VANISHING OF PRINCIPAL VALUE INTEGRALS ON SURFACES

Willem Veys
Abstract. Principal value integrals are associated to multi–valued rational differential
forms with normal crossings support on a non–singular algebraic variety. We prove theirvanishing on rational surfaces in the context of a conjecture of Denef–Jacobs. As an application we obtain a strong vanishing result for candidate poles of p–adic and motivic Igusa
zeta functions.

Introduction
0.1. Real and p–adic principal value integrals were first introduced by Langlands in the
study of orbital integrals [Lan1] [Lan2] [LS1] [LS2]. They are associated to multi–valued
differential forms on real and p–adic manifolds, respectively.
Let for instance X be a non–singular projective algebraic variety of dimension n overQp (the field of p–adic numbers). Denoting by ΩnX the vector space of rational differential
n–forms on X, take ω ∈ (ΩnX )⊗d defined over Qp ; we then write formally ω 1/d and consider
it as a multi–valued rational differential form on X.
We suppose that the support | div ω| of div ω has normal crossings (over Qp ) on X;
P
say Di , i ∈ S, are its irreducible components. Let div ω 1/d := d1 div ω = i∈S (αi − 1)Di ,where then the αi ∈ d1 Z. If ω 1/d has no logarithmic poles, i.e. if all αi 6= 0, the principal
Rvalue integral P V X(Qp ) |ω 1/d |p of ω 1/d on X(Qp ) is defined as follows. Cover X(Qp )by (disjoint) small enough open balls B on which there exist local coordinates x1 , . . . , xn
suchthat all Di are coordinate hyperplanes. Consider for each B the converging integral
R
s 1/d
|x
|p for s ∈ C with R(s) >> 0, take its meromorphic continuation to1 x2 · · · xn |p |ω
B
C and evaluate this in s = 0; then add all these contributions. One can check that theresult is independent of all choices.
In the real setting we proceed similarly but then we also need a partition of unity,
and we have to assume that ω 1/d has no integral poles, i.e. the αi ∈/ Z≤0 . Here theindependency result is somewhat more complicated; it was verified in detail in [Ja1].
0.2. Denef and Jacobs proved a vanishing result for real principal value integrals, andconjectured a similar statement in the p–adic case. In both cases let L(ω 1/d ) be thelocally constant sheaf of C–vector spaces on X \ | div ω| associated to ω 1/d . It has rank 1,
a non–zero section on a connected open being an analytic branch of ω 1/d multiplied witha complex number. (In the p–adic case we choose an embedding of Qp into C.)
1991 Mathematics Subject Classification. 14J26 (11S80 14E15 28B99 14F17).
Key words and phrases. Principal value integral, multi–valued differential form, surface, motivic zetafunction.

1

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WILLEM VEYS

Theorem [DJ, 1.1.4][Ja1]. Let X be a non–singular projective algebraic variety of(complex)dimension n, defined over R. If H n (X(C) \ | div ω|, L(ω 1/d )) = 0, thenR
P V X(R) |ω 1/d | = 0.
Conjecture [DJ, 1.2.2]. Let X be a non-singular projective algebraic
variety, definedRi
1/dover Qp . If H (X(C) \ | div ω|, L(ω )) = 0 for all i ≥ 0, then P V X(Qp ) |ω 1/d |p = 0.
(The authors are cautious and mention that perhaps one has to suppose also some good
reduction mod p and that all αi 6∈ Z.)
0.3. In [Ve7] we ‘upgraded’ p–adic principal value integrals to motivic ones, in thesame spirit as how motivic integration and motivic zeta functions were inspired by(usual) p–adic integration and p–adic Igusa zeta functions. See [DL2][DL3] or the surveys[DL4][Loo][Ve6] for these notions.
More precisely, let X be a non–singular algebraic variety (say over C) P
of dimension n
1/d
1/d
and ω
a multi–valued differential form on X. Let as above div(ω ) = i∈S (αi −1)Di
be a normal crossings divisor. We denote also DI◦ := (∩i∈I Di ) \ (∪`6∈I D` ) for I ⊂ S; so`
X = I⊂S DI◦ . If ω 1/d has no logarithmic poles, then the motivic principal value integralof ω 1/d on X is
Z
X
Y L−1
PV
ω 1/d = L−n[DI◦ ].
Lαi − 1X
I⊂S

i∈I

Here [·] denotes the class of a variety in the Grothendieck ring of algebraic varieties, andL := [A1 ], see (1.1). We refer to [Ve7, §2] for a motivation for this expression. Note for
instance that this is just the formula for the converging motivic integral associated to theQ–divisor div(ω 1/d ) if all αi > 0.
It is natural to ‘upgrade’ also the conjecture in (0.2) to this setting (maybeRalso assuming that all αi 6∈ Z) : if H i (X(C)\| div ω|, L(ω 1/d )) = 0 for all i ≥ 0, then P V X ω 1/d = 0.
0.4. In this paper we attack (the motivic version of) the conjecture for surfaces, more
precisely rational surfaces.
We note that, in dimension 2, this is the crucial class to study : there is no ‘classification’ for configurations | div ω| ⊂ X with all H i (X \ | div ω|, L(ω 1/d )) = 0 on a rationalsurface X. For non–rational surfaces it is conceivable that the conjecture can be approached through the classification of such configurations from [GP] and [Ve3]. We plan
to report on this later. (Also for the applications that we will prove here, the class ofrational surfaces is the crucial one, see (0.7).)
Our main theorem is as follows.
Theorem. Let X be a non–singular projective rational surface and ω 1/d a multi–valued
differential form on X without logarithmic poles (in particular | div ω| has normal crossings).
R(1) Suppose that B := | div ω| is connected. If χ(X \ B) ≤ 0, then P V X ω 1/d = 0.
(2) More generally, let B be any
normal crossings divisor satisfying B ⊃R connected
1/d| div ω|. If χ(X \ B) ≤ 0, then P V X ω= 0.

VANISHING OF PRINCIPAL VALUE INTEGRALS ON SURFACES

3

Statement (1) is somewhat weaker than the conjecture because of the connectivity condition. On the other hand it is clearly stronger since we assume only that χ(X \ B) ≤ 0,
instead of the vanishing of all H i (X \ B, L(ω 1/d )). (And we do not need the extra assumption αi 6∈ Z.) The generalization (2) is important in view of the applications on
motivic zeta functions and is natural in that context, see (0.8).
The important ingredients in our proof are the structure theorem of [Ve3] for suchconfigurations B ⊂ X with χ(X \ B) ≤ 0, and the notion of a more general principal
value integral when ω 1/d is allowed to have ‘some’ logarithmic poles, see (1.4).
0.5. Real and p–adic principal value integrals appear as coefficients of asymptotic expansions of oscillating integrals and fibre integrals, and as residues of poles of distributions|f |λ and p–adic Igusa zeta functions, respectively. See [Ja2, §1] for an overview and[AVG][De2][Ig1][Ig2][Ja3][Lae] for more details.
We have in particular that the cancelation of some given candidate pole of a p–adic
Igusa zeta function is essentially equivalent to the vanishing of an associated p–adicprincipal value integral. Here we are interested in the analogous phenomenon for poles ofthe motivic zeta function versus associated motivic principal value integrals.
0.6. Denef and Loeser [DL2] associated to a non–constant regular function f on a non–singular algebraic variety M of dimension n + 1 its motivic zeta function Zmot (f ; T ); here
T is a variable. They obtained the following formula for it in terms of an embedded
resolution h : Y → M of the hypersurface {f = 0}. Let Ej , j ∈ K, be the irreducible
components of h−1 {f = 0} and Nj and νj −1 the multiplicity of Ej in the divisors div(f ◦h)and div(h∗ dx), respectively, where dx is a local generator of the sheaf of (n + 1)–forms
on M . Denote EJ◦ := (∩j∈J Ej ) \ (∪`6∈J E` ) for J ⊂ K. Then
Zmot (f ; T ) = L−(n+1)

X[EJ◦ ]J⊂K

Y (L − 1)T Njj∈JLνj − T Nj.

Fix a ‘candidate pole’ Lνj /Nj of Zmot (f ; T ); see (3.4) for more explanations. In the genericsituation that νj /Nj 6= νi /Ni for all i 6= j, the cancelation of the candidate pole Lνj /Nj
is equivalent to the vanishing of ‘its residue’
L−n

X
I⊂Sj

[DI◦ ]Y L−1,
Lαi − 1i∈Iwhere Sj = {i ∈ K \ {j} | Ei intersects Ej }, Di := Ej ∩ Ei and αi = νi − (νj /Nj )Ni for
i ∈ Sj , and DI◦ is as usual.
looks like a motivic principal value integral on Ej , andR This1/dindeed it is equal to P V Ej ω , where ω 1/d is some Poincaré residue; see (3.5).
Suppose now that Ej is mapped to a point by h. Denote Bj := ∪i∈Sj Di . The famous
Monodromy Conjecture predicts more or less that (generically), if (−1)n χ(Ej \ Bj ) ≤ 0,then Lνj /Nj is no pole of Zmot (f ; T ), see (3.5).
0.7. When n = 2, we now can prove this expected cancelation of candidate poles as aconsequence of our Main Theorem. More precisely, we may suppose that h is a composition of blowing–ups as in [Hi]. Then the exceptional surface Ej is created during the

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WILLEM VEYS

resolution process h either as a projective plane by blowing up a point, or as a ruled
surface by blowing up a non–singular curve. Then, with Bj as above, we can show as
corollary of (0.4):
Theorem. Let χ(Ej \ Bj ) ≤ 0.
(1) If Ej is created by blowing up a point, then we have always that Lνj /Nj is no pole.
(2) If Ej is created by blowing up a rational curve, then Lνj /Nj is no pole whenever
Bj is connected.
In fact we obtain a somewhat stronger statement; see (0.8(i)) below. But first we wantto comment on this new result.
Concerning (1), the best general result up to now [Ve2] was the analogous cancelationin the context of p–adic Igusa zeta functions when the centre of the blowing up is apoint of multiplicity at most 4 on the strict transform of {f = 0}. This was achieved
using a lengthy classification of all possible configurations with χ(Ej \ Bj ) ≤ 0 under thismultiplicity restriction.
Concerning (2), it is important to note that non–connected intersection configurations
on Ej are very rare, see (3.7). And moreover, the case of rational centres is the crucial
one. When Ej is created by blowing up a curve of genus g ≥ 2 we already proved the
expected cancelation in [Ve2], and we can now also handle the case g = 1 completely by
combining [Ve2] and [Ve3] with recent work of Rodrigues [Ro2].
0.8. Remarks. (i) In fact we assume only that νj /Nj 6= νi /Ni for the i ∈ Sj , and weshow that ‘the contribution of Ej to the residue of Lνj /Nj ’ vanishes when expected.
(ii) Similar vanishing results hold in the context of much more general motivic zetafunctions, for instance those of [Ve4] associated to an effective Q–Cartier divisor on any
Q–Gorenstein threefold (instead of just a hypersurface on A3 ). On the other hand, (0.7)
specializes to the context of Hodge, topological and p–adic zeta functions.
(iii) Note that in (0.6) it is possible that some αi = 1, and thus that Bj (:= ∪i∈Sj Di ) !
| div ω 1/d |. So the more general setting of part (2) of our main theorem pops up naturally
in this context of poles of zeta functions.
0.9. We will work over the base field C of complex numbers. In §1 we introduce the moregeneral principal value integrals on surfaces, allowing ‘some’ logarithmic poles, which weneed for the proof of the main theorem in §2. Then in §3 we obtain as a corollary thecancelation of candidate poles for zeta functions.
Acknowledgement. We would like to remark that [ACLM1, §2] contributed to our inspiration for the proof of the main theorem. And we thank Bart Rodrigues for his usefulremarks and suggestions.
1. Generalized principal value integrals on surfaces
1.1. We first recall briefly the notion of Grothendieck ring of varieties and related constructions.

VANISHING OF PRINCIPAL VALUE INTEGRALS ON SURFACES

5(i) The Grothendieck ring K0 (Var) of complex algebraic varieties is the free abeliangroup generated by the symbols [V ], where V is a variety, subject to the relations [V ] =[V 0 ] if V is isomorphic to V 0 , and [V ] = [V \ W ] + [W ] if W is closed in V . Its ring
structure is given by [V ] · [W ] := [V × W ]. (This ring is quite mysterious; see [Po] for the
recent proof that it is not a domain.) Usually, one abbreviates L := [A1 ].
For the sequel we need to extend K0 (Var) with fractional powers of L and to localize.
Fix d ∈ Z>0 ; we consider
K0 (Var)[L−1/d ] :=K0 (Var)[T ](LT d − 1)(where L−1/d := T̄ ). We then localize this ring with respect to the elements Li/d − 1,i ∈ Z \ {0}. What we really need is the subring of this localization generated by K0 (Var),
L−1 and the elements (L − 1)/(Li/d − 1), i ∈ Z \ {0}; we denote this subring by Rd . (Wedo not know whether or not Rd has zero divisors.)(ii) For a variety V , we denote by hp,q (Hci (V, C)) the rank of the (p, q)–Hodge component in the mixed Hodge structure of the ith cohomology group with compact support of
V . The Hodge polynomial of V is
X¡X¢(−1)i hp,q (Hci (V, C)) up v q ∈ Z[u, v].
H(V ) = H(V ; u, v) :=
p,q

i≥0

Precisely by the defining relations of K0 (Var), there is a well–defined ring homomorphism
H : K0 (Var) → Z[u, v], determined by [V ] 7→ H(V ). It induces a ring homomorphism H
from Rd to the ‘rational functions in u, v with fractional powers’.
(iii) The topological Euler characteristic χ(V ) of a variety V satisfies χ(V ) = H(V ; 1, 1)
and we obtain a ring homomorphism χ : K0 (Var) → Z, determined by [V ] 7→ χ(V ). Since
χ(L) = 1, it induces a ring homomorphism χ : Rd → Q by declaring χ((L − 1)/(Li/d −1)) = i/d (see e.g. [DL2][Ve4] for similar constructions).
1.2. On surfaces we want to extend the notion of principal value integral of (0.3) in
two ways. First we will allow the differential form ω 1/d to have ‘some’ logarithmic poles.
Another somewhat technical generalization consists in considering a normal crossings
divisor whose support contains | div ω|. In view of the application on candidate poles ofzeta functions, this is natural; see (0.8(iii)) or (3.5). More precisely we fix the following
setting.
1.3. Let X be a non–singular projective surface, ω 1/d a multi–valued differential form on
X, and D = ∪i∈T Ci P
a normal crossings divisor on X satisfying D ⊃ | div ω|. We write
1/dformally div ω= i∈T (αi − 1)Ci , where thus αi = 1 if Ci 6⊂ | div ω|. We put thefollowing restriction on the possible logarithmic poles of ω 1/d .
If for i ∈ T we have αi = 0, then
(1) Ci is rational,(2) no C` that intersects Ci has α` = 0,
(3) in all but at most two intersection points of Ci with other C` the intersecting curveC` has α` = 1.
We call such a pair (D, ω 1/d ) allowed.

6

WILLEM VEYS

When αi = 0 the adjunction formula (KX + Ci ) · Ci = KCi and the restrictions above
easily yield that either two curves Ci1 and Ci2 with αi1 6= 1 6= αi2 intersect Ci and then
αi1 + αi2 = 0, or only one curve Ci1 with αi1 6= 1 intersects Ci and then αi1 = −1. (Hereand further on, when applying the adjunction formula we always consider div ω 1/d as arepresentative of KX .)1.4. Definition. Let X be a non–singularprojective surface and (D, ω 1/d ) an allowedPpair on X. We write div ω 1/d = i∈T (αi − 1)Ci as in (1.3) and CI◦ := (∩i∈I Ci ) \(∪`6∈I C` )for I ⊂ T as before. Furthermore for i ∈ T with αi = 0 we denote by Ci · Ci the self–intersection number of Ci and by Cj , j ∈ Ti (⊂ T ), the curves that intersect Ci . To
(D, ω 1/d ) we associate the invariantX

EX (D, ω 1/d ) :=[CI◦ ]

I⊂T∀i∈I:αi 6=0

X
Y L−1
Y L−1
+
(−C
·
C
)
,
i
i
Lα i − 1
Lαj − 1
i∈T
αi =0

i∈I

j∈Ti

Q
living in Rd . In the last sum the expression (−Ci · Ci ) j∈Ti LL−1
is the easiest to write
αj−1
down ‘uniformly’, but, since at most two of the αj are different from 1, this expression
boils down to the following.
(1) If Ci intersects two curves Ci1 and Ci2 with αi1 6= 1 6= αi2 , we get(−Ci · Ci )(L − 1)2(L − 1)2 Lαi1
(L − 1)2 Lαi2
=
(C
·
C
)
=
(C
·
C
)
.
i
i
i
i
(Lαi1 − 1)(Lαi2 − 1)
(Lαi1 − 1)2(Lαi2 − 1)2(2) If Ci intersects only one curve Ci1 with αi1 6= 1, we get(−Ci · Ci )L−1
= (Ci · Ci )L .
Lα i 1 − 1

Note. (i) In fact the Ci with αi = 1, i.e. those Ci not belonging to | div ω|, play no role inthe definition of EX (D, ω 1/d ): we could as well consider instead of T only {i ∈ T | Ci ⊂| div ω|}. So this invariant is really an invariant of ω 1/d only. However, for the sequel it
is useful to introduce it as above.
(ii) As a motivation for the expression for the contribution of Ci with αi =R 0: it isa kind of limit of the ‘total contribution of Ci ’ in the formula of (0.3) for P V X ω 1/d if
αi 6= 0. See e.g. [Ve5, 3.3].
Ci 1 Ci 2
Ci...

Ci 1 Ci 2

Cj
•P

h
←−

Figure 1

Ci

...

.....
..

Cj
.....
..

..........
..
... ..
... .....
..
.
.
..
...
..
...
.
.
..
..

C
•

VANISHING OF PRINCIPAL VALUE INTEGRALS ON SURFACES

7

1.5. Lemma. Let X be a non–singular projective surface, P ∈ X and h : X̃ → X
1/d
the blowing–up
P of X with centre P . Let (D, ω ) be allowed on X, writing as usual
1/d
div ω
= `∈T (α` − 1)C` . Then
(1) (h−1 D, h∗ ω 1/d ) is allowed on X̃,(2) EX̃ (h−1 D, h∗ ω 1/d ) = EX (D, ω 1/d ) except when (on X) there exists a curve Ci with
αi = 0 and curves Ci1 and Ci2 , intersecting Ci , with αi1 + αi2 = 0 and {αi1 , αi2 } 6={−1, 1}, such that P ∈ Ci , P 6∈ Ci1 and P 6∈ Ci2 . (And in this exceptional case we dohave inequality.)
Proof. The necessary (easy) computations are essentially in [ACLM1] and [Ve5, 3.5]. We
just illustrate the exceptional case, see Figure 1. Note that in this case, since (D, ω 1/d )is allowed, if there is another curve Cj passing through P , it must satisfy αj = 1. And
by Note (1.4(i)) we may as well assume that only Ci1 and Ci2 intersect Ci .
Let C denote the exceptional curve of h. Then h−1 D consists of the union of C andthe strict transforms of the C` . And since αi = 0 we havePthat C does not appear in
the divisor of h∗ ω 1/d . So we write formally div(h∗ ω 1/d ) = `∈T (α` − 1)C` + (α − 1)Cwith α = 1. We have to compare the contributions of Ci to EX (D, ω 1/d ) and of Ci ∪ C
to EX̃ (h−1 D, h∗ ω 1/d ). These are−(Ci · Ci )and−(Ci · Ci − 1)

(L − 1)2(Lαi1 − 1)(Lαi2 − 1)

(L − 1)2+ L,(Lαi1 − 1)(Lαi2 − 1)2

(L−1)
respectively. Their difference (Lαi1 −1)(L+ L is nonzero (in Rd ). (On the otherαi
2 −1)hand, when {αi1 , αi2 } = {−1, 1}, this difference would be −L + L = 0.) ¤2. Rational surfaces
2.1. We first summarize the structure theorem of [Ve3] and some of its refinements,
which will be the starting point of the proof of our main theorem. Remember that a
non–singular rational curve with self–intersection −1 is called a (−1)–curve.
Structure Theorem. Let X be a non–singular projective rational surface. Let B be aconnected normal crossings curve on X with χ(X \ B) ≤ 0. Assume that X does notcontain any (−1)–curve disjoint from B.
By [GP, Theorem 3] there is a dominant morphism ϕ : X \ B → P1 ; let h : X̃ → X be
the minimal morphism that resolves the indeterminacies of ϕ, considered as rational map
from X to P1 .
(1) Then there exists a connected curve B 0 ⊃ B with χ(X \ B 0 ) ≤ χ(X \ B) ≤ 0, suchthat the morphism ϕ ◦ h decomposes asgπX̃ −→ Σ −→ P1 ,

8

WILLEM VEYS

where g is a composition of blowing–downs with exceptional curve in h−1 B 0 , and π : Σ →P1 is a ruled surface; see Diagram 1. Moreover, h−1 B 0 has normal crossings in X̃.
(2) We can require the configuration g(h−1 B 0 ) ⊂ Σ to be one of the configurations inFigure 2.

...

...
C1

...

C1

C

.....................................................
.........................
............
...........
.......
.......
....
.....
.
....
....
........
........
.............
.
.
.
.
.
.
.
.
.
.
.
.
..........................................................................

C2
(a)

(b)

(c)

Figure 2
Here C1 and C2 are sections of π, C is a non–singular curve for which π|C : C → P1
has degree 2 (a ‘bisection’), and the other curves are fibres of π. The minimal number offibres in (a) and (b) is 2 and 1, respectively; in (c) there must pass a fibre through eachramification point of π|C , and we can have any number of other fibres.
(3) Irreducible curves C ⊂ h−1 B 0 , which are not components of h−1 B, occur only infibres of g. Moreover, any fibre of g contains at most one such curve C and½card(C ∩ h−1 B) = 1 if χ(Σ \ g(h−1 B 0 )) < 0card(C ∩ h−1 B) = 2 if χ(Σ \ g(h−1 B 0 )) = 0.
Note that always χ(Σ \ g(h−1 B 0 )) = 0 in cases (b) and (c).
Proof. Combine essentially (3.3), (3.5) and (4.3) in [Ve3]. ¤

 h
g. X̃
&

ϕ̃

Σ
X
y
&
.
ϕ
π
P1
Diagram 1

2.2. We will denote by p : Σ̃ → Σ the minimal embedded resolution of the configuration
D := g(h−1 B 0 ) ⊂ Σ in case (c) of the structure theorem. When ω 1/d is a multi–valued

VANISHING OF PRINCIPAL VALUE INTEGRALS ON SURFACES

9

differential form on Σ with D ⊃ | div ω|, we will slightly abuse the terminology of (1.3)
and say that (D, ω 1/d ) is allowed on Σ if (p−1 D, p∗ ω 1/d ) is allowed on Σ̃. In that case wealso put EΣ (D, ω 1/d ) := EΣ̃ (p−1 D, p∗ ω 1/d ).
Lemma. Let D := g(h−1 B 0 ) ⊂ Σ in the structure theorem and let ω 1/d be a multi-valued
differential form on Σ. Assume in case (c) that C is rational (this is equivalent to π|Chaving exactly two ramification points). If (D, ω 1/d ) is allowed on Σ, then EΣ (D, ω 1/d ) =0.
PProof. We write as usual D = ∪i∈T Ci and div ω 1/d = i∈T (αi − 1)Ci . Applying the
adjunction formula to a generic fibre of π yields in cases (a), (b) and (c) that α1 = −1,
α1 + α2 = 0 and α = 0, respectively. We first treat the cases (a) and (b).
If no αi = 0 this is well known and easily verified; see e.g. [Ve2] for a similar computation. The point is that the contribution of any fibre of π to EX (D, ω 1/d ) is zero. Now
if αi = 0 for some fibre Ci of π, the contribution of Ci is still zero, simply because its
self–intersection Ci · Ci = 0. This finishes already case (a).
In case (b) we are left with the following possibility: α1 = α2 = 0 and (omitting thepossible fibres C` with α` = 1) there is either only one fibre Ci ⊂ D with then necessarily
αi = −1, or there are two fibres Ci and Ci0 in D with αi + αi0 = 0 and αi 6= 0 6= αi0 . We
compute EX (D, ω 1/d ) in this last case:
L−1
L−1+ (L − 1) α0
αi
L −1
L i −1
2(L − 1)2(L − 1)
+
(−C
·
C
)
.
+ (−C1 · C1 ) α
2
200(L i − 1)(Lαi − 1)
(Lαi − 1)(Lαi − 1)

EX (D, ω 1/d ) =(L − 1)2 + (L − 1)

Since C1 · C1 = −C2 · C2 (see e.g. [Ha, Theorem V.2.17]) the last two terms cancel and,since αi + αi0 = 0, we obtain0

EX (D, ω1/dLαi +αi − 1) = (L − 1)
= 0.
0(Lαi − 1)(Lαi − 1)2

For case (c) we have to consider EΣ̃ (p−1 D, p∗ ω 1/d ). We denote the two curves in p−1 D
which intersect C(⊂ Σ̃) in P1 and P2 by C1 and C2 , respectively. See Figure 3.

Σ̃Σ................................................C................
....................

....
.........
.....

...........
......
...
.
...
.......
.
.
.
.
.
...............
.
.
.
.
..........................................................................

P1•.................

C10

•P2

Cj

p
←−

C100
C10
C

C20

C200

P
• 1
C1

Figure 3

..
..
..
.

Cj

..
..

...
...
..
..
... ...
.. ..
.....

C20
P2
•
C2

10

WILLEM VEYS

Claim. A fibre Cj in Σ not containing P1 or P2 must have αj = 1. Indeed, by allowedness,
if αj 6= 1 we must have α1 = α2 = 1 and also α` = 1 for possible other fibres C` on Σ. But
then the adjunction formula for C on Σ̃ yields αj = 0, contradicting the other requirement
for allowedness.
Consequently, for the computation of EΣ̃ (p−1 D, p∗ ω 1/d ) we can neglect possible fibres
not containing P1 or P2 , and we know that α1 + α2 = 0 and α1 6= 0 6= α2 . The adjunction
formula on Σ̃ for Ci0 and Ci00 yields αi0 = αi00 = αi2+1 for i = 1, 2. Using the structure of
Pic Σ one computes that the self–intersection C · C = 4 on Σ, see e.g. [Ve1, Remark 6.7].
Then clearly C · C = 0 on Σ̃.
So in the defining expression for EΣ̃ (p−1 D, p∗ ω 1/d ) we only need to sum the terms for
I ⊂ T with αi 6= 0 for all i ∈ I. A simple computation (or directly the formula [Ve5,Proposition 5.4]) yields
EΣ̃ (p−1 D, p∗ ω 1/d ) = L(L − 1) +

2
X
αi +1
L−1
2
(L
−
2
+
2(1
+
L
))
α
i − 1
L
i=1

αi +1
2
X
1
L 2
= L(L − 1)(1 +) + 2(L − 1)
Lαi − 1
Lα i − 1
i=1
i=1

2
X

1/2 L

= 0 + 2(L − 1)L= 2(L − 1)Lusing twice that α1 + α2 = 0.

1/2 (Lαα1 + 22

α1

α2

α1

α2

α1

− L 2 − L 2 + Lα 2 + 2(Lα1 − 1)(Lα2 − 1)

α1 +α22− 1)(L 2 + L 2 )= 0,(Lα1 − 1)(Lα2 − 1)¤

We are now ready to prove our vanishing theorem.
2.3. Theorem. Let X be a non–singular projective rational surface and ω 1/d a multi–
valued differential form on X without logarithmic poles. Let B = ∪i∈T Ci be a connectedcrossings divisor on X satisfying B ⊃ | div ω|. If χ(X \ B) ≤ 0, thenR normal
1/d
PV X ω
(= EX (B, ω 1/d )) = 0.
Remark. The generalization with B ⊃ | div ω| is not only needed for the applications in§3, but is already useful in the proof for B = | div ω|.
ghProof. We first explain our strategy. We will construct maps Σ ← X̃ → X as in the
Structure Theorem 2.1. If the exceptional situation of Lemma 1.5(2) does not occur inany blowing–up of g or h, then this lemma implies
EX (B, ω 1/d ) = EX̃ (h−1 B, ω 1/d ) = EX̃ (h−1 B 0 , ω 1/d ) = EΣ (g(h−1 B 0 ), ω 1/d ),where for simplicity we keep the notation ω 1/d on each surface (a rational differential
form is a birational notion anyway). By Lemma 2.2 this last expression vanishes (theVANISHING OF PRINCIPAL VALUE INTEGRALS ON SURFACES

11

configuration on Σ turns out to be allowed). The crucial point is that we will show thatindeed such an exceptional situation never occurs.
We will denote all curves appearing in the image of h−1 B 0 in any intermediate surface
by C` .
Step 1. The surface X does not contain any (−1)–curve disjoint from B.
Indeed, suppose that A is such a (−1)–curve disjoint from B. Applying the adjunction
formula for A on X would yield −2 = deg KA = A · A = −1.
So we can really apply Theorem 2.1 and we obtain the extended configuration B 0 ⊃ Bghon X and the maps Σ ← X̃ → X, using from now on all notations introduced in thattheorem.
Step 2. The exceptional situation of Lemma 1.5(2) does not occur for any constituting
blowing–up of h : X̃ → X.
Suppose that there is such a blowing–up b : X 00 → X 0 with centre Pi ∈ Ci , αi = 0, andtwo other curves Ci1 and Ci2 intersecting Ci outside Pi , satisfying αi1 + αi2 = 0 and{αi1 , αi2 } 6= {−1, 1}; see Figure 4. Denote by C the exceptional curve of b; recall thatα = 1.
..
.
Ci1
Ci2

P
• 1 ...
P• 2 ...

..
.

X0

P
• 1 ...
P• 2 ...

Ci 1
b
←−

Ci 2
Pi

Pi •
Ci (αi = 0)

X 00

•

C (α = 1)

Ci (αi = 0)
Figure 4

Since b is really needed to resolve the indeterminacies of ϕ (h is minimal), either C
itself or some other curve, created after a chain of blowing–ups starting with centre apoint of C, will project by g onto a section or bisection on Σ. Such a curve C 0 has
α0 ∈ Z>0 , and hence the configuration on Σ must be case (b). This also implies that Ci
cannot project onto a section or bisection on Σ, so it is either blown down during g or
becomes a fibre on Σ. Both possibilities yield that at some stage of g the curve Ci canintersect at most two other components of the total transform of B 0 .
So in order to reach that stage by blowing–downs starting from X̃, it is necessary thatat least for one of the points P1 , P2 or Pi (considered on Ci ⊂ X̃) the following holds:
h−1 B 0 \ {P` } has two connected components, and the one not containing Ci is a tree thatgets completely contracted before reaching that stage of g.

12

WILLEM VEYSThis cannot happen for Pi because, as explained above, either C itself or another curvein ‘its tree’ must project onto a section of Σ.
Consequently there is such a tree contracting to P1 or P2 during g, say to P1 . Now, since
αi = 0, the component C` of h−1 B through P1 (in X̃) still has α` = αi1 . Analogously,at the stage of g just before the last blowing–down needed to contract the tree, the ‘last’component C`0 through P1 still has α`0 = αi1 (6= 1). But, on the other hand, since C`0 now
gets contracted by that blowing–down, we must have α`0 = 1. This contradiction finishes
step 2.
Step 3. The exceptional situation of Lemma 1.5(2) does not occur for any constituting
blowing–up of g : X̃ → Σ.
(To be precise, in case (c) for Σ we consider only constituting blowing–ups of X̃ → Σ̃,using the notation of (2.2).) Suppose that there is such a blowing–up b : X 00 → X 0 , where
we use again the notations as in Figure 4, but this time ‘during g’. We consider two cases.
(I) Pi ∈ X 0 does not belong to any other component of the total transform of B 0 .
Here we consider two subcases.
(i) C ⊂ h−1 B in X̃.
Since Ci (⊂ X̃) must be blown down during h (all α` 6= 0 on X) and B has normal
crossings, at most two components of the total transform of B can intersect Ci justbefore that blowing–down. As we explained above, since αi1 6= 1 6= αi2 , the components
intersecting Ci in P1 and P2 cannot be blown down before Ci . So the tree that was
created during g, starting with the blowing–up with centre Pi , must be contracted again
by h. This contradicts the minimality of h.
(ii) C(⊂ X̃) is a component of (the strict transform of) B 0 \ B.
Because h−1 B is connected, each possible further blowing–up of g (after b) with centre
in C must have Pi as centre. Consequently
χ(X \ B 0 ) < χ(X \ B) ≤ 0
and Σ must be as in case (a) with at least three fibres. The strict transform in X̃ ofthe section C1 ⊂ Σ can intersect at most two other components of h−1 B. (Indeed, since
αi = 0, the morphism h cannot be the identity. Hence C1 , being the only non–fibre inΣ, must be created by the last blowing–up of h.) So C1 must intersect in X̃ at least one
component C 0 coming from B 0 \ B. But C 0 gets contracted during g; so in order to haveat least three intersections with C1 in Σ, C 0 must intersect in X̃ another component of
h−1 B. This contradicts Theorem 2.1(3).
(II) Pi ∈ X 0 also belongs to another component C 0 of the total transform of B 0 .
Since, by Lemma 1.5, (h−1 B, ω 1/d ) is allowed on X̃, we then have necessarily α0 = 1.
Moreover, we claim that Ci must already live on Σ, or on Σ̃ in case (c), where we use theVANISHING OF PRINCIPAL VALUE INTEGRALS ON SURFACES

13

notation of (2.2). Indeed, suppose that Ci is created by a blowing–up of g, or of X̃ → Σ̃in case (c); at that stage Ci can intersect at most two other components. Thus C 0 shouldbe created afterwards during g. But this situation is precisely the previous case (I), which
we already contradicted.
We now investigate the cases (a), (b) and (c) for Σ in our (hypothetical) situation.
Case (a) is clearly impossible. In case (b) Ci must be a section, and in case (c) Ci must
be the bisection in Σ, see Figure 5. (In case (c) Ci could be a priori another curve inΣ̃, but then we would have two intersecting curves with α` = 0 on Σ̃, implying the same
phenomenon on X̃.) Somewhat abusing notation in Figure 5, the curves Ci1 , Ci2 and C 0
in Σ could be different from those intersecting Ci in X 0 , but their α–coefficients are also
αi1 , αi2 and α0 (= 1), respectively.

(b)

Σ

P1 P2
•
•

Pi
•

Ci (αi = 0)

P10 P20••Pi0•

Ci0 (αi0 = 0)

Ci 1

C0

Ci2

Σ̃

Σ
Pi
•

(c)

(αi = 0)C
•P....................................................................
..........
..................
......
..........
...
......
i
.
.....
...
1 .............
......
.
.
.
.
.
.
.
...............
.
.
............................................................................

P•

2

Ci 1

p
←−

..
0 ..
..
..
..
...
...
...
..
...
..
.
..
..
...
..
...
...
.
.
.
.
.
.
i ..........

C

Ci
P
• 1
(αi = 0)

C0

Ci 2

P

•

P2
•

Figure 5

Case (b). On X̃ we have that Ci (being an exceptional curve of h) can intersect at mosttwo other components of h−1 B. So all but at most two intersections of Ci in Σ with fibres
C` of π : Σ → P1 must ‘split’ during g, i.e. Ci must eventually get separated from thoseC` by a blowing–up of g with centre in Ci and exceptional curve coming from B 0 \ B.
(The same is true for Ci0 .)In order to obtain such an exceptional curve Cj from B 0 \ B, intersecting Ci in P1 , weshould have simultaneously αj = 1 and αj = αi1 (6= 1), by the same argument as for theconclusion of step 2. The same contradiction works for P2 , P10 and P20 . Hence we must
separate Ci and Ci0 from C 0 with exceptional curves from B 0 \ B, intersecting Ci and Ci0
in Pi and Pi0 , respectively. But this contradicts the connectivity of h−1 B.
Case (c). Analogously we will have to separate C 0 from Ci , contradicting the connectivityof h−1 B.

14

WILLEM VEYSStep 4. (g(h−1 B 0 ), ω 1/d ) is allowed on Σ and EX (B, ω 1/d ) = EΣ (g(h−1 B 0 ), ω 1/d ) = 0.
By Lemma 1.5 we have EX (B, ω 1/d ) = EX̃ (h−1 B, ω 1/d ). Since also h−1 B 0 is a normal
crossings divisor and the curves C` in B 0 \ B have α` = 1, we have that also (h−1 B 0 , ω 1/d )is allowed on X̃ and EX̃ (h−1 B, ω 1/d ) = EX̃ (h−1 B 0 , ω 1/d ). It is easy to see that then necessarily (g(h−1 B 0 ), ω 1/d ) is allowed on Σ, and thus EX̃ (h−1 B 0 , ω 1/d ) = EΣ (g(h−1 B 0 ), ω 1/d ),again by Lemma 1.5(2). This last expression equals zero by Lemma 2.2. ¤2.4. Remark. Theorem 2.3 and its proof are quite subtle. In [Ve7, 3.4] we constructed
the following similar example. Let X = P2 and ω 1/2 a multi–valued differential form with| div ω| a (non–singular)
conic B. So ω 1/2 has no logarithmic poles on X and one easilyRcomputes that P V X ω 1/2 6= 0. Taking B 0 as the union of B and one tangent line toghB, we can construct Σ ← X̃ → X, where h is a composition of three blowing–ups andg a composition of two blowing–downs, all outside of X \ B 0 , such that g(h−1 B 0 ) ⊂ Σ
is case (b) in Theorem 2.1 withR exactly one fibre. Also on Σ we have that ω 1/2 has nologarithmic poles, but now P V Σ ω 1/2 = 0. In this example we have that χ(X \ B 0 ) = 0;
the only obstruction with the data of Theorem 2.3 is that here χ(X \ B) = 1 (> 0). Andin fact the ‘change’ in principal value integral is caused by g which consists precisely ofthe exceptional situation of Lemma 1.5(2).
2.5. Our vanishing theorem specializes to the level of Hodge polynomials or Euler characteristics. With the same notations as in Definition 1.4 we can introduce analogously
the invariantsX

EX (D, ω 1/d ) :=H(CI◦ )i∈I

I⊂T
∀i∈I:αi 6=0

and

Y

X

eX (D, ω 1/d ) :=Y uv − 1X
uv − 1
(−C
·
C
)
+
i
i
(uv)αi − 1(uv)αj − 1

χ(CI◦ )

I⊂T
∀i∈I:αi 6=0

Y 1
i∈I

αi

+

i∈T
αi =0

j∈Ti

X

Y 1
.
αj

(−Ci · Ci )

i∈T
αi =0

j∈Ti

Then with the same data as in Theorem 2.3 we obtain that EX (B, ω 1/d ) = eX (B, ω 1/d ) =0.
2.6. Let L be a p–adic field with valuation ring O, maximal ideal P and residue fieldO ∼1/d
are defined over L, we
P = Fq . We choose an embedding of L into C. When X and ω
can introduce analogouslyLEX(D, ω 1/d ) :=X
I⊂T∀i∈I:αi 6=0

card(CI◦ )Fq

Y q−1
Y q−1
X
+
(−C
·
C
)
,
i
i
q αi − 1
q αj − 1
i∈I

i∈T
αi =0

j∈Ti

where card(·)Fq denotes the number of Fq –rational points of the reduction mod P of CI◦ .
When ω 1/d has no logarithmic poles and if suitable conditions about good reduction mod

VANISHING OF PRINCIPAL VALUE INTEGRALS ON SURFACES

15

P are satisfied, a similar proof as for Denef’s formula for the p–adic Igusa zeta functionL[De1] yields thatEX
(D, ω 1/d ) is (up to a power of q) precisely the p–adic principal valueR
1/d
integral P V X(L) |ω |.
When we assume that the field L is ‘big enough’, the same data as in Theorem 2.3 willagain imply that this invariant vanishes. More precisely we want that all constructions inthis proof of the theorem can be done ‘over L’, for instance that all centres of blowing–ups
are L–rational points. This can always be achieved by taking a finite extension of a givenp–adic field.

3. Cancelation of candidate poles for zeta functions
3.1. Fix a polynomial f ∈ C[X1 , . . . , Xn+1 ]\C, or, more generally, a non–constant regularfunction f : M → A1 from a non–singular (n + 1)–dimensional variety M . Denef andLoeser [DL2] associated to f its motivic zeta function Zmot (f ; T ) ∈ M[[T ]], where M isthe localization of K0 (V ar) with respect to L. It describes the orders of all truncatedarcs on M along the hypersurface {f = 0}, and as in fact modeled on the classical p–
adic Igusa zeta function. (See also [DL4] or [Ve6] for an introduction to this topic.) We
just mention a formula for Zmot (f ; T ) in terms of an embedded resolution of {f = 0},
implying in particular that this invariant belongs to the localization of M[T ] with respect
to La − T b , where a, b ∈ Z>0 .
3.2. Theorem [DL2, 2.2.1]. Let h : Y → M be an embedded resolution of {f = 0}. Let
Ej , j ∈ K, be the irreducible components of h−1 {f = 0}. Denote by Nj the multiplicity
of Ej in div(f ◦ h) and by νj − 1 the multiplicity of Ej in div(h∗ dx), where dx is alocal generator of the sheaf of differential(n + 1)–forms on M . For J ⊂ K we put`◦
EJ := (∩j∈J Ej ) \ (∪`6∈J E` ); so Y = J⊂K EJ◦ . Then
Zmot (f ; T ) = L−(n+1)

X[EJ◦ ]J⊂K

Y (L − 1)T Njj∈JLνj − T Nj.

3.3. This zeta function specializes to the Hodge zeta function ZHod (f ; T ), replacing in
the formula above all classes of varieties in M by their Hodge polynomial, and further tothe ‘classical’ topological zeta function
ztop (f ; s) =X
J⊂Kχ(EJ◦ )Yj∈J

1
∈ Q(s)
νj + sNj

of [DL1]. We refer to [DL2] or [Ve6] for more details.
3.4. The famous monodromy conjecture, stated originally for the p–adic Igusa zetafunction, can be formulated for Zmot (f ; T ) as follows [DL2, 2.4]:

16

WILLEM VEYS

Zmot (f ; T ) belongs to the localization of M with respect to those La − T b , a, b ∈ Z>0 ,such that e2πia/b is an eigenvalue of the local monodromy on the Milnor fibre of f at somepoint of {f = 0}.
So if Lν/N is a pole of Zmot (f ; T ), then e2πiν/N is expected to be an eigenvalue of thelocal monodromy. Note however that one has to be careful with the notion of pole here,
the difficulty being that we do not know whether M is a domain. See e.g. [RV2] for a
precise definition. For the Hodge and topological zeta function the notion of pole is clear.
The (p–adic version of) the conjecture was proved in dimension two (n = 1) by Loeser[Loe1]; a simple proof in the motivic/Hodge/topological setting is in [Ro1]. It is still
open in general, with partial results in dimension three [Ve2][RV1][ACLM1] in and otherspecial cases [Loe2][ACLM2].
3.5. We keep using the notation of Theorem 3.2. Fix an exceptional component Ej ofh which is mapped to a point by h. It induces the candidate pole Lνj /Nj for Zmot (f ; T ),respectively (uv)νj /Nj for ZHod (f ; T ) and −νj /Nj for ztop (f ; s).
In order for the monodromy conjecture to hold, looking at A’Campo’s formula for themonodromy zeta function [A’C] one expects the following. Suppose we are in the genericcase that νj /Nj 6= νi /Ni for all i 6= j. If χ(Ej◦ ) = 0, maybe even if (−1)n χ(Ej◦ ) ≤ 0, then‘in general’ Lνj /Nj should not be a pole of Zmot (f ; T ).
Somewhat more precise, suppose only that νj /Nj 6= νi /Ni for all i ∈ Sj := {i ∈ K |Ei ∩ Ej 6= ∅}. Then, if (−1)n χ(Ej◦ ) ≤ 0, one expects that ‘in general’ Ej does notcontribute to the possible pole Lνj /Nj , which means that Lνj /Nj should not be a pole of

(∗)

X

1
Ln+1

[EI◦ ]j∈I⊂K

Y (L − 1)T Ni
i∈ILνi − T Ni.

We refer to e.g. [Ve2, §1] for a motivation. For n = 1 this expectation is true and is partof the proof of the monodromy conjecture [Loe1][Ro1]. Now, Lνj /Nj not being a pole of(∗) can be reformulated as ‘its residue is zero’, i.e.
(∗∗)

X
j∈I⊂K

[EI◦ ]Yi∈I\{j}L−1
Lαi − 1is zero, where αi := νi − (νj /Nj )Ni for i ∈ Sj . Now (∗∗) is a motivic principal value
integral on Ej . Indeed, let dx be a local generator of the sheaf of (n + 1)–forms on M
around the point h(Ej ). Then the Poincaré residue ω 1/d of (f ◦ h)−νj /Nj h∗ (dx) on Ej isPa multi–valued differential form on Ej with div ω 1/d = i∈Sj (αi − 1)(Ej ∩ Ei ). This iseasily verified with local coordinates,
see also [Ja3]. We thus have that (∗∗) is (up to aR
1/d
non–zero constant) equal to P V Ej ω .
Note however that by assumption all αi 6= 0, so ω 1/d indeed has no logarithmic poles,
but it is possible that some αi = 1. So in fact we land in a natural way in the moregeneral framework of (1.3) with D := ∪i∈Sj (Ej ∩ Ei ) ⊃ | div ω|, where the inclusion may
be strict, and (∗∗) is (essentially) EEj (D, ω 1/d ).

VANISHING OF PRINCIPAL VALUE INTEGRALS ON SURFACES

17

3.6. Let now n = 2. We may assume that h is constructed as a composition of blowing–
ups with non–singular centre. Fix as above a projective exceptional surface Ej which
is mapped to a point by h. The surface Ej was created during some blowing–up π of
(0)
(0)
(0)
the resolution process h as a surface Ej , where either Ej ∼= P2 or Ej is a ruled
surface, when the centre of π is a point or a curve, respectively. And then Ej is obtained(0)(0)
from Ej by a composition ϕ : Ej → Ej of (point) blowing–ups. Denote again D :=(0)∪i∈Sj (Ej ∩ Ei ). We have that D is the inverse image by ϕ of the intersection of Ej

with
(0)

the other components of the total inverse image of {f = 0} at the stage of h when Ej(0)was just created. In particular D is connected if and only if this intersection on Ej(0)connected. And it is thus always connected if E ∼= P2 .

isj3.7. By the considerations above Theorem 2.3 yields the following cancelation result forcandidate poles of the motivic zeta function. For ease of reference we recall the notations.
Let M be a threedimensional non–singular variety and f : M → A1 a non–constantregular function. Let h : Y → M be an embedded resolution of {f = 0}, constructed
as a composition of blowing–ups. Denote by Ej , j ∈ K, the irreducible components of
h−1 {f = 0} and let Nj , νj and EJ◦ be as in (3.2). Suppose that Ej is mapped to a pointby h and that νj /Nj 6= νi /Ni for all i ∈ Sj := {i ∈ K | Ej ∩ Ei 6= ∅}. Denote
Y
X
L−1
REj :=[EI◦ ],
Lα i − 1j∈I⊂Ki∈I\{j}‘the contribution of Ej to the residue of Lνj /Nj for Zmot (f ; T )’.
Theorem. Let χ(Ej◦ ) ≤ 0.
(1) If Ej is created by blowing up a point, then we have always REj = 0.
(2) If Ej is created by blowing up a rational curve, and if ∪i∈Sj (Ej ∩ Ei ) is connected,
then REj = 0.
Recall that in case (2) this connectivity is equivalent to the connectivity of the analogous(0)intersection configuration D(0) on the (rational) ruled surface Ej (3.6). Note then thatthe exceptions in (2), i.e. those Ej with a non–connected intersection configuration, arevery special! Indeed, non–connectivity of D(0) implies for instance that D(0) does not(0)contain any fibre of the ruled surface Ej . In the embedded resolution process this isquite rare.
There is a recent vanishing result of Rodrigues [Ro2] in this exceptional case. Whenχ(Ej◦ ) = 0, and assuming a minor extra condition, he classified all possible non–connected
D(0) with non–singular irreducible components, and verified that then again REj = 0.
3.8. The case where Ej is a rational surface is the most difficult one. There is noclassification of the possible intersection configurations on Ej with χ(Ej◦ ) ≤ 0; instead
we used our structure theorem 2.1. When Ej is created by blowing up a non–rational
curve, we already obtained a classification of the possible configurations with χ(Ej◦ ) ≤ 0
in [Ve2] and [Ve3].

18

WILLEM VEYS

Proposition. We use all notations of (3.7). Let χ(Ej◦ ) ≤ 0. If Ej is created by blowing
up a non–rational curve, then we have always REj = 0.
(0)

Proof. Let as above Ej be created as the ruled surface Ej

while blowing up a curve of(0)

genus g during h. When g ≥ 2, we classified the possible configurations on Ej with
χ(Ej◦ ) ≤ 0 in [Ve2, Proposition 5.13], and verified in [Ve2, Propositions 5.1 and 5.3] thatREj = 0 for them. (The calculation there is in the context of p–adic Igusa zeta functions,but is essentially the same in the motivic setting.)(0)
When g = 1, the possible configurations C of curves on the ruled surface Ej withχ(Ej \ C) ≤ 0 were classified in [Ve3, Theorem 6.5]. Again by [Ve2, Propositions 5.1 and5.3] we have that REj = 0, except for one annoying case (where REj 6= 0). More precisely,in this case the curves on the ruled surface consist of a number of disjoint elliptic curves,
where either one of the curves is not a section, or there are at least three curves. Now
recently Rodrigues showed in [Ro2] that in fact this configuration cannot occur in thecontext of exceptional surfaces in an embedded resolution. ¤3.9. Theorem 3.7 and Proposition 3.8 specialize to the analogous results in the context
of the Hodge and topological zeta functions of (3.3). They are also valid in the context
of p–adic Igusa zeta functions (see e.g. [De1][Ve2]) if the p–adic field L is assumed ‘big
enough’ as in (2.6), and if suitable conditions concerning good reduction mod P as in[De1] are satisfied. Alternatively, one can take a big enough number field F . Then ourvanishing results will be true for the Igusa zeta functions over all except a finite numberof completions L of F .
References[A’C]
N. A’Campo, La fonction zeta d’une monodromie, Comment. Math. Helv. 50 (1975), 233–248.
[ACLM1] E. Artal Bartolo, P. Cassou-Noguès, I. Luengo and A. Melle Hernández, Monodromy conjecture
for some surface singularities, Ann. Scient. Ec. Norm. Sup. 35 (2002), 605–640.
[ACLM2] E. Artal Bartolo, P. Cassou-Noguès, I. Luengo and A. Melle Hernández, Quasi-ordinary power
series and their zeta functions, math.NT/0306249.
[AVG]
V. Arnold, A. Varchenko and S. Goussein–Zadé, Singularités des applications différentiables
II, Editions Mir, Moscou, 1986.
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J. Denef, On the degree of Igusa’s local zeta function, Amer. J. Math. 109 (1987), 991–1008.
[De2]
J. Denef, Report on Igusa’s local zeta function, Sém. Bourbaki 741, Astérisque 201/202/203(1991), 359–386.
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J. Denef and F. Loeser, Motivic Igusa zeta functions, J. Alg. Geom. 7 (1998), 505–537.
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VANISHING OF PRINCIPAL VALUE INTEGRALS ON SURFACES
[Ha1]
[Ha2]
[Hi]
[Ig1]
[Ig2]
[Ja1]
[Ja2]
[Ja3]
[Lae]
[Lan1][Lan2][Loe1][Loe2]
[Loo]
[LS1]
[LS2]
[Po]
[Ro1]
[Ro2]
[RV1]
[RV2]
[Ve1]
[Ve2]
[Ve3]
[Ve4]
[Ve5]
[Ve6]

[Ve7]

19

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K.U.Leuven, Departement Wiskunde, Celestijnenlaan 200B, B–3001 Leuven, Belgium
E-mail address: wim.veys@wis.kuleuven.ac.be
http://www.wis.kuleuven.ac.be/algebra/veys.htm