ON ‘MAXIMAL’ POLES OF ZETA FUNCTIONS, ROOTS OF b-FUNCTIONS
AND MONODROMY JORDAN BLOCKS
A. MELLE-HERNÁNDEZ, T. TORRELLI AND WILLEM VEYS
Dedicated with admiration to C.T.C. Wall on the occasion of his seventieth birthday

Abstract. The main objects of study in this paper are the poles of several local zeta functions: theIgusa, topological and motivic zeta function associated to a polynomial or (germ of) holomorphic
function in n variables. We are interested in poles of maximal possible order n. In all known cases(curves, non-degenerate polynomials) there is at most one pole of maximal order n which is thengiven by the log canonical threshold of the function at the corresponding singular point.
For an isolated singular point we prove that if the log canonical threshold yields a pole oforder n of the corresponding (local) zeta function, then it induces a root of the Bernstein-Satopolynomial of the given function of multiplicity n (proving one of the cases of the strongest formof a conjecture of Igusa-Denef-Loeser). For an arbitrary singular point we show under the same
assumption that the monodromy eigenvalue induced by the pole has ‘a Jordan block of size n on
the (perverse) complex of nearby cycles’.

Introduction
0.1. Let f : X → C be a non-constant analytic function on an open part X of Cn . The ‘classical’complex zeta function associated to f is an integralZ
Zϕ (f ; s) :=|f (x)|2s ϕ(x)dx ∧ dx̄
X∞for s ∈ C with <(s) > 0, where ϕ is a C function with compact support on X. (Here and further
x = (x1 , · · · , xn ) and dx = dx1 ∧ · · · ∧ dxn .) One verifies that Zϕ (f ; s) is holomorphic in s. Either
by resolution of singularities [3], or using the Bernstein-Sato polynomial bf (s) of f [2], one can showthat it admits a meromorphic continuation to C. The second method also yields that each pole ofZϕ (f ; s) is a translate by a nonpositive integer of a root of bf (s). And moreover, for a root s0 of
bf (s), the order of s0 − m as pole of Zϕ (f ; s) is at most the multiplicity of s0 as root of bf (s) [13].
In particular a pole of (maximal) order n induces a root of multiplicity n.
0.2. Let now f : X → Qp be a non-constant (Qp -)analytic function on a compact open X ⊂ Qnp ,where Qp denotes the field of p-adic numbers. Let | · |p and |dx| denote the p-adic norm and theHaar measure on Qnp , normalized in the standard way. The p-adic integralZ
Zp (f ; s) :=|f (x)|sp |dx|,
Xagain defined for s ∈ C with <(s) > 0, is called the (p-adic) Igusa zeta function of f . Using resolutionof singularities Igusa [11],[12] showed that it is a rational function of p−s ; hence it also admits ameromorphic continuation to C. (Everything can be generalized to finite extensions of Qp .) There
are various ‘algebro-geometric’ zeta functions, related to the p-adic Igusa zeta functions: the motivic,
2000 Mathematics Subject Classification. Primary: 14B05, 32S25. Secondary: 11S80, 32S45.
Key words and phrases. Igusa and topological zeta function, Bernstein-Sato polynomial, monodromy, nearby cycle
complex, log canonical threshold.
The first author is partially supported by Spanish Contract MTM2007-67908-C02-02. The third author is partially
supported by the Fund of Scientific Research - Flanders (G.0318.06).
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A. MELLE-HERNÁNDEZ, T. TORRELLI AND WILLEM VEYS

Hodge and topological zeta functions. We recall the definition of the local and global version of the
topological zeta function.
Let f : (Cn , 0) → (C, 0) be a germ of a non-zero holomorphic function f (resp. f ∈ C[x1 , . . . , xn ]non-zero, f (0) = 0). Let B be an open ball centered at the origin. Let π : X → B (resp. π :X → Cn ) be an embedded resolution of (f −1 {0}, 0) (resp. f −1 {0}). We denote by Ei , i ∈ J, theirreducible components of π −1 (f −1 {0})red . Let Ni (resp. νi − 1) be the multiplicity of f ◦ π (resp. ofπ ∗ (dx1 ∧. . .∧dxn )) at a generic point of Ei . For I ⊂ J, we set EI := ∩i∈I Ei and EI◦ := EI \(∪j6∈I Ej ).
The local topological zeta function Ztop,0 (f, s) (resp. topological zeta function Ztop (f, s)) of f at
0 (resp. of f ) is the rational function defined by
XY
1∈ Q(s),(∗)
Ztop,0 (f, s) :=χ(EI◦ ∩ π −1 {0})
ν i + Ni s
I⊂J

Ztop (f, s) :=

i∈I

X
I⊂J

χ(EI◦ )Yi∈I

1
∈ Q(s),
νi + Ni s

respectively. In [8], Denef and Loeser proved that these rational functions are well-defined (they do
not depend on the resolution π), by expressing them as a kind of limit of p-adic Igusa zeta functions.
We just mention that the motivic zeta function specializes to the topological zeta function and tothe various p-adic Igusa zeta functions (for almost all p).
In this paper we study a piece of a remarkable conjecture of Igusa-Denef-Loeser, relating the polesof these zeta functions to roots of the Bernstein-Sato polynomial, modeled on the result for Zϕ (f ; s).
We will treat poles of (maximal possible) order n. For the topological zeta function it is clear thatthese occur if and only if there exist n different components Ei with the same quotient νi /Ni andhaving a non-empty intersection. For the other zeta functions the situation is analogous. For thatreason we formulate everything in terms of the ‘simplest’ zeta function, being the topological one.
Our results are however valid also for the other mentioned zeta functions.
Conjecture 1. The poles of Ztop,0 (f, s) are roots of the local Bernstein-Sato polynomial bf,0 (s).
Conjecture 2. The function bf,0 (s) · Ztop,0 (f, s) is a polynomial.
Conjecture 2 is a stronger version of Conjecture 1, saying that the order of a pole s0 of Ztop,0 (f, s)
is at most the multiplicity of s0 as root of bf,0 (s). For curves (n = 2) Conjecture 1 was proved
by Loeser [17]. In that paper he also verified Conjecture 2 for reduced f . For arbitrary n these
conjectures are still wide open.
0.3. There is a well known relation between roots of Bernstein-Sato polynomials and monodromy
eigenvalues of f . In particular, if s0 is a root of bf,0 (s), then exp(2πis0 ) is an eigenvalue of themonodromy acting on some cohomology group of the (local) Milnor fibre of f at some point of
the germ of f −1 {0} at 0 (equivalently; exp(2πis0 ) is a monodromy eigenvalue on the nearby cycle
complex ψf C). So the following conjecture, relating poles of Ztop,0 (f, s) to monodromy eigenvalues,
is implied by Conjecture 1.
Conjecture 3. If s0 is a pole of Ztop,0 (f, s), then exp(2πis0 ) is an eigenvalue of the local
monodromy acting on some cohomology group of the Milnor fibre of f at some point of the germ off −1 {0} at 0.
When (f −1 {0}, 0) is a germ of an isolated singularity, then a result of Varchenko [25] relates
the multiplicity of a root of bf,0 (s) to the size of the monodromy Jordan blocks for the associatedmonodromy eigenvalue. A root of multiplicity n corresponds essentially to a Jordan block of size n(see Theorem 3 for the precise formulation).
This is certainly not true in general for non-isolated singularities: for any homogeneous
Qn f itsmonodromy is finite and hence all Jordan blocks have size 1. And for instance when f = i=1 xN
iQN
we have that bf,0 (s) = j=1 (s − j/N )n . The ‘right’ generalization of Varchenko’s result should bestated in terms of the sub-complex ψf,λ C of the nearby cycle complex ψf C; see Section 1.

MAXIMAL POLES OF ZETA FUNCTIONS, ROOTS b-FUNCTIONS AND MONODROMY JORDAN BLOCKS

30.4. In this paper we investigate for an arbitrary f in n variables, assuming that its topologicalzeta function has a pole s0 of maximal order n, the implications concerning s0 being a root of bf (s)of multiplicity n, and concerning a possible associated monodromy Jordan block of size n. In aforthcoming paper we will study the case n = 2 more in detail, in particular for non-reduced f.
0.5. With the notation of 0.2 the log canonical threshold c0 (f ) of f at 0 (resp. c(f ) of f ) is definedasc0 (f ) :=min {νi /Ni },c(f ) :=min{νi /Ni },−1i∈J:0∈π(Ei )i∈J:f{0}∩π(Ei )6=∅

see e.g. Proposition 8.5 in [15]. It does not depend on the resolution π since e.g. −c(f ) (resp.
−c0 (f )) is the root closest to the origin of the Bernstein-Sato polynomial bf (s) (resp. bf,0 (s)) of f
(at 0), see e.g. Theorem 10.6 in [15]. (In fact by results of Lichtin and Kashiwara every root of bf (s), for some i ∈ J and some integer k ≥ 0, see Theorem 10.7 in [15].)
is of the form − νiN+ki
Clearly −c0 (f ) is the candidate pole of Ztop,0 (f, s) closest to the origin. The third author formulated the following.
Conjecture 4.
(1) Ztop,0 (f, s) has at most one pole of order n.
(2) If Ztop,0 (f, s) has in s0 a pole of order n, then s0 is the pole closest to the origin of Ztop,0 (f, s).
This conjecture is proved in case n = 2 by himself [26] and with Laeremans [16] when f is nondegenerate with respect to its Newton polyhedron and in these cases s0 = −c0 (f ) in (2).
Our main result is roughly as follows. Let f : (Cn , 0) → (C, 0) be a germ of a holomorphic function
with f (0) = 0 such that s0 = −c0 (f ) is a pole of order n of Ztop,0 (f, s). Denote λ := exp(2πis0 ).
Then the λ-characteristic subspace of the (n − 1)-th cohomology of the Milnor fibre of f at 0 has
a nonzero (2n − 2)-graded part of its weight filtration. Morally ‘λ has a Jordan block of size n on
the perverse sheaf ψf C’. See Theorem 2 and Corollary 1 for a precise formulation. The result of
Varchenko implies then:Theorem 1. Let f : (Cn , 0) → (C, 0) be a germ of a non-zero holomorphic function such that(f −1 {0}, 0) is a germ of an isolated hypersurface singularity. If s0 = −c0 (f ) is a pole of order n ofZtop,0 (f, s), then (s + c0 (f ))n divides the Bernstein-Sato polynomial bf,0 (s).
In such a case there exists an integer N ≥ 1 such that c0 (f ) = 1/N and either• N = 1 and (s + 1)n divides bf,0 (s), or• N > 1 and (s + 1/N )n (s + 2/N )n . . . (s + (N − 1)/N )n (s + 1)n divides bf,0 (s).
If Conjecture 4 is true, then Theorems 2 and 1 treat in fact ‘all’ poles of maximal order n. Our proof
uses a result of Saito (Proposition 1) and the ideas in the proof of the main result of van Doorn andSteenbrink [10]. In Theorem 4 we provide a global version for polynomials f and Ztop (f, s).
Acknowledgements. The authors would like to thank A. Dimca, M. Saito and J. Steenbrink for useful
discussions and comments about the paper.
1. Preliminaries
1.1. Monodromy. Let f be a holomorphic function on an n-dimensional complex manifold X.
Denote by Xt the hypersurface f −1 {t} for t ∈ C. Let x ∈ X0 and choose ε, η > 0 with η << ε << 1.
The restriction of f to {z ∈ X |z − x| ≤ ε, 0 < |f (z)| < η} is a C ∞ fibre bundle, the Milnor
fibration, whose typical fibre
Ff,x := {z ∈ X |z − x| ≤ ε, f (z) = δ}for 0 < δ < η

is called the Milnor fibre of f at x ∈ X0 . The Milnor fibre is endowed with the monodromyqautomorphism Mf,x which induces an automorphism, denoted by Mf,x, on the cohomology groups
H q (Ff,x , C).

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A. MELLE-HERNÁNDEZ, T. TORRELLI AND WILLEM VEYS

Following Deligne [6] one has a sheaf theoretic version of the previous constructions. Let D be asmall disk around the origin in C, D∗ := D \ {0} and D̃∗ the universal covering of D∗ . Consider the
preimage X ∗ of D∗ in X and denote by X̃ ∗ the fibre product X ∗ ×D∗ D̃∗ . Let i : X0 → X be theinclusion morphism and j : X̃ ∗ → X.
For the constructible sheaf CX on X and for any q ≥ 0, the nearby cycle sheaf Rq ψf CX :=∗ q
i R j∗ j ∗ CX is a constructible sheaf on X0 . The deck transformation (x, u) 7→ (x, u + 1) on X̃ ∗induces the action of a canonical monodromy automorphism T q on Rq ψf CX such that the vectorqspace (Rq ψf CX , T q )x with automorphism is canonically isomorphic to (H q (Ff,x , C), Mf,x).
In fact, working on the derived category of complexes with automorphisms and bounded constructible cohomology, the nearby cycle complex ψf CX on X0 is defined by ψf CX := i∗ Rj∗ j ∗ CX ,see [6]. Recall that the sheaf ψf CX [n − 1] is a perverse sheaf. The monodromy T on the shiftedperverse sheaf ψf CX admits a decomposition T = Ts Tu where Ts is semisimple and Tu is unipotent.
For λ ∈ C, letψf,λ CX = Ker (Ts − λ) ⊂ ψf CX .
There are also decompositions
ψf CX =

Lλ ψf,λ CX ,H q (Ff,x , C) =

L

λH

q

(Ff,x , C)λ ,such that the action of Ts on ψf,λ CX and on H (Ff,x , C)λ is the multiplication by λ ∈ C∗ . The
groups H q (Ff,x , C)λ ⊕ H q (Ff,x , C)λ̄ = Hq (ψf,λ CX )x ⊕ Hq (ψf,λ̄ CX )x have a canonical Mixed Hodge
structure, see e.g. [19],[20],[24]. Let W be the weight filtration of this canonical mixed Hodge
structure. The following proposition is proved by M. Saito (for a proof see (1.1.3) and Proposition
1.7 in [21]).
q

Proposition 1. [21] Let N be the logarithm of the unipotent part Tu of the monodromy T . If
n−1
(Ff,x , C)λ 6= 0, then N n−1 6= 0 on ψf,λ CX in the category of shifted perverse sheaves.
GrW
2n−2 H
Since X is smooth and n-dimensional, N n = 0 on the nearby cycle sheaf ψf CX [n − 1]. Thisq
implies that the Jordan blocks of the monodromy Mf,xon the cohomology groups H q (Ff,x , C) havesize ≤ q + 1 (see e.g. [9] and references there). In fact, it is also proved there that the support ofthe perverse sheaf N n−1 ψf CX [n − 1] is empty or 0-dimensional (see Proposition 0.5 in [9]).
1.2. Bernstein-Sato polynomials. Let X be a complex n-dimensional manifold, resp. smooth
algebraic variety, and let X0 be the hypersurface defined as the zero locus of a holomorphic function,
resp. regular function, f . Let DX be the ring of analytic, resp. algebraic, partial differential
operators associated to X.
The Bernstein-Sato polynomial (or b-function) bf (s) of f is the unique monic polynomial of lowest
degree satisfying
bf (s)f s = P f s+1 with P ∈ DX [s].
It exists at least locally, and globally if X is an affine algebraic variety [2],[4],[22]. Moreover theb-function of a regular function f and of its associated analytic function coincide. Restricting to thestalk at a point x ∈ X0 , one can also define the local b-function bf,x (s). If X is Stein, resp. affine,then bf (s) is the least common multiple of these local b-functions.
Let Rf be the set of the roots of bf (−s), and mα the multiplicity of α ∈ Rf . Then Rf ⊂ Q>0 ,and mα ≤ n because bf (s) is closely related to the monodromy on the nearby cycle sheaf ψf CX , seee.g. [14]. Set αf = min Rf ; this number coincides with the log canonical threshold, see [15],[19].
2. Monodromy on ψf CX and poles of zeta functions
2.1. We are interested in poles of maximal order n of Ztop,0 (f, s). Laeremans and the third author[16] proved that every pole of maximal order of Ztop,0 (f, s) (or of Ztop (f, s)) must be of the form
−1/N , for a positive integer N ≥ 1.

MAXIMAL POLES OF ZETA FUNCTIONS, ROOTS b-FUNCTIONS AND MONODROMY JORDAN BLOCKS

5Theorem 2. Let f : (Cn , 0) → (C, 0) be a germ of a non-zero holomorphic function. If s0 = −c0 (f )is a pole of order n of Ztop,0 (f, s), thenn−1
GrW
(Ff,0 , C)λ 6= 02n−2 Hfor λ := exp(2πis0 ).

In such a case, there exists an integer N ≥ 1, such that c0 (f ) = 1/N and either
n−1• N = 1 and GrW(Ff,0 , C)1 6= 0 or
2n−2 Hn−1• N > 1 and GrW(Ff,0 , C)exp(2πi(−j/N )) 6= 0 for all j with 1 ≤ j ≤ N .
2n−2 H

Proof. Assume s0 is a pole of (maximal) order n of Ztop,0 (f, s), then write s0 = −c0 (f ) = −1/N forn−1some integer N ≥ 1 and set λ := exp(2πi(−1/N )). To show that GrW(Ff,0 , C)λ 6= 0 we will
2n−2 Hadapt the proof of the main result of van Doorn and Steenbrink in [10], see also Varchenko [25].
Let B be an open ball centered at the origin. Let π : X → B be an embedded resolution of thegerm (f −1 {0}, 0) which is an isomorphism outside of the preimage of f −1 {0}. Set E := π −1 (f −1 {0})and ExT:= π −1 (0) and denote by Ei , i ∈ J, the irreducible components of E. For I ⊂ J, put also
EI := j∈I Ej .
By the definition of the local topological zeta function, see (∗), since s0 is a pole of order n ofZtop,0 (f, s), there exist n irreducible components E0 , . . . , En−1 of E and there is a point x̃0 ∈ ∩n−1
i=0 Eisuch that π(x̃0 ) = 0 and νi /Ni = 1/N for all 0 ≤ i ≤ n − 1.
We may assume that one of these irreducible components, called e.g. E0 , is a Kähler compact nonsingular variety. Otherwise we blow up X at x̃0 and get a new configuration of exceptional divisors
where the new exceptional divisor is a Kähler compact non-singular variety E0 with ν0 /N0 = 1/N ,and we can choose a ‘new’ x̃0 on this E0 satisfying the requirements above.
n−1(Ff,0 , C)λ one uses the fact that it is pure of type (n−1, n−1)
To describe the quotient GrW
2n−2 Hand therefore a quotient of the piece F n−1 of the Hodge filtration. These terms can be computed
using the relative logarithmic de Rham complex.
Let e be a common multiple of all multiplicities Ni , i ∈ J, and let C̃ be another copy of C. LetỸ be the normalization of the space Y obtained from X by the base change σ : C̃ → C : σ(t̃) = t̃e .
Let ρ : Ỹ → X and f˜ : Ỹ → C̃ be the natural projections maps. Let Di := ρ−1 (Ei ), i ∈ J, andset D := ρ−1 (E), this is nothing but D = ∪i∈J Di . Let Dx := ρ−1 (Ex ). For every I ⊂ J, let
DI := ρ−1 (EI ). The map DI → EI is a cyclic cover of degree gcd(Ni , i ∈ I).

Ỹ

ρ

f˜

%
Y = C̃ ×C X
! 
C̃

/' X
f ◦π
σ


/C

By the semi-stable reduction theorem, π and e can be chosen in such a way that Ỹ is smooth. The
divisor D = f˜−1 (0) is a reduced normal crossing divisor, see [24]. From [23], see also [24], there is
an isomorphism
H q (Ff,0 , C) ' Hq (Dx , Ω•Ỹ /C̃ (log D) ⊗ ODx ),so in particular GrpF H q (Ff,0 , C) ' H q−p (Dx , ΩpỸ /C̃ (log D) ⊗ ODx ). Then

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A. MELLE-HERNÁNDEZ, T. TORRELLI AND WILLEM VEYS

F n−1 H n−1 (Ff,0 , C) ' Grn−1
H n−1 (Ff,0 , C) ' H 0 (Dx , Ωn−1(log D) ⊗ ODx )FỸ /C̃
' H 0 (Dx , ΩnỸ (log D) ⊗ ODx ).
The following results can be deduced from Section 4 in [25]. For every ω ∈ H 0 (B, Ωn ) define thegeometrical weight g(ω) with respect to the resolution π as
ordEi (ω) + 1.
g(ω) := min
i∈J
Ni
−e
For every ω ∈ H 0 (B, Ωn ) with geometrical weight g(ω) ≤ 1, define R(ω) := f˜ N (πρ)∗ (ω). Then
R(ω) ∈ H 0 (Ỹ , ΩnỸ (log D)). Let σ(ω) be its Poincaré residue along Dx , that is σ(ω) is the restriction−eto Dx of f˜ N +1 (πρ)∗ (ω)/df˜. Then σ(ω) is an element in F n−1 H n−1 (Ff,0 , C) and the semisimple
part of the monodromy acts on σ(ω) as

Ts (σ(ω)) = exp(−2πig(ω))σ(ω).
ord

(ω)+1

Ei
, and else R(ω) is
The form R(ω) has a first order pole along Di if and only if g(ω) =Niregular along Di .
Consider the differential form η = dx1 ∧ . . . ∧ dxn . Since s0 = −c0 (f ) = −1/N does not dependon the resolution, g(η) = 1/N ≤ 1 does not depend on π. In fact 1/N is the minimum and thenR(η) has a first order pole along Di if and only if 1/N = νi /Ni and else R(η) is regular along Di .

◦Let D00 be one of the irreducible components of D0 . On the open subspace D00= D00 \(∪j6=0 Dj ),◦◦the restriction of the map ρ from D00 to E0 is an étale cover. The form σ(η) is not equal to zero on◦◦
D00
and is in fact a meromorphic (n−1)-form with logarithmic poles on D00 \D00
. Thus R(η) defines
0na non-zero element in H (Ỹ , ΩỸ (log D)). Notice that, according to Deligne’s theorem [5],[23],[25],◦the class of σ(η) in H n−1 (D00
, C) is a non-zero element since D00 is a projective manifold and◦
D00 \ D00 ⊂ D00 is a divisor with normal crossings.
On an adequate chart on Ỹ , with local coordinates y1 , . . . , yn , the function f˜ is given byf˜(y1 , . . . , yn ) = y1 y2 . . . yk .
Moreover, if y1 = 0 is the equation of the divisor D00 (and hence of D0 ) and yj = 0 gives the divisor
D00 ∩ Dj , thendy2
dyk
∧ ... ∧
∧ dyk+1 ∧ . . . ∧ dyn ,
y2
yk


ν
1
−
.
where q(y2 , . . . , yn ) is holomorphic, q(0) 6= 0 and aj = e Nj(i)Nj(i)σ(η) = q(y2 , . . . , yn ) · y2a2 . . . ykakdyn1
In particular, at a preimage P0 of x̃0 in Ỹ , R(η) can be written locally as u dy
y1 ∧ . . . ∧ yn , withu(0) 6= 0, because of the minimum. Considering for each n-fold point P on Dx the multiple residue
map ResP : H 0 (Ỹ , ΩnỸ (log D)) → C, we have in particular that ResP0 is surjective.

Let Vλ be the set of the n-fold points of D that are preimages of those n-fold points in ∪I EI for
which |I| = n, λNi = 1 for all i ∈ I, and at least one Ei , i ∈ I, is an irreducible component of Ex .
Then
L
n−1
GrW
(Ff,0 , C)λ ∼= Image( P ∈Vλ ResP : H 0 (Ỹ , ΩnỸ (log D)) → CVλ );2n−2 Hfor the isomorphism see [23] and [10].
n−1Since P0 ∈ Vλ and ResP0 is surjective then GrW(Ff,0 , C)λ 6= 0 which concludes the proof.
2n−2 Hn−1(Ff,0 , C)exp(2πi(−j/N )) 6= 0 also for 2 ≤ j ≤ N we argue as follows,
To show that GrW
2n−2 Hsee [10]. Since the weight filtration is defined over Q, it has a complex conjugation compatible
with the one of CVλ . Let AP0 = {g(ω) : ω ∈ H 0 (B, Ωn ), g(ω) ≤ 1 and ResP0 (R(ω)) 6= 0 }. Theng(η) = N1 ∈ AP0 . If 1/N < 1, then the complex conjugate of ResP0 (R(η)) in CVλ is an eigenvector

MAXIMAL POLES OF ZETA FUNCTIONS, ROOTS b-FUNCTIONS AND MONODROMY JORDAN BLOCKS

7of the semisimple part of the monodromy Ts for the eigenvalue λ̄ = exp(2πi(1/N )). In particular
there exists η̃ ∈ H 0 (B, Ωn ) such that g(η̃) = NN−1 ∈ AP0 . After the remark on page 230 in [10] one
can prove that {1/N, 2/N, . . . , (N − 1)/N, 1} ⊂ AP0 .

2.2. Using Proposition 1 one ‘morally’ obtains a Jordan block of size n in the category of shiftedperverse sheaves.
Corollary 1. Let f : (Cn , 0) → (C, 0) be a germ of a non-zero holomorphic function. If s0 = −c0 (f )is a pole of order n of Ztop,0 (f, s) and λ := exp(2πis0 ), then N n−1 6= 0 on ψf,λ CX in the categoryof shifted perverse sheaves.
3. Applications for isolated hypersurface singularities
3.1. Letf : (C , 0) → (C, 0) be a germ of a holomorphic function such that (f −1 {0}, 0) is a germ of
an isolated hypersurface singularity. The following result, by Varchenko [25] Theorem 1.4, relatesroots of the Bernstein-Sato polynomial bf,0 (s) and Jordan blocks of the algebraic monodromy. Letb̃f,0 (s) be the microlocal (or reduced) Bernstein-Sato polynomial defined by bf,0 (s) = (s + 1)b̃f,0 (s).
nn−1Theorem 3. [25] Let Mf,0be the algebraic monodromy action on the (n − 1)-th cohomologyn−1H(Ff,0 , C) of the Milnor fibre of f at the origin.
n−1(1) b̃f,0 (s) is divisible by (s − β)n if and only if β ∈ (−1, 0) and Mf,0
has a Jordan block ofsize n for the eigenvalue exp(2πi(β)).
n−1(2) b̃f,0 (s) is divisible by (s + 1 − α)n−1 , with α ∈ Z+ , if and only if α = 0 and Mf,0
has a
Jordan block of size n − 1 for the eigenvalue 1.

Proof of Theorem 1. (see introduction) The proof follows from Theorems 2 and 3, together withn−1
n−1
(Ff,0 , C)λ 6= 0 if and only if Mf,0the fact that for the eigenvalue λ = 1 (resp. λ 6= 1), GrW
2n−2 H
has a Jordan block of size n − 1 (resp. of size n), see [10] and [23].

Theorem 4. Let f ∈ C[z1 , . . . , zn ] be a polynomial such that f −1 {0} has only isolated singularities.
If s0 = −c(f ) is a pole of order n of Ztop (f, s), then (s+c(f ))n divides the Bernstein-Sato polynomial
bf (s).
In such a case there exists an integer N ≥ 1 such that c(f ) = 1/N and either• N = 1 and (s + 1)n divides bf (s), or• N > 1 and (s + 1/N )n (s + 2/N )n . . . (s + (N − 1)/N )n (s + 1)n divides bf (s).
Proof. The proof follows from the following two facts. First, a pole of order n of the local topologicalzeta function is also a pole of the global Ztop (f, s), and conversely, a pole of order n of Ztop (f, s) isa pole of some Ztop,x (f, s) at some point x ∈ f −1 {0}. Secondly bf (s) is the least common multipleof all local Bernstein-Sato polynomials bf,x (s).

3.2. Non-degenerate Newton polyhedron. For the notion of function with non-degenerate
Newton polyhedron we refer for instance to [16] or [1]. Remark that almost all polynomials are
non-degenerate with respect to their (either local or global) Newton polyhedron (see [1] p.151).
For such functions, Denef proved that a set of candidate poles of the corresponding zeta functions
is obtained from the (n − 1)-dimensional faces of the corresponding polyhedron, e.g. see [7]. Loeser[18] proved that under some additional conditions these candidate poles are roots of the BernsteinSato polynomial bf (s).
Corollary 2. Let f : (Cn , 0) → (C, 0) be a germ of a holomorphic function which defines a germ of
an isolated hypersurface singularity which is non-degenerate with respect to its Newton polyhedronat the origin. If s0 is a pole of order n of Ztop,0 (f, s) then (s + s0 )n divides bf,0 (s).

8

A. MELLE-HERNÁNDEZ, T. TORRELLI AND WILLEM VEYS

Proof. Under the hypothesis, Laeremans and the third author proved in Theorem 2.4 in [16] that
Ztop,0 (f, s) has at most one pole of order n. Moreover, if such a pole exists, then it is the pole closest
to the origin which coincides with −c0 (f ). Thus after Theorem 1 we get the result.

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Facultad de Matemáticas, Universidad Complutense, Plaza de Ciencias 3, E-28040, Madrid, Spain
Laboratoire Jean Alexandre Dieudonné, Université de Nice - Sophia Antipolis, Faculté des Sciences,
Parc Valrose, 06108 Nice Cedex 02, France
University of Leuven, Department of Mathematics, Celestijnenlaan 200 B, B-3001, Leuven (Heverlee),Belgium
E-mail address: amelle@mat.ucm.es, tristan.torrelli@laposte.net, wim.veys@wis.kuleuven.be