ARC SPACES, MOTIVIC INTEGRATION
AND STRINGY INVARIANTS

Willem Veys
Abstract. The concept of motivic integration was invented by Kontsevich to show thatbirationally equivalent Calabi-Yau manifolds have the same Hodge numbers. He constructeda certain measure on the arc space of an algebraic variety, the motivic measure, with thesubtle and crucial property that it takes values not in R, but in the Grothendieck ring ofalgebraic varieties. A whole theory on this subject was then developed by Denef and Loeser
in various papers, with several applications.
Batyrev introduced with motivic integration techniques new singularity invariants, thestringy invariants, for algebraic varieties with mild singularities, more precisely log terminal
singularities. He used them for instance to formulate a topological Mirror Symmetry test forpairs of singular Calabi-Yau varieties. We generalized these invariants to almost arbitrary
singular varieties, assuming Mori’s Minimal Model Program.
The aim of these notes is to provide a gentle introduction to these concepts. There exist
already good surveys by Denef-Loeser [DL8] and Looijenga [Loo], and a nice elementary
introduction by Craw [Cr]. Here we merely want to explain the basic concepts and first
results, including the p-adic number theoretic pre-history of the theory, and to provide
concrete examples.
The text is a slightly adapted version of the ‘extended abstract’ of the author’s talks at
the 12th MSJ-IRI ”Singularity Theory and Its Applications” (2003) in Sapporo. At the end
we included a list of various recent results.

Table of contents
1. Pre-history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Arc spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3. Motivic integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4. First applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5. Motivic volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6. Motivic zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7. Batyrev’s stringy invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
8. Stringy invariants for general singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
9. Miscellaneous recent results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1991 Mathematics Subject Classification. 14B05, 14E15, 14J17, 11S40, 32S05, 32S20 (14B10, 14C30,
14E30, 14J32, 11G25, 11S80, 32S45).
Key words and phrases. Arc spaces, jet spaces, motivic integration, motivic measure, motivic zetafunction, Grothendieck ring, stringy invariants, resolution of singularities, Minimal Model Program.

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WILLEM VEYS

1. Pre-history
1.1. Let f ∈ Z[x1 , · · · , xm ] and r ∈ Z>0 . A very general problem in number theory
is to compute the number of solutions of the congruence f (x1 , · · · , xm ) = 0 mod r (in(Z/rZ)m ). Thanks to the Chinese remainder theorem it is enough to consider the casewhere r is a power of a prime.
So we fix a prime number p and we investigate congruences modulo varying powers of
p. We denote by Fn the number of solutions of f (x1 , · · · , xm ) = 0 mod pn+1 .
1.2. Examples.
1. f1 = y − x2 . It should be clear that Fn = pn+1 .
2. f2 = x · y. Exercise : Fn = (n + 2)pn+1 − (n + 1)pn .
3. f3 = y 2 − x3 . We list Fn for small n : F0 = p,
F1 = p(2p − 1)
F2 = p2 (2p − 1)
F3 = p3 (2p − 1)
F4 = p4 (2p − 1)

F5 = p5 (p2 + p − 1)
F6 = p6 (p2 + p − 1)

F7 = p7 (2p2 − 1)
F8 = p8 (2p2 − 1)
F9 = p9 (2p2 − 1)
F10 = p10 (2p2 − 1)

F11 = p11 (p3 + p2 − 1)
F12 = p12 (p3 + p2 − 1).

Note that the plane curve {f1 = 0} is nonsingular, {f2 = 0} has the easiest curve
singularity, an ordinary node, and {f3 = 0} has a slightly more complicated singularity,
an ordinary cusp. It is in fact this cusp which is responsible for the at first sight not sonice behavior of the Fn for f3 .
More generally, the problem of the behavior of the Fn turns out to be non-obvious
precisely when {f = 0} has singularities.
1.3. We now know that, for any f ∈ Z[x1 , · · · , xm ], the Fn do satisfy the following‘regular’ behavior.
Conjecture [Borewicz,= Theorem [Igusa]. The generating formal series
P Shafarevich]nJp (T ) := Jp (f, T ) = n≥0 Fn T is a rational function in T . (In particular the Fn aredetermined by a finite number of them.)
Igusa showed this in 1975 [Ig1] using

R(1) a ‘translation’ of Jp (T ) into a p-adic integral (more precisely into Zm |f |sp |dx|,pwhich is now called Igusa’s local zeta function, and which is the ancestor of the motivic
zeta function of section 6),(2) an embedded resolution of singularities for {f = 0},(3) the change of variables formula for integrals.
(We will see later an analogue of this strategy in the theory of motivic integration.)
1.4. Examples (continuing 1.2).
p(easy).
1. Jp (f1 ; T ) = 1−pT
2T
2. Exercise : Jp (f2 ; T ) = 2p−1−p(1−pT )2 .

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS
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5

5

7

3

6

+(p −p )T −p T
3. Claim : Jp (f3 ; T ) = p 1+(p−1)T.
(1−p7 T 6 )(1−pT )1.5. We already want to mention another connection with singularity theory; the famous(still open) monodromy conjecture of Igusa relates the poles of Jp (T ) with eigenvalues of
local monodromy of f considered as a map f : Cn → C, see (6.8).
1.6. Before introducing arc spaces and motivic integration in the next sections, we present
a hopefully motivating analogy between this number theoretic setting and the geometricarc setting.
f ∈ Z[x1 , · · · , xm ]f ∈ C[x1 , · · · , xm ]solution of f = 0 over the ring
solution of f = 0 over the ring
Z/pn+1 Z ∼C[t]/(tn+1 ) ∼= Zp /pn+1 Zp , i.e.
= C[[t]]/(tn+1 ), i.e.
an m-tuple with coordinates of the form
an m-tuple with coordinates of the form
a0 + a1 p + ... + an pn (ai ∈ {0, 1, ..., p − 1}) a0 + a1 t + ... + an tn (ai ∈ C)(“n-jet” of {f = 0})
solution of f = 0 over Zp = lim Z/pn+1 Z,
←
i.e.
with
coordinates
of
the
form
P∞
i
n=0 ai psolution of f = 0 over C[[t]] = lim C[t]/(tn+1 ),
←
i.e.
with
coordinates
of
the
form
P∞
i
n=0 ai t(“arc” of {f = 0})

integrate over Zmpintegrate over L(Cm ) := { arcs of Cm }Warning. Here and further on we sometimes use other (better ?) normalizations than in
the original papers.

2. Arc spaces
Let X be an algebraic variety over C. (The theory of arc spaces and motivic integration
can be generalized to any field of characteristic zero, see e.g. [DL8].)
2.1. The space of arcs modulo tn+1 or space of n-jets on X is an algebraic variety Ln (X)over C such that{points of Ln (X) with coordinates in C} = {points of X with coordinates inC[t]}.
(tn+1 )For all n there are obvious ‘truncation maps’ πnn+1 : Ln+1 (X) → Ln (X), obtained by
reducing (n + 1)-jets modulo tn+1 , and more generally πnm : Lm (X) → Ln (X) for m ≥ n.
This description is somewhat informal, but is essentially what is needed. We now firstprovide examples and give the ‘exact’ definition later.

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WILLEM VEYS

2.2. Example. Let X = Cd . Then
(1)

(1)

(d)

(d)

(j)

n
(d) n
Ln (X) = {(a0 + a1 t + · · · + a(1)n t , · · · , a0 + a1 t + · · · + an t ), with all ai∼= C(n+1)d .

∈ C}

2.3. Example. Let X = {y 2 − x3 = 0}(0) L0 (X) = {(a0 , b0 ) ∈ C2 |b20 = a30 } = X.
(1)
L1 (X) = {(a0 + a1 t, b0 + b1 t) ∈ (C[t]/(t2 ))2 | (b0 + b1 t)2 = (a0 + a1 t)3 mod t2 }= {(a0 + a1 t, b0 + b1 t) ∈ (C[t]/(t2 ))2 | b20 = a30 and 2b0 b1 = 3a20 a1 }.
So we can consider L1 (X) as the (2-dimensional) algebraic variety in C4 with equations
b20 = a30 and 2b0 b1 = 3a20 a1 in the coordinates a0 , a1 , b0 , b1 . The map π01 : L1 (X) →L0 (X) = X is induced by the projection C4 → C2 : (a0 , a1 , b0 , b1 ) 7→ (a0 , b0 ).
The fibre of π01 above (0, 0) is {(0, a1 , 0, b1 )} ∼= C2 ; this corresponds to the fact thatthe tangent space to X at (0, 0) is the whole C2 . The fibre above (a0 , b0 ) 6= (0, 0) is theline in the (a1 , b1 )-plane with equation 2b0 b1 = 3a20 a1 , which corresponds to the tangent
line at X in (a0 , b0 ). In other words : L1 (X) is the tangent bundle T X, and π01 is thenatural projection T X → X.
(2) L2 (X) = {(a0 + a1 t + a2 t2 , b0 + b1 t + b2 t2 ) ∈ (C[t]/(t3 ))2 | (b0 + b1 t + b2 t2 )2 =(a0 + a1 t + a2 t2 )3 mod t3 } is given in C6 by the equations 2
b = a30 0
2b0 b1 = 3a20 a1
 2
b1 + 2b0 b2 = 3a0 a21 + 3a20 a2 .
Exercise. a) Verify the description of L2 (X) and note that the map π12 : L2 (X) → L1 (X)is not surjective. More precisely, the fibre of π02 above (0, 0) is {(0, a1 , a2 , 0, 0, b2 )} ∼= C3 ,
2
but its image by π1 is not the whole (a1 , b1 )-plane; it is just the line {b1 = 0}.
b) Compute L3 (X) and note that also π23 : L3 (X) → L2 (X) is not surjective.
c) However, above the nonsingular part of X = L0 (X) all considered maps πnn+1 :Ln+1 (X) → Ln (X) are fibrations with fibre C.
2.4. Some observations in the examples are easily seen to be satisfied in general.
(1) L0 (X) = X, L1 (X) = T X.
(2) If X is smooth of dimension d, then all πnn+1 are locally trivial fibrations (w.r.t. theZariski topology) with fibre Cd .
2.5. The space of arcs on X is an ‘algebraic variety of infinite dimension’ L(X) over C
such that{points of L(X) with coordinates in C} = {points of X with coordinates in C[[t]]}.

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

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We provide the ‘exact’ definition after continuing the examples. Now we have for all ntruncation maps πn : L(X) → Ln (X), obtained by reducing arcs modulo tn+1 .
2.6. Example. Let X = Cd . Then
L(X) = {(

∞
X

n
a(1)
n t ,··· ,

n=0

∞
X

n
(j)
a(d)
n t ), with all an ∈ C},

n=0

which can be considered as an infinite dimensional affine space.
2.7. Example. Let X = {y 2 − x3 = 0}. Then L(X) is given in the infinite dimensional
affine space with coordinates½

a0 , a1 , a2 , · · · , an , · · ·b0 , b1 , b2 , · · · , bn , · · ·by the infinite number of equations
 2
b0 = a30 2b b = 3a2 a
0 1
0 1
2

b1 + 2b0 b2 = 3a0 a21 + 3a20 a2




···

2.8. More precise definitions.
(i) The ‘base extension operation’ Y → Y ×C C[t]/(tn+1 ) is a covariant functor on the
category of complex algebraic varieties, and it has a right adjoint X → Ln (X). (Evenmore precisely we should say that we consider the reduced scheme Ln (X) associated tothis right adjoint scheme.) This says that, for any C-algebra R, the set of R-valued
points of Ln (X) is in natural bijection with the set of R[t]/(tn+1 )-valued points of X. In
particular, as we said in (2.1), the C-valued points of Ln (X) can be naturally identified
with the C[t]/(tn+1 )-valued points of X.
(ii) Then L(X) is the inverse limit lim Ln (X). (Technically, it is important here that←the truncation morphisms πnn+1 : Ln+1 (X) → Ln (X) are affine.) The K-valued points of
L(X), for any field K ⊃ C, are in natural bijection with the K[[t]]-valued points of X.
We mention the following result, attributed to Kolchin : if X is irreducible, then L(X)
is irreducible.
See [DL3] for more information.
2.9. When X is an affine variety, i.e. given by a finite number of polynomial equations,
one can describe equations for the Ln (X) and for L(X) as in Examples 2.3 and 2.7.
2.10. Some first natural and fundamental questions are how the Ln (X) and πn (L(X))
change with n. (For πn (L(X)) this was already considered by Nash [Na].) Note that

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Ln (X) describes by definition the n-jets on X, and πn (L(X)) those n-jets that can belifted to arcs on X.
This can be compared with the number theoretical setting of the previous section :there the question was how the solutions over Z/pn+1 Z changed with n, and we could
consider the same question for those solutions over Z/pn+1 Z that can be lifted to solutions
over Zp .
2.11. We now introduce the Grothendieck ring of algebraic varieties, which is the ‘best’framework to answer these questions, and which is moreover (essentially) the value ringfor motivic integration, to be explained in the next section.
Recall first two fundamental properties of the topological Euler characteristic χ(·) ∈ Z
on complex algebraic varieties :(1) χ(V ) = χ(Z) + χ(V \ Z) if Z is (Zariski-)closed in V ,(2) χ(V × W ) = χ(V ) · χ(W ).
A finer invariant satisfying these properties is the Hodge-Deligne polynomial H(·) =H(·; u, v) ∈ Z[u, v], given for an algebraic variety V of dimension d byd2dX
X
H(V ; u, v) :=( (−1)i hp,q (Hci (V, C)))up v q ,p,q=0 i=0

where hp,q (·) denotes the dimension of the (p, q)-component of the mixed Hodge structure. (When we would work over an arbitrary field of characteristic zero, we use anembedding into C of the field of definition of V . The up v q –coefficients of H(V ; u, v) do
not depend on the chosen embedding, since for a smooth projective V they are equal to(−1)p+q dim H q (V, ΩpV ).)Note that H(V ; 1, 1) = χ(V ).
The Grothendieck ring is the value ring of the ‘universal Euler characteristic’ on algebraic varieties.
Definition. (i) The Grothendieck group of (complex) algebraic varieties is the abeliangroup K0 (V arC ) generated by symbols [V ], where V is an algebraic variety, with therelations [V ] = [W ] if V and W are isomorphic, and [V ] = [Z] + [V \ Z] if Z is (Zariski-)closed in V .
(ii) there is a natural ring structure on K0 (V arC ) given by [V ] · [W ] := [V × W ].
— So by construction the map {Varieties over C} → K0 (V arC ) : V 7→ [V ] is indeed universal with respect to the two properties above. Of course we still loose some informationby this operation. For example X = {y 2 − x3 = 0} ⊂ A2 satisfies [X] = [A1 ]. Also, when
V → B is a locally trivial fibration with fibre F , then [V ] = [B] · [F ]. —(iii) Let C be a constructible subset of some variety V , i.e. a disjoint union of (finitely
many)
locally closed subvarieties Ai of V , then [C] ∈ K0 (V arC ) is well defined as [C] :=
P
[A
].
i
i
(iv) We denote 1:= [point], L := [A1 ] and MC := K0 (V arC )L the ring obtained from
K0 (V arC ) by inverting L.

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The rings K0 (V arC ) and MC are quite mysterious. For instance, it was shown only
recently that K0 (V arC ) is not a domain [Po], and it is still not known whether MC is adomain or not, or whether the natural map K0 (V arC ) → MC is injective.
Remark. There is an interesting alternative description of K0 (V arC ) as the abelian group,
generated by isomorphism classes [V ] of nonsingular projective varieties V , with therelations [∅] = 0 and [Ṽ ] − [E] = [V ] − [Z], where Ṽ → V is the blowing-up with centre
Z and exceptional variety E [Bi1].
2.12. We now answer the questions in (2.10). We will consider [Ln (X)] and [πn (L(X))]
in MC . For the latter we use a theorem of Greenberg [Gr], stating that πn (L(X)) is aconstructible subset of Ln (X).
Theorem [DL3][DL8]. The generating formal series
X
X
J(T ) :=[Ln (X)]T n and P (T ) :=[πn (L(X))]T n
n≥0

n≥0

in MC [[T ]] are rational, with moreover as denominators products of polynomials of theform 1 − La T b , where a ∈ Z and b ∈ Z>0 .
The proof uses motivic integration, which ‘explains’ why MC is needed instead of
K0 (V arC ); see section 3.
This result specializes to the analogous statement, replacing [·] by χ(·) or H(·). Note
for this that χ : K0 (V arC ) → Z and H : K0 (V arC ) → Z[u, v] obviously extend to1
χ : MC → Z and H : MC → Z[u, v][ uv]. When X = {f = 0} for some polynomial f , thestatement for J(T ) should be compared with Theorem 1.3 for Jp (T ) ! In this case, we will
outline a proof for J(T ) later. We just mention that the proof for P (T ) uses techniquesfrom logic, more precisely quantifier elimination.
2.13. Example. When X is smooth of dimension d, all Ln (X) = πn (L(X)) are locallytrivial over X with fibre Cnd . Hence
J(T ) = P (T ) =Xn≥0[X]Lnd T n =[X].
1 − Ld T

2.14. Example. Let X = {y 2 −x3 = 0}. The descriptions in Example 2.3 yield [L0 (X)] =[X] = L, [L1 (X)] = L2 + (L − 1)L = 2L2 − L, [L2 (X)] = L3 + (L − 1)L2 = 2L3 − L2 . We
claim that
1 + (L − 1)T + (L6 − L5 )T 5 − L7 T 6J(T ) = L,
(1 − L7 T 6 )(1 − LT )see section 6. (Compare with 1.4(3)!) The formula in [DL5, Proposition 10.2.1] yields
P (T ) =L + (1 − L)T − LT 2.
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WILLEM VEYS

2.15. Example. Let X = {xy = 0}. Exercise :(i) [Ln (X)] = (n + 2)Ln+1 − (n + 1)Ln . Then
J(T ) =

2L − 1 − L2 T.
(1 − LT )2(Compare again with Examples 1.2 and 1.4.)(ii) [πn (L(X))] = 2Ln+1 − 1. Then
P (T ) =2L − 1 − LT.
(1 − LT )(1 − T )2.16. [Mu1] To conclude this section, we relate some properties of the spaces of n-jets
on X to properties of X. Let X be irreducible of dimension d.
(i) The closure in Ln (X) of (π0n )−1 (Xreg ) is an irreducible component of Ln (X) of dimension d(n + 1).
(ii) Suppose that X is locally a complete intersection. Then
(1) Ln (X) is pure dimensional if and only if dim Ln (X) ≤ d(n + 1).
(2) Ln (X) is irreducible if and only if dim(π0n )−1 (Xsing ) < d(n + 1).
(3) If Ln+1 (X) is pure dimensional or irreducible, then so is Ln (X).
(4) If Ln (X) is irreducible for some n > 0, then X is normal.
(5) Ln (X) is irreducible for all n > 0 if and only if X has rational singularities.
(iii) When d = 1 we have for any n > 0 that Ln (X) is irreducible if and only if X is
nonsingular.

3. Motivic integration
This notion is due to Kontsevich [Ko] on nonsingular varieties. It has been further developed by Batyrev [Ba2][Ba3], and especially by Denef and Loeser [DL3][DL4][DL6][DL8],
with some improvements by Looijenga [Loo]. Probably the best way to view and understand it, is as being an analogue of p-adic integration.
Let in this section X be any algebraic variety of pure dimension d.
−1
3.1. A subset A of L(X) is called constructible or cylindric or a cylinder if A = πmC
for some m and some constructible subset C of Lm (X). These can be considered as‘reasonably nice’ subsets of the arc space L(X), being precisely all arcs obtained bylifting a nice subset of a jet space.
−1
3.2. Suppose that X is nonsingular. Then such a constructible subset A = πm
C satisfies
the property[πn (A)] = L(n−m)d [C] for all n ≥ m,

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

9nsince πm
: Ln (X) = πn (L(X)) → Lm (X) = πm (L(X)) is a locally trivial fibration with(n−m)dfibre C. We have in particular that the[πn (A)]
Lnd
are all equal in MC for n ≥ m.
For general X, a constructible set A ⊂ L(X) for which A ∩ L(Xsing ) = ∅ still satisfies(A)]
the property that the [πLnnd
stabilize for n big enough [DL3, Lemma 4.1]. More precisely
we have the following.
Definition. We call a set A ⊂ L(X) stable if for some m ∈ N we have−1(i) πm (A) is constructible and A = πm(πm (A)), and(ii) for all n ≥ m the projection πn+1 (A) → πn (A) is a piecewise trivial fibration withfiber Cd .
(So in particular A is constructible.)
Lemma [DL3]. If A ⊂ L(X) is constructible and A ∩ L(Xsing ) = ∅, then A is stable.
(A)]
Hence for such A it makes sense to consider limn→∞ [πLnnd∈ MC as an invariant of
A; it is called its naive motivic measure. Note that for nonsingular X the measure of
L(X) is just [X].
(A)]
3.3. For arbitrary constructible A ⊂ L(X) the sequence [πLnnd
will not stabilize.

Example. Let X = {xy = 0}. From Example 2.15 we see that
[πn (L(X))]
2Ln+1 − 11=
= 2L − n .
nd
n
L
L
L
This sequence ‘almost’ stabilizes (the singular point of X of course causes the trouble),and it would be nice to be able to consider 2L as the limit of this sequence.
This will indeed work in Kontsevich’s completed Grothendieck ring M̂C . This is bydefinition the completion of MC with respect to the decreasing filtration F m , m ∈ Z, of
MC , where F m is the subgroup of MC generated by the elements [S]Li with S an algebraic
variety and dim S − i ≤ −m. Note that this is indeed a ring filtration : F m · F n ⊂ F m+n .
MC
So M̂C = lim
.
←
− Fm
m

Continuing the example. Indeed in M̂C we have1[πn (L(X))]= 2L − lim n = 2L.
nd
n→∞ L
n→∞
L
lim

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Theorem [DL3]. Let A be a constructible subset of L(X). Then the limit[πn (A)]n→∞
Lndµ(A) := lim
exists in M̂C .

We call µ(A) the motivic measure of A. This yields a σ-additive measure µ on the Boolean
algebra of constructible subsets of L(X). Thus, given any sequence Ai , i ∈ Z>0 , of disjointPconstructible subsets in L(X), we have that µ(∪i Ai ) = i µ(Ai ) in M̂C .
Note. It is not known whether the natural map MC → M̂C is injective; its kernel is∩m∈Z F m . However, e.g. the topological Euler characteristic χ(·) and the Hodge-Deligne
polynomial H(·) factor through the image of MC in M̂C .
Remark. Let S ( X be a closed subvariety; it is not difficult to see that L(S) is nota constructible subset of L(X). It is possible to introduce more generally measurable
subsets of L(X), and to associate analogously a motivic measure (in M̂C ) to those subsets[Ba2][DL6]; we then have that such L(S) are measurable of measure zero.
3.4. We briefly compare with the p-adic case. Let M be a d-dimensional submanifoldof Zm
p , defined algebraically. Denote by |πn (M )| the cardinality of the image of M
under the natural truncation map πn : (Zp )m → (Zp /pn+1 Zp )m = (Z/pn+1 Z)m . Then|πn (M )|∈ Z[ p1 ] is constant for n big enough and is called the volume µp (M ) of M .
p(n+1)d
For a singular d-dimensional subvariety Z of Zm
p one defines its volume as µp (Z) :=lim²→0 µp (Z \ T² (Zsing )) ∈ R, where T² denotes a small tubular neighbourhood ‘of radius²’. Then by a Theorem of Oesterlé [Oe] we have, with analogous notation |πn (Z)|,|πn (Z)|.
n→∞ p(n+1)dµp (Z) = lim
Note the analogyp-adicmotivic

integrate overZmp

(C[[t]])mvalue rings

Z
Z[ p1 ]K0 (V arC )
MC

R

M̂C

The brilliant idea of Kontsevich was to use M̂C instead of R as a value ring for integration.
3.5. We can now consider in a natural way motivic integration. We do not treat themost general setting; the following suffices in practice. Let A ⊂ L(X) be constructible
and α : A → Z ∪ {+∞} a function with constructible fibres α−1 {n}, n ∈ Z. ThenZ
Xµ(α−1 {n})L−n
L−α dµ :=An∈Z

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

11

in M̂C , whenever the right hand side converges in M̂C . Then we say that L−α is integrable
on A. (This will always be the case if α is bounded from below.)
3.6. An important example of an integrable function is induced by an effective Cartierdivisor D on X, i.e. D is an (eventually non-reduced) subvariety of X which is locallygiven by one equation. Define ordt D : L(X) → N ∪ {+∞} : γ 7→ ordt fD (γ), wherefD is a local equation of D in a neighbourhood of the origin π0 (γ) of γ. Note e.g.
that (ordt D)(γ) = +∞ if and only if γ ∈ L(Dred ) and (ordt D)(γ) = 0 if and only if
π0 (γ) 6∈ Dred . One easily verifies that L−ordt D is integrable on L(X).
We note that (ordt D)−1 (+∞) = L(Dred ) is not constructible; it is however measurablewith measure zero.
Example. Take X = A1 and D the divisor associated to the function xN , i.e. the ‘origin
with multiplicity N ’.
Exercise. (i) N |(ordt D)(γ) for all γ ∈ L(A1 ) andµ({γ ∈ L(A1 ) | (ordt D)(γ) = iN }) =(ii)

R
L(A1 )

L−1
for all i ∈ N.
Li

N +1

L−1
L−ordt D dµ = (L−1)L= (L − 1) + L1+N.
LN +1 −1−1This example is the easiest case of the following very useful formula.
P
Proposition [Ba3][Cr]. Let X be nonsingular and D = i∈S Ni Di a normal crossings
divisor on X, i.e. all Di are nonsingular hypersurfaces intersecting transversely (andoccurring with multiplicity Ni ). Denote DI◦ := (∩i∈I Di ) \ (∪`6∈I D` ) for I ⊂ S; theDI◦ , I ⊂ S, form a natural locally closed stratification of X (note that D∅◦ = X \(∪`∈S D` )).
Then
Z
X
Y L−1
L−ordt D dµ =[DI◦ ].
L1+Ni − 1
L(X)
I⊂S

i∈I

3.7. The construction in (3.6) can be generalized as follows. Let I be a sheaf of ideals
on X. Then we define
ordt I : L(X) → N ∪ {+∞} : γ 7→ min ordt g(γ),
g

where the minimum is taken over g ∈ I in a neighbourhood of π0 (γ). Of course, when I
is the ideal sheaf of an effective Cartier divisor D, then ordt I = ordt D.
3.8. The most crucial ingredient in the theory of motivic integration is the change of variables formula or transformation rule for motivic integrals under a birational morphism.

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Theorem [DL3]. (i) Let h : Y → X be a proper birational morphism between algebraic
varieties X and Y , where Y is nonsingular. Let A ⊂ L(X) be constructible and α : A →Z ∪ {+∞} such that L−α is integrable on A. Then
Z

Z
−α

L

L−(α◦h)−ordt (Jach ) dµ.

dµ =
h−1 A

A

Here the ideal sheaf Jach is defined as follows. When also X is nonsingular, it is locallygenerated by the ‘ordinary’ Jacobian determinant with respect to local coordinates on
X and Y . For general X, the sheaf of regular differential d-forms h∗ (ΩdX ) is still a
submodule of ΩdY ; but now h∗ (ΩdX ) is not necessarily locally generated by one element.
Taking (locally) a generator ωY of ΩdY , each h∗ (ω) for ω ∈ ΩdX can be written as h∗ (ω) =gω ωY , and Jach is defined as the ideal sheaf which is (locally) generated by these gω .
(ii) When also X is nonsingular and α = ordt D for some effective divisor D on X,
we can rewrite the formula as follows :ZZ−ordt D

LA∗

L−ordt (h D+KY |X ) dµ.

dµ =
h−1 A

Here h∗ D is the pullback of D, i.e. locally given by the equation f ◦ h, if D is given by
the equation f . And KY |X is the relative canonical divisor, which is precisely the effective
divisor with equation the Jacobian determinant. Alternatively, KY |X = KY − h∗ KX
where K¦ denotes the (ordinary) canonical divisor, i.e. the divisor of zeros and poles of adifferential d-form.

Note. The birational morphism h above must be proper in order to induce a bijection
from L(Y ) to L(X) outside subsets of measure zero. More precisely, denoting by Exc
the exceptional locus of h, we have a bijection from L(Y ) \ L(Exc) to L(X) \ L(h(Exc)).
This is an easy consequence of the valuative criterion of properness [Har, Theorem II.4.7].
Exercise. Check the change of variables formula in the following special case : h is
the blowing-up of a nonsingular X in a nonsingular centre, A = L(X) and α is the zerofunction.

4. First applications
4.1. Here we mean by a Calabi-Yau manifold M of dimension d a nonsingular complete(=compact) algebraic variety, which admits a nowhere vanishing regular differential dform ωM . Alternative formulations of this last condition are that the first Chern class ofthe tangent bundle of M is zero, or that the canonical divisor KM of M is zero.

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

13

Theorem [Ko]. Let X and Y be birationally equivalent Calabi-Yau manifolds. Then[X] = [Y ] in M̂C .
Proof. Since X and Y are birationally equivalent there exist a nonsingular complete
algebraic variety Z and birational morphisms hX : Z → X and hY : Z → Y . By thedefinition of the motivic measure and the change of variables formula we have in M̂C :Z
ZZ−ordt KZ|X
[X] = µ(L(X)) =
1dµ =
L
dµ =
L−ordt KZ dµ,
L(X)

L(Z)

L(Z)

and of course [Y ] is given by the same right hand side.

¤This implies that birationally equivalent Calabi-Yau manifolds have the same HodgeDeligne polynomial, meaning that they have the same Hodge numbers. This result was
Kontsevich’s motivation to invent motivic integration !
The same proof gives the following more general result. Two nonsingular complete algebraic varieties are called K-equivalent if there exists a nonsingular complete algebraic variety Z and birational morphisms hX : Z → X and hY : Z → Y such that h∗X KX = h∗Y KY .
This is an important notion in birational geometry.
Theorem. Let X and Y be K-equivalent varieties. Then [X] = [Y ] in M̂C .
4.2. Let h : Y → X be a proper birational morphism between nonsingular algebraic
varieties. We assume that the exceptional locus Exc of h, i.e. the subvariety of Y
where h is not an isomorphism, is a normal crossings divisor. Let Ei , i ∈ S, be theirreducible components of Exc. The relative canonical divisor KY |X P
is supported on
Exc; let νi − 1 be the multiplicity of Ei in this divisor, so KY |X =i∈S (νi − 1)Ei .
◦Denoting EI := (∩i∈I Ei ) \ (∪`6∈I E` ) for I ⊂ S, we have
[X] =

X
I⊂S

[EI◦ ]

Y L−1
i∈I

Lνi − 1=X

[EI◦ ]

I⊂S

Y
i∈I

1
[Pνi −1 ]in M̂C . Indeed, by the change of variables formula we have again thatZ
[X] = µ(L(X)) =L−ordt KY |X dµ,L(Y )and then Proposition 3.6 yields the stated formula. Specializing to the topological Euler
characteristic yields the remarkable formula
χ(X) =XI⊂S

χ(EI◦ )

Y 1
i∈I

νi

,

which was first surprisingly obtained in [DL1], using p-adic integration and the Grothendieck-Lefschetz trace formula.

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5. Motivic volume
Here X is again any algebraic variety of pure dimension d.
5.1. Definition. The motivic volume of X is µ(L(X)) ∈ M̂C , thus the motivic measureof the whole arc space of X. Recall that µ(L(X)) = limn→∞ [πn (L(X))], and that it equals
Lnd[X] when X is nonsingular.
We computed in (3.3) the motivic volume of X = {xy = 0} as µ(L(X)) = 2L by thedefining limit procedure. For more complicated X, the following formula in terms of asuitable resolution of singularities is very useful.
5.2. Theorem [DL3]. Let h : Y → X be log resolution of X; i.e. h is a proper birational
morphism from a nonsingular Y such that the exceptional locus Exc of h is a normal
crossings divisor. Assume also that the image of h∗ (ΩdX ) in ΩdY is locally principal, i.e.
locally generated by one element.
Denote by Ei , i ∈ S, the irreducible components of Exc, and let ρi −1 be the multiplicity
along Ei of the divisor associated to h∗ (ΩdX ), i.e. the (effective) divisor locally given by
the zeroes of a generator of h∗ (ΩdX ). Finally, set EI◦ := (∩i∈I Ei ) \ (∪`6∈I E` ) for I ⊂ S.
Then
X
Y L−1
X
Y
1
◦
µ(L(X)) =[EI◦ ]
=
[E
]
I
ρ
ρ
i
L −1
[P i −1 ]
I⊂S

i∈I

I⊂S

i∈I

in M̂C ; in particular µ(L(X)) belongs to the subring of M̂C , obtained from (the imageof ) MC by inverting the elements 1 + L + · · · + Lj = [Pj ].
We will denote this subring by Mloc .
5.3. Example. Let X = {y 2 − x3 = 0} in A2 . We take A1 → X : u 7→ (u2 , u3 ) as a log
resolution. Since Ω1X is generated by dx and dy (subject to the relation 2ydy = 3x2 dx),
one easily verifies that h∗ Ω1X is generated by udu. Hence the image of h∗ Ω1X in Ω1Y isprincipal and we can apply Theorem 5.2.
Note that Exc = E1 = {0}, occurring with multiplicity 1 in the divisor of udu. So
ρ1 = 2 and1
L2µ(L(X)) = L − 1 + 1 =
.
[P ]
L+1
(Recall that [X] = L.)
5.4. Example. Let X = {z 2 = xy} in A3 .
Exercise. (i) Verify that µ(L(X)) = L2 . (The ‘obvious’ log resolution satisfies theassumption of Theorem 5.2, and the unique component E1 of the exceptional locus has
ρ1 = 2.)(ii) Note that also [X] = L2 ; this could be interpreted as the singularity of X being‘very mild’.

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

15

5.5. Exercise. Compute again that the motivic volume of X = {xy = 0} is 2L; now
using Theorem 5.2. (Note here that [X] = 2L − 1; one could say that the motivic volume
counts the double point twice.)
5.6. Recall that for nonsingular X its universal Euler characteristic [X] ∈ K0 (V arC )specializes to its Hodge-Deligne polynomial H(X) ∈ Z[u, v] and further to χ(X) ∈ Z.
Since χ(·) and H(·) factor through the image of MC in M̂C , they induce natural maps
χ : Mloc → Q and H : Mloc → Z[[u, v]]. Applying these specialization maps to themotivic measure of X yields new (numerical) singularity invariants, which generalize theusual χ(X) and H(X) for nonsingular X. Denef and Loeser call χ(µ(L(X))) the arc-Euler
characteristic of X.
For example the arc-Euler characteristic of {y 2 − x3 = 0} is 12 and the one of {xy = 0}
is 2.

6. Motivic zeta functions
In this section M is a nonsingular irreducible algebraic variety of dimension m, andf : M → C is a non-constant regular function.
6.1. For each n ∈ N the morphism f : M → A1 = C induces a morphism fn : Ln (M ) →Ln (A1 ). A point α ∈ Ln (A1 ) corresponds to an element α(t) ∈ K[t]/(tn+1 ) for somefield K ⊃ C; we denote as usual the largest e such that te divides α(t) by ordt α ∈{0, 1, · · · , n, +∞}. We set
Xn := {γ ∈ Ln (M ) | ordt fn (γ) = n}

for n ∈ N;

it is a locally closed subvariety of Ln (M ).
Exercise. Denote X := {f = 0}. Then [Xn ] = Lm [Ln−1 (X)] − [Ln (X)] for n ≥ 1, and[X0 ] = [M ] − [X].
Definition. The motivic zeta function of f : M → C is the formal power seriesZ(T ) :=

X

[Xn ](L−m T )nn≥0in MC [[T ]].
6.2. Considering
the exercise above, it is not a surprise that for X := {f = 0} the series
P
J(T ) = n≥0 [Ln (X)]T n and Z(T ) determine each other. Indeed, one easily verifies that
Z(Lm T ) − [M ].
J(T ) =
Lm T − 1

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WILLEM VEYS

6.3. The definition of Z(T ) is inspired by the p-adic Igusa zeta function, associated to apolynomial f ∈ Zp [x1 , · · · , xm ], which is defined asZ
Zp (s) :=|f (x)|sp |dx|Zmpfor s ∈ C with <(s) > 0. Recall that each z ∈ Zp \ {0} can be expressed as z = p` u with− ordp z` ∈ Z≥0 and u ∈ Z×
= p−` . To compare
p . One denotes ordp (z) := ` and |z|p := pwith 6.1, note that Zp (s) can be rewritten as
Zp (s) =X−nsvolume{x ∈ Zm
p | ordp f (x) = n}p

n≥0

=

1 X
#{x ∈ (Z/pn+1 Z)m | ordp f (x) = n}(p−m p−s )n .
pm
n≥06.4. Exercise. Write D for the (effective) divisor of zeros of f , i.e. D is “{f = 0} withmultiplicities”. ThenZ
L−ordt D dµ = Z(L−1 )L(M )in M̂C , meaning in particular that the substitution in the right hand side yields a welldefined element of M̂C .
6.5. As for the motivic volume, there is an important (similar) formula for Z(T ) in termsof a resolution.
Theorem [DL2]. Let h : Y → M be an embedded resolution of {f = 0}; i.e. h is
a proper birational morphism from a nonsingular Y such that h is an isomorphism on
Y \ h−1 {f = 0} and h−1 {f = 0} is a normal crossings divisor. Let Ei , i ∈ S, be theirreducible components of h−1 {f = 0}. For i ∈ S we denote by Ni the multiplicity of
Ei in the divisor of f ◦ h on Y , and by νi − 1 the multiplicity of Ei inPthe divisor ofmh∗ ω, where. (Equivalently : div(f ◦ h) = i∈S Ni Ei andP ω is a local generator of ΩM◦
KY |M = i∈S (νi − 1)Ei .) Set finally EI := (∩i∈I Ei ) \ (∪`6∈I E` ) for I ⊂ S. Then
Z(T ) =X
I⊂S

[EI◦ ]Y (L − 1)T Ni
i∈ILνi − T Ni;

in particular Z(T ) is rational and belongs more precisely to the subring of MC [[T ]] genN
erated by MC and the elements LνT−T N , where ν, N ∈ Z>0 .
6.6. Corollaries.
(i) In the special case that X = {f = 0} is a hypersurface this yields the statedrationality of J(T ) in (2.12).

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

17(ii) Let M = Am and f ∈ Z[x1 , · · · , xm ]. Then by a similar formula of Denef [De2]for the p-adic Igusa zeta functions Zp (s), Theorem 6.5 yields that Z(T ) specializes to theZp (s) for all p except a finite number. See [DL2] for a precise statement. Similarly J(T )specializes to Jp (T ) for all p except a finite number [DL8, Theorem 6.1].
(iii) For any f : M → C we now explain how Z(T ) specializes to the topological zetafunction of f . Using Theorem 6.5 and the notations there, we evaluate Z(T ) at T = L−s
for any s ∈ N; this yields the well-defined elements
X
I⊂S[EI◦ ]

Y
i∈I

X
Y
L−1
1
◦
[E
=
]
I
ν
+sN
ν
+sN
i
i
i
i −1 ]
L
−1
[P
I⊂S

i∈I

in (the image in M̂C of) the localization of MC with respect to the elements [Pj ]. Applying
the Euler characteristic specialization map χ(·) yields the rational numbers
X
I⊂S

χ(EI◦ )Yi∈I

1
νi + sNi

for s ∈ N. The topological zeta function Ztop (s) of f is the unique rational function in
one variable s admitting the values above for s ∈ N.
Without the specialization argument above it is not at all clear that Ztop (s) does notdepend on the chosen resolution h : Y → M . In fact Ztop (s) was first introduced in [DL1],
in terms of a resolution, and p-adic Igusa zeta functions and the Grothendieck-Lefschetz
trace formula were needed to prove independence of the chosen resolution.
6.7. We just mention that there is an important generalization of the motivic zetafunction, working over a relative and equivariant Grothendieck ring; it specializes by alimit procedure to objects in (an equivariant version of) MC , which are shown to be agood virtual motivic incarnation of the Milnor fibres of f at the points of {f = 0}. It isquite remarkable that a definitely non-algebraic notion as the Milnor fibre has such an
algebraic incarnation. See [DL2][DL7].
Moreover these objects satisfy a motivic Thom-Sebastiani Theorem, generalizing the
known results of Varchenko and Saito. See [DL4].
6.8. Monodromy Conjecture.
There is an intriguing conjectural relation between the poles of the topological zeta function and the eigenvalues of the local monodromy of f .
Monodromy conjecture. If s0 is a pole of Ztop (s), then e2πis0 is an eigenvalue ofthe local monodromy action on the cohomology of the Milnor fibre of f at some point of{f = 0}.
One can also state the analogous conjecture for the motivic zeta function, but thenone has to be careful with the notion of pole, see [RV2]. Alternatively, we can formulate
this monodromy conjecture for Z(T ) as follows, without mentioning poles [DL2] :N
Z(T ) belongs to the ring generated by MC and the elements LνT−T N , where ν, N ∈ Z>0νand e2πi N is an eigenvalue of the local monodromy as above.

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WILLEM VEYS

Actually, it was originally stated for the p-adic Igusa zeta function, being even more
remarkable, for then it relates number theoretical invariants of f ∈ Z[x1 , · · · , xm ] todifferential topological invariants of f , considered as function Cn → C.
The conjecture was shown by Loeser for M = A2 [Loe1]; a shorter proof in dimension
2 is in [Ro]. In dimension 3 there is a lot of ‘experimental evidence’ [Ve1], and by now
various special cases are proved [ACLM1][ACLM2][Loe2][RV1].
Example. Let M = A2 and f = y 2 − x3 .
Exercise. Compute, using Theorem 6.5,
L5 − L3 T + L3 T 2 − T 5
Z(T ) = L (L − 1)
(L5 − T 6 )(L − T )2

andZtop (s) =5 + 4s.
(5 + 6s)(1 + s)

(This is how we computed J(T ) in Example 2.14.) In particular, the poles of Ztop (s) are−1 and −5/6. On the other hand, it is well known that the monodromy eigenvalues of fπiπi
are 1, e 3 , and e− 3 . Hence the monodromy conjecture is indeed satisfied here.
Note. The previous example was too simple to exhibit the ‘typical’ situation. Each irνireducible component Ei in Theorem 6.5 induces a candidate-pole − N, and quite miraci
ulously, for a generic example with a lot of components Ei , ‘most’ of these candidates
cancel. This experimental fact is compatible with the monodromy conjecture, see [Ve1].
7. Batyrev’s stringy invariants
Using motivic integration, Batyrev [Ba1][Ba2] introduced new singularity invariants foralgebraic varieties with ‘mild’ singularities, more precisely with at worst log terminal
singularities. He used them for instance to formulate a topological mirror symmetry
test for singular Calabi-Yau varieties, to give a conjectural definition for stringy Hodge
numbers, and to prove a version of the McKay correspondence.
We first explain log terminal and related singularities; for this we need the Gorenstein
notion.
7.1. Let X be a normal algebraic variety of dimension d. In particular X is irreducible,
Xsing has codimension at least 2 in X, and X has a well defined canonical divisor KX(up to linear equivalence). One can view (a representative of) KX as the divisor of zeroes
and poles of a rational differential d-form on X; it is also the Zariski-closure of the usual
canonical divisor on Xreg .
When X is nonsingular, KX is a Cartier divisor, i.e. locally given by one equation.
This is not true in general.
Definition. A normal variety X is Gorenstein if KX is a Cartier divisor. Alternatively :X is Gorenstein if the rational differential d-forms on X, which are regular on Xreg , arelocally generated by one element.

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

19

Example. Let X = {z 2 = xy}; then those differential 2-forms are generated by dx∧dy
=
2z
dy∧dz
dx∧dz
= − y (which is indeed regular on Xreg ).
xThis notion is quite general; for instance all (normal) hypersurfaces and even complete
intersections are Gorenstein.
Note. In the literature one often uses the term Gorenstein alternatively for varieties X
for which all local rings OX,x (x ∈ X) are Gorenstein rings, and then the property that
KX is a Cartier divisor is called 1-Gorenstein.
7.2. We now introduce a certain ‘badness’ for singularities, in terms of numerical invariants of a resolution.
Let X be Gorenstein of dimension d. Take a log resolution π : Y → X of X and denoteby Ei , i ∈ S, the irreducible components of the exceptional locus Exc of h. We associate
as follows an integer ai to each Ei .
(1) Description with divisors. Since KX is Cartier, the pullback π ∗ KX makes sense andone can consider the relative canonical divisor KY |X = KY − π ∗ KX ,P
which is supported
on Exc. Then ai − 1 is the multiplicity of Ei in KY |X , i.e. KY |X = i∈S (ai − 1)Ei .
(2) Description with differential forms. Take a general point Qi of Ei and local coordinates y1 , y2 , · · · , yd around Qi such that the local equation of Ei is y1 = 0. Let ωi be a
local generator around π(Qi ) of the d-forms on X, which are regular on Xreg . (Such an
ωi exists by the Gorenstein property.) Then around Qi one can write π ∗ ωi as
π ∗ ωi = uy1ai −1 dy1 ∧ dy2 ∧ · · · ∧ dyd ,where u is regular and nonzero around Qi .
In general the ai ∈ Z, and when X is nonsingular they satisfy ai ≥ 2.
Terminology. One calls ai the log discrepancy of Ei with respect to X (and ai − 1 thediscrepancy).
Example. The standard log resolution of X = {z 2 = xy} has one exceptional curveE∼= P1 with log discrepancy a = 1.
7.3. We also have to consider a technical generalization: a normal variety X is called
Q-Gorenstein if rKX is Cartierfor some r ∈ Z>0 . Then the log discrepancies are definedP
analogously by KY |X = i∈S (ai − 1)Ei , which should be considered as an abbreviationPof rKY |X = rKY − rKX = i∈S r(ai − 1)Ei . Now the r(ai − 1) ∈ Z, and hence ai ∈ 1r Z.
Example. Let X be the quotient of A2 by the action of µ3 = {z ∈ C | z 3 = 1} given by(x, y) 7→ (²x, ²y) for ² ∈ µ3 . Concretely, X is given in A4 by the equations{u1 u3 − u22 = u2 u4 − u23 = u1 u4 − u2 u3 = 0} ,

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WILLEM VEYS

in particular it is not a complete intersection. Here KX is not Cartier; a representativeof KX is for example {u1 = u2 = u3 = 0}. However, 3KX is Cartier; a representative is{u1 = 0}.
The standard log resolution of X has one exceptional curve E ∼= P1 with log discrepancy2
a = 3.
A nice introduction to these notions is in [Re1].
7.4. Definition. (i) Let X be a Q-Gorenstein variety. Take a log resolution π : Y → X
of X; let Ei , i ∈ S, be the irreducible components of the exceptional locus of π with log
discrepancies ai . Then X is called terminal, canonical, log terminal and log canonical if
ai > 1, ai ≥ 1, ai > 0 and ai ≥ 0, respectively, for all i ∈ S.
One can show that these conditions do not depend on the chosen resolution.
(ii) We say that X is strictly log canonical if it is log canonical but not log terminal.
We should note that 0 is indeed the relevant ‘border value’ here; if some ai < 0 on somelog resolution, then one can easily construct log resolutions with arbitrarily negative ai .
The log terminal singularities should be considered ‘mild’, the singularities which are
not log canonical ‘general’, and the strictly log canonical ones as a special ‘border’ class.
7.5. Example. (1) When X is a surface (d = 2) terminal is equivalent to non-singular,
the canonical singularities are precisely the so-called ADE singularities or rational double
points, and the log terminal singularities are precisely the Hirzebruch-Jung or quotient
singularities.
(2) Let X = {xk1 + xk2 + · · · + xkd+1 = 0} in Ad+1 . The origin is the only singular pointof X, and the blowing-up with the origin as centre yields a log resolution π : Y → X of
X with exceptional locus consisting of one irreducible component E, which is isomorphic
to {xk1 + xk2 + · · · + xkd+1 = 0} ⊂ Pd .
Exercise. (i) The log discrepancy of E with respect to X is d + 1 − k.
(ii) X is log terminal, strictly log canonical, and not log canonical when k < d + 1, k =d + 1, and k > d + 1, respectively.
7.6. There are nice results of Ein, Mustaţǎ and Yasuda, relating the previous notions
with jet spaces.
Theorem [Mu1][EMY][EM]. Let X be a normal variety, which is locally a complete
intersection. Then X is terminal, canonical, and log canonical if and only if Ln (X) is
normal, irreducible, and equidimensional, respectively, for every n.
7.7. Definition. Let X be a log terminal algebraic variety. Take a log resolution
π : Y → X of X. Let Ei , i ∈ S, be the irreducible components of the exceptional locus ofπ with log discrepancies ai (∈ Q>0 ). Denote also EI◦ := (∩i∈I Ei ) \ (∪`6∈I E` ) for I ⊂ S.
(i) The stringy Euler number of X isX
Y 1.
est (X) :=χ(EI◦ )
ai
I⊂S

i∈I

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

21(ii) The stringy E-function of X isEst (X) :=X

H(EI◦ )

I⊂S

Y
i∈I

uv − 1.
(uv)ai − 1(iii) The stringy E-invariant of X isEst (X) :=X[EI◦ ]

I⊂S

Y L−1
i∈I

Lai − 1.

Remarks. (1) Clearly est (X) ∈ Q; Est (X) is a rational function in u, v (with ‘fractional
powers’), and Est (X) lives in a finite extension of M̂C . We have specialization maps
Est (X) 7→ Est (X) 7→ est (X).
(2) Strictly speaking, Batyrev defined and used only the levels (i) and (ii) [Ba2][Ba3].
When X is nonsingular, Est (X) = [X] (this is 4.2), and of course Est (X) = H(X) andest (X) = χ(X). So also these invariants are new singularity invariants, generalizing[·], H(·) and χ(·), respectively, for nonsingular X. (Just as the motivic volume and its
specializations. We give a comparing example in 7.11.)
7.8. The crucial point is that the defining expressions above do not depend on thechosen resolution. We indicate three different arguments, supposing for simplicity that
X is Gorenstein, i.e. the ai ∈ Z>0 .
(1) Let π : Y → X and π 0 : Y 0 → X be two log resolutions of X. By the formula of
Proposition 3.6 we have in fact
Z
X
Y L−1
◦
[EI ]
=
L−ordt KY |X dµ.
Lai − 1L(Y )I⊂S

So we must show thati∈I

R

L−ordt KY |X dµ =L(Y )R
L(Y 0 )
0L−ordt KY 0 |X dµ. To this end we takeσπa log resolution ρ : Z → X, dominating π and π ; i.e. we have ρ : Z → Y → X and
σ

0

π

0

ρ : Z → Y 0 → X. By the change of variables formula in (3.8) we haveZ
Z
Z
−ordt (σ ∗ KY |X +KZ|Y )−ordt KY |X
dµ =
L
dµ =
L−ordt (KZ|X ) dµ ,
L
L(Y )

L(Z)

L(Z)

and of course the same is true for the integral over L(Y 0 ).
This is essentially Batyrev’s proof.
(2) We can define Est (X) intrinsically, using motivic integration on X [Ya1][DL6]. Thereis an ideal sheaf IX on X such thatZ
Est (X) =Lordt IX dµ,
L(X)

22

WILLEM VEYS

using the setting of (3.5) and (3.7). More precisely, denoting by ωX the sheaf of differential
d-forms on X which are regular on Xreg , we have a natural map ΩdX → ωX whose image
is IX ωX . See [Ya1, Lemma 1.16].
(3) Using the Weak Factorization Theorem, see below, one essentially has to show thatthe defining expressions in (7.7) do not change after blowing-up Y in a nonsingular centre
which intersects ∪i∈S Ei transversely. This is straightforward.
7.9. Weak Factorization Theorem [AKMW][WÃl].
(1) Let φ : Y − → Y 0 be a proper birational map between nonsingular irreducible varieties, and let U ⊂ Y be an open set where φ is an isomorphism. Then φ can be factored
as follows into a sequence of blow–ups and blow–downs with smooth centres disjoint fromU.
There exist nonsingular irreducible varieties Y1 , . . . , Y`−1 and a sequence of birational
maps
φ1

φ2

φi−1

φi

φi+1

φ`−1

φ`

Y = Y0 − → Y1 − → · · · − → Yi−1 − → Yi − → · · · − → Y`−1 − → Y` = Y 0
where φ = φ` ◦ φ`−1 ◦ · · · ◦ φ2 ◦ φ1 , such that each φi is an isomorphism over U (we identify
U with an open in the Yi ), and for i = 1, . . . , ` either φi : Yi−1 − → Yi or φ−1: Yi − → Yi−1
iis the blowing–up at a nonsingular centre disjoint from U , and is thus a morphism.
(10 ) There is an index i0 such that for all i ≤ i0 the map Yi → Y is a morphism, andfor i ≥ i0 the map Yi → Y 0 is a morphism.
(2) If Y \ U and Y 0 \ U are normal crossings divisors, then the factorization above canbe chosen such that the inverse images of these divisors under Yi → Y or Yi → Y 0 are
also normal crossings divisors, and such that the centres of blowing–up of the φi or φ−1iintersect these divisors transversely.
Remark. (i) In [AKMW] and [WÃl] the theorem is stated for a birational map φ between
complete Y and Y 0 ; the generalization to proper birational maps between not necessarily
complete Y and Y 0 is mentioned by Bonavero [Bo].
(ii) In [AKMW, Theorem 0.3.1] the first claim of (2) is not explicitly stated, but canbe read off from the proof (see [AKMW, 5.9 and 5.10]).
7.10. Important Intermezzo. Using weak factorization instead of motivic integration,
we can define Est (X) in a localization of (a finite extension of) MC , which is a priori
finer than in (a finite extension of) M̂C , since we do not know whether the natural map
MC → M̂C is injective.
This remark also applies e.g. to (4.1), yielding [X] = [Y ] in the localization of MCwith respect to the [Pj ] instead of merely in M̂C .
7.11. Example. Let X = {xk1 + xk2 + · · · + xkd+1 = 0} ⊂ Ad+1 .
Exercise. We use the notation E of Example 7.5.
L−1(i) Est (X) = (L − 1)[E] + [E] Ld+1−k,−1L−1(ii) µ(L(X)) = (L − 1)[E] + [E] Ld −1 ,

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

23(iii) [X] = (L − 1)[E] + 1.
(Note also that (ii) and (iii) are consistent with Example 5.4.)
7.12. Applications.
(i) Topological mirror symmetry test for singular Calabi-Yau mirror pairs [Ba2].
(ii) A conjectural definition of stringy Hodge numbers for certain canonical Gorensteinvarieties [Ba2].
(iii) A proof of a version of the McKay correspondence [Ba3][DL6][Ya1].
(iv) A new birational invariant for varieties of nonnegative Kodaira dimension, assuming the Minimal Model Program [Ve2, (2.8)].

8. Stringy invariants for general singularities
In this section X is a Q-Gorenstein variety.
8.1. For a log resolution π : Y → X of X, we use the notations Ei and ai , i ∈ S, andEI◦ , I ⊂ S, as before. There are (at least) two natural questions concerning a possible
generalization of Batyrev’s stringy invariants beyond the log terminal case.
Question I. Suppose there exists at least one log resolution π : Y → X of X for which
all log discrepancies ai 6= 0. Is (e.g.)X

χ(EI◦ )

I⊂S

Y 1
i∈I

ai

independent of a chosen such resolution ?
This question is still open (a positive answer would yield a generalized stringy invariant
for those X admitting such a log resolution). Note that, when using the weak factorization
theorem to connect two such log resolutions by chains of blowing-ups, log discrepancies
on ‘intermediate varieties’ could be zero, obstructing an obvious attempt of proof.
Question II. Do there exist any kind of invariants, associated to all or ‘most’ QGorenstein varieties, which coincide with Batyrev’s stringy invariants if the variety is
log terminal ?
Concerning this question, we obtained the following result [Ve4]. We associated invariants to ‘almost all’ Q-Gorenstein varieties, more precisely to all Q-Gorenstein varieties
without strictly log canonical singularities, which do generalize Batyrev’s invariants forlog terminal varieties. (Note that in particular log discrepancies can be zero in a log
resolution of a non log canonical variety !)• To construct these invariants we have to assume Mori’s Minimal Model Program (infact the relative and log version).
• As in the previous section, we can work on any level : χ(·), H(·), and [·]. For simplicity
we treat here just the roughest level χ(·); the other levels are analogous.

24

WILLEM VEYS

8.2. We associate to any Q-Gorenstein X without strictly log canonical singularities arational function zst (X; s) in one variable s, the stringy zeta function of X. It will turnout that for log terminal X, this rational function is in fact a constant and equal toest (X).
We just present the main idea of our construction. The ‘pragmatic’ idea is to split thelog discrepancies ai of a log resolution π : Y → X as ai = νi +Ni such that (νi , Ni ) 6= (0, 0)
for all i, and to define zst (X; s) asX

χ(EI◦ )

I⊂S

Y
i∈I

1
∈ Q(s).
νi + sNi

This is done in a geometrically meaningful way via factoring π through a certain ‘partial
resolution’ p : X m → X of X, which is called a relative log minimal model of X. This isa natural object in the (relative, log) Minimal Model Program; important here is that it
is not unique and that X m can have certain mild singularities. (Its existence is the key
point in this Program and this is for the moment proved only in dimensions 2 and 3.)
For the specialists : p is a proper birational morphism, X m is Q-factorial, the pair
(X m , E m ) is divisorial log terminal, and KX m + E m is p-nef, where E m denotes the
reduced exceptional divisor of p. References for these notions are e.g. in [KM][KMM][Ma].
ph

We consider the factorization π : Y → X m → X. In general h is only a birational map(maybe not everywhere defined), but we suppose for the moment that it is a morphism.
We justify this later. Denoting as usual by Ei , i ∈ S, the irreducible components of the
exceptional divisor of π, we let Eim , i ∈ S m , be the images in X m of those Ei which‘survive’ in X m , i.e. which are not contracted by h to varieties of smaller dimension.
ThenX
X
a i E i = KY +E i − π ∗ KX
i∈S

i∈S

= KY +|Xi∈S

Ei − h∗ (KX m +
{z

X
i∈S m

Eim ) + h∗ (KX m +}XEim ) − h∗ p∗ KX .

i∈S m

|

{z

(1)

}

(2)

P
Both (1) and (2) are divisors onP
Y , supported on ∪i∈S Ei . We write (1) as i∈S νi Ei ;all νi ≥ 0 because the pair (X m , i∈S m Eim ) has only mild singularities (more precisely,
because it is divisorial log terminal). We can rewrite (2) asXXai Eim );h∗ (KX m +Eim − p∗ KX ) = h∗ (
i∈S m

i∈S m

and it isP
well known that all ai , i ∈ S m , are non-positive
P (more precisely, this follows since
KX m + i∈S m Eim is p-nef). So we can write (2) as i∈S Ni Ei where all Ni ≤ 0.
With these definitions of νi and Ni we indeed have ai = νi +Ni for i ∈ S, with moreoverνi ≥ 0 and Ni ≤ 0. One can show that, if X has no strictly log canonical singularities,
the situation νi = Ni = 0 cannot occur.

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

25

When X is log terminal, the morphism p : X m → X has no exceptional divisors, so
S m = ∅, all Ni = 0 and νi = ai , and as promised zst (X; s) = est (X).
In fact we FIRST choose a relative log minimal model p : X m → X of X, we secondly
choose a log resolution h : Y → X m of the pair (X m , E m ), where E m is the reduced
exceptional divisor of p, and then we put π := p ◦ h.
The point is again that zst (X; s) is independent of both choices, for which a crucial
ingredient is the Weak Factorization Theorem.
8.3. Theorem [Ve4]. Let X be any surface without strictly log canonical singularities.
Then
lim zst (X; s) ∈ Q.
s→1

(Recall that this is non-obvious since some ai can be zero. The clue is that if ai = 0,then Ei must be rational and must intersect exactly once or twice other components; thisthen easily implies the cancelling of νi + sNi in the denominator of zst (X; s).) So we candefine in dimension 2 a generalized stringy Euler number est (X) as the limit above forany such surface X. In fact we constructed this generalized est (X) in [Ve3] by a ‘direct’approach.
•

E•

•

...

•

(1)

•
• ...
•
•
(i)
(i)
.. E E (i)
E
E
ri −1
2
1
.

(i)

...

(k)

...
......
......
......
.
.
.
.
.
.....
......
......
......
.
.
.
.
.
...
.....
......
..........
......
......
(i)......
......
......
ri......
......
......
......
......
......
...

•

..
.

•

•

•

•

Figure 1

8.4. Example [Ve3]. Let P ∈ X be a normal surface singularity with dual graph of its
minimal log resolution π : X → S as in Figure 1. There is a central curve E with genus gand self-intersection number −κ, and all other curves are rational. Each attached chain(i)(i)
E1 − · · · − Eri is determined by two co-prime numbers ni and qi , which are the absolute
(i)
(i)
(i)
(i)
value of the determinant of the intersection matrix of E1 , . . . , Eri and E1 , . . . , Eri −1 ,respectively. Finally, we denote by d the absolute value of the determinant of the totalintersection matrix of π −1 P . This is a quite large class of singularities; it includes allweighted homogeneous isolated complete intersection singularities, for which the numbers{g; κ; (n1 , q1 ), · · · , (nk , qk )} are called the Seifert invariants of the singularity.
If P ∈ X is not strictly log canonical, thenkX1ni ) + χ(X \ {P }) ,est (X) = lim zst (X; s) = (2 − 2g − k +
s→1
a
i=1

26

WILLEM VEYS

where

Pk
Qk
k
X
2 − 2g − k + i=1 n1i
1i=1 nia==(2 − 2g − k +
)
Pk qi
d
n
i
κ − i=1 ni
i=1is the log discrepancy of E.
We note that some other log discrepancies might be zero. A particular example is theso-called triangle singularity, given by g = 0, κ = 1, k = 3 and r1 = r2 = r3 = 1. So,
concretely, there is a central rational curve with self-intersection −1 to which three otherrational curves are attached. Then a = −1 and the three other log discrepancies are zero,and est (X) = 1 − (n1 + n2 + n3 ) + χ(X \ {P }).
When such P ∈ X is a weighted homogeneous isolated hypersurface singularity, thisgeneralized stringy Euler number appears in some Taylor expansion associated to it,studied by Némethi and Nicolaescu [NN].
8.5. Example. [Ve4] Here we mention a concrete example of a threefold singularity
P ∈ X, having an exceptional surface with log discrepancy zero in a log resolution, andsuch that nevertheless lims→1 zst (X; s) ∈ Q, i.e. such that the evaluation zst (X; 1) makes
sense.
Let X be the hypersurface {x4 + y 4 + z 4 + t5 = 0} in A4 ; its only singular point is
P = (0, 0, 0, 0). We sketch the following constructions in Figure 2; we denote varieties
and their strict transforms by the same symbol.
The blowing-up π1 : Y1 → X with centre P is already a resolution of X (Y1 is smooth).
Its exceptional surface E1 is the affine cone over the smooth projective plane curve C ={x4 + y 4 + z 4 = 0}. Let π2 : Y2 → Y1 be the blowing-up with centre the vertex Q of thiscone, and exceptional surface E2 ∼= P2 . Then E1 ⊂ Y2 is a ruled surface over C whichintersects E2 in a curve isomorphic to C. The composition π = π1 ◦ π2 is a log resolutionof P ∈ X, and one easily verifies that the log discrepancies are a1 = 0 and a2 = −1; inparticular P ∈ X is not log canonical.
Now E1 ⊂ Y2 can be contracted (more precisely one can check that the numericalequivalence class of the fibre of the ruled surface E1 is an extremal ray). Let h : Y2 → X mdenote this contraction, and let π = p ◦ h. As the notation suggests, one can verify that
KX m + E2 is p-nef, implying that (X m , E2 ) is a relative log minimal model of P ∈ X.
Denoting as usual
KY2 = h∗ (KX m + E2 ) + (ν1 − 1)E1 + (ν2 − 1)E2

andh∗ (a2 E2 ) = N1 E1 + N2 E2

we have clearly that ν2 = 0 and N2 = −1, and one computes that ν1 = 51 and N1 = − 15 .
So
χ(C)
χ(E1 \ C) χ(E2 \ C)+++ χ(X \ {P })(ν1 + sN1 )(ν2 + sN2 )ν1 + sN1
ν2 + sN2
−4
−4
7
13
= 1 1
+ 1 1 ++ χ(X \ {P }) =+ χ(X \ {P }) ,−ss
( 5 − 5 s)(−s)
5 − 5szst (X; s) =yielding lims→1 zst (X; s) = zst (X; 1) = 13 + χ(X \ {P }).

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

E1 ...................................................

Y2

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

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

C

π2................

E2...
......
......
...........
.
.
.
.
.
.
.
.
.
.
...

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

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

h

E

Y1

1
.............................................
...
.......
.......
..
.
.
...........
......
... ....................................... ....
.
...
.
...
...
...
...
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.
.
.....
.
.
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.....
...............
.
.
.
.
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.
.
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.
.
.
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.
.
.
.
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........... ....
...
.......
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.
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..........................

27

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

• Q

.....
.....
.....
..... 1
.....
.....
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...........
..

E

π

2
....................................................
................
.........
.........
.......
..............................
.
.
.
.
.
.......
.
.....
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.
.
.
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.................................................

C

p ..........

π

Xm

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

•P
X

Figure 2

8.6. Question. Let X be a Q-Gorenstein variety of arbitrary dimension without strictly
log canonical singularities. When islim zst (X; s) ∈ Q ?

s→1

9. Miscellaneous recent results
Here we gather a collection of various results, which were obtained after the redaction of
the survey paper [DL8].
• Aluffi noticed in [Al1] that the Euler characteristic formula in (4.2) implies interesting
similar statements about Chern-Schwartz-MacPherson classes. Then in [Al2] he studies
the birational behavior of Chern classes with respect to the ‘motivic integration philosophy’. There he also introduces stringy Chern classes of log terminal varieties, which wasdone simultaneously by de Fernex, Lupercio, Nevins and Uribe in [dFLNU].
• Bittner [Bi2] calculated the relative dual of the motivic nearby fibre and constructed anearby cycle morphism on the level of the Grothendieck group of varieties.
• More exotic motivic measures are introduced by Bondal, Larsen and Lunts [BLL] andDrinfeld [Dr].

28

WILLEM VEYS

• Using arc spaces and motivic integration, Budur [Bu] relates the Hodge spectrum of a
hypersurface singularity to its jumping numbers (which come from multiplier ideals).
• Campillo, Delgado and Gusein-Zade [CDG1][CDG2][CDG3], and Ebeling and GuseinZade [EG1][EG2] studied filtrations on the ring of germs of functions on a germ of acomplex variety, defined by arcs on the singularity. An important technique is integration
with respect to the Euler characteristic over the projectivization of the space of function
germs; this notion is similar to (and inspired by) motivic integration.
• Cluckers and Loeser [CL1][CL2][CL3] built a more general theory for relative motivic
integrals, avoiding moreover the completion of Grothendieck rings. These integrals specialize to both ‘classical’ and arithmetic motivic integrals.
More ‘relative theory’ is in [Ni3].
• Dais and Roczen obtained formulas for the stringy Euler number and stringy E-function
for some special classes of singularities [Da][DR].
• Now available are the ICM 2002 survey [DL9] and the recent expository paper of Hales[Hal3] on the theory of arithmetic motivic measure of Denef and Loeser [DL5]. Related
work is in [DL10] and [Ni3].
• In [dSL] du Sautoy and Loeser associate motivic zeta functions to a large class of infinite
dimensional Lie algebras.
• Ein, Lazarsfeld, Mustaţǎ and Yasuda have various other papers about spaces of jets,
relating them for instance to singularities of pairs, in particular to the log canonicalthreshold, and to multiplier ideals [ELM][Mu2][Ya2].
• Koike and Parusiński [KP] associated motivic zeta functions to real analytic function
germs and showed that these are invariants of blow-analytic equivalence. Fichou [Fi]obtained similar results in the context of Nash funcion germs. Both constructions areuseful for classification issues.
• Gordon [Go] introduced a motivic analogue of the Haar measure for the (non locallycompact) groups G(k((t))), where G is a reductive algebraic groups, defined over analgebraically closed field k of characteristic zero.
• Guibert [Gui] computed the motivic zeta function associated to irreducible plane curve
germs, yielding a new proof of the formula expressing the spectrum in terms of the Puiseux
data. Here he studied also a motivic zeta function for a family of functions and related
it with the Alexander invariants of the family; this is used to obtain a formula for theAlexander polynomial of a plane curve.
• Guibert, Loeser and Merle [GLM1] introduced iterated motivic vanishing cycles andproved a motivic version of a conjecture of Steenbrink concerning the spectrum of hypersurface singularities.
• Gusein-Zade, Luengo and Melle Hernández [GLM2] treat integration over spaces of
non-parametrized arcs and introduce motivic versions of the classical monodromy zetafunction. They indicate a formula connecting the motivic zeta function with this monodromy zeta function.
• Arithmetic motivic integration in the context of p-adic orbital integrals and transfer
factors is considered by Gordon and Hales in [GH] and [Hal2]. An introduction to thistheory is [Hal1].

ARC SPACES, MOTIVIC INTEGRATION AND STRINGY INVARIANTS

29• Ishii and Kollár [IK] found counter examples in dimensions at least 4 to the Nashproblem, which relates irreducible components of the space of arcs through a singularity
to exceptional components of a resolution. (And they proved it in general for toric
singularities.) Reguera [Reg] showed in any dimension that the Nash problem is equivalent
to the so-called wedge problem.
For a toric variety, Ishii [Is] described precisely the relation between arc families andvaluations, and obtained the answer to the embedded version of the Nash problem.
• Ito produced an alternative proof that birational smooth minimal models have equal
Hodge numbers [It1], and that Batyrev’s stringy E-function is well defined [It2], using
p-adic Hodge theory.
• Kapranov [Ka] introduced another motivic zeta function as the generating series for
motivic measures of varying n-fold symmetric products of a fixed variety. Larsen andLunts [LL1][LL2] determined for which surfaces this is a rational function over K0 (V arC ).
It is not known whether it is always a rational function over MC . See also [DL10, §7]
and [BDN].
• For toric surfaces, Lejeune-Jalabert and Reguera [LR] and Nicaise [Ni1] computed an
explicit formula for the series P (T ) and J(T ), respectively. This last paper also contains
a sufficient condition for the equality of P (T ) and the arithmetic Poincaré series of atoric singularity, which is always satisfied in the surface case. A counter example for thisequality in dimension 3 is given.
In [Ni2] Nicaise provides a concrete formula for P (T ) if the variety has an embedded
resolution of a simple form; this yields a short proof of the formula for toric surfaces.
• Loeser [Loe3] studied the behavior of motivic zeta functions of prehomogeneous vector
spaces under castling transformations; he deduced in particular how the motivic Milnor
fibre and the Hodge spectrum at the origin behave under such transformations.
• In [Ni4] Nicaise establishes the motivic zeta function as a Weil zeta function of the rigid
Milnor fibre.
• Sebag [Se1][Se2] studied motivic integration and motivic zeta functions in the context
of formal schemes. Loeser and Sebag [LS] developed a theory of motivic integration forsmooth rigid varieties, obtained a motivic Serre invariant, and provided new geometricbirational invariants of degenerations of algebraic varieties.
• The author introduces motivic principal value integrals and investigates their birational
behavior in [Ve5].
• Vojta provides in [Vo] a general reference for jet spaces and jet differentials (at the levelof EGA), using Hasse-Schmidt higher differentials.
• Yasuda [Ya1][Ya3] introduced twisted jets and arcs over Deligne-Mumford stacks andstudied then motivic integration over them. As applications he obtained a McKay correspondence for general orbifolds (see also [LP]), and a common generalization of thestringy E-function and the orbifold cohomology.
• Yokura [Yo] constructs Chern-Schwartz-MacPherson classes on pro-algebraic varieties
and relates this to the motivic measure.

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WILLEM VEYS

References[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.
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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