SEIBERG-WITTEN INVARIANTS AND SURFACE SINGULARITIES II (SINGULARITIES WITH GOOD ℂ^{∗}-ACTION)

ANDRÁS NÉMETHI AND LIVIU I. NICOLAESCU

Abstract. We determine the Reidemeister-Turaev sign-refined torsion (associated with any spin^{c} structure) of a Seifert 3-manifold with negative orbifold Euler number and genus zero. In particular, we obtain all the Seiberg-Witten invariants of such a manifold.

As a main application, we verify the conjecture formulated in [28] for any normal surface singularity which admits a good ℂ^{∗}-action. This result connects the Seiberg-Witten invariant of the link (associated with a certain “canonical” spin^{c} structure) with the geometric genus of the singularity, provided that the link is a rational homology sphere. As an application, we find a topological interpretation of the generalized Batyrev’s

stringy invariant (in the sense of Veys) associated with such a singularity.

1. Introduction

The main motivation of the present article is [28], where the authors formulated a very general conjecture which relates the topological and the analytical invariants of a complex normal surface singularity whose link is a rational homology sphere.

Let (X, 0) be a normal two-dimensional analytic singularity. From a topological point of view, it is completely characterized by its link M, which is an oriented 3-manifold. Moreover, by a result of Neumann [30], any decorated resolution graph of (X, 0) carries the same information as M. A property of (X, 0) will be called topological if it can be determined from M, or equivalently, from any resolution graph of (X, 0). For example, for a given resolution, if we take the canonical divisor K, and the number #V of irreducible components of the exceptional divisor, then K^{2} + #V is independent of the choice of the resolution, hence it is an invariant of the link M (cf. 2.4).

Let us recall some definitions regarding the analytical structure of (X, 0). Consider the line bundle Ω_{X\{0}}^{2} of holomorphic 2-forms on X \{0}. If this line bundle is holomorphically trivial then we say that (X, 0) is Gorenstein. If one of its powers is holomorphically trivial then we say that (X, 0) is ℚ-Gorenstein. Let π : X̃ → X be a resolution over a sufficiently small Stein representative X of the germ (X, 0). Then p_{g} := dim H^{1}(X̃, OX̃) is finite and independent of the choice of π. It is called the geometric genus of (X, 0).

Our goal is to investigate in which instances one can express the geometric genus in terms of topological invariants of the link. By the conjecture [28], if the link of a ℚ-Gorenstein singularity is a rational homology sphere, then it determines p_{g} (by the precise formula 1.1(∗)). The conjecture has grown out partly from the work of Artin on rational singularities [2, 3], and of Laufer, Wagreich, S. S.-T. Yau and the first author on elliptic singularities [18, 43, 27]; on the other hand, from the work of Fintushel and Stern [10], respectively of Neumann and Wahl [32] connecting the signature of the Milnor fiber with the Casson invariant of the link, provided that the link is an integral homology sphere. The conjecture formulated in [32] about this connection had a crucial influence on our project. For more details and historical data the reader is invited to consult [28]. (For a related set of conjectures, see also [33].)

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

14E15, 32S45, 57M25.

Key words and phrases. (links of) normal surface singularities, (ℚ-)Gorenstein singularities, singularities with ℂ^{∗}-action, geometric genus, Seiberg-Witten invariants of ℚ-homology spheres, Reidemeister-Turaev torsion, Casson-Walker invariant.

The first author is partially supported by NSF grant DMS-0088950; the second author is partially supported by NSF grant DMS-0071820.

In the conjecture [28], the main new ingredient is the Seiberg-Witten invariant of the link M associated with a “canonical” spin^{c} structure. We recall (see [28], and the references therein, or 2.2–2.3 below) that the Seiberg-Witten invariant associates to any spin^{c} structure σ of M a rational number sw_{M}^{0}(σ). In [28] we introduced a “canonical” spin^{c} structure σ_{can} of M as follows. The (almost) complex structure on X \ {0} induces a natural spin^{c} structure on X \{0}. Then σ_{can}, by definition, is its restriction to M. The point is that σ_{can} depends only on the topology of M. By the result of the second author [35], sw_{M}^{0}(σ) can be expressed in terms of the Casson-Walker invariant and the sign refined Reidemeister-Turaev torsion of M associated with σ (see 2.3). (If M is integral homology sphere then σ_{can} is the unique spin^{c} structure of M, and −sw_{M}^{0}(σ_{can}) is the Casson invariant of M.)

In this article we prove the conjecture [28] for singularities with good ℂ^{∗}-action. A complex affine algebraic variety X admits a ℂ^{∗}-action if and only if the affine coordinate ring A admits a grading A = ⊕_{k}A_{k}. Following Orlik-Wagreich, we say that the action is good if A_{k} = 0 for k < 0 and A_{0} = ℂ. This means that the point 0 corresponding to the maximal ideal ⊕_{k>0}A_{k} is the only fixed point of the action. Additionally, we assume that (X, 0) is normal. (Notice also that by [29], if a normal surface singularities admits a good ℂ^{∗} action and its link is a rational homology sphere then it is ℚ-Gorenstein.)

For these singularities we prove:

1.1. Theorem. Let (X, 0) be a normal surface singularity with a good ℂ^{∗}-action whose link is a rational homology sphere. Then

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

In particular, if (X, 0) is Gorenstein and has a smoothing with Milnor fiber F, then its signature σ(F) satisfies: −sw_{M}^{0}(σ_{can}) = σ(F)/8.

The last statement follows from (∗) and from the well-known formula 8p_{g} + σ(F) + K^{2} + #V = 0 (valid for smoothings of Gorenstein singularities), see e.g. [22]. The proof of (∗) is based, in part, on Pinkham’s formula [36] for p_{g} expressed in terms of the Seifert invariants of the link (cf. also with Dolgachev’s work about weighted homogeneous singularities; see e.g. [6]). On the other hand, in the proof we use the formulae for K^{2}+#V and the Reidemeister-Turaev torsion determined in [28], and a formula for the Casson-Walker invariant proved in [20]. The Fourier transform of Reidemeister-Turaev torsion is very closely related to the (equivariant) Poincaré series associated with the graded coordinate ring of the universal abelian cover of (X, 0). In the proof we also borrowed some techniques used by Neumann in his investigation of this Poincaré series [29]. Nevertheless, our proof and results provide new interpretations of several of the coefficients of the Poincaré series, which were beyond the reach in the pre-Seiberg-Witten period (cf. also with section 4).

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

1.2. Corollary. Assume that the link of a normal surface singularity (X, 0) is a rational homology sphere Seifert 3-manifold. If (X, 0) is rational, or minimally elliptic, or Gorenstein elliptic, then the identity (∗) holds.

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

The paper is organized as follows. In section 2 we review the needed definitions and results. For a more complete picture and list of references the reader is invited to consult [28]. The first theorem of section 3 determines the Reidemeister-Turaev torsion of M (associated with an arbitrary spin^{c} structure) in terms of the Seifert invariants of M. This result, via 2.3(1) and 2.6(5), provides the complete Seiberg-Witten invariant of a Seifert manifold with e < 0. This is really remarkable (even independently of theorem 1.1), since, in general, Seiberg-Witten computations are difficult (see e.g. [26], [35] or [28] for the Seifert case) and only sporadic cases were completely clarified. The next result 3.2 connects four topological invariants of the link: the Reidemeister-Turaev torsion, the Casson-Walker invariant, the Dolgachev-Pinkham invariant DP_{M} (which is the topological candidate for p_{g}), and finally K^{2} + #V (which can be identified with the Gompf invariant, cf. 2.4). This result implies theorem 1.1 via Pinkham’s result [36] (cf. 2.6(11)).

Finally, in section 4, we analyze more closely the relationship with some of the coefficients of the Laurent expansions appearing in this article (e.g. of the Poincaré series), provided that (X, 0) is a complete intersection or hypersurface singularity. In the second case, using some results of K. Saito [38] and W. Ebeling [9], we also give a topological/geometrical interpretation of (generalized) Batyrev’s stringy Euler number [4] (as it is generalized by W. Veys in [41]). (This suggests a possible connection with Arnold’s strange duality and with the mirror symmetry of K3 surfaces, see the comments in [9] and [7].)

2. Preliminaries

2.1. The canonical spin^{c} structure of M. Let (X, 0) be a normal surface singularity. Its link M is a compact oriented 3-manifold. In this article we assume that M is a rational homology sphere, and we write H := H_{1}(M, ℤ).

The almost complex structure on X \ {0} determines a spin^{c} structure on X \ {0}, whose restriction σ_{can} ∈ Spin^{c}(M) to M depends only on the oriented C^{∞} type of M [28].

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

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

Then for each spin^{c} structure σ on M one defines the (σ, g, η)-Seiberg-Witten monopoles. For a generic parameter u, the Seiberg-Witten invariant sw_{M}(σ, u) is the signed monopole count. This integer depends on the choice of the parameter u and thus it is not a topological invariant. To obtain an invariant of M, one needs to alter this monopole count by the Kreck-Stolz invariant KS_{M}(σ, u), cf. [21] (or see [17] for the original “spin version”). Then, by [5, 21, 23], the rational number

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

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

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

𝒯_{M,σ} = ∑_{h∈H} 𝒯_{M,σ}(h) h ∈ ℚ[H]

the sign refined Reidemeister-Turaev torsion associated with σ (see [40]). We think of 𝒯_{M,σ} as a function H → ℚ given by h ↦ 𝒯_{M,σ}(h). Let λ(M) be the Casson-Walker invariant of M normalized as in [20, §4.7]. Then by a result of the second author [35], one has:

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

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

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

It is known that 𝒯̂_{M,σ}(1) = 0.

2.4. M as a plumbed manifold. Let π : X̃ → X be a resolution of the singular point 0 ∈ X such that the exceptional divisor E := π^{−1}(0) is a normal crossing divisor with irreducible components {E_{v}}_{v∈V}. Let Γ(π) be the dual resolution graph associated with π decorated with the self intersection numbers {E_{v} · E_{v}}_{v}. Γ(π) can be identified with a plumbing graph, and M with a plumbed 3-manifold constructed from Γ(π). Since M is a rational homology sphere, Γ(π) is a tree and any E_{v} is rational. Denote by δ_{v} the degree of any vertex v (i.e. #{w : E_{w} · E_{v} = 1}).

Let D_{v} be a small transversal disc to E_{v} in X̃. In fact, ∂D_{v} can be considered as the generic fiber of the S^{1}-bundle over E_{v} used in the plumbing construction of M. Consider the elements g_{v} := [∂D_{v}] (v ∈ V) in H. It is not difficult to verify that they generate H.

Next, we define the canonical cycle Z_{K} of (X, 0) associated with the resolution π. This is a rational cycle Z_{K} = ∑_{v∈V} r_{v}E_{v}, r_{v} ∈ ℚ, supported by the exceptional divisor E, and defined by (the adjunction formula):

Z_{K} · E_{v} = E_{v} · E_{v} + 2 for any v ∈ V.

Since the matrix {E_{v} ·E_{w}}_{v,w} is nondegenerate, this system has a unique (rational) solution. We write K^{2} for Z_{K} · Z_{K}, and #V for the number of irreducible components of E. Then K^{2} + #V does not depend on the choice of the resolution π, it is an invariant of M.

[One can define on M a canonical contact structure ξ_{can} induced by the natural almost complex structure on T M ⊕ ℝ_{M} with c_{1}(ξ_{can}) torsion (see e.g. [15, p. 420]). On the other hand, in [14], Gompf associates with such a contact structure ξ an invariant θ_{M}(ξ). It turns out, that θ_{M}(ξ_{can}) = K^{2} + #V − 2 (see [28, 4.8]).]

If (X, 0) is a normal surface singularity with a good ℂ^{∗}–action, then M is a Seifert 3manifold, and the minimal resolution graph is star-shaped. Hence, it is convenient to express the topological invariants of M in terms of the Seifert invariants. In the next subsections we recall briefly some definitions, notations and needed properties.

2.5. The Seifert invariants. [16, 29, 31] Consider a Seifert fibration π : M → Σ. In our situation, since M is a rational homology sphere, the base space Σ has genus zero (Σ ≈ S^{2}).

Consider a set of points {x_{i}}_{i=1}^{ν} in such a way that the set of fibers {π^{−1}(x_{i})}_{i} contains the set of singular fibers. Set O_{i} := π^{−1}(x_{i}). Let D_{i} be a small disc in X containing x_{i}, Σ′ := Σ \ ∪_{i}D_{i} and M′ := π^{−1}(Σ′). Now, π : M′ → Σ′ admits sections, let s : Σ′ → M′ be one of them. Let Q_{i} := s(∂D_{i}) and let H_{i} be a circle fiber in π^{−1}(∂D_{i}). Then in H_{1}(π^{−1}(D_{i}), ℤ) one has H_{i} ∼ α_{i}O_{i} and Q_{i} ∼ −β_{i}O_{i} for some integers α_{i} > 0 and β_{i} with (α_{i}, β_{i}) = 1. The set {(α_{i}, β_{i})}_{i=1}^{ν} constitute the set of (unnormalized) Seifert invariants. The number

e := − ∑ β_{i}/α_{i}

is called the orbifold Euler number of M. M is a link of singularity if and only if e < 0. Replacing the section by another one, a different choice changes each β_{i} within its residue class modulo α_{i} in such a way that the sum e = − ∑_{i}(β_{i}/α_{i}) is constant.

The set of normalized Seifert invariants {(α_{i}, ω_{i})}_{i=1}^{ν} are defined as follows. Write

e = b + ∑ ω_{i}/α_{i}  (2)

for some integer b, and 0 ≤ ω_{i} < α_{i} with ω_{i} ≡ −β_{i} (mod α_{i}). Clearly, these properties define {ω_{i}}_{i} uniquely. Notice that b ≤ e < 0. In the sequel we assume ν ≥ 3. (Recall that for cyclic quotient singularities 1.1.(∗) was verified in [28].)

For each i, consider the continued fraction α_{i}/ω_{i} = b_{i1} − 1/(b_{i2} − 1/(⋯ − 1/b_{iν_{i}}) ⋯). Then (a possible) plumbing graph of M is a star-shaped graph with ν arms. The central vertex has decoration b and the arm corresponding to the index i has ν_{i} vertices decorated by −b_{i1}, …, −b_{iν_{i}} (the vertex decorated by −b_{i1} is connected by the central vertex).

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

H = ab⟨gv̄_{0}, gv̄_{1}, … gv̄_{ν} | g−b ; v̄_{0} = ∏_{i=1}^{ν} gω_{i} ; v̄_{i}, gv̄_{0} = gα_{i} ; v̄_{i} for all i⟩.  (3)

Let α := lcm(α_{1}, …, α_{ν}). The order of the group H and the order o of the subgroup ⟨gv̄_{0}⟩ can be determined by (cf. [29]):

|H| = α_{1} ⋯ α_{ν}|e|, o := |⟨gv̄_{0}⟩| = α|e|.  (4)

2.6. Invariants computed from the plumbing graph. In the sequel we will also use Dedekind sums. They are defined as follows [37]. Let ⌊x⌋ be the integer part, and {x} := x − ⌊x⌋ the fractional part of x. Then

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

where ((x)) denotes the Dedekind symbol

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

Assume that M is a Seifert manifold with e < 0 and genus zero. Then one has the following formulae for some of its invariants.

• The Casson-Walker invariant. [20] (6.1.1):

−\frac{24}{|H|}λ(M) = \frac{1}{e}(2 − ν + ∑_{i=1}^{ν} \frac{1}{α_{i}^{2}} ) + e + 3 + 12 ∑_{i=1}^{ν} s(β_{i}, α_{i}).  (5)

• K^{2} + #V. [28] (5.4):

K^{2} + #V = \frac{1}{e}(2 − ν + ∑_{i=1}^{ν} \frac{1}{α_{i}} )^{2} + e + 5 + 12 ∑_{i=1}^{ν} s(β_{i}, α_{i}).  (6)

• The coefficient r_{0} of Ev̄_{0} in Z_{K} is (see e.g. [28], 5.2 and 5.5):

r_{0} = 1 + \frac{1}{e}(2 − ν + ∑_{i=1}^{ν} \frac{1}{α_{i}} ).  (7)

R := r_{0} − 1 is called the “exponent of (X, 0)”, and −R the “log discrepancy of Ev̄_{0}”. Some authors (see e.g. [29]) prefer to use the notation χ_{M} := 2 − ν + ∑_{i=1}^{ν}(1/α_{i}) = eR as well.

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

P̂_{χ}(t) := \frac{(t^{α}χ(gv̄_{0}) − 1)^{ν−2}}{∏_{i=1}^{ν} (t^{α/α_{i}} χ(gv̄_{i}) − 1)}.  (8)

Recall that Spin^{c}(M) is an H-torsor. For an arbitrary σ ∈ Spin^{c}(M) take the unique h_{σ} ∈ H so that h_{σ} · σ_{can} = σ. Then, by [28, 5.7, 5.8], one has:

𝒯̂_{M,σ}(χ̄) = χ̄(h_{σ}) · lim_{t→1} P̂_{χ}(t) for any χ ∈ Ĥ \ {1}.  (9)

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

DP_{M} := ∑_{l≥0} max (0 , −1 + lb − ∑_{i=1}^{ν} ⌊ \frac{−lω_{i}}{α_{i}} ⌋ ).  (10)

Assume that (X, 0) is a normal surface singularity with a good ℂ^{∗}–action (see e.g. [36]) such that its link M is a rational homology sphere. Then, by [36], (5.7):

p_{g}(X, 0) = DP_{M}.  (11)

3. The main results

In [28] (cf. 2.6(8-9)) we determined the Fourier transforms 𝒯̂_{M,σ}(χ) (χ ≠ 1) of the Reidemeister-Turaev torsion of M in terms of a regularization limit. The first theorem of this section provides 𝒯_{M,σ} (for any spin^{c} structure σ) in terms of the Seifert invariants of M.

3.1. Theorem. Assume that M is a Seifert 3-manifold with e < 0 and genus zero. Fix an arbitrary spin^{c} structure σ ∈ Spin^{c}(M) characterized by h_{σ} · σ_{can} = σ. Write h_{σ} as

ga_{0} ; v̄_{0}ga_{1} ; v̄_{1} ⋯ ga_{ν} ; v̄_{ν} for some integers a_{0}, a_{1}, …, a_{ν}. Finally, define ã := α · (a_{0} + ∑_{i} a_{i}/α_{i}). Then

\frac{1}{|H|} ∑_{χ∈Ĥ} χ̄(h_{σ})P̂_{χ}(t) = ∑_{l∈ℤ} max ( 0 , 1 + a_{0} − lb + ∑_{i=1}^{ν} ⌊ \frac{−lω_{i} + a_{i}}{α_{i}} ⌋ ) t^{ol+ã}.

(Here, there exists an integer m, such that for l < m the contribution in the above sum is zero.) Therefore,

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

In particular, for σ = σ_{can} or for h_{σ} = 1, one can take a_{0} = a_{1} = ⋯ a_{ν} = ã = 0, hence

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

Proof. Notice that there is a “mysterious” similarity between our formula (8)-(9) for the Fourier transform of the Reidemeister-Turaev torsion, and the formula [29, 4.2] of W. Neumann of the Poincaré series of the graded affine ring associated with the universal abelian cover of (X, 0). The idea of the next proof is based on Neumann’s computation of this graded ring in [29], page 241.

Using the identity gα_{i} ; v̄_{i} = gv̄_{0} in H (cf. 2.5(3)), first write χ̄(h_{σ})P̂_{χ}(t) as

χ̄(h_{σ}) · (1 − t^{α}χ(gv̄_{0}))^{−2} ∏_{i} \frac{1 − ( t^{α/α_{i}}χ(gv̄_{i}))^{α_{i}}}{1 − t^{α/α_{i}} χ(gv̄_{i})}

= χ̄(h_{σ})( ∑_{s_{0}=0}^{∞} (1 + s_{0})χ(gv̄_{0})^{s_{0}} t^{αs_{0}}) · ∏_{i} ∑_{s_{i}=0}^{α_{i}−1} t^{s_{i}α/α_{i}}(χ(gv̄_{i}))^{s_{i}}

= ∑ (1 + s_{0}) t^{αs_{0}+∑_{i} αs_{i}/α_{i}} χ(gs_{0}−a_{0} ; v̄_{0}gs_{1}−a_{1} ; v̄_{1} ⋯ gs_{ν}−a_{ν} ; v̄_{ν}),

where the (unmarked) sum is over s_{0} ≥ 0 and 0 ≤ s_{i} < α_{i} for each i. But ∑_{χ∈Ĥ} χ(h) is non-zero only if h = 1, and in that case it is |H|. Using the group structure 2.5(3) one gets that all the relations in H have the form

gl_{1}+···+l_{ν}−lb ; v̄_{0} ∏_{i} g−ω_{i}l−α_{i}l_{i} ; v̄_{i} = 1,

where l_{1}, …, l_{ν} and l are integers. Therefore, gs_{0}−a_{0} ; v̄_{0}gs_{1}−a_{1} ; v̄_{1} ⋯ gs_{ν}−a_{ν} ; v̄_{ν} = 1 if and only if s_{0} = a_{0} + l_{1} + ⋯ + l_{ν} − lb and s_{i} = a_{i} − ω_{i}l − α_{i}l_{i} (1 ≤ i ≤ ν) for some integers l_{1}, …, l_{ν}, l. Since 0 ≤ s_{i} < α_{i} one obtains that

l_{i} = ⌊ \frac{−lω_{i} + a_{i}}{α_{i}} ⌋.

In particular,

1 + s_{0} = 1 + a_{0} − lb + ∑_{i} ⌊ \frac{−lω_{i} + a_{i}}{α_{i}} ⌋,

and only those integers l are allowed for which this number 1 + s_{0} is ≥ 1. If h_{σ} = 1, it is easy to see that this cannot happen for l < 0. Indeed, for l < 0

−lb + ∑_{i} ⌊ \frac{−lω_{i}}{α_{i}} ⌋ ≤ −lb + ∑_{i} \frac{−lω_{i}}{α_{i}} = −le < 0.

The exponent α(s_{0} + ∑_{i} s_{i}/α_{i}) of t is −lαe + ã = lo + ã by 2.5(4). Finally recall that 𝒯̂_{M,σ}(1) = 0 (cf. 2.3), hence 𝒯_{M,σ}(1) follows from the Fourier inversion 2.3 and 2.6(9). □

Theorem 1.1 is the consequence of the following key identity.

3.2. Theorem. Let M be a Seifert 3–manifold with e < 0 of genus zero (i.e. M is a rational homology 3-sphere). Then the invariants 𝒯_{M,σ_{can}}(1), λ(M), K^{2} + #V and DP_{M} are connected by the following identity:

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

The proof of is carried out in several steps.

3.3. Corollary. 3.1 implies the following:

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

Proof. Use 2.6(10), 3.1 and the identity max(0, x) − max(0, −x) = x. □

On the right hand side we have a difference of two series, both having poles of order two at t = 1. The next results provide their Laurent series at t = 1. In fact, we prefer to expand the series in terms of the powers of t^{o} − 1 (instead of t − 1).

3.4. Proposition. Recall the definition of χ_{M} after 2.6(7). Then

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

with lim_{t→1} R(t) = 0.

Proof. The first step is as in [29], page 241. The left hand side of 3.4, via 2.5(2) becomes

∑_{l≥0} ( − le + \frac{χ_{M}}{2} ) t^{ol} + ∑_{i=1}^{ν} ∑_{l≥0} ( − \{ \frac{−lω_{i}}{α_{i}} \} + \frac{α_{i} − 1}{2α_{i}} ) t^{ol}.

Evidently

∑_{l≥0} ( − le + \frac{χ_{M}}{2} ) t^{ol} = \frac{−et^{o}}{(1 − t^{o})^{2}} + \frac{χ_{M}/2}{1 − t^{o}},

which gives the non-holomorphic part. The second contribution is a sum over 1 ≤ i ≤ ν. For each fixed i, write l = α_{i}m + q with m ≥ 0 and 0 ≤ q < α_{i}. Using the notation ∑_{q} := ∑_{q=0}^{α_{i}−1} and ∑′_{q} := ∑_{q=1}^{α_{i}−1}, the i^{th} summand is

∑_{q} ( − \{ \frac{−qω_{i}}{α_{i}} \} + \frac{α_{i} − 1}{2α_{i}} ) ∑_{m≥0} t^{oα_{i}m+oq} = \frac{∑_{q} ( − \{ \frac{−qω_{i}}{α_{i}} \} + \frac{α_{i}−1}{2α_{i}} ) t^{oq}}{1 − t^{oα_{i}}} .

Separating the two cases q = 0 and q > 0, and using the definition of the Dedekind symbol and the identity {−x} = 1 − {x} for x ∉ ℤ, this is transformed into

A(t) := \frac{\frac{α_{i}−1}{2α_{i}} + ∑′_{q} ((\frac{qω_{i}}{α_{i}} ))t^{oq} − \frac{1}{2α_{i}} (t^{o} + t^{2o} + ⋯ + t^{(α_{i}−1)o})}{1 − t^{oα_{i}}}.

By L’Hospital theorem (and by some simplifications):

lim_{t→1} A(t) = −∑_{q}′((\frac{qω_{i}}{α_{i}} ))\frac{q}{α_{i}} + \frac{α_{i} − 1}{4α_{i}}.

Since ∑′_{q}((qω_{i}/α_{i})) = 0 and ω_{i} ≡ −β_{i} (mod α_{i}), the result follows from the definition of the Dedekind symbol and the Dedekind sums. □

3.5. Remarks. (Cf. also with [29], page 242.) In fact,

∑_{l≥0} ( 1 − lb + ∑_{i=1}^{ν} ⌊ \frac{−lω_{i}}{α_{i}} ⌋ ) t^{ol} = \frac{−e}{(t^{o} − 1)^{2}} + \frac{−e − χ_{M}/2}{t^{o} − 1} + ∑_{i=1}^{ν} \frac{1}{α_{i}} ∑_{ξ∈ℤ_{α_{i}}}′ \frac{1}{(1 − ξ)(1 − ξ^{ω_{i}}t^{o})},

where the last sum is over ξ^{α_{i}} = 1, ξ ≠ 1. Since we do not need this statement now, we will skip its proof. The interested reader can prove it easily using the expression A(t) above and the property (16c) of the Dedekind symbol from [37], page 14.

3.6. Proposition. P̂_{1}(t)/|H| has the following Laurent expansion:

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

where lim_{t→1} Q(t) = 0 and

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

Proof. First notice that one has the following Taylor expansion;

\frac{t^{γ} − 1}{t^{τ} − 1} = \frac{γ}{τ} + \frac{γ}{2oτ} (γ − τ) · (t^{o} − 1) + \frac{γ}{oτ} (γ − τ)(\frac{2γ − τ}{12o} − \frac{1}{4}) · (t^{o} − 1)^{2} + · · · .

Now, use this formula ν + 2 times in the expression

P̂_{1}(t) = \frac{1}{(t^{o} − 1)^{2}} · (\frac{t^{o} − 1}{t^{α} − 1})^{2} · ∏_{i} \frac{t^{α} − 1}{t^{α/α_{i}} − 1}.

A long (but elementary) computation, together with 2.5(4), gives the result. □

Proof of 3.2. Apply 2.6(5) and (6), respectively 3.3, 3.4 and 3.6. The verification is elementary. □

3.7. Corollary. Assume that (X, 0) is Gorenstein and admits a smoothing with Milnor fiber F. Then the topological Euler characteristic χ(F) of F satisfies:

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

(Notice that because of [13], the first Betti number of F is zero. Hence χ(F) = 1 + μ, where μ is the Milnor number of the smoothing.)

Proof. By generalization of Steenbrink [39] of Laufer’s formula [19] one has μ = 12p_{g} +K^{2} + #V. Then use 1.1 (or 3.2) and 2.6(5-6). □

Assume that in 3.7 one has |H| = 1. Then in the identity 𝒯_{M,σ_{can}}(1) = 0, and −1/e = α = ∏_{i} α_{i}, cf. 2.5(4). Moreover, in this case, by [29], (X, 0) is a Brieskorn-Hamm complete intersection X(α_{1}, …, α_{ν}). For such singularities the Milnor number is computed in [12](3.10.b). This formula agrees with our identity (simplified for |H| = 1).

4. Examples

4.1. The Poincaré series. Let (X, 0) be a normal singularity with a good ℂ^{∗}-action and affine graded coordinate ring A = ⊕_{k}A_{k}. Then its Poincaré series is defined by p_{(X,0)}(t) = ∑_{k} dim(A_{k})t^{k}. The point is that the last expression from 3.1 involves exactly p_{(X,0)}(t^{o}), provided that the genus of (X, 0) is zero. More precisely (cf. [29], page 241):

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

Moreover, if (X_{ab}, 0) denotes the universal abelian cover of (X, 0), then p_{(X_{ab},0)}(t) = P̂_{1}(t) (cf. [29], page 240). Therefore, 3.1 reads as follows:

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

Notice that for many special families, the Poincaré series p_{(X,0)}(t) is computed very explicitly, see e.g. [42]. For p_{(X_{ab},0)}(t) = P̂_{1}(t) one can use the expression 2.6(8) which provides it in terms of the Seifert invariants of M.

Notice that both Poincaré series are rational function with pole of order 2 at t = 1 (the Laurent expansion of P̂_{1}(t) is given in 3.6). In the next examples we will emphasize the first three terms of the corresponding Laurent expansions.

4.2. Complete intersections. The above formula provides p_{(X,0)}(t) in terms of the Seifert invariants. Nevertheless, if (X, 0) ⊂ (ℂ^{n}, 0) is a complete intersection with weights q_{1}, …, q_{n} (where gcd(q_{1}, …, q_{n}) = 1) and degrees d_{1}, …, d_{n−2}, then one also has (cf. e.g. with [42]):

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

Similarly as in 3.6, one can determine its Laurent expansion at t = 1:

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

where lim_{t→1} U(t) = 0 and

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

By (†), the coefficients of (t^{o} − 1)^{k} (k = −2, −1) of p_{(X,0)}(t^{o}) and P̂_{1}(t)/|H| should agree. By this (and by 3.6), we recover two identities well-known by specialists ([8], or [42] for the hypersurface case):

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

where R is the “exponent of (X, 0)”, see 2.6(7). On the other hand, the interpretation of the coefficients of (t^{o} − 1)^{0} as the Reidemeister-Turaev torsion is the novelty of the present article:

𝒯_{M,σ_{can}}(1) = −e · F − E,

where F is given above and E in 3.6.

4.3. Hypersurface singularities. Assume that (X, 0) ⊂ (ℂ^{3}, 0) is a hypersurface weighted homogeneous singularity. Then (†) can be rewritten in a rather surprising form.

Denote by Δ(t) the characteristic polynomial of the algebraic monodromy operator. By a recent result of W. Ebeling [9], Δ(t) is in a very subtle relationship with p_{(X,0)}(t). In order to explain this, we need to recall K. Saito’s duality [38].

Fix an integer h > 0 and assume that ϕ(t) is a rational function of the form

ϕ(t) = ∏_{m|h}(1 − t^{m})^{χ_{m}}.  (∗)

Then K. Saito in [38] defined a dual rational function (with respect to the integer h) by

ϕ^{∗}(t) = ∏_{k|h}(1 − t^{k})^{−χ_{h/k}}.

In this article we wish to eliminate the dependency of the duality on the integer h, and we will use the following principle. If ϕ(t) is a rational function of the form ∏_{m}(1 − t^{m})^{χ_{m}}, then we take h = h(ϕ) := lcm{m : χ_{m} ≠ 0}, and we define ϕ^{∗}(t) using this h = h(ϕ). It is not difficult to prove that if gcd{m : χ_{m} ≠ 0} = 1 then h(ϕ^{∗}) = h(ϕ) and ϕ^{∗∗}(t) = ϕ(t).

Let M be the link of (X, 0) with Seifert invariants as above. Following Ebeling, we write

ψ(t) := (1 − t)^{2−ν} ∏_{i=1}^{ν}(1 − t^{α_{i}}),

and ϕ(t) := p_{(X,0)}(t) · ψ(t). Notice that Δ(t) always can be written in the form (∗) with gcd{m : χ_{m} ≠ 0} = 1 (in fact, χ_{1} ≠ 0, cf. [1]). Then by [9], ϕ also has form (∗), and

Δ(t) = ϕ^{∗}(t) or Δ^{∗}(t) = ϕ(t).

[In [9] it is not explicitly said that h = h(ϕ), but it can be verified using [25].]

This has the following connection with our result. Notice that ψ^{∗}(t) = P̂_{1}(t), hence by (†):

𝒯_{M,σ_{can}}(1) = lim_{t→1} [ϕ(t^{o})ψ^{−1}(t^{o}) − \frac{1}{|H|}ψ^{∗}(t)] = lim_{t→1} [ϕ(t^{o}) − \frac{1}{|H|}ψ^{∗}(t)ψ(t^{o})] · ψ^{−1}(t^{o}).

Since

lim_{t→1} ψ^{−1}(t^{o})(t^{o} − 1)^{2} = lim_{t→1} \frac{(t^{o} − 1)^{ν}}{∏_{i}(t^{α_{i}o} − 1)} = ∏_{i} \frac{1}{α_{i}} = \frac{|e|}{|H|},

via Ebeling’s result and ψ^{∗}(t) = P̂_{1}(t) one gets:

𝒯_{M,σ_{can}}(1) = \frac{|e|}{|H|} · lim_{t→1} \frac{Δ^{∗}(t^{o}) − P̂_{1}(t)P̂_{1}^{∗}(t^{o})/|H|}{(t^{o} − 1)^{2}}.  (‡)

Similarly as in 3.6, one can write the following Taylor expansion at t = 1 (with respect to t^{o} − 1):

\frac{1}{|H|}P̂_{1}(t)P̂_{1}^{∗}(t^{o}) = \frac{1}{|H|}(\frac{1 − t^{α}}{1 − t^{o}} )^{ν−2}·∏_{i} \frac{1 − t^{oα_{i}}}{1 − t^{α/α_{i}}} = |H|+\frac{1}{2}|H|(R+2−ν+_{∑i}α_{i})(t^{o}−1)+· · · .

If we examine our list in 2.6, we realize that the numerical expression 2 − ν + ∑ α_{i} is not involved in them. But, it is involved in the stringy Euler number computation of W. Veys [41](6.4). More precisely, let e_{st} denote Batyrev’s stringy Euler characteristic of (X, 0) (cf.

12 András Némethi and Liviu I. Nicolaescu

[4]) as it is generalized by Veys in [41]. Then, for our germ (X, 0) (provided that it is not strictly log canonical), [41](6.4.ii) reads as

e_{st} = −(2 − ν + ∑ α_{i})/R.

The stringy Euler characteristic can be defined by motivic integration or using the resolution of (X, 0), and people asked for a possible geometrical/topological interpretation of it.

Notice that by (‡), the first two terms of the corresponding expansions agree. The equality of the first term implies Δ^{∗}(1) = |H| (a fact which is not very deep, it can also be deduces from the definition of Saito’s duality and from Δ(1) = |H|). But the equality of the second terms provides the above wanted interpretation:

\frac{dΔ^{∗}}{dt} (1) = |H| · R · (1 − e_{st})/2.

Finally, the next Taylor coefficient of Δ^{∗}(t) involves 𝒯_{M,σ_{can}}(1).

We end with the following remark. It is interesting to compare the Taylor expansions

\frac{Δ^{∗}(t)}{|H|} = 1 + \frac{R}{2}(1 − e_{st})(t − 1) + ⋯ and \frac{Δ(t)}{|H|} = 1 + \frac{μ}{2}(t − 1) + · · · .

Here μ is the Milnor number of (X, 0) (and the second expansion follows easily from [1]). This shows that the Milnor number, respectively R(1 − e_{st}), correspond to each other via some duality (which extends Arnold’s strange duality).

For a different application of Ebeling’s result [9] (in the spirit of the present article), see [24].

References

[1] N. A’Campo: La fonction zeta d’une monodromy, Com. Math. Helvetici 50 (1975), 233-248.

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

[3] On isolated rational singularities of surfaces, Amer. J. of Math., 88(1966), 129-136.

[4] V. Batyrev: Stringy Hodge numbers of varieties with Gorenstein canonical singularities, Proc. Taniguchi Symposium 1997, In “Integrable Systems and Algebraic Geometry, Kobe/Kyoto, 1997”, World Sci. Publ. (1999), 1-32.

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

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

[7] I. V. Dolgachev: Mirror symmetry for lattice polarized K3 surfaces, J. Math. Sci., 81 (1996), 2599-2630.

[8] I. V. Dolgachev: private communication.

[9] W. Ebeling: Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity, Manuscripta Math., 107 (2002), 271-282.

[10] R. Fintushel and R. Stern: Instanton homology of Seifert fibered homology three spheres, Proc. London Math. Soc., 61(1990), 109-137.

[11] M. Furuta, B. Steer: Seifert fibered homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points, Adv. in Math., 96(1992), 38-102.

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

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

[14] R.E. Gompf: Handlebody construction of Stein surfaces, Ann. of Math., 148 (1998), 619-693.

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

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

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

Seiberg-Witten invariants and surface singularities II 13

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

[19] H. B. Laufer: On μ for surface singularities, Proc. of Symp. in Pure Math., 30, 45-49 (1977).

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

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

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

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

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

[25] J. Milnor and P. Orlik: Isolated singularities defined by weighted homogeneous polynomials, Topology, 9 (1970), 385-393.

[26] T. Mrowka, P. Ozsvath, B. Yu: Seiberg-Witten monopoles on Seifert fibered spaces, Comm. Anal. and Geom., 5(1997), 685-791.

[27] A. Némethi: “Weakly” elliptic Gorenstein singularities of surfaces, Invent. math. 137 (1999), 145-167.

[28] A. Némethi, L.I. Nicolaescu: Seiberg-Witten invariants and surface singularities, preprint math.AG/0111298. Geometry and Topology, 6(2002), 369-328.

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

[30] A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Transactions of the AMS, 268(1981), 299-344.

[31] W.D. Neumann and F. Raymond: Seifert manifolds, plumbing, μ-invariant and orientation reserving maps, Algebraic and Geometric Topology (Proceedings, Santa Barbara 1977), Lecture Notes in Math. 664, 161-196.

[32] W. Neumann and J. Wahl: Casson invariant of links of singularities, Comment. Math. Helvetici, 65(1990), 58-78.

[33] Universal abelian covers of surface singularities, math.AG/0110167.

[34] L.I. Nicolaescu: Finite energy Seiberg-Witten moduli spaces on 4-manifolds bounding Seifert fibrations, Comm. Anal. Geom. 8(2000), 1027-1096.

[35] : Seiberg-Witten invariants of rational homology spheres, math.DG/0103020.

[36] H. Pinkham: Normal surface singularities with ℂ^{∗} action, Math. Ann. 117(1977), 183-193.

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

[38] K. Saito: Duality for regular systems of weights: a précis. In: Topological Field Theory, Primitive Forms and Related Topics (M. Kashiwara, A. Matsuo, K. Saito, I. Satake, eds.), Progress in Math., 160, Birkhäuser, 1998, 379-426.

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

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

[41] W. Veys: Stringy invariants of normal surfaces, arXiv:math.AG/0205293.

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

[43] S. S.-T. Yau: On maximally elliptic singularities, Transact. AMS, 257(1980), 269-329.

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

   E-mail address: nemethi@math.ohio-state.edu
   URL: http://www.math.ohio-state.edu/ ~nemethi/

University of Notre Dame, Notre Dame, IN 46556

   E-mail address: nicolaescu.1@nd.edu
   URL: http://www.nd.edu/ ~lnicolae/