MONODROMY EIGENVALUES AND POLES OF ZETA FUNCTIONSTHOMAS CAUWBERGS AND WILLEM VEYS

Abstract. We study the poles of the topological and related zeta functions associatedto a polynomial f and a suitable differential form, and establish a strong link between
these poles and monodromy eigenvalues of f .

Introduction
Archimedean zeta functions. Let f : X → C be a non-constant analytic function on
an open part X of Cn . We consider C ∞ functions ϕ with compact support on X and thecorresponding differential forms ω = ϕdx ∧ dx̄. Here and further x = (x1 , . . . , xn ) anddx = dx1 ∧ · · · ∧ dxn . For such f the integral
Z
Z(f, ω; s) :=|f (x)|2s ω,Xwhere s ∈ C with Re(s) > 0, has been the object of intensive study. One verifies that Z(f, ω; s)is holomorphic in s. Either by resolution of singularities [Ati70], [BG69], or by the theory ofBernstein polynomials [Ber72], one can show that it admits a meromorphic continuation to C,
and that all its poles are among the translates by Z<0 of a finite number of rational numbers.
Combining results of Barlet [Bar84b], Kashiwara [Kas83] and Malgrange [Mal83], the polesof (the extended) Z(f, ω; s) are strongly linked to the eigenvalues of (local) monodromy atpoints of f −1 {0}; see §1 for the concept of monodromy.
√
Theorem 0.1. (1) If s0 is a pole of Z(f, ω; s) for some diffential form ω, then exp(2π −1s0 )is a monodromy eigenvalue of f at some point of f −1 {0}.
(2) If λ is a monodromy eigenvalue of f at a point of f −1√{0}, then there exists a differential
form ω and a pole s0 of Z(f, ω; s) such that λ = exp(2π −1s0 ).
There are also more precise local versions in a neighbourhood of a pointRof f −1 {0}. Similar
results hold for a real analytic function f : X(⊂ Rn ) → R and integrals X∩{f >0} f s ϕdx; werefer to e.g. [Bar84a], [Bar85], [Bar04], [BM93], [JM98]. Nowadays such (complex or real)
integrals are also called Archimedean zeta functions.
Topological and related zeta functions. There are similar integrals over p-adic fields,
called (p-adic) Igusa zeta functions or non-Archimedean zeta functions, whose meromorphic
continuation was proved by Igusa using resolution of singularities [Igu75]. Strongly related
Both authors were supported by the KU Leuven grant OT/11/069. The first author was also supported by
DFG Sonderforschungsbereich 701 at Bielefeld University.
1

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THOMAS CAUWBERGS AND WILLEM VEYS

to these p-adic zeta functions, there are various ‘algebro-geometric’ zeta functions: the motivic, Hodge and topological zeta functions. (We have for instance that the motivic zetafunction specializes to the various p-adic Igusa zeta functions, for all but finitely many p).
In this algebro-geometric context one rather considers polynomial functions f and algebraic
differential forms ω.
Let now f : Cn → C be a non-constant polynomial function and ω a regular (algebraic)differential n-form on Cn . The topological zeta function associated to f and ω is describedin terms of an embedded resolution of f −1 {0} ∪ div ω. Fix such an embedded resolution
π : X → Cn , where we assume that π is an isomorphism outside the inverse image off −1 {0} ∪ div ω. We denote by Ei , i ∈ I, the irreducible components of the inverse image
π −1 (f −1 {0}∪div ω), and by Ni and νi −1 the multiplicities of Ei in the divisor of π ∗ f and π ∗ ω,◦
respectively. We put EJ◦ := (∩j∈J Ej ) \ (∪`∈J/ E` ) for J ⊂ I, in particular E∅ = X \ (∪i∈I Ei ).
Note that the EJ◦ form a stratification of X in locally closed subsets. Finally, we denote byχ(·) the topological Euler characteristic.
Definition 0.2. The (global) topological zeta function of (f, ω) and its local version at b ∈ Cn
are
X
Y
1
,
χ(EJ◦ )
Ztop (f, ω; s) :=νj + sNj
J⊂I

j∈J

and
Ztop,b (f, ω; s) :=XJ⊂Iχ(EJ◦ ∩ π −1 {b})

Y
j∈J

1
,
νj + sNj

respectively, where s is a variable.
They were introduced by Denef and Loeser in [DL92] for the ‘standard‘ ω, i.e., for ω = dx.
Their original proof that these expressions do not depend on the chosen resolution proceeds
by describing them as a kind of limit of p-adic Igusa zeta functions. Later they obtained
them as a specialization of the intrinsically defined motivic zeta functions [DL98]. Anothertechnique is applying the Weak Factorization Theorem [AKMW02], [Wlo03] to compare twodifferent resolutions.
Note that the numbers −νi /Ni , i ∈ I, form a complete list of candidate poles for the topologicalzeta function associated to f and ω. (For the Archimedean zeta functions, there exist similarcomplete lists of candidate poles in terms of embedded resolutions.)

The finer variants, Hodge and motivic zeta functions, involve, instead of Euler characteristics,
Hodge polynomials and classes in the Grothendieck ring of varieties, respectively. Concerning
Hodge and motivic zeta functions, we refer to e.g. [DL98], [Rod05], [Vey06a] for versions with
ω = dx, and to [ACLM02], [ACLM05], [Vey01], [Cau14] involving more general ω. We mention
that, in contrast with topological zeta functions, Hodge and motivic zeta functions can be
defined intrinsically as formal power series with coefficients determined by the behaviour of
the arcs on Cn with respect to their intersection with f −1 {0} and with div ω. Then one
shows that they are rational functions by proving explicit formulae as above in terms of anembedded resolution.

MONODROMY EIGENVALUES AND POLES OF ZETA FUNCTIONS

3

Towards a generalized monodromy conjecture. It is very natural to investigate thevalidity of the analogous statements of Theorem 0.1 in the context of the non-Archimedean
and related zeta functions. Here we focus on the topological zeta function, because this is the‘most difficult’ one in the context of our main result, see Remark 2.2. First of all, there is thefamous Monodromy Conjecture (originally stated for the p-adic Igusa zeta function), being
an analog of Theorem 0.1(1) in the special case where ω = dx. Without loss of generality, we
assume from now on that f (0) = 0.
√
Conjecture 0.3. [DL92] If s0 is a pole of Ztop,0 (f, dx; s), then exp(2π −1s0 ) is a monodromy
eigenvalue of f : Cn → C at a point of f −1 {0}, close to 0.
This conjecture was proved for n = 2 by Loeser [Loe88]. There are by now various otherpartial results as [ACLM02], [ACLM05], [BV13], [BMT11], [Cau14], [LVP11], [LV09], [Vey93],[Vey06b]. In contrast with the Archimedean case, such a statement is certainly not true forarbitrary√ ω. There are plenty of examples involving a pole s0 of Ztop,0 (f, ω; s), such thatexp(2π −1s0 ) is not a monodromy eigenvalue of f . Concerning Theorem 0.1(2), the second
author investigated analogous statements in [Vey07]. He showed for instance the following.
Theorem 0.4. Let f : (Cn , 0) → (C, 0) be a nonzero polynomial function (germ). Let λ bea monodromy eigenvalue of f at 0. Then there exists a differential n-form √ω and a point−1b ∈ f {0}, close to 0, such that Ztop,b (f, ω; s) has a pole s0 satisfying exp(2π −1s0 ) = λ.
If f −1 {0} has an isolated singularity at 0, then we can take 0 itself as point b.
The zeta functions associated to f and the constructed ω in the theorem above can have otherpoles that do not induce monodromy eigenvalues of f . So for those zeta functions the analog
of Theorem 0.1(1) is (unfortunately) not true. Then, in [Vey07], the following questions wereraised, aiming at a deeper analog of Theorem 0.1.
Question 1. Given f , does there exist some set of distinguished differential forms ω suchthat(1) for every monodromy eigenvalue λ of f , there exists a distinguished differential n-formω and a√point b ∈ f −1 {0}, close to 0, such that Ztop,b (f, ω; s) has a pole s0 satisfying
exp(2π −1s0 ) = λ;√(2) for all distinguished ω and for all poles s̃ of Ztop,b (f, ω; s), we have that exp(2π −1s̃)
is a monodromy eigenvalue of f ?
Question 2. Does there exist a set as in Question 1, such that moreover the standard form
ω = dx is distinguished?
Of course, Question 2 is quite ambitious because a positive answer would be a lot stronger
than the (in arbitrary dimension) still wide open monodromy conjecture. For n = 2 this was
realized by Némethi and the second author, first for analytically irreducible germs [NV10]
and later for arbitrary curve germs [NV12]. The first author then showed in [Cau14] that theset proposed by the second author and Némethi also works for the motivic zeta function.
The aim of this paper is to give an affirmative answer to Question 1, in full generality forarbitrary n. Statements and proofs are only given for the topological zeta function, buteverything is also valid for the motivic and Hodge zeta function, see Remark 2.2 for moredetails. More precisely, we show the following.

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THOMAS CAUWBERGS AND WILLEM VEYS

Theorem 0.5. Let f : (Cn , 0) → (C, 0) be a non-zero polynomial function (germ).
(1) Let λ be a monodromy eigenvalue of f at 0. Then there exists a differential n-formω and a√point b ∈ f −1 {0}, close to 0, such that Ztop,b (f, ω; s) has a pole s0 satisfying
exp(2π −1s0 ) = λ, and such that s̃ ∈ Z for all poles s̃ of Ztop,b (f, ω; s) different froms0 .
(2) If f −1 {0} has an isolated singularity at 0, then we can take 0 itself as point b in (1).
√Since exp(2π −1Z) = {1}, and 1 is always a monodromy eigenvalue of f , the set of differential
forms in Theorem 0.5(1), for varying λ, satisfies the requirements of Question 1.
1. Prerequisites
1.1. Monodromy. Let f : Cn → C be a non-constant polynomial function satisfying f (b) =0. Let B ⊂ Cn be a small enough ball with centre b; the restriction f |B is a topological fibration over a small enough pointed disc D ⊂ C\{0} with centre 0. The fibre Fb of this fibration
is called the (local) Milnor fibre of f at b; see e.g. [Mil68]. The counterclockwise generator of
the fundamental group of D induces an automorphism of the cohomologies H q (Fb , C), whichis called the (local) monodromy of f at b. By a monodromy eigenvalue of f at b we mean aneigenvalue of the monodromy action on a least one of the H q (Fb , C). It is well known that
H q (Fb , C) = 0 for q > n, and that all monodromy eigenvalues are roots of unity.
Let Pq (t) denote the characteristic polynomial of the monodromy action on H q (Fb , C). If
Q N
f = j fj j is the decomposition of f in irreducible components and d := gcdj Nj , thenP0 (t) = td −1. When b is an isolated singularity of f −1 {0}, then H q (Fb , C) = 0 for q 6= 0, n−1;
and P0 (t) = t − 1.
Definition 1.1. The monodromy zeta function ζf,b (t) of f at b is the alternating product of
all characteristic polynomials Pq (t):ζf,b (t) :=

n−1
Y

q

Pq (t)(−1) .

q=0(Note that there are also other conventions, see for example [A’C75], [AGV88].) In particular,
for an isolated singularity the knowledge of ζf,b (t) and of Pn−1 (t) are equivalent. In general,
the following result says that ‘knowing all eigenvalues is equivalent to knowing all zeroes andpoles of all monodromy zeta functions’.
Lemma 1.2. [Den93, Lemma 4.6] Let f : Cn → C be a non-constant polynomial function. If
λ is a monodromy eigenvalue of f at b ∈ f −1 {0}, then there exists P ∈ f −1 {0} (arbitrarilyclose to b) such that λ is a zero or pole of the monodromy zeta function of f at P .
Also the monodromy zeta function can be described in terms of an embedded resolution.
Theorem 1.3 (A’Campo’s formula [A’C75]). Take an embedded resolution π : X → Cn off −1 {0} (that is an isomorphism outside the inverse image of f −1 {0}). Denote by Ei , i ∈ I,

MONODROMY EIGENVALUES AND POLES OF ZETA FUNCTIONS

5

the irreducible components of the inverse image π −1 (f −1 {0}), and by Ni the multiplicity of
Ei in the divisor of π ∗ f . We put Ei◦ := Ei \ ∪`6=i E` for i ∈ I. Then we haveY◦−1
ζf,b (t) =(tNi − 1)χ(Ei ∩π {b}) .
i∈I1.2. ‘Curvettes’ in higher dimension. We briefly recall a construction in [Vey07], generalizing the notion of curvette in dimension two.
Let X0 be a smooth quasi-projective (complex) variety of dimension n and let
π

π

π

m
X0 ←1 X1 ←2 · · · ←Xm(1)

be a composition π of m blowing-ups πi at a smooth irreducible centre Zi−1 (⊂ Xi−1 ), having
normal crossings with the exceptional locus of π1 ◦ · · · πi−1 . Denote the exceptional locus of
πi , as well as its consecutive strict transforms, by Ei .
Proposition 1.4. [Vey07, Proposition 3.2 and 3.3] One can construct consecutively for i =1, . . . , m a smooth hypersurface Cj on Xj such that
(1) Cj has normal crossing with E1 ∪ . . . ∪ Ej , with (the strict transform of ) previously
created C1 , . . . , Cj−1 , and with the next centre of blowing-up Zj (and such that Zj 6⊂
Cj ).
(2) Given another hypersurface H on Xm , having normal crossings with E1 ∪ · · · ∪ Em ,we can choose C1 , . . . , Cm such that furthermore H and all Ei and Ci form a normal
crossings divisor on Xm .
P(3) Denote C j = π(Cj ) ⊂ X0 and π ∗ C j = m
i=1 aji Ei + Cj for j = 1, . . . , m. Then thedeterminant of the matrix (aij ) ∈ Zm×m is equal to 1.
2. Results
Let f : (Cn , 0) → (C, 0) be a non-zero polynomial function (germ). We will use an appropriate
embedded resolution of f −1 {0}, for which we first fix notation.
2.1. Notation. Let f : X0 (⊂ Cn ) → C be a relevant representative of f , in the sense thatsome embedded resolution of f −1 {0} ⊂ X0 only has exceptional components that intersect theinverse image of 0. Take such an embedded resolution π : Xm → X0 , which is a compositionof m blowing-ups as in (1).
• Let Bi , i ∈ I s , be the irreducible components of the strict transform of f −1 {0}. We
may and will assume that these are pairwise disjoint. Let also Ei , i ∈ I e = {1, . . . , m},be the exceptional components of π.
• Define Ni , i ∈ I e , and Mi , i ∈ I s , such thatXX
div(π ∗ f ) =Ni Ei +M i Bi .
i∈I e

i∈I s

Define ∆i , i ∈ I e , such that
Kπ = div(π ∗ dx) =

X
i∈I e

∆i Ei .

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THOMAS CAUWBERGS AND WILLEM VEYS

• Introduce Ci , i ∈ I e = {1, . . . , m}, by taking them as in Proposition 1.4. (The extra
hypersurface H considered in Proposition 1.4(2) is f −1 {0}.) Let gi be a defining
polynomial for Ci = π(Ci ). Recall thatX
π ∗ Cj =aji Ei + Cji∈I e

with det(aij ) = 1.
• Let fi , i ∈ I s , be a defining polynomial for Bi = π(Bi ), and denoteXπ ∗ Bj =bji Ei + Bj .
i∈I e

Theorem 2.1. Let f : (Cn , 0) → (C, 0) be a non-zero polynomial function (germ). Let λ be azero or pole of the monodromy zeta function ζf,0 (t) of f at 0. Then there√ exists a differential
n-form ω such that Ztop,0 (f, ω; s) has a pole s0 satisfying λ = exp(2π −1s0 ) and such thats̃ ∈ Z for all poles s̃ of Ztop,0 (f, ω; s) different from s0 .
Proof. We first treat the obvious case where π is the identity. This means that f is of theform f1M1 , with f1−1 {0} smooth at 0, since we assumed that the componentsof the strict√M1transform are disjoint. Then ζf,0 (t) = P0 (t) = t − 1. Taking λ = exp −2π −1 Ma1 , witha ∈ {1, 2, . . . , M1 }, the n-form ω = f1a−1 dx does the job, since Ztop,0 (f, ω; s) = 1/(a + sM1 ).
From now on, we assume that π is not the identity. Then every point in π −1 {0} belongs tosome exceptional component. This implies that Ei◦ ∩ π −1 {0} = ∅ for i ∈ I s , and consequently
the product in Theorem 1.3 is in fact over i ∈ I e . Hence there exist by assumption somei0 ∈ I e and a ∈ {1, . . . , Ni0 } such that


√
a
and
χ(Ei◦0 ∩ π −1 {0}) 6= 0.
λ = exp −2π −1
Ni0
We will look for adequate mi ∈ Z≥0 , i ∈ I e , and ni ∈ Z≥0 , i ∈ I s , such that the differential
form
!
!
Y n
Y m
ω=
fi i
gi i dx
i∈I s

i∈I e

satisfies the desired properties. Define νi (∈ Z≥0 ), i ∈ I e , such thatX
X
X
div(π ∗ ω) =(νi − 1)Ei +mj Cj +
ni Bi .
i∈I e

j∈I e

i∈I s

We can also write
div(π ∗ ω) =X

∆i Ei +i∈I e

X

mj π ∗ Cj +X

nj π ∗ Bj ,

j∈I s

j∈I e

and thus(2)νi − 1 = ∆i +

X
j∈I e

mj aji +Xj∈I snj bjifor i ∈ I e .

MONODROMY EIGENVALUES AND POLES OF ZETA FUNCTIONS

7

Then we obtain the following expression for the topological zeta function:
X
χ(EJ◦ e ,J s ,Le ∩ π −1 {0})(3) Ztop,0 (f, ω; s) =J e ⊂I e ,J s ⊂I s ,Le ⊂I e
J e ∪J s ∪Le 6=∅

Y
Y
1
1
1
,
Ni s + νi
M i s + ni + 1
1 + mi
e
s
e

Y
i∈J

i∈J

i∈L

where
! 
EJ◦ e ,J s ,Le =

\

\

Ei ∩

i∈J e

Bi ∩

i∈J s

\

Ci

\

i∈Le


[

Ei ∪

i6∈J e[i6∈J sBi ∪[Ci  .

i6∈Le

Note that[

Ei◦0 = Ei0 \ i∈I e \{i0 }

Ei ∪

[

◦
Bi  = ∪Le ⊂I e E{ie.
0 },∅,Li∈I sDefine N = lcm ({Ni , i ∈ I e } ∪ {Mi , i ∈ I s }), let p be coprime to N and let q ∈ Z such thatpq ≡ 1 (mod N ).
The remainder of the proof consists in putting conditions on the mi and ni , and arguingthat they can be satisfied. First, we assume that ni ≡ −1 mod pN for all i ∈ I s . Next,
we put also on the mi , i ∈ I e , congruence conditions mod pN . Using (2) and the fact thatdet(aij ) = 1, we do this indirectly by putting such conditions on the νi . More precisely, we
assume that νi ≡ 0 mod pN if i 6= i0 , and that νi0 ≡ 1 + pq(a − 1) mod pN . This implies
that νi0 ≡ a mod Ni0 and that p does not divide νi0 .
ν

Hence s0 = − Nii0 will never be an integer divisible by p. And on the other hand, all other0
candidate poles, i.e., elements of the set

 

νi
ni + 1es−| i ∈ I \ {i0 } ∪ −|i∈I ,Ni
Mi
are integers divisible byconcludeis a candidate pole of order at most one with that s0 √
 p. We√athe property λ = exp −2π −1 Ni = exp 2π −1s0 , and with residue
0(4) R =

1
Ni0

X

χ(EJ◦ e ∪{i0 },J s ,Le ∩ π −1 {0})J e ⊂I e \{i0 },J s ⊂I s ,Le ⊂I e

!
Y
Y
1
1
1
.
Ni s0 + νi
Mi s 0 + n i + 1
1 + mi
e
s
e

Y
i∈J

i∈J

i∈L

We will now prove that we can take mi and ni such that R is not zero, meaning that s0 isreally a pole.
• Assume first that

◦
χ(E{i
∩ π −1 {0}) = χ (Ei◦0 \ ∪i∈I e Ci ) ∩ π −1 {0} 6= 0.
0 },∅,∅

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THOMAS CAUWBERGS AND WILLEM VEYS

Note that the denominators occurring in the other terms of R are polynomials in theni , i ∈ I s , and mi , i ∈ I e , that are all not constant. Indeed, the expression Ni s0 + νi
is equal to
X
X
Ni
Ni
Ni
(aji −(bji −(∆i0 + 1) +aji0 ) mj +bji0 ) nj ,∆i + 1 −
Ni0
N
N
i0
i0
e
s
j∈I

j∈I

see (2), and at least one of the coefficients aji − NNii aji0 is non-zero because the rows
0of the matrix (aij ) are linearly independent. The other expressions Mi s0 + ni + 1 and1 + mi are clearly not constant.
We conclude that R is never identically zero as rational function in the mi and ni ,and then there exist (positive) ni , i ∈ I s , and mi , i ∈ I e , such that R 6= 0.
◦• If however χ(E{i∩ π −1 {0}) = 0, this expression for R could be identically zero.
0 },∅,∅To remedy the problem, we adapt our choice of ω, as in the proof of [Vey07, Theorem
3.5]. Namely, we take some t ≥ 2 and introduce for each i ∈ I e different hypersurfaces
Ci,j , j ∈ {1, . . . , t}, as in Proposition 1.4, with defining polynomial gi,j for π(Ci,j ).
Then we choose adequate ni and mi,j such that the form

!
t
Y n
YY
m
fi i 
ω=gi,ji,j  dxi∈I si∈I e j=1

satisfies the desired properties. In fact, proceeding analogously as before, we have toshow that the residue of the candidate pole s0 is not identically zero as rational function in the ni and mi,j . This follows from the fact that the relevant Euler characteristic
in this case,χ (Ei◦0 \ ∪i∈I e ,1≤j≤t Ci,j ) ∩ π −1 {0} ,is nonzero for t large enough. We refer for details to the last part of the proof of[Vey07, Theorem 3.5].

2.2. Proof of Theorem 0.5. Part (1) is immediate from Theorem 2.1 and Lemma 1.2. Part(2) is shown by the same argument as in the proof of [Vey07, Theorem 3.6].
Remark 2.2. The results of Theorem 0.5 are also true for the Hodge and motivic zetafunction. The point is that, for a given f and ω, the motivic zeta function specializes to the
Hodge zeta function, which in turn specializes to the topological zeta function. In particular,a pole of the topological zeta function will induce a pole of the other two. The converse is
not clear.
The main aspect of the problem that we treated here is, given a monodromy eigenvalue λ off , find a form ω such that the zeta function associated to f and ω has a pole ‘inducing λ’.
We proved this result on the level of the topological zeta function, implying the analogous
result for the ‘finer’ zeta functions.
A priori there could be a problem concerning the other poles of the finer zeta functions.
Indeed, in general a given candidate pole −νi /Ni for the topological/Hodge/motivic zetafunction associated to f and ω could be a pole for the Hodge/motivic zeta function, whilst it
is not a pole for the topological zeta function. However, in the proof of Theorem 2.1 we sawthat not only s̃ ∈ Z for all poles of Ztop,0 (f, ω; s) different from s0 , but also for all candidate
poles −νi /Ni different from s0 . Hence, all these candidate poles for the Hodge/motivic zeta

MONODROMY EIGENVALUES AND POLES OF ZETA FUNCTIONS

9function induce the monodromy eigenvalue 1, and we do not need to worry whether they arepoles or not.

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Thomas Cauwbergs, Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501
Bielefeld, Germany
E-mail address: cauwbergs@math.uni-bielefeld.deWillem Veys, University of Leuven, Department of Mathematics, Celestijnenlaan 200 B, B-3001
Leuven, Belgium
E-mail address: wim.veys@wis.kuleuven.be