Contemporary Mathematics

Poincaré series associated with surface singularities

András Némethi

Dedicated to Lê Dũng Tráng on his 60th birthday

Abstract. We unify and generalize formulas obtained by Campillo, Delgado and Gusein-Zade in their series of articles (see e.g. [?, ?, ?]). Positive results are established for rational and minimally elliptic singularities. By examples and counterexamples we also try to find the ‘limits’ of these identities. Connections with the Seiberg-Witten Invariant Conjecture and Semigroup Density Conjecture are discussed.

1. Introduction

In a long series of articles, Campillo, Delgado and Gusein-Zade established an identity of type Z = P, see e.g. [?, ?, ?] (the interested reader is invited to consult all their other articles in this subject as well). The identity, in different articles had different versions and targeted slightly different objects. One of the versions was valid for rational normal surface singularities (X, o) and for one of its fixed resolutions π; Z was a topological invariant expressed from the combinatorics of the dual graph, while P was a multi-variable Poincaré series associated with a filtration given by the divisorial valuations of the irreducible exceptional divisors. In the equivariant setting, an action of a finite abelian group was present too. Finally, in a relative version, they considered an embedded curve C into the rational singularity (X, o), Z was an Alexander type topological invariant expressed from an embedded resolution graph, while P the Poincaré series of a filtration associated with the evaluations provided by the normalization of C.

The authors provide independent proofs for the different cases; the proofs are based on the structure of arc-spaces and an ‘euler-characteristic summation’ (suggested by the motivic integration).

The present article has several goals.

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

2000 Mathematics Subject Classification. Primary 32C35, 32S05, 32S25, 32S45.

Key words and phrases. plumbed 3-manifolds, rational homology spheres, surface singularities, universal abelian covers, Poincaré series, Hilbert series, Campillo–Delgado–Gusein-Zade formula, Seiberg–Witten invariant conjecture, rational singularities, minimally elliptic singularities, weighted homogeneous singularities, splice-quotient singularities, superisolated singularities.

The author is partially supported by OTKA grants.

First, we create an uniform language to incorporate all the above different versions (the absolute, equivariant and relative) in only one setting. Moreover, we show that the statement of the absolute version (the ‘Main Identity’ ??) stated for all possible good resolutions implies all the other versions. All the statements are formulated for any normal surface singularity (with rational homology sphere link); the formulation involves heavily the universal abelian cover of (X, o). Then we establish the identities for some families of singularities (see below). In the presentation we embed the statement and the proof into the classical singularity theory: the presentation is in the spirit of the work of Artin and Laufer. In fact, one of the crucial arguments is the vanishing theorem of Röhr (a Laufer, or Grauert-Riemenschneider type vanishing). In this way, the present proof not only recovers the result of Campillo, Delgado and Gusein-Zade for rational singularities, but automatically extends it to the minimally elliptic singularities as well. The article also establishes a new realization of the topological invariant Z, expressed in terms of (analytic) euler-characteristics of cycles supported by the exceptional locus, which spans as a bridge the topological and analytic invariants (see (??)). Its formula might have an independent interest as well.

In fact, the new unifying language also enables us to formulate the relative version also in a much higher generality (compared with [?]).

We would like to stress, that although in the case of rational singularities it is unimportant which resolution we choose, for all other singularities this is crucial. For minimally elliptic singularities we establish our positive results for the minimal (good) resolution, and by examples we show pathologies valid for non-minimal ones.

The machinery also allows us to reduce the number of variables. Surprisingly, if we simplify (rather much) the identities, e.g. we consider the reduced identities with one variable, even by these simplified versions we recover famous classical/older (in general, hardly non-trivial) results already present in the literature. Our wish is to present some of them (regarding e.g. weighted homogeneous, superisolated or splice-quotient singularities). The relation with the Seiberg-Witten conjecture [?] and the Semigroup Density conjecture (regarding the existence of unicuspidal rational projective plane curves) [?] is striking (cf. (??)). (In fact, this connection was the motivation of the author in the subject.)

Finally, by examples, we try to establish the ‘limits’ of the identity Z = P. We provide counterexamples and explicit methods which provide examples with Z ≠ P, showing serious geometric reasons which obstruct the identity, in general.

2. Notations and preliminaries

2.1. The resolution. Let (X, o) be an isolated complex analytic normal surface singularity. Let π: X̃ → X be a good resolution with exceptional set E with irreducible components E_{j}, j ∈ V := {1, …, s}. Let Γ denote its dual resolution graph (for details see e.g. [?]). For any j ∈ V we write δ_{j} for the valency of j in Γ.

We assume that the link Σ of X is a rational homology sphere, i.e. Γ is a connected tree and E_{j} ≈ ℙ^{1} for every j.

Set L := H_{2}(X̃, ℤ) and L′ := H_{2}(X̃, Σ, ℤ). These groups are free with bases the classes of E_{i} and their duals E_{i}^{∗}. Here, we prefer the sign convention (−E_{i}^{∗}, E_{j}) = δ_{ij} (the Kronecker delta function). The matrix I of the inclusion L ↪ L′ in the basis {E_{i}}_{i} of L and {−E_{i}^{∗}}_{i} of L′ is exactly the intersection matrix (E_{i}, E_{j})_{i,j}.

We denote the group L′/L ≅ H_{1}(Σ, ℤ) by H and set [l′] for the class of l′ ∈ L′. Let |H| and Ĥ denote its order and Pontrjagin dual Hom(H, ℂ^{∗}), respectively. There is a natural isomorphism θ : H → Ĥ given by [l′] ↦ e^{2πi(l′,·)}. Sometimes we also write d := |H|.

2.2. Positive cones and representatives. We write L_{ℚ} for the group of rational cycles L ⊗ ℚ. The form (·, ·) has a natural extension to L_{ℚ}.

A rational cycle x = ∑_{j} r_{j}E_{j} ∈ L_{ℚ} is called effective, denoted by x ≥ 0, if r_{j} ≥ 0 for all j. Their cone is denoted by L_{ℚ,e}, while L′_{e} := L_{ℚ,e} ∩ L′ and L_{e} := L_{ℚ,e} ∩ L.

A rational cycle x ∈ L_{ℚ} is called numerical effective if (x, E_{j}) ≥ 0 for all j. Their cone is denoted by NE_{ℚ}. We also write (for Lipman’s cones) S_{ℚ} := −NE_{ℚ}, S′ := S_{ℚ} ∩ L′ and S := S_{ℚ} ∩ L. Notice that both S′ and S are semigroups, S′ is generated over ℕ by the vectors E_{j}^{∗}. One can show (using the fact that I is negative definite) that E_{j}^{∗} ∈ L′_{e} for all j.

Define the ‘unit cube’ in L′ by

Q := {r ∈ L′ : r = ∑_{j} r_{j}E_{j} with r_{j} ∈ [0, 1)}.

For any h ∈ H, we denote the unique representative of H from Q by r_{h}.

2.3. The ring ℤ[[t^{L′}]]. Consider the notations of (??). We denote by ℤ[[t^{±1}]] the ring of formal power series ℤ[[t_{1}^{±1}, …, t_{s}^{±1}]] (where s = |V|). In fact, it is convenient to consider a larger ring as well, the ring ℤ[[t^{±1/d}]] = ℤ[[t_{1}^{±1/d}, …, t_{s}^{±1/d}]] of formal power series in variables t_{j}^{±1/d}. ℤ[[t^{±1/d}]] has a natural sub-ring associated with the resolution π. Namely, ℤ[[t^{L′}]] is the ℤ-linear combinations of monomials of type

t^{l′} = t_{1}^{l′_{1}} ⋯ t_{s}^{l′_{s}}, where l′ = ∑_{j} l′_{j}E_{j} ∈ L′.

This admits several sub-rings, e.g. ℤ[[t^{L′_{e}}]], or ℤ[[t^{S′}]], generated by monomials t^{l′} with l′ ∈ L′_{e}, or l′ ∈ S′ respectively.

In fact, ℤ[[t^{S′}]] is a usual formal power series ring in variables t^{E_{j}^{∗}}. More precisely, any of its element has the form

(2.3.1)  Φ(f)(t) := f(t^{E_{1}^{∗}}, …, t^{E_{s}^{∗}}), where f(x_{1}, …, x_{s}) ∈ ℤ[[x]].

Definition 2.3.2. Any series S(t) = ∑_{l′} a_{l′}t^{l′} ∈ ℤ[[t^{L′}]] decomposes in a unique way as

(2.3.3)  S = ∑_{h∈H} S_{h}, where S_{h} = ∑_{[l′]=h} a_{l′}t^{l′}.

S_{h} is called the h-component of S.

Lemma 2.3.4. Consider F (t) := Φ(f)(t) for some f ∈ ℤ[[x]] as in (??). Then

F_{h}(t) = \frac{1}{|H|} · ∑_{ρ∈Ĥ} ρ(h)^{−1} · f(ρ([E_{1}^{∗}])t^{E_{1}^{∗}}, …, ρ([E_{s}^{∗}])t^{E_{s}^{∗}}).

Proof. First notice that ∑_{h} F_{h} = F, hence it is enough to show that F_{h} is an h-component. Indeed, if ∏ x_{j}^{n_{j}} is a monomial of f, and l′ := ∑ n_{j}E_{j}^{∗}, then (1/d) ∑_{ρ} ρ(−h)ρ([l′])t^{l′} is t^{l′} if [l′] = h and zero otherwise. □

3. The topological/combinatorial invariant

3.1. The rational functions ζ(t). Our main topological object is the rational function (in variables x_{j} = t^{E_{j}^{∗}}), or its Taylor expansion at the origin (cf. ??):

(3.1.1)  ζ(t) := Φ(z)(t), where z(x) := ∏_{j∈V}(1 − x_{j})^{δ_{j}−2}.

Notice that by (??), its h-component is

(3.1.2)  ζ_{h}(t) := \frac{1}{|H|} · ∑_{ρ∈Ĥ} ρ(h)^{−1} · ∏_{j∈V} (1 − ρ([E_{j}^{∗}])t^{E_{j}^{∗}})^{δ_{j}−2}.

ζ_{h} is the ‘multivariable’ version of the rational function used by [?, ?, ?] in the Reidemeister-Turaev computation of Σ, by [?] in the computation of the geometric genus of splice-quotient singularities, by [?, ?] in the proof of the Neumann-Wahl Casson Invariant Conjecture for splice-quotients; see also [?]. The multivariable version appears in [?]. See also the comments of (??).

3.2. The CDG-series. In their series of articles, Campillo, Delgado and Gusein-Zade (see e.g. [?, ?]) considered the next infinite series. In fact, it tautologically equals the Taylor expansion at the origin of ζ. One starts with the following coefficients: for any n ≥ 0 and δ ≥ 0, set χ_{δ,n} as the coefficient of x^{n} in the Taylor expansion at the origin of the the function (1 − x)^{δ−2}. Then, with the notation l′ = ∑_{j} n_{j}E_{j}^{∗} ∈ S′, set

(3.2.1)  Z(t) := ∑_{l′∈S′} ( ∏_{j∈V} χ_{δ_{j},n_{j}}) · t^{l′}.

Since Z = Φ(∑_{n≥0}(∏_{j} χ_{δ_{j},n_{j}})x^{n}) = Φ(z), one gets:

Proposition 3.2.2. The Taylor expansion at the origin (in variables t^{E_{i}^{∗}}) of ζ is the series Z(t).

3.3. Next, we present a new (to the best of the author’s knowledge), and more subtle appearance of ζ(t). It uses the euler-characteristic of cycles supported by E, hence (as we will see) it creates the right bridge connecting topology with sheaf-theoretical invariants. In order to state it, we need some preparation.

As usual, let KX̃ ∈ L′ be the canonical cycle associated with π. It is identified by the equations (KX̃ + E_{j}, E_{j}) = −2 for any j. The (Riemann-Roch) eulercharacteristic χ(l) = −(l, l + KX̃)/2 can be extended to any l′ ∈ L_{ℚ} by χ(l′) = −(l′, l′ + KX̃)/2. Clearly,

(3.3.1)  χ(l′_{1} + l′_{2}) = χ(l′_{1}) + χ(l′_{2}) − (l′_{1}, l′_{2}).
For any subset I ⊂ V we write E_{I} := ∑_{j∈I} E_{j}.

Definition/Lemma 3.3.2. (1) For any fixed l′ ∈ L′ and subset I ⊂ V, there is a unique minimal subset J(l′, I) ⊂ V which contains I, and such that

(3.3.3)  there is no j ∈ V \ J(l′, I) with (E_{j}, l′ + E_{J(l′,I)}) > 0.

(2) J(l′, I) can be found by the following algorithm: one constructs a sequence {I_{m}}_{m=0}^{k} of subsets of V, with I_{0} = I, I_{m+1} = I_{m} ∪ {j(m)}, where the index j(m) is determined as follows. Assume that I_{m} is already constructed.

If I_{m} satisfies (??) we stop and m = k. Otherwise, there exists at least one j with (E_{j}, l′ + E_{I_{m}}) > 0. Take j(m) one of these j and continue the algorithm with I_{m+1}. Then I_{k} = J(l′, I).

Proof. For (??) notice that if J_{1} and J_{2} satisfies the wished requirement (??) of J(l′, I) then J_{1} ∩ J_{2} satisfies too. The part (??) is a version of the well-known Laufer algorithm, compare e.g. with (??). □

Theorem 3.3.4.

(3.3.5)  ∑_{l′∈S′} ∑_{I}(−1)^{|I|+1}χ(l′ + E_{J(l′,I)})t^{l′} = Z(t),

where the sum ∑_{I} runs over all the subsets I of V. In other words, with the notation l′ = ∑_{j} n_{j}E_{j}^{∗}, one has

(3.3.6)  ∑_{n≥0} ∑_{I}(−1)^{|I|+1}χ(l′ + E_{J(l′,I)})x^{n} = z(x)

(where x^{n} = x_{1}^{n_{1}} ⋯ x_{s}^{n_{s}}), or, for any l′ = ∑_{j} n_{j}E_{j}^{∗}

∑_{I}(−1)^{|I|+1}χ(l′ + E_{J(l′,I)}) = ∏_{j} χ_{δ_{j},n_{j}}.

Proof. In the proof we use induction over s = |V|. If s = 1 then the identity is elementary, whose verification is left to the reader. Hence, we assume s > 1. Fix a vertex in V = {1, …, s} corresponding to the index s (after a possible reordering) so that δ_{s} = 1. Let Γ_{0} := Γ \ {s}, and let j_{0} be that vertex of Γ_{0} which is adjacent to s in Γ.

Let f_{Γ}(x) denote the left hand side of (??). We wish to show that f_{Γ}(x) = z_{Γ}(x). Let x_{0} := (x_{1}, ⋯, x_{s−1}). The induction start with the identity

(3.3.7)  z_{Γ}(x) = z_{Γ_{0}}(x_{0}) · \frac{1 − x_{j_{0}}}{1 − x_{s}} .

We will establish similar identity for f_{Γ}. For this we will use the notation l′_{0} := ∑_{j<s} n_{j}E_{j}^{∗,Γ_{0}} for any l′ = ∑_{j} n_{j}E_{j}^{∗} (here E_{j}^{∗,Γ_{0}} is the dual of E_{j} in Γ_{0}). Notice that for any Z supported in Γ_{0} one has

(3.3.8)  (l′, Z) = (l′_{0}, Z) and (−E_{j_{0}}^{∗,Γ_{o}}, Z) = (E_{s}, Z).

First, we fix some l′ ∈ S′ and a subset I ⊂ V with s ∉ I. If s ∈ J(l′, I), we may assume (cf. the notations of (??)) that I_{k−1} = J(l′, I) \ s. Since (l′, E_{s}) ≤ 0, δ_{s} = 1 and (l′ + E_{I_{k−1}}, E_{s}) > 0, we get that, in fact, (l′ + E_{I_{k−1}}, E_{s}) = 1. Hence χ(l′ + E_{I_{k}}) = χ(l′ + E_{I_{k−1}}). In other words, if s ∉ I, then

(3.3.9)  χ(l′ + E_{J(l′,I)}) = χ(l′ + E_{J(l′,I)\s}).

Using E_{J(l′,I)\s} = E_{J(l′_{0},I)}^{Γ_{0}}, (??) and (??), (??) can be rewritten in the following identity, where in the right hand side all invariants are considered in Γ_{0}:

(3.3.10)  χ(l′ + E_{J(l′,I)}^{Γ}) − χ(l′) = χ(l′_{0} + E_{J(l′,I)}^{Γ_{0}}) − χ(l′_{0}).

Next, fix again l′ ∈ S′ and take I ⊂ V with s ∈ I. By (??) one has J(l′, I) = J^{Γ_{0}}(l′_{o}, I \ s) ∪ s. Using this, (??) and (??) with a computation we get

(3.3.11) χ(l′ + E_{J(l′,I)}^{Γ}) − χ(l′ + E_{s}) = χ(l′_{0} − E_{j_{0}}^{∗,Γ_{0}} + E_{J(l′_{0},I\s)}^{Γ_{0}}) − χ(l′_{0} − E_{j_{0}}^{∗,Γ_{0}}).

Since for any constant c, one has ∑_{I:I∌s}(−1)^{|I|+1}c = ∑_{I:I∋s}(−1)^{|I|+1}c = 0, the identities (??) and (??) read as

∑_{n≥0} ∑_{I∌s}(−1)^{|I|+1}χ(l′ + E_{J(l′,I)})x^{n} = f_{Γ_{0}}(x_{0}) · ∑_{n_{s}≥0} x_{s}^{n_{s}};

∑_{n≥0} ∑_{I∋s}(−1)^{|I|+1}χ(l′ + E_{J(l′,I)})x^{n} = −f_{Γ_{0}}(x_{0})x_{j_{0}} · ∑_{n_{s}≥0} x_{s}^{n_{s}}.

Hence f_{Γ}(x) = f_{Γ_{0}}(1 − x_{j_{0}}) ∑_{n_{s}≥0} x_{s}^{n_{s}}. □

4. The analytic invariant

4.1. The setup. We start with a normal surface singularity (X, o), and we fix one of its good resolutions π. In the sequel L and L′ will stay for the corresponding lattices associated with π. Moreover, L (respectively L_{ℚ}) will also be identified with integral (rational) divisorial cycles supported by E.

We denote by c : (Y, o) → (X, o) the universal abelian cover of (X, o). Let Ỹ be the normalized pull-back of X̃ by c, π_{Y} : Ỹ → Y the pull-back of π, and c̃: Ỹ → X̃ the induced finite map making the diagram commutative. We write c^{∗} for the pull-back of integral/rational cycles.

By [?, (3.3)] or [?, (4.2.3)], c^{∗}(l′) is an integral cycle for any l′ ∈ L′.

Definition 4.1.1. We define a filtration on the local ring of holomorphic functions O_{Y,o}: for any l′ ∈ L′, we set

(4.1.2)  F(l′) := {f ∈ O_{Y,o} | div(f ∘ π_{Y}) ≥ c^{∗}(l′)}.

Notice that the natural action of H on Y induces an action on O_{Y,o} which keeps F(l′) invariant. Therefore, H acts on O_{Y,o}/F(l′) too. For any l′ ∈ L′, let 𝔥_{[l′]}(l′) be the dimension of the θ([l′])-eigenspace (O_{Y,o}/F(l′))_{θ([l′])}. Then one defines the Hilbert series H(t) by

(4.1.3)  H(t) := ∑_{l′∈L′} 𝔥_{[l′]}(l′)t^{l′}.

Remark 4.1.4. Campillo, Delgado and Gusein-Zade in [?, ?] use two other series as well. Here we present their relationship with the above Hilbert series. Set

(4.1.5)  L(t) := ∑_{l′∈L′} dim_{ℂ}(F(l′)/F(l′ + E))_{θ([l′])} · t^{l′} ∈ ℤ[[t^{L′}]],

where E = ∑_{j} E_{j} as above. Since F(l′) = F(l′ + E_{j}) if (l′, E_{j}^{∗}) < 0, it is easy to see that L(t) ∏_{j}(t_{j} − 1) is an element of ℤ[[t^{L′_{e}}]]. Hence the next infinite power series is well-defined:

(4.1.6)  P (t) := −\frac{L(t) ∏_{j}(t_{j} − 1)}{1 − t^{E}} = −L(t) ∏_{j}(t_{j} − 1) · ∑_{k≥0} t^{kE} ∈ ℤ[[t^{L′_{e}}]].

Notice that 𝔥_{h}(l′) = 0 for l′ ≤ 0. This and the obvious relation

H(t)(1 − t^{E}) = t^{E} · L(t)

show that

(4.1.7)  P (t) = −H(t) · ∏_{j}(1 − t_{j}^{−1}) in ℤ[[t^{L′_{e}}]].

Apparently, taking P instead of H, one loses some analytic information of H (because ∏_{j}(1 − t_{j}^{−1}) is a zero divisor). Nevertheless, in [?] there is an argument that, in fact, P and H contain the same amount of information (without providing a formula for the inversion).

In the next lemma, we show that this can be done very explicitly. In order to state the expression, first we have to notice that, in fact, P (t) is supported on S′ (this is an automatically consequence of the discussion from (??), but can be verified directly as well).

Lemma 4.1.8. [?] Write P (t) = ∑_{l′∈S′} 𝔭(l′)t^{l′} for its coefficients. Then:

H(t) = P (t) · ∑_{a∈L, a≱0} t^{−a}.

In other words,

(4.1.9)  𝔥_{[l′]}(l′) = ∑_{a∈L, a≱0} 𝔭(l′ + a).

(The elements of S′ have the property that if one of the coefficients tends to infinity then all of them do, hence the sum in (??) is finite.) For the hint of the proof see (??).

4.2. Reformulation of H(t) and P (t). The integers 𝔥_{h}(l′) can be reinterpreted in terms of sheaf-cohomology. We start with the following result (for the definition of r_{h} see (??)):

Proposition 4.2.1.

(1) [?, (3.7)] [?, (4.2.9)] [?, (3.1)] For any h ∈ H there exists a divisor D_{h} on X̃ numerically equivalent to r_{h} (i.e. c_{1}(OX̃(D_{h})) = r_{h}) such that

c̃_{∗}OỸ = ⨁_{h∈H} OX̃(−D_{h}),

where the last sum is an H-eigenspace decomposition: OX̃(−D_{h}) is the θ(h)-eigenspace of c̃_{∗}OỸ. Moreover, the Chern class c_{1} : P ic(X̃) → L′, D ↦ c_{1}(OX̃(D)), admits a group section d_{1} : L′ → P ic(X̃) given by d_{1}(l + r_{h}) = OX̃(l + D_{h}) (where l ∈ L is identified with an integral cycle).

(2) [?, (4.2)] [?, (4.3.3)] For any l′ ∈ L′ there exists a unique minimal s(l′) ∈

S′ such that l′ ≤ s(l′) and [l′] = [s(l′)].

s(l′) can be computed by the following (Laufer) algorithm. One constructs a ‘computation sequence’ x_{0}, …, x_{k} ∈ L with x_{0} = 0 and x_{m+1} = x_{m}+E_{j(m)}, where the index j(m) is determined by the following principle. Assume that x_{m} is already constructed. If l′ +x_{m} ∈ S′ then one stops and k = m. Otherwise, there exists at least one j with (l′ + x_{m}, E_{j}) > 0. Take for j(m) one of these j, and continue with x_{m+1}. The algorithm stops after a finitely many steps, and l′ + x_{k} = s(l′).

Since for any f ∈ O_{Y,o} that part of div(f ∘ π_{Y}) which is supported by the exceptional divisor of π_{Y} is in the Lipman cone of Ỹ, we get that F(l′) = F(s(l′)).

On the other hand, for any l′ > 0 (in particular for any non-zero s(l′)), one has the exact sequence

(4.2.2)  0 → OỸ (−c^{∗}(l′)) → OỸ → O_{c^{∗}(l′)} → 0.

If we consider (??) for some l′ = l + r_{h} > 0 (l ∈ L), then by [?, (3.2)] the corresponding θ(h)-eigenspaces constitute the exact sequence

(4.2.3)  0 → OX̃(−D_{h} − l) → OX̃(−D_{h}) → O_{l}(−D_{h}) → 0.

In particular, we get

Corollary 4.2.4. For any l′ = l + r_{h} > 0 one has:

𝔥_{h}(l′) = dim \frac{H^{0}(X̃, OX̃(−D_{h}))}{H^{0}(X̃, OX̃(−D_{h} − l))}.

Example 4.2.5. Since D_{0} = 0, for h = 0 we obtain:

H_{0}(t) = ∑_{l∈L} dim \frac{O_{X,o}}{\{f ∈ O_{X,o} : div(f ∘ π) ≥ l\}} t^{l}.

This is the Poincaré (Hilbert) series of O_{X,o} associated with the divisorial filtration of π. For some of its qualitative properties and interesting examples, see e.g. [?].

Therefore, for any l′ = l + r_{h} > 0, from (??), one has:

(4.2.6)  𝔥_{h}(l′) = χ(O_{l}(−D_{h})) + h^{1}(−D_{h}) − h^{1}(−D_{h} − l),

where h^{1}(D) denotes the dimension of H^{1}(OX̃(D)). It is also convenient to write h^{1}(l′) for h^{1}(d_{1}(l′)), cf. (??)(??), e.g. h^{1}(−D_{h}−l) = h^{1}(−l′). Moreover, χ(O_{l}(−D_{h})) = χ(l′) − χ(r_{h}) for any l′ = l + r_{h}. Hence, (??) reads as

(4.2.7)  𝔥_{h}(l′) + h^{1}(−l′) = χ(l′) − χ(r_{h}) + h^{1}(−r_{h}).

Notice that the right hand side is a quadratic function in l. Since 𝔥_{h}(l′) = 𝔥_{h}(s(l′)) we also get

(4.2.8)  𝔥_{h}(l′) = χ(s(l′)) − χ(r_{h}) − h^{1}(−s(l′)) + h^{1}(−r_{h}).

Hence, by (??),

(4.2.9)  P (t) = ∑_{l′} ∑_{I}(−1)^{|I|+1} ( χ(s(l′ + E_{I})) − h^{1}(−s(l′ + E_{I})) ) t^{l′}.

Assume that l′ ∉ S′. Then there exists j_{0} ∈ V with (l′, E_{j_{0}}) > 0. Therefore, for any subset I ⊂ V \ {j_{0}} the Laufer algorithm applied for l′ + E_{I} may start adding E_{j_{0}}, hence s(l′ + E_{I}) = s(l′ + E_{I∪j_{0}}). In particular, these two terms cancel each other in the sum of the above expression of P. In particular, the first sum in (??), in fact, runs over l′ ∈ S′.

Moreover, analysing the computation sequence {x_{i}}_{i} connecting l′ + E_{I} with s(l′ + E_{I}), at each step one has

χ(x_{m}) − h^{1}(−x_{m}) = χ(x_{m+1}) − h^{1}(−x_{m+1}).

Since l′ + E_{J(l′,I)} is one of the elements of this sequence, we get:

Corollary 4.2.10.

P (t) = ∑_{l′∈S′} ∑_{I}(−1)^{|I|+1} ( χ(s(l′ + E_{I})) − h^{1}(−s(l′ + E_{I})) ) t^{l′}

= ∑_{l′∈S′} ∑_{I}(−1)^{|I|+1} ( χ(l′ + E_{J(l′,I)}) − h^{1}(−l′ + E_{J(l′,I)}) ) t^{l′}

= ∑_{l′∈S′} ∑_{I}(−1)^{|I|+1} ( χ(l′ + E_{I}) − h^{1}(−l′ + E_{I}) ) t^{l′}.

Via (??), the second identity reads as

Corollary 4.2.11.

P (t) = Z(t) + ∑_{l′∈S′} ∑_{I}(−1)^{|I|} h^{1}(−l′ + E_{J(l′,I)}) t^{l′}.

I.e., P (t) = Z(t) if and only if ∑_{I}(−1)^{|I|}h^{1}(−l′ + E_{J(l′,I)}) = 0 for all l′ ∈ S′.

Remark 4.2.12. By (??)

𝔭(l′) = ∑_{I}(−1)^{|I|+1} dim \frac{H^{0}(O(−r_{h}))}{H^{0}(O(−l′ − E_{I}))} = ∑_{I}(−1)^{|I|+1} dim \frac{H^{0}(O(−l′))}{H^{0}(O(−l′ − E_{I}))}.

Using the very last expression for 𝔭(l′), one can show that for any l ∈ L_{≥0}, one has (cf. [?]):

dim \frac{H^{0}(X̃, O(−l′))}{H^{0}(X̃, O(−l′ − l))} = ∑_{a∈L_{≥0}, a≱l} 𝔭(l′ + a).

The proof uses induction over l, one compares the above expressions for l and l+E_{i} (for some i). Then, taking l′ = r_{h}, one gets the proof of (??).

5. The Main Identity.

5.1. In the next paragraph, we formulate an identity which connects the topological invariant Z with the analytic invariant P, which is conjecturally valid for ‘some’ normal surface singularities.

Definition 5.1.1. Consider a normal surface singularity (X, o) with rational homology sphere link. We say that (X, o) and its resolution π satisfy the Main Identity if

(5.1.2)  Z(t) = P (t).

Example 5.1.3. Consider the cyclic quotient singularity whose minimal resolution π has only one irreducible component E with self-intersection −p (p ≥ 2). Then H = ℤ_{p}, Y = ℂ^{2}, O_{Y,o} = ℂ{z_{1}, z_{2}}. The action of H on ℂ{z} is given by h ∗ z_{i} = θ(h)(E^{∗})z_{i}. Fix the generator g := [−E^{∗}] of H and set ξ := e^{2πi/p}. Then the action of g is given by g ∗ z_{i} = ξz_{i}. Fix ϕ ∈ Ĥ with ϕ(g) = ξ^{q} for some 0 ≤ q < p. Then ϕ = θ(h) with h := [qE^{∗}]. Moreover r_{h} = qE^{∗} = (q/p)E.

Ỹ is just the blow up of Y in one point with exceptional divisor Ẽ a (−1)-curve. Since c^{∗}(E) = pẼ, we get that F(l′) consists of all the monomials with degree ≥ pl′. Therefore, for l′ = l + q/p (l ∈ L), (F(l′)/F(l′ + E))_{ϕ} can be identified with the vector space of monomials of degree pl + q. Hence

P_{h}(t) = ∑_{l≥0}(1 + q + pl)t^{l+ \frac{q}{p}} , and P (t) = ∑_{k≥0}(1 + k)t^{k/p}.

On the other hand, Z(t) is equal to the same sum by its very definition.

Remark 5.1.4. Notice that if for some analytic structure and resolution Z = P holds, then, by (??), the Hilbert function H can be recovered from the resolution graph of π.

The Main Identity for the component h = 0 (cf. ??) was proved in [?] for any rational singularity and any resolution, see also [?] for an equivariant version, valid for rational singularities, formulated in a different way. Here we prove it in the following situations:

Theorem 5.1.5. The Main Identity is true in the following cases:

(1) (X, o) is rational, and π is arbitrary resolution,

(2) (X, o) is minimally elliptic singularity whose minimal resolution is good, and π is this minimal resolution,

The present proof emphasizes the importance of some vanishing theorems valid for these singularities, and also the numerical behavior of the ‘computation sequences’ associated with the Artin cycles.

Proof of (??). Let p_{g} be the geometric genus of (X, o). Then (X, o) is rational iff p_{g} = 0, and (X, o) is minimally elliptic iff p_{g} = 1 and (X, o) is Gorenstein [?]. We write Z_{min} ∈ L for the Artin cycle, the unique minimal element of S \ 0.

By [?, 1.7], one gets h^{1}(−s(l′)) = 0 except when (X, o) is minimally elliptic and s(l′) = 0 (in the rational case this vanishing was proved by several authors, see e.g., [?, 12.1], [?, 4.1.1]). On the other hand, s(l′) = 0 if and only if l′ ∈ −L_{e}. Therefore, the second term in (??) is

( ∑_{l′} h^{1}(−s(l′))t^{l′} ) · ∏_{j}(1 − t_{j}^{−1}) = ( ∑_{l∈−L_{e}} p_{g}t^{l} ) · ∏_{j}(1 − t_{j}^{−1}) = p_{g}t^{0}.

Hence, (??) (or (??)) reads as

(5.1.6)  P (t) = p_{g}t^{0} + ∑_{l′∈S′} ∑_{I}(−1)^{|I|+1}χ(s(l′ + E_{I})) · t^{l′}.

Remark 5.1.7. Notice that (??) (as a consequence of the above vanishing result) already shows that P (t) is topological (i.e. it can be determined from Γ).

In general, the above vanishing is not valid (even for ‘very large’ l). E.g., for Gorenstein elliptic singularities with p_{g} > 1, h^{1}(OX̃(−kZ_{min})) = p_{g} − 1 > 0 for any k > 0 [?, 2.21].

Moreover, in general, P is not topological, it might depend on the choice of the analytic structure (supported by a fixed topological type), see e.g. (??).

In the sum (??), let us separate l′ = 0. In this case, since s(E_{I}) = 0 for I = ∅, and = Z_{min} otherwise, we get ∑_{I}(−1)^{|I|+1}χ(s(E_{I})) = χ(Z_{min}) = 1 − p_{g}. In particular

(5.1.8)  P (t) = t^{0} + ∑_{l′∈S′\0} ∑_{I}(−1)^{|I|+1}χ(s(l′ + E_{I})) · t^{l′}.

Now we focus on the sum in the right-hand side of this identity. If we run the Laufer algorithm (??)(??) for l′ + E_{I}, then for any m we have

(5.1.9)  χ(l′ + x_{m+1}) = χ(l′ + x_{m}) + 1 − (l′ + x_{m}, E_{j(m)}).

Remark 5.1.10. Take h = 0. If in the Laufer algorithm (??)(??) one starts with l = E, one gets for s(l) the Artin cycle Z_{min} [?]. During these steps, in the rational case one has all the time (E + x_{m}, E_{j(m)}) = 1 [?]. On the other hand, in the minimally elliptic case one has all the time (E + x_{m}, E_{j(m)}) = 1 excepting in the very last step, when (E + x_{m}, E_{j(m)}) = 2; moreover, for any prescribed E_{i^{∗}} one may arrange the computation sequence in such a way, that in this very last step j(k − 1) = i^{∗}, cf. [?].

Lemma 5.1.11. Assume that (X, o) and π is as in (??). The for any l′ ∈ S′ \ 0 and I ⊂ V one has

χ(s(l′ + E_{I})) = χ(l′ + E_{J(l′,I)}).

Proof. Write l′ = l + r_{h}. In the Laufer algorithm of l′ + E_{I} we may start the sequence with cycles of type l′ + E_{I_{m}}, where I_{m+1} = I_{m} ∪ {i_{m}} and i_{m} ∉ I_{m} (i.e. with the computation sequence of J(l′, I), cf. (??)(??)), till we hit l′ + E_{J(l′,I)}. Then, as a second turn, we continue to add cycles E_{j} with j ∈ J(l′, I) till we get the smallest cycle z ∈ L′ with z ≥ l′ + E_{J(l′,I)} and (z, E_{j}) ≤ 0 for all j ∈ J(l′, I). Let Z^{∗} be the (sum of the) Artin cycles of (the connected components of) J(l′, I). Clearly z ≤ l′ + Z^{∗}. Moreover, the steps in this second turn may be considered as a part of the computation of the Artin cycle Z^{∗}.

Assume J(l′, I) = V. Then z = s(l′+E_{I}). Moreover χ(z) = χ(l′+E_{J(l′,I)}). This follows from (??) and (??): in the second turn we have all the time (l′+x_{m}, E_{j(m)}) = 1 except if the graph is elliptic, z = l′ + Z_{min}, and we are in the very last step. In this case, we may assume that the last added cycle E_{j} is one prescribed with (l′, E_{j}) < 0, hence (z − E_{j}, E_{j}) = 1 too.

Assume that J(l′, I) ≠ V. Then, by [?], J(l′, I) supports a rational singularity, hence by (??) and (??), χ(z) = χ(l′ + E_{J(l′,I)}). If z = s(l′ + E_{I}) then we are done. Otherwise, there exists an index j ∈ V \ J(l′, I) with (z, E_{j}) > 0 (∗). Notice that automatically (E_{J(l′,I)}, E_{j}) > 0 (otherwise we would have (z, E_{j}) = (l′, E_{j}) ≤ 0, a contradiction). By the definition of J(l′, I), we have (l′ + E_{J(l′,I)}, E_{j}) ≤ 0, hence (l′, E_{j}) ≤ −(E_{J(l′,I)}, E_{j}) < 0. This together with (∗) implies (z − l′, E_{j}) ≥ 2. Since the computation sequence {x_{m}}_{m} can be considered as the beginning of the computation sequence of Z_{min}, this can happen only if the graph is elliptic, (z − l′, E_{j}) = 2, z + E_{j} = Z_{min} + r_{h} = s(l′ + E_{I}) and z = l′ + Z^{∗}. But in this case one also has 0 < (z, E_{j}) = 2 + (l′, E_{j}) < 2, hence (z, E_{j}) = 1 which imply χ(z) = χ(z + E_{j}). □

Finally, (??) follows from (??), (??) and (??) (and the fact that the coefficient of t^{0} in Z(t) is 1). □

6. The Reduced Identities

6.1. For any fixed resolution π, in the main identities (??) one takes a variable t_{j} for each exceptional divisor E_{j} of π. Since, in general, π is not minimal, (and the set of all exceptional divisors of an arbitrary resolution supports no intrinsic geometric meaning), it is natural to create the possibility to reduce the number of variables, or replace L and L′ by some more intrinsic (geometrically defined) lattices. E.g., one might take only the vertices corresponding to the nodes of the graph, or corresponding to the essential exceptional divisors only, or pull-backs/images of lattices provided by some geometric/universal constructions.

Here, we make explicit that situation when one reduces the number of variables. To start with, we define the corresponding topological and analytical series with reduced number of variables, then we formulate the corresponding ‘Reduced Identities’, and we prove that the ‘Main Identity’ implies all the reduced ones. Finally, in the next section, we interpret some classical and recent relations of the singularity theory as reduced identities.

We fix (X, o) and the resolution π as in (??). Let U be a non-empty subset of V. Our plan is to define formal series in variables {t_{j}}_{j∈U}, still denoted by t. It is convenient to define the projection pr_{U} : L′ → L′ by pr_{U}(∑_{j} l′_{j}E_{j}) = ∑_{j∈U} l′_{j}E_{j}. Correspondingly, we write

t^{pr_{U} (l′)} = ∏_{j∈U} t_{j}^{l′_{j}} = t^{l′}|_{t_{j}=1 for all j∉U}.

Although, in the non-reduced case, t^{l′} codifies completely the θ(h)-eigenspace which provides the coefficient of t^{l′} (via h = [l′]), this is not true anymore for t^{pr_{U} (l′)}. Hence, in the reduced context we lose this correspondence, and we do not consider h-components in the sense of (??).

Definition 6.1.1. For any h ∈ H set

(6.1.2)  ζ_{h}^{U}(t) := \frac{1}{|H|} · ∑_{ρ∈Ĥ} ρ(h)^{−1} · ∏_{j∈V} (1 − ρ([E_{j}^{∗}])t^{pr_{U} (E_{j}^{∗})})^{δ_{j}−2},

or, equivalently,

(6.1.3)  ζ_{h}^{U}(t) := ζ_{h}(t)|_{t_{j}=1 for all j∉U}.

Having in mind (??), we will identify ζ_{h}^{U} and Z_{h}^{U} (where, one might define Z_{h}^{U}(t) := Z_{h}(t)|_{t_{j}=1 for all j∉U}).

Next, we define the analytic invariants. For any l′_{U} ∈ pr_{U}(L′) ⊂ L′ set

F^{U}(l′_{U}) := {f ∈ O_{Y,o} | div(f ∘ π_{Y}) ≥ c^{∗}(l′_{U})}.

Moreover, for any l′_{U} ∈ pr_{U}(L′) and h ∈ H, let 𝔥_{h}^{U}(l′_{U}) be the dimension of the θ(h)-eigenspace (O_{Y,o}/F^{U}(l′_{U}))_{θ(h)}. Then set

H_{h}^{U}(t) := ∑_{l′_{U} ∈pr_{U} (L′)} 𝔥_{h}^{U}(l′_{U}) · t^{l′_{U}} ,

L_{h}^{U}(t) := ∑_{l′_{U} ∈pr_{U} (L′)} dim (\frac{F^{U}(l′_{U})}{F^{U}(l′_{U} + E_{U})})_{θ(h)} · t^{l′_{U}} ,

P_{h}^{U}(t) := −\frac{L_{h}^{U}(t) ∏_{j∈U}(t_{j} − 1)}{1 − t^{E_{U}}}.

Then, again,

(6.1.4)  P_{h}^{U}(t) = −H_{h}^{U}(t) · ∏_{j∈U}(1 − t_{j}^{−1}).

Definition 6.1.5. We say that (X, o) and the resolution π satisfy the Reduced Identity for h ∈ H, corresponding to the subset U ⊂ V, if

(6.1.6)  Z_{h}^{U}(t) = P_{h}^{U}(t) ( in ℤ[[t_{j}^{1/d}, j ∈ U]], or in ℤ[[t^{pr_{U} (L′)}]]).

We say that (X, o) and π satisfy the Reduced Identity (for U) if (??) is true for all h ∈ H.

Although, by (??), Z^{U} can be obtained by a simple substitution from Z, the same statement is not immediate for P^{U}. Nevertheless, we have the following:

Theorem 6.1.7.

(6.1.8)  P_{h}^{U}(t) = P_{h}(t)|_{t_{j}=1 for all j∉U}.

In particular, if for some singularity (X, o), resolution π and element h ∈ H the Main Identity (??) is true, then for any non-empty U ⊂ V the Reduced Identity (for (X, o), π and h) is also true.

Proof. We prove (??) by descending induction over the cardinality of U. Set Ū := U \ i_{0} such that Ū ≠ ∅. Then, using the above identities, it is enough to verify

(6.1.9)  L_{h}^{U}(t)(t_{i_{0}} − 1)|_{t_{i_{0}} =1} = LŪ ; h(t).

Let L_{U} be the lattice generated by {E_{j}}_{j∈U}. Then L_{h}^{U}(t) · t^{−pr_{U} (r_{h})} has the form ∑_{l∈L_{U}} a_{h}^{U}(l) t^{l}. This can be rewritten as

∑_{l̄∈LŪ} tl̄ ∑_{k∈ℤ} a_{h}^{U}(l̄+ kE_{i_{0}}) t_{i_{0}}^{k}.

By definition,

a_{h}^{U}(l̄+ kE_{i_{0}}) = aŪ ; h(l̄) for k ≪ 0.

Also, for any l ∈ L there exists an integer n such that if for some s ∈ S one has s ≥ l and s_{i_{0}} ≥ n, then automatically s ≥ l + E too. Therefore

a_{h}^{U}(l̄+ kE_{i_{0}}) = 0 for k ≫ 0.

Hence

∑_{k∈ℤ} a_{h}^{U}(l̄+ kE_{i_{0}}) t_{i_{0}}^{k}(t_{i_{0}} − 1)|_{t_{i_{0}} =1} = aŪ ; h(l̄).

□

7. Examples, counterexamples

7.1. Theorem (??) connects the Main Identity with some non-trivial results, already present in the literature. In the beginning of this section we list some of them. Then, we list some examples when Z ≠ P, trying to catch the limits of the Main Identity.

In this section, we focus on the situation when U contains only one element, say j_{0}. Notice that even in this case, the verification (or disprove) of the identity (??) for some specific analytic structures can be hardly non-trivial.

The unique variable will be denoted by t; notice that P_{h}^{U}(t) = L_{h}^{U}(t). For the trivial element 0 ∈ H, the filtration is the divisorial (valuation) filtration associated with E_{j_{0}} on O_{X,o}, and L_{0}^{U}(t) is the Poincaré series of the associated graded algebra.

7.2. Example (weighted homogeneous singularities). Assume that (X, o) is the germ at the unique singular point of an affine space X which admits a good ℂ^{∗} action (and we also assume that it is not a cyclic quotient singularity), and let π be its minimal good resolution. Then O_{X,o} admits a grading ⊕_{k≥0}A_{k} with Poincaré series p_{X,o}(t) = ∑_{k} dim A_{k}t^{k}. The link of such a singularity is a Seifert 3-manifold. In fact, the dual graph of π is star-shaped, which can be identified by the Seifert invariants of the link as well. Assume that U contains only one element, namely the central exceptional divisor E_{cen} of π.

For such a singularity, Pinkham in [?] proved that L_{0}^{U}(t) = p_{X,o}(t) and computed this Poincaré series in terms of the Seifert invariants of the link. On the other hand, [?, (5.2)] identifies this expression with ζ_{0}(t). In particular, for such a singularity, the identity P_{0}^{U} = Z_{0}^{U} follows.

The equivariant version P_{h}^{U} = Z_{h}^{U} is proved by Neumann in [?, §4] (for a more general argument, see (??)). From [?] one also obtains that ∑_{h} Z_{h}^{U}(t^{c}) is the Poincaré series p_{Y,o}(t) of the universal abelian cover (Y, o) (where c is the order of [E_{cen}^{∗}] in H).

7.3. Example (splice-quotient singularities). Assume that (X, o) is a splicequotient singularity (introduced and intensively studied by Neumann and Wahl, see e.g. [?]). We point here in short what we mean by this.

We fix a possible resolution graph Γ. (It should satisfy some combinatorial properties which are not essential in the present discussion.) Then, using Γ, one can write equations (up to an equisingular deformation) for the universal abelian cover (Y, o) of (X, o) and for the action of H on (Y, o). These equations determine the analytic structure of (X, o). Then, the resolution used in all our discussions is exactly that resolution π_{Γ} of (X, o) which has dual graph Γ (it is uniquely determined by Γ). Notice that there is always a linkage between the analytic structures constructed by this procedure and the graph used in the construction. The construction gives hope that the analytic invariants of the singularity might be determined combinatorially from the dual resolution graph, if one writes them (exactly) from π_{Γ}. Otherwise, this might not be the case, and the topology-analytic linkage might be lost, see also (??).

We recall that splice-quotient singularities include all the rational singularities (where Γ can be arbitrary), all the minimally elliptic singularities (where Γ has the property that the support of the minimal elliptic cycle is not proper — e.g., for all minimally good resolutions) [?], and also those singularities which admit good ℂ^{∗} action (and Γ is the minimal good resolution) [?].

Let us fix a graph Γ and consider the associated splice-quotient singularity and the resolution π_{Γ}. Set U = {j_{0}}, where j_{0} is one of the nodes of Γ (i.e. δ_{j_{0}} ≥ 3). Then the identity Z_{h}^{U} = P_{h}^{U} was proved by Okuma in [?, §3]. (The same is true if j_{0} is not a node, but is not an end-vertex either, and H = 1, see [?].)

In other words, by Okuma’s result, the Main Identity reduced to any individual node is true (in π_{Γ}). This is improved in [?], where the Main Identity is proved in its whole generality. (Notice that this very general result includes theorem (??) too, nevertheless the proofs are very different, and the proof of (??) might illuminate some aspects — e.g. the relevance with vanishing theorems —, which are less transparent in [?].)

7.4. Example (hypersurface singularities). This is a small digression about Poincaré polynomial computations of (not necessarily weighted homogeneous) hypersurface singularities when the filtration can be connected with some weights.

Fix three positive integers w = (w_{1}, w_{2}, w_{3}), and consider on O := O_{ℂ^{3},o} the graded ring structure ⊕_{e≥0}O(e), where the three coordinates have degrees deg(z_{i}) = w_{i}. Here O(e) is the vector space of monomials of w-degree e. Fix f ∈ O and define (X, o) := {f = 0} ⊂ (ℂ^{3}, o). Write f = f_{d} + f_{d+1} + ⋯, where f_{k} is homogeneous of w-degree k, and f_{d} ≠ 0. Let q : O → O_{X,o} be the natural projection.

We will analyze different filtrations {F(l)}_{l≥0} of the ℂ-algebra O_{X,o}. As usual, for any filtration F, we write Gr_{l}^{F} := F(l)/F(l+1), and P^{F}(t) := ∑_{l≥0} dim(Gr_{l}^{F})t^{l} for its Poincaré series.

First, consider the filtration G defined by G(l) := q(⊕_{e≥l}O(e)). The next isomorphism can be easily verified and is left to the reader:

Lemma 7.4.1.

Gr_{l}^{G} ≃ \frac{O(l)}{f_{d} · O(l − d)}.

In particular, P^{G}(t) agrees with the Poincaré series of O/(f_{d}) with weights w, namely P^{G}(t) = (1 − t^{d})/ ∏_{i}(1 − t^{w_{i}}).

Next, we assume that f is irreducible (this is automatically satisfied if (X, o) is normal). Let K be the field of fraction of O_{X,o}. We are interested in filtrations determined by a valuation ν : K^{∗} → ℤ with the property ν(z_{i}) = w_{i}: we set F(l) := {g ∈ O_{X,o} : g ≠ 0, ν(g) ≥ l} ∪ {0}. Clearly, from the definition of the valuations, we have G(l) ⊂ F(l) for all l. But the equality, in general, fails.

Remark 7.4.2. ν may appear as follows. Consider ϕ, the w-weighted blow up of (X, o), and let E be its strict transform. (One might even take any resolution or modification π which dominates ϕ, and identify E with its π-strict transform.) Assume that E is irreducible. Then the divisorial valuation ν of E provides such a filtration F.

In the next lemma we restrict ourself to the case w_{3} = 1 — this is enough in the main application in (??); we invite the reader to formulate the corresponding statement for more general weights (when one must handle an additional cyclic group action).

Lemma 7.4.3. Assume w_{3} = 1 and f_{d} irreducible (but f_{d} ≠ z_{3}). Consider the filtration F as in (??). Then F = G. In particular, P^{F}(t) = (1 − t^{d})/ ∏_{i}(1 − t^{w_{i}}).

Proof. We may do the computations in a chart of the weighted blow up ϕ : Z → ℂ^{3}. We fix the chart (z_{1}, z_{2}, z_{3}) ↦ (α^{w_{1}}β, α^{w_{2}}γ, α) of ϕ. Then the strict transform X̃ of X has local equations {f_{d}(p) + αf_{d+1}(p) + ⋯ = 0}, while the exceptional curve E ⊂ X̃ is {f_{d}(p) = α = 0}, where p = (β, γ, 1).

We wish to show that F(l) ⊂ G(l). For this, take g with q(g) ∈ F(l), and write g as a sum of w-homogeneous elements g_{k} + g_{k+1} + · · ·. If k ≥ l then q(g) ∈ G(l), hence assume that k < l. By substitution, ϕ^{∗}(g) = α^{k}h, where h := g_{k}(p) + αg_{k+1}(p) + · · ·. Since the vanishing order of ϕ^{∗}(g) is ≥ l, h must vanishes along E, hence there exists local functions u and v in (α, β, γ) such that h = uf_{d}(p) + vα. Taking α = 0, we get a function ω(β, γ) such that g_{k}(β, γ, 1) = ω(β, γ) · f_{d}(β, γ, 1). We interpret this in coordinates z. The homogenized identity has the form g_{k}(z) = ω̃(z) · f_{d}(z) · z_{3}^{a} for some a ∈ ℤ. Since f_{d} is irreducible, we obtain f_{d}|g_{k}. Then g′ := g−g_{k}+(g_{k}/f_{d})(−f +f_{d}) has the property that q(g′) = q(g) but its k-homogeneous term vanishes. By induction, g can be replaced by g′′ with q(g′′) = q(g) and g′′ ∈ ⊕_{e≥l}O(e), hence q(g) ∈ G(l). □

Example 7.4.4. Assume that (X, o) is the zero set of the unimodal exceptional hypersurface singularity of type E_{12}: f_{a} = z_{1}^{2} + z_{2}^{3} + z_{3}^{7} + ayz^{5}. One may verify that (X, o) is minimally elliptic. Let π be its minimal good resolution, whose dual graphs is:

[figure: −7; −1; −2; −3; E_{1}; Γ :]

Let E_{cen} be the node, and U = {E_{cen}}. Then Z^{U} = P^{U} by (??) if a = 0, or by (??), in general. Here we would like to show the same identity for U = {E_{1}} (see the picture). Notice that H is trivial. By a computation:

(7.4.5)  Z^{U}(t) = \frac{t^{6} − 1}{(t^{3} − 1)(t^{2} − 1)(t − 1)}.

On the other hand, P^{U} can be computed by (??): the weights are w = (3, 2, 1), hence P^{U} = Z^{U} follows.

Notice that π is not minimal, hence (??) cannot be applied. The graph of all minimally elliptic singularities (graphs) whose minimal resolution is not good are listed in [?]. Those which provide rational homology sphere links are similar with the above example, and for all of them by similar method on can show that Z = P. (This fact follows also from [?], where Z = P is proved for splice-quotients, since this applies for minimally elliptic singularities and their minimal good resolutions, cf. (??).)

7.5. Example (superisolated singularities). Let (X, o) be a hypersurface super-isolated singularity (a family introduced by I. Luengo in [?]). More precisely, (X, o) = ({f = 0}, o) ⊂ (ℂ^{3}, o), where f = f_{d} + f_{d+1} + ⋯, f_{k} is homogeneous of degree k, and the projective curve {f_{d+1} = 0} ⊂ ℂℙ^{2} does not meet the singular points of C := {f_{d} = 0} ⊂ ℂℙ^{2}. We assume d ≥ 2 and that C is irreducible. The link of (X, o) is a rational homology sphere if and only if C is rational and cuspidal (i.e. all the singular points (C, p_{i})_{i=1}^{ν} of C are locally irreducible). In the sequel we assume these facts too.

The minimal resolution of (X, o) can be obtained by a single blow up of the maximal ideal. The exceptional divisor (corresponding to the tangent cone) is isomorphic to C. In general, this is not a good resolution, one has to resolve the singularities of C as well. Hence, the minimal good resolution graph can be obtained from the minimal good embedded resolution graphs Γ_{i} of the plane curve singularities (C, p_{i})_{i=1}^{ν} by adding one extra vertex v_{C} (corresponding to C), and for all i connecting v_{C} by one edge to the (−1)-curve of Γ_{i}. π will stay for this minimal good resolution and U = {v_{C}}.

All the needed invariants regarding the next discussion (excepting P^{U}) were computed in [?], [?] and [?]. We invite the reader to consult these articles for more details. We will consider the h = 0 case only.

We write Δ(t) for the product of the characteristic polynomials (associated with the monodromy) of all local plane curve singularities (C, p_{i}) ⊂ (ℂ^{2}, p_{i}). One may verify that H = ℤ_{d} and

Z_{0}^{U}(t) = \frac{1}{d} ∑_{ξ^{d}=1} \frac{Δ(ξt^{1/d})}{(1 − ξt^{1/d})^{2}} .

On the other hand, the Poincaré series (associated with the divisorial filtration of the projectivised tangent cone) is given by (??), namely

P_{0}^{U}(t) = \frac{1 − t^{d}}{(1 − t)^{3}} .

Following [?], we define

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

Clearly, the Reduced Identity (for U = {C} and h = 0) is equivalent with the vanishing of N(t). In [?, (2.4)] is proved that N(t) is a symmetric polynomial (i.e. N(t) = t^{d−3} · N(1/t)) with integral coefficients and with N(0) = 0. Moreover,

N(1) = sw(Σ) − (K2 ; X̃ + s)/8 − p_{g}(X, o),

where sw(Σ) is the Seiberg–Witten invariant of Σ associated with the canonical spin^{c}-structure, KX̃ is the canonical class of X̃, s = |V| as above, and p_{g}(X, o) is the geometric genus of (X, o). Notice that the vanishing of N(1) is exactly the subject of the ‘Seiberg–Witten invariant conjecture’, formulated by L. Nicolaescu and the author, cf. [?, ?].

In particular, all superisolated hypersurface singularities which are counterexamples for the Seiberg–Witten invariant conjecture provide examples when the Reduces Identity (hence the Main Identity too) is not valid. This cannot happen if d ≤ 4, but may appear starting from d ≥ 5. The complete list of cases when N(t) is non-zero for 5 ≤ d ≤ 6, and more examples with d = 7 and even with arbitrary high d are provided in [?, (2.7)-(2.8)]. In all known cases (by the author) N(t) ≠ 0 may occur only if the number ν of singular points of C is ≥ 2.

In fact, in [?] it is conjectured that N(t) has only non-positive coefficients. This conjecturally provides a very strong criterion for the existence of cuspidal rational planes curves with singularities with prescribed topological types.

For unicuspidal curves (i.e. when ν = 1), in [?] is proved that N(t) has nonnegative coefficients. In particular, the above conjecture about N(t) transforms into the conjecture which predicts the vanishing of N(t). In fact, this is also equivalent with the Seiberg-Witten invariant conjecture for (X, o). Its validity was verified for all situation when the logarithmic Kodaira dimension κ̄ of ℂℙ^{2} \ C is < 2, and for κ̄ = 2 in all the examples known (by the authors of [?]).

Example 7.5.2. Here is the case C_{4} from [?] with d = 5, where C has two singularities with multiplicity sequence [3] and [2_{3}]. The dual resolution graph of f is presented below. Let U = {C}, where C is the (−31)-curve. In this case N(t) = −2t, hence the Reduced Identity is not satisfied.

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

7.6. Example (a Newton non-degenerate singularity). Let (X, o) be the zero set of the germ f = z_{1}^{3} +z_{2}(z_{3}^{2} +z_{1}^{2}z_{2} +z_{2}^{6}) (or any germ with the same Newton diagram; in fact f is a member of the bimodal singularity Q_{2,1}). Let π be its minimal resolution with dual graph:

[figure: −2; −2; −2; −2; −5; E_{1}; Γ :; −2]

Γ can be obtained, e.g., by a toric resolution described in [?]. By this, there is a 1-1 correspondence between the faces of the Newton diagram and the nodes of Γ (see also [?]). By this correspondence, the exceptional divisor (node) E_{1} corresponds to the face Δ determined by the monomials of z_{2}(z_{3}^{2} + z_{1}^{2}z_{2} + z_{2}^{6}). The positive primitive normal vector of this face is (5, 2, 6), whose entries agree with the vanishing orders of the three coordinates on E_{1}.

The group H is ℤ_{12}, if ρ is a generator of Ĥ then the possible values ρ([E_{i}^{∗}]) are given as follows (where ξ is a primitive 12–root of unity):

[figure: ξ^{7}; ξ^{8}; ξ^{10}; ξ; ξ^{4}; ξ^{2}]

Our goal is to compare three series, namely Z_{0}^{U}(t), P_{0}^{U}(t) (for h = 0 and U = {E_{1}}), and P^{G}(t) associated with the filtration G constructed in (??), associated with weights w = (5, 2, 6), corresponding to the face Δ.

First, notice that Γ is a minimally elliptic graph, and the resolution is minimal, hence P (t) = Z(t) by (??). In particular, P^{U} = Z^{U} too. Z_{0}^{U} can be read from the graph, namely it is the following Fourier–Dedekind sum (over the 12–roots ξ of unity):

Z_{0}^{U}(t^{12}) = \frac{1}{12} ∑_{ξ} \frac{(1 − ξ^{2}t^{52})(1 − ξ^{8}t^{40})}{(1 − ξ^{7}t^{26})(1 − ξt^{26})(1 − ξ^{10}t^{20})(1 − ξ^{4}t^{8})}.

Hence, by a computation, we get:

P_{0}^{U}(t) = Z_{0}^{U}(t) = \frac{(1 − t^{15})(1 − t^{12})}{(1 − t^{13})(1 − t^{5})(1 − t^{2})(1 − t^{6})}.

Finally, notice that the series P^{G} provided by the face Δ : 5z_{1} + 2z_{2} + 6z_{3} = 14 is

P^{G}(t) = \frac{1 − t^{14}}{(1 − t^{5})(1 − t^{2})(1 − t^{6})}.

This also shows that the divisorial filtration F(l) (of E_{1}) is strictly larger than the filtration G(l) (for some l), see (??) for the terminology. Let us analyze the case l = 12 and 13 more closely. In the case of G the weight 14 of the equation is higher, so Gr^{G}(l) is the same as the corresponding vector spaces computed for the polynomial ring O. We get a basis {z_{3}^{2}, z_{3}z_{2}^{3}, z_{2}^{6}, z_{1}^{2}z_{2}} for Gr^{G}(12), and {z_{1}z_{2}z_{3}, z_{1}z_{2}^{4}} for Gr^{G}(13). On the other hand, for F we notice that along E_{1}, the expression h := z_{3}^{2} + z_{1}^{2}z_{2} + z_{2}^{6} equals z_{1}^{3}/z_{2} which has order of vanishing 13. Hence h provides a relation in Gr^{F}(12) = Gr^{G}(12)/(h = 0), while h is a new element in Gr^{F}(13) = Gr^{G}(13)⊕ℂ⟨h⟩. This fits with the corresponding coefficients of the Poincaré series.

In particular, the ‘naive’ series P^{G} associated with a face Δ of a Newton diagram does not agree (in general) with the Poincaré series P_{0}^{U} provided by the valuation of the exceptional divisor corresponding to Δ (although they agree for weighted homogeneous singularities). Moreover, although both P^{G} and Z_{0}^{U} are topological/combinatorial, and both are very natural, and one might hope that they should coincide, in general, this is not the case. In general, Z_{0}^{U} keeps more information from the Newton boundary (from the complement of Δ too). The author knows no direct formula of Z_{0}^{U} directly from the combinatorics of the Newton diagram.

Finally notice that in this example the ‘leading term’ z_{2}(z_{3}^{2} + z_{1}^{2}z_{2} + z_{2}^{6}) is not irreducible, causing the anomaly G ≠ F^{U}. This shows that the irreducibility restriction in (??) is necessary indeed.

7.7. In the next example we provide a general method to find singularities with Z ≠ P. In fact, as this construction suggests, the Main (Reduced) Identity may hold only for very special, rigid analytic structures, and only for those resolutions which ‘fit with some resolution properties of the analytic type’ (see also (??) and (??)).

7.8. Example (for Z ≠ P). Consider the following resolution graph Γ

[figure: −2; −2; −2; −2; −2; −2; −3; −2; −2; E_{1}]

It is realized e.g. by {z_{1}^{2} + z_{2}^{3} + z_{3}^{11} = 0}, but one might take any analytic structure supported by Γ. Notice that H = 1.

First we consider the (minimal) resolution π with the above graph. Since Γ is minimally elliptic, Z = P by (??). Next, we concentrate on the reduced case U = {E_{1}}. The identity Z^{U} = P^{U} can be verified independently of (??) as well: Z_{π}^{U} is equal to the right hand side of (??) by a computation; while P_{π}^{U} follows from (??) with weights (3, 2, 1). In fact, the point what we wish to make, is not about the resolution π.

Let Γ_{b} be a non-minimal graph obtained from Γ by blowing up a regular point of E_{1}, and let E_{2} be the new exceptional divisor. From topological point of view, all the possible choices of the center of this blow up are equivalent. But this is not the case from analytic point of view. In order to explain this we need a definition: For any resolution ϕ, Z_{max} denotes the divisorial part of ϕ^{∗}(m_{X,o}), where m_{X,o} is the maximal ideal of O_{X,0}.

Let us analyze closer the resolution π. Since the singularity is minimally elliptic one has Z_{max} = Z_{min}. But, since Z_{min}^{2} = −1 and the multiplicity is 2, π^{∗}(m_{X,o}) has a unique base point Q; in fact, π^{∗}(m_{X,o}) = O(−Z_{min}) · m_{Q}, where m_{Q} is the maximal ideal of Q (see [?]). Since E_{1} is the only component with (Z_{min}, E_{1}) negative, we get that Q is on the regular part of E_{1}.

Let π^{gen} (respectively π^{Q}) be the resolution obtained from π by blowing up a generic point (≠ Q) of E_{1} (respectively Q). In the first case Z_{max} = Z_{min}, while in the second case Z_{max} = Z_{min} + E_{2}. Notice that the multiplicity of E_{2} in Z_{min} is 1.

Therefore, considering the reduced situation for the graph Γ_{b} and U = {E_{2}}, we get the following: the coefficient of t in the series P_{π^{gen}}^{E_{2}}(t) is non-zero, while in P_{π^{Q}}^{E_{2}}(t) is 0. Since (computed in Γ_{b})

Z^{E_{2}}(t) = \frac{t^{6} − 1}{(t^{3} − 1)(t^{2} − 1)^{2}} ,

we get that Z_{π^{gen}}^{E_{2}} ≠ P_{π^{gen}}^{E_{2}}. In fact, one can show that the divisorial filtration associated with E_{1} in π agrees with the divisorial filtration associated with E_{2} in π^{gen}, hence P_{π^{gen}}^{E_{2}} = P_{π}^{E_{1}} (which agrees with the expression from (??)).

Example 7.8.1. Let us analyse the above examples from the point of view of splice-quotient singularities. In the resolution π the minimally elliptic structure is compatible with the splice-quotient analytic structure constructed from the graph of π (they are equisingular). On the other hand, on π^{gen}, there are two analytic structures (both minimally elliptic), which are ‘not compatible’, they are ‘different’. The first is the pull-back of the splice-quotient analytic structure constructed by the graph of π, the other is the splice-quotient structure constructed in π^{gen}. The point is that the first is not even splice-quotient in π^{gen} (e.g., it does not admit all the ‘end curves’ by its very construction). The splice-quotient analytic structure of π^{gen} is the same with a pulled back splice-quotient if and only if the blow up happened at the base point, i.e. in a very special point.

Hence, blowing up in a wrong point a splice-quotient singularity we can get a non-splice-quotient one.

The same is true for the property Z = P.

Example 7.8.2. The interested reader may construct (or find in the literature) easily pairs of analytic structures of singularities supported by the same topological type, and fixed resolution graphs (even good minimal ones) such that the two cycles Z_{max} are nor equal. By the same argument as above, we get that the Main Identity cannot hold for both analytic structures.

Notice that analyzing the maximal ideal cycle as above we target the very first coefficient of Z(t). Evidently, there are many more subtle analytic information (e.g. basepoints of other linear systems) which may obstruct the identity Z = P.

8. The Relative Identities

8.1. The setup. Let (X, o) be a normal surface singularity and (C, o) ⊂ (X, o) be a reduced curve-germ on it. We fix a good embedded resolution π : X̃ → X of C ⊂ X, which satisfies the following restriction:

(8.1.1)  \{ \begin{matrix}π has no irreducible exceptional divisor intersected by \\ two different components of the strict transform of C. \end{matrix}

L and L′ denote the corresponding lattices associated with π. Furthermore, we consider the universal abelian cover c : (Y, o) → (X, o) and the maps c̃ and π_{Y} as in (??). For any j ∈ V, {k(j)} stays for the index set of the irreducible exceptional divisors of π_{Y} which are projected by c̃ onto E_{j}.

Let C̃ be the π-strict transform of C, U^{C} ⊂ V be the set of those irreducible exceptional divisors which intersect C̃, and C̃_{j} (j ∈ U^{C}) the component of C̃ which intersects E_{j}. The irreducible components of c^{−1}(C̃_{j}) are denoted by {C̄_{j,α}}_{α}.

Definition 8.1.2. We define the following filtration of O_{Y,o}, associated with π and the pair (C, o) ⊂ (X, o), and indexed by l′ ∈ L′:

(8.1.3) F^{C}(l′) :=

\{f ∈ O_{Y,o} : \begin{matrix}div(f ∘ π_{Y})_{k(j)} ≥ c^{∗}(l′)_{k(j)} & for any k(j) with j ∉ U^{C} \\ (div(f ∘ π_{Y}) − c^{∗}(l′), C̄_{j,α}) ≥ 0 & for any α with j ∈ U^{C} \end{matrix} \}.

Above, in the second line, the intersection multiplicity (·, ·) takes values in ℕ ∪ ∞, (D, C̄_{j,α}) being ∞ if D contains C̄_{j,α} as a component.

(Above, for j ∈ U^{C}, even if along the exceptional divisor the divisor div(f ∘ π_{Y}) is not larger than c^{∗}(l′), its intersection multiplicity with C̄_{j,α} can be larger if one of its non-compact components has a large intersection multiplicity.)

Similarly as in the non-relative version, one denotes by 𝔥_{h}^{C}(l′), ([l′] = h) the dimension of the θ(h)-eigenspace of O_{Y,o}/F^{C}(l′), and one defines H^{C}, L^{C} and P^{C} by similar formulas as in the non-relative case. E.g.:

(8.1.4)  P^{C}(t) = −H^{C}(t) · ∏_{j}(1 − t_{j}^{−1}) in ℤ[[t^{L′_{e}}]].

In the combinatorial setting one does the following modification. Define δ_{j}^{C} as δ_{j} for any j ∉ U^{C}, otherwise set δ_{j}^{C} := δ_{j} + 1. Then one defines ζ^{C} and Z^{C} similarly as the corresponding non-relative invariants, but using δ_{j}^{C} instead of δ_{j}. By the same argument as in (??), the identity ζ^{C} = Z^{C} follows.

Definition 8.1.5. We say that the pair (C, o) ⊂ (X, o) and the resolution π satisfies the Relative Identities if

Z^{C}(t) = P^{C}(t).

Theorem 8.1.6. Assume that (X, o) and \underline{all} its good resolutions satisfy the ‘Main Identities’ (??). Then, for any curve-germ (C, o) ⊂ (X, o) and good resolution π as in (??), the ‘Relative Identities’ are satisfied too.

Proof. By the definitions, F(l′) ⊂ F^{C}(l′) for any l′ ∈ L′. Moreover, if f ∈ F^{C}(l′), then either f ∈ F(l′), or the π_{Y} -strict transform of {f = 0} intersects c^{−1}(C̃).

We fix an integer n > 0, and we will focus on those coefficients of P^{C}(t) = ∑_{l′} 𝔭^{C}(l′)t^{l′} for witch l′_{j} ≤ n for any j ∈ V. For two series A(t) = ∑ a(l′)t^{l′} and B(t) = ∑ b(l′)t^{l′} ∈ ℤ[[t^{L′_{e}}]] we write A ≡^{n} B if a(l′) = b(l′) for all l′ ∈ L′ with l′_{j} ≤ n for all j. Our goal is to show that P^{C} ≡^{n} Z^{C} for any arbitrarily chosen n.

Let π̄ be the resolution obtained from π by blowing up the intersection of any strict transform component C̃_{j} (j ∈ U^{C}), with the exceptional divisor in m > 0 infinitezimally closed points. In this way we introduce, for each j ∈ U^{C}, a string of new irreducible exceptional divisors E_{j1}, …, E_{jm}, where all of them are (−2) curves excepting the last one E_{jm} which is a (−1) curve, E_{j1} intersects E_{j} and δ_{jm} = 1. Let V̄ be the set of irreducible exceptional divisors of π̄, let E be the union of components of type E_{jm}, (j ∈ U^{C}), and finally Ū := (V \ U^{C}) ∪ E ⊂ V̄.

In the next paragraphs we will compare F^{C} (associated with π) with F Ū (associated with π̄). There is a natural identification of their t-variables: the variables corresponding to V \U^{C} = Ū \E are identified, and also for any j ∈ U^{C} the variable corresponding to E_{j} with the variable which corresponds to E_{jm}. According to this correspondences, for any l′ = ∑_{j} l′_{j}E_{j} ∈ L′ we write

l̄′ = ∑_{j∈V\U^{C}} l′_{j}E_{j} + ∑_{j∈U^{C}} l′_{j}E_{jm} ∈ L̄′.

Notice also that [E_{jm}^{∗}] = [E_{j}^{∗}] in H for any j ∈ U^{C}.

Next, we formulate two properties used later in the proof. (1) uses the numerical behaviour of the intersection multiplicity with respect to a blow up (of a smooth point), and the fact that the strict transforms of f and C after sufficiently many blow ups can be separated. (2) follows from standard arithmetical properties of negative definite graph. Their detailed verification is left to the reader. The statements are: for any fixed n > 0, there exists an integer m(n) > 0 such that π̄ constructed with m ≥ m(n) satisfies the following two properties:

(1) for any l′ ∈ L′ with l′_{j} ≤ n (j ∈ V) one has F_{π}^{C}(l′) = F Ū ; π̄(l̄′);
(2) −(E_{jm}^{∗}, E_{jm}^{∗})π̄ > n.

For any n, fix such a π̄. Then, by (??), P_{π}^{C} ≡^{n} P Ū ; π̄. But P Ū ; π̄ = ZŪ ; π̄ by the assumption that (X, o) satisfies the main identities for π̄, hence by (??) it satisfies the reduced identities too. But ZŪ ; π̄ and Z_{π}^{C} differ only by the contribution given by E_{jm} (in the first one), hence using (??) and the fact that all these series are in ℤ[[t^{L′_{e}}]], we have ZŪ ; π̄ ≡^{n} Z_{π}^{C}. In particular, P^{C} ≡^{n} Z^{C} for any n, hence they are equal. □

The next fact follows from (??) and (??). It generalizes the main result of [?] (here we have a variable even for j ∉ U^{C}; see also (??)).

Corollary 8.1.7. Any curve (C, o) on a rational surface singularity (X, o), and any resolution π satisfy the Relative Identities.

Remark 8.1.8. The analog of (??) for minimal elliptic singularities is not true, see (??).

9. The Reduced Relative Identities

9.1. Assume that we are in the situation of (??). By a similar construction as in §?? we can reduce the number of variables, and we can formulate the ‘Reduced Relative Identities’ for any non-empty subset U ⊂ V. Moreover, repeating the proof of (??) (combined with (??)) we get

Proposition 9.1.1. If the Main Identities are satisfied for (X, o) and for all the resolutions π, then the Reduced Relative Identities are satisfied for any (C, o) ⊂ (X, o) and U ⊂ V as well.

In the sequel, t denotes the multivariable {t_{j}}_{j∈U}.

In this section we consider that case when U is exactly the subset U^{C} defined in (??), namely, is the index set of irreducible exceptional divisors intersected by the strict transform of C. As we will see, for this special U, both invariants Z^{C,U} and P^{C,U} are independent of the resolution, and can be related with important classical invariants.

9.2. The topological part. Let K_{C} := C ∩ Σ be the link of C in the link Σ of X. Set G := H_{1}(Σ \ K_{C}, ℤ). Then G/T ors(G) is isomorphic with ℤ^{r}, where r is the number of irreducible components of C (= |U^{C}|). Let τ : G → ℤ^{r} be this natural morphism. Moreover, for any character ρ ∈ Ĥ, let ρ^{e} ∈ Ĝ be its extension via the natural map G → H induced by the inclusion.

Then, the Milnor-Reidemeister torsion [?] of Σ \ K_{c} associated with the representation (τ, ρ^{e}) can be identified (up to the natural ambiguities of the torsion) with

(9.2.1)  Ẑ_{ρ}(t) = ∏_{j∈V}(1 − ρ([E_{j}^{∗}])t^{pr_{U} (E_{j}^{∗})})^{δ_{j}^{C}−2},

where this expression is written in terms of a resolution π which satisfies (??). (Since this fact will be not used later, we omit its proof.) As such, Ẑ_{ρ}(t) depends only on the embedding K_{C} ⊂ Σ and on ρ, and it is independent of the resolution chosen (a fact, which can be verified by a direct computation as well).

E.g., when Σ is an integral homology sphere, then ρ = 1, and Ẑ_{1}(t) is the Alexander polynomial of K_{C} ⊂ Σ, see [?].

Notice that for any h ∈ H one has

Z_{h}^{C}(t) = \frac{1}{|H|} ∑_{ρ} ρ(h)^{−1} Ẑ_{ρ}(t).

Therefore, {Z_{h}^{C}}_{h} and {Ẑ_{ρ}}_{ρ} correspond by Fourier transform.

Example 9.2.2. There is an important specialization of Ẑ_{ρ}(t). Assume that f : (X, o) → (ℂ, o) is a non-zero analytic function-germ. For simplicity, we will assume that it defines an isolated singularity. Set C := {f = 0}. Let π be an embedded good resolution of C ⊂ X with (??), and denote by m_{j} the order of vanishing of f ∘π along E_{j} (j ∈ V). Since (div(f ∘ π), E_{j}^{∗}) = 0, we get that m_{j} = − ∑_{i∈U}(E_{i}^{∗}, E_{j}^{∗}), or in other words t^{m_{j}} can be obtained by specialization

(9.2.3)  t^{m_{j}} = t^{E_{j}^{∗}}|_{t_{i}↦t for all i∈U}.

Consider the Milnor fibration arg(f) : Σ\K_{C} → S^{1} of f with Milnor fiber F. Let ℂ_{ρ} be the flat-bundle associated with representation ρ above Σ. Then on can consider the algebraic monodromy action induced by the Milnor fibration on H_{∗}(F, ℂ_{ρ}) (with twisted coefficients), let ζ_{ρ}(t) be its zeta function. Then, one can prove (similarly as A’Campo’s formula for the usual monodromy zeta-function, see [?]) that

ζ_{ρ}(t) = Ẑ_{ρ}(t)|_{t_{i}↦t for all i∈U}.

In particular, if C is irreducible, then ζ_{ρ}(t) = Ẑ_{ρ}(t). If ρ = 1, then ζ_{ρ} is just the usual monodromy zeta function of f.

9.3. The analytical part. Consider the setup of (??). Set C^{Y} := c^{−1}(C). Then c : C^{Y} → C is a branched covering (regular over C \ o) with Galois group H.

First, we show that P^{C,U} depends only on the covering c : C^{Y} → C. Indeed, for any l′ ∈ pr_{U}(L′), (??) reads as

(9.3.1) F^{C,U}(l′) := {f ∈ O_{Y,o} : (div(f ∘ π_{Y}), C̄_{j,α}) ≥ l′_{j} for any α and j ∈ U}.

Therefore, if f ∈ I(C^{Y}), the ideal of C^{Y} in O_{Y,o}, then (div(f ∘ π_{Y}), C̄_{j,α}) = ∞ for any component C̄_{j,α}, hence I(C^{Y}) ⊂ F^{C,U}(pr_{U}(l′)) for any l′ ∈ L′. In particular, any of the series H, L or P associated with the pair (C, U) depends only on the induced filtration of O_{C^{Y}} = O_{Y,o}/I(Y^{C}) and the action of H on it. On the other hand, the filtration also can be recovered intrinsically from C^{Y}. Let {C_{j}}_{j∈U} be the irreducible decomposition of C, let {C_{j,α}}_{α} be the irreducible decomposition of c^{−1}(C_{j}). Notice that the restriction of π_{Y} to C̄_{j,α} is, in fact, the normalization n_{j,α} of C_{j,α}. We fix a local coordinate s_{j,α} in C̄_{j,α}, and we define the valuation v_{j,α} on O_{C^{Y}} by v_{j,α}(f) being the multiplicity of n_{j,α} ∘ f, as a power series in s_{j,α}. Then, for any l′ ∈ pr_{U}(L′), one has

(9.3.2)  F^{C,U}(l′) := {f ∈ O_{C^{Y}} : v_{j,α}(f) ≥ l′_{j} for any α and j ∈ U}.

For any l′ ∈ pr_{U}(L′) and h ∈ H, let 𝔥_{h}^{C,U}(l′) be the dimension of the θ(h)-eigenspace of (O_{C^{Y}} /F^{C,U}(l′))_{θ(h)}. Then

H_{h}^{C,U}(t) = ∑_{l′∈pr_{U} (L′)} 𝔥_{h}^{C,U}(l′) · t^{l′}.

One can describe L_{h}^{C,U} and P_{h}^{C,U} similarly.

If h = 0, then the situation is more simpler: H_{0}^{C,U}, L_{0}^{C,U} and P_{0}^{C,U} depend only on the curve C. Indeed, let v_{j} be the valuation of O_{C} defined via the normalization n_{j} : C̄_{j} → C_{j}. Then, for any l ∈ L_{U} := pr_{U}(L) set

F(l) := {f ∈ O_{C} : v_{j}(f) ≥ l_{j} for any j ∈ U}.

Then

H_{0}^{C,U}(t) = ∑_{l∈L_{U}} dim(O_{C}/F(l)) · t^{l}.

Example 9.3.3. The covering c : C^{Y} → C, in general, is not trivial. Moreover, c cannot be recovered form the curve C and the abstract group H only: the covering c : C^{Y} → C preserves some information from the universal abelian cover c : Y → X. E.g., let (X, o) be the hypersurface singularity given by {x^{2} + y^{2} − z^{4} = 0}, and C be the smooth curve on X given by {x = y − z^{2} = 0}. (If π is the minimal resolution of X, then π^{−1}(C) is a normal crossing divisor, and the strict transform of C intersects the component with δ_{j} = 2.) Then c : C^{Y} → C corresponds to the representation π_{1}(C \ o) = ℤ → ℤ_{4} given by 1 ↦ ˆ2.

9.4. The identity Z^{C,U}(t) = P^{C,U}(t) (or any of its reduced identities) is hardly non-trivial, and, in fact, is rather special and surprising if it holds.

Nevertheless, (??) and (??) provide the next result for rational singularities (which is the main result of [?]):

Corollary 9.4.1. Z_{h}^{C,U}(t) = P_{h}^{C,U}(t) holds for any h ∈ H and for any curve C embedded into a rational singularity (X, o).

Let us return back to the first sentence of (??), and analyze it more explicitly for h = 0. By the above discussion,

Z_{0}^{C,U}(t) = \frac{1}{|H|} ∑_{ρ} Ẑ_{ρ}(t)

is a subtle topological invariant of the embedding C ⊂ X (or K_{C} ⊂ Σ), while P_{0}^{C,U}(t) depends only on (analytic type) of the (abstract) curve C.

The validity of the identity Z_{0}^{C,U} = P_{0}^{C,U} (say, for some family of normal surface singularities X) can be translated as follows: the analytic type of an abstract curve C imposes that if C is embedded into some X (from that family), the Alexander invariant of the embedding is independent of the embedding.

E.g., if a smooth irreducible curve C can be embedded into X then Z_{0}^{C,U} = P_{0}^{C,U} implies that the Alexander invariant of the embedding should be ∑_{k≥0} t^{k} = 1/(1−t) (the Alexander invariant of the trivial knot in S^{3}, or the Poicaré polynomial of ℂ[s], where deg(s) = 1).

Example 9.4.2. Consider the situation from (??) (when X is rational). Then (written in the minimal resolution)

P (t) = \frac{1}{4} ∑_{ξ^{4}=1} \frac{1 − ξ^{2}t}{(1 − ξt^{1/2})(1 − ξ^{3}t^{1/2})}.

After a computation this, indeed, equals 1/(1 − t).

Example 9.4.3. The statement of (??) is not true if we consider arbitrary embeddings into minimally elliptic singularities, even if the curve C is smooth irreducible (unless C is embedded in a very special way).

Consider the minimally elliptic singularity X = {x^{2} +y^{3} +z^{7} = 0}. Its minimal good resolution graph is Γ from (??) whose notations we will use. Let C be the irreducible curve {z = x^{2} + y^{3} = 0}. The minimal good resolution resolves the pair C ⊂ X as well, the strict transform cuts E_{1}. Hence, Z(t), which in this case equals also the monodromy zeta function of z : X → ℂ is (t^{6} − 1)/(t^{3} − 1)(t^{2} − 1) = (t^{2} − t + 1)/(1 − t), hence equals the monodromy zeta function of the plane curve singularity x^{2} + y^{3} (i.e., of C embedded differently), and equals also the Poincaré series P (t) (which in this case is the same as the generating function of the semigroup {0, 2, 3, 4 ⋯ } ⊂ ℕ of C). Hence Z^{C,U} = P^{C,U}.

Consider now another irreducible curve C′ in X which has the same embedded topological type as C, i.e. its strict transform cuts transversally E_{1}. E.g., one of the possibilities is to take C′ := \{y = z^{2}, x = iz^{3}\sqrt{1 + z}\}. (Notice that the parametrization z ↦ (iz^{3}\sqrt{1 + z}, z^{2}, z) has leading terms of weights (3, 2, 1), which fits with the weights considered for E_{1} in (??).) Clearly, Z^{C′,U} is as before, hence equals (t^{2} − t + 1)/(1 − t). But, since C′ is smooth, P^{C′,U}(t) = 1/(1 − t). Hence Z^{C′,U} ≠ P^{C′,U}. This also shows that the non-reduced, relative identity also fails in general (cf. ??).

This construction is not special only for this case. (E.g., by a very small modification, one may repeat the same argument for (??) too.) Definitely, the main difference between C and C′ is that one of them contains the basepoint Q of π^{∗}(m_{X,o}) the other one not. (Compare with the discussion from (??).) Hence, possibilities to construct such pairs appear in abundance.

References

1. N. A’Campo, La fonction zeta d’une monodromie, Comment. Math.Helv. 50 (1975), 233–248.

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

3. , The cohomology of line bundles of splice-quotient singularities, manuscript in preparation.

4. , Invariants of Newton non-degenerate surface singularities, Compositio Math. 143 (2007), 1003–1036.

5. A. Campillo, F. Delgado, and S. M. Gusein-Zade, Poincaré series of a rational surface singularity, Invent. Math. 155 (2004), no. 1, 41–53. MR2025300 (2004j:14042)

6. , Poincaré series of curves on rational surface singularities, Comment. Math. Helv. 80 (2005), 95–102.

7. , Universal abelian covers of rational surface singularities and multi-index filtrations,

    (2007), arXiv:0706.4062.

8. S.D. Cutkosky, J. Herzog, and A. Reguera, Poincaré series of resolutions of surface singularities, Transactions of AMS 356 (2003), no. 5, 1833–1874.

9. D. Eisenbud and W. Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, vol. 110, Princeton University Press, Princeton, NJ, 1985. MR817982 (87g:57007)

10. J. Fernández de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández, and A. Némethi, On rational cuspidal projective plane curves, Proc. London Math. Soc. (3) 92 (2006), no. 1, 99–138. MR2192386 (2007a:14031)

11. , On rational cuspidal curves, open surfaces and local singularities, Singularity theory, World Sci. Publ., Hackensack, NJ, 2007, pp. 411–442. MR2342921

12. H. B. Laufer, On rational singularities, Amer. J. Math. 94 (1972), 597–608. MR0330500 (48 13. , On minimally elliptic singularities, Amer. J. Math. 99 (1977), no. 6, 1257–1295.

MR0568898 (58 #27961)

14. J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279.

15. I. Luengo, The μ-constant stratum is not smooth, Invent. Math. 90 (1987), no. 1, 139–152.

16. I. Luengo-Velasco, A. Melle-Hernández, and A. Némethi, Links and analytic invariants of superisolated singularities, J. Algebraic Geom. 14 (2005), no. 3, 543–565. MR2129010 (2005m:32057)

17. J.W. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–428.

18. A. Némethi, Line bundles associated with normal surface singularities,

arXiv:math.AG/0310084, published in [?].

19. , “Weakly” elliptic Gorenstein singularities of surfaces, Invent. Math. 137 (1999), no. 1, 145–167. MR1703331 (2000e:32037)

20. , Invariants of normal surface singularities, Real and complex singularities, Contemp. Math., vol. 354, Amer. Math. Soc., Providence, RI, 2004, pp. 161–208.

21. , Graded roots and singularities, Singularities in geometry and topology, World Sci. Publ., Hackensack, NJ, 2007, pp. 394–463. MR2311495

22. A. Némethi and L.I. Nicolaescu, Seiberg-Witten invariants and surface singularities, Geom. Topol. 6 (2002), 269–328 (electronic). MR1914570 (2003i:14048)

23. , Seiberg-Witten invariants and surface singularities. II. Singularities with good ℂ^{∗}- action, J. London Math. Soc. (2) 69 (2004), no. 3, 593–607. MR2050035 (2005g:14070)

24. , Seiberg-Witten invariants and surface singularities: splicings and cyclic covers, Selecta Math. (N.S.) 11 (2005), no. 3-4, 399–451. MR2215260

25. A. Némethi and T. Okuma, On the Casson invariant conjecture of Neumann-Wahl, to appear in J. of Algebraic Geometry; arXiv:math.AG/0610465.

26. , The Seiberg–Witten invariant conjecture for splice-quotients, to appear in Journal of London Math. Soc.

27. W. D. Neumann, Abelian covers of quasihomogeneous surface singularities, Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 233–243. MR713252 (85g:32018)

28. W. D. Neumann and J. Wahl, Complete intersection singularities of splice type as universal abelian covers, Geom. Topol. 9 (2005), 699–755 (electronic). MR2140991 (2006i:32037)

29. M. Oka, On the resolution of the hypersurface singularities, Complex analytic singularities, Adv. Stud. Pure Math., vol. 8, North-Holland, Amsterdam, 1987, pp. 405–436. MR894303 (88m:32023)

30. T. Okuma, The geometric genus of splice-quotient singularities, to appear in Trans. AMS;

arXiv:math.AG/0610464.

31. , Universal abelian covers of rational surface singularities, J. London Math. Soc. (2) 70 (2004), no. 2, 307–324. MR2078895 (2005e:14006)

32. , Universal abelian covers of certain surface singularities, Math. Ann. 334 (2006), no. 4, 753–773. MR2209255

33. H. Pinkham, Normal surface singularities with C^{∗} action, Math. Ann. 227 (1977), no. 2, 183–193. MR0432636 (55 #5623)

34. A. Röhr, A vanishing theorem for line bundles on resolutions of surface singularities, Abh. Math. Sem. Univ. Hamburg 65 (1995), 215–223.

Rényi Institute of Mathematics, 1053 Budapest, Reáltanoda u. 13–15, Hungary

 E-mail address: nemethi@renyi.hu