_{1} : P ic(X̃) → L′ is an isomorphism. Moreover, using (6.1.3) and (6.1.4)(c), one has h^{1}(L) = −(l′, l_{l′})−χ(l_{l′}). In particular, h^{1}(L) depends only on Γ and it is independent of the analytic structure of (X, 0).

6.2. Path cohomology and upper bounds for h^{1}(L).

6.2.1. For the next result, we start with the following set of data and notations: L ∈ P ic(X̃), l′ := c_{1}(L), k := K−2l′ (cf. 2.2.5). We consider a ‘path’ γ: {x_{i}}_{i=0}^{t}, where x_{0} = 0, x_{t} ∈ l′−K−L_{ℚ,ne}, and x_{i+1} = x_{i} ± E_{j(i)} for some j(i) ∈ J (0 ≤ i < t).

Using the exact sequence 0 → L ⊗ O(−x_{t}) → L → L|_{x_{t}} → 0, and the Grauert-Riemenschneider vanishing (6.1.2), we get h^{1}(L) = h^{1}(L|_{x_{t}}) (this motivates the corresponding restriction for x_{t}). In the next proposition the ‘symbol’ h^{1}(L|_{x_{0}}) will stand for zero.

6.2.2. Proposition. With the above notations, for any 0 ≤ i < t with x_{i+1} > x_{i} one has:

h^{1}(L|_{x_{i+1}}) − h^{1}(L|_{x_{i}}) ≤ \{ \begin{matrix}−Δ_{i} + M^{g_{j(i)}}(−Δ_{i}) & if Δ_{i} < 0, \\ M^{g_{j(i)}}(Δ_{i}) & if Δ_{i} ≥ 0, \end{matrix}

where Δ_{i} := χ_{k}(x_{i+1}) − χ_{k}(x_{i}). If x_{i+1} < x_{i} then h^{1}(L|_{x_{i+1}}) − h^{1}(L|_{x_{i}}) ≤ 0.

In particular, adding all these inequalities, we get a topological upper bound for h^{1}(L).

6.2.3. Example. Assume that g = 0 and γ is increasing. Since M^{0}(n) = 0 for all n ≥ 0, we get

h^{1}(L) ≤ ∑_{i=0}^{t−1} max{ 0 , χ_{k}(x_{i}) − χ_{k}(x_{i+1}) }.

Proof of (6.2.2). Assume that x_{i+1} > x_{i} (the other case is trivial). Write M_{i} for the line bundle L ⊗ O_{E_{j(i)}}(−x_{i}) on E_{j(i)}. From the cohomological exact sequence

⋯ → H^{1}(E_{j(i)}, M_{i}) → H^{1}(L|_{x_{i+1}}) → H^{1}(L|_{x_{i}}) → 0

we have to estimate h^{1}(M_{i}). Notice that χ(M_{i}) = Δ_{i} by (2.2.5). Hence, if Δ_{i} ≥ 0, then h^{1}(M_{i}) ≤ M^{g_{j(i)}}(Δ_{i}), by the very definition of M^{g}(n). Assume that Δ_{i} < 0. Then, by Serre duality

h^{1}(M_{i}) = −Δ_{i} + h^{0}(M_{i}) = −Δ_{i} + h^{1}(M_{i}^{−1}(K + E_{j(i)})) ≤ −Δ_{i} + M^{g_{j(i)}}(−Δ_{i}).

6.2.4. Remark. Assume that we add another term x_{t+1} = x_{t} + E_{j(t)} to the sequence {x_{i}}_{i=1}^{t} with similar restriction x_{t+1} ∈ l′ − K − L_{ℚ,ne}. Then deg_{E_{j(t)}}M_{t} > 2g_{j(t)} − 2, Δ_{t} ≥ g_{j(i)} and M^{g_{j(t)}}(Δ_{t}) = h^{1}(M_{t}) = 0. Therefore, even if one continues the sequence arbitrarily long inside of l′ − K − L_{ℚ,ne}, nothing will be changed (e.g. the upper bound accumulates no more contribution). Sometimes we will just say and write that x_{t} = ∞, which means that x_{t} is in the ‘right’ region l′ − K − L_{ℚ,ne}.

Next we reinterpret (6.2.2) in terms of path cohomology. Let P be the set of paths with x_{0} = 0 and x_{t} = ∞, in the sense of (6.2.4). Moreover, consider the weight functions {w_{q}}_{q} associated with (Γ, k) as in (3.2.1), and write ℍ^{0}(γ; Γ, k) for ℍ^{0}(γ, w). Then from (6.2.2) and (3.5.2) we get

6.2.5. Corollary. For any γ ∈ P one has h^{1}(L) ≤ eu (ℍ^{0}(γ; Γ, k)). Hence

h^{1}(L) ≤ min_{γ∈P} eu (ℍ^{0}(γ; Γ, k)).

6.2.6. Remark. Recall that by (3.5.4) one has: min_{γ∈P} eu (ℍ^{0}(γ; Γ, k)) ≤ eu (ℍ^{0}(Γ, k)).

6.2.7. Example. If Γ is almost rational (cf. 4.3.1), a consequence of the results of [12] is that

min_{γ∈P} eu (ℍ^{0}(γ; Γ, k)) = eu (ℍ^{0}(Γ, k)),

and, in fact, the minimum min_{γ∈P} is realized by an increasing path. The point is that ker U ∈ ℍ^{0}(Γ, k) admits ‘representative’ lattice points which are totally ordered (with respect to <) sitting on an increasing path. In fact, ℍ^{0}(Γ, k) is determined in [12] from the values of χ_{k} along this path.

6.2.8. Example. The situation from (6.2.7), in general, is not true. I.e., one may have

min_{γ∈P} eu (ℍ^{0}(γ; Γ, k)) < eu (ℍ^{0}(Γ, k)),

i.e., the path cohomology may provides a strict better upper bound for h^{1}(L) than the lattice cohomology (cf. 6.2.6). To see this, construct Γ with c = g = 0 as follows. Let E and E′ be two vertices, both with self–intersection −14, and connected by an edge. Attach to both of them two–two cusps as in (4.4.1–4.4.2). Take k = K. Then χ(Z_{min}) = χ(3Z_{min}) = −3 and m_{K} = χ(2Z_{min}) = −4. By a computation

ℍ^{0}(Γ, K) = T_{−8}^{+} ⊕ T_{−6}(1)^{6} ⊕ T_{0}(1)^{2},

where the generators in degree zero are 0 and −K, in degree −4 is 2Z_{min}, while in degree −3 the cycles Z_{min}, L, R, L′, R′, 3Z_{min}. Here, the cycles L and R are symmetric with respect to the natural symmetry compatible with E ↔ E′, for both Z_{min} < L, R < 2Z_{min}, but L and R are not comparable by <. Hence, when one travels from Z_{min} to 2Z_{min} by a Laufer type path, then one has to make a choice (left-right) to pass through L or R, but one doesn’t have to touch both of them. The situation is similar with L′ and R′ which sit between 2Z_{min} and 3Z_{min}. Hence, it turns out that the module for a minimal increasing path (with end–point at K, or at ∞) is

ℍ^{0}(γ_{min}, K) = T_{−8}^{+} ⊕ T_{−6}(1)^{4} ⊕ T_{0}(1)^{2},

which has eu two less than ℍ^{0}(Γ, K).

6.2.9. Example. We may ask how sharp is the topological upper bound (6.2.5). Although it is not very easy to provide abundant examples for h^{1}(L), for the geometric genus p_{g} := h^{1}(OX̃) more examples are available. In this case, in many graphs the inequality (6.2.5) is optimal, i.e. the topological upper bound is realized by the p_{g} of some analytic structure. Nevertheless, this is not the case all the time. For the graph Γ discussed in (4.4.1), both the lattice and path cohomologies provide the same upper bound p_{g} ≤ 4 (cf. 6.2.6). On the other hand, by a (not simple) line of arguments, one finds out that there is no analytic structure supported on this topological type with p_{g} = 4 (p_{g} = 3 can be realized by a splice type complete intersection). The reader may decide if this example is ‘generic’ or ‘pathological’.

(Note that p_{g} ≥ h^{1}(Z_{min}) = 1 − χ(Z_{min}), hence p_{g} ≥ 2 for any analytic structure, while p_{g} = 3 for any Gorenstein structure.)

7. The Seiberg–Witten invariant conjecture.

7.1. Line bundles on X̃ revisited.

7.1.1. The bundles OX̃(l′). Start with the data of (6.1.1) and assume that M is rational homology sphere. The ‘exponential exact sequence’ 0 → ℤ → O → O^{∗} → 0 on X̃ induces the exact sequence

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

For any l ∈ L one has c_{1}(OX̃(l)) = l. Hence l ↦ OX̃(l) is a group section of c_{1} above the subgroup L of L′. Since L′/L is torsion, and H^{1}(OX̃) = ℂ^{p_{g}} is torsion–free, this can be extended in unique way to a group section s : L′ → P ic(X̃) of c_{1}. We write OX̃(l′) for s(l′).

7.1.2. Relation with coverings. The next theorem (7.1.3) illuminates a different aspect of the line bundles OX̃(l′). 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 p : Z → X̃ with Z normal and p finite [5]. The (reduced) branch locus of p 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 that Z might have are cyclic quotient singularities.

7.1.3. Theorem. Consider the finite covering p : Z → X̃, and set Q ⊂ L′ as in (2.2.4). Then the H-eigenspace decomposition of p_{∗}O_{Z} has the form:

p_{∗}O_{Z} = ⊕_{χ∈Ĥ}L_{χ},

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

The proof is based on a similar statement of Kollár valid for cyclic coverings, see e.g. [6], §9. For details, see [13, 15] or [22].

7.2. The conjectured identities.

7.2.1. The next expected property is a generalization of the conjecture of [16], where only the case of canonical spin^{c}–structure was considered. The generalization to any spin^{c}–structure appeared in [13], where it was formulated for any ℚ–Gorenstein singularity (with rational homology sphere link). The article [10] shows that we cannot expect the validity of the identities in this generality. Nevertheless, we expect that it is true for a large class of normal surface singularities (subclass of ℚ-Gorenstein singularities with rational homology sphere links). In the next paragraphs we will present two (equivalent) versions.

In this section we assume that the link M of (X, 0) is a rational homology sphere. We fix a good resolution π : X̃ → X with s := #J. Also, we set

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

(For notations, see (2.2.4) and (6.1.4).) One can verify that L′_{e} ⊂ ⋃_{l′∈Q} −l′ + L_{e} ⊂ 𝕃′.

7.2.2. Property A. Consider an arbitrary l′ ∈ 𝕃′ and define a characteristic element by k := K − 2l′ ∈ Char. Then, we say that (X, 0) satisfies Property A if

(1)  h^{1}(OX̃(l′)) = −sw(M, [k]) − \frac{k^{2} + s}{8}.

7.2.3. Remark. 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 (6.1.4)(c) shows that (7.2.2)(1) for L and L ⊗ OX̃(−l) are equivalent.

In fact, consider any set of representatives {l′}_{l′∈R} (R ⊂ 𝕃′) of the classes H, i.e. {l′+L}_{l′∈R} = H. Then the above argument applied for elements from R shows that the validity of the property (7.2.2) follows from the verification of (1) for line bundles L with c_{1}(L) ∈ R. The possibility R = −Q is emphasized by (7.1.3) and will be exploited in the second version of the property.

7.2.4. Universal abelian cover. Let (X_{ab}, 0) be the universal abelian cover of (X, 0) with its natural H-action. Namely, (X_{ab}, 0) is the unique normal singularity with a finite projection (X_{ab}, 0) → (X, 0), regular over X \ 0 corresponding to the abelianization map π_{1}(X \ 0) = π_{1}(M) → H. Then the space Z considered in (7.1.2–7.1.3) is a partial resolution of (X_{ab}, 0) with only cyclic quotient singularities. The geometric genus p_{g}(X_{ab}, 0) of (X_{ab}, 0) can be computed as the dimension of H^{1}(Z, O_{Z}), but this space has a natural eigenspace decomposition ⊕_{χ∈Ĥ}H^{1}(Z, O_{Z})_{χ} too. Hence one may consider the invariants

p_{g}(X_{ab}, 0)_{χ} := dim_{ℂ} H^{1}(Z, O_{Z})_{χ} (for any χ ∈ Ĥ).

Notice that (7.1.3) reads as

p_{g}(X_{ab}, 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 for H, by (7.2.3) the previous Property A (7.2.2) is equivalent with the following.

7.2.5. Property B. For any h ∈ H consider k := K + 2l′_{e}(h) ∈ Char. Then for any h ∈ H

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

7.3. Examples.

7.3.1. Example. Property A (hence B too) is true for any rational singularity. Indeed, by (7.2.3), we can assume that l′ = l′_{ne}(h) for some h. Then, by (6.1.3), h^{1}(OX̃(l′)) = 0. On the other hand, by [12], −sw(M, [k]) = (k_{r}^{2} + s)/8, where k_{r} = K + 2l̄′_{ne}(−l′ + L). Since l̄′_{ne}(−l′ + L) = −l′_{ne}(l′ + L) = −l′ one gets k_{r} = k. Hence the right hand side of (7.2.2)(1) is also vanishing.

This proof also shows that for (X, 0) rational, and for any h ∈ H, one has

p_{g}(X_{ab}, 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_{ab}, 0) is rational if and only if χ(l̄′_{ne}(h)) = χ(l′_{e}(h)) for all h ∈ H. One can find rational graphs whose universal abelian covers are not rational, a fact which stresses the differences between the ‘liftings’ l′_{e}(h) and l̄′_{ne}(h).

7.3.2. Example. Splice quotients. The validity of Property A for rational singularities (cf. 7.3.1), the surgery formulas of [3] regarding the Seiberg–Witten invariants, and the result of Okuma from [23] lead in [3] to the verification of Property A for all splice quotients. (The case of trivial line bundle was verified earlier in [19].) Splice quotient singularities were introduced by Neumann and Wahl (see e.g. [21]), they include all the rational, minimal elliptic singularities, and all singularities which admit a good ℂ^{∗}–action.

Assume now that (X, 0) is a splice quotient, and additionally, its topological type is also almost rational. Set l′ ∈ 𝕃′ and k = K − 2l′ as in Property A. Then Property A, (5.2.3) and (6.2.7) read as

h^{1}(OX̃(l′)) = eu(ℍ^{0}(Γ, k)) = min_{γ∈P} eu(ℍ^{0}(γ; Γ, k)),

which (by 6.2.5) is a topological upper bound for h^{1}(L), where L is an any bundle with c_{1}(L) = l′.

In particular, if Γ is almost rational, and the topological type admits a splice quotient analytic structure, then the geometric genus of the splice quotient analytic structure (which satisfies Property A) is an upper bound for the geometric genera of all the possible analytic structures.

7.3.3. Example. One can find even hypersurface singularities when Property A is not true for p_{g} (i.e. for l′ = 0). Such examples are provided in [10] by super–isolated singularities. In the examples of [10](4.1), p_{g} is strict higher then the expected value −sw(M, [K)] − (K^{2} + s)/8. Now, using our previous discussions, this phenomenon can be explained as follows.

In general, in the light of Conjecture (5.2.4), Property A/B is equivalent to

(1)  p_{g} = eu(ℍ^{∗}(Γ, k)) = − min χ_{K} +_{∑q}(−1)^{q} rank_{ℤ} ℍ_{red}^{q}(Γ, K).

On the other hand, the inequalities from subsection (6.2) read as

(2)  p_{g} ≤ min_{γ∈P} eu (ℍ^{0}(γ; Γ, K)) ≤ eu(ℍ^{0}(Γ, K)) = − min χ_{K} + rank_{ℤ} ℍ_{red}^{0}(Γ, K).

Assume that three things are happening simultaneously: (a) in (2) the second inequality is equality,

(b) for some analytic structure the first inequality in (2) is sharp (hence p_{g} = eu(ℍ^{0}(Γ, K))), and (c) ℍ_{red}^{q} ≠ 0 for q ≥ 2, creating the situation eu(ℍ^{0}) > eu(ℍ^{∗}). Then Property A fails, and in fact p_{g} > eu(ℍ^{∗}(Γ, k)) for that analytic structure.

This is the case for all the examples of (4.1)[10].

Let us analyse a little bit more the case C_{4} of [10]. The corresponding graph Γ is

[figure: −2; −1; −31; −1; −3; −4; −2; −2; −2; −2]

In this case ℍ^{0}(Γ, K) = T_{−10}^{+} ⊕ T_{−10}(3) ⊕ T_{0}(1)^{2}, hence eu(ℍ^{0}) = 10, but eu(ℍ^{∗}) = 8. (Strictly speaking, the author verified that −sw(M, [K])−(K^{2} +s)/8 = 8, cf. (5.2.4).) Hence the topological bound given by (2) is p_{g} ≤ 10. This topological type admits two, very natural, but rather different analytic structures. The first is the super–isolated hypersurface singularity mentioned above: it has p_{g} = 10 [10]. On the other hand, there is a splice quotient singularity which satisfies Property A, hence with p_{g} = 8 [19]. This is the ℤ_{5}–factor of the complete intersection {z_{1}^{3} + z_{2}^{4} + z_{3}^{5}z_{4} = z_{3}^{7} + z_{4}^{2} + z_{1}^{4}z_{2} = 0} ⊂ (ℂ^{4}, 0) by the diagonal action (α^{2}, α^{4}, α, α) (α^{5} = 1).

Therefore, in general, the geometric genus of those analytic structures which satisfy Property A is not ‘extremal’ (in contrast with the almost rational case (7.3.2)). In [3], Property A is reformulated completely in terms of the analytic structure (independently of any Seiberg–Witten type theory).

[23] suggests that in the heart of the its validity there is a cohomological vanishing result.

References

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

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

[3] Braun, G. and Némethi, A.: Surgery formula for Seiberg–Witten invariants of negative definite plumbed 3-manifolds, arXiv:0704.3145.

[4] Gompf, R.E. and Stipsicz, I.A.: An Introduction to 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, vol. 20, Amer. Math. Soc., 1999.

[5] Grauert, H. and Remmert, R.: Komplexe Räume, Math. Ann. 136 (1958), 245-318.

[6] Kollár, János: Shafarevich Maps and Automorphic Forms, Princeton University Press, Princeton, 1995.

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

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

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

[10] Luengo-Valesco, I.; Melle-Hernández, A. and Némethi, A.: Links and analytic invariants of superisolated singularities, Journal of Algebraic Geometry, 14 (2005), 543-565.

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

[12] Némethi, A.: On the Ozsváth-Szabó invariant of negative definite plumbed 3-manifolds, Geometry and Topology 9 (2005), 991-1042.

[13] Némethi, A.: Line bundles associated with normal surface singularities, arXiv:math.AG/0310084.

[14] Némethi, A.: On the Heegaard Floer homology of S_{−p/q}^{3}(K), arXiv:math.GT/0410570.

[15] Némethi, A.: Graded roots and singularities, Singularities in geometry and topology, 394–463, World Sci. Publ., Hackensack, NJ, 2007.

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

[17] Némethi, A. and Nicolaescu, L.I.: Seiberg-Witten invariants and surface singularities II (singularities with good C^{∗}-action), Journal of London Math. Soc. (2) 69, 2004, 593-607.

[18] Némethi, A. and Nicolaescu, L.I.: Seiberg-Witten invariants and surface singularities: Splicings and cyclic covers, Selecta Mathematica, New series, Vol. 11, Nr. 3-4 (2005), 399-451.

[19] Némethi, A. and Okuma, T.: The Seiberg–Witten invariant conjecture for splice–quotients, submitted.

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

[21] Neumann, W. and Wahl, J.: Complete intersection singularities of splice type as universal abelian covers, Geom. Topol., 9 (2005), 699-755.

[22] Okuma, T.: Universal abelian covers of rational surface singularities, J. London Math. Soc., (2) 70, (2004), 307-324.

[23] Okuma, T.: The geometric genus of splice quotient singularities, arXiv:math.AG/0610464, to appear in Trans. AMS.

[24] Ozsváth, P.S. and Szabó, Z.: Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math., (2) 159 (2004), no. 3, 1027–1158.

[25] Ozsváth, P.S. and Szabó, Z.: On the Floer homology of plumbed three-manifolds, Geom. Topol., 7 (2003), 185–224 (electronic).

[26] Ozsváth, P.S. and Szabó, Z.: Holomorphic triangle invariants and the topology of symplectic fourmanifolds, Duke Math. J., 121 (2004), no. 1, 1–34.

[27] Reid, M.: Canonical 3-folds, Algebraic Geomtry Angers, 1979 (A. Beauville, editor) 273-310; Sijthoff & Noordhoff, 1980.

[28] Rustamov, R.: A surgery formula for renormalized Euler characteristic of Heegaard Floer homology, math.GT/0409294.

[29] Tomari, M.: A p_{g}–formula and elliptic singularities, Publ. R.I.M.S. Kyoto University, 21, 297-354, 1985.

[30] Tomari, M. and Watanabe, Kei-ichi: Filtered rings, Filtered Blowing–Ups, Normal Two–Dimensional Singularities with “Star–Shaped” Resolution, Publ. R.I.M.S. Kyoto University, 25, 681-740, 1989.

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

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

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