GENERALIZED MONODROMY CONJECTURE IN DIMENSIONTWOANDRÁS 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). 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 topologicalzeta 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 alongeach splice component. For plane curves we show the following three guiding properties: (1) if s0 is any pole of Z(f, ω; s) with ω allowed, then exp(2πis0 ) 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 s0 .
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 mostfertilizing conjectures in singularity theory. It relates poles of Igusa/motivic/topological zetafunctions to monodromy eigenvalues. For instance, for a local analytic isolated singularityf : (Cn , 0) → (C, 0) it predicts that if s0 is a pole of the local topological zeta function of
f , then exp(2πis0 ) is an eigenvalue of the local monodromy operator acting on H ∗ (F0 , C),
where F0 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 (Cn , 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, 27, 31], 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 zetafunction and the monodromy operator; the existent proofs of the particular cases basically2000 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.
1

A. Némethi and W. Veys

2compute both sides independently (using their special properties) and compare the two ﬁnal
data.
A possible way to ﬁnd a more conceptual understanding and tools is to extend theconjecture to a larger class. This leads us to the replacement of (Cn , 0) with a singularspace, and of the standard diﬀerential form with some generalization of it. Although bothtypes of generalizations are obstructed (see the next subsections), the main target of thepresent 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, inorder 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 : (Cn , 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 insome 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 withthe original germ f and with a set of analytic diﬀerential n-forms ω living in (Cn , 0).
We now describe these zeta functions; they are deﬁned in terms of an embedded resolution
π : X̃ → Cn of f −1 (0) ∪ div(ω). We denote by Ei , i ∈ S, the irreducible components(exceptional divisors and strict transforms) of the inverse image π −1 (f −1 (0) ∪ div(ω)) andby Ni and νi − 1 the multiplicities of Ei in the divisor of π ∗ f and π ∗ ω, respectively. We◦
put EI◦ := (∩i∈I Ei ) \ (∪j ∈I/ Ej ) for I ⊂ S. Hence, the EI constitute a stratiﬁcation of X̃ inlocally closed subsets.
Deﬁnition 1.2.1. The (local) topological zeta function of (f, ω) at 0 ∈ Cn is∑∏1
Z(f, ω; s) :=χ(EI◦ ∩ π −1 (0)),νi + sNi
I⊂Si∈I

where s is a variable.
This deﬁnition extends naturally the deﬁnition of Denef and Loeser from [7] valid for the
standard form ω = dx1 ∧ · · · ∧ dxn . 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 intrinsicallydeﬁned motivic zeta functions [8]. Another technique is applying the Weak Factorization
Theorem [4, 33] 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, 30];
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 it is hard to characterize those which survive.
1.2.2. We recall that the ‘classical’ monodromy conjecture predicts the implication
s0 is a pole of Z(f, ω; s) ⇒ exp(2πis0 ) is a monodromy eigenvalue of f,where ω is the standard form dx1 ∧ · · · ∧ dxn . The point is that for arbitrary analytic
diﬀerential forms ω this implication is in general false. Even more, in [32] the second
author showed that every given monodromy eigenvalue of f is induced by a pole s0 of some

Generalized Monodromy Conjecture

3

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 theselection of forms with compatibility properties with the monodromy operator.
Partly initiated in [32] (see also [20]) we propose the following program.
Goal 1.2.3. Deﬁne/identify a collection of allowed analytic forms ω (depending on f ) suchthat(1) if s0 is any pole of Z(f, ω; s), where ω is allowed, then exp(2πis0 ) is a monodromy
eigenvalue of f ,(2) the standard form ω = dx1 ∧ · · · ∧ dxn is allowed,(3) every monodromy eigenvalue of f is obtained as in (1) for some allowed ω and somes0 .
A few remarks are in order. First, note that (1) and (2) imply the classical Monodromy
Conjecture. Furthermore, (1) and (3) combined show that the set of eigenvalues of fcoincides 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 thatthe topological zeta function is a kind of avatar of the p-adic Igusa zeta function, whichis 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 zetafunction, 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 theanalogous statements of the goals above. Roughly, if s0 is any pole of the zeta function off and any compactly supported C ∞ form ω, then exp(2πis0 ) is a monodromy eigenvalue off , 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 theintroduction of [32]. 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 decompositionin 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 asinspiration 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 thegoals (1)–(2)–(3) for this f . First we must identify the relevant (generalized) topologicalzeta function for such a pair (X, f ) and for an analytic form ω. In fact, in all our combinatorial arguments, we will replace (f, ω) by two Weil divisors (for details and motivation
see subsection (1.5.1)).

4

A. Némethi and W. Veys

Of course, in order to have a well–deﬁned analogue of part (2) of Goal 1.2.3, we needto 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 automaticallyguaranteed, for example, if the link is an integral homology sphere.
Nevertheless, we will need some further combinatorial restrictions. There is an exampleof Rodrigues [25] indicating that the ‘naive’ extension of the Monodromy Conjecture to theGorenstein singular setting might be obstructed. This example produces a set of integers{νi }i and {Ni }i associated with the exceptional divisors (or, with vertices of the plumbing
graph) which topologically are not obstructed to be the multiplicities of the standard form
and of an analytic germ f , and they produce a counterexample to the Monodromy Conjecture. In (7.4.25) we even construct another such example involving a Gorenstein surface
singularity with unimodular dual graph. One of the following two possibilities can solve
this situation in order to have a chance for a positive continuation: either we impose some
additional topological restrictions which eliminate any such counterexample, or we try toshow that the analytic realization of the analytic germ (in the presence of the Gorenstein
structure) guarantees these additional needed topological restrictions. The second possibility looks very diﬃcult and is hopeless with the present tools of the theory, and it is not thegoal of the present article to attack it. Therefore, in order to have an extended version, westay with the ﬁrst possibility.
The additional restriction we impose, in fact, is very natural; it is a modiﬁcation of thesemigroup condition of Neumann and Wahl [21], 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). This also emphasizes a subtle connection
between the semigroup condition and the Monodromy Conjecture.
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 the semigroupcondition.
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 thefunctions and forms (and their analytic realizations will be not involved). In subsection(1.5.1) we give some details about the formalism of zeta functions associated with Weildivisors, and in (1.5.3) we motivate the deﬁnition of allowed forms. An ambient germ withintegral 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 andfor 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 semigroupcondition of Neumann and Wahl in the presence of a divisor. Section 3 treats the conceptof 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 derivea 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 restrictionand 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 eigenvalueof 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

Generalized Monodromy Conjecture5(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 isnecessary to have Goal (2) and (3), respectively, (at least in any topological treatment),
while the discussion from 7.4.25 shows that if we drop the IHS assumption about the linkof the ambient space we need to treat 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 tosingular ambient varieties X (instead of Cn ), see for example [30], and speciﬁcally forsurfaces [25, 26, 29]. 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 ) andW := div(ω) on (Cn , 0), and not on the actual function f and form ω. In terms of thesedivisors, the numerical data Ni and νi − 1 above are given as the multiplicities of Ei in thedivisors π ∗ F and Kπ + π ∗ W , respectively, where Kπ 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 Kπ exists. Both conditions are satisﬁed when (X, 0)
is an arbitraryfor Kπ see (2.3.2)).
∑ normal surface germ (for the pullback see (2.2.2), while ∑Let F := j∈J Nj Ej be an eﬀective non-zero Weil divisor, and W := j∈J (νj − 1)Ej an
arbitrary Weil divisor on X, where Ej , j ∈ J, are (ﬁnitely many) irreducible Weil divisors.
We only require that (Nj , νj ) ̸= (0, 0) for j ∈ J, that is, a component Ej 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 Ei , i ∈ S, the irreducible components of its inverse image, and by Ni and νi − 1 themultiplicities of Ei in the divisors π ∗ F and Kπ + π ∗ W , respectively. Note that the Ni andνi of exceptional components Ei are in general rational numbers. The (local) topologicalzeta function of (F, W ) at 0 ∈ X, denoted as Z(F, W ; s), is deﬁned by the same formula asin 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 Ni and νi 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 graphof 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 anothernovelty 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 zetafunction, and that those of valency at least 3 (corresponding to the nodes in the spliceA. Némethi and W. Veys6

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 associatedwith 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 d1 − xd2 )(y d1 + xd2 ), where d1 > 1, d2 > 1
and gcd(d1 , d2 ) = 1. Below is its star-shaped splice diagram with node E, where the dashedarrows 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).
k1 − 1
sH
*
*

d
1
HH
s 


H
s d2 Hj
j k2 − 1Hi2 − 1  
i1 − 1 Then N = 2d1 d2 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) =(tN − 1)2.
(tN/d1 − 1)(tN/d2 − 1)

These are all N -th roots of unity that are not simultaneously (N/d1 )-th and (N/d2 )-th rootsof unity; in other words all exp(2πi Nu ) for which d1 - u or d2 - u.
(ii) Using (2.4) one has
1
Z(f, ω; s) =
ν + sN

(

d1 d2
1
1
−2 +
+
+
+
i1
i2
k1 + s k2 + s),

where (see (2.3.5))(1.5.4)

ν = d1 d2 (k1 + k2 − 2) + d2 i1 + d1 i2 .

We now investigate the candidate pole s0 := −ν/N of the zeta function. If s0 is a pole oforder one, one easily veriﬁes that its residue R is not identically zero as function in the four
variables i1 , i2 , k1 , k2 . Hence s0 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 k1 and k2 if
i1 = d1 and i2 = d2 , 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 dℓ | ν if and only if dℓ | iℓ(by (1.5.4)). We call W allowed if the following condition on i1 and i2 is satisﬁed: if d1 |i1
and d2 |i2 , then i1 = d1 and i2 = d2 . (There are no conditions on k1 and k2 .) Therefore, for
W allowed, if exp(2πis0 ) is not a root of Λ, then d1 | i1 and d1 | i2 , hence R = 0.
Moreover with this deﬁnition the divisor W = 0, corresponding to i1 = i2 = k1 = k2 = 1,
is clearly allowed (Goal (2)), and Goal (3) is also satisﬁed. Indeed, ﬁx a root exp(2πi Nu ) of
Λ(t). Since the numbers d1 and d2 are coprime, there exist integers k1 , k2 , i1 , i2 (all positive

Generalized Monodromy Conjecture

7

if we desire so) such that ν in (1.5.4) satisﬁes ν ≡ u mod N . The restrictions on the given
u imply that d1 - i1 or d2 - i2 . Hence the constructed W is allowed.
One can carry out without too much eﬀort a similar analysis for any plane curve germf , or, more generally, for any function f on an IHS germ for which the splice diagram isstar-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, forthem 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’ deﬁnitionof 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 isthat 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 = Gπ (X) associated with the irreducible components {Ei }i of E: each Ei
determines a vertex of G with genus decoration [g(Ei )] and self–intersection (Ei , Ei ), while
the edges of G correspond to intersection points Ei ∩ Ej , cf. [11]. Since X is normal, G
is connected. Let I(G) be the negative deﬁnite intersection form (Ei , Ej )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(Ei ) = 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 morecondensed form via its splice diagram Γ = Γπ (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 ishomeomorphic to G, but it has no vertices of valency 2. Its vertices are either nodes (ofvalency ≥ 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 decorationsof Γ are as follows. At each node v of Γ one inserts a weight dve on each incident edge e. Let
Gve be the connected component of G\{v} ‘in the direction of e’, then dve := det(−I(Gve )).
It is proved in [9, Ch. V] that the decorated graphs G and Γ determine each other.
The decorations {dve }v,e of Γ satisfy the next compatibility conditions: (a) dve ≥ 1,(b) {dve }e are pairwise coprime integers for any ﬁxed node v,(2.1.1)(c) any ‘edge determinant’ qe is positive.

8

A. Némethi and W. Veys

Part (c) means the following: for any ﬁxed edge e with end–nodes v and w, let the decorations at v be dve and {dvei }i , and similarly dwe and {dwe′j }j at w. Then (see [9, §24]):∏∏(2.1.2)
qe := dve dwe −dvei
dwe′j > 0.
ij

If π is the minimal good resolution, then Γ also is minimal, in the sense that all thedecorations dve 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 (dependingon 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 weregard 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 theintegers {dve }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 ℓvw 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 dv be the product of edge weights adjacent to v.
For any node v and adjacent edge e, let Γve be the connected component of Γ \ {v} inthe direction of e. We say, following [21], that the node v and adjacent edge e satisfy the
semigroup condition if
dv is in the semigroup generated by the ℓvw , where w is a boundary vertex of Γ in Γve .
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 Gπ (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
Ei is codiﬁed by an arrowhead supported by that vertex of G which corresponds to Ei .
Additionally, each vertex and arrowhead inherits a multiplicity decoration, the vanishingorder of f ◦ π along the corresponding irreducible divisor.
Clearly, the divisor div(f ◦ π) on X̃ is a principal divisor, hence (div(f ◦ π), Ei ) = 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, thisproperty 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, Ei ) = 0 for any i. If we start with an arbitrary

Generalized Monodromy Conjecture

9

Weil divisor F ′ on X and π is an embedded resolution of the pair (X, F ′ ), then there isan unique P –divisor F on X̃ whose arrow-multiplicities agree with the multiplicities of thecomponents 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 theirmultiplicities. This pair is denoted by Gπ (X, F ).
In all of our topological–combinatorial discussions, we regard Gπ (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 = Γπ (X, f ) associated with (X, f ) (or more generally, Γπ (X, F )associated with Gπ (X, F )) is constructed similarly as above, but now the nodes are thosevertices 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 Gπ (X, F ) with valency ≥ 3 (including theedges 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 oris missing. Next, assume that the arrowhead a of Gπ (X, F ) is supported by a vertex v of G
of valency 2. This means that v is a boundary vertex of Gπ (X) with an arrowhead. If weforget about the arrowheads and we determine Γ from Gπ (X), then v becomes a boundary
vertex of Γπ (X). Then, reconsidering the arrowheads, this boundary vertex in Γπ (X, F )is replaced by an arrowhead. Summed up: in minimal diagrams, all the arrowheads ofΓπ (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 whichcorresponds 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 equivalentdiagrams:
H..HHs d . =(for any d ≥ 1)

H..HHs d
.

s

-



Also, Γπ (X, F ) inherits the multiplicity of each arrowhead and node from Gπ (X, F ) (withthe 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 Γπ (X, F ). Moreover, we call special edges those connecting two
nodes, denoted by E s . The arrowheads will be denoted by AF , the multiplicities by (Nw ),w ∈ V ∪ AF .
Again, we can regard Γπ (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 Na > 0 for alla ∈ AF , then some analytic realization exists, cf. [9, § 24].)

A. Némethi and W. Veys

102.2.2. 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 multiplicitiesof 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 ℓva
as the product of the edge weights that are adjacent to, but not on, the path from v to a.
Then the multiplicity Nv of any vertex v is given by, cf. [9, §10],∑(2.2.3)Nv =Na ℓva .
In particular, if F ′ =

a∈AF

∑

′
′
a Na Fa is a Weil divisor on X with the Fa irreducible, and {Fa }a∈AF
are the strict transforms and Fv = Ev the exceptional
curves, then the pullback F = π ∗ (F ′ )∑of F ′ is represented in the splice diagram by v∈AF ∪V Nv Fv .

2.2.4. The splice diagram Γπ (X, F ) also satisﬁes the compatibility conditions (2.1.1).
If one deletes all the arrowheads and decorations of Γπ (X, F ) associated with F we recovera possible (maybe non–minimal) splice diagram of X. Nevertheless, by this simpliﬁcation,some of the nodes might disappear.
2.2.5. Semigroup condition for Γπ (X, F ). In the presence of a divisor, the semigroupcondition (2.1.4) will be modiﬁed as follows. We say that Γπ (X, F ) satisﬁes the semigroupcondition if all pairs (v, e) satisfy (2.1.4), provided that v is a node with adjacent edge e,and the connected part of Γπ (X, F ) \ {v} in the direction of e contains no arrowheads.
Remark 2.2.6. (1) The splice diagram associated with the minimal embedded resolutionof 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 thissub–diagram are not sitting on geodesic paths connecting two arrowheads of the diagram.
v1

s

a1

vr−1

v2

s

a2

s

...

ar−1

vr

s

ar

p1

p2

pr−1

s

s

s

. 
s
H ..
pr H
H
s

For this part, the semigroup condition follows from the positivity of the edge determinants
and the fact (used literately 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 generalargument, see (7.1).)(2) If F = 0, we recover the semigroup condition of (2.1.4). On the other hand, thesemigroup condition of Γπ (X, F ), and of the diagram obtained from Γπ (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 Γπ (X) does not satisfy the semigroup condition, but Γπ (X, F ) forsome 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 Γπ (X) satisﬁes the semigroup condition, butΓπ (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.
−2

s

−1

s

−13

s

s −3 ?

−1

s

s −3

−2

s

s

2 s7
3

1 s1

7 s2
3

s

?

s

s

Generalized Monodromy Conjecture

11

Then Γπ (X, F ) does not satisfy the semigroup condition at the central node, although if wedelete the arrowhead then Γπ (X) does since that node disappears.
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 anembedded 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. Thenclassically one considers ω0 = dx ∧ dy (for some local coordinates (x, y) of (X, 0)), or, moregenerally, ω = gω0 for some local analytic germ g on (X, 0). In this case the pullbackω̃ := π ∗ (ω) is clearly holomorphic on X̃.
• Generalizing to the singular setting, we assume that (X, 0) is Gorenstein and that ω0 is
a nowhere vanishing holomorphic 2–form on X \ {0}. Then, for any holomorphic germ g on
X, the pullback ω̃ := π ∗ (gω0 ) 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) = Kπ + 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 anembedded resolution of supp(F ′ ) ∪ supp(W ′ );∑′′(2) if W ′ :=k (ik − 1)Wk is the irreducible decomposition of W , then the strict′
transform of each Wk either is identical with the strict transform Fa of one of theFa′ , or, it intersects a boundary component of Gπ (X, F ). Moreover, each boundary
component can intersect at most one of the strict transforms of the Wk′ .
We denote the P –divisor π ∗ (W ′ ) by W . In our diagrams, the strict transform Wk of
a Wk′ that intersects Ei will be denoted by a dashed arrowhead, attached to the vertex
corresponding to Ei . If Wk agrees with one of the Fa , then the dashed arrowhead doubles
the associated ordinary arrowhead, while if a boundary vertex does not support any ordinary
arrowhead but supports a Wk , then its dashed arrow is attached to this vertex. We will
denote by AW the dashed arrowheads of W . The ‘double’ arrowheads are given by AF ∩AW .
Again, we will identify the following diagrams.
HH
.

s
.. H




-

=

HH
.. Hs 1
. 


s

-

Finally, we have to add to the decorations the multiplicities of the components of Kπ +W .
By technical (and traditional) reasons, the multiplicity of a dashed arrowhead is denoted
as above by ia − 1, while the multiplicity of a vertex v by νv − 1. Recall that, in general, we
do not impose for the integers ia and νv to be positive. We denote this enriched diagramas Γπ (X, F, W ).
Note that in the three diﬀerent levels Γπ (X), Γπ (X, F ) and Γπ (X, F, W ), the valencies
of a ﬁxed vertex v of Γ are not the same; these three valencies will be denoted by δv , δv′ andδv′′ respectively.
2.3.1. In order to identify the coeﬃcients νv − 1 of Kπ + W , we only must describe thecoeﬃcients of the Ev in Kπ , since those of W are as in (2.2.2).

A. Némethi and W. Veys

12

The divisor K = Kπ is determined in the graph Gπ (X) by the adjunction relations(K + E, Ei ) = δi − 2,

(2.3.2)where here δi is the valency of the vertex i in Gπ (X). Let L be the lattice H2 (X̃, Z)
with the intersection form I(G). Then for each vertex i of G one can deﬁne Ei∗ ∈ Lwith (Ei∗ , Ej ) = −δij (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 − δi )Ei∗ . Since all the valency 2 vertices of G are irrelevant, the relation
descends naturally to the level of Γ = Γπ (X) (this means that the multiplicities of K + E
along the vertices of Γ are those given on the right):∑(2 − δw )Ew∗ .
(2.3.3)
K +E =w∈V∗
Each Ew , considered as divisor supported on E, together with a non–compact irreducible
divisor intersecting Ew 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 ofa divisor F , along any node of Γπ (X, F )) is∑(2 − δw )ℓvw .
(2.3.4)w∈V

In the presence of the divisor W ′ =(2.3.5)

νv =

∑

w∈V

∑

′
a∈AW (ia − 1)Wa on X, again (2.2.3) then yields(2 − δw )ℓvw +∑(ia − 1)ℓva .

a∈AW

We warn the reader about the following fact. The sum (2.3.4) is associated with Γπ (X).
Therefore, in the applications later, even if we start with some Γπ (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 verticesof Γπ (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 diagramΓπ (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 [28] 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 Γπ (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 zetafunction 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 zetafunctions 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 Na and ia − 1 to all a ∈ AF ∪ AW , that is, we put
Na = 0 for a ∈ AW \ AF and ia = 1 for a ∈ AF \ AW .
Deﬁnition 2.4.1. Let Γ = Γ(F, W ) be such a diagram. We require for each a ∈ AF ∪ AW
that (Na , ia ) ̸= (0, 0). For a node v, let (AF ∪ AW )v and Bv 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 dva andGeneralized Monodromy Conjecture

13

dvw , respectively. For any w ∈ Bv , let iw − 1 be the decoration of the dashed arrowhead
supported by w; if such an arrowhead does not exist then set iw = 1. For a special edgee, let v and w denote its end vertices, and qe the edge determinant (2.1.2). Then the zetafunction Z(Γ) of the diagram Γ is∑Z(Γ) = Z(Γ; s) :=

v∈N


1
2 − δv′′ +νv + sNv

∑ dvw
w∈Bv



∑

+

dva 
ia + sNa

iwa∈(AF ∪AW )v∑qe.
+(νv + sNv )(νw + sNw )se∈E

3. Splicing the diagrams and their invariants
3.1. Splicing the diagrams. The main advantage of the splice diagrams is that theydescribe 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 {Nw }w associated with a principal divisor f is classical,
it was developed in [9].
It is easy to see that any P –divisor follows the same formula. On the other hand, therules for the numbers {νv }v are slightly more involved, and we were not able to ﬁnd themin the literature (though, see the ‘simpler’ situation considered in [20]). In this subsection,
we will present these splice formulae.
Subsection (3.1.1) treats the case Γ = Γπ (X), (3.1.2) the case Γπ (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 vL and vR(left/right) is the operation which replaces the left diagram Γ into the two diagrams ΓL andΓR (containing vL and vR respectively), with two new end vertices v̄L and v̄R . (All the otherparts of the diagrams are kept unmodiﬁed.) If either d or d′ is 1, then the corresponding
leg in ΓL or ΓR can be deleted in the minimal representation, but we prefer to keep it. In
this way, the valency of the vertices vL and vR stays unmodiﬁed. Moreover, when we equipsuch a leg later with an arrowhead, it cannot be deleted.
′1
Hd..HHs d
.
dn

vL

d′

d1
sH...

H′ H
dn′

e

vR

splicing-′

1
Hd..HHs d
.
dn

s

vL

v̄L

s

d′

d
.1
s
HH..
′H
dn′

v̄R

vR

(The diagrams ΓL and ΓR correspond again to dual graphs of certain IHS normal surface
singularities. If Γ represents M = S 3 , then both ΓL and ΓR represent S 3 , see (2.1.3).)
3.1.2. Splicing Γ(F ). Next, we analyze the behavior of the multiplicity system determinedby a function f or a P –divisor F . Let A := AF be the index set of arrowheads; it can be
written as a disjoint union AL ∪ AR , according to the position of the arrowheads. First,
assume that both AL and AR are non–empty. If a ∈ AL , then let ℓea be the product of theedge weights, all of them in ΓL , that are adjacent to, but not on, the path from vR to a.
Symmetrically, one deﬁnes the integers ℓea for a ∈ AR . Then Γ(F ) has the following splice
decomposition:

A. Némethi and W. Veys

14

′

1
Hd
..HHs d
.
(N )
d
n

vL

d′

d
1
sH ...
(N ′ ) H′ H
d ′splicing-n1
Hd..HHs d .
(N )
d
n

vR

e

(M )

vL

d′1
′

d

s ...
(M ′ ) 
H(N ′ ) H′ H
dn′vR

where (cf. [9, (10.6)])(3.1.3)M=∑

Na ℓea

and

M′ =

a∈AR

∑

Na ℓea .

a∈AL

If all the arrowheads of Γ are in one side, say AL = ∅, then one has the new situation′

1
Hd..HHs d.dn (N )

vL

d′

d1


s ...
H
(N ′ ) H′ H
d ′splicing-

n

e

vR

1
Hd
..HHs d .
dn (N )

(M )

vL

(M )

s

d′

.1
s
H ..
(N ′ ) H′ H
d ′

d′

n

vR

where M is computed by the same formula as in (3.1.3).
We denote the ‘total’ inherited divisors on ΓL and ΓR by FL and FR , respectively. They
can be identiﬁed with P –divisors of the diagrams ΓL and ΓR , 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 ωor, more generally, K + W for some P –divisor W . Since the splicing of W is covered by theprevious 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 ΓL and ΓR . We wish to compare the divisors of the pullbacks of theforms ω0,Γ with those of ω0,ΓL and ω0,ΓR — if they exist analytically; and, more generallyKΓ with KΓL and KΓR (a combinatorial, always well–posed question). For any v ∈ V(ΓL ),∗we denote the dual basis element computed in ΓL by Ev,Γ. Moreover, we separate theLvertices of ΓL inherited from Γ: we set V̄(ΓL ) := V(ΓL ) \ {v̄L }, cf. the notation of (3.1.1),
and similarly for ΓR . Let us rewrite (2.3.3) into∑∑
(3.1.6)(K + E)Γ =(2 − δv )Ev∗ +(2 − δv )Ev∗ .
v∈V̄(ΓL )v∈V̄(ΓR )Recall that Ev∗ behaves as a P –divisor associated with one arrowhead supported on v.
∗(i.e., the|So, by (2.2.3), for any v ∈ V̄(ΓL ) the restrictions satisfy Ev∗ |V̄(ΓL ) = Ev,ΓL V̄(ΓL )multiplicities along V̄(ΓL ) agree). Hence()∑(K + E)Γ |V̄(ΓL ) = (K + E)ΓL − Ev̄∗L ,ΓL +(2 − δv )Ev∗ |V̄(ΓL ) .
v∈V̄(ΓR )Clearly, all multiplicities of both EΓ and EΓL along V̄(ΓL ) are one, hence they cancel:()∑(3.1.7)
KΓ |V̄(ΓL ) = KΓL − Ev̄∗L ,ΓL +(2 − δv )Ev∗ |V̄(ΓL ) .
v∈V̄(ΓR )Generalized Monodromy Conjecture

15

Using again (2.2.3), the sum can be replaced by a P –divisor of ΓL . Indeed, set∑(3.1.8)i = ie,L :=(2 − δv )ℓev ,v∈V̄(ΓR )where ℓev (for any v ∈ V̄(ΓR )) is the product of the edge weights of Γ, all of them in ΓR , thatare adjacent to, but not on, the path from vL to v. Furthermore, let GL be the P –divisor
on ΓL determined by one arrowhead with multiplicity one supported on v̄L . Then (2.2.3)
and (3.1.7) imply
KΓ |V̄(ΓL ) = (KΓL + (i − 1)GL ) |V̄(ΓL ) .

(3.1.9)If the forms above exist, and if GL is the pullback divisor of a function gL , then we have()
(3.1.10)div(π ∗ ω0,Γ )|V̄(ΓL ) = div πL∗ (gLi−1 · ω0,ΓL ) |V̄(ΓL ) .
Obviously, there is a symmetric identity for the restriction on V̄(ΓR ). On diagrams we have1
Hd..HHs d
.
dn

′

d′

d1
sH...
H′
dH′

splicing-n1
Hd..HHs d
.
dn

s

s

?

i−1

?

i′ − 1

d′

d′

.1
s
HH..
d′ H
′
n

When we incorporate also the divisor W , the equations (3.1.8) and (3.1.9) respectively
extend to∑∑(ia − 1)ℓea ,(3.1.11)i :=(2 − δv )ℓev +v∈V̄(ΓR )a∈AW,R

where ℓea is the product of the edge weights of Γ, all of them in ΓR , that are adjacent to,
but not on, the path from vL to the corresponding dashed arrow in ΓR , and()(3.1.12)(KΓ + W )|V̄(ΓL ) = KΓL + (i − 1)GL + WL+ |V̄(ΓL ) ,where WL+ is induced by W on ΓL , as in (3.1.2). We will reserve the notation WL rather
for (i − 1)GL + WL+ , in order to have the expression(KΓ + W )|V̄(ΓL ) = (KΓL + WL ) |V̄(ΓL ) .

(3.1.13)

Note that (usually) the vertices v̄L and v̄R 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 AF,L and AF,R non–empty, the above diagram
modiﬁes into
Hd1..HHs d
.
(N )
d
n

′

d′

d1

sH...
(N ′ ) H′ H

splicing-dn′

Hd1..HHs d .
(N
)
dn

i−1
(M )

d′1
′

d

i′ −′1 

s ...
H
(M )
(N ′ ) H′ H
dn′where M and M ′ are determined as in (3.1.3). If AF,L = ∅, then one has
1
Hd..HHs d
.
(N )
d
n

′

d′

d1

sH...
(N ′ ) H′ H
dn′splicing-

1
Hd..HHs d - i − 1
.
(M )
(N )
d
n

d′1
′

d

s
s ...
H
(M )
(N ′ ) H′ H
dn′
?
′
i −1

A. Némethi and W. Veys

16

Remark 3.1.14. (1) Assume that we start with W = 0. Then both WL+ and WR+ are zero,but in general WL and WR are not. Furthermore, even if W is eﬀective (in particular, if
W = 0), the induced divisors WL and/or WR are in general not eﬀective.
In a diﬀerent language: even if we start with the ‘standard 2–from’ ω0 (instead of themore general gω0 ) — like in the traditional framework of, say, topological or motivic zetafunctions —, once an inductive splice–decomposition argument is used, we are forced toenlarge the class of our forms: in (3.1.10) the standard form decomposes in the splicecomponent into a generalized form of type gω0 . Also, one can see that even if we start witha holomorphic form, the forms on the splice components, usually, are not holomorphic (thatis, they are meromorphic).
(2) Even if we have a precise analytic realization of a diagram Γ, it is not clear what therelations are connecting this analytic structure and the eventual analytic realizations of ΓLand ΓR . In general, there is no analytic construction known by the authors which would
deﬁne a natural analytic structure with topology ΓL 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 asΓπ (X, f, ω), after splicing the two combinatorial packages might not be realized analytically(for example, the ‘correction term’ (i − 1)GL 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 thefunctions and (meromorphic) forms will exist.
Example 3.1.15. Consider the following diagram Γ(F ), compare also with Remark 2.2.6(4).
s

(3)

v1

v0

v1′

2 s7
(6) 3

1 s1
(1)

7 s2
(6) 3

(2)

(1)

(2)

s

?

s

(3)

s

We denote the nodes by v1 , v0 and v1′ , and their ν–numbers by ν1 , ν0 and ν1′ . Then
ν1 = ν1′ = −13 and ν0 = −2. Splicing the diagram Γ(F, W = 0) we get the three star–shaped
subgraphs Γ1 , Γ0 and Γ′1 :ν1 = −13

s

(3)

2 s7
(6) 3

ν1′ = −13

ν0 = −2

- −2

−2 

(1)

s

1 s1
(1)

−2 
(1)

- −2

?

(2)

7 s2
(6) 3

s

(1)

s

(3)

(2)

The zeta function Z(Γ; s) associated with Γ(F, W = 0), cf. (2.4.1), isZ(Γ; s) = 2

4
1
1
1
+
(−1 +
)+2
.
6s − 13 s − 2
s+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 vL and vR :Hd1..HHs d. (N )
d
n ν−1

′

d1

sH...
(N ′ ) H
′H
′
d′

ν −1

dn′

splicing-

Hd1..HHs d . (N ) dn ν − 1

i−1
(M )

i′ −′1
(M )

d′1
′

d



s ...
H
(N ′ ) H
ν′ − 1d′nH′

Generalized Monodromy Conjecture

17

If AF,L = ∅, then replace this diagram with the adapted one as in the last diagram of(3.1.5). In this case of AF,L = ∅ one always has M ′ = 0.
Note that, to be able to deﬁne Z(ΓL ) and Z(ΓR ), we need that the condition (Na , ia ) ̸=(0, 0) for all arrowheads is also valid after splicing, see (2.4.1). For the moment we justassume 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 ‘rightleg’ of ΓL to Z(ΓL ) and the contribution of the ‘left leg’ of ΓR to Z(ΓR ), minus an easycorrection term, as shown below. This then yields a simple splicing formula for topologicalzeta 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′ − ( nj=1 dj )( nj=1 d′j ) for the edge determinant of e. If AF,L = ∅, set M ′ = 0.
Then we have the equality(3.2.2)qdd′=+(ν + sN )(ν ′ + sN ′ )(ν + sN )(i + sM ) (ν ′ + sN ′ )(i′ + sM ′ )1−.
(i + sM )(i′ + sM ′ )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{{∏∏ν = ( nj=1 dj )i + di′
N = ( nj=1 dj )M + dM ′
and
∏ ′
∏ ′
ν ′ = ( nj=1 dj )i′ + d′ i.
N ′ = ( nj=1 d′j )M ′ + d′ M
Hence (as polynomials in s){∏( nj=1 dj )(i + sM ) = ν + sN − d(i′ + sM ′ )(3.2.3)∏ ′( nj=1 d′j )(i′ + sM ′ ) = ν ′ + sN ′ − d′ (i + sM ).
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 forZ(Γ).
Theorem 3.2.4. (1) Consider the splicing of the diagram Γ as in the last diagram of(3.1.5). Again, if AF,L = ∅, set M ′ = 0. Then
Z(Γ) = Z(ΓL ) + Z(ΓR ) −1.
(i + sM )(i′ + sM ′ )(2) The contribution of vL in Z(Γ) has − Nν as a pole of order 2 if and only if the contributionof vL in Z(ΓL ) has − Nν as a pole of order 2.
(3) Suppose that − Nν is not a pole of order 2 of Z(Γ). Then the contributions of vL to theresidue of Z(Γ) and to the residue of Z(ΓL ) at − Nν 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(ΓL ) or Z(ΓR ).

18

A. Némethi and W. Veys

(2) First note that the coeﬃcient of any expression 1/(ν + sN )2 in the topological zetafunction formula is always positive, hence there are no cancelations among them. Consequently, the statement follows immediately from the linear dependency considerations inLemma 3.2.1.
(3) The diﬀerence between the contributions to both residues is
(
)
1
q
d
−
,
N ν ′ + sN ′ i + sM
evaluated in s = − Nν , and this is zero because of (3.2.2).



Example 3.2.5. With the notations of Example 3.1.15 one has
1 (
7 )
1 (
1 )
Z(Γ1 ; s) = Z(Γ′1 ; s) =4+
, Z(Γ0 ; s) =−3+.
6s − 13s−1
s−2
s+1
One veriﬁes that indeed, according to Proposition 3.2.4(1),1Z(Γ; s) = Z(Γ1 ; s) + Z(Γ0 ; s) + Z(Γ′1 ; s) − 2,(−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 F0 be its Milnor ﬁber,
hi : Hi (F0 , C) → Hi (F0 , C) the algebraic monodromy (i = 0, 1), ∆i (t) := det(tI − hi ) the
characteristic polynomial of hi , and ﬁnally, ζ(t) = ∆1 /∆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 = Γπ (X, f ) as follows:∏′(3.3.1)
ζ(t) =(tNv − 1)δv −2 ,v∈V(Γ)

where, for each vertex v, Nv denotes its multiplicity and δv′ its valency in Γπ (X, f ). The
zeta–function is ‘almost’ multiplicative with respect to the splice decomposition. In orderto have a uniform statement, we consider the Alexander polynomial (in one variable), cf. [9,(12.1)], as follows:{
ζ(t)if #A(Γπ (X, f )) ≥ 2,
(3.3.2)
Λ(t) :=∆1 (t) if #A(Γπ (X, f )) = 1.
The formula (3.3.1) provides Λ(t) too, since, if #A = 1, then Λ = ζ · ∆0 , and ∆0 (t) =a is the multiplicity of the unique arrowhead a. (In general, F0 has dconnected components, hence ∆0 (t) = td − 1, where d := gcda∈A (Na ).) In particular, Λ(t)can be recovered from the diagram Γ = Γπ (X, f ); let us write ΛΓ (t) for this expression.
Clearly, ΛΓ (t) depends only on the divisor of f , hence the above formula deﬁnes ΛΓ (t)for any P –divisor F and Γ(F ). Moreover, assume that the splice diagram Γ(F ) has the
splice decomposition ΓL and ΓR 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.
tNa − 1, where N

Proposition 3.3.3.
ΛΓ (t) = ΛΓL (t) · ΛΓR (t).
Example 3.3.4. With the notation of Example 3.1.15 one has ΛΓ1 (t) = ΛΓ′1 (t) = t2 − t + 1,ΛΓ0 (t) = 1, and their product is indeed Λ(t) = ∆1 (t) = (t2 − t + 1)2 .

Generalized Monodromy Conjecture

19

The advantage (at least in the present paper) of Λ compared with ζ is that Λ is apolynomial, hence we do not have to deal with possible cancelations of the roots and polesin the multiplicative formula of Proposition 3.3.3.
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 classof 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 ourallowed 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, wewill 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 ofdecorated dashed arrows a∈AW (ia − 1)Wa , i.e., the associated P –divisor W , is allowed forΓ (or, the diagram Γ(F, W ) is allowed), if the following conditions are satisﬁed:(1) ia ̸= 0 for a ∈ AW \ AF , that is, (Na , ia ) ̸= (0, 0) for all a ∈ AF ∪ AW .
(2) Suppose that Γ is star-shaped. Let the central node be connected to n boundary
vertices whose supporting edges have decorations {dℓ }nℓ=1 , and with r other incident
edges connecting with arrowheads, doubled by dashed arrows or not (r ≥ 1 always).
s - i1 − 1
H
YH
d
1
.. H
.
s
. HH..
dn Hs in − 1r arrowheads
which might be doublearrowsn boundary vertices

Then the decorations i1 − 1, . . . , in − 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 dℓ |iℓ for at least n − 1 indexes ℓ ∈ {1, . . . , n}, then iℓ = dℓ for at leastn − 1 indexes ℓ ∈ {1, . . . , n};• r = 2 : if dℓ |iℓ for all 1 ≤ ℓ ≤ n, then iℓ = dℓ 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 ia for the arrowheads given by a ∈ AF , that is, those associated to the other r edges.
(b) The value iℓ = 1 is possible. It corresponds to no dashed arrow, or formally to adashed arrow with decoration zero. Also, for the boundary vertices from the right, if dℓ = 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 (withdℓ = 1).
(c) One can formulate the restrictions in (2) simultaneously for all r as follows:
if dℓ |iℓ for at least n + r − 2 indexes ℓ ∈ {1, . . . , n}, then iℓ = dℓ for atleast n + r − 2 of the indexes ℓ. (This assumption is empty if r ≥ 3.)

A. Némethi and W. Veys

20(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 Weildivisor on it. A Weil divisor W ′ of (X, 0) is allowed for the pair (X, F ′ ) if there exists anembedded resolution π : X̃ → X of F ′ such that the diagram Γπ (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 :=
Γπ (X, F, W ) is allowed. Let h : X̃1 → X̃ be a blowing-up in some point of X̃. Then the
diagram Γ1 := Γπ◦h (X, F1 , W1 ) (with obvious notations) is also allowed.
Proof. We only have to investigate the spliced star-shaped subgraphs of Γ1 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 aredone because then Γ1 = Γ. We are left with the following two cases for P .
Case 1. P is a point of Ej◦ (that is, a generic point of Ej ), where the vertex corresponding
to Ej has valency 2 in Gπ (X, F, W ) (so it does not occur explicitly in Γ):
H..HHsp
. 


p′


sH...
blowup
HH

H..HHsp
. 


v



p′ 
s ..

q q′
s
1

w

s - 0

HH.
H

(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’ nodesin Γ and Γ1 are the same. Moreover, the new spliced star-shaped subgraph of Γ1 around v
satisﬁes the deﬁnition of allowedness. Indeed, in both cases r = 1 or r = 2, the fact that thedecorations associated to the boundary vertex w satisfy d1 = i1 = 1 ﬁnishes the veriﬁcation.
Case 2. P is a point of Ev◦ , where v is a node in Γ.
In this case the only novelty in Γ1 is an extra edge at the node v supporting a boundary
vertex, again with edge decoration dℓ = 1 and associated number iℓ = 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 someΓπ (X, F, W ) associated with some non–minimal π. Consider for example the situation
s

p

*

sH
HH
j

blowups?q − 1

q sd
1

p


*

sHHH
j

s - i−1

where π is obtained from the minimal embedded resolution πmin 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 πmin in a component Ej 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 νv /Nv it has a global integral (i − 1)-shift. In particular, in such a situation (having

Generalized Monodromy Conjecture

21

connections with monodromy in mind), we might take i − 1 = 0 without restricting thegenerality 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:
i1 − 1 

s

(3)

2 s7
(6) 3
(2)

s

1 s1
(1)

?

(1)

7 s2
(6) 3
(2)

?

s

s

(3)

- i′1 − 1?

i′2 − 1i2 − 1The splice decomposition provides:
s
?

2 s7
3

s

i1 − 1-i − 1(1)i0 − 1 

1 s1- i′0 − 1i′ − 1

(1)

7 s2
3

s

?

(1)

?

s
?

i′1 − 1?

i′2 − 1i2 − 1In the above picture i − 1 = i′0 − 1 = −2 + 3(i′1 − 1) + 2(i′2 − 1) and i′ − 1 = i0 − 1 =−2 + 3(i1 − 1) + 2(i2 − 1).
Assume that W is allowed. This imposes the following numerical conditions.
• In the middle graph Γ0 we impose: either 3i1 + 2i2 = 7 or 3i′1 + 2i′2 = 7.
• In Γ1 one gets: if 2 | i1 or 3 | i2 then either 2 = i1 or 3 = i2 . Note that if 3i1 + 2i2 = 7
then 2 - i1 and 3 - i2 . There is a symmetric restriction in Γ′1 too: if 2 | i′1 or 3 | i′2 theneither 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 anallowed W for Γ(F ) is that the induced WL and WR should be allowed for ΓL (FL ) andΓR (FR ), respectively. This is almost clear from the nature of the deﬁnitions.
There is potentially a problem when (say) AF,L = ∅, since then a new dashed arrowhead
at a boundary vertex of ΓR (FR ) 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 wewill verify that (1) will be automatically satisﬁed.
Lemma 4.2.1. When AF,L = ∅ and ia ̸= 0 for all a ∈ AW,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 ΓL it is suﬃcient to prove that i′ ̸= 0 when
ΓL is star-shaped. Let i1 − 1, . . . , in − 1 be the multiplicitiesof the dashed arrowheads along∏
the edges with decorations d1 , . . . , dn . Denote D := nℓ=1 dℓ . Then, using (3.1.11),
i′ = (1 − n)D +

n
∑
D
ℓ=1

dℓ

iℓ .

Suppose that i′ = 0. Then dℓ divides dDℓ iℓ , hence divides iℓ too, for all ℓ. By the deﬁnitionof allowedness we then know that iℓ = dℓ for at least n − 1 of the iℓ ; say for i2 , . . . , in . Thus
0 = i′ = (1 − n)D + (n − 1)D + Di1 /d1

A. Némethi and W. Veys22and hence i1 = 0, contradicting the assumptions. For the second statement, note thatia = 1 for any a ∈ AF \ AW , while ia ̸= 0 for a ∈ AW \ AF since W is allowed.

A crucial question in our setting is the converse: can we ‘extend’ an allowed W ♭ on
ΓR (FR ) to Γ(F ), that is, can we construct an allowed W on Γ(F ) for which WR = W ♭ ?
It is enough to study this question when ΓL (FL ) is star-shaped, since we can then proceedfurther 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 ΓL (FL ) is star-shaped. Then the mapΨ : {allowed W for Γ(F )} → {allowed W ♭ for ΓR (FR )} : W 7→ WR
is surjective if there is at least one arrowhead in Γ(F ) on the right of e, or the number ofarrowheads on the left of e is diﬀerent from one and two.
Proof. Take an allowed W ♭ for ΓR (FR ). Let i′ −1 be the decoration of the dashed arrowheadat the left of vR .
Let i1 − 1, . . . , in − 1 (n ≥ 2) be the (still to be determined) decorations of the dashedarrowheads 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 vL for ΓL (FL ). Note that j isﬁxed 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 argumentin the proof of Lemma 4.2.1.)
i1 − 1 in − 1 s d1
H..HHs d .
s dn


j−1

i′ − 1



d′

d′

1
sH...

H′
dH
′
n

vL

vR

(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 := nℓ=1 dℓ . Then, cf. (3.1.11), we are searching for i1 , . . . , in with
(4.2.3)i′ = (1 − n)D +

n
∑
D
ℓ=1

dℓ

iℓ .

Since gcdℓ {D/dℓ } = 1, we know that there exist i1 , . . . , in ∈ Z satisfying (4.2.3). We have toverify that this can be done compatibly with the restrictions on the iℓ and j, when exactly
one or two (ordinary) arrowheads are among the legs in ΓL (FL ).
Note ﬁrst that, by (4.2.3), we have for any ℓ = 1, . . . , n that dℓ |i′ if and only if dℓ |iℓ . With
the given assumptions on the arrowheads in Γ(F ), we encounter two cases.
(1) There is exactly one arrowhead on ΓL (FL ), and it coincides with the dashed (j − 1)–arrowhead. Then we suppose that dℓ |i′ for at least n − 1 of the iℓ , say for i1 , . . . , in−1 .
(Otherwise nothing has to be veriﬁed.)(2) There are exactly two arrowheads on ΓL (FL ), and they coincide with the dashed(j − 1)– and (in − 1)–arrowheads. Then we suppose that dℓ |i′ for ℓ = 1, . . . , n − 1.
In each of these cases we take iℓ = dℓ for ℓ = 1, . . . , n − 1 and then, in order to satisfy(4.2.3), we take in given by i′ = Din /dn . This way we thus constructed an allowed W forΓ(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 ΓL (FL ), namely when they coincide with the dashedGeneralized Monodromy Conjecture23(in−1 − 1)– and (in − 1)–arrowheads. To verify allowedness we have to suppose that d|j anddℓ |i′ for ℓ = 1, . . . , n − 2. Then, in order to get an allowed extension we should have thatd = j, but this is not true in general.
Similarly, assume that ΓL (FL ) has exactly one arrowhead which coincides with the dashed(in − 1)–arrowhead. Then in the situation d|j, dℓ |i′ for ℓ ≤ n − 2 but dn−1 - i′ one gets anallowed 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)either d - j or d = j;the only arrowhead coincides with the dashed (in − 1)–arrowheadand dℓ |i′ for ℓ ≤ n − 1.

(a)
(b)

(b) Let Γ(F ) be an arbitrary diagram. Let ΓA be that minimal connected subdiagramof Γ which contains those nodes which either support at least one arrowhead of F , or sit ona (geodesic) path connecting two arrowheads of F , and those boundary vertices which aresupported by these nodes. The connected components of Γ \ ΓA are denoted by {Γj }j∈J ,and each Γj is connected to ΓA at the vertex vj of ΓA . (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 nodeof ΓA 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 Γj can beextended ‘away from vj ’. In order to extend it ‘in the direction of vj ’ 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 toany 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 orr = 2 ordinary arrowheads, equipped with decorations as below. We assume that iℓ = dℓ forℓ = 1, . . . , n − 1 (and in ̸= 0) if r = 1, and iℓ = dℓ for ℓ = 1, . . . , n if r = 2 . Thus W isallowed. When r = 2, we assume also that − Nνvv is not a pole of order 2 of Z(S), that is,
k1
k2
νv
νv
νv
Nv ̸= N1 and Nv ̸= N2 . Then, in all the above situations, − Nv is not a pole of Z(S).
i1 − 1 sH
1 p1 - k − 1..HdH1.s - (N1)
 dnin − 1  s

i1 − 1 in − 1 

k1 − 1
sH
p1 *
(N1 )
*

1

..HdH
s
. 
H
sdn p2Hj
j k2 − 1
H

v

v

(N2 )

Proof. We consider ﬁrst the case r = 2. By (2.3.5) and (2.2.3) we have, with D =(5.1.2)νv =n∑
Dp1 p2ℓ=1dℓiℓ − nDp1 p2 + Dp2 k1 + Dp1 k2

νv = D(p2 k1 + p1 k2 )ℓ dℓ ,and Nv = D(p2 N1 + p1 N2 ),respectively. With our assumptions this simpliﬁes to(5.1.3)∏and Nv = D(p2 N1 + p1 N2 ).

A. Némethi and W. Veys24

The residue of − Nνvv is (up to a factor Nv )
(

−n+

n
∑
dℓ
ℓ=1

iℓ

+

)
p1
p2
p1
p2
+
=
+
.
νv
νv
k1 + sN1 k2 + sN2 (s=− Nv ) k1 − Nv N1 k2 − Nνvv N2

1 k2
This expression being zero is equivalent to Nνvv = pp22Nk11 +p+p1 N2 , which follows from (5.1.3).
When r = 1, we have by (2.3.5) and (2.2.3) thatn∑Dp1
(5.1.4)νv =iℓ − (n − 1)Dp1 + Dk1 and Nv = DN1 ,dℓℓ=1

simplifying with our assumptions to
p1(5.1.5)νv = D( in + k1 )dn

and Nv = DN1 .

k1
Thus Nνvv ̸= N
(since in ̸= 0) and − Nνvv is not a pole of order 2. The fact that its residue is1
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 s0 is a pole of the topological zeta function
Z(f, W ; s), then exp(2πis0 ) is a monodromy eigenvalue of f at some point of {f = 0} (inone 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 := Γπ (X, f, W ). We consider three subcases.
(1) There is a component Fa of {f = 0} such that s0 = − Niaa . Then exp(2πis0 ) is aneigenvalue of f at a point of Fa close to 0 (since all Na -th roots of unity are eigenvalues atsuch a point).
(2) There is no Fa as in (1) and let s0 be a pole of order 1 of Z(f, W ; s). Then s0 = − Nνvv
for some node v, such that the contribution of v to the residue of Z(f, W ; s) at s0 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 s0 is exactly this contribution.
k1 − 1
sH
p1 *
(N1 )
*

d
1
..
..HH
s

.H .
s dn prHj
j kr − 1H
in − 1  
i1 − 1 v(r arrowheads)(Nr )From (3.3) we compute(tNv − 1)r+n−2
ζS (t) = ∏n,
Nv /dℓ − 1)ℓ=1 (t∏and by (2.3.5), denoting D := nℓ=1 dℓ , we have
(5.1.7)(5.1.8)

νv =

r
n
∑
D ∏
(
pj )iℓ + D · (some integer).
dℓ
ℓ=1

j=1

We distinguish three possibilities for r in order to show that exp(2πis0 ) is always a rootof ΛS (t).
• (r ≥ 3) Then, via (5.1.7), exp(2πis0 ) is clearly a root of ΛS (t) = ζS (t).
• (r = 2) Suppose that exp(2πis0 ) is not a root of ΛS (t) = ζS (t). Then, by (5.1.7), νdvℓmust be an integer for all ℓ = 1, . . . , n. By (5.1.8) this is equivalent to dℓ |iℓ for all these ℓ.

Generalized Monodromy Conjecture25

By allowedness we conclude that then iℓ = dℓ for all ℓ. Lemma 5.1.1 then contradicts thatthe residue at s0 is non-zero.
• (r = 1) Suppose that exp(2πis0 ) is not a root of ΛS (t) = ∆S (t). Then, analogously, forat least n − 1 of the numbers i1 , . . . , in we have that νdvℓ must be an integer, or, equivalently,
dℓ |iℓ . Then by allowedness iℓ = dℓ for n − 1 of these numbers, and again Lemma 5.1.1
contradicts that the residue at s0 is non-zero.
We conclude that exp(2πis0 ) is indeed always a root of ΛS (t), hence of ΛΓ (t), and thusthat exp(2πis0 ) is a monodromy eigenvalue of f at 0.
(3) There is no Fa as in (1) and let s0 be a pole of order 2. This implies that s0 = − Nνvv =− Nνww for two nodes v and w, connected by a special edge e. Equivalently, cf. (3.2.4),(5.1.9)s0 = −

νv
k
=−
Nv
N

for the central node v and an arrowhead with decorations (Na , ia ) = (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 s0 = − Nνww
only. Either at least one node w in C has at least three attached (ordinary) arrowheads
in the induced star-shaped subgraph Sw after splicing, and then exp(2πis0 ) is a root ofΛSw (t) = ζSw (t); or all nodes w in C have exactly one or two attached arrowheads in Sw .
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∑n Dp1νv
k1ℓ=1 dℓ iℓ − (n − 1)Dp1 + Dk1
=
=
.
Nv
DN1
N1
∑
This simpliﬁes to nℓ=1 dDℓ iℓ − (n − 1)D = 0, implying that dℓ |iℓ for all ℓ = 1, . . . , n. By
allowedness we then have iℓ = dℓ for say ℓ = 2, . . . , n. Hence the previous equality reduces
to i1 = 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 νv /Nv = k1 /N1 ̸= k2 /N2 . Now (2.2.3) and (2.3.5),
see also (5.1.2), together with (5.1.9) yield∑n Dp1 p2
νv
k1ℓ=1 dℓ iℓ − nDp1 p2 + Dp2 k1 + Dp1 k2==.
Nv
D(p2 N1 + p1 N2 )N1
Suppose that exp(2πis0 ) is not a root of ΛSv (t). By the same arguments as in case (2) we
get that dℓ |iℓ for all ℓ = 1, . . . , n. By allowedness we now have that iℓ = dℓ for all ℓ, andthen the previous equality reduces to
k1
p2 k1 + p1 k2=.
p 2 N 1 + p 1 N2
N1
k1
k2
This is equivalent to N=N(= −s0 ), contradicting that v is an extremity of C.
1
2
We conclude that exp(2πis0 ) is a root of ΛSv (t) for some node v, and hence also ofΛΓ (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.

26

A. Némethi and W. Veys

Start with a diagram Γ(F ) and set the possible ‘eigenvalues of the monodromies at
diﬀerent points and in diﬀerent homologies’:∪
Eig := {λ : ∆1 (λ) = 0} ∪{λ : λNa = 1}.
a∈AF

Then, for any allowed W and pole s0 of the zeta function Z(F, W ; s), we have thatexp(2πis0 ) 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)(t2 − t +1). Theorem 5.1.6
says that if W is allowed and s0 is a pole of Z(s), then exp(2πis0 ) is 1 or a primitive 6-throot 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 ν1 = −13 + 21(i1 −1) + 14(i2 − 1) + 18(i′1 − 1) + 12(i′2 − 1), which is congruent with 3i1 + 2i2 modulo Nv1 = 6.
We wish to get, for example, ν1 ≡ 3 ( mod 6), hence we might take i1 = 1 and i2 = 6.
Assume also that i′1 = i′2 = 1. Then ν1 = 57, ν0 = 8 and ν1′ = 47. Then computing the zetafunction we realize that s0 = −57/6 is a pole, but exp(2πis0 ) = −1 ̸∈ Eig.
(b) It is not hard to ﬁnd divisors W which are not allowed, but such that neverthelesstheir 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 i1 = 1, i2 = 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 ν1 = 6I ′ − 15, ν0 = I ′ − 3 and ν1′ = 7I ′ − 24. Moreover, the zetafunction Z(s) is)(2−1 + I ′ /(i′1 i′2 )
1
1
1
1
+
+
−
1
+
+
+
.
6I ′ − 15 + 6s
7I ′ − 24 + 6s
I′ − 3 + s
s + 1 6I ′ − 15 + 6s 7I ′ − 24 + 6s
In particular, s0 = (15−6I ′ )/6 is a candidate pole of Z(s), such that exp(2πis0 ) = −1 ̸∈ Eig.
The point is that, for any I ′ , the residue of this candidate pole is zero, that is, this is afake candidate pole, not a pole. In particular, for I ′ = 7 we get no contradiction, and forsome 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 i1 = 1, i2 = 3 and I ′ = 7. ThenZ has a pole s0 with exp(2πis0 ) = exp(−2πi/6).
The other root exp(2πi/6) of t2 − t + 1 can be realized by the following allowed form.
Consider i1 = 1, i2 = 1 and I ′ = 7. Then ν1 = −1, ν0 = 0, ν1′ = 1 and
1
1 (
7 )
12
4
−
+
−1+ ′ ′ +.
Z(s) =
6s − 1 s + 1 6s + 1
i1 i2(6s − 1)(6s + 1)
One immediately veriﬁes that all candidate poles are indeed poles. Hence, in fact, the polesof 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).

Generalized Monodromy Conjecture27

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 (muchshorter) 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 ofthis condition, just using their standard properties.
6.1. In this subsection we verify that the standard form is allowed, and hence we reprovethe ‘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) = (C2 , 0).
Proof. We use the minimal embedded resolution π : X̃ → X of f and show that the diagramΓ := Γπ (X, F = div(f ), W = 0) is allowed. Note that thus the decorations ia − 1 = 0 for
all a ∈ AW .
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 ia (= 1),
hence the allowedness condition is satisﬁed again.
It is well known that other star-shaped subdiagrams can arise from at most one connectedpart of Γ, that has the following form.
vr−1
vr 
v1
v2

a
ar 
a
a
r−1
1
2
s
s
s
s
sH ...
...
pr−1
p1
p2
prHH
s

s

s

s

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

a1 s p1s

ak s pkssjk − 1?

sjr − 1?

ar sH...
prHH
s

Here the leg with decoration pr is optional. If it does not occur, we put formally pr = 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 ak > ak−1 pk−1 pk for
k = 2, . . . , r.
For the ﬁrst diagram the allowedness is automatically satisﬁed.
In the spliced star-shaped subdiagrams around the vertices vk , 2 ≤ k ≤ r, it is a priori
possible that ak |jk for some k. If furthermore jk ̸= ak , the allowedness condition would beviolated. We will show however that −ak < jk < 0 for k = 2, . . . , r, and this will ﬁnish theproof.
More precisely, we verify by induction that jk < 0 and |jk | < ak /pk for k = 2, . . . , r. Firstby (3.1.8) we have that j2 = p1 + a1 − a1 p1 . Hence j2 < 0 and |j2 | < a1 p1 < a2 /p2 .
Take now k ∈ {2, . . . , r − 1}. In this case (3.1.11) yields
jk+1 = pk jk + ak − ak pk .

A. Némethi and W. Veys28

The induction hypothesis says that jk < 0 and |pk jk | < ak . Consequently also jk+1 < 0 and|jk+1 | < ak pk < ak+1 /pk+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 [25] for a direct proof).
Corollary 6.1.2. For any plane curve singularity f , if s0 is a pole of the topological zetafunction Z(f ; s), then exp(2πis0 ) 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 theextension 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 ton boundary vertices and r arrowheads whose supporting edges have decorations {dℓ }nℓ=1 and{pℓ }rℓ=1 , respectively. Here r ≥ 1, n ≥ 0, and r + n ≥ 3.
k1 − 1
sH
p1 *
(N1 )
*

d
1
..
..HH
s

. H .
s dn prHj
j kr − 1H
in − 1  
i1 − 1 (Nr )v

Let λ be a root of the Alexander polynomial ΛΓ (t) of Γ. Then there exist inﬁnitely
many (even inﬁnitely many eﬀective) allowed P -divisors W for Γ (corresponding to thedecorated dashed arrows on the diagram) admitting a pole s0 of the topological zeta function
Z(Γ(F, W ); s), such that exp(2πis0 ) = λ.
∏∏Proof. Denote D := nℓ=1 dℓ and P := rℓ=1 pℓ . For any P -divisor W , that is, for any set ofdecorations k1 , . . . , kr , i1 , . . . , in (with the iℓ ̸= 0), we have that Z(s) := Z(Γ(F, W ); s) has
the form
)
(
n
r
∑
dℓ ∑
pℓ
1
2−r−n++,
ν + sNiℓkℓ + sNℓ
ℓ=1

ℓ=1

where
(6.2.2)

N =D

r
∑
P
ℓ=1

pℓ

Nℓ

and ν = D

r
∑
P
ℓ=1

pℓ

kℓ + P

n
∑
D
ℓ=1

dℓ

iℓ − (r + n − 2)DP.

If the candidate pole −ν/N is equal to some kℓ /Nℓ , 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 +

n
∑
dℓ
ℓ=1

iℓ

+

r
∑
ℓ=1

pℓ
.
kℓ − (ν/N )Nℓ

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

Generalized Monodromy Conjecture29

We consider three cases (depending on the value of r) for the roots of ΛΓ (t), which is
given by(tN − 1)r+n−2∏n,N/dℓ − 1)ℓ=1 (texcept when r = 1, where we must multiply this expression by tN1 − 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/dℓ )-th roots of unity
simultaneously for all ℓ = 1, . . . , n. In other words all exp(2πi Nu ) for which u ̸≡ 0 mod dℓ
for at least one dℓ .
• (r = 1) Then its roots are all N -th roots of unity that are not (N/dℓ )-th roots of unity
simultaneously for (at least) n − 1 indexes ℓ = 1, . . . , n. In other words all exp(2πi Nu ) for
which u ̸≡ 0 mod dℓ for at least two diﬀerent dℓ .
Fix a root λ = exp(2πi Nu ) of ΛΓ (t). Since the numbers p1 , . . . , pr , d1 , . . . , dn are pairwise
coprime, there exist integers k1 , . . . , kr , i1 , . . . , in (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 dℓ - iℓ for at least one or at least two indexes ℓ, respectively. Hence in each case theconstructed W is allowed for Γ.
Since we can choose the numbers k1 , . . . , kr , i1 , . . . , in 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 s0 = −ν/Nis then a pole of Z(s) satisfying exp(2πis0 ) = λ.
Remark 6.2.3. Assume that above r = 1 with p1 = 1 and n = 2. Set s0 = −ν/N . The
fact that λ = exp(2πis0 ) is a root of ΛΓ (t) is equivalent to d1 - i1 and d2 - i2 . On the other
hand, R = 0 if and only if (d1 − i1 )(d2 − i2 ) = 0. Hence, if λ is a root, then s0 is a pole ofZ(s) (for any allowed W ); see also (3.4) in [20].
6.2.4. Now we return to plane curve singularities and we target that subdiagram of theminimal 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 = Γπ (X, F ) of a plane curve germ, determining a P –divisor F . Herer ≥ 2 and the leg with decoration pr is optional.
Fix k ∈ {1, . . . , r − 1}, consider the star-shaped subdiagram Sk = Γ(Fk ) around vk andﬁx a root λ of the Alexander polynomial ΛSk (t). Then there exist inﬁnitely many allowed
divisors Wk for Sk , such that if (Nk , νk − 1) denote the decorations of vk as above associatedwith Sk and Wk , then(0) s0 = −νk /Nk is a pole of Z(Fk , Wk ; s),(1) exp(2πis0 ) = λ, and(2) Wk can be extended to an allowed divisor on the whole diagram Γ.
The subdiagram of Γ is
s

v1
a1 s
p1

v2
a2 s
p2

s

s

vr−1
vr 

a
a
r−1
r
s
sH ... . . .

...
pr−1
prHH
s

s

with spliced star–shaped subdiagrams (where we insert the wanted Wk ’s too)A. Némethi and W. Veys

30

s

1−1
a1 s ℓ...
p1?

j1 − 1s -i1 − 1k −1
ak s ℓ( pNak ) . . . s
k k
pk

s
?

s -

jk − 1

ik − 1?

jr − 1ar s  ...
HprHHs -ir − 1

Proof. From Lemma 6.2.1 we know that inﬁnitely many Wk satisfying (0) and (1) exist, but
in order to satisfy also (2), we will specify choices. Write λ as λ = exp(2πi(u/Nk )), where(since it is a root of ΛSk (t)) ak - u and pk - u. We choose the decorations ik , jk and ℓk suchthat
νk = ik ak + jk pk + (ℓk − 1)ak pk ≡ u(mod Nk ),and, moreover(6.2.6)0 < ik ak + jk pk − ak pk < ak pk .

Note that this is possible since ak pk |Nk , and that we can choose ℓk freely mod Nk /(ak pk ).
By Remark 6.2.3 we know that −νk /Nk is a pole of Z(Fk , Wk ; s).
We claim that we can choose inductively {im , ℓm } for m = k + 1, . . . , r − 1 such that|jm+1 | < am pm for m = k, . . . , r − 1. Then for these m this yields |jm+1 | < am pm < am+1
and thus am+1 - jm+1 . By Remark 4.2.4 this ensures that we can extend Wk further toobtain an allowed divisor W on the whole diagram Γ.
We now prove the claim. By (3.1.11) we have
(6.2.7)ℓm = im+1 + pm+1 (ℓm+1 − 1)for m = k, . . . , r − 2, and(6.2.8)

jm+1 = im am + jm pm − am pm

for m = k, . . . , r − 1. In particular, we know already from (6.2.6) that |jk+1 | < ak pk . Whensome {im , ℓm } is constructed we take each time im+1 and ℓm+1 in (6.2.7) such that 1 ≤im+1 ≤ pm+1 . Then it follows from (6.2.8) and the inductive argument (and the positivityof the edge determinant) that indeed |jm+1 | < am pm 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 pointof {f = 0}. Then there exist inﬁnitely many allowed P -divisors W for (X, div(f )), and foreach of them a pole s0 of the topological zeta function Z(f, W ; s) such that exp(2πis0 ) = 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∈AF Na Fa , this means that λ is a Na -th root of unity for some Na .
Fix such an a ∈ AF ; so λ = exp(2πi(−u/Na )) for some (ﬁxed) u ∈ {1, . . . , Na }.
HH.. Hs d - ia − 1
. 
(Na )


v

Consider the star-shaped subdiagram Sv = Γ(Fv ) around the vertex v, to which thearrowhead a is attached. Choose an allowed P -divisor Wv for Sv with decoration ia ≡ u
mod Na , such that moreover ia /Na ̸= ia′ /Na′ for all (eventual) other arrowheads a′ on Sv .

Generalized Monodromy Conjecture

31(Here we have inﬁnitely many such choices.) Then either −ia /Na is a pole of order two ofZ(Fv , Wv ; s), or the contribution of a to this zeta function isd(νv + sNv )(ia + sNa )and hence the residue of −ia /Na is nonzero.
We surely can extend Wv 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 ia . (For this see the proof of Proposition 4.2.2;
with the notation of that proof, from ΓR only i′ was used.) If −ia /Na is not a pole of order
two and ia /Na would be equal to some value νw /Nw , w ∈ (N ∪ AF ) \ {a}, then we can add
to ia some multiple of Na in order to avoid this. This way we are sure that −ia /Na is apole of Z(f, W ; s).
(2) Suppose now that λ is not as in (1); hence it is a root of the Alexander polynomialΛΓ(F ) (t). By Proposition 3.3.3 there is at least one node w ∈ N such that λ is a root ofΛSw (t), where Sw = Γ(Fw ) is the star-shaped subdiagram around w. By Lemma 6.2.1 and
Proposition 6.2.5 there exist inﬁnitely many allowed P -divisors Ww for Sw , and for each ofthem a pole s0 of Z(Fw , Ww ; s) such that exp(2πis0 ) = λ and Ww can be extended to anallowed divisor W on the whole diagram Γ(F ). Indeed, the possible obstruction to extend
Ww , as described in Remark 4.2.4, is removed in Proposition 6.2.5.
If s0 is a pole of order two of Z(Fw , Ww ; 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 thefollowing situation occurs: s0 is not a pole of order two, s0 = −νw /Nw 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 s0 ) Rw ̸= 0, and w∈N ′ Rw = 0. In this case s0 is nota pole of Z(s), although it is a pole of several Z(Fw , Ww ; s).
i1 − 1(N1 ) YH
Y

d1..HHs d .
in − 1 

 dn
(Nn )

ΓR
i−1
(N )

k−1 



sH ...

HH

Take an ‘extreme’ node v ∈ N ′ (the node of the left diagram above), meaning that it isa boundary vertex of the full subdiagram of Γ generated by N ′ . Consider the star-shaped
subdiagram Sv 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 ΓR , obtained after splicing Γ along e.
(It is not necessary to have arrowheads at all the legs of Sv , as it is indicated in the above
diagram; in those∏cases we put formally Nℓ = 0 or N = 0.)
Denote D := nj=1 dj . By (2.3.5) and (3.1.11) (or from (6.3.2) below) we know that νv∑
depends only on i and k = nj=1 (D/dj )ij − (n − 1)D, and not on the actual (separate)
values of i1 , . . . , in . The residue-contribution Rv though depends on these values.
We claim that we can modify i1 , . . . , in keeping k (and hence νv ) ﬁxed, but changing Rv ,such that the newly created divisor (determined by these new i1 , . . . , in ) has the following
properties: it agrees with the old P –divisor on ΓR , and can be extended from Sv ∪ ΓR further‘to the left’ to a new allowed P -divisor W̃ on the whole diagram.
∑In that way the new value becomes v∈N ′ Rv ̸= 0. If ‘on the left’ there are no nodes
v ′ with ‘new value’ νv′ /Nv′ = s0 , we are done
∑ since then this sum is the (total) residue ofs0 for Z(s). We are still done if the sum of v∈N ′ Rv and all new residue-contributions ofthese 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.

A. Némethi and W. Veys

32(i) Since there are at least two nodes in N ′ , we can always assume that there is an ordinary
arrowhead at the leg of Sv 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 ΓR ‘unconditionally’ to the left.
More precisely, the speciﬁc situation/problem of Proposition 6.2.5 will never occur; we canalways simply follow the procedure in the proof of Proposition 4.2.2.
(ii) We must be sure that we can modify i1 , . . . , in while extending from ΓR , considering
the extension procedure described in the proof of Proposition 4.2.2. In this procedure there
is no room to modify i1 , . . . , in only if (after renumbering) iℓ must be chosen as iℓ = dℓ forℓ ≤ n − 1. But in this case we would have that Rv = 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)νv =n∑d(D/dj )ij + Di − (n − 1)dD = dk + Di

j=1

and
Nv =

n
∑

d(D/dj )Nj + DN.

j=1

We have further thatNv Rv = 1 − n +

n
∑
j=1

dj
d
+
.
ij − (νv /Nv )Nji − (νv /Nv )N

Case n ≥ 3. We replace the triple (i1 , i2 , i3 ) by (i1 + xd1 , i2 + yd2 , i3 − (x + y)d3 ) wherex, y ∈ Z. Then νv does not change, but the three corresponding terms in Nv Rv are replaced
by
d1
d2
d3
+
+
.
i1 + xd1 − (νv /Nv )N1 i2 + yd2 − (νv /Nv )N2 i3 − (x + y)d3 − (νv /Nv )N3
It is easy to see that this expression is not constant as function in x and y; hence we canchoose appropriate x and y in Z such that the ‘new’ Rv is diﬀerent from the original one.
(Note that divisibility of iℓ by dℓ does not change, so we don’t destroy allowedness.) We
then extend this new P -divisor from Sv ∪ ΓR further to an allowed divisor on the wholediagram.
Case n = 2. We replace the pair (i1 , i2 ) by (i1 + xd1 , i2 − xd2 ) where x ∈ Z. Again νv does
not change, and now the two corresponding terms in Nv Rv are replaced by
d1
d2
+
.
i1 + xd1 − (νv /Nv )N1 i2 − xd2 − (νv /Nv )N2
When this expression is not constant in x, we conclude as above. It is constant in x if andonly if it is identically zero if and only if
νv
d1 i2 + d2 i1 −(d1 N2 + d2 N1 ) = 0.
Nv
Suppose this identity holds. Then the formulas for νv and Nv above easily yield thatνv /Nv = (i − d)/N . But then the (original) Nv Rv would be equal to−1 +contradicting the assumption.

d= 0,i − (νv /Nv )N

Generalized Monodromy Conjecture

33

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 torecall 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 edgee (as in the following diagram, see also (3.1.5)), such that AF,L = ∅, d′ is in the semigroup
Se generated by {lew }w where the index w runs over all the boundary vertices of ΓL . (Forthe deﬁnition of lew see (3.1.2).)
HH
.. Hs
. 




e

d′ 
sH...
H

H

This condition appears naturally in the context of splice quotient singularities, introduced
by Neumann and Wahl [21, 22]. For some special diagrams this condition is automaticallysatisﬁed. For example, if Γ represents a rational germ (which in the context of IHS germs isequivalent with the fact that Γ represents either the smooth or the E8 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 (automaticallyGorenstein) singularity. These facts follow from the ‘End Curve Theorem’ [23, 24].
It is convenient to denote the subsemigroup of N generated by g1 , . . . gt by S⟨g1 , . . . , gt ⟩.
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 d1 , . . . , dn be pairwise coprime positive integers, and denote D :=∏n
j=1 dj . Then the following two facts hold.
(a) There exist no positive integers mj such that
n
∑
j=1

mj

D
= (n − 1)D.
dj(b) If d ∈ S⟨D/d1 , . . . , D/dn ⟩, d > 0 and d|D, then (D/dj ) | d for some j ∈ {1, . . . , n}.
Proof. (a) In such an equality we would have that dj |mj for all j = 1, . . . , n. But then theleft hand side would be at least nD. In (b), by assumption, we can write d in the form
(7.2.2)

d=

n
∑
j=1

mj

D
,
dj

∏
where all mj are nonnegative integers, and also d = nj=1 d¯j with d¯j |dj for all j. Since∏
the dj are pairwise coprime, (7.2.2) shows that d¯j |mj for all j. Writing d as d¯j ℓ̸=j d¯ℓ ,we conclude that d divides mj D/dj for all j. If at least two of the numbers mj would be
nonzero, say m1 ̸= 0 and m2 ̸= 0, we obtain the contradictionDDd ≥ m1 + m2 ≥ d + d.
d1
d2

A. Némethi and W. Veys

34

Hence exactly one mj is nonzero, implying then that (D/dj ) | d.

Recall that a diagram Γ is called minimal if all the decorations dve are strictly greaterthan 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) andminimal in the above sense. Let e be an edge connecting two nodes such that AF,L = ∅, andset i′ − 1 the multiplicity of the induced dashed arrowhead at vR after splicing Γ along e asin (3.1.5).
ΓL
ΓR
d1
HH
e
.
.  splicing- H.HHs
.. Hs
d′ 
d′ 
′−1 
s
s
s
.
..
.
i
(M
)
. 
. 
H
H .



dn

HH

HH

(1) Then i′ < 0.
(2) Assume that the semigroup condition is satisﬁed in Γ(F ) (at least for the edge e andthe edges eL sitting in ΓL ). (This means that d′ is in the semigroup Se generated by {lew }wwhere the index w runs over all the boundary vertices of ΓL , and there are similar inclusions
for all edges eL of ΓL .) Then −i′ ̸∈ Se . Hence, d′ - i′ .
∏Proof. (1) Set D := nj=1 dj . We proceed by induction on the number of nodes in ΓL .
∑Suppose ﬁrst that vL is the only node of ΓL . By (3.1.8) we have that i′ = nj=1 D/dj −(n − 1)D. When n = 2, ∑this is d1 + d2 − d1 d2 and thus negative. When n > 2, then byminimality dj ≥ 2, hence
1/dj ≤ n/2 < n − 1, therefore i′ < 0 again.
We suppose now that ΓL contains at least two nodes. From (3.1.8) we can write i′ as


n
∑
∑
D

(2 − δw )ℓej w  − (n − 1)D,
i′ =
dj
j=1

w∈Vj

where for j = 1, . . . , n the set Vj consists of the vertices of ΓL connected (geodesically) to vL
through the edge ej with weight dj , and ℓej w is the product of all the decorations adjacentto, but not on, the path from w to ej . For all j this sum is either equal to 1 (when ej ends
at a boundary vertex), or negative by induction. Since at least one sum is negative, we
conclude that i′ < 0.
(2) Denote by BL and NL the boundary vertices and nodes, respectively, in ΓL . We willshow the following claim. Let∑∑(2 − δw )ℓew ,mw ℓew +
I :=
w∈BL

w∈NL

where all mw ∈ Z>0 . Then I ̸= 0 and, if I < 0, then −I ̸∈ Se .
Since i′ = I when all mw = 1, and i′ < 0 by part (1), the statement then follows.
We now prove the claim, again by induction on the cardinality of NL . If NL = {vL } thenn∑DI=mj − (n − 1)D,dj
j=1∑
and this is nonzero by Lemma 7.2.1(a). If I < 0 and −I ∈ Se , then nj=1 kj dDj = (n − 1)Dfor some positive integers kj , contradicting again Lemma 7.2.1(a).
Let now NL have at least two elements. Suppose again that I < 0 and −I ∈ Se . Thenanalogously we get that∑∑kw ℓew =(δw − 2)ℓew
w∈BL

w∈NL

Generalized Monodromy Conjecture

35

for some positive integers kw . We separate vL (with δvL = n + 1 and ℓevL = D) on the righthand side, and rewrite this equality as


n
∑D ∑
∑

kw ℓej w +(2 − δw )ℓej w  = (n − 1)D,(7.2.4)

dj
(j)
(j)
j=1

(j)

w∈BL

w∈NL

(j)

where BL ∪ NL = Vj . By induction the square bracket is non–zero. Moreover, (7.2.4)
shows that dj divides the j-th square bracket for all j. Applying the induction hypothesison all these terms (and the assumption dj ∈ Sej ) 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 kw replaced by the original mw . 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 ΓL , 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 Riemannsurface. Let its ﬁrst Betti number be µ (the Milnor number). Clearly, µ is even. Since ΓLis 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 Se isexactly 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 ΓL (as our (2) does)
one needs some restriction about ΓL : for example, if we have two nodes, the second one in
V1 , and d1 = 1, then Se = 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 thatthe minimal embedded resolution diagram Γπ (X, F ) satisﬁes the semigroup condition. Thenthe divisor W = 0 is allowed for the pair (X, F ).
Proof. Denoting by π this minimal embedded resolution, we will show that the diagramΓ = Γπ (X, F, W = 0) is allowed. Note that thus the decorations ia − 1 = 0 for all a ∈ AW .
Recall again that on any star-shaped subdiagram of Γ without boundary vertex theallowedness 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 ia (= 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′ − 1is the decoration of the constructed dashed arrow attached to that boundary vertex, andd′ 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 toguarantee the allowedness of W = 0.

A. Némethi and W. Veys

36

Corollary 7.3.3. Let (X, 0) be a Gorenstein IHS germ, with nowhere vanishing 2-form ω0
on X \ {0}. Let f be a function germ on (X, 0) such that the minimal embedded resolution
diagram Γπ (X, f ) satisﬁes the semigroup condition.
If s0 is a pole of the topological zeta function Z(f ; s) = Z(div(f ), W = 0; s), thenexp(2πis0 ) 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 [25], Rodrigues showed without requiring the semigroupcondition that even in the singular setting exp(2πis0 ) is a monodromy eigenvalue of fprovided that the pole s0 satisﬁes s0 ≤ 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 allits 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 ofplane 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 basedon the following technical proposition.
Proposition 7.4.1. Let (X, 0) be an IHS germ and f an analytic function on X, such thatthe minimal embedded resolution diagram Γ = Γπ (X, f ) satisﬁes the semigroup condition.
With the notation of Remark 4.2.4(b), let us ﬁx a node v0 in ΓA , and another node vm notin ΓA at ‘distance’ m ≥ 1 from v0 . Consider the diagram Γ0 given below obtained from Γ
by cutting via splice–decomposition all the nodes not sitting on the geodesic path connecting
v0 with vm . Here the legs with decorations d0,2 , . . . , d0,n0 are optional, and all boundary
vertices are either original boundary vertices of Γ, or are obtained after splicing.
vkv0
H.. HHs d0,1
. Ad0,n0 Ad0,2
s . . . As...

dk s dk,1dk,nk A dk,2

vm
...

 A
s . . . As

dm s dm,1dm,nm A dm,2s A
s . . . As

Splice this diagram at the edge (vk−1 vk ), 0 < k ≤ m, and denote that splice component
which contains vk , . . . , vm by Γk = Γk (Fk ). Let the decoration of the dashed arrowhead notin Γ1 , associated with the splicing along (v0 v1 ), be i0,1 − 1 (see the picture below).
Fix a root λ of the Alexander polynomial ΛΓm (t) associated with the star–shaped diagramΓm . Then there exist inﬁnitely many allowed divisors Wm for Γm , such that if (Nm , νm − 1)denote the decorations of vm as above associated with Γm and Wm , then(1) s0 = −νm /Nm is a pole of Z(Fm , Wm ; s), with exp(2πis0 ) = λ, and(2) Wm extends to an allowed divisor on Γ1 , such that d0,1 - i0,1 .
Moreover, inﬁnitely many of these allowed (extended) Wm on Γ1 , 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.

Generalized Monodromy Conjecture37

7.4.2. We ﬁx notations for the wanted Wm and its extensions on the spliced star–shaped
subdiagrams of Γ1 . Moreover, we also consider the decoration i0,1 which is part of a potential
extension to Γ0 , but it is completely determined by the extension on Γ1 .
v0

v1

HH
0,1 − 1.. Hs d0,1 s i.Ad0,n0 Ad0,2s . . . As

i1 − 1s

d1 s d1,1

d1,n1 A d1,2

i1,1 − 1

s -

...

 A
s . . . As?

i1,n1 − 1?

i1,2 − 1∏ k∏ k
We also set Dk := nℓ=1
dk,ℓ and Dk∗ := nℓ=2
dk,ℓ (1 ≤ k ≤ m). Note that nk ≥ 2.
The semigroup condition for Γ implies that for any k > 0 one has∗ D⟨D⟩D∗ Dk+1 Dk∗ Dk+1
Dk Dk∗ Dk+1
k+2
k
(7.4.3)dk−1,1 ∈ S,...,,,..., k,,···dk,2dk,nk dk+1,2
dk+1,nk+1
dk+2,2The wanted divisor will be constructed by induction. From (6.2.1) we know that inﬁnitely
many Wm , even inﬁnitely many eﬀective Wm , satisfying (1) exist (see also 7.4.7). Here Wm
identiﬁes νm by(7.4.4)

νm = im−1,1 dm + im Dm .

Then, we analyze how an allowed divisor Wk+1 from Γk+1 can be extended over Γk . Alongthis procedure we will use the following identities ‘around vk ’ satisﬁed by any extension:∑(7.4.5)
ik+1 = −(nk − 1)dk Dk∗ + ik Dk∗ +ik,ℓ dk Dk∗ /dk,ℓ ,ℓ≥2(7.4.6)

ik−1,1 = −(nk − 1)Dk +∑ik,ℓ Dk /dk,ℓ .

ℓ≥1

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 Wk is an allowed divisor on Γk for some k ≥ 1. The decorations of Wk will
distinguish the nodes as follows. For some k ′ ∈ {k, . . . , m}, the node vk′ is called ﬂexible if
there are at least two indexes ℓ ∈ {1, . . . , nk′ } for which dk′ ,ℓ - ik′ ,ℓ . For k ′ ∈ {k, . . . , m − 1},
if vk′ is not ﬂexible, but dk′ ,ℓ = ik′ ,ℓ for all ℓ ∈ {2, . . . , nk′ } 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 Wm which satisﬁes (1) and is ﬂexible at vm . We
search for im , im,1 , . . . , i1,nm such that they satisfy the allowedness at vm , (7.4.4), (7.4.6) fork = m, and exp(−2πiνm /Nm ) is root of the Alexander polynomial ΛΓm (t). Since Dm | Nm ,the last condition implies that
(7.4.8)dm,ℓ - νm for at least two indexes ℓ ∈ {1, . . . , nm }.

We proceed as follows. For any νm with (7.4.8) we ﬁnd im−1,1 and im satisfying (7.4.4).
mThis is possible since gcd(dm , Dm ) = 1. Then we ﬁnd integers {im,ℓ }nℓ=1satisfying (7.4.6).
Dm
This, again, is possible since gcdℓ ( dm,ℓ ) = 1. Since dm,ℓ | νm ⇔ dm,ℓ | im−1,1 ⇔ dm,ℓ | im,ℓ ,by (7.4.8) we have that dm,ℓ - im,ℓ for at least two indexes, hence Wm is ﬂexible at vm .

A. Némethi and W. Veys38

7.4.9. Next we analyze the possibilities how one can extend divisors. Consider an allowed
divisor Wk+1 on Γk+1 (1 ≤ k < m). Note that it also determines ik,1 by (7.4.6). Extending
over vk means that we already know everything over Γk+1 and ik,1 , and we are searchingfor ik and {ik,ℓ }ℓ≥2 which satisfy the allowedness condition at vk and the identity (7.4.5).
The divisor Wk+1 and the decorations of Γ0 (F ) contain all the divisibility information,
like dk,ℓ divides ik,ℓ or not, for any extension Wk on Γk . Indeed, ik,1 and dk,1 are determinedby Γ0 and Wk+1 , and the divisibility conditions dk,ℓ - ik,ℓ (ℓ ≥ 2) are determined by (7.4.5),
since dk,ℓ - ik,ℓ ⇔ dk,ℓ - ik+1 . Hence, several crucial divisibility properties of an extension
Wk on Γk are already decided at the level of its restriction Wm on Γm . This makes theinductive construction of Wk , staring from Wm ‘global’ and diﬃcult.
In order to guarantee the existence of such an extension Wk , we will use two types ofcriteria: Wk+1 satisﬁes either dk,1 - ik,1 or Dk∗ | ik+1 (see (4.2.5)).
If Dk∗ | ik+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 vk will be rigid: dk,ℓ =ik,ℓ for ℓ ≥ 2. Moreover, (7.4.5) and (7.4.6) read as
ik = ik+1 /Dk∗ and ik−1,1 = ik,1 Dk∗ .

(7.4.10)

If dk,1 - ik,1 , then for any extension Wk the node vk is either ﬂexible or rigid (and the typeis decided already at the level of Wk+1 ); vk is rigid if and only if additionally Dk∗ | ik+1 , thecase discussed before. If vk 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 Wk at the closest ﬂexible node’, itwill satisfy the inductive criteria dk−1,1 - ik−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 Wk on Γk inductively as follows.
Assume ﬁrst that vk is rigid, but at least one node of (Γk , Wk ) is ﬂexible. Let k ′ > k bethat ﬂexible node for which vk , . . . , vk′ −1 are all rigid. We modify Wk such that we keep
unmodiﬁed the restriction on Wk′ +1 and ik′ ,1 . We ﬁx some ℓ ≥ 2 such that dk′ ,ℓ - ik′ ,ℓ .
Then we replace ik′ ,ℓ into ik′ ,ℓ + tdk′ ,ℓ , t ∈ Z, but keep all other ik′ ,ℓ ’s. Moreover, modify
ik′ 7→ ik′ − tdk′ too. Then ik′ +1 and ik′ ,1 will stay ﬁxed.
This is the set of modiﬁcations we will refer to, and for strict divisors we impose thefollowing properties. First, we assume that for all the possible modiﬁcations, the value
ik′ is multiple of Dk∗′ −1 . Then, all these modiﬁcations can be extended by a rigid vk′ −1
to Γk′ −1 . Then we run again all the modiﬁcations (at vk′ ) and we assume that for all ofthem Dk∗′ −2 | ik′ −1 . Then, again, all of them can be extended. We continue this, at thevery end asking Dk∗ | ik+1 for all the modiﬁcations. If all these conditions are satisﬁed for
Wk then in all its modiﬁcations W̄k the nodes vk , . . . , vk′ −1 will be rigid, and we call Wk
strict. The strictness guarantees that when we run all the modiﬁcations at the level of vk′ ,all the divisors can be extended to some Wk . (Otherwise it might happen that for somemodiﬁcation and at some vertex both dk′′ ,1 - ik′′ ,1 and Dk∗′′ | ik′′ +1 fail.)
From (7.4.10) we get(7.4.12)

ik =

ik′
,
∗
Dk · · · Dk∗′ −1and

ik−1,1 = ik′ −1,1 Dk∗ · · · Dk∗′ −1 .

Since ik′ −1,1 7→ ik′ −1,1 + tDk′ , the modiﬁcations induce(7.4.13)

ik 7→ ik −

tdk′
,
∗
Dk · · · Dk∗′ −1and

ik−1,1 7→ ik−1,1 + tDk∗ · · · Dk∗′ −1 Dk′ .

Generalized Monodromy Conjecture

39

If vk is ﬂexible then Wk is strict by deﬁnition. In fact the above discussion is valid in thiscase too with k ′ = k. In particular, the set of modiﬁcations is given by ik,ℓ 7→ ik,ℓ + tdk,ℓfor the chosen ℓ and keeping the other ik,ℓ ’s, ik 7→ ik − tdk , ik−1,1 7→ ik−1,1 + tDk .
If we run the above modiﬁcation for the divisors Wm constructed in (7.4.7), then νm stays
stable, hence if the restriction of some Wk to Γm satisﬁes (1), then all its modiﬁcations keepsatisfying (1).
In our procedure we consider only strict allowed divisors. They will be constructed inductively starting from the strict divisors Wm constructed in (7.4.7). The inductive statement
we prove is the following: for any 1 ≤ k ≤ m there exists a strict allowed divisor Wk on Γksatisfying dk−1,1 - ik−1,1 , and (1) on Γm .
The proof of the inductive step breaks into two parts.
(a) If the above properties are true for some strict Wk+1 on Γk+1 then deﬁnitely it canbe extended to an allowed divisor Wk , but this is not necessarily strict. We prove that by agood choice of one of its modiﬁcations, that divisor has a strict extension (not necessarily
satisfying dk−1,1 - ik−1,1 ).
(b) If Wk 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 dk−1,1 - ik−1,1 .
Note that part (b) provides the main inductive statement for m = 1 too. Indeed, by(7.4.7) a strict divisor Wm 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 Wk+1 is a strict divisor on Γk+1 satisfying (1) and dk,1 - ik,1 . We consider
all the modiﬁcations W̄k+1 of Wk+1 as in (7.4.11), and we distinguish the next two cases.
First, suppose that there is no W̄k+1 (with or without dk,1 - ik,1 ) for which Dk∗ - ik+1 .
Then we extend Wk+1 by a rigid node. The extended divisor Wk will be strict.
Second, we assume that there exist some W̄k+1 with Dk∗ - ik+1 . The problem is that it
might happen that in the new situation dk,1 - ik,1 fails, and the extension is not guaranteed.
We claim that the two conditions dk,1 - ik,1 and Dk∗ - ik+1 can be obtained simultaneously
by some W̄k+1 . Then we extend this new W̄k+1 to get a strict Wk with ﬂexible vk .
Let us prove now the above claim.
Recall that dk,1 - ik,1 . If Dk∗ - ik+1 for Wk+1 then we are done. Similarly, if dk,1 - ik,1 forW̄k+1 then again we are done. Otherwise, by (7.4.13) we must havedk′∗
(7.4.15)
dk,1 - δ := Dk+1
· · · Dk∗′ −1 Dk′ and Dk∗ - ∆ := ∗.
Dk+1 · · · Dk∗′ −1
We consider the modiﬁcations for t = 1, 2, 3. Then either we get a wanted pair or we will
have simultaneously{{{{
dk,1 | ik,1 − 3δdk,1 - ik,1 − 2δdk,1 | ik,1 − δdk,1 - ik,1
Dk∗ - ik+1 + 3∆.
Dk∗ | ik+1 + 2∆
Dk∗ - ik+1 + ∆Dk∗ | ik+1
This implies dk,1 | 2δ and Dk∗ | 2∆. This together with (7.4.15) implies that both dk,1 andDk∗ should be even. This is not possible since dk,1 and Dk∗ are relative prime.
7.4.16. Finally we prove part (b) of the inductive step (7.4.11).
Assume that Wk is a strict allowed divisor on Γk such that its restriction satisﬁes (1). If
vk is rigid we will use all the notations of (7.4.11), where vk′ is the closest ﬂexible node to
vk . In fact, these notations can also be used when vk is ﬂexible, with the convention k ′ = k.
We have to show that for some modiﬁcation of Wk one has dk−1,1 - ik−1,1 . We assume
that this is not the case, that is, for all modiﬁcations of Wk at vk′ one has(7.4.17)dk−1,1 | ik−1,1 = (ik′ −1,1 + tDk′ ) · Dk∗ · · · Dk∗′ −1 ,

A. Némethi and W. Veys

40

and we wish to get a contradiction.
∏
For ℓ ∈ {2, . . . , nk } set aℓ := gcd(dk−1,1 , dk,ℓ ), A∗ := ℓ≥2 al and d′k−1,1 := dk−1,1 /A∗ .
About d′k−1,1 we wish to prove two facts. First, clearly∗d′k−1,1 | (ik′ −1,1 + tDk′ ) · Dk+1
· · · Dk∗′ −1 .

(7.4.18)
The second one is(7.4.19)d′k−1,1 ∈ S

⟨D

k+1

dk+1,2,...,

Dk+1
dk+1,nk+1

,

∗ D
⟩
Dk+1
D∗ Dk+2
k+2, . . . , k+1,··· .
dk+2,2dk+2,nk+2

The semigroup involved above is the semigroup associated with that diagram which is
obtained from Γ0 by deleting the star–shaped subdiagram around vk . (In fact, (7.4.18) can
also be interpreted in this way.) The proof of (7.4.19) runs as follows. Write∑∑ Dk+1∑ Dk+2Dk∗
dk−1,1 =mℓ+ Dk∗ ·nℓ
+ Dk∗ Dk+1·
n′ℓ
+ ··· .
dk,ℓ
dk+1,ℓdk+2,ℓ
ℓ≥2

ℓ≥2

ℓ≥2

Then aℓ | mℓ , hence A∗ | mℓ Dk∗ /dk,ℓ too, for any ℓ ≥ 2. In particular, d′k−1,1 belongs to∗∗S⟨dk,1 , Dk+1 /dk+2,1 , . . . , Dk+1 /dk+1,nk+1 , Dk+1
Dk+2 /dk+2,1 , . . . , Dk+1
Dk+2 /dk+2,nk+2 , · · · ⟩.

But dk,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 dk−1,1 can be continued. In the second step we set∗ ). Dividing consecutively by the corresponding divisor ofd′′k−1,1 := d′k−1,1 / gcd(d′k−1,1 , Dk+1
Dk∗ , . . . , Dk∗′ −1 , from dk−1,1 we get d¯k−1,1 with the following properties:(7.4.20)(7.4.21)d¯k−1,1 | ik′ −1,1 + tDk′ , or, equivalently, d¯k−1,1 divides both ik′ −1,1 and Dk′ ,
⟨D ′
Dk′ ⟩
k
d¯k−1,1 ∈ S,...,.
dk′ ,1dk′ ,nk′

Now (7.4.21) together with d¯k−1,1 | Dk′ , via Lemma 7.2.1(b), implies that for some ℓ0 ∈{1, . . . , nk′ } one has Dk′ /dk′ ,ℓ0 | d¯k′ −1,1 . This with d¯k−1,1 | ik′ −1,1 implies that Dk′ /dk′ ,ℓ0 |ik′ −1,1 . Then the formula (7.4.6) for ik′ −1,1 implies that dk′ ,ℓ | ik′ ,ℓ for ℓ ∈ {1, . . . , nk′ } \ {ℓ0 },which contradicts the fact that vk′ is ﬂexible.
7.4.22. Now we verify that the above construction provides inﬁnitely many divisors Wk ateach step k. Indeed, when k = m then in (7.4.7) there are inﬁnitely many possibilities for
im,1 to realize a desired Wm . Furthermore, in the extension procedure, this initially chosen
Wm is modiﬁed, but the original im,1 is never touched.
Moreover, we can obtain this way inﬁnitely many (extended) eﬀective divisors Wm . Indeed, in (7.4.7) we can choose the value νm freely modulo Nm , in particular positive andlarge enough with respect to all decorations along the edges of Γ. Then also im−1,1 andmfurther {im,ℓ }nℓ=1
can be chosen ‘large’. In fact, if νm is large enough, then all constructed(and modiﬁed) multiplicities along dashed arrows, on Γ1 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 thatthe minimal embedded resolution diagram Γπ (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 s0 of the
topological zeta function Z(f, W ; s) such that exp(2πis0 ) = 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.
Generalized Monodromy Conjecture

41

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 toany allowed divisor.
We re-consider Example 3.1.15, but taking F as the divisor corresponding to the uniquearrowhead with multiplicity N (instead of multiplicity 1).
i1 − 1 

s

(3N )

2 s7
(6N ) 3

1 s1
(N )

7 s2
(6N ) 3

(2N )

(N )

(2N )

s

?

?

s

s

(3N )

- i′1 − 1?

i′2 − 1i2 − 1

Now we have that the Alexander polynomial is Λ(t) = ∆1 (t) = (t2N − tN + 1)2 . We takemore speciﬁcally N = 7, then N1 = 42, and we pick λ := exp(2πi(−5/42)) ∈ Eig.
Recall that, cf. Example 5.1.11, ν1 = −78+7I +6I ′ , where I = 3i1 +2i2 and I ′ = 3i′1 +2i′2 .
We search for an allowed W such that −ν1 /42 is a pole of Z(F, W ; s) and ν1 ≡ 5 (mod 42).
This last condition is equivalent to I ≡ 5 (mod 6) and I ′ ≡ 1 (mod 7). But for W to beallowed 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 v1′ cannot induce λ by the symmetric argument,
and that the node v0 cannot induce primitive 42-th roots of unity.
Example 7.4.25.
(a) We recall Example (3.5) of B. Rodrigues [25]. Consider the following resolution graph(the right graph below):
−2

s

−1

−6

s −4

s −3

s

s

−2

s

(6)

s

(12)

s

s(3)

(3)

s

(5)

s

- (7)

s(1)

It is easy to verify that it is a numerically Gorenstein elliptic graph with length of theelliptic sequence two (for terminology, see e.g. [18]). It was not mentioned in [25], but
this graph can be realized by a hypersurface isolated singularity with multiplicity 3 andgeometric genus 2 (see also [34], case (24) in Table 4). We consider the nowhere vanishingform ω0 on X \ {0}. The computation in [25] shows that, for the indicated Weil divisor F ,the zeta function Z(F, ω0 ; s) has s0 = 1/3 as a simple pole, but that exp(2πi/3) is not aroot of the involved Alexander polynomial.
Note that this 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 associatewith it a splice diagram (by the very similar way as in (2.1)), and one can impose thesemigroup condition in the same way as above read from the splice diagram, cf. [21]. For
example, the ‘culpable’ node with decoration −6 (which provides the counterexample to the‘naively generalized’ Monodromy Conjecture) satisﬁes the semigroup condition, since thedeterminant 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 pointis 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 generalizationof 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 providing a counterexample to the‘naively generalized’ Monodromy Conjecture. Here is one, again, in a combinatorial setting.
The form is the standard Gorenstein form, whose analytic realization can easily be checked;

A. Némethi and W. Veys

42

the function–multiplicities are listed on the second diagram, where the analytic realizationof the function is not guaranteed, and N is a positive integer. The example shows that in our
combinatorial arguments from this section the semigroup assumption cannot be eliminated.
−2

s

−1

−7

s −3

s −3

s

−2

s

s

(9N ) (18N )ss(3N ) (2N )s(6N )ss .- (1)@ ..
N arrows
s(N )
@
R (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∆1 (t) =(t9N + 1)(t2N − 1)N −1 (t − 1).
(t3N + 1)(tN − 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: thefamily 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 withthe 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 s0 of the topological zeta function
Z(g, W ; s), exp(2πis0 ) 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 ΛΓ(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 toobserve that the deﬁnition of the semigroup condition associated with Γπ (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 thatchoice, we had in mind the analytic realization of the divisor W too, that is, the applicabilityof the main results. More precisely, in general, it is a rather hard question to determine theanalytic 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. [21, 22]. In is worth
to mention that the analytic realization of these germs is guaranteed by an arithmeticalproperty of the graph Γ (see End Curve Theorem in [23, 24]), which is nothing else but thesemigroup 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, thesemigroup condition too will be satisﬁed (compare also with subsection 7.1). This supports

Generalized Monodromy Conjecture

43

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)).
References[1] A’Campo, N.: La fonction zêta d’une monodromie, Comment. Math. Helv. 50, 1975, 233–248.
[2] Artal Bartolo, E., Cassou-Noguès, P., Luengo, I. and Melle Hernández, A.: Monodromy conjecture for some surface singularities Ann. Scient. Ec. Norm. Sup. 35, 2002, 605–640.
[3] Artal Bartolo, E., Cassou-Noguès, P., Luengo, I. and Melle Hernández, A.: Quasi-ordinary
power series and their zeta functions, Mem. Amer. Math. Soc. 178 no 841, 2005.
[4] Abramovich, D., Karu, K., Matsuki, K. and Wlodarczyk, J.: Toriﬁcation and factorization of
birational maps, J. Amer. Math. Soc. 15, 2002, 531–572.
[5] Barlet, D.: Contribution eﬀective de la monodromie aux développements asymptotiques, Ann.
Scient. Ec. Norm. Sup. 17, 1984, 293–315.
[6] Budur N., Mustaţa M. and Teitler, Z.: The Monodromy Conjecture for hyperplane arrangements, Geom. Dedicata 153, no 1, 2011, 131-137.
[7] Denef J. and Loeser F.: Caractéristiques d’Euler–Poincaré, fonctions zêta locales, et modiﬁcations analytiques, J. Amer. Math. Soc. 5, 1992, 705–720.
[8] Denef J. and Loeser F.: Motivic Igusa zeta functions, J. Alg. Geom. 7, 1998, 505–537.
[9] Eisenbud, D. and Neumann, W.: Three-Dimensional Link Theory and Invariants of Plane
Curve Singularities, Ann. of Math. Studies 110, Princeton University Press, 1985.
[10] Kashiwara, M.: Vanishing cycles sheaves and holomorphic systems of diﬀerential equations,Lecture Notes in Math. 1016, 1983, Springer Verlag.
[11] Laufer, H.B.: Normal two–dimensional singularities, Annals of Math. Studies 71, Princeton
University Press 1971.
[12] Lemahieu A. and Van Proeyen, L.: Monodromy conjecture for non–degenerate surface singularities, Trans. of AMS 363, 2011, 4801–4829.
[13] Lemahieu A. and Veys W.: Zeta functions and monodromy for surfaces that are general for a
toric idealistic cluster, Intern. Math. Res. Notices ID rnn122, 2009, 52p.
[14] Loeser, F.: Fonctions d’Igusa p–adiques et polynômes de Bernstein, Amer. J. Math. 110, 1988,
1–22.
[15] Loeser, F.: Fonctions d’Igusa p–adiques, polynômes de Bernstein, et polyèdres de Newton, J.
Reine Angew. Math. 412, 1990, 75–96.
[16] Luengo-Valesco, I.; Melle-Hernández, A. and Némethi, A.: Links and analytic invariants ofsuperisolated singularities, J. Alg. Geom. 14, 2005, 543-565.
[17] Malgrange, B.: Polynômes de Bernstein–Sato et cohomologie évanescente, Astérisque 101/102,
1983, 243–267.
[18] Némethi, A.: “Weakly” Elliptic Gorenstein singularities of surfaces, Inventiones math. 137,
1999, 145-167.
[19] Némethi, A., Neumann, W.D. and Pichon, A.: Principal analytic link theory in homology spherelinks, Topology of algebraic varieties and singularities, 377-387, Contemp. Math. 538, Amer.
Math. Soc., Providence, RI, 2011.
[20] Némethi, A. and Veys, W.: Monodromy eigenvalues are induced by poles of zeta functions: theirreducible curve case, Bull. London Math. Soc. (2010); doi: 10.1112/blms/bdp 128, 11p.
[21] Neumann, W.D. and Wahl, J.: Universal abelian covers of surface singularities, Trends insingularities, Birkhäuser, 2002, A. Libgober and M. Tibăr Eds., 181–190.
[22] Neumann, W. and Wahl, J.: Complex surface singularities with integral homology sphere links,
Geom. Topol. 9, 2005, 757–811 (electronic).
[23] Neumann, W. and Wahl, J.: The End Curve Theorem for normal complex surface singularities,
J. European Math. Soc. 12, 2010, 471–503.
[24] Okuma, T.: Another proof of the end curve theorem for normal surface singularities, J. Math.
Soc. Japan 62, no 1, 2010, 1-11.
[25] Rodrigues, B.: On the monodromy conjecture for curves on normal surfaces, Math. Proc. Cambridge Phil. Soc. 136, 2004, 313–324.

44

A. Némethi and W. Veys[26] Rodrigues, B. and Veys, W.: Poles of zeta functions on normal surfaces, Proc. London Math.
Soc. 87, 2003, 164–196.
[27] Veys, W.: Poles of Igusa’s local zeta function and monodromy, Bull. Soc. Math. France 121,
1993, 545–598.
[28] Veys, W.: Zeta functions for curves and log canonical models, Proc. London Math. Soc. 74,
1997, 360–378.
[29] Veys, W.: The topological zeta function associated to a function on a normal surface germ,
Topology 38, 1999, 527–548.
[30] Veys, W.: Zeta functions and ‘Kontsevich invariants’ on singular varieties, Canadian J. Math.
53, 2001, 834–865.
[31] Veys, W.: Vanishing of principal value integrals on surfaces, J. Reine Angew. Math. 598, 2006,
139-158.
[32] Veys, W.: Monodromy eigenvalues and zeta functions with diﬀerential forms, Adv. Math. 213,
2007, no. 1, 341–357.
[33] Wlodarczyk, J.: Combinatorial structures on toroidal varieties and a proof of the weak factorization theorem, Invent. Math. 154, 2003, 223–331.
[34] Yau, S. S.-T.: Hypersurface weighted dual graphs of normal singularities of surfaces, Amer. J.
of Math. 101 (4) 1979, 761–812.
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