On monodromy for a class of surfaces
Ann Lemahieu a , Willem Veys b
a Fachbereich Mathematik, Universität Kaiserslautern, 67653 Kaiserslauternb Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven

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Abstract
In this note we present a result on the monodromy conjecture for surfaces that are generic with respect to a toricidealistic cluster. To cite this article: A. Lemahieu, W. Veys, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
RésuméSur la monodromie pour une certaine classe de surfaces. On présente dans cette note un résultat sur laconjecture de monodromie pour les surfaces qui sont génériques pour un amas torique idéalistique. Pour citer cet
article : A. Lemahieu, W. Veys, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Version française abrégée
En 1992 Denef et Loeser ont introduit la fonction zêta topologique. Un analogue de la conjecture demonodromie, déjà établie pour la fonction zêta d’Igusa, était né pour cette nouvelle fonction zêta. Loeser
a demontré la conjecture de monodromie pour les courbes et pour les polynômes qui sont nondégénéréspour leur polyèdre de Newton et qui satisfont des conditions numériques ([L1,L2]). Artal Bartolo, CassouNoguès, Luengo, Melle Hernández ([A-B,C-N,Lu,M-H]), Rodrigues ([Ro]) et Veys ([Ve1]) ont également
traité de cette conjecture mystérieuse.
Soient f : (Cd , 0) → (C, 0) le germe d’une fonction holomorphe et π : X → Cd une résolution plongée dessingularités de f −1 {0}. On note (Ej )j∈S les composantes irréductibles de π −1 (f −1 {0}) et on note respectivement les multiplicités de Ej dans les diviseurs de f ◦π et de π ∗ (dx1 ∧. . .∧dxd ) sur X par Nj et par νj −1.
Pour I ⊂ S on pose EI := ∩i∈I Ei et EI◦ := EI \ (∪j ∈Ià/ Ej ). La fonction
P zêta topologique locale
Q associée1
f est la fonction rationnelle en une variable complexe Ztop,f (s) := I⊂S χ(EI◦ ∩ π −1 {0}) i∈I Ni s+ν.
i
Denef et Loeser ont montré que la fonction zêta topologique ne dépend pas de la résolution plongée choisie ([De,L1]). Supposons maintenant que f est une fonction en trois variables et que la surface f = 0 a
Email addresses: lemahieu@mathematik.uni-kl.de (Ann Lemahieu), wim.veys@wis.kuleuven.be (Willem Veys).
Preprint submitted to the Académie des sciences

October 3, 2007

exactement une singularité isolée 0. Alors on peut supposer que π est un isomorphisme sur le complément
de l’image inverse de l’origine. Soient (Ej )j∈J:={1,··· ,r} les composantes irréductibles exceptionnelles deπ −1 {0}. Dans l’article [A’C] A’Campo montre que la fonction zêta de monodromie ζf de f à 0 peut être
Qr◦écrite comme ζf (t) = j=1 (1 − tNj )χ(Ej ) . Sous cette description les valeurs propres de la monodromiede f sont les zéros de ζf et la conjecture de monodromie devient ‘Si s est un pôle de Ztop,f , alors e2πis
est un zéro de ζf ’. Fixons maintenant un candidat pôle s := −νj /Nj , j ∈ J, de Ztop,f . Nous écrivonsνj /Nj comme a/b avec a et b premiers entre eux et nous définissons l’ensemble
P Jb := {j ∈ J | b|Nj }. La
formule d’A’Campo implique que e2πis est un zéro de ζf si et seulement si j∈Jb χ(Ej◦ ) 6= 0.
On considère les surfaces génériques pour lesquelles il existe une résolution plongée des singularités réaliséepar l’éclatement d’une constellation torique qui est la constellation des points de base d’un idéal monomial
de support fini.
2
Ej ∼
=P

R

b

Eγ ∩ Ej

Eα ∩ EjbQ

Pb
Eβ ∩ Ej

Le contexte des constellations toriques en dimension 3 impliqueque la configuration dans Ej ∼= P2 est comme dans la Figure
1. Les seules intersections possibles avec d’autres composantes
exceptionelles sont les droites de coordonnées Eα ∩ Ej , Eβ ∩ Ej
et Eγ ∩ Ej et les points P , Q et R sont les seuls points de Ej ∼=P2 dans lesquelles il est permis d’éclater.

Figure 1P
En général, il peut y avoir beaucoup d’annulations qui font que j∈Jb χ(Ej◦ ) = 0. On obtient que χ(Ej◦ ) <0 si et seulement si la configuration dans Ej ∼= P2 consiste en (au moins trois) droites - éventuellement
exceptionelles - qui passent toutes par le même point. Ce résultat nous permet de démontrer :ThéorèmeSi χ(Ej◦ ) > 0, alors e−2πiνj /Nj est une valeur propre de la monodromie de f .

Comme application on peut vérifier la conjecture de monodromie pour ‘la plupart’ des pôles.
Corollaire Si −νj /Nj est un candidat pôle de Ztop,f d’ordre 1 qui est un pôle, alors e−2πiνj /Nj est une
valeur propre de la monodromie de f .
1. Introduction
In 1992 Denef and Loeser introduced the topological zeta function. The monodromy conjecture, firststated for the Igusa zeta function, got an analogue for this new zeta function. Loeser proved the conjecture for curves and for polynomials that are nondegenerate with respect to their Newton polyhedronand that satisfy some numerical conditions ([L1,L2]). Also Artal Bartolo, Cassou-Noguès, Luengo, Melle
Hernández ([A-B,C-N,Lu,M-H]), Rodrigues ([Ro]) and Veys ([Ve1]) provided results about this mysterious conjecture. We explain in this note by geometric arguments a phenomenon concerning calculation ofmonodromy and the monodromy conjecture for surfaces that are generic with respect to a 3-dimensional
toric idealistic cluster .
Let f : (Cd , 0) → (C, 0) be the germ of a holomorphic function and let π : X → Cd be an embedded
resolution of singularities of f −1 {0}. We write (Ej )j∈S for the irreducible components of π −1 (f −1 {0})and we denote by Nj and νj − 1 the multiplicities of Ej in the divisor on X of f ◦ π and π ∗ (dx1 ∧ . . . ∧ dxd )respectively. The couples (Nj , νj )j∈S are called the numerical data of the embedded resolution (X, π). For
I ⊂ S we denote also EI := ∩i∈I Ei and EI◦ := EI \ (∪j ∈I
associ/ Ej ). The local
P topological zeta function
Q
1
ated to f is the rational function in one complex variable Ztop,f (s) := I⊂S χ(EI◦ ∩ π −1 {0}) i∈I Ni s+ν
.
iIndeed, Denef and Loeser proved that this expression does not depend on the chosen embedded resolution2

(see [De,L1]). From now on suppose that f is a function in three variables and that the surface f = 0
has only one isolated singularity 0. Then we may suppose that π is an isomorphism outside the inverse
image of the origin. Say that the (Ej )j∈J:={1,··· ,r} are the irreducible exceptional components of π −1 {0}.
In [A’C] A’Campo showed that the zeta function of monodromy ζf of f at 0 can then be written asQ◦ζf (t) = rj=1 (1 − tNj )χ(Ej ) . Under this description, the eigenvalues of monodromy of f are the zeroes ofζf and the monodromy conjecture becomes ‘If s is a pole of Ztop,f , then e2πis is a zero of ζf ’. Now fix acandidate pole s := −νj /Nj , j ∈ J, of Ztop,f . We write νj /Nj as a/b such that a and b are coprime and
2πis
we define theis a zero of ζf if
P set Jb :=◦ {j ∈ J | b|Nj }. It follows from A’Campo’s formula that eand only if j∈Jb χ(Ej ) 6= 0.
To introduce the class of surfaces we work with, we first say something about clusters. For details we refer
to [C,G-S,L-J]. Let X be a nonsingular variety of dimension d ≥ 2 and Z be obtained from X by a finite
succession of point blowing-ups. A point Q ∈ Z is infinitely near to a point P ∈ X if P is in the imageof Q. Let Q0 ∈ X =: X0 . A constellation with origin Q0 is a finite sequence C := {Q0 , Q1 , · · · , Qr−1 } ofpoints such that each Qj is infinitely near to Q0 and is a point on the variety Xj obtained by blowing
up Qj−1 in Xj−1 , j ∈ {1, · · · , r − 1}. For each Qi ∈ C, denote the exceptional divisor of the blowing-up
in Qi and its strict transform at any stage by Ei . If Qj ∈ Ei , then Qj is proximate to Qi ; we write
j → i. A d-dimensional toric constellation with origin Q0 is a constellation C := {Q0 , Q1 , · · · , Qr−1 }such that each Qj is a 0-dimensional orbit in the toric variety Xj obtained by blowing up Qj−1 in Xj−1 ,1 ≤ j ≤ r − 1. A tree with a root such that each vertex has at most d following adjacent vertices is calleda d-ary tree. There is a natural bijection between the set of d-dimensional toric constellations with origin
and the set of finite d-ary trees with a root, with the edges labeled with positive integers not greaterthan d, such that two edges with the same source have different labels. The labels indicate in which affine
chart the points are created. A cluster A := (C, m) consists of a constellation C := {Q0 , · · · , Qr−1 } and asequence m := (m0 , · · · , mr−1 ) of nonnegative integers. The idea of clusters is to express that a system ofhypersurfaces is passing through the points of the constellation with (at least) the given multiplicities. An
ideal I in OX,Q0 is finitely supported if I is primary for the maximal ideal of OX,Q0 and if there exists a
constellation C of X such that IOXr is an invertible sheaf. An infinitely near point Q of Q0 is a base pointof I if Q belongs to the constellation with the minimal number of points, say r, with the above property.
The canonical map Xr → X associated to the constellation of base points of a finitely supported ideal I
to X is an embedded resolution of the subvariety of (X, Q0 ) defined by a general enough element in I.
We will consider generic surfaces for which there exists an embedded resolution of singularities realised by
the blowing-up of a toric constellation with origin that is the constellation of base pointsPof a finitely supported monomial ideal. In general there can be a lot of cancellations which make that j∈Jb χ(Ej◦ ) = 0.
We study when χ(Ej◦ ) < 0 for some j ∈ J. We obtain that χ(Ej◦ ) < 0 if and only if the configuration in
Ej ∼= P2 consists of (at least three) lines - possibly exceptional - that are all going through the same point.
Using this result we prove:
TheoremIf χ(Ej◦ ) > 0, then e−2πiνj /Nj is an eigenvalue of monodromy of f .

As an application we can verify the assertion in the monodromy conjecture for ‘most’ poles.
Corollary If −νj /Nj is a candidate pole of Ztop,f of order 1 that is a pole, then e−2πiνj /Nj is aneigenvalue of monodromy of f .
The results presented in this note are shown by geometric arguments and cover unknown cases of themonodromy conjecture. In a forthcoming paper we will actually prove the monodromy conjecture in thiscontext. That proof is completely combinatorial.
32. Determination of the cases in which appears a negative χ(Ej◦ )Let I ⊂ C[x, y, z] be a finitely supported monomial ideal and let S be a surface given by a generalelement in I. We will write Ŝ for its strict transform, whatever the stage is. We will denote the curves
Ej ∩ Ŝ ⊂ Ej ∼= P2 by Cj , j ∈ J, and their strict transforms by Ĉj .
Lemma 2.1 x, y or z can not be a factor in the equation of Cj .
Proof. Suppose that x is a factor in the equation of Cj . Then the monomial ideal I would not be finitely
supported. Indeed, at the moment that, say x appears as factor in the equation of Cj , the strict transformof Cj in some affine chart takes the form xgx + zgz , for some polynomials gx , gz in C[x, y, z]. It is now
obvious that the ideal can not be principalised by blowing up a toric constellation.

Lemma 2.2 The curve Cj ⊂ P2 is generically reducible if and only if its equation is of the form h(m, n) =0 with h a homogeneous polynomial in two variables of degree at least 2, and m and n two monomials.
Proof. The claim follows from [Bo,Deb,Na, Thm. 1.2].
Notice also that the setting of toric constellations in dimension 3 implies that the configuration in Ej ∼= P2is as in Figure 1. The only possible intersections with other exceptional components are the coordinatelines Eα ∩ Ej , Eβ ∩ Ej and Eγ ∩ Ej and the points P , Q and R are the only points in Ej ∼= P2 in which
it is allowed to blow up.
Lemma 2.3 If the equation of Cj contains three variables, then χ(Ej◦ ) ≥ 0.
Proof. Suppose that χ(Ej◦ ) < 0. We split cases according to whether Cj is irreducible or not.
(i) If the curve Cj is irreducible, then χ(Cj ) ≤ 2 and if χ(Ej◦ ) < 0, the configuration in Ej should consistof Cj and two lines such that these three curves intersect in exactly one point, say P . Moreover χ(Cj )should then be equal to 2. When the degree d of Cj is at least 2 and if one of the lines would not be aprincipal tangent line to Cj in P , then this line would intersect Cj in another point. This is a contradiction. Hence, Cj should be analytically reducible in P , and consequently χ(Cj ) < 2. So suppose now thatdeg(Cj ) = 1. In an affine chart where one can see P , one can write Eα ↔ x = 0, Eβ ↔ y = 0 and then
Cj should have an equation of the form cx x + cy y = 0 with cx and cy complex numbers. However, theequation of Cj contains the three variables.
(ii) Suppose now that the curve Cj is reducible. Lemma 2.2 implies that the equation of Cj is of the formh(m, n) = 0 with h a homogeneous polynomial in two variables of degree at least 2, and m and n two
monomials. Then by Lemma 2.1 one easily verifies that (m, n) must be of the form (xa y b , z a+b ). We may
suppose moreover that a and b are coprime (otherwise we change h). The decomposition in irreducible
Qk
components of Cj is then i=1 (xa y b − ci z a+b ) with the ci ∈ C∗ . As the curves xa y b − ci z a+b = 0 are
rational with Euler characteristic 2, we get χ(Ej◦ ) = 0 or χ(Ej◦ ) = 1, depending on the position and thenumber of exceptional components that are present.

We continue to search for cases in which appears an exceptional component Ej for which χ(Ej◦ ) < 0.
It follows by Lemma 2.1 that the equation of Cj can not contain just one variable. The remaining caseto investigate is when the equation of Cj contains exactly two variables, say x and y. As Cj then has ahomogeneous equation in two variables, say of degree m, it follows that the curve Cj consists of m lines
having exactly one point in common. From Lemma 2.1 it follows that xm and y m certainly appear in theequation of Cj . The point Qj is then contained in a subcluster of the form

4

m
b

3

Qt

m
b

Qt+1

3

m

3

b

m
b

Qj

Qt+2

3

m
b

m’

3

Ql

b

(1)

Ql+1

In this chain Qt is the point with the lowest level for which an edge with label 3 is leaving and that
also has multiplicity m. We suppose that Ql is the point in the chain with the highest level for which its
multiplicity is equal to m. The point Qj can be equal to Qt and Ql+1 can be absent. As Cj is of degreem, the point Qj has multiplicity m on the surface. Obviously the multiplicity of Qj+1 is also m.
We will now study χ(Ej◦ ).
- If Qj = Qt , then the configuration in Ej ∼= P2 is as in Figure 2. If the exceptional line Eγ ∩ Ej is◦present, then χ(Ej ) is always equal to 0. Suppose now that Eγ ∩ Ej does not appear. This means that
Qt is moreover the point with the lowest level in the chain from which an edge with label 3 is leaving.
If Qj is the origin of the constellation, then χ(Ej◦ ) = 2 − m. If there is exactly one point, say Qα , forwhich j → α, then χ(Ej◦ ) = 1 − m. Finally, if there exist two points, say Qα and Qβ , for which j → α
and j → β, then χ(Ej◦ ) = −m.
- If j ∈ {t + 1, · · · , l − 1}, then the configuration in Ej is as in Figure 3 and then χ(Ej◦ ) = 0.
Eγ ∩ Ej2
Ej ∼=P

Ej ∩ Ej−1

2
Ej ∼
=P

Eα ∩ Ej

Eα ∩ Ej

Eβ ∩ Ej

Eβ ∩ Ej

Figure 2

Figure 3

Conclusion: χ(Ej◦ ) < 0 if and only if the configuration in Ej ∼= P2 consists of (at least three) lines possibly exceptional - that are all going through the same point.
3. Application
Lemma 3.1 Let χ(Et◦ ) < 0 such that we are in the situation (1).
(i) If a set Jb contains the index t, then it also contains the indices in {t + 1, · · · , l}.
νl(ii) If N= dc with c and d coprime, then t ∈/ Jd .
l
Proof. If we denote the numerical data of Et by (N, ν), then, independently of the number of points
Qs for which t → s, one easily computes that the numerical data for i ∈ {t + 1, · · · , l} are Ei ((i − t +1)N, (i − t + 1)ν − (i − t)). Now the first assertion follows immediately. To see the second claim, supposethat t ∈ Jd . Then d | N which implies that l−t+1|(l−t+1)ν−(l−t). This contradiction closes the proof. 
Theorem 3.2 If χ(Ej◦ ) > 0, then e−2πiνj /Nj is an eigenvalue of monodromy of f .
Proof. Suppose that Ej is an exceptional component such that χ(Ej◦ ) > 0. To prove that e−2πiνj /Nj is
an eigenvalue of monodromy of f , we show that e−2πiνj /Nj is a zero of ζf . We write νj /Nj as a/b with aand b coprime. If Jb does not contain an index t for which χ(Et◦ ) < 0, then there is nothing to verify. So
suppose now that χ(Et◦ ) < 0 and that t ∈ Jb . From Lemma 3.1 it follows that Ej 6= El and that l ∈ Jb .
We will show that χ(Et◦ ) + χ(El◦ ) ≥ 0. Let us therefore study the configuration in El ∼= P2 .
m
m−1
m
The equation of Cl−1 is of the form c0 x + c1 x
y + · · · + cm y , with ck ∈ C (0 ≤ k ≤ m) and c0 and
cm different from 0. It follows from Lemma 2.1 that the equation of Cl is then equal to c0 xm + c1 xm−1 y +· · · + cm y m + zg(x, y, z) for some polynomial g ∈ C[x, y, z], g 6= 0. In particular we have that Cl cannot
be of the form as in Lemma 2.2 and thus that it is irreducible. Hence χ(Cl ) ≤ 2.
5

El ∩ El−1

2
El ∼
=P

El ∩ El−1

P

2
El ∼
=P

2
El ∼
=P

b

Eβ ∩ ElEα ∩ ElEl ∩ EαEl−1 ∩ El

Figure 4

Figure 5

Figure 6

If Qt is the origin of the constellation, then the configuration in El ∼= P2 is like in Figure 4. The curve◦
Cl can be singular. We get χ(El ) = χ(El ) − (χ(Cl ) + (χ(El−1 ) − m)) ≥ 3 − (2 + (2 − m)) = m − 1 andχ(Et◦ ) + χ(El◦ ) ≥ (2 − m) + (m − 1) = 1. Suppose that there exists exactly one point, say Qα , for whicht → α. Then χ(El◦ ) is minimal if Cl and El ∩ Eα intersect only in one point, as in Figure 5. Again we
find χ(El◦ ) ≥ m − 1. Then χ(Et◦ ) + χ(El◦ ) ≥ (1 − m) + (m − 1) = 0. When there exist two points, say Qα
and Qβ , to which Qt is proximate, then χ(El◦ ) is minimal when Eα , Eβ and Cl intersect in exactly one
point, say in P , see Figure 6. Moreover, to have χ(El◦ ) = m − 1, one should also require that χ(Cl ) = 2.
If m ≥ 2, then Eα ∩ El and Eβ ∩ El are principal tangent lines to Cl in P but then χ(Cl ) ≤ 1. If m = 1,
then the equation of Cl must be x + cy y + cz z = 0 for some cy , cz ∈ C∗ . But then Cl intersects the threecoordinate lines in three different points and χ(El◦ ) = 1. It follows that we alwaysP
have that χ(El◦ ) ≥ m◦◦
and thus χ(Et ) + χ(El ) ≥ −m + m = 0. This study permits us to conclude that i∈Jb χ(Ei◦ ) > 0. 
Corollary 3.3 If −νj /Nj is a candidate pole of Ztop,f of order 1 that is a pole, then e−2πiνj /Nj is aneigenvalue of monodromy of f .
Proof. In [Ve2] it is shown that then there exists an exceptional component Ek such that νk /Nk = νj /Nj
and such that χ(Ek◦ ) > 0. The result follows now immediately from Theorem 3.2.
These results are particular for this context. In [Ve1] are given many examples where appear negative
χ(Ej◦ ) while the configuration in Ej = P2 does not consist of lines. It also often happens that positive
χ(Ej◦ ) does not imply that e−2πiνj /Nj is an eigenvalue of monodromy of f .
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[A-B,C-N,Lu,M-H] E. Artal-Bartolo, P. Cassou-Noguès, I. Luengo, A. Melle-Hernández, Monodromy conjecture for somesurface singularities, Ann. Scient. Ec. Norm. Sup. 35 (2002), 605-640.
[Bo,Deb,Na] A. Bodin, P. Dèbes, S. Najib, Irreducibility of hypersurfaces, preprint.
[C,G-S,L-J] A. Campillo, G. Gonzalez-Sprinberg, M. Lejeune-Jalabert, Clusters of infinitely near points, Math. Ann. 306(1996), 169-194.
[De,L1] J. Denef and F. Loeser, Caractéristique d’Euler-Poincaré, fonctions zêta locales et modifications analytiques, J.
Amer. Math. Soc. 5, 4 (1992), 705-720.
[De,L2] J. Denef and F. Loeser, Motivic Igusa zeta functions, J. Alg. Geom. 7 (1998), 505-537.
[L1] F. Loeser, Fonctions d’Igusa p-adiques et polynômes de Bernstein. Amer. J. Math. 110 (1988), 1-22.
[L2] F. Loeser, Fonctions d’Igusa p-adiques, polynômes de Bernstein, et polyèdres de Newton, J. reine angew. Math. 412(1990), 75-96.
[Ro] B. Rodrigues, On the monodromy conjecture for curves on normal surfaces, Math. Proc. of the Cambridge PhilosophicalSociety 136 (2004), 313-324.
[Ve1] W. Veys, Poles of Igusa’s local zeta function and monodromy, Bull. Soc. Math. France 121 (1993), 545-598.
[Ve2] W. Veys, Vanishing of principal value integrals on surfaces, J. Reine Angew. Math. 598 (2006), 139-158.

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