GENERALIZED MONODROMY CONJECTURE IN DIMENSION TWO

ANDRÁS NÉMETHI AND WILLEM VEYS

Abstract. The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimension two, incorporating zeta functions with differential forms and targeting all monodromy eigenvalues, and also considering singular ambient spaces. That is, we treat in a conceptual unity the poles of the (generalized) topological zeta function and the monodromy eigenvalues associated with an analytic germ f : (X, 0) → (ℂ, 0) defined on a normal surface singularity (X, 0). First, we show that the ‘naive’ extension fails. The article targets the ‘right’ extension in the case when the link of (X, 0) is a homology sphere. As a first step, we prove a splice decomposition formula for the topological zeta function Z(f, ω; s) for any f and analytic differential form ω, which will play the key technical localization tool in the later definitions and proofs.

Then, we define a set of ‘allowed’ differential forms via a local restriction along each splice component. For plane curves we show the following three guiding properties: (1) if s_{0} is any pole of Z(f, ω; s) with ω allowed, then exp(2πis_{0}) is a monodromy eigenvalue of f, (2) the ‘standard’ form is allowed, (3) every monodromy eigenvalue of f is obtained as in (1) for some allowed ω and some s_{0}.

For general (X, 0) we prove (1) unconditionally, and (2)–(3) under an additional (necessary) assumption, which generalizes the semigroup condition of Neumann– Wahl. Several examples illustrate the definitions and support the basic assumptions.

1. Introduction

1.1. The Monodromy Conjecture of Igusa, Denef and Loeser [7, 8] is one of the most fertilizing conjectures in singularity theory. It relates poles of Igusa/motivic/topological zeta functions to monodromy eigenvalues. For instance, for a local analytic isolated singularity f : (ℂ^{n}, 0) → (ℂ, 0) it predicts that if s_{0} is a pole of the local topological zeta function of f, then exp(2πis_{0}) is an eigenvalue of the local monodromy operator acting on H^{∗}(F_{0}, ℂ), where F_{0} is the Milnor fiber of f. In the definition of the topological zeta function not only some invariants of the local germ f are codified, but in a subtle way also some numerical data of the standard differential form of (ℂ^{n}, 0) lifted to an embedded resolution of f.

The conjecture was proved for n = 2 by Loeser (originally in the context of p-adic Igusa zeta functions) in [14]. There are by now various other partial results, e.g. [2, 3, 6, 12, 13, 15, 28, 32], nevertheless the conjecture resists to all attacks (even for n = 3). The main obstacle is the lack of a conceptual bridge connecting the two invariants, the topological zeta

2000 Mathematics Subject Classification. Primary. 14B05, 32S40, 14H20; Secondary. 32S05, 14H50, 14J17, 32S25.

Key words and phrases. Monodromy conjecture, topological zeta function, monodromy, surface singularity, plane curve singularities, resolution graphs, semigroup condition, splice diagrams, splice decomposition.

The first author is partially supported by OTKA Grants. The second author is partially supported by FWO–Flanders project G.0318.06.

2 A. Némethi and W. Veys

function and the monodromy operator; the existent proofs of the particular cases basically compute both sides (using their special properties) and compare the two final data.

A possible way to find a more conceptual understanding and tools is to extend the conjecture to a larger class. This leads us to the replacement of (ℂ^{n}, 0) with a singular space, and of the standard differential form with some generalization of it. Although both types of generalizations are obstructed (see the next subsections), the main target of the present article is to find the right such extension when the ambient space is 2–dimensional. Since the two types of generalizations are independent, and have rather different effects, in order to understand their nature, at the first discussion we separate them.

1.2. Extending the differential form. There is a more direct motivation for the generalization of the standard form. It is easy to see on explicit examples that for any fixed germ f : (ℂ^{n}, 0) → (ℂ, 0), and by considering the ‘classical’ topological zeta function, not all the eigenvalues of the monodromy operator are realized; actually quite few eigenvalues are obtained this way (in general). Hence, for any fixed f, it is natural to try to extend in some way this set of poles, such that the same procedure would yield all eigenvalues of f. We expect that such a construction could reveal the conceptual bridge mentioned above. A natural way to extend poles is using the local topological zeta functions associated with the original germ f and with a set of analytic differential n-forms ω living in (ℂ^{n}, 0).

We now describe these zeta functions; they are defined in terms of an embedded resolution π : X̃ → ℂ^{n} of f^{−1}(0) ∪ div(ω). We denote by E_{i}, i ∈ S, the irreducible components (exceptional divisors and strict transforms) of the inverse image π^{−1}(f^{−1}(0) ∪ div(ω)) and by N_{i} and ν_{i} − 1 the multiplicities of E_{i} in the divisor of π^{∗}f and π^{∗}ω, respectively. We put E_{I}^{∘} := (∩_{i∈I}E_{i}) \ (∪_{j∉I}E_{j}) for I ⊂ S. Hence, the E_{I}^{∘} constitute a stratification of X̃ in locally closed subsets.

Definition 1.2.1. The (local) topological zeta function of (f, ω) at 0 ∈ ℂ^{n} is

Z(f, ω; s) := ∑_{I⊂S} χ(E_{I}^{∘} ∩ π^{−1}(0)) ∏_{i∈I} \frac{1}{ν_{i} + sN_{i}} ,

where s is a variable.

This definition extends naturally the definition of Denef and Loeser from [7] valid for the standard form ω = dx_{1} ∧ ⋯ ∧ dx_{n}. Their original proof that the corresponding expression does not depend on the chosen resolution is by describing it as a kind of limit of p-adic Igusa zeta functions. Later they obtained the statement as a specialization of the intrinsically defined motivic zeta functions [8]. Another technique is applying the Weak Factorization Theorem [4, 34] to compare two different resolutions. For arbitrary ω one can proceed analogously. Hence Z(f, ω; s) is a well–defined invariant of the pair (f, ω).

In the literature similar generalizations are already present, see for example [2, 3, 31]; however they are subject to the restriction supp(div(ω)) ⊂ f^{−1}(0). In the present article we release this condition. (In the original context of p-adic Igusa zeta functions, see e.g. [14, III 3.5].)

Although Z(f, ω; s) is a sum of ‘local’ contributions, in this sum many local candidate poles cancel, and usually is hard to characterize those which survive.

1.2.2. We recall that the ‘classical’ monodromy conjecture predicts the implication

s_{0} is a pole of Z(f, ω; s) ⇒ exp(2πis_{0}) is a monodromy eigenvalue of f,

Generalized Monodromy Conjecture 3

where ω is the standard form dx_{1} ∧ ⋯ ∧ dx_{n}. The point is that for arbitrary analytic differential forms ω this implication is in general false. Even more, in [33] the second author showed that every given monodromy eigenvalue of f is induced by a pole s_{0} of some Z(f, ω; s), but in general that zeta function has other ‘bad’ poles, not inducing eigenvalues. This shows, that one can indeed generate a lot of poles, but their relationship with the eigenvalues is uncontrolled. The next program targets exactly this uncertainty via the selection of forms with compatibility properties with the monodromy operator.

Partly initiated in [33] (see also [21]) we propose the following program.

Goal 1.2.3. Define/identify a collection of allowed analytic forms ω (depending on f) such that

(1) if s_{0} is any pole of Z(f, ω; s), where ω is allowed, then exp(2πis_{0}) is a monodromy eigenvalue of f,

(2) the standard form ω = dx_{1} ∧ ⋯ ∧ dx_{n} is allowed,

(3) every monodromy eigenvalue of f is obtained as in (1) for some allowed ω and some s_{0}.

A few remarks are in order. First, note that (1) and (2) imply the classical Monodromy Conjecture. Furthermore, (1) and (3) combined shows that the set of eigenvalues of f coincides with the set exp(2πiP), where P runs over all the poles of the zeta functions of f and all allowed forms.

Note also that the ‘size’ of the wished allowed forms is obstructed by both conditions (1) and (3). A larger set is obstructed more by (1), while if this set is too small then it may not realize in (3) all eigenvalues. In particular, its construction really requires a conceptual understanding of the geometry of the pole–eigenvalue bridge mentioned above.

Let us briefly support our goal by comparing with a more classical context. Recall that the topological zeta function is a kind of avatar of the p-adic Igusa zeta function, which is the meromorphic continuation of a p-adic integral associated to a p-adic function germ f (with complex parameter/variable s). We could rephrase the above also for that zeta function, again with complex parameter/variable s. For the analogous complex integral associated to a complex function germ f, and involving compactly supported C^{∞} forms ω, there are general theorems by Malgrange [17], Kashiwara [10] and Barlet [5], claiming the analogous statements of the goals above. Roughly, if s_{0} is any pole of the zeta function of f and any compactly supported C^{∞} form ω, then exp(2πis_{0}) is a monodromy eigenvalue of f, and all eigenvalues are obtained this way. Our ‘standard form’ can be compared with a C^{∞} form that is non-vanishing. For a detailed explanation and comparison, we refer to the introduction of [33]. However, the ‘exact comparison’ in general fails, and it is still hidden what the analogue of compactly supported C^{∞} forms is in the holomorphic category.

One of the main results of this paper is an identification of a set of allowed forms realizing the goals (1)–(2)–(3) above for n = 2, that is for an arbitrary plane curve germ f.

Our technique is to consider the so–called splice diagram of f, and its splice decomposition in star–shaped pieces. It is not difficult (and reasonably conceptual) to define the allowed forms and realize our goal when the diagram of f itself is star–shaped; we then use this as inspiration for the general case, identifying allowedness ‘locally’, that is, on all star–shaped subdiagrams, cf. section 4.

1.3. Extension to a singular ambient space. An important new feature in this paper is generalizing (1.2.3) to a singular setting. More precisely, we will consider an analytic function germ f defined on a normal surface germ (X, 0) and study the analogue of the goals (1)–(2)–(3) for this f. First we must identify the relevant (generalized) topological zeta function for such a pair (X, f) and for an analytic form ω. Later, in all our combinatorial argument, we will replace (f, ω) by two Weil divisors (for details and motivation see subsection (1.5.1)).

Of course, in order to have a well–defined analogue of part (2) of Goal 1.2.3, we need to consider Gorenstein germs, which guarantees the existence of a ‘standard form’. If the ambient space is two-dimensional, the Gorenstein condition simplifies at topological/combinatorial level to the numerically Gorenstein condition, which is guaranteed, for example, if the link is an integral homology sphere.

Our first claim is that the ‘naive’ extension of the Monodromy Conjecture to the singular setting in general fails. In (3.3.5) we construct a Gorenstein surface singularity (X, 0) with standard form ω_{0}, and an analytic germ f on (X, 0), such that the zeta function of (X, f, ω_{0}) has a pole which does not induce a monodromy eigenvalue. This is the first example in the literature pointing out the failure of the Monodromy Conjecture extended blindly from the smooth situation to the Gorenstein setting. More precisely, this example shows that in a ‘naive’ extension of Goal 1.2.3 to the general (Gorenstein) singular setting, parts (1) and (2) might contradict each other. Therefore, some restrictions regarding the pair (X, f) are necessary!

Our second main result extends the combinatorial definition of the allowed forms to the singular surface case when the link of the ambient space is an integral homology sphere, and establishes for them Goal (1) unconditionally, and (2) and (3) under a natural topological restriction on (X, f). The restriction is a modification of the semigroup condition of Neumann and Wahl [22], adapted to the present situation and to ‘divisors supported on a graph’. This condition is automatically satisfied if the ambient space is smooth (and in several other cases too).

Allowed forms are again defined via the same local picture of the splice components. We emphasize that the definition of the allowed forms, the generalized semigroup condition, and the whole proof of Goal 1.2.3 is combinatorial: one uses only the Weil divisors of the functions and forms (and their analytic realizations will not be involved). In subsection (1.5.1) we give some details about the formalism of zeta functions associated with Weil divisors, and in (1.5.3) we motivate the definition of allowed forms. An ambient germ with integral homology sphere link will be abbreviated by IHS germ.

1.4. Here is the plan of the paper and the list of the most important new results.

In the next section we recall the classical notion of splice diagrams for surface germs and for functions/divisors on them, and we incorporate in the picture also differential forms (and their generalizing divisors). Here we also introduce the extension of the semigroup condition of Neumann and Wahl in the presence of a divisor. Section 3 treats the concept of splicing of these diagrams. The relevant ‘splice formulae’ are well known for functions on surface germs; we develop them for differential forms. Then we use these formulae to derive a splice formula for the topological zeta function, see Theorem 3.2.4. In section 4 we define our allowed forms/divisors and investigate their crucial properties concerning restriction and extension along (sub)diagrams. In section 5 we show that any pole of the zeta function associated to a function f and any allowed form/divisor induces a monodromy eigenvalue of f (first goal). The second goal (standard form is allowed) and third goal (any eigenvalue is induced by a pole of a zeta function of an allowed form) are proved for plane curve germs (unconditionally) in section 6. Finally their generalized versions for functions on IHS germs (under the semigroup condition) are proven in section 7.

Additionally, we list several examples in order to make the manuscript more readable, and in order to emphasize the role of several key points in definitions or about needed restrictions. For example, Examples 4.1.7 and 7.4.24 show that the semigroup condition is necessary to have Goal (2) and (3), respectively, while the discussion from 7.4.25 shows that if we drop the IHS assumption about the link of the ambient space we need to introduce a much stronger (and presumably more technical) notion replacing the semigroup condition.

1.5. Some more details and motivations. Here we present the key motivations for the major restrictions and constructions of the article as a separated guide.

1.5.1. We first explain what the topological zeta function is on a singular ambient surface associated with two Weil divisors.

There have been various generalizations of topological and motivic zeta functions to singular ambient varieties X (instead of ℂ^{n}), see for example [31], and specifically for surfaces [26, 27, 30]. Before introducing the ones we will use, note that the zeta function Z(f, ω; s) of Definition 1.2.1 depends in fact only on the effective divisors F := div(f) and W := div(ω) on (ℂ^{n}, 0), and not on the actual function f and form ω. In terms of these divisors, the numerical data N_{i} and ν_{i} − 1 above are given as the multiplicities of E_{i} in the divisors π^{∗}F and K_{π} + π^{∗}W, respectively, where K_{π} is the relative canonical divisor of π.

Therefore, it is natural to associate in a singular setting a topological zeta function to two Weil divisors on X, in terms of an embedded resolution π of the union of their supports. For this we should be in a situation where there is a natural notion of pullback of Weil divisors, and where the relative canonical divisor K_{π} exists. Both conditions are satisfied when (X, 0) is an arbitrary normal surface germ (for the pullback see (2.2.2), while for K_{π} see (2.3.2)). Let F := ∑_{j∈J} N_{j}E_{j} be an effective non-zero Weil divisor, and W := ∑_{j∈J}(ν_{j} − 1)E_{j} an arbitrary Weil divisor on X, where E_{j}, j ∈ J, are (finitely many) irreducible Weil divisors. We only require that (N_{j}, ν_{j}) ≠ (0, 0) for j ∈ J, that is, a component E_{j} that appears in W with multiplicity −1 must appear in the support of F.

Let π : X̃ → X be an embedded resolution of supp(F) ∪ supp(W). We denote again by E_{i}, i ∈ S, the irreducible components of its inverse image, and by N_{i} and ν_{i} − 1 the multiplicities of E_{i} in the divisors π^{∗}F and K_{π} + π^{∗}W, respectively. Note that the N_{i} and ν_{i} of exceptional components E_{i} are in general rational numbers. The (local) topological zeta function of (F, W) at 0 ∈ X, denoted as Z(F, W; s), is defined by the same formula as in Definition 1.2.1. It is straightforward to verify that this expression does not depend on the chosen resolution π.

In our study below we will assume that the link of X is an integral homology sphere; then in particular all N_{i} and ν_{i} are integers. In such a context, more useful for us will be a formula for Z(F, W; s) in terms of the splice diagram associated to (F, W), see (2.4). Roughly, the splice diagram is obtained from the dual minimal embedded resolution graph of supp(F) ∪ supp(W) by collapsing the strings to edges and modifying the decorations by a system of data which describes more trustworthily the needed linking numbers.

This description allows us to determine a splice formula showing the ‘almost additivity’ of Z(F, W; s) with respect to the splice decomposition of the diagrams. This is another novelty of the article, which becomes a crucial tool in the main proofs.

1.5.2. (Restrictions regarding W.) It is an easy fact that in the resolution graph exceptional components of valency 1 or 2 do not contribute to the actual poles of the zeta function, and that those of valency at least 3 (corresponding to the nodes in the splice diagram) in general do contribute to the poles. Since precisely those last components contribute to the monodromy eigenvalues, it is very reasonable to restrict from the start the support of the desired allowed W as follows. The map π should be also an embedded resolution of f^{−1}(0) ∪ supp(W), and more precisely supp(W) should consist only of components coinciding with components of f and components whose strict transform intersects the exceptional locus in a component of valency 1; moreover such a component of valency 1 must intersect at most one component of W. In this way, one does not create new exceptional components of valency at least 3 and does not transform those of valency 1 or 2 into components of valency 3, what would probably create undesired new poles. This restriction has a similar formulation in terms of splice diagrams too.

1.5.3. Next, we give the idea of the definition of allowed forms/divisors associated with a fixed function f (or Weil divisor F), with some motivation. It will be illustrated via the plane curve germ given by the function f = (y^{d_{1}} − x^{d_{2}})(y^{d_{1}} + x^{d_{2}}), where d_{1} > 1, d_{2} > 1 and gcd(d_{1}, d_{2}) = 1. Below is its star-shaped splice diagram with node E, where the dashed arrows indicate the support of the strict transform of W = div(ω), and the decorations along these arrows are the multiplicities of its components (for details, see section § 2).

[figure: d_{1}; d_{2}; k_{1} − 1; k_{2} − 1; i_{2} − 1; i_{1} − 1]

Then N = 2d_{1}d_{2} is the vanishing order of f along the node E.

(i) By A’Campo’s formula, the monodromy eigenvalues of f are, besides the trivial eigenvalue 1, precisely the roots of the polynomial

Λ(t) = \frac{(t^{N} − 1)^{2}}{(t^{N/d_{1}} − 1)(t^{N/d_{2}} − 1)}.

These are all N-th roots of unity that are not simultaneously (N/d_{1})-th and (N/d_{2})-th roots of unity; in other words all exp(2πi\frac{u}{N}) for which d_{1} ∤ u or d_{2} ∤ u.

(ii) Using (2.4) one has

Z(f, ω; s) = \frac{1}{ν + sN} (−2 + \frac{d_{1}}{i_{1}} + \frac{d_{2}}{i_{2}} + \frac{1}{k_{1} + s} + \frac{1}{k_{2} + s}) ,

where (see (2.3.5))

(1.5.4)  ν = d_{1}d_{2}(k_{1} + k_{2} − 2) + d_{2}i_{1} + d_{1}i_{2}.

We now investigate the candidate pole s_{0} := −ν/N of the zeta function. If s_{0} is a pole of order one, one easily verifies that its residue R is not identically zero as function in the four variables i_{1}, i_{2}, k_{1}, k_{2}. Hence s_{0} is a pole of Z(s) as soon as the algebraic equation R = 0 is not satisfied. It is also straightforward to compute that R is identically zero in k_{1} and k_{2} if i_{1} = d_{1} and i_{2} = d_{2}, and that generally R is not identically zero otherwise.

(iii) With respect to Goal (1), if we wish to put only ‘necessary’ restrictions to realize it, the following is a very natural choice for allowed W. Note first that d_{ℓ} | ν if and only if d_{ℓ} | i_{ℓ} (by (1.5.4)). We call W allowed if the following condition on i_{1} and i_{2} is satisfied: if d_{1}|i_{1} and d_{2}|i_{2}, then i_{1} = d_{1} and i_{2} = d_{2}. (There are no conditions on k_{1} and k_{2}.) Therefore, for W allowed, if exp(2πis_{0}) is not a root of Λ, then d_{1} | i_{1} and d_{1} | i_{2}, hence R = 0.

Moreover with this definition the divisor W = 0, corresponding to i_{1} = i_{2} = k_{1} = k_{2} = 1, is clearly allowed (Goal (2)), and Goal (3) is also satisfied. Indeed, fix a root exp(2πi\frac{u}{N}) of Λ(t). Since the numbers d_{1} and d_{2} are coprime, there exist integers k_{1}, k_{2}, i_{1}, i_{2} (all positive if we desire so) such that ν in (1.5.4) satisfies ν ≡ u mod N. The restrictions on the given u imply that d_{1} ∤ i_{1} or d_{2} ∤ i_{2}. Hence the constructed W is allowed.

One can carry out without too much effort a similar analysis for any plane curve germ f, or, more generally, for any function f on an IHS germ for which the splice diagram is star-shaped, identifying allowed forms ω/divisors W satisfying our Goal 1.2.3, see (4.1.1). For arbitrary f the situation is at first sight combinatorially hopeless. Nevertheless, for them we use the concept of splicing of a general splice diagram into star-shaped building blocks, and we ask that the ‘restriction’ of the desired W to any such star-shaped building block satisfies the ‘natural’ (already identified) conditions of allowedness.

Though conceptually appealing, it is not clear from the start that such a ‘local’ definition of allowedness will do the job, a priori it is even not obvious that allowed forms/divisors exist on arbitrary diagrams. It will turn out that there are plenty of them, and at the end the proof of the first goal will be (combinatorially) quite conceptual.

We still want to mention one important point regarding the introduction of Weil divisors discussed in (1.5.1). In the singular setting, in order to deal with our goals concerning the zeta functions associated to a given function f, by our inductive splicing strategy, we consider the restriction of the divisor of f to the star-shaped building blocks; the point is that usually the analytic realization of them is a difficult issue deviating from the original main objective. This shows that the introduction of Weil divisors is even necessary.

2. Splice diagrams associated with normal surface singularities

2.1. Splice diagrams of surface–germs. Let (X, 0) be the germ of a complex normal surface singularity, and M be its link. It is well–known that M is an oriented plumbed 3–manifold, and any dual resolution graph might serve as a plumbing graph for M. Let π : X̃ → X be a good resolution, that is, the exceptional divisor E := π^{−1}(0) is a normal crossings divisor on the smooth complex surface X̃. The topology of π is codified in the dual graph G = G_{π}(X) associated with the irreducible components {E_{i}}_{i} of E: each E_{i} determines a vertex of G with genus decoration [g(E_{i})] and self–intersection (E_{i}, E_{i}), while the edges of G correspond to intersection points E_{i} ∩ E_{j}, cf. [11]. Since X is normal, G is connected. Let I(G) be the negative definite intersection form (E_{i}, E_{j})_{i,j}. By plumbing construction one recovers from G both M and (the C^{∞}–type of) (X̃, E).

We recall that M is a rational homology sphere if and only if G is a tree and g(E_{i}) = 0 for all i. Moreover, M is an integral homology sphere if additionally det(−I(G)) = 1.

If M is an integral homology sphere, then G can equivalently be codified in a more condensed form via its splice diagram Γ = Γ_{π}(X), cf. [9]. The diagram Γ is the tree obtained from G by replacing each maximal string of G by a single edge. Hence, Γ is homeomorphic to G, but it has no vertices of valency 2. Its vertices are either nodes (of valency ≥ 3) or ends/boundary vertices (of valency 1); they correspond to the nodes/rupture vertices and boundary vertices of G with valencies ≥ 3 and 1, respectively. The decorations of Γ are as follows. At each node v of Γ one inserts a weight d_{ve} on each incident edge e. Let G_{ve} be the connected component of G\{v} ‘in the direction of e’, then d_{ve} := det(−I(G_{ve})). It is proved in [9, Ch. V] that the decorated graphs G and Γ determine each other.

The decorations {d_{ve}}_{v,e} of Γ satisfy the next compatibility conditions:

(2.1.1)  \{ \begin{matrix}(a) d_{ve} ≥ 1, \\ (b) \{d_{ve}\}_{e} are pairwise coprime integers for any fixed node v, \\ (c) any ‘edge determinant’ q_{e} is positive. \end{matrix}

Part (c) means the following: for any fixed edge e with end–nodes v and w, let the decorations at v be d_{ve} and {d_{ve_{i}}}_{i}, and similarly d_{we} and {d_{we′_{j}}}_{j} at w. Then (see [9, §24]):

(2.1.2)  q_{e} := d_{ve}d_{we} − ∏_{i} d_{ve_{i}} ∏_{j} d_{we′_{j}} > 0.

If π is the minimal good resolution, then Γ also is minimal, in the sense that all the decorations d_{ve} are strictly greater than 1, provided that e connects v with a boundary vertex. If G, or Γ, is not minimal, then such a restriction does not hold. By ‘splice calculus’, one can delete such an edge with decoration 1 and the supported boundary vertex, getting a new equivalent diagram. All these equivalent diagrams represent the same 3–manifold M. The nodes (and the corresponding star–shaped subgraphs around them) correspond exactly to the (minimal or non–minimal) Jaco–Shalen–Johannson decomposition of M (depending on the minimality of Γ), each star–shaped subgraph describing a Seifert piece.

2.1.3. The diagram Γ (or G) contains the same amount of information as the link M, hence working with it we disregard completely the analytic structure of (X, 0). In the sequel we regard Γ as an abstract splice diagram which satisfies (2.1.1). In fact, any such diagram can be realized by some singularity link, but the corresponding analytic structure(s) can be hard to determine and are irrelevant from the point of view of many invariants.

In this correspondence, in fact, there is an ‘easy case’, namely when (X, 0) is smooth and M is the 3–sphere S^{3}: this is happening if and only if in any (maybe non–minimal) splice diagram which represents them the following fact holds: for any node v at most two of the integers {d_{ve}}_{e} can be strictly greater than 1 (that is, any Seifert piece is an S^{3}).

2.1.4. Semigroup condition for Γ. It is convenient to introduce some other combinatorial invariants of a splice diagram Γ as well. If v and w are two vertices of Γ, we set ℓ_{vw} for the product of the edge weights that are adjacent to, but not on, the path from v to w. Furthermore, for each node v, let d_{v} be the product of edge weights adjacent to v.

For any node v and adjacent edge e, let Γ_{ve} be the connected component of Γ \ {v} in the direction of e. We say, following [22], that the node v and adjacent edge e satisfy the semigroup condition if

d_{v} is in the semigroup generated by the ℓ_{vw}, where w is a boundary vertex of Γ in Γ_{ve}.

By definition, a splice diagram Γ satisfies the semigroup condition, if all pairs (v, e) as above satisfy the semigroup condition.

2.2. Splice diagrams of function–germs/divisors. Assume that f : (X, 0) → (ℂ, 0) is the germ of an analytic function on an IHS germ (X, 0). If π : X̃ → X is an embedded resolution of the pair (X, f^{−1}(0)), then the topology of f is described by the embedded resolution graph of f. This consists of the dual graph G_{π}(X, f) of the exceptional divisors decorated by the self–intersections, and supplemented by the following data: each irreducible component of the strict transform intersecting an irreducible exceptional divisor E_{i} is codified by an arrowhead supported by that vertex of G which corresponds to E_{i}. Additionally, each vertex and arrowhead inherits a multiplicity decoration, the vanishing order of f ∘ π along the corresponding irreducible divisor.

Clearly, the divisor div(f ∘ π) on X̃ is a principal divisor, hence (div(f ∘ π), E_{i}) = 0 for any i. This (and the fact that det(I(G)) ≠ 0) shows that all the multiplicities of the strict transforms of f = 0 determine div(f ∘ π) completely (compare with (2.2.3)). In fact, this property identifies div(f ∘ π) as the pullback of the divisor f = 0 on X.

More generally, a divisor F on X̃ supported on E and on some noncompact transversal slices of E is called P –divisor of X̃ if (F, E_{i}) = 0 for any i. If we start with an arbitrary Weil divisor F′ on X and π is an embedded resolution of the pair (X, F′), then there is an unique P –divisor F on X̃ whose arrow-multiplicities agree with the multiplicities of the components of F′. This F will be called the pullback of F′. On the other hand, if F is a P - divisor on X̃, projecting down its noncompact components (by keeping their multiplicities) we get a Weil divisor F′ on X such that F is the pullback of F′. Hence, F and F′ determine each other (thus we will sometimes write F for both of them).

A P –divisor is codified in the graph G similarly as the principal divisors via its arrowheads and multiplicity system, and it is uniquely determined by the arrowheads and their multiplicities. This pair is denoted by G_{π}(X, F).

In all of our topological–combinatorial discussions, we regard G_{π}(X, F) as a combinatorial object, a plumbing representation of a pair (M, M ∩ F′). We do not ask the analytic realization of F′ as a principal divisor (and even if we do in some discussions, we always consider analytic realizations and not algebraic ones).

2.2.1. The splice diagram Γ = Γ_{π}(X, f) associated with (X, f) (or more generally, Γ_{π}(X, F) associated with G_{π}(X, F)) is constructed similarly as above, but now the nodes are those vertices which have valency ≥ 3 including the edges supporting arrowheads. Moreover, the new splice diagram contains arrowheads and multiplicity decorations as well (see [9]).

For the arrowheads of the minimal splice diagram we use the following principle. If an arrowhead a of G is supported by a vertex v of G_{π}(X, F) with valency ≥ 3 (including the edges supporting arrowheads) then v becomes a node of Γ and a becomes an arrowhead of Γ supported by v. The weight of the edge at v supporting such an arrowhead is either 1 or is missing. Next, assume that the arrowhead a of G_{π}(X, F) is supported by a vertex v of G of valency 2. This means that v is a boundary vertex of G_{π}(X) with an arrowhead. If we forget about the arrowheads and we determine Γ from G_{π}(X), then v becomes a boundary vertex of Γ_{π}(X). Then, reconsidering the arrowheads, this boundary vertex in Γ_{π}(X, F) is replaced by an arrowhead. Summed up: in minimal diagrams, all the arrowheads of Γ_{π}(X, F) are supported by nodes, nevertheless they have two different interpretations: if the weight of an edge (at the node v) supporting an arrowhead is ≥ 2 then the corresponding strict transform intersects the corresponding boundary curve of G, while if the weight is 1 (or it is missing) then the strict transform intersects that exceptional component which corresponds to the node. (In the language of knots: weight ≥ 2 gives a special Seifert fiber, while weight 1 a generic Seifert fiber in the corresponding Seifert piece.)

Nevertheless, sometimes we also allow non–minimal representations (which appear naturally when we splice the diagrams). Namely, the following calculus provides equivalent diagrams:

[figure: ⋮; d; =; (for any d ≥ 1); ⋮; d]

Also, Γ_{π}(X, F) inherits the multiplicity of each arrowhead and node from G_{π}(X, F) (with the same geometric interpretation). In the case of principal divisors, f defines an isolated singularity if and only if all arrowhead–multiplicities are 1.

We will use the notation V for the vertices, N for the nodes, B for the boundary vertices and E for the edges of Γ_{π}(X, F). Moreover, we call special edges those connecting two nodes, denoted by E^{s}. The arrowheads will be denoted by A_{F}, the multiplicities by (N_{w}), w ∈ V ∪ A_{F}.

Again, we can regard Γ_{π}(X, F) as an abstract graph, we do not ask about the analytic realization of the pair (X, F) (although, if the graph satisfies (2.1.1) and N_{a} > 0 for all a ∈ A_{F}, then some analytic realization exists, cf. [9, § 24].)

2.2.2. Recall that for any P –divisor the multiplicities of the arrowheads (and the combinatorics of the splice diagram without the other multiplicities) determine all the multiplicities of the vertices. (In the case of the graph G, this is done via I(G)^{−1}.) In the language of Γ one has the following. Let v be a fixed vertex, and let a be an arrowhead. Then define ℓ_{va} as the product of the edge weights that are adjacent to, but not on, the path from v to a. Then the multiplicity N_{v} of any vertex v is given by, cf. [9, §10],

(2.2.3)  N_{v} = ∑_{a∈A_{F}} N_{a}ℓ_{va}.

In particular, if F′ = ∑_{a} N_{a}F′_{a} is a Weil divisor on X with the F′_{a} irreducible, and {F_{a}}_{a∈A_{F}} are the strict transforms and F_{v} = E_{v} the exceptional curves, then the pullback F = π^{∗}(F′) of F′ is represented in the splice diagram by ∑_{v∈A_{F} ∪V} N_{v}F_{v}.

2.2.4. The splice diagram Γ_{π}(X, F) also satisfies the compatibility conditions (2.1.1). If one deletes all the arrowheads and decorations of Γ_{π}(X, F) associated with F we recover a possible (maybe non–minimal) splice diagram of X. Nevertheless, by this simplification, some of the nodes might disappear.

2.2.5. Semigroup condition for Γ_{π}(X, F). In the presence of a divisor, the semigroup condition (2.1.4) will be modified as follows. We say that Γ_{π}(X, F) satisfies the semigroup condition if all pairs (v, e) satisfy (2.1.4), provided that v is a node with adjacent edge e, and the connected part of Γ_{π}(X, F) \ {v} in the direction of e contains no arrowheads.

Remark 2.2.6. (1) The splice diagram associated with the minimal embedded resolution of a plane curve singularity always satisfies the semigroup condition. Indeed, there is only one sub–diagram (see below), where the condition is not satisfied trivially: the nodes of this sub–diagram are not sitting on geodesic paths connecting two arrowheads of the diagram.

[figure: a_{1}; a_{2}; a_{r−1}; a_{r}; p_{1}; p_{2}; p_{r−1}; p_{r}; …; ⋮; v_{1}; v_{2}; v_{r−1}; v_{r}]

For this part, the semigroup condition follows from the positivity of the edge determinants and the fact (used iteratively several times) that for coprime positive integers a and p, each integer larger than ap belongs to the semigroup generated by a and p. (For a more general argument, see (7.1).)

(2) If F = 0, we recover the semigroup condition of (2.1.4). On the other hand, the semigroup condition of Γ_{π}(X, F), and of the diagram obtained from Γ_{π}(X, F) by deleting the information regarding F, are independent. This fact is exemplified next. Note that in the semigroup condition the position of the arrowheads is important, while the multiplicity system of F is irrelevant, hence we will omit the multiplicities from the next diagrams.

(3) It is possible that Γ_{π}(X) does not satisfy the semigroup condition, but Γ_{π}(X, F) for some F does. Take for instance any P -divisor F with enough arrowheads such that Γ \ {v} contains an arrowhead in the direction of e for each pair (v, e).

(4) On the other hand, it is possible that Γ_{π}(X) satisfies the semigroup condition, but Γ_{π}(X, F) does not, due to the appearance of the new nodes which support the arrowheads. Take for example the following resolution graph and the corresponding splice diagram

[figure: −2; −1; −13; −1; −2; −3; −3; 2; 7; 1; 7; 1; 2; 3; 3]

Then Γ_{π}(X, F) does not satisfy the semigroup condition at the central node, although if we delete the arrowhead then Γ_{π}(X) does since that node disappears.

In the next embedded resolution graph and splice diagram, the arrows are determined by the restrictions of the functions z and z^{2} + y on X = {x^{2} + y^{3} + z^{7} = 0}.

[figure: −2; −1; −9; z; −1; −2; −3; z^{2} + y; z^{2} + y; 2; 7; 1; 3; 1; z; 2; 3]

One sees that Γ_{π}(X, F) does not satisfy the semigroup condition. Nevertheless, if we delete at least one arrowhead, then the remaining diagram does (for this, if we delete z^{2} + y, we have to transform the graph into a minimal one).

2.3. Splice diagrams and differential forms. We still consider an IHS singularity (X, 0), and a function germ f or a nonzero effective Weil divisor F′ on X. We fix an embedded resolution π : X̃ → X of (X, f^{−1}(0)) or (X, F′) as in (2.2).

Next, we also wish to incorporate in the picture a differential (meromorphic) 2–form ω or a Weil divisor W. The basic models for us are the following situations.

• Assume that (X, 0) is smooth, hence f determines a plane curve singularity. Then classically one considers ω_{0} = dx ∧ dy (for some local coordinates (x, y) of (X, 0)), or, more generally, ω = gω_{0} for some local analytic germ g on (X, 0). In this case the pullback ω̃ := π^{∗}(ω) is clearly holomorphic on X̃.

• Generalizing to the singular setting, we assume that (X, 0) is Gorenstein and that ω_{0} is a nowhere vanishing holomorphic 2–form on X \ {0}. Then, for any holomorphic germ g on X, the pullback ω̃ := π^{∗}(gω_{0}) has a meromorphic extension over the exceptional curve. If (X, 0) is rational then it is holomorphic, otherwise it might have poles.

We will enrich the diagrams with the vanishing orders of div(ω̃) along the corresponding irreducible divisors. Again, this only depends on div(ω̃), not on ω̃ itself; so we rather incorporate from the start a Weil divisor W′ on X. In the special cases above div(ω) = div(g) = W′ and div(ω̃) = K_{π} + π^{∗}W′. Also, it will be natural in our context to restrict the possible support of W′; we assume that the following facts hold:

(1) π is also an embedded resolution of supp(F′) ∪ supp(W′);

(2) if W′ := ∑_{k}(i_{k} − 1)W′_{k} is the irreducible decomposition of W′, then the strict transform of each W′_{k} either is identical with the strict transform F_{a} of one of the F′_{a}, or, it intersects a boundary component of G_{π}(X, F). Moreover, each boundary component can intersect at most one of the strict transforms of the W′_{k}.

We denote the P –divisor π^{∗}(W′) by W. In our diagrams, the strict transform W_{k} of a W′_{k} that intersects E_{i} will be denoted by a dashed arrowhead, attached to the vertex corresponding to E_{i}. If W_{k} agrees with one of the F_{a}, then the dashed arrowhead doubles the associated ordinary arrowhead, while if a boundary vertex does not support any ordinary arrowhead but supports a W_{k}, then its dashed arrow is attached to this vertex. We will denote by A_{W} the dashed arrowheads of W. The ‘double’ arrowheads are given by A_{F} ∩A_{W}.

Again, we will identify the following diagrams.

[figure: ⋮; 1; =; ⋮]

Finally, we have to add to the decorations the multiplicities of the components of K_{π} +W. By technical (and traditional) reasons, the multiplicity of a dashed arrowhead is denoted as above by i_{a} − 1, while the multiplicity of a vertex v by ν_{v} − 1. Recall that, in general, we do not impose for the integers i_{a} and ν_{v} to be positive. We denote this enriched diagram as Γ_{π}(X, F, W).

Note that in the three different levels Γ_{π}(X), Γ_{π}(X, F) and Γ_{π}(X, F, W), the valencies of a fixed vertex v of Γ are not the same; these three valencies will be denoted by δ_{v}, δ′_{v} and δ′′_{v} respectively.

2.3.1. In order to identify the coefficients ν_{v} − 1 of K_{π} + W, we only must describe the coefficients of the E_{v} in K_{π}, since those of W are as in (2.2.2).

The divisor K = K_{π} is determined in the graph G_{π}(X) by the adjunction relations

(2.3.2)  (K + E, E_{i}) = δ_{i} − 2,

where here δ_{i} is the valency of the vertex i in G_{π}(X). Let L be the lattice H_{2}(X̃, ℤ) with the intersection form I(G). Then for each vertex i of G one can define E_{i}^{∗} ∈ L with (E_{i}^{∗}, E_{j}) = −δ_{ij} (the negative of the Kronecker delta), i.e. the sign–modified dual basis of L. (Since det(−I(G)) = 1, they are well–defined.) Therefore, (2.3.2) reads as K + E = ∑_{i} (2 − δ_{i})E_{i}^{∗}. Since all the valency 2 vertices of G are irrelevant, the relation descends naturally to the level of Γ = Γ_{π}(X) (this means that the multiplicities of K + E along the vertices of Γ are those given on the right):

(2.3.3)  K + E = ∑_{w∈V} (2 − δ_{w})E_{w}^{∗}.

Each E_{w}^{∗}, considered as divisor supported on E, together with a non–compact irreducible divisor intersecting E_{w} in a smooth point of E, form a P –divisor. Hence, we can use (2.2.3) and we obtain that the multiplicity of K + E along any node v of Γ (or in the presence of a divisor F, along any node of Γ_{π}(X, F)) is

(2.3.4)  ∑_{w∈V} (2 − δ_{w})ℓ_{vw}.

In the presence of the divisor W′ = ∑_{a∈A_{W}} (i_{a} − 1)W′_{a} on X, again (2.2.3) then yields

(2.3.5)  ν_{v} = ∑_{w∈V} (2 − δ_{w})ℓ_{vw} + ∑_{a∈A_{W}} (i_{a} − 1)ℓ_{va}.

We warn the reader about the following fact. The sum (2.3.4) is associated with Γ_{π}(X). Therefore, in the applications later, even if we start with some Γ_{π}(X, F), those ordinary arrowheads associated with F with weight of their supporting edge greater than 1 should be replaced by boundary vertices (hence the summation index V should be the set of vertices of Γ_{π}(X)). This is valid for the left sum of (2.3.5) too.

2.3.6. Notation. In all the next combinatorial formulas associated with a splice diagram Γ_{π}(X, F, W), the resolution π or the geometric source of the diagram is irrelevant.

In particular, in the sequel Γ(F, W) means a splice diagram with two P –divisors F and W. If W = 0 then we just write Γ(F). The divisor W will always be linked with K (determined in Γ by (2.3.1)) in the expression K + W.

2.4. Topological zeta functions of diagrams. In [29] the second author derived a formula for the topological zeta function of a plane curve germ f in terms of its so-called relative log canonical model; this can be interpreted as being in terms of Γ_{π}(X, f) (where π is minimal). The same proof yields a similar formula in our more general context of (2.3), with the divisors F′ and W′ on the germ (X, 0). In fact one associates in this way a zeta function to a decorated diagram Γ(F, W), cf. (2.3.6). We want to formalize this, since our technique to study the topological zeta function is in fact a ‘splicing formula’ for zeta functions of decorated splice diagrams.

Therefore, let us consider a decorated splice diagram Γ(F, W) as in (2.3.6). Moreover, it is convenient to associate multiplicities N_{a} and i_{a} − 1 to all a ∈ A_{F} ∪ A_{W}, that is, we put N_{a} = 0 for a ∈ A_{W} \ A_{F} and i_{a} = 1 for a ∈ A_{F} \ A_{W}.

Definition 2.4.1. Let Γ = Γ(F, W) be such a diagram. We require for each a ∈ A_{F} ∪ A_{W} that (N_{a}, i_{a}) ≠ (0, 0). For a node v, let (A_{F} ∪ A_{W})_{v} and B_{v} be the (ordinary and/or dashed) arrowheads and boundary vertices, respectively, attached at v. Denote the weight at v on the incident edge in the direction of such an arrowhead a or boundary vertex w by d_{va} and d_{vw}, respectively. For any w ∈ B_{v}, let i_{w} − 1 be the decoration of the dashed arrowhead supported by w; if such an arrowhead does not exist then set i_{w} = 1. For a special edge e, let v and w denote its end vertices, and q_{e} the edge determinant (2.1.2). Then the zeta function Z(Γ) of the diagram Γ is

\begin{matrix}Z(Γ) = Z(Γ; s) := ∑_{v∈N} \frac{1}{ν_{v} + sN_{v}} (2 − δ′′_{v} + ∑_{w∈B_{v}} \frac{d_{vw}}{i_{w}} + ∑_{a∈(A_{F} ∪A_{W})_{v}} \frac{d_{va}}{i_{a} + sN_{a}} ) \\ + ∑_{e∈E^{s}} \frac{q_{e}}{(ν_{v} + sN_{v})(ν_{w} + sN_{w})}. \end{matrix}

3. Splicing the diagrams and their invariants

3.1. Splicing the diagrams. The main advantage of the splice diagrams is that they describe in an ideal way the splice (non–minimal JSJ–) decomposition of the 3–manifold M into its Seifert pieces: while doing this operation, the decorations follow rather simple rules. The behavior of the multiplicities {N_{w}}_{w} associated with a principal divisor f are classical, see e.g. [9]. It is easy to see that any P –divisor follows the same formula. On the other hand, the rules for the numbers {ν_{v}}_{v} are slightly more involved, and we were not able to find them in the literature (though, see the ‘simpler’ situation considered in [21]). In this subsection, we will present these splice formulae.

Subsection (3.1.1) treats the case Γ = Γ_{π}(X), (3.1.2) the case Γ_{π}(X, f) and its generalization Γ(F), while (3.1.5) the case Γ(F, W).

3.1.1. Splicing Γ. First, recall that splicing along the edge e with end–nodes v_{L} and v_{R} (left/right) is the operation which replaces the left diagram Γ into the two diagrams Γ_{L} and Γ_{R} (containing v_{L} and v_{R} respectively), with two new end vertices v̄_{L} and v̄_{R}. (All the other parts of the diagrams are kept unmodified.) If either d or d′ is 1, then the corresponding leg in Γ_{L} or Γ_{R} can be deleted in the minimal representation, but we prefer to keep it. In this way, the valency of the vertices v_{L} and v_{R} stays unmodified. Moreover, when we equip such a leg later with an arrowhead, it cannot be deleted.

[figure: ⋮; ⋮; v_{L}; v_{R}; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; e; splicing; ⋮; ⋮; v_{L}; v_{R}; v̄_{L}; v̄_{R}; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}]

(The diagrams Γ_{L} and Γ_{R} correspond again to dual graphs of certain IHS normal surface singularities. If Γ represents M = S^{3}, then both Γ_{L} and Γ_{R} represent S^{3}, see (2.1.3).)

3.1.2. Splicing Γ(F). Next, we analyze the behavior of the multiplicity system determined by a function f or a P –divisor F. Let A := A_{F} be the index set of arrowheads; it can be written as a disjoint union A_{L} ∪ A_{R}, according to the position of the arrowheads. First, assume that both A_{L} and A_{R} are non–empty. If a ∈ A_{L}, then let ℓ_{ea} be the product of the edge weights, all of them in Γ_{L}, that are adjacent to, but not on, the path from v_{R} to a. Symmetrically, one defines the integers ℓ_{ea} for a ∈ A_{R}. Then Γ(F) has the following splice decomposition:

[figure: ⋮; ⋮; v_{L}; v_{R}; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; e; (N); (N′); splicing; ⋮; ⋮; v_{L}; v_{R}; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; (N); (N′); (M); (M′)]

where (cf. [9, (10.6)])

(3.1.3)  M = ∑_{a∈A_{R}} N_{a}ℓ_{ea} and M′ = ∑_{a∈A_{L}} N_{a}ℓ_{ea}.

If all the arrowheads of Γ are in one side, say A_{L} = ∅, then one has the new situation

[figure: ⋮; ⋮; v_{L}; v_{R}; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; e; (N); (N′); splicing; ⋮; ⋮; v_{L}; v_{R}; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; (N); (N′); (M); (M)]

where M is computed by the same formula as in (3.1.3).

We denote the ‘total’ inherited divisors on Γ_{L} and Γ_{R} by F_{L} and F_{R}, respectively. They can be identified with P –divisors of the diagrams Γ_{L} and Γ_{R}, respectively.

Remark 3.1.4. When F = div(f) is a plane curve germ, the left and right graphs above correspond again to dual graphs associated to plane curve germs. In particular, the arrowheads with multiplicities M and M′ correspond to components of these new germs.

3.1.5. Splicing Γ(F, W). Finally, let us analyze the behavior of the divisor of a 2–form ω or, more generally, K + W for some P –divisor W. Since the splicing of W is covered by the previous step (valid for any P –divisor), we have to understand what happens to K only.

Let us consider the splicing of Γ along e as in (3.1.1). Any invariant associated with Γ has its analogue for Γ_{L} and Γ_{R}. We wish to compare the divisors of the pullbacks of the forms ω_{0,Γ} with those of ω_{0,Γ_{L}} and ω_{0,Γ_{R}} — if they exist analytically; and, more generally K_{Γ} with K_{Γ_{L}} and K_{Γ_{R}} (a combinatorial, always well–posed question). For any v ∈ V(Γ_{L}), we denote the dual basis element computed in Γ_{L} by E_{v,Γ_{L}}^{∗}. Moreover, we separate the vertices of Γ_{L} inherited from Γ: we set V̄(Γ_{L}) := V(Γ_{L}) \ {v̄_{L}}, cf. the notation of (3.1.1), and similarly for Γ_{R}. Let us rewrite (2.3.3) into

(3.1.6)  (K + E)_{Γ} = ∑_{v∈V̄(Γ_{L})} (2 − δ_{v})E_{v}^{∗} + ∑_{v∈V̄(Γ_{R})} (2 − δ_{v})E_{v}^{∗}.

Recall that E_{v}^{∗} behaves as a P –divisor associated with one arrowhead supported on v. So, by (2.2.3), for any v ∈ V̄(Γ_{L}) the restrictions satisfy E_{v}^{∗}|_{V̄(Γ_{L})} = E_{v,Γ_{L}}^{∗}|_{V̄(Γ_{L})} (i.e., the multiplicities along V̄(Γ_{L}) agree). Hence

(K + E)_{Γ}|_{V̄(Γ_{L})} = ( (K + E)_{Γ_{L}} − E_{v̄_{L},Γ_{L}}^{∗} + ∑_{v∈V̄(Γ_{R})} (2 − δ_{v})E_{v}^{∗} )|_{V̄(Γ_{L})}.

Clearly, all multiplicities of both E_{Γ} and E_{Γ_{L}} along V̄(Γ_{L}) are one, hence they cancel:

(3.1.7)  K_{Γ}|_{V̄(Γ_{L})} = ( K_{Γ_{L}} − E_{v̄_{L},Γ_{L}}^{∗} + ∑_{v∈V̄(Γ_{R})} (2 − δ_{v})E_{v}^{∗} )|_{V̄(Γ_{L})}.

Using again (2.2.3), the sum can be replaced by a P –divisor of Γ_{L}. Indeed, set

(3.1.8)  i = i_{e,L} := ∑_{v∈V̄(Γ_{R})} (2 − δ_{v})ℓ_{ev},

where ℓ_{ev} (for any v ∈ V̄(Γ_{R})) is the product of the edge weights of Γ, all of them in Γ_{R}, that are adjacent to, but not on, the path from v_{L} to v. Furthermore, let G_{L} be the P –divisor on Γ_{L} determined by one arrowhead with multiplicity one supported on v̄_{L}. Then (2.2.3) and (3.1.7) imply

(3.1.9)  K_{Γ}|_{V̄(Γ_{L})} = (K_{Γ_{L}} + (i − 1)G_{L}) |_{V̄(Γ_{L})}.

If the forms above exist, and if G_{L} is the pullback divisor of a function g_{L}, then we have

(3.1.10)  div(π^{∗}ω_{0,Γ})|_{V̄(Γ_{L})} = div (π_{L}^{∗}(g_{L}^{i−1} · ω_{0,Γ_{L}})) |_{V̄(Γ_{L})}.

Obviously, there is a symmetric identity for the restriction on V̄(Γ_{R}). On diagrams we have

[figure: ⋮; ⋮; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; splicing; ⋮; ⋮; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; i − 1; i′ − 1]

When we incorporate also the divisor W, the equations (3.1.8) and (3.1.9) respectively extend to

(3.1.11)  i := ∑_{v∈V̄(Γ_{R})} (2 − δ_{v})ℓ_{ev} + ∑_{a∈A_{W,R}}(i_{a} − 1)ℓ_{ea},

where ℓ_{ea} is the product of the edge weights of Γ, all of them in Γ_{R}, that are adjacent to, but not on, the path from v_{L} to the corresponding dashed arrow in Γ_{R}, and

(3.1.12)  (K_{Γ} + W)|_{V̄(Γ_{L})} = (K_{Γ_{L}} + (i − 1)G_{L} + W_{L}^{+}) |_{V̄(Γ_{L})},

where W_{L}^{+} is induced by W on Γ_{L}, as in (3.1.2). We will reserve the notation W_{L} rather for (i − 1)G_{L} + W_{L}^{+}, in order to have the expression

(3.1.13)  (K_{Γ} + W)|_{V̄(Γ_{L})} = (K_{Γ_{L}} + W_{L}) |_{V̄(Γ_{L})}.

Note that (usually) the vertices v̄_{L} and v̄_{R} have valency 2 (counting all the arrowheads), hence in the formulas considered in the next sections they will be irrelevant.

In the presence of a P –divisor F, with both A_{F,L} and A_{F,R} non–empty, the above diagram modifies into where M and M′ are determined as in (3.1.3). If A_{F,L} = ∅, then one has Remark 3.1.14. (1) Assume that we start with W = 0. Then both W_{L}^{+} and W_{R}^{+} are zero, but in general W_{L} and W_{R} are not. Furthermore, even if W is effective (in particular, if W = 0), the induced divisors W_{L} and/or W_{R} are in general not effective.

[figure: ⋮; ⋮; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; (N); (N′); splicing; ⋮; ⋮; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; (N); (N′); i − 1; i′ − 1; (M); (M′)]

[figure: ⋮; ⋮; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; (N); (N′); splicing; ⋮; ⋮; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; (N); (N′); i′ − 1; (M); (M); i − 1]

In a different language: even if we start with the ‘standard 2–from’ ω_{0} (instead of the more general gω_{0}) — like in the traditional framework of, say, topological or motivic zeta functions —, once an inductive splice–decomposition argument is used, we are forced to enlarge the class of our forms: in (3.1.10) the standard form decomposes in the splice component into a generalized form of type gω_{0}. Also, one can see that even if we start with a holomorphic form, the forms on the splice components, usually, are not holomorphic (that is, they are meromorphic).

(2) Even if we have a precise analytic realization of a diagram Γ, it is not clear what the relations are connecting this analytic structure and the eventual analytic realizations of Γ_{L} and Γ_{R}. In general, there is no analytic construction known by the authors which would define a natural analytic structure with topology Γ_{L} starting from the original (X, 0).

In the presence of functions and forms the situation becomes even more difficult. In that case it might happen that even if we know that Γ(F, W) is analytically realized as Γ_{π}(X, f, ω), after splicing the two combinatorial packages might not be realized analytically (for example, the ‘correction term’ (i − 1)G_{L} is maybe not the divisor of a function).

Nevertheless, if Γ represents S^{3}, i.e. if any realization of Γ is smooth, then all the functions and (meromorphic) forms will exist.

Example 3.1.15. Consider the following diagram Γ(F), compare also with Remark 2.2.6(4).

[figure: 2; 1; 7; 7; 1; 2; 3; 3; (3); (6); (1); (6); (3); (2); (1); (2); v_{1}; v_{0}; v′_{1}]

We denote the nodes by v_{1}, v_{0} and v′_{1}, and their ν–numbers by ν_{1}, ν_{0} and ν′_{1}. Then ν_{1} = ν′_{1} = −13 and ν_{0} = −2. Splicing the diagram Γ(F, W = 0) we get the three star–shaped subgraphs Γ_{1}, Γ_{0} and Γ′_{1}:

[figure: 2; 1; 7; 7; 1; 2; 3; 3; (3); (6); (1); (6); (3); (2); (1); (2); ν_{1} = −13; ν_{0} = −2; ν′_{1} = −13; (1); −2; −2; −2; (1); −2]

The zeta function Z(Γ; s) associated with Γ(F, W = 0), cf. (2.4.1), is

Z(s) = 2\frac{4}{6s − 13} + \frac{1}{s − 2}(−1 + \frac{1}{s + 1}) + 2\frac{1}{(s − 2)(6s − 13)}.

3.2. Splicing the topological zeta function. We will analyze the splicing behavior of the topological zeta function of a graph Γ(F, W). Let us consider again the splicing of the diagram Γ(F, W) along the edge e as in (3.1.5), where we insert the relevant integers (N, ν − 1) for both vertices v_{L} and v_{R}:

[figure: ⋮; ⋮; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; (N); ν − 1; (N′); ν′ − 1; splicing; ⋮; ⋮; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; (N); ν − 1; (N′); ν′ − 1; i − 1; i′ − 1; (M); (M′)]

If A_{F,L} = ∅, then replace this diagram with the adapted one as in the last diagram of (3.1.5). In this case of A_{F,L} = ∅ one always has M′ = 0.

Note that, to be able to define Z(Γ_{L}) and Z(Γ_{R}), we need that the condition (N_{a}, i_{a}) ≠ (0, 0) for all arrowheads is also valid after splicing, see (2.4.1). For the moment we just assume this. In the context of allowed divisors we will show in (4.2.1) that it is always true.

The contribution of e to Z(Γ) turns out to be the sum of the contribution of the ‘right leg’ of Γ_{L} to Z(Γ_{L}) and the contribution of the ‘left leg’ of Γ_{R} to Z(Γ_{R}), minus an easy correction term, as shown below. This then yields a simple splicing formula for topological zeta functions. We start with the following numerical relation.

Lemma 3.2.1. We use the notation of (3.1.5), as indicated on the diagram above, and put also q := dd′ − (∏_{j=1}^{n} d_{j})(∏_{j=1}^{n′} d′_{j}) for the edge determinant of e. If A_{F,L} = ∅, set M′ = 0. Then we have the equality

\begin{matrix}\frac{q}{(ν + sN)(ν′ + sN′)} = \frac{d}{(ν + sN)(i + sM)} + \frac{d′}{(ν′ + sN′)(i′ + sM′)} \\ −\frac{1}{(i + sM)(i′ + sM′)} . \end{matrix} ; (3.2.2)

Moreover, if two of the pairs (ν, N), (ν′, N′), (i, M) and (i′, M′) are linearly dependent, then any other choice of two pairs are also linearly dependent.

Proof. The equations (2.2.3) and (3.1.3), respectively (2.3.5) and (3.1.11), imply

\begin{cases}N = (∏_{j=1}^{n} d_{j})M + dM′ \\ N′ = (∏_{j=1}^{n′} d′_{j})M′ + d′M \end{cases} and \begin{cases}ν = (∏_{j=1}^{n} d_{j})i + di′ \\ ν′ = (∏_{j=1}^{n′} d_{j})i′ + d′i. \end{cases}

Hence (as polynomials in s)

(3.2.3)  \begin{cases}(∏_{j=1}^{n} d_{j})(i + sM) = ν + sN − d(i′ + sM′) \\ (∏_{j=1}^{n′} d′_{j})(i′ + sM′) = ν′ + sN′ − d′(i + sM). \end{cases}

Multiplying the left and right hand sides of (3.2.3), and using the defining formula of q, we obtain

(ν +sN)(ν′ +sN′)−d(ν′ + sN′)(i′ + sM′)−d′(ν + sN)(i + sM)+q(i+sM)(i′ +sM′) = 0.

This is clearly equivalent to (3.2.2). The linear dependency statements follow easily from (3.2.3), using that q ≠ 0. □

One of the main new results of the article is the next splice decomposition formula for Z(Γ).

Theorem 3.2.4. (1) Consider the splicing of the diagram Γ as in the last diagram of (3.1.5). Again, if A_{F,L} = ∅, set M′ = 0. Then

Z(Γ) = Z(Γ_{L}) + Z(Γ_{R}) − \frac{1}{(i + sM)(i′ + sM′)}.

(2) The contribution of v_{L} in Z(Γ) has −\frac{ν}{N} as a pole of order 2 if and only if the contribution of v_{L} in Z(Γ_{L}) has −\frac{ν}{N} as a pole of order 2.

(3) Suppose that −\frac{ν}{N} is not a pole of order 2 of Z(Γ). Then the contributions of v_{L} to the residue of Z(Γ) and to the residue of Z(Γ_{L}) at −\frac{ν}{N} are exactly the same.

Proof. (1) This is a direct consequence of (3.2.2), since the other contributions to Z(Γ) appear in a disjoint way as the other contributions to Z(Γ_{L}) or Z(Γ_{R}).

(2) First note that the coefficient of any expression 1/(ν + sN)^{2} in the topological zeta function formula is always positive, hence there are no cancelations among them. Consequently, the statement follows immediately from the linear dependency considerations in Lemma 3.2.1.

(3) The difference between the contributions to both residues is

\frac{1}{N} (\frac{q}{ν′ + sN′} − \frac{d}{i + sM} ) ,

evaluated in s = −\frac{ν}{N}, and this is zero because of (3.2.2). □

Example 3.2.5. With the notations of Example 3.1.15 one has

Z(Γ_{1}; s) = Z(Γ′_{1}; s) = \frac{1}{6s − 13}(4 + \frac{7}{s − 1}), Z(Γ_{0}; s) = \frac{1}{s − 2}( − 3 + \frac{1}{s + 1}).

One verifies that indeed, according to Proposition 3.2.4(1),

Z(Γ; s) = Z(Γ_{1}; s) + Z(Γ_{0}; s) + Z(Γ′_{1}; s) − 2\frac{1}{(−1)(s − 1)},

and in the sum Z(Γ; s) the pole s = 1 ‘disappears’.

3.3. Splicing the monodromy zeta function and Alexander polynomial. Let f : (X, 0) → (ℂ, 0) be the germ of a holomorphic function as in (2.2), let F_{0} be its Milnor fiber, h_{i} : H_{i}(F_{0}, ℂ) → H_{i}(F_{0}, ℂ) the algebraic monodromy (i = 0, 1), Δ_{i}(t) := det(tI − h_{i}) the characteristic polynomial of h_{i}, and finally, ζ(t) = Δ_{1}/Δ_{0} the monodromy zeta function associated with f at 0.

It is well–known, cf. [9, (11.3)], that the zeta function can be computed from the splice diagram Γ = Γ_{π}(X, f) as follows:

(3.3.1)  ζ(t) = ∏_{v∈V(Γ)} (t^{N_{v}} − 1)^{δ′_{v}−2},

where, for each vertex v, N_{v} denotes its multiplicity and δ′_{v} its valency in Γ_{π}(X, f). The zeta–function is ‘almost’ multiplicative with respect to the splice decomposition. In order to have a uniform statement, we consider the Alexander polynomial (in one variable), cf. [9, (12.1)], as follows:

(3.3.2)  Λ(t) := \{ \begin{matrix}ζ(t) & if #A(Γ_{π}(X, f)) ≥ 2, \\ Δ_{1}(t) & if #A(Γ_{π}(X, f)) = 1. \end{matrix}

The formula (3.3.1) provides Λ(t) too, since, if #A = 1, then Λ = ζ · Δ_{0}, and Δ_{0}(t) = t^{N_{a}} − 1, where N_{a} is the multiplicity of the unique arrowhead a. (In general, F_{0} has d connected components, hence Δ_{0}(t) = t^{d} − 1, where d := gcd_{a∈A}(N_{a}).) In particular, Λ(t) can be recovered from the diagram Γ = Γ_{π}(X, f); let us write Λ_{Γ}(t) for this expression.

Clearly, Λ_{Γ}(t) depends only on the divisor of f, hence the above formula defines Λ_{Γ}(t) for any P –divisor F and Γ(F). Moreover, assume that the splice diagram Γ(F) has the splice decomposition Γ_{L} and Γ_{R} as in (3.1.2), without considering or asking any analytic realization. Then, analyzing the splice decompositions of (3.1.2) and the formula (3.3.1), we easily get the following.

Proposition 3.3.3.

Λ_{Γ}(t) = Λ_{Γ_{L}}(t) · Λ_{Γ_{R}}(t).

Example 3.3.4. With the notation of Example 3.1.15 one has Λ_{Γ_{1}}(t) = Λ_{Γ′_{1}}(t) = t^{2} −t+1, Λ_{Γ_{0}}(t) = 1, and their product is indeed Λ(t) = Δ_{1}(t) = (t^{2} − t + 1)^{2}.

The advantage (at least in the present paper) of Λ compared with ζ is that Λ is a polynomial, hence we do not have to deal with possible cancelations of the roots and poles in the multiplicative formula of Proposition 3.3.3.

Example 3.3.5. Counterexample to the ‘naive’ generalization of the Monodromy Conjecture. This is a modification of Example (3.5) of B. Rodrigues [26], where only the combinatorial part was treated letting open the analytic realization. The proof of the analytic realization is based on the theory of elliptic singularities, the details will appear in [20]. The authors thank T. Okuma for his valuable advices.

Consider the following resolution graph (the right graph below):

[figure: −2; −1; −6; −2; −4; −3; ⋮; (6); (12); (3); (5); (3); 7 arrows; (1); (1); (1)]

It is easy to verify that it is a numerically Gorenstein elliptic graph with length of the elliptic sequence two (for terminology, see e.g. [18]). It can be realized by a hypersurface isolated singularity with multiplicity 3 and geometric genus 2 (see also [35], case (24) in Table 4). We consider the nowhere vanishing form ω_{0} on X \ {0}. It is harder to verify that on this Gorenstein space–germ the multiplicity system and strict transforms indicated on the right graph can be realized by an analytic function germ f : (X, 0) → (ℂ, 0).

Using the definition of the topological zeta function or the (adapted) formula from [29] one easily computes that

Z(f, ω_{0}; s) = \frac{−21s^{2} + 335s − 120}{5(s + 1)(3s − 1)(12s − 5)}.

In particular, it has s_{0} = 1/3 as a simple pole. On the other hand, exp(2πi/3) is not a root of the monodromy operator acting on the first homology of the Milnor fiber whose characteristic polynomial is

Δ_{1}(t) = (t^{6} + 1)(t^{5} − 1)^{6}.

4. Allowed forms/P –divisors

4.1. Definition. In the original setting of a plane curve f we want to pin down a class of 2-forms ω such that we can realize the goals of (1.2.3) from the introduction. More generally, starting with an effective divisor F on a IHS germ (X, 0), we look for an appropriate class of Weil divisors W.

From the point of view of splicing, our definition of allowed forms/divisors below is quite natural. It is not difficult to identify a natural class of divisors W that do the job on a ‘basic building block’, i.e. a star-shaped graph. We use this as guideline to identify our allowed divisors on a general graph, just demanding that we obtain allowed divisors on all star-shaped subgraphs after (repeated) splicing.

Again, the restriction is combinatorial, depending only on the splicing graph; hence, we will treat allowed P –divisors W of graphs Γ(F).

Definition 4.1.1. Let Γ = Γ(F) be a diagram as in (2.2), see also (2.3.6). The set of decorated dashed arrows ∑_{a∈A_{W}} (i_{a} − 1)W_{a}, i.e., the associated P –divisor W, is allowed for Γ (or, the diagram Γ(F, W) is allowed), if the following conditions are satisfied:

(1) i_{a} ≠ 0 for a ∈ A_{W} \ A_{F}, that is, (N_{a}, i_{a}) ≠ (0, 0) for all a ∈ A_{F} ∪ A_{W}.

(2) Suppose that Γ is star-shaped. Let the central node be connected to n boundary vertices whose supporting edges have decorations {d_{ℓ}}_{ℓ=1}^{n}, and with r other incident edges connecting with arrowheads, doubled by dashed arrows or not (r ≥ 1 always).

[figure: ⋮; ⋮; d_{1}; d_{n}; i_{1} − 1; i_{n} − 1; r arrowheads; which might be doublearrows; n boundary vertices]

Then the decorations i_{1} − 1, …, i_{n} − 1 of the dashed arrows at these boundary vertices are subject to the following restrictions provided that r = 1 or r = 2.

• r = 1 : if d_{ℓ}|i_{ℓ} for at least n − 1 indexes ℓ ∈ {1, …, n}, then i_{ℓ} = d_{ℓ} for at least n − 1 indexes ℓ ∈ {1, …, n};

• r = 2 : if d_{ℓ}|i_{ℓ} for all 1 ≤ ℓ ≤ n, then i_{ℓ} = d_{ℓ} for all these indexes ℓ.

(3) For arbitrary Γ we require that the induced decorations on each star-shaped subdiagram of Γ, obtained after repeated splicing as in (3.1.5), satisfy the restrictions (2).

Remark 4.1.2. (a) In (2) there are thus no conditions on the decorations i_{a} for the arrowheads given by a ∈ A_{F}, that is, those associated to the other r edges.

(b) The value i_{ℓ} = 1 is possible. It corresponds to no dashed arrow, or formally to a dashed arrow with decoration zero. Also, for the boundary vertices from the right, if d_{ℓ} = 1 for some ℓ, and the corresponding leg is not represented as above, but with its minimal diagram as in (2.3), then the above definition applies for these dashed arrowheads too (with d_{ℓ} = 1).

(c) One can formulate the restrictions in (2) simultaneously for all r as follows:

if d_{ℓ}|i_{ℓ} for at least n + r − 2 indexes ℓ ∈ {1, …, n}, then i_{ℓ} = d_{ℓ} for at least n + r − 2 of the indexes ℓ. (This assumption is empty if r ≥ 3.)

(d) We assumed implicitly in (2) that n ≥ 1. When n = 0 the conditions are empty.

(e) A priori it is not clear at all that there exist allowed W on a general graph Γ(F). We will construct plenty of them later.

Definition 4.1.3. Let (X, 0) be an IHS surface germ, and F′ a (non-zero) effective Weil divisor on it. A Weil divisor W′ of (X, 0) is allowed for the pair (X, F′) if there exists an embedded resolution π : X̃ → X of F′ such that the diagram Γ_{π}(X, F, W) is allowed.

This notion is well defined, in the sense that it is invariant under ‘extra’ blowing-ups, as shown below.

Proposition 4.1.4. We use the notation from (4.1.3). Suppose that the diagram Γ := Γ_{π}(X, F, W) is allowed. Let h : X̃_{1} → X̃ be a blowing-up in some point of X̃. Then the diagram Γ_{1} := Γ_{π∘h}(X, F_{1}, W_{1}) (with obvious notations) is also allowed.

Proof. We only have to investigate the spliced star-shaped subgraphs of Γ_{1} that are new or different with respect to Γ. If the centre P of the blowing-up h is either a point of a boundary curve, or an intersection point of two components (exceptional or strict transform), we are done because then Γ_{1} = Γ. We are left with the following two cases for P. Case 1. P is a point of E_{j}^{∘} (that is, a generic point of E_{j}), where the vertex corresponding to E_{j} has valency 2 in G_{π}(X, F, W) (so it does not occur explicitly in Γ):

[figure: ⋮; ⋮; p; p′; blowup; p; p′; q; q′; ⋮; ⋮; 1; 0; v; w]

(Above we did not insert the information about F; and one of the nodes of the diagram before blowup can be replaced by a boundary vertex with or without dashed arrows.)

It is not difficult to verify that the spliced star-shaped subgraphs around the ‘old’ nodes in Γ and Γ_{1} are the same. Moreover, the new spliced star-shaped subgraph of Γ_{1} around v satisfies the definition of allowedness. Indeed, in both cases r = 1 or r = 2, the fact that the decorations associated to the boundary vertex w satisfy d_{1} = i_{1} = 1 finishes the verification. Case 2. P is a point of E_{v}^{∘}, where v is a node in Γ.

In this case the only novelty in Γ_{1} is an extra edge at the node v supporting a boundary vertex, again with edge decoration d_{ℓ} = 1 and associated number i_{ℓ} = 1. This again does not affect the allowedness condition for the star-shaped subgraph around v. □

Remark 4.1.5. It is possible that a divisor W′ on (X, 0) is not allowed in the diagram associated with the minimal embedded resolution π of (X, F′), but is allowed in some Γ_{π}(X, F, W) associated with some non–minimal π. Consider for example the situation

[figure: p; blowup; p; 1; i − 1; q − 1; q; d]

where π is obtained from the minimal embedded resolution π_{min} by composing with one blowing-up, and i ∈ ℤ_{>1}. The component of the strict transform of W′ with multiplicity i − 1 intersects the exceptional divisor of π_{min} in a component E_{j} of valency 2, and this was not permitted, cf. (2.3).

Remark 4.1.6. Assume that F has only one arrowhead with multiplicity 1. Then for any W, the dashed arrowhead with multiplicity i − 1 which doubles the arrowhead of F has an almost irrelevant geometric contribution. Indeed, its only effect is the following: in any ratio ν_{v}/N_{v} it has a global integral (i − 1)-shift. In particular, in such a situation (having connections with monodromy in mind), we might take i − 1 = 0 without restricting the generality of the discussion.

Example 4.1.7. Let us continue the discussion of Example 3.1.15. Having in mind Remark 4.1.6, the general form W will have the following dashed arrowheads: The splice decomposition provides:

[figure: 2; 1; 7; 7; 1; 2; 3; 3; (3); (6); (1); (6); (3); (2); (1); (2); i_{1} − 1; i′_{1} − 1; i_{2} − 1; i′_{2} − 1] [figure: 2; 1; 7; 7; 1; 2; 3; 3; (1); (1); i − 1; i_{0} − 1; i′_{0} − 1; (1); i′ − 1; i_{1} − 1; i′_{1} − 1; i_{2} − 1; i′_{2} − 1]

In the above picture i − 1 = i′_{0} − 1 = −2 + 3(i′_{1} − 1) + 2(i′_{2} − 1) and i′ − 1 = i_{0} − 1 = −2 + 3(i_{1} − 1) + 2(i_{2} − 1).

Assume that W is allowed. This imposes the following numerical conditions.

• In the middle graph Γ_{0} we impose: either 3i_{1} + 2i_{2} = 7 or 3i′_{1} + 2i′_{2} = 7.

• In Γ_{1} one gets: if 2 | i_{1} or 3 | i_{2} then either 2 = i_{1} or 3 = i_{2}. Note that if 3i_{1} + 2i_{2} = 7 then 2 ∤ i_{1} and 3 ∤ i_{2}. There is a symmetric restriction in Γ′_{1} too: if 2 | i′_{1} or 3 | i′_{2} then either 2 = i′_{1} or 3 = i′_{2}.

In particular, the zero form W = 0 is not allowed.

4.2. Restricting and extending allowed divisors. Consider the splicing of a given diagram Γ(F) along a special edge e as in (3.1.2). A basic idea in the definition of an allowed W for Γ(F) is that the induced W_{L} and W_{R} should be allowed for Γ_{L}(F_{L}) and Γ_{R}(F_{R}), respectively. This is almost clear from the nature of the definitions.

There is potentially a problem when (say) A_{F,L} = ∅, since then a new dashed arrowhead at a boundary vertex of Γ_{R}(F_{R}) is created, and it could have associated decoration i′ = 0, see the last picture in (3.1.5). Indeed, in Definition 4.1.1, part (3), we asked for each star– shaped subdiagram to satisfy condition (2), but we didn’t ask (1). In the next lemma we will verify that (1) will be automatically satisfied.

Lemma 4.2.1. When A_{F,L} = ∅ and i_{a} ≠ 0 for all a ∈ A_{W,L}, then i′ ≠ 0. In particular, allowed divisors on a graph always ‘restrict’ to allowed divisors on spliced subdiagrams.

Proof. By induction on the number of nodes in Γ_{L} it is sufficient to prove that i′ ≠ 0 when Γ_{L} is star-shaped. Let i_{1} −1, …, i_{n} −1 be the multiplicities of the dashed arrowheads along the edges with decorations d_{1}, …, d_{n}. Denote D := ∏_{ℓ=1}^{n} d_{ℓ}. Then, using (3.1.11),

i′ = (1 − n)D + ∑_{ℓ=1}^{n} \frac{D}{d_{ℓ}} i_{ℓ}.

Suppose that i′ = 0. Then d_{ℓ} divides \frac{D}{d_{ℓ}} i_{ℓ}, hence divides i_{ℓ} too, for all ℓ. By the definition of allowedness we then know that i_{ℓ} = d_{ℓ} for at least n − 1 of the i_{ℓ}; say for i_{2}, …, i_{n}. Thus

0 = i′ = (1 − n)D + (n − 1)D + Di_{1}/d_{1}

and hence i_{1} = 0, contradicting the assumptions. For the second statement, note that i_{a} = 1 for any a ∈ A_{F} \ A_{W}, while i_{a} ≠ 0 for a ∈ A_{W} \ A_{F} since W is allowed. □

A crucial question in our setting is the converse: can we ‘extend’ an allowed W^{♭} on Γ_{R}(F_{R}) to Γ(F), that is, can we construct an allowed W on Γ(F) for which W_{R} = W^{♭}?

It is enough to study this question when Γ_{L}(F_{L}) is star-shaped, since we can then proceed further inductively.

Proposition 4.2.2. Let Γ(F) be an arbitrary diagram as in (2.2). Splice Γ(F) along a special edge e such that Γ_{L}(F_{L}) is star-shaped. Then the map

Ψ : {allowed W for Γ(F)} → {allowed W^{♭} for Γ_{R}(F_{R})} : W ↦ W_{R}

is surjective if there is at least one arrowhead in Γ(F) on the right of e, or the number of arrowheads on the left of e is different from one and two.

Proof. Take an allowed W^{♭} for Γ_{R}(F_{R}). Let i′−1 be the decoration of the dashed arrowhead at the left of v_{R}.

Let i_{1} − 1, …, i_{n} − 1 (n ≥ 2) be the (still to be determined) decorations of the dashed arrowheads of some W, that we want to construct in the pre-image of W^{♭} by Ψ. Let j − 1 be the decoration of the new induced dashed arrowhead at v_{L} for Γ_{L}(F_{L}). Note that j is fixed in the sense that it is uniquely determined in terms of W^{♭} by the formula (3.1.11).

(And, when there are no arrowheads on the right of e, we have that j ≠ 0 by the argument in the proof of Lemma 4.2.1.)

[figure: ⋮; ⋮; v_{L}; v_{R}; d; d′; d′_{1}; d′_{n′}; d_{1}; d_{n}; j − 1; i′ − 1; i_{n} − 1; i_{1} − 1]

(In this diagram we did not insert the information regarding the divisor F, the corresponding arrowhead positions might determine different cases, see below.)

Denote D := ∏_{ℓ=1}^{n} d_{ℓ}. Then, cf. (3.1.11), we are searching for i_{1}, …, i_{n} with

(4.2.3)  i′ = (1 − n)D + ∑_{ℓ=1}^{n} \frac{D}{d_{ℓ}} i_{ℓ}.

Since gcd_{ℓ}{D/d_{ℓ}} = 1, we know that there exist i_{1}, …, i_{n} ∈ ℤ satisfying (4.2.3). We have to verify that this can be done compatibly with the restrictions on the i_{ℓ} and j, when exactly one or two (ordinary) arrowheads are among the legs in Γ_{L}(F_{L}).

Note first that, by (4.2.3), we have for any ℓ = 1, …, n that d_{ℓ}|i′ if and only if d_{ℓ}|i_{ℓ}. With the given assumptions on the arrowheads in Γ(F), we encounter two cases.

(1) There is exactly one arrowhead on Γ_{L}(F_{L}), and it coincides with the dashed (j − 1)– arrowhead. Then we suppose that d_{ℓ}|i′ for at least n − 1 of the i_{ℓ}, say for i_{1}, …, i_{n−1}. (Otherwise nothing has to be verified.)

(2) There are exactly two arrowheads on Γ_{L}(F_{L}), and they coincide with the dashed (j − 1)– and (i_{n} − 1)–arrowheads. Then we suppose that d_{ℓ}|i′ for ℓ = 1, …, n − 1. In each of these cases we take i_{ℓ} = d_{ℓ} for ℓ = 1, …, n − 1 and then, in order to satisfy (4.2.3), we take i_{n} given by i′ = Di_{n}/d_{n}. This way we thus constructed an allowed W for Γ(F) that ‘restricts’ to W^{♭}. □

Remark 4.2.4. (a) Consider in the proof above the excluded cases. First assume that there are exactly two arrowheads on Γ_{L}(F_{L}), namely when they coincide with the dashed (i_{n−1} − 1)– and (i_{n} − 1)–arrowheads. To verify allowedness we have to suppose that d|j and d_{ℓ}|i′ for ℓ = 1, …, n − 2. Then, in order to get an allowed extension we should have that d = j, but this is not true in general.

Similarly, assume that Γ_{L}(F_{L}) has exactly one arrowhead which coincides with the dashed (i_{n} − 1)–arrowhead. Then in the situation d|j, d_{ℓ}|i′ for ℓ ≤ n − 2 but d_{n−1} ∤ i′ one gets an allowed extension only if d = j.

These cases motivate the restrictions of (4.2.2). This discussion shows the following Addendum to Proposition 4.2.2: with the above notations, in the following cases an extension is still possible:

(4.2.5)

(a) either d ∤ j or d = j;

(b) the only arrowhead coincides with the dashed (i_{n} − 1)–arrowhead and d_{ℓ}|i′ for ℓ ≤ n − 1.

(b) Let Γ(F) be an arbitrary diagram. Let Γ_{A} be that minimal connected subdiagram of Γ which contains those nodes which either support at least one arrowhead of F, or sit on a (geodesic) path connecting two arrowheads of F, and those boundary vertices which are supported by these nodes. The connected components of Γ \ Γ_{A} are denoted by {Γ_{j}}_{j∈J}, and each Γ_{j} is connected to Γ_{A} at the vertex v_{j} of Γ_{A}. (For example, if Γ is the minimal diagram of a plane curve singularity, then |J | ≤ 1.) Then one has the following facts.

• Any allowed P –divisor supported on a star–shaped sub–diagram centered at any node of Γ_{A} can be extended by (4.2.2) to an allowed P –divisor of the whole Γ(F). In particular,

any Γ(F) always admits allowed divisors W.

• Any allowed P –divisor on a star–shaped diagram centered at a vertex v in Γ_{j} can be extended ‘away from v_{j}’. In order to extend it ‘in the direction of v_{j}’ one needs some extra conditions (like in (4.2.5)).

5. Allowed forms/divisors induce eigenvalues

5.1. In this section we prove that the poles of the topological zeta function associated to any Γ(F) and allowed divisor W provide eigenvalues for the monodromy zeta function.

Lemma 5.1.1. Let S be the star-shaped diagram as in Definition 4.1.1 with r = 1 or r = 2 ordinary arrowheads, equipped with decorations as below. We assume that i_{ℓ} = d_{ℓ} for ℓ = 1, …, n − 1 (and i_{n} ≠ 0) if r = 1, and i_{ℓ} = d_{ℓ} for ℓ = 1, …, n if r = 2 . Thus W is allowed. When r = 2, we assume also that −\frac{ν_{v}}{N_{v}} is not a pole of order 2 of Z(S), that is, \frac{ν_{v}}{N_{v}} ≠ \frac{k_{1}}{N_{1}} and \frac{ν_{v}}{N_{v}} ≠ \frac{k_{2}}{N_{2}} . Then, in all the above situations, −\frac{ν_{v}}{N_{v}} is not a pole of Z(S).

[figure: ⋮; v; p_{1}; d_{1}; d_{n}; (N_{1}); k_{1} − 1; i_{n} − 1; i_{1} − 1; ⋮; v; p_{1}; p_{2}; d_{1}; d_{n}; (N_{1}); k_{1} − 1; (N_{2}); k_{2} − 1; i_{n} − 1; i_{1} − 1]

Proof. We consider first the case r = 2. By (2.3.5) and (2.2.3) we have, with D = ∏_{ℓ} d_{ℓ},

(5.1.2) ν_{v} = ∑_{ℓ=1}^{n} \frac{Dp_{1}p_{2}}{d_{ℓ}} i_{ℓ} − nDp_{1}p_{2} + Dp_{2}k_{1} + Dp_{1}k_{2} and N_{v} = D(p_{2}N_{1} + p_{1}N_{2}),

respectively. With our assumptions this simplifies to

(5.1.3)  ν_{v} = D(p_{2}k_{1} + p_{1}k_{2}) and N_{v} = D(p_{2}N_{1} + p_{1}N_{2}).

The residue of −\frac{ν_{v}}{N_{v}} is (up to a factor N_{v})

( − n + ∑_{ℓ=1}^{n} \frac{d_{ℓ}}{i_{ℓ}} + \frac{p_{1}}{k_{1} + sN_{1}} + \frac{p_{2}}{k_{2} + sN_{2}} )|_{(s=− \frac{νv}{Nv})} = \frac{p_{1}}{k_{1} − \frac{ν_{v}}{N_{v}} N_{1}} + \frac{p_{2}}{k_{2} − \frac{ν_{v}}{N_{v}} N_{2}} .

This expression being zero is equivalent to \frac{ν_{v}}{N_{v}} = \frac{p_{2}k_{1}+p_{1}k_{2}}{p_{2}N_{1}+p_{1}N_{2}} , which follows from (5.1.3). When r = 1, we have by (2.3.5) and (2.2.3) that

(5.1.4)  ν_{v} = ∑_{ℓ=1}^{n} \frac{Dp_{1}}{d_{ℓ}} i_{ℓ} − (n − 1)Dp_{1} + Dk_{1} and N_{v} = DN_{1},

simplifying with our assumptions to

(5.1.5)  ν_{v} = D(\frac{p_{1}}{d_{n}} i_{n} + k_{1}) and N_{v} = DN_{1}.

Thus \frac{ν_{v}}{N_{v}} ≠ \frac{k_{1}}{N_{1}} (since i_{n} ≠ 0) and −\frac{ν_{v}}{N_{v}} is not a pole of order 2. The fact that its residue is zero is a similar easy computation as above. □

Theorem 5.1.6. Let (X, 0) be an IHS germ, and f an analytic function on X. Let W be an arbitrary allowed divisor for (X, f). If s_{0} is a pole of the topological zeta function Z(f, W; s), then exp(2πis_{0}) is a monodromy eigenvalue of f at some point of {f = 0} (in one of the homology groups).

Proof. Fix an embedded resolution π of f such that W is allowed for it. We use the usual notation associated to Γ := Γ_{π}(X, f, W). We consider three subcases.

(1) There is a component F_{a} of \{f = 0\} such that s_{0} = −\frac{i_{a}}{N_{a}}. Then exp(2πis_{0}) is an eigenvalue of f at a point of F_{a} close to 0 (since all N_{a}-th roots of unity are eigenvalues at such a point).

(2) There is no F_{a} as in (1) and let s_{0} be a pole of order 1 of Z(f, W; s). Then s_{0} = −\frac{ν_{v}}{N_{v}} for some node v, such that the contribution of v to the residue of Z(f, W; s) at s_{0} is nonzero. Consider after repeated splicing the induced star-shaped diagram S around v; by Proposition (3.2.4) we know that the residue of Z(S) at s_{0} is exactly this contribution.

[figure: ⋮; ⋮; v; p_{1}; p_{r}; d_{1}; d_{n}; (N_{1}); k_{1} − 1; (N_{r}); k_{r} − 1; i_{n} − 1; i_{1} − 1; (r arrowheads)]

From (3.3) we compute

(5.1.7)  ζ_{S}(t) = \frac{(t^{N_{v}} − 1)^{r+n−2}}{∏_{ℓ=1}^{n}(t^{N_{v}/d_{ℓ}} − 1)},

and by (2.3.5), denoting D := ∏_{ℓ=1}^{n} d_{ℓ}, we have

(5.1.8)  ν_{v} = ∑_{ℓ=1}^{n} \frac{D}{d_{ℓ}} (∏_{j=1}^{r} p_{j})i_{ℓ} + D · (some integer).

We distinguish three possibilities for r in order to show that exp(2πis_{0}) is always a root of Λ_{S}(t).

• (r ≥ 3) Then, via (5.1.7), exp(2πis_{0}) is clearly a root of Λ_{S}(t) = ζ_{S}(t).

• (r = 2) Suppose that exp(2πis_{0}) is not a root of Λ_{S}(t) = ζ_{S}(t). Then, by (5.1.7), \frac{ν_{v}}{d_{ℓ}} must be an integer for all ℓ = 1, …, n. By (5.1.8) this is equivalent to d_{ℓ}|i_{ℓ} for all these ℓ. By allowedness we conclude that then i_{ℓ} = d_{ℓ} for all ℓ. Lemma 5.1.1 then contradicts that the residue at s_{0} is non-zero.

• (r = 1) Suppose that exp(2πis_{0}) is not a root of Λ_{S}(t) = Δ_{S}(t). Then, analogously, for at least n − 1 of the numbers i_{1}, …, i_{n} we have that \frac{ν_{v}}{d_{ℓ}} must be an integer, or, equivalently, d_{ℓ}|i_{ℓ}. Then by allowedness i_{ℓ} = d_{ℓ} for n − 1 of these numbers, and again Lemma 5.1.1 contradicts that the residue at s_{0} is non-zero.

We conclude that exp(2πis_{0}) is indeed always a root of Λ_{S}(t), hence of Λ_{Γ}(t), and thus that exp(2πis_{0}) is a monodromy eigenvalue of f at 0.

(3) There is no F_{a} as in (1) and let s_{0} be a pole of order 2. This implies that s_{0} = −\frac{ν_{v}}{N_{v}} = −\frac{ν_{w}}{N_{w}} for two nodes v and w, connected by a special edge e. Equivalently, cf. (3.2.4),

(5.1.9)  s_{0} = −\frac{ν_{v}}{N_{v}} = −\frac{k}{N}

for the central node v and an arrowhead with decorations (N_{a}, i_{a}) = (N, k) in some starshaped spliced subdiagram of Γ.

Let C be the connected part of Γ containing v and consisting of nodes w with s_{0} = −\frac{ν_{w}}{N_{w}} only. Either at least one node w in C has at least three attached (ordinary) arrowheads in the induced star-shaped subgraph S_{w} after splicing, and then exp(2πis_{0}) is a root of Λ_{S_{w}}(t) = ζ_{S_{w}}(t); or all nodes w in C have exactly one or two attached arrowheads in S_{w}.

So we are left with this second possibility. We distinguish two subcases for such a node v. For the two diagrams and notations, see just before the proof of Lemma 5.1.1.

• (r = 1) Using (2.2.3) and (2.3.5), see also (5.1.4), we have from (5.1.9) that

\frac{ν_{v}}{N_{v}} = \frac{∑_{ℓ=1}^{n} \frac{Dp_{1}}{d_{ℓ}} i_{ℓ} − (n − 1)Dp_{1} + Dk_{1}}{DN_{1}} = \frac{k_{1}}{N_{1}} .

This simplifies to ∑_{ℓ=1}^{n} \frac{D}{d_{ℓ}} i_{ℓ} − (n − 1)D = 0, implying that d_{ℓ}|i_{ℓ} for all ℓ = 1, …, n. By allowedness we then have i_{ℓ} = d_{ℓ} for say ℓ = 2, …, n. Hence the previous equality reduces to i_{1} = 0, contradicting that W is allowed. Hence this case cannot occur.

• (r = 2) We may assume also that v is an ‘extremity’ of C, that is, that v is only connected to one other node of C, say ν_{v}/N_{v} = k_{1}/N_{1} ≠ k_{2}/N_{2}. Now (2.2.3) and (2.3.5), see also (5.1.2), together with (5.1.9) yield

\frac{ν_{v}}{N_{v}} = \frac{∑_{ℓ=1}^{n} \frac{Dp_{1}p_{2}}{d_{ℓ}} i_{ℓ} − nDp_{1}p_{2} + Dp_{2}k_{1} + Dp_{1}k_{2}}{D(p_{2}N_{1} + p_{1}N_{2})} = \frac{k_{1}}{N_{1}} .

Suppose that exp(2πis_{0}) is not a root of Λ_{S_{v}}(t). By the same arguments as in case (2) we get that d_{ℓ}|i_{ℓ} for all ℓ = 1, …, n. By allowedness we now have that i_{ℓ} = d_{ℓ} for all ℓ, and then the previous equality reduces to

\frac{p_{2}k_{1} + p_{1}k_{2}}{p_{2}N_{1} + p_{1}N_{2}} = \frac{k_{1}}{N_{1}} .

This is equivalent to \frac{k_{1}}{N_{1}} = \frac{k_{2}}{N_{2}} (= −s_{0}), contradicting that v is an extremity of C.

We conclude that exp(2πis_{0}) is a root of Λ_{S_{v}}(t) for some node v, and hence also of Λ_{Γ}(t). □

Remark 5.1.10. Although we formulated the previous Theorem 5.1.6 for an analytic function f, it has a purely combinatorial version (with the same proof) valid for diagrams.

Start with a diagram Γ(F) and set the possible ‘eigenvalues of the monodromies at different points and in different homologies’:

Eig := {λ : Δ_{1}(λ) = 0} ∪ ⋃_{a∈A_{F}} {λ : λ^{N_{a}} = 1}.

Then, for any allowed W and pole s_{0} of the zeta function Z(F, W; s), we have that exp(2πis_{0}) belongs to Eig.

Example 5.1.11. Let us continue the main Example 4.1.7 further.

Using (3.3.4) we get that the eigenvalues are the roots of (t−1)(t^{2} −t+1). Theorem 5.1.6 says that if W is allowed and s_{0} is a pole of Z(s), then exp(2πis_{0}) is 1 or a primitive 6-th root of unity.

(a) First we show that we can find easily non–allowed forms W such that the corresponding pole will not provide an eigenvalue. This proves that some kind of restriction regarding the divisors W is necessary.

Consider in (3.1.15) a general form W, not necessarily allowed. Then ν_{1} = −13 + 21(i_{1} − 1) + 14(i_{2} − 1) + 18(i′_{1} − 1) + 12(i′_{2} − 1), which is congruent with 3i_{1} + 2i_{2} modulo N_{v_{1}} = 6. We wish to get, for example, ν_{1} ≡ 3 ( mod 6), hence we might take i_{1} = 1 and i_{2} = 6. Assume also that i′_{1} = i′_{2} = 1. Then ν_{1} = 57, ν_{0} = 8 and ν′_{1} = 47. Then computing the zeta function we realize that s_{0} = −57/6 is a pole, but exp(2πis_{0}) = −1 ∉ Eig.

(b) It is not hard to find divisors W which are not allowed, but such that nevertheless their poles provide only eigenvalues. Take for example the trivial form W = 0. Then by (4.1.7) it is not allowed, but using the expression for Z(s) from (3.1.15) we can conclude that the poles provide eigenvalues.

(c) In order to emphasize the subtlety of the statement of Theorem 5.1.6 (and of its proof) we ‘will try to find a counterexample’ of this fact. Namely, let us take i_{1} = 1, i_{2} = 3 and write I′ for the expression 3i′_{1} + 2i′_{2}. Note that W is allowed exactly when I′ = 7, cf. (4.1.7).

By a computation ν_{1} = 6I′ − 15, ν_{0} = I′ − 3 and ν′_{1} = 7I′ − 24. Moreover, the zeta function Z(s) is

\frac{2}{6I′ − 15 + 6s} + \frac{−1 + I′/(i′_{1}i′_{2})}{7I′ − 24 + 6s} + \frac{1}{I′ − 3 + s}( − 1 + \frac{1}{s + 1} + \frac{1}{6I′ − 15 + 6s} + \frac{1}{7I′ − 24 + 6s}).

In particular, s_{0} = (15−6I′)/6 is a candidate pole of Z(s), such that exp(2πis_{0}) = −1 ∉ Eig.

The point is that, for any I′, the residue of this candidate pole is zero, that is, this is a fake candidate pole, not a pole. In particular, for I′ = 7 we get no contradiction, and for some other special choices of I′ we get plenty of non–allowed forms for which all the poles provide eigenvalues. E.g., for I′ ≡ 0 ( mod 6) all the poles are integers.

(d) Let us consider the allowed form as in (c) given by i_{1} = 1, i_{2} = 3 and I′ = 7. Then Z has a pole s_{0} with exp(2πis_{0}) = exp(−2πi/6).

The other root exp(2πi/6) of t^{2} − t + 1 can be realized by the following allowed form. Consider i_{1} = 1, i_{2} = 1 and I′ = 7. Then ν_{1} = −1, ν_{0} = 0, ν′_{1} = 1 and

Z(s) = \frac{4}{6s − 1} − \frac{1}{s + 1} + \frac{1}{6s + 1}( − 1 + \frac{7}{i′_{1}i′_{2}} ) + \frac{12}{(6s − 1)(6s + 1)}.

One immediately verifies that all candidate poles are indeed poles. Hence, in fact, the poles of this unique zeta function hit all eigenvalues of Eig.

6. Plane curves

In this section we treat the case when (X, 0) is smooth, that is, f is a plane curve singularity. We will fix some local coordinates (x, y) of (X, 0).

Although the results of the section 7 generalize some of the statements of the present section, we prefer to provide some details in this particular case too, since some of the (much shorter) arguments might be of interest for specialists of plane curve germs. Moreover, we also show how the classical situation (W′ = 0) is included in our general treatment.

There is another reason to separate the plane curve case. The proof of the abundance of the allowed forms for general (X, 0) (which allows to realize all the monodromy eigenvalues) will be proved under a technical assumption regarding Γ(F) (namely, the semigroup condition). Although this condition is satisfied by splice diagrams of plane curve singularities, cf. Remark 2.2.6(1), it is natural to see how the case of plane curves runs independently of this condition, just using their standard properties.

6.1. In this subsection we verify that the standard form is allowed, and hence we reprove the ‘classical monodromy conjecture’ for plane curve singularities.

Proposition 6.1.1. The standard differential form dx ∧ dy (corresponding to the divisor W′ = 0 on X) is allowed for any (plane) curve singularity f on (X, 0) = (ℂ^{2}, 0).

Proof. We use the minimal embedded resolution π : X̃ → X of f and show that the diagram Γ := Γ_{π}(X, F = div(f), W = 0) is allowed. Note that thus the decorations i_{a} − 1 = 0 for all a ∈ A_{W}.

Recall that on any star-shaped subdiagram of Γ without boundary vertex the allowedness condition is trivially satisfied. If a star-shaped subdiagram contains exactly one boundary vertex, which by assumption is a boundary vertex of Γ too (before the splice operation), then the corresponding leg decoration (being > 1) does not divide the associated i_{a}(= 1), hence the allowedness condition is satisfied again.

It is well known that other star-shaped subdiagrams can arise from at most one connected part of Γ, that has the following form.

[figure: a_{1}; a_{2}; a_{r−1}; a_{r}; p_{1}; p_{2}; p_{r−1}; p_{r}; …; ⋮; v_{1}; v_{2}; v_{r−1}; v_{r}]

This has the following type of splice sub–diagrams (where 2 ≤ k ≤ r − 1):

[figure: a_{1}; a_{k}; a_{r}; p_{1}; p_{k}; p_{r}; ⋮; j_{k} − 1; j_{r} − 1]

Here the leg with decoration p_{r} is optional. If it does not occur, we put formally p_{r} = 1. When r = 1 it must occur, otherwise we are in the situation discussed just before. We will use the positivity of the edge determinants, saying in this case that a_{k} > a_{k−1}p_{k−1}p_{k} for k = 2, …, r.

For the first diagram the allowedness is automatically satisfied.

In the spliced star-shaped subdiagrams around the vertices v_{k}, 2 ≤ k ≤ r, it is a priori possible that a_{k}|j_{k} for some k. If furthermore j_{k} ≠ a_{k}, the allowedness condition would be violated. We will show however that −a_{k} < j_{k} < 0 for k = 2, …, r, and this will finish the proof.

More precisely, we verify by induction that j_{k} < 0 and |j_{k}| < a_{k}/p_{k} for k = 2, …, r. First by (3.1.8) we have that j_{2} = p_{1} + a_{1} − a_{1}p_{1}. Hence j_{2} < 0 and |j_{2}| < a_{1}p_{1} < a_{2}/p_{2}.

Take now k ∈ {2, …, r − 1}. In this case (3.1.11) yields

j_{k+1} = p_{k}j_{k} + a_{k} − a_{k}p_{k}.

The induction hypothesis says that j_{k} < 0 and |p_{k}j_{k}| < a_{k}. Consequently also j_{k+1} < 0 and |j_{k+1}| < a_{k}p_{k} < a_{k+1}/p_{k+1}. □

This together with Theorem 5.1.6 give an alternative proof of the classical monodromy conjecture for curves (see [14] for the original proof of a stronger result in the context of p-adic zeta functions, and [26] for a direct proof).

Corollary 6.1.2. For any plane curve singularity f, if s_{0} is a pole of the topological zeta function Z(f; s), then exp(2πis_{0}) is a monodromy eigenvalue of f at some point of {f = 0}.

6.2. A technical lemma. In the remaining part of this section we show that any monodromy eigenvalue of a given f can be generated by poles of different Z(f, W) with W allowed. The proof is given in several steps. This subsection contains two technical partial steps, Lemma 6.2.1 and Proposition 6.2.5, targeting those subdiagrams from where the extension of the allowed forms is harder (compare with (4.2.4)(b)). The main result is Theorem (6.3.1).

We formulate and prove the following lemma in the context of an arbitrary diagram Γ(F) (corresponding to an effective divisor on an IHS germ (X, 0)). The proof in the plane curve case is not easier, and we will need the general statement in the next section as well.

Lemma 6.2.1. Suppose that Γ = Γ(F) is star-shaped. Let the central node be connected to n boundary vertices and r arrowheads whose supporting edges have decorations {d_{ℓ}}_{ℓ=1}^{n} and {p_{ℓ}}_{ℓ=1}^{r}, respectively. Here r ≥ 1, n ≥ 0, and r + n ≥ 3.

[figure: ⋮; ⋮; v; p_{1}; p_{r}; d_{1}; d_{n}; (N_{1}); k_{1} − 1; (N_{r}); k_{r} − 1; i_{n} − 1; i_{1} − 1]

Let λ be a root of the Alexander polynomial Λ_{Γ}(t) of Γ. Then there exist infinitely many (even infinitely many effective) allowed P -divisors W for Γ (corresponding to the decorated dashed arrows on the diagram) admitting a pole s_{0} of the topological zeta function Z(Γ(F, W); s), such that exp(2πis_{0}) = λ.

Proof. Denote D := ∏_{ℓ=1}^{n} d_{ℓ} and P := ∏_{ℓ=1}^{r} p_{ℓ}. For any P -divisor W, that is, for any set of decorations k_{1}, …, k_{r}, i_{1}, …, i_{n} (with the i_{ℓ} ≠ 0), we have that Z(s) := Z(Γ(F, W); s) has the form

\frac{1}{ν + sN} (2 − r − n + ∑_{ℓ=1}^{n} \frac{d_{ℓ}}{i_{ℓ}} + ∑_{ℓ=1}^{r} \frac{p_{ℓ}}{k_{ℓ} + sN_{ℓ}} ) ,

where

(6.2.2)  N = D ∑_{ℓ=1}^{r} \frac{P}{p_{ℓ}} N_{ℓ} and ν = D ∑_{ℓ=1}^{r} \frac{P}{p_{ℓ}} k_{ℓ} + P ∑_{ℓ=1}^{n} \frac{D}{d_{ℓ}} i_{ℓ} − (r + n − 2)DP.

If the candidate pole −ν/N is equal to some k_{ℓ}/N_{ℓ}, then it is a pole of order two. Otherwise, we consider its residue, which is (up to a factor N) equal to

R := 2 − r − n + ∑_{ℓ=1}^{n} \frac{d_{ℓ}}{i_{ℓ}} + ∑_{ℓ=1}^{r} \frac{p_{ℓ}}{k_{ℓ} − (ν/N)N_{ℓ}} .

One easily verifies that this expression is not identically zero as function in the r + n(≥ 3) variables i_{ℓ} and k_{ℓ}. Hence −ν/N is a pole of Z(s) as soon as the algebraic equation R = 0 is not satisfied.

We consider three cases (depending on the value of r) for the roots of Λ_{Γ}(t), which is given by

\frac{(t^{N} − 1)^{r+n−2}}{∏_{ℓ=1}^{n}(t^{N/d_{ℓ}} − 1)},

except when r = 1, where we must multiply this expression by t^{N_{1}} − 1.

• (r ≥ 3) Then its roots are all the N-th roots of unity.

• (r = 2) Then its roots are all N-th roots of unity that are not (N/d_{ℓ})-th roots of unity simultaneously for all ℓ = 1, …, n. In other words all exp(2πi\frac{u}{N}) for which u ≢ 0 mod d_{ℓ} for at least one d_{ℓ}.

• (r = 1) Then its roots are all N-th roots of unity that are not (N/d_{ℓ})-th roots of unity simultaneously for (at least) n − 1 indexes ℓ = 1, …, n. In other words all exp(2πi\frac{u}{N}) for which u ≢ 0 mod d_{ℓ} for at least two different d_{ℓ}.

Fix a root λ = exp(2πi\frac{u}{N}) of Λ_{Γ}(t). Since the numbers p_{1}, …, p_{r}, d_{1}, …, d_{n} are pairwise coprime, there exist integers k_{1}, …, k_{r}, i_{1}, …, i_{n} (all positive if we desire so) such that ν in (6.2.2) satisfies ν ≡ u mod N. When r = 2 or r = 1, the restrictions on the given u imply that d_{ℓ} ∤ i_{ℓ} for at least one or at least two indexes ℓ, respectively. Hence in each case the constructed W is allowed for Γ.

Since we can choose the numbers k_{1}, …, k_{r}, i_{1}, …, i_{n} freely mod N, it is clear that we can find infinitely many such sets (in ℤ or in ℤ_{>0}) that satisfy R ≠ 0, and hence s_{0} = −ν/N is then a pole of Z(s) satisfying exp(2πis_{0}) = λ. □

Remark 6.2.3. Assume that above r = 1 with p_{1} = 1 and n = 2. Set s_{0} = −ν/N. The fact that λ = exp(2πis_{0}) is a root of Λ_{Γ}(t) is equivalent to d_{1} ∤ i_{1} and d_{2} ∤ i_{2}. On the other hand, R = 0 if and only if (d_{1} − i_{1})(d_{2} − i_{2}) = 0. Hence, if λ is a root, then s_{0} is a pole of Z(s) (for any allowed W); see also (3.4) in [21].

6.2.4. Now we return to plane curve singularities and we target that subdiagram of the minimal splice diagram whose star–shaped components after splicing have only one ordinary arrowhead; see also Remark 4.2.4(b).

Proposition 6.2.5. Consider the subdiagram (given below) of the minimal embedded resolution diagram Γ = Γ_{π}(X, F) of a plane curve germ, determining a P –divisor F. Here r ≥ 2 and the leg with decoration p_{r} is optional.

Fix k ∈ {1, …, r − 1}, consider the star-shaped subdiagram S_{k} = Γ(F_{k}) around v_{k} and fix a root λ of the Alexander polynomial Λ_{S_{k}}(t). Then there exist infinitely many allowed divisors W_{k} for S_{k}, such that if (N_{k}, ν_{k} −1) denote the decorations of v_{k} as above associated with S_{k} and W_{k}, then

(0) s_{0} = −ν_{k}/N_{k} is a pole of Z(F_{k}, W_{k}; s),

(1) exp(2πis_{0}) = λ, and

(2) W_{k} can be extended to an allowed divisor on the whole diagram Γ.

The subdiagram of Γ is

[figure: a_{1}; a_{2}; a_{r−1}; a_{r}; p_{1}; p_{2}; p_{r−1}; p_{r}; …; ⋮; v_{1}; v_{2}; v_{r−1}; v_{r}; …]

with spliced star–shaped subdiagrams (where we insert the wanted W_{k}’s too)

[figure: a_{1}; a_{k}; a_{r}; p_{1}; p_{k}; p_{r}; ⋮; j_{k} − 1; j_{r} − 1; ℓ_{1} − 1; i_{1} − 1; ℓ_{k} − 1; ( \frac{N_{k}}{p_{k}a_{k}}); i_{k} − 1; i_{r} − 1; j_{1} − 1; …; …]

Proof. From Lemma 6.2.1 we know that infinitely many W_{k} satisfying (0) and (1) exist, but in order to satisfy also (2), we will specify choices. Write λ as λ = exp(2πi(u/N_{k})), where (since it is a root of Λ_{S_{k}}(t)) a_{k} ∤ u and p_{k} ∤ u. We choose the decorations i_{k}, j_{k} and ℓ_{k} such that

ν_{k} = i_{k}a_{k} + j_{k}p_{k} + (ℓ_{k} − 1)a_{k}p_{k} ≡ u (mod N_{k}),

and, moreover

(6.2.6)  0 < i_{k}a_{k} + j_{k}p_{k} − a_{k}p_{k} < a_{k}p_{k}.

Note that this is possible since a_{k}p_{k}|N_{k}, and that we can choose ℓ_{k} freely mod N_{k}/(a_{k}p_{k}). By Remark 6.2.3 we know that −ν_{k}/N_{k} is a pole of Z(F_{k}, W_{k}; s).

We claim that we can choose inductively {i_{m}, ℓ_{m}} for m = k + 1, …, r − 1 such that |j_{m+1}| < a_{m}p_{m} for m = k, …, r − 1. Then for these m this yields |j_{m+1}| < a_{m}p_{m} < a_{m+1} and thus a_{m+1} ∤ j_{m+1}. By Remark 4.2.4 this ensures that we can extend W_{k} further to obtain an allowed divisor W on the whole diagram Γ.

We now prove the claim. By (3.1.11) we have

(6.2.7)  ℓ_{m} = i_{m+1} + p_{m+1}(ℓ_{m+1} − 1)

for m = k, …, r − 2, and

(6.2.8)  j_{m+1} = i_{m}a_{m} + j_{m}p_{m} − a_{m}p_{m}

for m = k, …, r − 1. In particular, we know already from (6.2.6) that |j_{k+1}| < a_{k}p_{k}. When some {i_{m}, ℓ_{m}} is constructed we take each time i_{m+1} and ℓ_{m+1} in (6.2.7) such that 1 ≤ i_{m+1} ≤ p_{m+1}. Then it follows from (6.2.8) and the inductive argument (and the positivity of the edge determinant) that indeed |j_{m+1}| < a_{m}p_{m} for all m = k + 1, …, r − 1. □

6.3. Now we are ready to prove the theorem regarding the abundance of the allowed forms.

Theorem 6.3.1. Let (X, 0) be a smooth surface germ and f an analytic function on X, determining a (plane) curve singularity. Let λ be a monodromy eigenvalue of f at a point of {f = 0}. Then there exist infinitely many allowed P -divisors W for (X, div(f)), and for each of them a pole s_{0} of the topological zeta function Z(f, W; s) such that exp(2πis_{0}) = λ.

Proof. Let Γ(F) be the diagram of the minimal embedded resolution of f.

(1) Suppose first that λ is a monodromy eigenvalue at a point b ∈ {f = 0}, with b ≠ 0. Writing div(f) = ∑_{a∈A_{F}} N_{a}F_{a}, this means that λ is a N_{a}-th root of unity for some N_{a}. Fix such an a ∈ A_{F}; so λ = exp(2πi(−u/N_{a})) for some (fixed) u ∈ {1, …, N_{a}}.

[figure: ⋮; d; v; i_{a} − 1; (N_{a})]

Consider the star-shaped subdiagram S_{v} = Γ(F_{v}) around the vertex v, to which the arrowhead a is attached. Choose an allowed P -divisor W_{v} for S_{v} with decoration i_{a} ≡ u mod N_{a}, such that moreover i_{a}/N_{a} ≠ i_{a′}/N_{a′} for all (eventual) other arrowheads a′ on S_{v}. (Here we have infinitely many such choices.) Then either −i_{a}/N_{a} is a pole of order two of Z(F_{v}, W_{v}; s), or the contribution of a to this zeta function is

\frac{d}{(ν_{v} + sN_{v})(i_{a} + sN_{a})}

and hence the residue of −i_{a}/N_{a} is nonzero.

We surely can extend W_{v} to an allowed divisor W on the whole of Γ(F) (see Remark 4.2.4), and doing so we do not use the value i_{a}. (For this see the proof of Proposition 4.2.2; with the notation of that proof, from Γ_{R} only i′ was used.) If −i_{a}/N_{a} is not a pole of order two and i_{a}/N_{a} would be equal to some value ν_{w}/N_{w}, w ∈ (N ∪ A_{F}) \ {a}, then we can add to i_{a} some multiple of N_{a} in order to avoid this. This way we are sure that −i_{a}/N_{a} is a pole of Z(f, W; s).

(2) Suppose now that λ is not as in (1); hence it is a root of the Alexander polynomial Λ_{Γ(F)}(t). By Proposition 3.3.3 there is at least one node w ∈ N such that λ is a root of Λ_{S_{w}}(t), where S_{w} = Γ(F_{w}) is the star-shaped subdiagram around w. By Lemma 6.2.1 and Proposition 6.2.5 there exist infinitely many allowed P -divisors W_{w} for S_{w}, and for each of them a pole s_{0} of Z(F_{w}, W_{w}; s) such that exp(2πis_{0}) = λ and W_{w} can be extended to an allowed divisor W on the whole diagram Γ(F). Indeed, the possible obstruction to extend W_{w}, as described in Remark 4.2.4, is removed in Proposition 6.2.5.

If s_{0} is a pole of order two of Z(F_{w}, W_{w}; s), then it is a pole of order two of Z(s) = Z(f, W; s) too, hence we are done. Otherwise, there is a potential problem when the following situation occurs: s_{0} is not a pole of order two, s_{0} = −ν_{w}/N_{w} for a subset N′ ⊂ N containing at least two nodes w of Γ(F), such that for each w ∈ N′ the local residuecontribution (to the total residue of s_{0}) R_{w} ≠ 0, and ∑_{w∈N′} R_{w} = 0. In this case s_{0} is not a pole of Z(s), although it is a pole of several Z(F_{w}, W_{w}; s).

[figure: ⋮; ⋮; d; Γ_{R}; d_{1}; d_{n}; i − 1; (N); k − 1; i_{1} − 1; (N_{1}); i_{n} − 1; (N_{n})]

Take an ‘extreme’ node v ∈ N′ (the node of the left diagram above), meaning that it is a boundary vertex of the full subdiagram of Γ generated by N′. Consider the star-shaped subdiagram S_{v} of Γ = Γ(F, W) around v, where the edge e with decoration d is in the direction of the other nodes in N′, and the diagram Γ_{R}, obtained after splicing Γ along e. (It is not necessary to have arrowheads at all the legs of S_{v}, as it is indicated in the above diagram; in those cases we put formally N_{ℓ} = 0 or N = 0.)

Denote D := ∏_{j=1}^{n} d_{j}. By (2.3.5) and (3.1.11) (or from (6.3.2) below) we know that ν_{v} depends only on i and k = ∑_{j=1}^{n}(D/d_{j})i_{j} − (n − 1)D, and not on the actual (separate) values of i_{1}, …, i_{n}. The residue-contribution R_{v} though depends on these values.

We claim that we can modify i_{1}, …, i_{n} keeping k (and hence ν_{v}) fixed, but changing R_{v}, such that the newly created divisor (determined by these new i_{1}, …, i_{n}) has the following properties: it agrees with the old P –divisor on Γ_{R}, and can be extended from S_{v} ∪Γ_{R} further ‘to the left’ to a new allowed P -divisor W̃ on the whole diagram.

In that way the new value becomes ∑_{v∈N′} R_{v} ≠ 0. If ‘on the left’ there are no nodes v′ with ‘new value’ ν_{v′}/N_{v′} = s_{0}, we are done since then this sum is the (total) residue of s_{0} for Z(s). We are still done if the sum of ∑_{v∈N′} R_{v} and all new residue-contributions of these v′ is nonzero. Otherwise, we repeat the argument, replacing v in the claim by such a new (extreme) node v′. This process must stop by finiteness of the diagram.

(3) We now prove the claim. We start with two observations.

(i) Since there are at least two nodes in N′, we can always assume that there is an ordinary arrowhead at the leg of S_{v} with decoration d, that is, that N ≠ 0. Then, with the terminology of Remark 4.2.4(b), we can extend any P -divisor on Γ_{R} ‘unconditionally’ to the left. More precisely, the specific situation/problem of Proposition 6.2.5 will never occur; we can always simply follow the procedure in the proof of Proposition 4.2.2. (ii) We must be sure that we can modify i_{1}, …, i_{n} while extending from Γ_{R}, considering the extension procedure described in the proof of Proposition 4.2.2. In this procedure there is no room to modify i_{1}, …, i_{n} only if (after renumbering) i_{ℓ} must be chosen as i_{ℓ} = d_{ℓ} for ℓ ≤ n − 1. But in this case we would have that R_{v} = 0 (see Lemma 5.1.1), contradicting our assumption.

By (2.3.5), (3.1.11) and (2.2.3) we have

(6.3.2)  ν_{v} = ∑_{j=1}^{n} d(D/d_{j})i_{j} + Di − (n − 1)dD = dk + Di

and

N_{v} = ∑_{j=1}^{n} d(D/d_{j})N_{j} + DN.

We have further that

N_{v}R_{v} = 1 − n + ∑_{j=1}^{n} \frac{d_{j}}{i_{j} − (ν_{v}/N_{v})N_{j}} + \frac{d}{i − (ν_{v}/N_{v})N} .

Case n ≥ 3. We replace the triple (i_{1}, i_{2}, i_{3}) by (i_{1} + xd_{1}, i_{2} + yd_{2}, i_{3} − (x + y)d_{3}) where x, y ∈ ℤ. Then ν_{v} does not change, but the three corresponding terms in N_{v}R_{v} are replaced by

\frac{d_{1}}{i_{1} + xd_{1} − (ν_{v}/N_{v})N_{1}} + \frac{d_{2}}{i_{2} + yd_{2} − (ν_{v}/N_{v})N_{2}} + \frac{d_{3}}{i_{3} − (x + y)d_{3} − (ν_{v}/N_{v})N_{3}} .

It is easy to see that this expression is not constant as function in x and y; hence we can choose appropriate x and y in ℤ such that the ‘new’ R_{v} is different from the original one. (Note that divisibility of i_{ℓ} by d_{ℓ} does not change, so we don’t destroy allowedness.) We then extend this new P -divisor from S_{v} ∪ Γ_{R} further to an allowed divisor on the whole diagram. Case n = 2. We replace the pair (i_{1}, i_{2}) by (i_{1} + xd_{1}, i_{2} − xd_{2}) where x ∈ ℤ. Again ν_{v} does not change, and now the two corresponding terms in N_{v}R_{v} are replaced by

\frac{d_{1}}{i_{1} + xd_{1} − (ν_{v}/N_{v})N_{1}} + \frac{d_{2}}{i_{2} − xd_{2} − (ν_{v}/N_{v})N_{2}} .

When this expression is not constant in x, we conclude as above. It is constant in x if and only if it is identically zero if and only if

d_{1}i_{2} + d_{2}i_{1} − \frac{ν_{v}}{N_{v}} (d_{1}N_{2} + d_{2}N_{1}) = 0.

Suppose this identity holds. Then the formulas for ν_{v} and N_{v} above easily yield that ν_{v}/N_{v} = (i − d)/N. But then the (original) N_{v}R_{v} would be equal to

−1 + \frac{d}{i − (ν_{v}/N_{v})N} = 0,

contradicting the assumption. □

Remark 6.3.3. For plane curve singularity germs f, the associated allowed W in Theorem 6.3.1 are always divisors of differential forms ω. From the proof of the more general Theorem 7.4.23, we will see that there exist moreover infinitely many effective allowed divisors W doing the job in the theorem, corresponding here in the plane curve case to holomorphic differential forms ω.

7. Diagrams Γ(F) with the semigroup condition

7.1. The semigroup condition. Let us fix a diagram Γ(F). The reader is invited to recall the definition of the semigroup condition associated with Γ(F) from (2.2.5).

The semigroup condition of Γ(F) is equivalent with the following property: for any edge e (as in the following diagram, see also (3.1.5)), such that A_{F,L} = ∅, d′ is in the semigroup S_{e} generated by {l_{ew}}_{w} where the index w runs over all the boundary vertices of Γ_{L}. (For the definition of l_{ew} see (3.1.2).)

[figure: ⋮; ⋮; d′; e]

This condition appears naturally in the context of splice quotient singularities, introduced by Neumann and Wahl [22, 23]. For some special diagrams this condition is automatically satisfied. For example, if Γ represents a rational germ (which in the context of IHS germs is equivalent with the fact that Γ represents either the smooth or the E_{8} germ), and the diagram is not necessarily minimal and F is arbitrary, then Γ(F) has the semigroup condition. Another case is when Γ is minimal and it represents a minimally elliptic (automatically Gorenstein) singularity. These facts follow from the ‘End Curve Theorem’ [24, 25].

It is convenient to denote the subsemigroup of ℕ generated by g_{1}, … g_{t} by S⟨g_{1}, …, g_{t}⟩.

7.2. Preliminary arithmetical properties. Here we gather some arithmetical properties which will be useful in the proofs of the main results of this section (listed in the next subsection).

Lemma 7.2.1. Let d_{1}, …, d_{n} be pairwise coprime positive integers, and denote D := ∏_{j=1}^{n} d_{j}. Then the following two facts hold.

(a) There exist no positive integers m_{j} such that

∑_{j=1}^{n} m_{j} \frac{D}{d_{j}} = (n − 1)D.

(b) If d ∈ S⟨D/d_{1}, …, D/d_{n}⟩, d > 0 and d|D, then (D/d_{j}) | d for some j ∈ {1, …, n}.

Proof. (a) In such an equality we would have that d_{j}|m_{j} for all j = 1, …, n. But then the left hand side would be at least nD. In (b), by assumption, we can write d in the form

(7.2.2)  d = ∑_{j=1}^{n} m_{j} \frac{D}{d_{j}} ,

where all m_{j} are nonnegative integers, and also d = ∏_{j=1}^{n} d̄_{j} with d̄_{j}|d_{j} for all j. Since the d_{j} are pairwise coprime, (7.2.2) shows that d̄_{j}|m_{j} for all j. Writing d as d̄_{j} ∏_{ℓ≠j} d̄_{ℓ}, we conclude that d divides m_{j}D/d_{j} for all j. If at least two of the numbers m_{j} would be nonzero, say m_{1} ≠ 0 and m_{2} ≠ 0, we obtain the contradiction

d ≥ m_{1} \frac{D}{d_{1}} + m_{2} \frac{D}{d_{2}} ≥ d + d.

Hence exactly one m_{j} is nonzero, implying then that (D/d_{j}) | d. □

Recall that a diagram Γ is called minimal if all the decorations d_{ve} are strictly greater than 1, provided that e connects the node v with a boundary vertex.

Proposition 7.2.3. Let Γ(F) be a splice diagram as in (2.2) (hence with W′ = 0) and minimal in the above sense. Let e be an edge connecting two nodes such that A_{F,L} = ∅, and set i′ − 1 the multiplicity of the induced dashed arrowhead at v_{R} after splicing Γ along e as in (3.1.5).

[figure: ⋮; ⋮; d′; e; splicing; ⋮; d′; i′ − 1; (M); ⋮; Γ_{R}; Γ_{L}; d_{1}; d_{n}]

(1) Then i′ < 0.

(2) Assume that the semigroup condition is satisfied in Γ(F) (at least for the edge e and the edges e_{L} sitting in Γ_{L}). (This means that d′ is in the semigroup S_{e} generated by {l_{ew}}_{w} where the index w runs over all the boundary vertices of Γ_{L}, and there are similar inclusions for all edges e_{L} of Γ_{L}.) Then −i′ ∉ S_{e}. Hence, d′ ∤ i′.

Proof. (1) Set D := ∏_{j=1}^{n} d_{j}. We proceed by induction on the number of nodes in Γ_{L}.

Suppose first that v_{L} is the only node of Γ_{L}. By (3.1.8) we have that i′ = ∑_{j=1}^{n} D/d_{j} − (n − 1)D. When n = 2, this is d_{1} + d_{2} − d_{1}d_{2} and thus negative. When n > 2, then by minimality d_{j} ≥ 2, hence ∑ 1/d_{j} ≤ n/2 < n − 1, therefore i′ < 0 again.

We suppose now that Γ_{L} contains at least two nodes. From (3.1.8) we can write i′ as

i′ = ∑_{j=1}^{n} \frac{D}{d_{j}} ( ∑_{w∈V_{j}}(2 − δ_{w})ℓ_{e_{j}w}) − (n − 1)D,

where for j = 1, …, n the set V_{j} consists of the vertices of Γ_{L} connected (geodesically) to v_{L} through the edge e_{j} with weight d_{j}, and ℓ_{e_{j}w} is the product of all the decorations adjacent to, but not on, the path from w to e_{j}. For all j this sum is either equal to 1 (when e_{j} ends at a boundary vertex), or negative by induction. Since at least one sum is negative, we conclude that i′ < 0.

(2) Denote by B_{L} and N_{L} the boundary vertices and nodes, respectively, in Γ_{L}. We will show the following claim. Let

I := ∑_{w∈B_{L}} m_{w}ℓ_{ew} + ∑_{w∈N_{L}}(2 − δ_{w})ℓ_{ew},

where all m_{w} ∈ ℤ_{>0}. Then I ≠ 0 and, if I < 0, then −I ∉ S_{e}.

Since i′ = I when all m_{w} = 1, and i′ < 0 by part (1), the statement then follows.

We now prove the claim, again by induction on the cardinality of N_{L}. If N_{L} = {v_{L}} then

I = ∑_{j=1}^{n} m_{j} \frac{D}{d_{j}} − (n − 1)D,

and this is nonzero by Lemma 7.2.1(a). If I < 0 and −I ∈ S_{e}, then ∑_{j=1}^{n} k_{j} \frac{D}{d_{j}} = (n − 1)D for some positive integers k_{j}, contradicting again Lemma 7.2.1(a).

Let now N_{L} have at least two elements. Suppose again that I < 0 and −I ∈ S_{e}. Then analogously we get that

∑_{w∈B_{L}} k_{w}ℓ_{ew} = ∑_{w∈N_{L}}(δ_{w} − 2)ℓ_{ew}

for some positive integers k_{w}. We separate v_{L} (with δ_{v_{L}} = n + 1 and ℓ_{ev_{L}} = D) on the right hand side, and rewrite this equality as

(7.2.4)  ∑_{j=1}^{n} \frac{D}{d_{j}} [ ∑_{w∈B_{L}^{(j)}} k_{w}ℓ_{e_{j}w} + ∑_{w∈N_{L}^{(j)}}(2 − δ_{w})ℓ_{e_{j}w}] = (n − 1)D,

where B_{L}^{(j)} ∪ N_{L}^{(j)} = V_{j}. By induction the square bracket is non–zero. Moreover, (7.2.4) shows that d_{j} divides the j-th square bracket for all j. Applying the induction hypothesis on all these terms (and the assumption d_{j} ∈ S_{e_{j}}) yields that all these square brackets are positive. But this contradicts Lemma 7.2.1(a).

We still have to show that I ≠ 0. Assuming that I = 0 yields the same expression as in (7.2.4), with the k_{w} replaced by the original m_{w}. And then we obtain a contradiction by the same argument. □

Remark 7.2.5. Let us give the ‘Milnor number interpretation’ of the statement (7.2.3)(1). Consider the splice diagram Γ_{L}, but replace the multiplicity M of the unique arrowhead by 1. This represents a fibrable knot; let S be its fiber. It is a connected punctured Riemann surface. Let its first Betti number be μ (the Milnor number). Clearly, μ is even. Since Γ_{L} is minimal and non–empty, μ ≠ 0 (its proof is basically our proof of (1)). On the other hand, by [1], i′ = χ(S) = 1 − μ, hence i′ < 0.

The second part also has some ‘classical’ interpretation. Start again with the fact −i′ = μ − 1, and assume that the above diagram represents a plane curve singularity. Then S_{e} is exactly the semigroup S of the plane curve, and it is a classical fact that μ − 1 is the largest integer not in S.

The point is that in any generalization of μ − 1 ∉ S for more general Γ_{L} (as our (2) does) one needs some restriction about Γ_{L}: for example, if we have two nodes, the second one in V_{1}, and d_{1} = 1, then S_{e} = ℕ.

7.3. W = 0 is allowed.

Theorem 7.3.1. Let F be a (nonzero) effective divisor on an IHS germ (X, 0), such that the minimal embedded resolution diagram Γ_{π}(X, F) satisfies the semigroup condition. Then the divisor W = 0 is allowed for the pair (X, F).

Proof. Denoting by π this minimal embedded resolution, we will show that the diagram Γ = Γ_{π}(X, F, W = 0) is allowed. Note that thus the decorations i_{a} − 1 = 0 for all a ∈ A_{W}.

Recall again that on any star-shaped subdiagram of Γ without boundary vertex the allowedness condition is trivially satisfied. If a star-shaped subdiagram contains a boundary vertex, that is an original boundary vertex of Γ, then the corresponding leg decoration (being > 1 by minimality) does not divide the associated i_{a}(= 1).

If a star-shaped subdiagram contains a boundary vertex, that is created after splicing, that diagram looks like the right diagram in the statement of Proposition 7.2.3, where i′ −1 is the decoration of the constructed dashed arrow attached to that boundary vertex, and d′ is the corresponding edge weight. Since we showed in Proposition 7.2.3 that d′ ∤ i′, the allowedness condition is verified in this case too (hence everywhere) by Addendum (4.2.5). □

Example 7.3.2. Recall that in Example 4.1.7 we presented a minimal diagram Γ(F) for which W = 0 is not allowed. Hence some kind of restriction is indeed necessary in order to guarantee the allowedness of W = 0.

Corollary 7.3.3. Let (X, 0) be a Gorenstein IHS germ, with nowhere vanishing 2-form ω_{0} on X \ {0}. Let f be a function germ on (X, 0) such that the minimal embedded resolution diagram Γ_{π}(X, f) satisfies the semigroup condition.

If s_{0} is a pole of the topological zeta function Z(f; s) = Z(div(f), W = 0; s), then exp(2πis_{0}) is a monodromy eigenvalue of f at some point of {f = 0}.

Proof. Immediate from Theorem 7.3.1 and Theorem 5.1.6. □

Remark 7.3.4. In Theorem 2.2 of [26], Rodrigues showed without requiring the semigroup condition that even in the singular setting exp(2πis_{0}) is a monodromy eigenvalue of f provided that the pole s_{0} satisfies s_{0} ≤ 0. We wish to emphasize that this is a rather strong assumption in the context of singular ambient spaces. Indeed, if we consider a minimal resolution of a non–canonical surface singularity, then the canonical cycle K is nef, and all its coefficients ν − 1 are (strictly) negative, hence the corresponding values −ν/N are all non-negative. We can create negative poles when we have to blow up the minimal resolution in order to get a good resolution of (X, F), so by subgraphs which behave like graphs of plane curve singularities. Usually there are only a few poles like this (although, for plane curves all of them are negative by the very same argument).

7.4. All eigenvalues are realized by poles. The main result of the subsection is based on the following technical proposition.

Proposition 7.4.1. Let (X, 0) be an IHS germ and f an analytic function on X, such that the minimal embedded resolution diagram Γ = Γ_{π}(X, f) satisfies the semigroup condition. With the notation of Remark 4.2.4(b), let us fix a node v_{0} in Γ_{A}, and another node v_{m} not in Γ_{A} at ‘distance’ m ≥ 1 from v_{0}. Consider the diagram Γ_{0} given below obtained from Γ by cutting via splice–decomposition all the nodes not sitting on the geodesic path connecting v_{0} with v_{m}. Here the legs with decorations d_{0,2}, …, d_{0,n_{0}} are optional, and all boundary vertices are either original boundary vertices of Γ, or are obtained after splicing.

[figure: d_{0,1}; d_{0,n_{0}}; d_{0,2}; d_{m,1}; d_{m,n_{m}}; d_{m,2}; d_{m}; d_{k,1}; d_{k,n_{k}}; d_{k,2}; d_{k}; ⋮; v_{0}; v_{k}; v_{m}; …; …; …; …; …]

Splice this diagram at the edge (v_{k−1}v_{k}), 0 < k ≤ m, and denote that splice component which contains v_{k}, …, v_{m} by Γ_{k} = Γ_{k}(F_{k}). Let the decoration of the dashed arrowhead not in Γ_{1}, associated with the splicing along (v_{0}v_{1}), be i_{0,1} − 1 (see the picture below).

Fix a root λ of the Alexander polynomial Λ_{Γ_{m}}(t) associated with the star–shaped diagram Γ_{m}. Then there exist infinitely many allowed divisors W_{m} for Γ_{m}, such that if (N_{m}, ν_{m} −1) denote the decorations of v_{m} as above associated with Γ_{m} and W_{m}, then

(1) s_{0} = −ν_{m}/N_{m} is a pole of Z(F_{m}, W_{m}; s), with exp(2πis_{0}) = λ, and

(2) W_{m} extends to an allowed divisor on Γ_{1}, such that d_{0,1} ∤ i_{0,1}. Moreover, infinitely many of these allowed (extended) W_{m} on Γ_{1}, as well as their further extensions (in the sense of Remark 4.2.4(b)) on the whole of Γ, may be chosen to be effective.

Note that the above additional non–divisibility property (2) is the key assumption in the Addendum (4.2.5).

Proof. We proceed in several steps. During the proof λ is fixed.

7.4.2. We fix notations for the wanted W_{m} and its extensions on the spliced star–shaped subdiagrams of Γ_{1}. Moreover, we also consider the decoration i_{0,1} which is part of a potential extension to Γ_{0}, but it is completely determined by the extension on Γ_{1}.

[figure: …; ⋮; i_{0,1} − 1; d_{0,1}; d_{0,n_{0}}; d_{0,2}; v_{0}; d_{1,1}; d_{1,n_{1}}; d_{1,2}; d_{1}; v_{1}; …; i_{1} − 1; i_{1,1} − 1; i_{1,n_{1}} − 1; i_{1,2} − 1; …]

We also set D_{k} := ∏_{ℓ=1}^{n_{k}} d_{k,ℓ} and D_{k}^{∗} := ∏_{ℓ=2}^{n_{k}} d_{k,ℓ} (1 ≤ k ≤ m). Note that n_{k} ≥ 2. The semigroup condition for Γ implies that for any k > 0 one has

(7.4.3)  d_{k−1,1} ∈ S⟨\frac{D_{k}}{d_{k,2}} , …, \frac{D_{k}}{d_{k,n_{k}}} , \frac{D_{k}^{∗}D_{k+1}}{d_{k+1,2}} , …, \frac{D_{k}^{∗}D_{k+1}}{d_{k+1,n_{k+1}}} , \frac{D_{k}^{∗}D_{k+1}^{∗}D_{k+2}}{d_{k+2,2}} , ⋯ ⟩

The wanted divisor will be constructed by induction. From (6.2.1) we know that infinitely many W_{m}, even infinitely many effective W_{m}, satisfying (1) exist (see also 7.4.7). Here W_{m} identifies ν_{m} by

(7.4.4)  ν_{m} = i_{m−1,1}d_{m} + i_{m}D_{m}.

Then, we analyze how an allowed divisor W_{k+1} from Γ_{k+1} can be extended over Γ_{k}. Along this procedure we will use the following identities ‘around v_{k}’ satisfied by any extension:

(7.4.5)  i_{k+1} = −(n_{k} − 1)d_{k}D_{k}^{∗} + i_{k}D_{k}^{∗} + ∑_{ℓ≥2} i_{k,ℓ}d_{k}D_{k}^{∗}/d_{k,ℓ},

(7.4.6)  i_{k−1,1} = −(n_{k} − 1)D_{k} + ∑_{ℓ≥1} i_{k,ℓ}D_{k}/d_{k,ℓ}.

In this procedure we need a deeper understanding of the extensions (compared with (4.2)), and we need to consider divisors with some special properties, we will call them ‘strict’. The decorations of their nodes satisfy some additional conditions, as it is explained next.

Assume that W_{k} is an allowed divisor on Γ_{k} for some k ≥ 1. The decorations of W_{k} will distinguish the nodes as follows. For some k′ ∈ {k, …, m}, the node v_{k′} is called flexible if there are at least two indexes ℓ ∈ {1, …, n_{k′}} for which d_{k′,ℓ} ∤ i_{k′,ℓ}. For k′ ∈ {k, …, m − 1}, if v_{k′} is not flexible, but d_{k′,ℓ} = i_{k′,ℓ} for all ℓ ∈ {2, …, n_{k′}} then it is called rigid. Note that not all non–flexible nodes are rigid (see the cases discussed in (4.2.4)).

In this proof the nodes of all allowed divisors will be either flexible or rigid.

7.4.7. First we construct an allowed divisor W_{m} which satisfies (1) and is flexible at v_{m}. We search for i_{m}, i_{m,1}, …, i_{1,n_{m}} such that they satisfy the allowedness at v_{m}, (7.4.4), (7.4.6) for k = m, and exp(−2πiν_{m}/N_{m}) is root of the Alexander polynomial Λ_{Γ_{m}}(t). Since D_{m} | N_{m}, the last condition implies that

(7.4.8)  d_{m,ℓ} ∤ ν_{m} for at least two indexes ℓ ∈ {1, …, n_{m}}.

We proceed as follows. For any ν_{m} with (7.4.8) we find i_{m−1,1} and i_{m} satisfying (7.4.4). This is possible since gcd(d_{m}, D_{m}) = 1. Then we find integers {i_{m,ℓ}}_{ℓ=1}^{n_{m}} satisfying (7.4.6). This, again, is possible since gcd_{ℓ}(\frac{D_{m}}{d_{m,ℓ}}) = 1. Since d_{m,ℓ} | ν_{m} ⇔ d_{m,ℓ} | i_{m−1,1} ⇔ d_{m,ℓ} | i_{m,ℓ}, by (7.4.8) we have that d_{m,ℓ} ∤ i_{m,ℓ} for at least two indexes, hence W_{m} is flexible at v_{m}.

7.4.9. Next we analyze the possibilities how one can extend divisors. Consider an allowed divisor W_{k+1} on Γ_{k+1} (1 ≤ k < m). Note that it also determines i_{k,1} by (7.4.6). Extending over v_{k} means that we already know everything over Γ_{k+1} and i_{k,1}, and we are searching for i_{k} and {i_{k,ℓ}}_{ℓ≥2} which satisfy the allowedness condition at v_{k} and the identity (7.4.5).

The divisor W_{k+1} and the decorations of Γ_{0}(F) contain all the divisibility information, like d_{k,ℓ} divides i_{k,ℓ} or not, for any extension W_{k} on Γ_{k}. Indeed, i_{k,1} and d_{k,1} are determined by Γ_{0} and W_{k+1}, and the divisibility conditions d_{k,ℓ} ∤ i_{k,ℓ} (ℓ ≥ 2) are determined by (7.4.5), since d_{k,ℓ} ∤ i_{k,ℓ} ⇔ d_{k,ℓ} ∤ i_{k+1}. Hence, several crucial divisibility properties of an extension W_{k} on Γ_{k} are already decided at the level of its restriction W_{m} on Γ_{m}. This makes the inductive construction of W_{k}, staring from W_{m} ‘global’ and difficult.

In order to guarantee the existence of such an extension W_{k}, we will use two types of criteria: W_{k+1} satisfies either d_{k,1} ∤ i_{k,1} or D_{k}^{∗} | i_{k+1} (see (4.2.5)).

If D_{k}^{∗} | i_{k+1} then it has no flexible extension (but, it might happen that it has several allowed extensions); we take always that unique extension for which v_{k} will be rigid: d_{k,ℓ} = i_{k,ℓ} for ℓ ≥ 2. Moreover, (7.4.5) and (7.4.6) read as

(7.4.10)  i_{k} = i_{k+1}/D_{k}^{∗} and i_{k−1,1} = i_{k,1}D_{k}^{∗}.

If d_{k,1} ∤ i_{k,1}, then for any extension W_{k} the node v_{k} is either flexible or rigid (and the type is decided already at the level of W_{k+1}); v_{k} is rigid if and only if additionally D_{k}^{∗} | i_{k+1}, the case discussed before. If v_{k} is flexible, then the extension is not unique, it can be modified if it is necessary (and we will do this intensively).

Next, we have to check if the extension has one of the two criteria which guarantee the further extension. We show that if we ‘modify W_{k} at the closest flexible node’, it will satisfy the inductive criteria d_{k−1,1} ∤ i_{k−1,1}, provided that the tower of extensions was carefully constructed from the beginning. The careful choice of the sequence of flexible/rigid nodes and the family of modifications is described in the next part.

7.4.11. We define the class of strict allowed divisors W_{k} on Γ_{k} inductively as follows.

Assume first that v_{k} is rigid, but at least one node of (Γ_{k}, W_{k}) is flexible. Let k′ > k be that flexible node for which v_{k}, …, v_{k′−1} are all rigid. We modify W_{k} such that we keep unmodified the restriction on W_{k′+1} and i_{k′,1}. We fix some ℓ ≥ 2 such that d_{k′,ℓ} ∤ i_{k′,ℓ}.

Then we replace i_{k′,ℓ} into i_{k′,ℓ} + td_{k′,ℓ}, t ∈ ℤ, but keep all other i_{k′,ℓ}’s. Moreover, modify i_{k′} ↦ i_{k′} − td_{k′} too. Then i_{k′+1} and i_{k′,1} will stay fixed.

This is the set of modifications we will refer to, and for strict divisors we impose the following properties. First, we assume that for all the possible modifications, the value i_{k′} is multiple of D_{k′−1}^{∗}. Then, all these modifications can be extended by a rigid v_{k′−1} to Γ_{k′−1}. Then we run again all the modifications (at v_{k′}) and we assume that for all of them D_{k′−2}^{∗} | i_{k′−1}. Then, again, all of them can be extended. We continue this, at the very end asking D_{k}^{∗} | i_{k+1} for all the modifications. If all these conditions are satisfied for W_{k} then in all its modifications W̄_{k} the nodes v_{k}, …, v_{k′−1} will be rigid, and we call W_{k} strict. The strictness guarantees that when we run all the modifications at the level of v_{k′}, all the divisors can be extended to some W_{k}. (Otherwise it might happen that for some modification and at some vertex both d_{k′′,1} ∤ i_{k′′,1} and D_{k′′}^{∗} | i_{k′′+1} fail.)

From (7.4.10) we get

(7.4.12)  i_{k} = \frac{i_{k′}}{D_{k}^{∗} ⋯ D_{k′−1}^{∗}} , and i_{k−1,1} = i_{k′−1,1}D_{k}^{∗} ⋯ D_{k′−1}^{∗}.

Since i_{k′−1,1} ↦ i_{k′−1,1} + tD_{k′}, the modifications induce

(7.4.13)  i_{k} ↦ i_{k} − \frac{td_{k′}}{D_{k}^{∗} ⋯ D_{k′−1}^{∗}} , and i_{k−1,1} ↦ i_{k−1,1} + tD_{k}^{∗} ⋯ D_{k′−1}^{∗}D_{k′}.

If v_{k} is flexible then W_{k} is strict by definition. In fact the above discussion is valid in this case too with k′ = k. In particular, the set of modifications is given by i_{k,ℓ} ↦ i_{k,ℓ} + td_{k,ℓ} for the chosen ℓ and keeping the other i_{k,ℓ}’s, i_{k} ↦ i_{k} − td_{k}, i_{k−1,1} ↦ i_{k−1,1} + tD_{k}.

If we run the above modification for the divisors W_{m} constructed in (7.4.7), then ν_{m} stays stable, hence if the restriction of some W_{k} to Γ_{m} satisfies (1), then all its modifications keep satisfying (1).

In our procedure we consider only strict allowed divisors. They will be constructed inductively starting from the strict divisors W_{m} constructed in (7.4.7). The inductive statement we prove is the following: for any 1 ≤ k ≤ m there exists a strict allowed divisor W_{k} on Γ_{k} satisfying d_{k−1,1} ∤ i_{k−1,1}, and (1) on Γ_{m}.

The proof of the inductive step breaks into two parts.

(a) If the above properties are true for some strict W_{k+1} on Γ_{k+1} then definitely it can be extended to an allowed divisor W_{k}, but this is not necessarily strict. We prove that by a good choice of one of its modifications, that divisor has a strict extension (not necessarily satisfying d_{k−1,1} ∤ i_{k−1,1}).

(b) If W_{k} is strict and its restriction satisfies (1), then it can be replaced (by the above moves) by another strict divisor which satisfies both (1) and d_{k−1,1} ∤ i_{k−1,1}.

Note that part (b) provides the main inductive statement for m = 1 too. Indeed, by (7.4.7) a strict divisor W_{m} with (1) exists, which by (b) can be replaced by a wanted one.

7.4.14. Here we prove part (a) of the inductive step (7.4.11).

Assume that W_{k+1} is a strict divisor on Γ_{k+1} satisfying (1) and d_{k,1} ∤ i_{k,1}. We consider all the modifications W̄_{k+1} of W_{k+1} as in (7.4.11), and we distinguish the next two cases.

First, suppose that there is no W̄_{k+1} (with or without d_{k,1} ∤ i_{k,1}) for which D_{k}^{∗} ∤ i_{k+1}. Then we extend W_{k+1} by a rigid node. The extended divisor W_{k} will be strict.

Second, we assume that there exist some W̄_{k+1} with D_{k}^{∗} ∤ i_{k+1}. The problem is that it might happen that in the new situation d_{k,1} ∤ i_{k,1} fails, and the extension is not guaranteed. We claim that the two conditions d_{k,1} ∤ i_{k,1} and D_{k}^{∗} ∤ i_{k+1} can be obtained simultaneously by some W̄_{k+1}. Then we extend this new W̄_{k+1} to get a strict W_{k} with flexible v_{k}.

Let us prove now the above claim.

Recall that d_{k,1} ∤ i_{k,1}. If D_{k}^{∗} ∤ i_{k+1} for W_{k+1} then we are done. Similarly, if d_{k,1} ∤ i_{k,1} for W̄_{k+1} then again we are done. Otherwise, by (7.4.13) we must have

(7.4.15)  d_{k,1} ∤ δ := D_{k+1}^{∗} ⋯ D_{k′−1}^{∗}D_{k′} and D_{k}^{∗} ∤ Δ := \frac{d_{k′}}{D_{k+1}^{∗} ⋯ D_{k′−1}^{∗}} .

We consider the modifications for t = 1, 2, 3. Then either we get a wanted pair or we will have simultaneously

\{ \begin{matrix}d_{k,1} ∤ i_{k,1} \\ D_{k}^{∗} | i_{k+1} \end{matrix}  \{ \begin{matrix}d_{k,1} | i_{k,1} − δ \\ D_{k}^{∗} ∤ i_{k+1} + Δ \end{matrix}  \{ \begin{matrix}d_{k,1} ∤ i_{k,1} − 2δ \\ D_{k}^{∗} | i_{k+1} + 2Δ \end{matrix}  \{ \begin{matrix}d_{k,1} | i_{k,1} − 3δ \\ D_{k}^{∗} ∤ i_{k+1} + 3Δ. \end{matrix}

This implies d_{k,1} | 2δ and D_{k}^{∗} | 2Δ. This together with (7.4.15) implies that both d_{k,1} and D_{k}^{∗} should be even. This is not possible since d_{k,1} and D_{k}^{∗} are relative prime.

7.4.16. Finally we prove part (b) of the inductive step (7.4.11).

Assume that W_{k} is a strict allowed divisor on Γ_{k} such that its restriction satisfies (1). If v_{k} is rigid we will use all the notations of (7.4.11), where v_{k′} is the closest flexible node to v_{k}. In fact, these notations can also be used when v_{k} is flexible, with the convention k′ = k.

We have to show that for some modification of W_{k} one has d_{k−1,1} ∤ i_{k−1,1}. We assume that this is not the case, that is, for all modifications of W_{k} at v_{k′} one has

(7.4.17)  d_{k−1,1} | i_{k−1,1} = (i_{k′−1,1} + tD_{k′}) · D_{k}^{∗} ⋯ D_{k′−1}^{∗},

and we wish to get a contradiction.

For ℓ ∈ {2, …, n_{k}} set a_{ℓ} := gcd(d_{k−1,1}, d_{k,ℓ}), A^{∗} := ∏_{ℓ≥2} a_{l} and d′_{k−1,1} := d_{k−1,1}/A^{∗}. About d′_{k−1,1} we wish to prove two facts. First, clearly

(7.4.18)  d′_{k−1,1} | (i_{k′−1,1} + tD_{k′}) · D_{k+1}^{∗} ⋯ D_{k′−1}^{∗}.

The second one is

(7.4.19)  d′_{k−1,1} ∈ S⟨\frac{D_{k+1}}{d_{k+1,2}} , …, \frac{D_{k+1}}{d_{k+1,n_{k+1}}} , \frac{D_{k+1}^{∗}D_{k+2}}{d_{k+2,2}} , …, \frac{D_{k+1}^{∗}D_{k+2}}{d_{k+2,n_{k+2}}} , ⋯ ⟩.

The semigroup involved above is the semigroup associated with that diagram which is obtained from Γ_{0} by deleting the star–shaped subdiagram around v_{k}. (In fact, (7.4.18) can also be interpreted in this way.) The proof of (7.4.19) runs as follows. Write

d_{k−1,1} = ∑_{ℓ≥2} m_{ℓ} \frac{D_{k}}{d_{k,ℓ}} + D_{k}^{∗} · ∑_{ℓ≥2} n_{ℓ} \frac{D_{k+1}}{d_{k+1,ℓ}} + D_{k}^{∗}D_{k+1}^{∗} · ∑_{ℓ≥2} n′_{ℓ} \frac{D_{k+2}}{d_{k+2,ℓ}} + · · · .

Then a_{ℓ} | m_{ℓ}, hence A^{∗} | m_{ℓ}D_{k}^{∗}/d_{k,ℓ} too, for any ℓ ≥ 2. In particular, d′_{k−1,1} belongs to

S⟨d_{k,1}, D_{k+1}/d_{k+2,1}, …, D_{k+1}/d_{k+1,n_{k+1}}, D_{k+1}^{∗}D_{k+2}/d_{k+2,1}, …, D_{k+1}^{∗}D_{k+2}/d_{k+2,n_{k+2}}, ⋯ ⟩.

But d_{k,1} is in the semigroup generated by the others, cf. (7.4.3), thus we get (7.4.19).

The step how we get d′_{k−1,1} from d_{k−1,1} can be continued. In the second step we set d′′_{k−1,1} := d′_{k−1,1}/ gcd(d′_{k−1,1}, D_{k+1}^{∗}). Dividing consecutively by the corresponding divisor of D_{k}^{∗}, …, D_{k′−1}^{∗}, from d_{k−1,1} we get d̄_{k−1,1} with the following properties:

(7.4.20) d̄_{k−1,1} | i_{k′−1,1} + tD_{k′}, or, equivalently, d̄_{k−1,1} divides both i_{k′−1,1} and D_{k′},
(7.4.21)  d̄_{k−1,1} ∈ S⟨\frac{D_{k′}}{d_{k′,1}} , …, \frac{D_{k′}}{d_{k′,n_{k′}}} ⟩.

Now (7.4.21) together with d̄_{k−1,1} | D_{k′}, via Lemma 7.2.1(b), implies that for some ℓ_{0} ∈ {1, …, n_{k′}} one has D_{k′}/d_{k′,ℓ_{0}} | d̄_{k′−1,1}. This with d̄_{k−1,1} | i_{k′−1,1} implies that D_{k′}/d_{k′,ℓ_{0}} | i_{k′−1,1}. Then the formula (7.4.6) for i_{k′−1,1} implies that d_{k′,ℓ} | i_{k′,ℓ} for ℓ ∈ {1, …, n_{k′}}\{ℓ_{0}}, which contradicts the fact that v_{k′} is flexible.

7.4.22. Now we verify that the above construction provides infinitely many divisors W_{k} at each step k. Indeed, when k = m then in (7.4.7) there are infinitely many possibilities for i_{m,1} to realize a desired W_{m}. Furthermore, in the extension procedure, this initially chosen W_{m} is modified, but the original i_{m,1} is never touched.

Moreover, we can obtain this way infinitely many (extended) effective divisors W_{m}. Indeed, in (7.4.7) we can choose the value ν_{m} freely modulo N_{m}, in particular positive and large enough with respect to all decorations along the edges of Γ. Then also i_{m−1,1} and further {i_{m,ℓ}}_{ℓ=1}^{n_{m}} can be chosen ‘large’. In fact, if ν_{m} is large enough, then all constructed (and modified) multiplicities along dashed arrows, on Γ_{1} and further on the whole of Γ, will be positive. This ends the proof of Proposition 7.4.1. □

Theorem 7.4.23. Let (X, 0) be an IHS germ and f an analytic function on X, such that the minimal embedded resolution diagram Γ_{π}(X, f) satisfies the semigroup condition. Let λ be a monodromy eigenvalue of f at a point of {f = 0}. Then there exist infinitely many (effective) allowed P -divisors W for (X, div(f)), and for each of them a pole s_{0} of the topological zeta function Z(f, W; s) such that exp(2πis_{0}) = λ.

Proof. The proof of Theorem 6.3.1 is still valid here, replacing the use of Proposition 6.2.5 by Proposition 7.4.1. □

Example 7.4.24. We provide an example where the semigroup condition is not satisfied, and where a given eigenvalue cannot be induced by a pole of a zeta function associated to any allowed divisor.

We re-consider Example 3.1.15, but taking F as the divisor corresponding to the unique arrowhead with multiplicity N (instead of multiplicity 1).

[figure: 2; 1; 7; 7; 1; 2; 3; 3; (3N); (6N); (N); (6N); (3N); (2N); (N); (2N); i_{1} − 1; i′_{1} − 1; i_{2} − 1; i′_{2} − 1]

Now we have that the Alexander polynomial is Λ(t) = Δ_{1}(t) = (t^{2N} − t^{N} + 1)^{2}. We take more specifically N = 7, then N_{1} = 42, and we pick λ := exp(2πi(−5/42)) ∈ Eig.

Recall that, cf. Example 5.1.11, ν_{1} = −78+7I+6I′, where I = 3i_{1}+2i_{2} and I′ = 3i′_{1}+2i′_{2}. We search for an allowed W such that −ν_{1}/42 is a pole of Z(F, W; s) and ν_{1} ≡ 5 (mod 42). This last condition is equivalent to I ≡ 5 (mod 6) and I′ ≡ 1 (mod 7). But for W to be allowed we need (cf. Example 4.1.7) that I = 7 or I′ = 7. So such an allowed divisor W does not exist. Note that also the node v′_{1} cannot induce λ by the symmetric argument, and that the node v_{0} cannot induce primitive 42-th roots of unity.

Example 7.4.25.

(a) Via Example 3.3.5 we proved that unrestricted generalization of the Monodromy Conjecture fails. Nevertheless, that example does satisfy the (analogue of) the semigroup condition. Let us explain what we mean by this. Even if a graph is not unimodular, one can associate with it a splice diagram (by the very similar way as in (2.1)), and one can impose the semigroup condition in the same way as above read from the splice diagram, cf. [22]. For example, the ‘culpable’ node with decoration −6 (which provides the counterexample to the ‘naively generalized’ Monodromy Conjecture) satisfies the semigroup condition, since the determinant of the (−2, −1, −4) string is 2 which is included in the semigroup generated by 2 and 4. Nevertheless, for this graph the combination of Goal (1) and (2) fails. The point is that this graph is not unimodular, hence our main result does not apply to it.

This also shows that in our discussions the IHS–restriction is essential: any generalization of our main results to non–IHS germs requires the replacement of the semigroup condition by a much stronger assumption.

(b) One can ask if there is any unimodular graph which plays the role of Example 3.3.5. Here is a combinatorial example. The form is the standard Gorenstein form, whose analytic realization can easily be checked; the function is given on the second diagram whose analytic realization we do not know (N is a positive integer). The example shows that in our combinatorial arguments from this section the semigroup assumption cannot be eliminated.

[figure: −2; −1; −7; −2; −3; −3; ⋮; (9N); (18N); (3N); (2N); (6N); N arrows; (N); (1); (1)]

Clearly, the semigroup condition at the vertex with decoration −7 is not satisfied. (Indeed, the determinant of the (−2, −1, −3) string is 1, which is not an element of S⟨2, 3⟩.) By a computation one gets that 7/3N is a pole of Z(s), but exp(14πi/3N) is not a root of

Δ_{1}(t) = \frac{(t^{9N} + 1)(t^{2N} − 1)^{N−1}(t − 1)}{(t^{3N} + 1)(t^{N} − 1)}.

7.5. Final remarks. (a) (The definition of allowed forms revisited.) There is a crucial feature regarding the definition of the allowed divisors: it does not use the multiplicity system of the divisor F, only its support. This has the following positive output: the family of allowed divisors can be defined uniformly for all divisors F with the same support, and all the results we prove are valid uniformly for all these divisors F (or, functions f with the same support). To exemplify, let us rewrite Theorem 5.1.6 in the following way.

Theorem 7.5.1. Let (X, 0) be an IHS germ, and F′ a reduced Weil divisor on X. Consider an allowed divisor W associated with (X, F′). Then, for any function g which has set–theoretical vanishing set g^{−1}(0) = F′, and any pole s_{0} of the topological zeta function Z(g, W; s), exp(2πis_{0}) is a monodromy eigenvalue of g at some point of {g = 0}.

Note that the zeta–function Z(F, W; s) and the Alexander polynomial Λ_{Γ(F)}(t) do depend essentially on the multiplicities of F.

The above new version (7.5.1) is definitely a much stronger statement than the original (5.1.6). The interested reader is invited to rewrite all the other results, especially Theorems 7.3.1 and 7.4.23 in the corresponding new versions. Of course, in order to do this, we have to observe that the definition of the semigroup condition associated with Γ_{π}(X, F) too depends only on the support of F.

(b) (The restriction (2.3)(2) of W revisited.) The restriction (2.3)(2) (see also (1.5.2)) was very convenient in the computations of arithmetical and numerical invariants, and additionally created a strong link between the supports of F and W. Moreover, in that choice, we had in mind the analytic realization of the divisor W too, that is, the applicability of the main results. More precisely, in general, it is a rather hard question to determine the analytic realization of some topologically identified arrowheads/divisors. For example [19] shows that simultaneous realization of some arrowheads is strongly obstructed. On the other hand, there is a ‘natural’ family of analytic singularities for which the analytic realization of the class of arrowheads considered in (2.3)(2) (arrowheads supported by boundary vertices) is automatically guaranteed. This is the class of ‘splice singularities’, cf. [22, 23]. In is worth to mention that the analytic realization of these germs is guaranteed by an arithmetical property of the graph Γ (see End Curve Theorem in [24, 25]), which is nothing else but the semigroup condition (2.1.4).

In this way, the simultaneous appearance of the restriction (2.3)(2) regarding the divisors W, and of the semigroup condition might be natural. Moreover, for a considerably large class of examples, when the analytic realization of all the forms W is guaranteed, the semigroup condition too will be satisfied (compare also with subsection 7.1). This supports strongly the results of this section, and motivates once again the semigroup condition, showing that its appearance is not just a technical necessity (compare also with (7.4.25)(b)).

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

  E-mail address: nemethi@renyi.hu

K.U.Leuven, Dept. Wiskunde, Celestijnenlaan 200B, 3001 Leuven, Belgium

  E-mail address: wim.veys@wis.kuleuven.be