THE MOTIVIC ZETA FUNCTION
ANGELA ORTEGA
A. Notes of the talk in the Forschungsseminar ”Algebraische Geometrie”, winter term 2008/09.
1. E G 
Let S be an algebraic variety over a filed k. Let VarS be the category of
S-varieties and K0 (VarS ) the Grothendieck ring generated by the classes of
isomophisms [X], X ∈ VarS and the usual relations (cf. talk 1). Denote
L := [A1k × S] ∈ K0 (VarS ), where the structure of S-variety is given by the
projection to the second factor and MS := K0 (VarS )[L]−1 the ring obtained
from K0 (VarS ) by inverting [L]. When S = Spec (k) we denote K0 (Vark )
(resp. Mk ) instead of K0 (VarS ) (resp. MS ) .
Consider G an affine algebraic group and X ∈ VarS .
Definition 1.1. A morphism G × X → X is a good G-action if it is an action
and every orbit is contained in an affine open set.
For example, a representation of G on a k- vector space is a good action.
Definition 1.2. For a fixed variety S with G-action define the G-equivariant
Grothendieck ring K0G (VarS ) as the group generated by isomorphism classes
of S-varieties with good G-action with the relations:
• [X, G] = [Y, G] if there is an S-isomorphism X ' Y compatible with
the G-action.
• [X, G] = [Y, G] + [X\Y, G], where Y ⊂ X closed subset with G-action
induced by one on X.
• [X × V, G] = [X × AnS , G] , V is a n-dimensional affine space over S
with any good G-action and AnS is taken with the trivial action.
And the ring structure is given by
[X, G] · [Y, G] = [X ×S Y, G]
Remark 1.3. The last relation in the definition is equivalent to the relation
which declares that every finite dimensional representation ρ of G has the
same class as the trivial representation of the same degree, i.e., the class of
Ldeg(ρ) .
1
2
When A is a constructible
subset of X stable under the G-action we can
P
define [A, G] := i [Ai , G], where A is the disjoint union of a finite number of
locally closed subvarieties Ai . Observe that the class [A, G] does not depend
on the decomposition of A. In the case where the action of S is trivial the
product of K0 (VarS ) makes K0G (VarS ) a K0 (VarS )-algebra. When G is finite
the map [X] 7→ [G\X] =: X from K0G (VarS ) in K0 (VarS ) is well defined as a
consequence of the following Lemma ([3, Lemma 5.1]).
Lemma 1.4. Let be given a representation of a finite abelian group G on a k-vector
space V of dimension n. Then the class of V in K0 (Vark ) is Ln .
Proof : Let V =
L
χ∈Ĝ Vχ
be the decomposition in eigenspaces of the G-
action. Given I ⊂ Ĝ, denote by VI = {v = ⊕vχ : vχ , 0 iff χ ∈ I} . We have a
natural projection V I → Πχ∈I P(Vχ ). This has a structure of a torus bundle,
where the torus in question is a quotient of G|I|
m by a finite subgroup. So the
|I|
class of V I in Mk is (L − 1) times the class of Πχ∈I P(Vχ ). Since V|I| has also
that structure, the classes of V I and VI coincide in Mk . Hence the same is
for V and V.
2
We will denote [X/S] (resp. [X/S, G]) for an element in K0 (VarS ) (resp. in
K0G (VarS )) to recall that X is a S-variety.
1.1. Case G = µ̃. Consider µn the group of all the n-th roots of the unity (in
some fixed algebraic closure of k). This group has a structure of algebraic
variety since µn =Spec(k[x]/(xn − 1)). We have the maps µnd → µn , x 7→ xd
which makes of µn n a projective system. Define µ̂ := lim µn , which is not a
←−−
variety.
Definition 1.5. Let X be a S-variety. A good µ̂-action on X is an action of µ̂
on X which factors through a good µn -action (in the sense of 1.1) for some
n.
µ̂
Remark 1.6. The equivariant Grothendieck group K0 (VarS ) under good µ̂actions is also called the monodromic Grothendieck group.
2. T    X
In this section we will assume char(k) = 0 and X a reduced separated scheme
of finite type over k.
Definition 2.1. For all n ∈ N define the space of arcs modulo tn+1 on X,
Ln (X) as the algebraic variety over k whose K-rational points for any field
K ⊃ k are the K[t]/tn+1 K[t]-rational points of X.
3
is an affine variety defined by the equations fi (x) = 0,
For example, if X ⊂ Am
k
with x = (x1 , . . . , xm ), i = 1, . . . , r, Ln (X) is defined by the equations in the
variables a0 , . . . , an expressing that
fi (a0 + a1 t + · · · + an tn ) ≡ 0
mod, tn+1
i = 1, . . . , r.
In particular, L0 (X) = X and L1 (X) is the tangent bundle of X.
Definition 2.2. Define the arc space L(X) = lim Ln (X), which is a reduced
←−−
separated scheme over k. Thus, the K-rational points of L(X) are the K~trational points of X called K-arcs on X.
Remark 2.3. In general, L(X) is not of finite type (is a algebraic variety of
“infinite dimension”).
For any n, and m > n we have natural morphisms
πn : L(X) → Ln (X)
and
πm
n : Lm (X) → Ln (X)
given by truncation.
By a theorem of Greenberg, given X an algebraic variety over k it exists a
constant c > 0 such that for all n and K ⊃ k
πn (L(X)(K)) = πcn
n (Lcn (X)(K)),
this implies that πn (L(X)) is a constructible set of Ln (X). If X is smooth
one can take c = 1 and πn is surjective. In any case, πm
n is a locally trivial
(m−n) dim X
fibration with fiber Ak
.
3. M  
We consider X a non-singular irreducible algebraic variety over k of dimension d and f : X → A1k a non constant morphism. Let X0 = f −1 (0). For every
n ≥ 1 integer, f induces a map fn : Ln (X) → L(A1k ). Any point α ∈ L(A1k )
(resp. Ln (A1k )) yields a K-rational point for some K ⊃ k then yields to a
power series α(t) ∈ K~t (resp. α(t) ∈ K~t/tn+1 ). Define the function
ordt : Ln (A1k ) → N ∪ {∞}
by assigning to α(t), ordt (α) := { the largest e : te divides α(t)}. Set
X1 := {φ ∈ Ln (X) : ordt fn (φ) = n}.
This is a locally closed subvariety of Ln (X). Since πn0 (Xn ) ⊂ X0 for n ≥ 1, Xn
is an X0 -variety. Now, consider the morphism
f n : Xn → Gm,k = A1k − {0}
defined by sending φ to the coefficient of tn in fn (φ) . There is a natural
action of Gm,k on Xn given by a · φ(t) = φ(at), where φ(t) is the vector of
power series corresponding to φ (in some local coordinate system). Since
f n (a · φ) = an f n (φ) it follows that f n is a locally trivial fibration.
4
−1
Denote by Xn,1 := f n (1). Note that the action of Gm,n on Xn induces a good
action of µn , and hence of µ̂, on Xn,1 . Since f n is a locally trivial fibration,
the X0 -variety Xn,1 ⊂ Xn and the action of µn on it, determines completely
Xn and the morphism
( f n , πn0 ) : Xn → Gm,k × X0 .
One can verify that Xn as Gm,k × X0 -variety is isomorphic to (Xn,1 × Gm,k )/µn
with the action a(φ, b) = (aφ, a−1 b).
Definition 3.1. The motivic zeta function of f : X → A1k is the power series
µ̂
over MX defined by
0
Z(T) :=
X
[Xn,1 /X0 , µ̂]L−nd Tn .
n≥1
We define the naive motivic zeta function of f as the power series over MX0
by
X
Znaive (T) :=
[Xn /X0 ]L−nd Tn .
n≥1
Remark 3.2. Power series like the motivic zeta functions or the series
X
X
J(T) :=
[Ln (X)]Tn or P(T) :=
[πn (L(X)]Tn
n≥0
n≥0
are called “motivic ” because they specialize to power series over the
Grothendieck group K0 (Motk ) over the category of Chow motives over k.
3.1. A formula for the motivic zeta function. Consider the non constant
morphism f : X → A1k , with X0 = f −1 (0). Let (Y, h) be a resolution of f , that
is, Y a non singular irreducible algebraic variety over k and h : Y → X a
proper morphism such that :
'
(i) Y\h(−1) (X0 ) −→ X\X0
(ii) h−1 (X0 ) has only normal crossings as subvarieties of Y.
P
Let i∈J ni Ei be the divisor of zero of f ◦ h and νi − 1 the multiplicity of Ei in
the divisor h∗ dx, where dx is a local non vanishing volume form at any point
of h(Ei ), i.e., a local generator of the sheaf of differential forms of maximal
degree. For i ∈ J, I ⊂ J consider the non singular varieties:
[
\
[
E0i := Ei \
E j , EI =
Ei , E0I := EI \
Ej
j,i
i∈I
j∈J\I
e0 of E0 with Galois
Let mI = gcd(Ni )i∈I . Define an unramified Galois cover E
I
I
group µmI as follows. On a suitable Zariski open U ⊂ Y we can write
f ◦ h = ugmI , with u an unit on U an g : U → A1k morphism. The restriction
e0 above E0 ∩ U is defined by
of E
I
I
{(z, p) ∈ A1k × (E0I ∩ U) : zmI = u−1 (p)}.
5
Note that E0I can be covered by such open sets U of Y. Gluing together the
e0 ∩ U we obtain the cover E
e0 of E0 . More precisely, if xi are local
covers E
I
I
I
coordinates defining Ei , the function f ◦ h can be written in a neighborhood
i
U as uΠi∈I xN
, where u is a unit. If yi are other local coordinates in a
i
i
neighborhood V, with ηi units and f ◦ h is written as vΠi∈I yN
, with v unit
i
Ni
0
then u = vΠηi and the gluing of the coverings above EI ∩ U ∩ V is given
by the isomorphism
{(z, p) ∈ A1k × (E0I ∩ U ∩ V) : zmI = u−1 (p)} ' {(z, p) : zmI = v−1 (p)},
via (z, p) 7→ (Πi∈I ηαi i z, p) with αi =
Ni
mI .
e0 has a natural µm The covering E
I
I
e0 over E0 .
action which induces a good µ̂-action on E
I
I
µ̂
Theorem 3.3. With the previous notations the following relation holds in MX ~T:
0
Z(T) =
X
e0 /X0 , µ̂]
(L − 1)|I|−1 [E
I
∅,I⊂J
Y
L−νi TNi
i∈I
1 − L−νi TNi
.
Using Lemma 1.4 one can deduce the following Corollary.
Corollary 3.1. The following relation holds in MX0 ~T:
X
Y L−νi TNi
Znaive (T) =
(L − 1)|I|−1 [E0I /X0 ]
.
1 − L−νi TNi
i∈I
∅,I⊂J
In particular, this shows that Z(T) and Znaive (T) are rational.
4. N  
A subset A ⊂ L(X) is called constructible if A = π−1
n (C) with C ⊂ Ln (X)
constructible for some n ≥ 0. Assume L(X) of pure relative dimension d.
Definition 4.1. The subset A ⊂ L(X) is stable if, (1) for some n ∈ N
πn (A) ⊂ Ln (X) is constructible and A = π−1 πn (A) and, (2) for all m ≥ n,
πm+1 (A) → πm (A) is a piecewise trivial fibration (that is, trivial relative to a
decomposition into subvarities) with fiber an affine space of dimension d.
Notice that the smoothness of X implies the condition (2). If A is a stable
subset dim πm (A) − md is independent of the choice m > n and the same
is true for the class [πm (A)]Lmd ∈ Mk ; then [πn (A)]L−(n+1)d , considered as
element in Mk stabilizes for for n big enough and
µ̃(A) := lim [πn (A)]L−(n+1)d ∈ Mk
n→∞
is the naive motic mesure of A. It follows from [1, Lemma 4.1] the this is still
true even when X is singular. When θ : A → Mk is a map with finite image
6
whose fibers are stable sets of L(X) one can define
Z
X
θdµ̃ :=
cµ̃(θ−1 (c)).
A
c∈Im θ
One have the following change of variables formula.
Theorem 4.2. Let h : Y → X be a morphism of algebraic varieties over k. Suppose
that h is birational and proper. Let A be stable and suppose that ordt Jach is bounded
on h−1 (A) ⊂ L(Y). Then
Z
µ̃(A) =
Lordt Jach dµ̃.
h−1 (A)
Here ordt Jach (y) denotes de t-order of the Jacobian of h at y. When X and Y
are non singular this is the function ordt of the determinant of the Jacobian
matrix of h at y with respect to any local coordinates on X and on Y.
The proof of the formula in the Theorem 3.3 consist of the explicit calculation
µ̂
of [Xn,1 /X0 , µ̂] ∈ MX for each n. In Mk we have the equality
0
[Xn,1 ] = L(n+1)d µ̃(π−1
n (Xn,1 )).
Applying the changing of variables formula of Theorem 4.2 to the resolution
of singularities h : Y → X, one can check that [Xn,1 ] is equal to an integral
over a stable subset of L(Y). The integral can be explicitely calculated since
f ◦ h is locally a monomial, thus one obtains an expression for [Xn,1 ] as an
element of Mk . Taking into account the µn -action on Xn,1 and the natural
map Xn,1 → X0 , one also have a similar formula for [Xn,1 /X0 , µ̂] which yields
the Theorem 3.3
R
[1] J. Denef, F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration ,
Invent. Math. 135 (1999), 201–232.
[2] J. Denef, F. Loeser, Geometry on arc spaces of algebraic varities, European Congress of
Mathematics Vol. I (Barcelona, 2000), 327–348, Progr. Math., 201, Birkhäuser, Basel,
2001.
[3] E. Looijenga, Motivic mesures, in Séminaire Bourbaki, exposé 874, Mars 2000.