THE SEIBERG-WITTEN INVARIANT CONJECTURE FOR SPLICE–QUOTIENTS

ANDRÁS NÉMETHI AND TOMOHIRO OKUMA

Abstract. Splice-quotient singularities were introduced recently and studied intensively by W. Neumann and J. Wahl. For such a singularity we prove that the geometric genus can be recovered from the topology of the singularity, namely from the Seiberg-Witten invariant (associated with the canonical spin^{c} structure) of the link. This answers positively the conjecture formulated by the first author and L. Nicolaescu.

1. Introduction

In this note (X, o) denotes an analytic normal surface singularity over the complex numbers. We write Σ for its link, and we assume that it is a rational homology sphere. In [14] L. Nicolaescu and the first author formulated a conjecture which aims to recover the most important discrete analytic invariant of (X, o), namely the geometric genus p_{g}, from the topology of (X, o), i.e. from the link Σ. The involved topological invariants of Σ are the Seiberg-Witten invariant sw(Σ) (associated with the canonical spin^{c}–structure of Σ), and a certain invariant K^{2} + s, which usually is read from a dual resolution graph Γ of (X, o) (for details, see 2.1 and section 4).

1.1. Conjecture. [14] If the analytic structure of (X, o) is “nice”, then

p_{g} = sw(Σ) − (K^{2} + s)/8.

In the original version [14] the conjecture was formulated for all ℚ–Gorenstein singularities; thanks to examples of [8] now we know that we have to impose a stronger analytic restriction.

If Σ is an integral homology sphere then sw(Σ) equals the Casson invariant λ(Σ), and in that case the above identity was conjectured by Neumann and Wahl for isolated complete intersections [19]. This ‘Casson invariant conjecture’ was verified for Brieskorn hypersurface singularities by Fintushel and Stern [5], for Brieskorn complete intersection independently by Neumann–Wahl [19] and Fukuhara–Matsumoto–Sakamoto [6], for suspension hypersurface singularities by Neumann–Wahl [19], and for generalization of suspensions, involving iterative cyclic coverings, by Collin–Saveliev [1, 2, 3]. Recently, for special splice type singularities (with all nodes ‘in a line’) was verified by Neumann–Wahl in [21], while for arbitrary splice singularities (with integral homology sphere links) in [17] — we will come back to this later.

2 A. Némethi and T. Okuma

In the general case, when Σ is a rational homology sphere, the above identity is true for rational singularities (by [11, 12], cf. also [13]), for elliptic Gorenstein singularities (by combination of [9] and [13]), for singularities with good ℂ^{∗}–action (by [15]), for suspension hypersurface singularities f(x, y) + z^{n} with f irreducible (by [16]). Some other sporadic cases can be found in [14], and the interested reader may also consult the equivariant version in [11, 12]. For a review of related problems (‘how can one recover analytic invariants from topology?’) see [10, 12].

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

Key words and phrases. normal surface singularities, splice-quotient singularities, graded rings, Poincaré or Hilbert series, geometric genus, Seiberg-Witten invariants, Turaev torsion, Casson-Walker invariant, Neumann-Wahl conjecture, Seiberg-Witten invariant conjecture.

The first author is partially supported by NSF grant DMS-0605323, Marie Curie grant and OTKA grants; the second author by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.

The goal of the present article is to prove the above conjecture for splice-quotients. These singularities were introduced recently by Neumann and Wahl [20, 21, 22]. Splicequotients include rational and minimal elliptic singularities (cf. [24]), and also those singularities which admit a good ℂ^{∗}–action, cf. [18].

1.2. Theorem. p_{g} = sw(Σ) − (K^{2} + s)/8 for any splice-quotient singularity.

The identity was already verified by Stevens in [28] for a few special splice-quotients (in connection with his study of superisolated singularities and their abelian covers). As we mentioned, in [17] we have clarified the case when the link is integral homology sphere; that proof was based on the (‘almost’) additivity property of p_{g} with respect to the splice decomposition of Σ — a speciality of integral homology spheres, and important property for its own sake.

The present general proof follows a different idea and line of arguments. In fact, it differs also from the proof of all the previous cases, i.e. it is not based on the independent computation of both sides of the identity, not even on a precise additivity formula for the involved invariants. What we establish is the special behaviour (pattern) of the involved invariants with respect to a ‘series of surgeries’, parametrized by an integer n. Namely, we show that each invariant (applied for these surgery graphs) is a rational function in variable n. Since for any n sufficiently large the corresponding graph is ‘less complicated’, the result follows by induction (which starts with the minimal rational singularities). This provides a conceptual understanding of the identity (1.2) whose message is the following. Consider a family of singularities which is closed to the surgery operation which replaces any self-intersection number on the dual resolution graph by a smaller integer. If for this family the involved invariants in (1.2) behave according to the above pattern then for that family the identity (1.2) holds. (In this article, in fact, we check that the invariants K^{2} + s and λ(Σ), associated with any graph Γ, follow this pattern).

The organization of the article is the following. In section 2 we recall some facts about splice-quotients and formulate theorem (2.9), which provides a generalization to the case of splice-quotients of some formulas of Dolgachev and Pinkham [4, 26]. (Dolgachev and Pinkham via these formulas recovered the geometric genus and the affine coordinate ring of singularities with good ℂ^{∗} actions.) The next section contains the proof of theorem (2.9). In section 4 we recall how one can recover form the resolution graph (or, from Σ) the topological invariants sw and K^{2} + s. Section 5 contains the ‘rational identities’ associated with the series of surgeries mentioned in the previous paragraph and the proof of the main theorem. The most involved identity is verified in the last section 6.

Acknowledgements. The second author thanks Marie Curie Fellowship for supporting his visit at the Rényi Institute of Mathematics, Budapest, Hungary, where the work on this paper was started. He is also grateful to the members of the Rényi Institute for the warm hospitality.

2. Preliminaries and splice-quotient singularities.

This section contains the needed notations about complex analytic surface singularities and some additional facts about splice-quotients.

The geometric genus of splice quotients 3

2.1. Notations. Let (X, o) be a germ of a normal complex surface singularity and π: X̃ → X a good resolution, i.e. the exceptional divisor E := π^{−1}(o) has only simple normal crossings. Let {E_{v}}_{v∈V} denote the set of irreducible components of E. We assume that the link Σ of (X, o) is a rational homology sphere, i.e. E is a tree of rational curves. The group of cycles is defined by L := ∑_{v∈V} ℤE_{v}. With the induced intersection form I(E) := (E_{v} · E_{w})_{v,w∈V} it forms a lattice. Write (a_{vw}) = −I^{−1}. The dual lattice L^{∗} consists of rational cycles D ∈ L ⊗ ℚ such that (D · E_{v}) ∈ ℤ for any v. It is generated by the elements E_{v}^{∗} := ∑_{w∈V} a_{vw}E_{w}. Notice that L^{∗}/L can be identified with H_{1}(Σ, ℤ); this group, for simplicity, will be denoted by H. |H|, respectively Ĥ, denote the order and the Potrjagin dual Hom(H, ℂ^{∗}) of H.

We define the integers b_{v}, e_{v}, ℓ_{vw} and m_{vw} (v, w ∈ V) by

b_{v} = E_{v}^{2}, ℓ_{vw} = |H|a_{vw}, e_{v} = |H|/gcd{ℓ_{vw} | w ∈ V} and m_{vw} = e_{v}a_{vw}.

Let δ_{v} = (E − E_{v}) · E_{v} be the number of irreducible components of E intersecting E_{v}. A curve E_{v} (or its index v) is called an end (resp. a node) if δ_{v} = 1 (resp. δ_{v} ≥ 3). We denote by E (resp. N) the set of indices of ends (resp. nodes). A connected component of E − E_{v} is called a branch of v. Sometimes it is convenient to replace E by its dual resolution graph, which will be denoted by Γ. We write det Γ for det(−I(Γ)).

Finally, let K be the canonical class on X̃. By the adjunction relations (K·E_{v})+E_{v}^{2}+2 = 0, hence numerically it can be determined from Γ as an element of L^{∗}.

2.2. Splice-quotients. Splice-quotient singularities were introduced by Neumann and Wahl (see [20, 21, 22]). Any splice-quotient is associated with a dual resolution graph Γ which satisfies two combinatorial properties: the semigroup and congruence conditions.

The second author in [24, 25] have introduced a different (but equivalent) description based on ‘monomial cycles’; here we will follow this terminology.

2.3. Monomial cycles and the monomial condition. An element of the semigroup ∑_{w∈E} ℤ_{≥0}E_{w}^{∗}, where ℤ_{≥0} is the set of nonnegative integers, is called a monomial cycle.

We say that E (or Γ) satisfies the monomial condition if for any node v and any branch B of v, there exists a monomial cycle D such that D − E_{v}^{∗} is an effective cycle supported on B. (This is equivalent to the semigroup and congruence conditions of Neumann-Wahl by [22, §13]).

In our proof we will need the fact that the monomial condition is closed under some graph-modifications.

2.4. Lemma. Let Γ be a graph which satisfies the monomial condition, and let v be one of its nodes with decoration b_{v} = E_{v}^{2}. Then the following facts hold:

(1) Let Γ \ v be the maximal subgraph of Γ which does not contain v. Then all its connected components satisfy the monomial condition.

(2) For any n ∈ ℤ_{≥0} consider the new graph Γ_{n} obtained from Γ by replacing the decoration b_{v} of v by the integer b_{v} − n. Then Γ_{n} also satisfies the monomial condition.

Proof. Fix a component Γ′ of Γ \ v, and let w be one of its nodes. Let Γ′′ be the branch of w (considered in Γ) which contains v. Then there exists an effective integral cycle C supported in Γ′′ such that C+E_{w}^{∗} is monomial in Γ. For (1) let E_{w}^{∗}(Γ′) be the corresponding dual cycle computed in Γ′, and C|Γ′ be the restriction of C on Γ′. Then C|Γ′ + E_{w}^{∗}(Γ′) is monomial in Γ′. The case of all the other branches relevant for (1) are trivial. For (2) consider another component Γ̃′ of Γ \ v and an effective cycle C̃ supported by this branch such that C̃ + E_{v}^{∗} is monomial in Γ. Then C + nC̃ + E_{w}^{∗}(Γ_{n}) is monomial in Γ_{n}. □

2.5. In the sequel we assume that Γ is a graph which satisfies the monomial condition. Then, cf. [25], one writes (in a very similar way as in the Neumann-Wahl case) #E − 2 equations (the so called splice equations) in (ℂ^{#E}, o) which define an isolated complete intersection admitting an action of H free off the origin. Moreover, the quotient singularity admits Γ as a possible resolution graph. This singularity is called a splice-quotient associated with Γ. For more details see [20, 21, 22] and [25].

All the possible singularities defined in this way form an equisingular family (cf. [24, 4.3] and [22, 10.1]), hence the geometric genus of a splice-quotient singularity depends only on Γ, and it will be denoted by p_{g}(Γ).

Next, we will recall an inductive formula for p_{g}(Γ) proved by the second author in [25]. In order to do this, we need some preparation.

2.6. Hilbert series. Let Γ be as above. We assume that it has at least one node (otherwise (1.2) is already proved). To each node v one associates (cf. [25, 3.8)]) a formal power series H_{v}(t) ∈ ℂ[[t]] by

H_{v}(t) := \frac{1}{|H|} · ∑_{χ∈Ĥ} ∏_{w∈V} (1 − χ([E_{w}^{∗}]) t^{m_{vw}} )^{δ_{w}−2},

where [E_{w}^{∗}] is the class of E_{w}^{∗} in H = L^{∗}/L.

2.7. Definition. Periodic constant. Let F (t) = ∑_{i≥0} a_{i}t^{i} ∈ ℂ[[t]] be a formal power series. Set P_{F} (n) = ∑_{i=0}^{n−1} a_{i}. Suppose that P_{F} (kn) is a polynomial function of n for some k ∈ ℕ. Then for any other k′ ∈ ℕ satisfying the same property, the constant terms of P_{F} (kn) and P_{F} (k′n) are the same. We call this constant the periodic constant of F (t) and denote it by F |_{pc}.

2.8. Theorem. [25] Let Γ and v be as in (2.6). Let {Γ′_{i}}_{i} denote the connected components of Γ \ v (cf. 2.4). Then H_{v}(t) admit a periodic constant and the following identity holds:

p_{g}(Γ) = H_{v}|_{pc} + ∑_{i} p_{g}(Γ′_{i}).

The next theorem provides a more precise (qualitative) description of the Hilbert series H_{v} which generalizes the automorphic form description of O_{X,o} and p_{g}(X, o) for a singularity (X, o) with a good ℂ^{∗}-action, given by Dolgachev and Pinkham [4, 26]. (In that case Γ is star-shaped, and if one takes v the central vertex then the above theorem reads as p_{g} = H_{v}|_{pc}.) Below, for any cycle D = ∑_{w∈V} a_{w}E_{w}, we write m_{w}(D) = a_{w}.

2.9. Theorem. (1) H_{v}(t) has the form

H_{v}(t) = ∑_{l≥0} b_{v,l}t^{e_{v}l} for some b_{v,l} ∈ ℂ.

(2) There exist cycles {D_{v,l}}_{l≥0} so that (a) b_{v,l} = h^{0}(E_{v}, O_{E_{v}} (−D_{v,l})) for l ≥ 0;

(b) h^{1}(E_{v}, O_{E_{v}} (−D_{v,l})) = 0 for l ≫ 0;

(c) m_{v}(D_{v,l}) = l;

(d) D_{v,l} is independent of E_{v}^{2}.

The proof is separated in Section 3. (In fact, as we will see, the cycles D_{v,l} satisfy some additional properties as well: they are the unique cycles satisfying some universal properties of the graph.)

3. Generalization of Dolgachev–Pinkham formulae

In this section we will prove Theorem 2.9. Assume that X is a splice-quotient singularity. We fix a good resolution X̃ with dual graph Γ and also one of its vertices v corresponding to the curve E_{v}. First, we recall that the power series H_{v}(t) (cf. 2.6), in fact, is the Hilbert series of a graded ring associated with a certain filtration of O_{X,o} associated with v.

Let ℂ{z} := ℂ{z_{w}; w ∈ E} be the convergent power series ring in #E variables. Notice that H acts diagonally on ℂ{z} by [E_{u}^{∗}] ∗ z_{w} = exp(−2πia_{uw})z_{w} (u ∈ V, w ∈ E). For a monomial cycle D = ∑_{w∈E} α_{w}E_{w}^{∗}, we set z(D) = ∏_{w∈E} z_{w}^{α_{w}} ∈ ℂ{z}. Let R denote the subring of ℂ{z} consisting of those series f which are series in z(D)’s with D ∈ L. Notice that, in fact, R coincides with the invariant subring ℂ{z}^{H}. Then we have the following (see §2.2–2.3 of [25]):

3.1. Theorem. There exists a surjective homomorphism φ: R → O_{X,o} of ℂ-algebras such that for any monomial cycle D ∈ L the exceptional part of the divisor divX̃(φ(z(D))) coincides with D.

3.2. Filtrations. Let F_{R,l} be the ideal of R generated by power series in z(D)’s with D ≥ lE_{v}. Then we define a filtration {F_{l}}_{l≥0} of O_{X,o} by F_{l} = φ(F_{R,l}). Let R = ⨁_{l≥0} R_{l}, where R_{l} = F_{l}/F_{l+1}.

The next theorem implies part (1) of Theorem 2.9.

3.3. Theorem. One has the following facts:

(1) F_{l} = H^{0}(OX̃(−lE_{v})) for every l ∈ ℤ_{≥0}.

(2) H_{v}(t) = ∑_{l≥l}(dim R_{l})t^{e_{v}l}.

Proof. It follows from (3.6–3.8) of [25] (note that F_{l} coincides with I_{e_{v}l}^{1} of [25]). □

3.4. For l ∈ ℤ_{≥0}, let M_{l} denote the minimal cycle, which is anti-semi-nef with respect to v, (i.e., M_{l} · E_{w} ≤ 0 for every w ∈ V \ {v}), and such that M_{l} − lE_{v} is an effective cycle supported on E − E_{v}. By (3.3)(1) and by the argument of [11, 4.2], for every l ≥ 0 we obtain

F_{l} = H^{0}(OX̃(−M_{l})) and F_{l+1} = H^{0}(OX̃(−M_{l} − E_{v})).

Therefore, R_{l} is isomorphic to the image of H^{0}(OX̃(−M_{l})) → H^{0}(O_{E_{v}} (−M_{l})).

3.5. Definition. A cycle D ∈ L is said to be semi-monomial with respect to v if it is anti-semi-nef with respect to v (cf. 3.4), and D · E_{w} = 0 for w ∈ V \ (E ∪ {v}). Let D_{l} be the set of semi-monomial cycles D such that D − lE_{v} is an effective cycle supported on E − E_{v} for every D ∈ D_{l}. For each branch C of v, there exists a monomial integral cycle F supported on C, such that F − lE_{v} is anti-nef “on C” (cf. Proposition 5.1 of [22]). Thus D_{l} ≠ ∅. For w ∈ V and l ∈ ℤ_{≥0}, set α_{w,l} = min {m_{w}(D) | D ∈ D_{l} }. We define a cycle D_{l} by D_{l} := ∑_{w∈V} α_{w,l}E_{w}. (Notice, that in general, D_{l} ∉ D_{l}.) Clearly, H^{0}(OX̃(−D_{l})) ⊂ F_{l}.

3.6. We will show that D_{l} satisfies all the properties of D_{v,l} listed in Theorem 2.9(2). The property (c) immediately follows from the definition of D_{l}. Since the set D_{l} is completely determined by l and branches of E_{v}, the property (d) follows too. For (b), it is sufficient to show that D_{l} ·E_{v} ≤ 0 for l ≫ 0. Notice that e_{v}E_{v}^{∗} ∈ L and m_{v}(e_{v}E_{v}^{∗}) = m_{vv}. Therefore D_{km_{vv}} = ke_{v}E_{v}^{∗} for every k ∈ ℤ_{≥0}. In order to see this, notice that ke_{v}E_{v}^{∗} ∈ D_{km_{vv}} and ke_{v}E_{v}^{∗} ≤ D for any D ∈ D_{km_{vv}}.

Using D_{km_{vv}} = ke_{v}E_{v}^{∗}, by the definition of D_{l} one gets D_{km_{vv}+j} ≤ ke_{v}E_{v}^{∗} + D_{j} for 0 ≤ j ≤ e_{v} − 1. Since the multiplicities of these cycles along E_{v} are the same,

D_{km_{vv}+j} · E_{v} ≤ −ke_{v} + D_{j} · E_{v}.

This proves (b). Part (a) follows from (3.8) below. First, we verify the following lemma.

3.7. Lemma. The natural map H^{0}(OX̃(−D_{l})) → H^{0}(O_{E_{v}} (−D_{l})) is surjective.

Proof. First notice that there exists a semi-monomial cycle D ∈ D_{l} such that D · E_{v} = D_{l} · E_{v}. Indeed, fix a branch C and let E_{w} be that curve on C which has E_{w} · E_{v} > 0. Then take a monomial cycle F_{C} on C with m_{w}(F_{C}) = α_{w,l} such that (lE_{v} + F_{C}) · E_{w} = 0 if w ∉ E; this exists since α_{w,l} is realized for some D′ ∈ D_{l}. (Indeed, take D′ ∈ D_{l} so that α_{w,l} = m_{w}(D′), and set for F_{C} that part of D′ which is supported on C.) Then set D := lE_{v} + ∑_{C} F_{C}.

We may assume d := −D · E_{v} ≥ 0 (otherwise 3.7 is clear). Fix two branches C_{i} of v (i = 1, 2). By the monomial condition, there exists an effective cycle F_{i} supported on C_{i} such that E_{v}^{∗} + F_{i} is a monomial cycle. Let

Λ = {λ = (λ_{1}, λ_{2}) ∈ (ℤ_{≥0})^{2} | λ_{1} + λ_{2} = d }.

Then D_{λ} := D + λ_{1}F_{1} + λ_{2}F_{2} is monomial for any λ ∈ Λ. By (3.1), the set {φ(z(D_{λ}))}_{λ∈Λ} is a subset of H^{0}(OX̃(−D_{l})), and clearly those functions generate H^{0}(O_{E_{v}} (−D_{l})). □

3.8. Theorem. We have the natural isomorphism R_{l} ≅ H^{0}(O_{E_{v}} (−D_{l})).

Proof. By (3.4), we only need to show that the image of the natural map

r: H^{0}(OX̃(−M_{l})) → H^{0}(O_{E_{v}} (−M_{l}))

is H^{0}(O_{E_{v}} (−D_{l})). Assume that D is a monomial cycle such that D ≥ lE_{v}; i.e., φ(z(D)) ∈ F_{l}. If r(φ(z(D))) ≠ 0, then D − M_{l} is an effective cycle supported in E − E_{v}. Thus it follows from the definition of D_{l} that φ(z(D)) ∈ H^{0}(OX̃(−D_{l})). By (3.1), the image of r is r(H^{0}(OX̃(−D_{l}))). Hence the claim follows from (3.7). □

4. Some general computational facts about sw and K^{2} + s.

4.1. Let Σ be the link of a singularity, or the plumbed 3–manifold associated with a negative definite plumbing graph. We fix such a graph Γ (which can be the dual graph of a good resolution), and we write s := #V for the number of vertices V.

4.2. The Casson-Walker invariant. Let λ(Σ) denote the Casson-Walker invariant of Σ, normalized as in [7, (4.7)]. Then from the surgery formulas of Lescop [7], or from the thesis of Raţiu [27], one has:

− \frac{24}{|H|} λ(Σ) = ∑_{v∈V} b_{v} + 3s + ∑_{v∈V} (δ_{v} − 2)a_{vv}.

4.3. The Reidemeister-Turaev torsion. The next discussion is valid for any spin^{c} structure of Σ, but here we will consider only the canonical one (cf. e.g. [14]). We consider the Reidemeister-Turaev torsion ∑_{h∈H} 𝒯_{Σ}(h) h ∈ ℚ[H] associated with the canonical spin^{c} structure of Σ (see [30]). Then we identify the Seiberg-Witten invariant of Σ associated with the canonical spin^{c} structure (cf. [23]) with:

(1)  sw(Σ) = − \frac{1}{|H|} λ(Σ) + 𝒯_{Σ}(1).

By [14], 𝒯_{Σ}(1) can be determined from the graph Γ via Fourier transform as follows.

First, fix a non-trivial character χ ∈ Ĥ \ {1}. Then find a vertex v_{χ} ∈ V so that either χ([E_{v_{χ}}^{∗}]) ≠ 1, or it has an adjacent vertex u with χ([E_{u}^{∗}]) ≠ 1. Then set:

(2)  𝒯̂_{Σ}(χ̄) = lim_{t→1} ∏_{w∈V} (1 − χ([E_{w}^{∗}]) t^{m_{vχw}} )^{δ_{w}−2}.

(The limit always exists.) Corresponding to the trivial character, set 𝒯̂_{Σ}(1) = 0. Then

𝒯_{Σ}(1) = \frac{1}{|H|} ∑_{χ∈Ĥ} 𝒯̂_{Σ}(χ).

4.4. K^{2} + s. For the definition of K see (2.1). By [14] one has:

K^{2} + s = ∑_{v∈V} b_{v} + 3s + 2 − ∑_{v,w} (δ_{v} − 2)(δ_{w} − 2)a_{vw}.

4.5. In the sequel we also write 𝒯_{Γ} and λ(Γ) for 𝒯_{Σ}(1), respectively λ(Σ).

5. ‘Rational identities’ and the proof of Theorem (1.2).

5.1. The graph Γ_{n}. Let Γ be an arbitrary (negative definite) plumbing graph, and fix one of its nodes 0 ∈ V. For any positive integer n we construct a new graph Γ_{n} as follows. Γ_{n} differs from Γ only by the decoration of the vertex 0, here the decoration b_{0} of 0 in Γ is replaced by b_{0} − n in Γ_{n}. By (2.4), if Γ satisfies the monomial condition then Γ_{n} also does.

5.2. Identity for |H| = det(Γ). By linear algebra:

det(Γ_{n}) = det(Γ) + n det(Γ \ 0) = det(Γ)(1 + n · a_{00}),

since a_{00} = m_{0}(E_{0}^{∗}) = det(Γ \ 0)/ det(Γ). There are similar identities for the determinants of subgraphs (minors of I). E.g.:

(1)  det(Γ_{n} \ v) = \{ \begin{matrix}det(Γ \ v) + n det(Γ \ \{v, 0\}) & if v ≠ 0; \\ det(Γ \ v) & if v = 0. \end{matrix}

5.3. Identity for K^{2} + s. Denote the canonical cycle of Γ (resp of Γ_{n}) by K (resp. by K(Γ_{n})). Then, by a computation, K(Γ_{n}) = K − xE_{0}^{∗}, where

x = n · \frac{1 + m_{0}(K)}{1 + n · a_{00}} .

Moreover,

(K(Γ_{n})^{2} + s) − (K^{2} + s) = −n + n · \frac{(1 + m_{0}(K))^{2}}{1 + n · a_{00}} = −n + \frac{n · det(Γ)}{det(Γ_{n})} (1 + m_{0}(K))^{2}.

5.4. Identity for λ/|H|. Using the identity from (4.2), and (5.2)(1), one can deduce:

− \frac{24 · λ(Γ_{n})}{det(Γ_{n})} = − \frac{24 · λ(Γ)}{det(Γ)} − n + \frac{n}{det(Γ_{n})} · i(Γ),

where

i(Γ) = a_{00} ∑_{v}(2 − δ_{v}) det(Γ \ v) − ∑_{v≠0}(2 − δ_{v}) det(Γ \ {v, 0})

is a constant (independent of n).

5.5. Theorem. There is a rational function R(n) in variable n, such that R(0) = 0 and for any n = ke_{0} (k ∈ ℤ_{≥0}) one has:

(𝒯_{Γ_{n}} − H_{Γ_{n},0}|_{pc}) − (𝒯_{Γ} − H_{Γ,0}|_{pc}) = R(n).

The proof is separated in section 5.

Proof of Theorem (1.2). Our goal is to prove for any graph Γ (which satisfies the monomial condition) the vanishing of the combination of invariants (cf. also with 2.5):

V_{Γ} := 𝒯_{Γ} − \frac{λ(Γ)}{|H|} − \frac{K^{2} + s}{8} − p_{g}(Γ).

The proof goes by induction over the number of bad vertices of Γ: A vertex v of Γ is bad if −b_{v} < δ_{v}. A graph has no bad vertices if and only if it is minimal rational. In this case V_{Γ} = 0 by [11, 12]. By induction, we assume that the identity is true for any graph (with the monomial condition) with r − 1 bad vertices, and we take a graph Γ with exactly r bad vertices, and which satisfies the monomial condition.

Let 0 ∈ V be one of the bad vertices, and construct the graph Γ_{n} as in (5.1). Then by (2.8)

V_{Γ} := 𝒯_{Γ} − \frac{λ(Γ)}{|H|} − \frac{K^{2} + s}{8} − H_{Γ,0}|_{pc} − ∑_{i} p_{g}(Γ′_{i}).

The above identities (5.3-5.4-5.5) imply that there exists a rational function R(n) with R(0) = 0 such that

V_{Γ_{n}} = R(n) + V_{Γ} for any n ∈ e_{0}ℤ_{≥0}.

Since for n sufficiently large Γ_{n} has only r − 1 bad vertices, we get that R(n) + V_{Γ} = 0 for infinitely many values n. Therefore, R(n) + V_{Γ} is identically zero, in particular its value at 0, namely V_{Γ}, is zero too. □

6. The proof of Theorem (5.5).

6.1. Let Γ and 0 ∈ V and Γ_{n} as in (5.1). The main difficulty to prove identities, similar to those from (5.2-5.3-5.4), for the invariants 𝒯 and H_{v}|_{pc}, lies in the fact that the structure of the group H(Γ_{n}) is modified rather drastically for an arbitrary n. Therefore, in this section, we will assume that n = ke_{0} for some non-negative integer k, where e_{0} is defined in (2.1), and in fact it is the order of [E_{0}^{∗}] in H. In particular, det(Γ_{n})/ det(Γ) = 1+na_{00} = 1 + km_{00} ∈ ℤ.

Regarding the group H (associated with Γ) we consider the cyclic subgroup G generated by [E_{0}^{∗}], and its quotient group Q := H/G. Clearly, Q̂ = {χ ∈ Ĥ : χ([E_{0}^{∗}]) = 1}. Similarly, we denote by H_{n}, G_{n} and Q_{n} the corresponding groups associated with Γ_{n}. Since ℓ_{v0}(Γ_{n}) = ℓ_{v0} for any v, one gets that e_{0}(Γ_{n})/e_{0} = det(Γ_{n})/ det(Γ). Moreover, there is a natural epimorphism α : H_{n} → H given by [E_{v}^{∗}(Γ_{n})] ↦ [E_{v}^{∗}] (with cyclic kernel). α induces a monomorphism α̂ : Ĥ ↪ Ĥ_{n} which maps Q̂ isomorphically onto Q̂_{n}; in the sequel we will identify these two subgroups.

In the sequel it is convenient to write in (4.3)(2) 𝒯̂_{Γ}(χ) instead of 𝒯̂_{Σ}(χ), provided that Σ is the plumbed 3–manifold of Γ (cf. 4.5).

6.2. Notation. We consider two invariants I and J, associated with (resolution) graphs. We fix a graph Γ, and consider the series Γ_{n} as above. We write I ≡ J, if there exists a rational function R(n) in variable n (and with coefficients depending on Γ) such that R(0) = 0 and I(Γ_{n}) − J(Γ_{n}) = R(n) for n ∈ e_{0}ℤ_{≥0}. (If J(Γ_{n}) is a rational function in n, then the above definition is equivalent to the identity I(Γ_{n}) = J(Γ_{n}) · R̃(n) for some rational function R̃(n) with R̃(0) = 1 and for n ∈ e_{0}ℤ_{≥0}.)

6.3. Lemma. For any non-trivial character χ ∈ Q̂ = Q̂_{n} one has 𝒯̂_{Γ_{n}}(χ) ≡ 𝒯̂_{Γ}(χ).

Proof. Since the limit in (4.3)(2) exists one has S(χ) := ∑(δ_{w} − 2) ≥ 0, where the sum is over {w : χ([E_{w}^{∗}]) = 1}. This is the same for Γ and Γ_{n}. If S(χ) > 0 then 𝒯̂_{Γ_{n}}(χ) = 𝒯̂_{Γ}(χ) = 0. Assume that S(χ) = 0. Then, by a relation similar to (5.2)(1), m_{v_{χ}w}(Γ_{n}) has the form am_{v_{χ}w} + nb_{v_{χ}w}, such that a and b_{v_{χ},w} are independent of n. Hence

\frac{𝒯̂_{Γ_{n}}(χ)}{𝒯̂_{Γ}(χ)} = lim_{t→1} ∏_{w} ( \frac{1 − χ([E_{w}^{∗}]) t^{am_{vχw}+nb_{vχw}}}{1 − χ([E_{w}^{∗}]) t^{m_{vχw}}} )^{δ_{w}−2} = ∏ ( \frac{m_{v_{χ}w} + nb_{v_{χ}w}/a}{m_{v_{χ}w}} )^{δ_{w}−2},

where the last product is over {w : χ([E_{w}^{∗}]) = 1}. The last expression is a rational function R̃(n) with R̃(0) = 1. □

Proof of Theorem (5.5). For any χ ∉ Q̂ one can take v_{χ} = 0. Notice also that in (4.3)(2) and (2.6) one can replace t by any of its integral powers t^{c} (i.e. {m_{0w}}_{w} by {cm_{0w}}_{w}). We will do this with c = e_{0}(Γ_{n})/e_{0} = 1 + km_{00} in the case of Γ. Therefore, with the notation

P_{Γ,0}(t) := \frac{1}{|H|} · ∑_{χ∈Q̂} ∏_{w} ( 1 − χ([E_{w}^{∗}])t^{m_{0w}} )^{δ_{w}−2}

one gets that

𝒯_{Γ} − H_{Γ,0}|_{pc} = \frac{1}{|H|} · ∑_{χ∈Q̂} 𝒯̂_{Γ}(χ) + lim_{t→1} (H_{Γ,0}(t^{c}) − P_{Γ,0}(t^{c}) − H_{Γ,0}(t^{c})|_{pc}).

Set

S_{Γ,0}(t) := H_{Γ,0}(t) − ∑_{l≥0} h^{1}(E_{0}, O_{E_{0}}(−D_{0,l}))t^{e_{0}l},

cf. (2.9). Since the last sum is finite (by 2.9(2-b)), one obtains

𝒯_{Γ} − H_{Γ,0}|_{pc} = \frac{1}{|H|} · ∑_{χ∈Q̂} 𝒯̂_{Γ}(χ) + lim_{t→1} (S_{Γ,0}(t^{c}) − P_{Γ,0}(t^{c}) − S_{Γ,0}(t^{c})|_{pc}).

Therefore, if we write

Δ := (𝒯_{Γ_{n}} − H_{Γ_{n},0}|_{pc}) − (𝒯_{Γ} − H_{Γ,0}|_{pc}),

by (6.3) one has

Δ ≡ lim_{t→1} (S_{Γ_{n},0}(t) − P_{Γ_{n},0}(t) − S_{Γ_{n},0}(t)|_{pc} − S_{Γ,0}(t^{c}) + P_{Γ,0}(t^{c}) + S_{Γ,0}(t^{c})|_{pc}).

Notice also that by Riemann–Roch, and (2.9)(2):

S_{Γ,0}(t^{c}) = ∑_{l≥0} ( 1 − (E_{0} · D_{0,l})_{Γ} )t^{ce_{0}l} and S_{Γ_{n},0}(t) = ∑_{l≥0} ( 1 − (E_{0} · D_{0,l})_{Γ_{n}} )t^{ce_{0}l},

and

S_{Γ_{n},0}(t) − S_{Γ,0}(t^{c}) = ∑_{l≥0} nlt^{ce_{0}l}.

Since the periodic constant of this sum is zero, one gets

Δ ≡ lim_{t→1} (∑_{l≥0} nlt^{ce_{0}l} − P_{Γ_{n},0}(t) + P_{Γ,0}(t^{c}) ).

Since

P_{Γ_{n},0}(t) = \frac{det(Γ)}{det(Γ_{n})} P_{Γ,0}(t) = P_{Γ,0}(t)/c,

one has

(1)  Δ ≡ P^{∗}, where P^{∗} := lim_{t→1} ( \frac{nt^{ce_{0}}}{(1 − t^{ce_{0}})^{2}} − \frac{1}{c} P_{Γ,0}(t) + P_{Γ,0}(t^{c}) ).

Since in (4.3)(2) the limit exists for any χ ≠ 1 (hence the corresponding rational function has no pole at 1), |H| · P_{Γ,0}(t) has the same Laurent-part as ∏_{w}(1 − t^{m_{0w}})^{δ_{w}−2}, hence it has a pole of order two (cf. also with [14, A.7]), and it can be written as

P_{Γ,0}(t) = \frac{A}{(1 − t^{e_{0}})^{2}} + \frac{B}{1 − t^{e_{0}}} + C + D(t),

for some constants A, B, C ∈ ℂ and lim_{t→1} D(t) = 0. By Taylor expansion

\frac{1 − t^{ce_{0}}}{1 − t^{e_{0}}} = c − \frac{c − 1}{2} (1 − t^{ce_{0}}) − \frac{c^{2} − 1}{12c} (1 − t^{ce_{0}})^{2} + · · · .

Since in (1) the limit exists, A must be a_{00}^{−1} and the limit is

P^{∗} = \frac{c − 1}{c} ( C + \frac{1}{2} B − \frac{c − 5}{12} A).

Since c − 1 = na_{00} one gets Δ ≡ P^{∗} ≡ 0. □

References

[1] Collin, O.: Equivariant Casson invariant for knots and the Neumann-Wahl formula, Osaka J. Math., 37 (2000), 57-71.

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

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

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

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

[6] Fukuhara, S.; Matsumoto, Y. and Sakamoto, K.: Casson’s invariant of Seifert homology 3-spheres Math. Ann. 287 (1990), no. 2, 275–285.

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

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

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

[10] Némethi A.: Invariants of normal surface singularities, Proceedings of the Conference: Real and Complex Singularities, San Carlos, Brazil, August 2002; Contemporary Mathematics, 354, 2004, 161-208.

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

[12] Némethi, A.: Graded roots and singularities, to appear in the proceedings volume of the Institute of Trieste.

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

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

[15] 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.

[16] 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.

[17] Némethi, A. and Okuma T.: On the Casson invariant conjecture of Neumann-Wahl, arXiv:math.AG/0610465.

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

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

[20] Neumann, W. and Wahl, J.: Universal abelian covers of surface singularities, Trends in singularities, 181-190, Trends Math. Birkhäuser, Basel, 2002.

[21] Neumann, W. and Wahl, J.: Complex surface singularities with integral homology sphere links, Geometry and Topology, 9 (2005), 757-811.

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

[23] Nicolaescu, L.I.: Seiberg-Witten invariants of rational homology spheres, Commun. Contemp. Math. 6 (2004), no. 6, 833-866.

[24] Okuma, T: Universal abelian covers of certain surface singularities, Math. Ann. 334 (2006), 753-773.

[25] Okuma, T: The geometric genus of splice-quotient singularities, arXiv:math.AG/0610464.

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

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

[28] Stevens, J.: Universal abelian covers of superisolated singularities, arXiv:math.AG/0601669.

[29] Tomari, Masataka 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.

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

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

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

Rényi Institute of Mathematics, Budapest, Hungary

   E-mail address: nemethi@renyi.hu

Department of Education, Yamagata University, Yamagata 990-8560, Japan.

   E-mail address: okuma@e.yamagata-u.ac.jp