THE MONODROMY CONJECTURE FOR ZETAFUNCTIONS ASSOCIATED TO IDEALS IN DIMENSION
TWO
Lise Van Proeyen and Willem Veys ∗Abstract
The monodromy conjecture states that every pole of the topological(or related) zeta function induces an eigenvalue of monodromy. Thisconjecture has already been studied a lot. However in full generality
it is proven only for zeta functions associated to polynomials in two
variables.
In this article we work with zeta functions associated to an ideal.
First we work in arbitrary dimension and obtain a formula (like theone of A’Campo) to compute the ‘Verdier monodromy’ eigenvalues
associated to an ideal. Afterwards we prove a generalized monodromy
conjecture for arbitrary ideals in two variables.
2000 Mathematics Subject Classification. 14E15, 32S40, 14H20.

1

Introduction

Classically the invariants called topological, motivic and p-adic Igusa zetafunction are associated to one polynomial f over C, over an arbitrary fieldof characteristic zero and over a p-adic field, respectively. There are fascinating conjectures relating their poles with the roots of the Bernstein-Satopolynomial (also called b-function) of f and with the eigenvalues of the local Milnor monodromy of f in points of {f = 0}, up to now proven in fullgenerality only for polynomials in two variables.
One associates similarly in a natural way all these functions to several
polynomials or to an ideal. Not so obvious is the notion of Bernstein-Satopolynomials associated to several polynomials or an ideal. There is a construction of Sabbah [Sa] and more recently also of Budur, Mustaţǎ and Saito[BMS]. Concerning monodromy, the classical construction of local Milnor∗
K.U.Leuven, Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium,
email: Lise.VanProeyen@wis.kuleuven.be, Wim.Veys@wis.kuleuven.be. The research waspartially supported by the Fund of Scientific Research - Flanders (G.0318.06).

1

fibre does not generalize to arbitrary maps (f1 , . . . , fr ). However, there is a
notion of ‘Verdier monodromy’ in this general context [Ver].
In this paper we prove a relation between the poles of these zeta functions
associated to an arbitrary ideal in two variables and the ‘Verdier monodromy
eigenvalues’ of this ideal, generalizing the result for one polynomial.
We now provide more details, focussing on the topological zeta function.
Let I = (f1 , . . . , fr ) be a nontrivial ideal in C[x1 , . . . , xn ] and Y = V(I) its
associated subscheme of AnC . We assume that Y contains the origin.
We first fix notation to define the topological zeta function. Take aprincipalization ψ : X̃ → An of I. By this we mean that ψ is a properbirational map from a nonsingular variety X̃ such that the total transformψ ∗ I is a principal ideal with support a simple normal crossings divisor, andmoreover that the exceptional locus of ψ is contained in the support of ψ ∗ I.
If I has components of codimension one, we can write this total transform
as a product of two (principal) ideals: the support of the first one is the
exceptional locus, where the support of the second one is formed by theirreducible components of the total transform that are not contained in the
exceptional locus. This second ideal is the ‘weak transform’ of I.
Note that we use the word ‘principalization’, where other authors may
also use log-principalization,log-resolution or monomialization.
P
Let Ẽ = i∈J Ni Ei denote the divisor of ψ ∗ I, i.e. its irreducible components are the Ei , i ∈ J, occurring with multiplicity Ni . (Alternatively,
−1
one
ψ be
P can say that ψ Y = Ẽ.) Let the relative canonical divisor of∗ (dx ∧
(ν
−
1)E
,
i.e.
ν
−
1
is
the
multiplicity
of
E
in
the
divisor
of
ψ
i
i
i
1
i∈J i
. . . ∧ dxn ). Finally put EI◦ := (∩i∈I Ei )\(∪l6∈I El ) for I ⊂ J; these EI◦ form anatural locally closed stratification of X̃. (Note that E∅◦ = X̃\ ∪l∈J El .) The(local) topological zeta function of I at 0 is
XY
1
Ztop,I (s) :=χ(EI◦ ∩ ψ −1 {0})∈ Q(s).
νi + sNi
I⊂J

i∈I

There is a global version replacing EI◦ ∩ ψ −1 {0} by EI◦ .

When r = 1, Denef and Loeser showed in [DL1] that the expression
above does not depend on the chosen principalization (which for r = 1 isjust an embedded resolution) by writing it as a limit of p-adic Igusa zetafunctions. Alternatively, they obtained it later in [DL2] as a specializationof the motivic zeta function. This can be generalized to arbitrary r, seee.g. [VZ, (2.4)]. Still another possibility for arbitrary r is to use the Weak
Factorization Theorem of WÃlodarczyk et al. [AKMW] to compare two principalizations. At any rate, observe that a complete list of possible poles ofZtop,I (s) is given by the − Nνii , i ∈ J.
For r = 1, say I = (f ), there are the following intriguing conjectures[DL1]. (They were originally formulated for the p-adic Igusa zeta function,2

which is a certain p-adic integral, and partially motivated by analogous
statements that are true for a similar complex integral. See [I], in particularsection 5.4.)
Conjecture 1. If s0 is a pole of Ztop,f (s), then s0 is a root of the (local)
Bernstein-Sato polynomial bf,0 (s) of f.
Conjecture 2. (Monodromy Conjecture.) If s0 is a pole of Ztop,f (s), thene2πis0 is an eigenvalue of the local monodromy action on some cohomology group of the Milnor fibre of f at some point of {f = 0} close
to 0.
Note that Conjecture 1 implies Conjecture 2 since for any root s0 of bf,0 (s) wehave that e2πis0 is such a monodromy eigenvalue [M]. For n = 2 Conjecture 1
was proved by Loeser [L1]. He also verified it for non-degenerate polynomialssatisfying extra assumptions [L2]. (There is a more elementary proof of
Conjecture 2 for n = 2 by Rodrigues [R].) Concerning Conjecture 2, there
are various partial results, mainly for n = 3, by Artal, Cassou-Noguès,
Luengo and Melle [ACLM1], [ACLM2], and Lemahieu, Rodrigues and thesecond author [Vey2], [RV], [Vey3], [LV1], [LV2].
Budur, Mustaţǎ and Saito introduced a Bernstein-Sato polynomial associated to an arbitrary ideal I = (f1 , . . . , fr ) ⊂ C[x1 , . . . , xn ]. (Their polynomial coincides with a polynomial that appears in [G].)
Still for arbitrary r, the notion of local Milnor fibre is in general not
well-defined. There is however the following construction of Verdier. To any
constructible complex of sheaves F • on An , he associates a similar complex
on CY An , the normal cone of Y = V(I) in An . This complex is called thespecialization of F • . It is moreover equipped with a ‘canonical monodromy
operator’. In particular when I = (f ) it turns out that this specializationof C• is in some sense equivalent to the usual complex of nearby cycles on
Y = V(f ), and that the two monodromy notions essentially correspond.
A nice feature of the Bernstein-Sato polynomial of [BMS] is that for anyof its roots s0 , we have that e2πis0 is a ‘Verdier monodromy eigenvalue’,see [BMS, Corollary 2.8]. This thus generalizes the implication for r = 1
mentioned above.
It is natural to ask also for arbitrary I if poles of the topological zetafunction of I are always roots of its Bernstein-Sato polynomial, rememberingof course that this question turned out to be very difficult already for r = 1.
In the special case of a monomial ideal I this was verified in [HMY] by
Howald, Mustaţǎ and Yuen (for the p-adic Igusa zeta function). Maybe
more accessible, do the poles of the topological zeta function of an arbitrary
ideal I induce monodromy eigenvalues in the sense of Verdier? The main3

result of this paper is to provide an affirmative answer to the last question
for arbitrary ideals in two variables.
The plan of the paper is as follows. We work over the base field ofcomplex numbers. In §2 we explain the construction of the specialization
functor of Verdier. In arbitrary dimension we show in §3 a formula for the‘Verdier monodromy eigenvalues’ of an ideal I in terms of a principalizationof the ideal, in the same spirit as A’Campo’s formula [A] for the eigenvaluesof one function f in terms of an embedded resolution of Y = V(f ). Notethat this is a priori not obvious; for r = 1 the complex of nearby cycles lives
on Y but for r > 1 the specialized complex of Verdier lives not on Y but on
CY An . In §4 we prove the ‘generalized Monodromy Conjecture’ for ideals in
two variables, and finally in §5 we provide some examples.
Acknowledgements: The authors want to thank J. Schürmann for theinteresting conversations and explanations.

2

The specialization functor of Verdier

Let I be a coherent ideal sheaf on a variety X. Consider the associatedsubscheme Y = V(I) of X. We construct the blow-up π : B = BlY X → X
of X in Y and denote by E the inverse image π −1 (Y ).
For every e ∈ E we want to study the zeta function of monodromy of theideal I. To define what monodromy is in this context, we will need notions
as the normal cone of Y in X and the specialization functor of Verdier.
The normal cone of Y in X is a cone over Y defined as
CY X = SpecOY (⊕n≥0 I n /I n+1 ) ,see e.g. [F, B.6]. Interesting to notice is that the projectivization P (CY X)
of the normal cone is exactly the exceptional variety E of the blowing-up of
X in Y (with its non-reduced scheme structure). We can identify the ‘locus
of vertices’ of this cone with Y . So we have an embedding j : Y → CY X
and a projection p : CY X → Y. These maps satisfy p ◦ j = IdY . Moreover,
we have an action of C∗ on the normal cone, coming from the graduationon ⊕n≥0 I n /I n+1 . The locus of vertices is the scheme of the fixed points of
the action and the morphisms j and p commute with it. Starting with apoint e ∈ (CY X)\Y, we define the ruler through e as the orbit of e under
the action of C∗ (with its reduced scheme structure), so every ruler can be
identified with C∗ .
The construction of the normal cone is functorial in the following sense.
Suppose we have a map f : X 0 → X of schemes and two subschemes Y ⊂ X

4and Y 0 ⊂ X 0 such thatY0
↓ f |Y 0
Y

,→
,→

X0
↓f
X

is a cartesian diagram, then we can associate a map C(f ) : CY 0 X 0 → CY X
to f. In particular, the map π : B → X has an associated map C(π) :CE B → CY X. This last one has the following interesting properties which
we will use further on. If we restrict the map C(π) to the ‘punctured’ cones,
C 0 (π) : (CE B)\E → (CY X)\Y, this is an isomorphism. Likewise, the mapof projectivized cones C 00 (π) : P (CE B) → P (CY X) is well-defined and is an
isomorphism. Moreover, the following diagram is commutative.
∼E←− P (CE B) ←− (CE B)\E



yπ

◦

Y

←− P (CY X) ←− (CY X)\Yoy C 00 (π)

oy C 0 (π)

◦

(2.1)

The (canonical) deformation to the normal cone of Y in X is a schemeX̂ with a morphism π̂ : X̂ → A1 such that π̂ −1 (A1 \ {0}) ∼= X × (A1 \ {0})1
and the fibre over 0 ∈ A is the normal cone CY X. This construction can befound in [F, Chapter 5]. So we obtain the following commutative diagram.
CYX −→

◦
y
0

−→

j

X × (A1 \ {0})

◦
y

X̂


yπ̂

←-

A1

←-

j0

A1 \ {0}We introduce now the specialization functor of Verdier that associates(by means of the previous diagram) to a constructible complex of sheaves
F • on X a constructible complex of sheaves on CY X.
Note. By a constructible complex of sheaves (of C-vector spaces) G • on ascheme Z we mean an object in the full subcategory Dcb (Z) of the derived
category D(Z); its cohomology sheaves H n (G • ) are thus constructible andmoreover zero for |n| >> 0. (A sheaf on Z is called constructible if there isa Zariski-locally closed stratification of Zred such that the restriction of thesheaf to each stratum is locally constant for the complex topology.)
In [Ver] the specialization of F • is defined as
SpY \X (F • ) := ψπ̂ (j! pr1∗ F • ),where π̂ and j are as before, pr1 is the projection of X × (A1 \ {0}) on thefirst factor and ψπ̂ is the nearby cycle functor of Deligne [Del] (see also [Di,5section 4.2]). In [Ver, p.356-357], Verdier defines a canonical transformationof monodromy on the complex of sheaves SpY \X (F • ).
This specialization functor has a number of important properties, fromwhich we will give the ones that we need in this article. These properties
are stated in [Ver, sections 8 and 10].
(SP1) Monodromy. For every constructible complex of sheaves F • , thecomplex SpY \X (F • ) is monodromic. This means that SpY \X (F • ) is
locally constant (with respect to the usual complex topology) on every
ruler of CY X.
(SP2) Proper direct image. Suppose we have a cartesian diagram
Y 0 ,→
↓
Y ,→

X0
↓f
X

where f is proper. Let C(f ) : CY 0 X 0 → CY X be the morphism associated to f. Then C(f ) is proper and for each constructible complex
F • on X 0 , the natural morphism
SpY \X (Rf∗ F • ) → RC(f )∗ (SpY 0 \X 0 (F • ))is an isomorphism.
(SP6) Normalization. Suppose that Y is a principal divisor with equation f = 0. Then the morphism p × C(f ) : CY X → Y × A1 is an
isomorphism, where p : CY X → Y is the projection. Notice that
C{0} A1 = A1 , so C(f ) is indeed a map CY X → A1 . With this isomorphism, we can define a section for z ∈ C\{0} :sz : Y→ CY Xy 7→ (p × C(f ))−1 (y, z).
Then we have for each constructible complex of sheaves F • on X an
isomorphisms∗z (SpY \X (F • )) → ψf (F • ).
This isomorphism is compatible with the two monodromy operations
defined on both sheaves on Y, in the sense that they are each other’s
opposite. (We refer to [Del] or [Di, section 4.2] for a description of the‘classical’ monodromy on ψf (F • ).)(SP7) Perversity. The specialization functor transforms perverse sheaves
into perverse sheaves.

6

3

The zeta function of monodromy

As we said before, Verdier defined a canonical transformation of monodromyon the complex of sheaves SpY \X (C•X ), which we will denote by M. For
each m ∈ Z and y ∈ (CY X)\Y, we have an automorphism Mym on the
stalk H m (SpY \X (C•X )y ). Eigenvalues of monodromy are eigenvalues of these
vector space automorphisms.
We now define the zeta function of monodromy ZI,e (t) of I for a pointe ∈ E. Choose an arbitrary point e0 on (CY X)\Y that is mapped to e by
the projectivization. Then ZI,e (t) is the (finite) productYmdet(Id − t Mem0 )(−1) .
Z(SpY \X (C•X ))(e0 ) :=m∈Z

Note that this is the usual notion of a zeta function for M and e0 , and that,by (SP1), this definition is independent of the choice of e0 .
In the next section, we use the zeta function of monodromy to prove themonodromy conjecture in dimension two. We will prove that a pole s0 of the
topological zeta function induces a zero or a pole e2πis0 of the monodromy
zeta function for some point e ∈ E. This implies that this number e2πis0 is
an eigenvalue of monodromy.
Remark 3.1. Note that it is also true that each eigenvalue of monodromy isa zero or a pole of the monodromy zeta function for some point e ∈ E. Since
we know from (SP7) that SpY \X (C•X ) is a (shifted) perverse sheaf, we cancopy the proof of [Den, Lemma 4.6] to obtain this result.
Now we prove a generalization of the formula of A’Campo [A, Theorem
3] to the case of ideals.
Theorem 3.2. Let I be a sheaf of ideals on a variety X. Let Y = V(I) be
the associated subscheme of X and suppose that Sing (X) ⊂ Y. Let π : B =BlY X → X be the blow-up of X in Y and ψ : X̃ → X a principalizationof I. Define ϕ : X̃ → B as the unique morphism such that ψ = π ◦ ϕ. We
denote by E the inverse image π −1 (Y ) and Ẽ = ψ −1 (Y ). We use Ei , i ∈ J,
for the irreducible components of Ẽ and Ni for the according multiplicitiesin Ẽ. Put Ei◦ = Ei \ ∪j∈J, j6=i Ej . For a point e ∈ E, the zeta function ofmonodromy is equal toY◦−1
ZI,e (t) =(1 − tNj )χ(Ej ∩ϕ (e)) .
j∈JRemark 3.3. (1) Note that the principalization ψ factorizes through theblow-up π since the inverse image ψ −1 (Y ) is a Cartier subscheme in X̃.
(2) When I is principal we can consider π as the identity and thus
ψ = ϕ. Then ZI,e (t) is the usual hypersurface zeta function of monodromy
for e ∈ E = Y and we recover A’Campo’s formula.
7

Proof. Fix e ∈ E and choose an element e0 ∈ (CY X)\Y that is mapped toe by the projectivization map.
The restriction of π is an isomorphism B \E → X \Y, which implies that
Rπ∗ C•B |X\Y = C•X |X\Y . Since SpY \X (F • )|(CY X)\Y only depends on F|X\Y(see [Ver, p.354]), we can deduce that
SpY \X (C•X )|(CY X)\Y = SpY \X (Rπ∗ C•B )|(CY X)\Yand
Z(SpY \X (C•X ))(e0 ) = Z(SpY \X (Rπ∗ C•B ))(e0 ).
Because we have the cartesian diagram
E ,→
↓
Y ,→

B
↓π
X

we can use (SP2) to write that∼

SpY \X (Rπ∗ C•B ) −→ RC(π)∗ (SpE\B (C•B ))is an isomorphism, or that
Z(SpY \X (Rπ∗ C•B ))(e0 ) = Z(SpE\B (C•B ))(C(π)−1 (e0 )).
(Note that the restriction of C(π) to (CE B)\E is an isomorphism, as saidin §2.)
There exists an open Ve around e such that E ∩ Ve is the zero locus
of one nonzerodivisor f. So we can use (SP6) to see that locally, we have
an isomorphism p × C(f ) : CE B → E × A1 . Take z ∈ C\{0} such that(p × C(f ))(C(π)−1 (e0 )) = (e, z). This is possible, using the commutative
diagram (2.1). We can conclude that we have an isomorphisms∗z (SpE\B (C•B )) −→ ψf (C•B ).
This isomorphism is ‘compatible’ with the monodromy actions, so
Z(SpE\B (C•B ))(sz (e)) = Z(ψf (C•B ))(e).
The right hand side is the monodromy zeta function of the map f : Ve ⊂B → C in the point e. Because we already have an embedded resolution of E
in B, namely the map ϕ : X̃ → B, we can use the formula of A’Campo [A,Theorem 3]. (This formula was originally proven for functions on a smoothvariety. But one can check that for instance the proof that Dimca gives in[Di, Corollary 6.1.15] is still valid in this more general context where thehypersurface contains the singular locus of the ambient variety.) We obtain
Y◦−1
Z(ψf (C•B ))(e) =(1 − tNj )χ(Ej ∩ϕ (e)) .
j∈J8For this last equality, we need that the multiplicities of the irreducible components Ei of Ẽ in the divisor ϕ−1 (E) are precisely the Ni . Indeed:XẼ =Ni Ei = (π ◦ ϕ)−1 (Y ) = ϕ−1 (π −1 Y ) = ϕ−1 (E).
i∈J

Putting all these equalities of zeta functions together, proves our theorem.

4

The Monodromy Conjecture

From now on, we will work in dimension 2, but we will use the same notation
as before. So let I ⊂ C[x, y] be an ideal satisfying {0} ⊂ Supp I. Put
Y = V (I) the subscheme of X = C2 defined by I. Let π : B = BlY X → X
be the blowing-up of X in Y and ψ : X̃ → X be the minimal principalizationof I. The map ϕ : X̃ → B is defined such that Pψ = π ◦ ϕ. We denote−1−1
E = π (Y ) as divisor on B, and Ẽ = ψ (Y ) = i∈J Ni Ei as divisor on
X̃. Here the Ei (Ni , νi ) are the irreducible components of Ẽ, together with
their numerical data.
Theorem 4.1. (Generalized Monodromy Conjecture.) If − Nν is a pole ofthe local topological zeta function of an ideal I ⊂ C[x, y], then there exists aνpoint y ∈ E such that e−2πi N is an eigenvalue of monodromy in y.
We first mention some results that will be useful in the proof.
For an ideal I = (f1 , . . . , fr ) ⊂ C[x, y] we can look at the linear system{λ1 f1 + . . . + λr fr = 0 | λi ∈ C for i = 1, . . . , r}. A total generic curve of
I is a general element of this linear system. Now we determine whether
there are common components among the fi and put them together. So
we can write I = (h)(f10 , . . . , fr0 ), where (f10 , . . . , fr0 ) is a finitely supported
ideal. A generic curve of the ideal I is a general element of the linear system{λ1 f10 + . . . + λr fr0 = 0 | λi ∈ C for i = 1, . . . , r}. Notice that the definitionof the (total) generic curve depends on the choice of generators we use torepresent the ideal.
Lemma 4.2. If an irreducible component Ei of Ẽ is contracted by ϕ, thenthere is no intersection between Ei and the strict transform of a genericcurve of the ideal I in X̃.
Proof. If there is an intersection, the strict transform of a generic curve of
I in B contains the point ϕ(Ei ). But if we denote fei for the strict transformof fi0 in B, the linear system {λ1 fe1 + . . . + λr fer = 0 | λi ∈ C for i = 1, . . . , r}should be base point free (see e.g. [H, Example 7.17.3]).

9

Lemma 4.3. Let (f1 , . . . , fl ) be an ideal in C[x, y] and E0 (N, ν) an exceptional curve of the principalization (together with the numerical data).
Suppose E0 intersects n times the strict transform of a generic curve and mtimes other components E1 (N1 , ν1 ), . . . , Em (Nm , νm ) of the principalization.
Put αi = νi − Nν Ni for i = 1, . . . , m. Then we havePmνn1.
i=1 αi = m − 2 + N , where the left hand side is zero when m = 0,
and
2. −1 ≤ αi < 1 for i ∈ {1, . . . , m}. Moreover, αi = −1 only occurs whenm = 1.
The first equality can be found in [VV, §3]. It is a reformulation of therelation between the numerical data proved by Loeser in [L1]. The second
statement is [VV, Proposition 3.1].
In [VV], we gave a complete list of five conditions in which a candidate
pole is indeed a pole of the local topological zeta function of an ideal indimension two. Using Lemma 4.3, it is not difficult to see that the conditions
given in numbers 2, 3 and 4 of [VV, Theorem 4.2] are equivalent to thecondition of being intersected by the strict transform of a generic curve. So
we can reformulate this theorem as follows.
Theorem 4.4. Let I ⊂ C[x, y] be an ideal satisfying 0 ∈ Supp (I) andψ : X̃ → C2 the minimal principalization of I in a neighbourhood of 0. Let
E• (N• , ν• ) be the components of the support of the total transform ψ ∗ I with
their associated numerical data.
The rational number s0 is a pole of the local topological zeta function of I if
and only if at least one of the following conditions is satisfied:
1. s0 = − N1 for a component E0 (N, ν) of the support of the weak transform of I;
2. s0 = − Nν for E0 (N, ν) an exceptional curve that has non-empty intersection with the strict transform of a generic curve of the ideal I;
3. s0 = − Nν for E0 (N, ν) an exceptional curve that intersects at least
three times other components.
Now we are ready to prove the monodromy conjecture for ideals in dimension 2. We were inspired by [R], where Rodrigues gives an elementary
proof of the monodromy conjecture for curves on normal surfaces.
Proof of Theorem 4.1. Choose a pole s0 of the local topological zeta functionof the ideal I and take a, d ∈ Z satisfying s0 = − ad and gcd(a, d) = 1.
Suppose that there exists an irreducible component Em (Nm , νm ) withd | Nm that is not contracted to a point by ϕ : X̃ → B. Choose a point10◦ ) such that ϕ−1 (y) is a finite set of points. Then there exists fory ∈ ϕ(Emevery j ∈ J a nonnegative integer kj (with km 6= 0) such thatY
ZI,y (t) =(1 − tNj )kj ,j∈Jfrom which we see that e2πis0 is a zero of the zeta function of monodromy
in y, so it is an eigenvalue of monodromy.
On the other hand, suppose that every Ei (Ni , νi ) that satisfies d | Ni is
contracted to a point by ϕ. This implies (by using Lemma 4.2 and Theorem
4.4) that there exists an index m ∈ J, with s0 = − Nνmm , such that Em is
exceptional and Em intersects at least three times other components of Ẽ.
Fix such a m and take y = ϕ(Em ). Define T ⊂ J such that ϕ−1 {y} = ∪i∈T Ei .
To prove that e2πis0 is a zero or a pole of the zeta function of monodromy
in y, it is enough to show thatX
χ(Ei◦ ) < 0.
i∈T, d|Ni

First notice that the strict transform of a generic curve intersects at least
one exceptional component, so not every exceptional curve can be contracted
and it is impossible that d divides Ni for every i ∈ J. We define
[
Ei .
M̃ :=
i∈T, d|Ni

Every connected component of M̃ contains at least one irreducible component that intersects a component Ek (Nk , νk ) outside M̃ . If d | Nk , thenEk is contracted by ϕ and since Ek has a non-empty intersection with M̃ ,this also implies that k ∈ T . So d - Nk . Moreover, the irreducible component000of M̃ that intersects Ek needs to intersect a second component
Pr Ek (Nk , νk )with d - Nk0 . We can deduce this from the formula κNl = i=1 Ni , where
El is an exceptional curve intersecting r times other components Ei , fori = 1, . . . , r, and where −κ denotes
S the self-intersection number of El .
Every connected component i∈I0 Ei of M̃ satisfiesX
χ(Ei◦ ) ≤ 0.
i∈I0Because such a component has at least two externalintersections, this is aS
direct consequence ofP
the following fact: if ri=1 Ei is a tree consisting ofrational curves, then ri=1 χ(Ei \(∪j=1,...,r, j6=i Ej )) = 2. See e.g. [R, Lemma
2.2].
S
Now fix the connected component M = i∈IM Ei of M̃ that contains
Em . For this one, we prove the strict inequalityX
χ(Ei◦ ) < 0.
i∈IM

11

If this sum would equal 0, then M is a tree of rational curves that intersects
precisely two times with components Ej outside M̃ . From a previous argument, we know that these intersections are on one curve Es with s ∈ IM .
There are two possibilities.
Es = Em . The component Em will intersect at least one other curve
Ej (Nj , νj ). This curve belongs to M, so d | Nj and the number νj − ad Nj
is an integer. The only possibility is 0 (see Lemma 4.3), so this impliesνthat − Njj = − ad . We know that Ej is contracted to a point, so it is
not intersected by the strict transform of a generic curve. From therelations between the numerical data (see also Lemma 4.3), we candeduce that Ej intersects at least one other component of M and we
can repeat this process infinitely many times. This leads to a contradiction.
Es 6= Em . Now we can start for each of the (at least three) componentsthat intersect Em the same procedure of constructing a series of curves
in M. Only one of these can eventually stop, because there is only oneof these series that has intersections outside of M.
This ends the proof.
Remark 4.5. Theorem 4.1 is also true for the motivic and the Hodge zetafunction of I, and for p-adic Igusa zeta functions associated to several polynomials in two variables, because the necessary condition of [VV, Theorem
4.2] to be a pole is still valid for these zeta functions. We refer to [VV, §6]
for definitions and more explanation.

5

Some examples(1) I = (x3 y, x6 + y 4 ) ⊂ C[x, y]. The intersection diagram of the principalization X̃ is as in Figure 1, where the dashed line denotes the strict
transform of a generic curve.
Figure 1: Intersection diagram of the principalization of (x3 y, x6 + y 4 ).
E3 (6, 4)
E2 (5, 3)

E1 (4, 2)
We see that the poles of the local topological zeta function are − 12 and − 32 .
12

The blowing-up B = BlY C2 is given by
Proj

C[x, y][A, B](Ax3 y − B(x6 + y 4 )),see e.g. [EH, Section IV.2.1]. The exceptional variety consists of only one
irreducible curve, say E. The map ϕ : X̃ → B contracts E2 to a point a on
E. It maps E1 and E3 surjectively on E. The restriction to E3 is one-to-one,
but the restriction to E1 is three-to-one.
For a point e on E different from a, we compute that the monodromy
zeta function is
ZI,e (t) = (1 − t4 )3 (1 − t6 ).
4πi

We can immediately check that e−πi and e− 3 are zeroes of this function.
Since − 12 and − 23 were the only poles of the local topological zeta function,
this illustrates the monodromy conjecture.
(2) I = (x4 , xy 2 , y 3 ) ⊂ C[x, y]. The intersection diagram of the principalization (with numerical data) is given in Figure 2. We can easily deducethat the poles of Ztop,I (s) are − 32 and − 58 .
Figure 2: Intersection diagram of the principalization of (x4 , xy 2 , y 3 ).
E2 (4, 3)
E3 (8, 5)

E1 (3, 2)
The blowing-up B = BlY A2 is given by
B = ProjC[x, y][A, B, C](y 2 A + x3 B, xC + yB, B 3 x + AC 2 )and the exceptional variety E consists of two projective lines E 0 and E 00 . The
map ϕ : X̃ → B maps E1 and E3 one-to-one onto E 0 and E 00 , respectively.
It contracts E2 to a point a on E 00 .
We choose a point x ∈ E 0◦ and a point y ∈ E 00◦ \{a}. For these points
the monodromy zeta functions are
ZI,x (t) = 1 − t3 ,ZI,y (t) = 1 − t8 .

So we conclude that the two poles of the topological zeta function induce
4πi
5πi
two eigenvalues of monodromy e− 3 and e− 4 .
13Remark 5.1. Note that this example is a monomial ideal. As mentioned
in the introduction, the monodromy conjecture had already implicitly beenverified in this case.
(3) I = (x3 y, x3 − y 2 ) ⊂ C[x, y]. In Figure 3 we can see that Ztop,I (s)has poles in − 56 and − 98 .
Figure 3: Intersection diagram of the principalization of (x3 y, x3 − y 2 ).
E4 (7, 6)
E3 (6, 5)
E6 (9, 8)
E1 (2, 2)

E5 (8, 7)

E2 (3, 3)

We know that
B = BlY X = Proj

C[x, y][A, B].
(A(x3 − y 2 ) − Bx3 y)

The blowing-up has one irreducible exceptional component E. The map ϕ :X̃ → B maps E6 surjectively on E and all the other exceptional componentsof the principalization are contracted to a point a on E. We fix a pointx ∈ E ◦ , x 6= a, and look at the following two monodromy zeta functions:
ZI,a (t) =(1 − t2 )(1 − t3 ),1 − t6

ZI,x (t) = 1 − t9 .

We can see that the two poles of the topological zeta function − 56 and − 89
5πi
16πigive rise to eigenvalues of monodromy e− 3 and e− 9 .

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