Germs of arcs on singular algebraic varieties and motivic integration

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Germs of arcs on singular algebraic varieties and motivic integration Jan Denef 1 , Franc¸ois Loeser 2;3 1 University of Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium () 2 Centre de Mathe matiques, Ecole Polytechnique, F-91128 Palaiseau (URA 169 du CNRS) () 3 Institut de Mathe matiques, Universite P. et M. Curie, Case 82, 4 place Jussieu, F-75252 Paris Cedex 05 (UMR 9994 du CNRS) Oblatum 19-XII-1996 & 6-III-98 / Published online: 14 October 1998 1. Introduction Let k be a field of characteristic zero. We denote by M the Gro- thendieck ring of algebraic varieties over k (i.e. reduced separated schemes of finite type over k). It is the ring generated by symbols ?S?, for S an algebraic variety over k, with the relations ?S? ? ?S 0 ? if S is isomorphic to S 0 ; ?S? ? ?S n S 0 ? ? ?S 0 ? if S 0 is closed in S and ?S S 0 ? ? ?S??S 0 ?.

  • ?s

  • semi-algebraic condition

  • over z

  • variety over

  • l?x ?

  • let ‘

  • ?w ?

  • k??t??? ?


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Invent. math. 135, 201–232 (1999)
DOI 10.1007/s002229800880
Germs of arcs on singular algebraic varieties
and motivic integration
1 2;3Jan Denef , Francois Loeser
1University of Leuven, Department of Mathematics, Celestijnenlaan 200B,
B-3001 Leuven, Belgium (Jan.Denef@wis.kuleuven.ac.be)
2Centre de Mathematiques, Ecole Polytechnique, F-91128 Palaiseau
(URA 169 du CNRS) (loeser@math.polytechnique.fr)
3Institut de Mathematiques, Universite P. et M. Curie, Case 82,
4 place Jussieu, F-75252 Paris Cedex 05 (UMR 9994 du CNRS)
Oblatum 19-XII-1996 & 6-III-98 / Published online: 14 October 1998
1. Introduction
Let k be a field of characteristic zero. We denote by M the
Grothendieck ring of algebraic varieties over k (i.e. reduced separated
schemes of finite type over k). It is the ring generated by symbols�S�,
0for S an algebraic variety over k, with the relations�S��� S� if S
0 0 0 0is isomorphic to S;�S��� Sn S��� S� if S is closed in S and
0 0�S S��� S��S�. Note that, for S an algebraic variety over k, the
0 0mapping S7!�S� from the set of closed subvarieties of S extends
uniquely to a mapping W7!�W� from the set of constructible subsets
0 0 0of S toM, satisfying �W[ W��� W��� W��� W\ W�. We set
1 �1L :�� A� andM :� M�L �.WedenotebyM�T� thesubringoflock loc
�1a bM ��T�� generated byM �T� and the series�1� L T � with a inloc loc
Z and b in Nnf0g.
Let X be an algebraic variety over k. We denote by L�X� the
scheme of germs of arcs on X. It is a scheme over k and for any field
extension k K there is a natural bijection
L�X��K ’Mor �Spec K��t��; X�;k-schemes
between the set of K-rational points of L�X� and the set of
K��t��rationalpointsof X (calledthesetofgermsofarcswithcoe cientsin
K on X).Moreprecisely,theschemeL�X�isdefinedastheprojective
limit L�X� :� limL�X�, in the category of k-schemes, of then

schemesL�X�; n2N, representing the functorn´
202 J. Denef, F. Loeser
n�1R7!Mor �Spec R�t�=t R�t�; X�;k-schemes
definedonthecategoryof k-algebras.(Thus,forany k-algebra R,the
setof R-rationalpointsofL�X�isnaturallyidentifiedwiththesetofn
n�1R�t�=t R�t�-rational points of X.) The existence of L�X� is welln
known, cf. [B-L-R] p. 276, and the projective limit exists since the
transition morphisms are a ne. We shall denote by p the canonicaln
morphism L�X !L�X� corresponding to truncation of arcs. Inn
the present paper, the schemes L�X� andL�X� will always ben
considered with their reduced structure. Note that the set-theoretical
image p�L�X��isaconstructiblesubsetofL�X�,asfollowsfroman n
theorem of M. Greenberg [G], see (4.4) below. These constructible
sets p�L�X�� were first studied by J. Nash in [N], in relation withn
Hironaka’sresolutionofsingularities.Theyarealsoconsideredinthe
papers [L-J], [H].
The following result is the first main result of the paper. It is
ananalogueoftherationalityofthePoincareseriesassociatedtothe
p-adic points on a variety proved in [D1].
Theorem 1.1. Let X be an algebraic variety over k. The power series
1X
nP�T� :� �p�L�x���T ;n
n�0
considered as an element of M ��T��, is rational and belongs toloc
M�T� .loc
The proof of the theorem is given in section 5 and uses two main
ingredients. The first one is a result of J. Pas [P] on quantifier
elimination for semi-algebraic sets of power series in characteristic
zero. The second one is M. Kontsevich’s marvellous idea of motivic
integration [K]. More precisely, M. Kontsevich introduced
mbthe completionM ofM with respect to the filtration F M ,loc loc
m �iwhere F M is the subgroup ofM generated by f S�L ji�loc loc
dim S mg, and defined, for smooth X, a motivic integration on
bL�X� with values intoM. This is an analogue of classical p-adic
integration. In the present paper we extend Kontsevich’s
construction to semi-algebraic subsets of L�X�; with X any pure
dimensional algebraic variety over k, not necessarily smooth. For such an
X, let B be the set of all semi-algebraic subsets of L�X�. We
conbstruct in section 3 a canonical measure l : B! M. This allows us
to define integrals´

Germs of arcs on singular algebraic varieties and motivic integration 203
Z
�aL dl
A
for A in B and a : A!Z[f1g a simple function which is
bounded from below. (Semi-algebraic subsets of L�X� and simple
functions are defined in section 2.) The properties of this motivic
integration, together with resolution of singularities and the result of
bPas, su ce to prove the rationality of the image of P�T� in M��T��.
To prove the rationality of P�T�, considered as an element of
M ��T��, one needs a more refined argument based on Lemma 2.8loc
and the use of an obvious lifting l~�A� in M of l�A�, when A is aloc
stable semi-algebraic subset ofL�X� (a notion defined in section 2).
For an algebraic variety X, it is natural to consider its motivic
volume l�L�X��.Insection 6,wegiveexplicitformulasfor l�L�X��
in terms of certain special resolutions of singularities of X (which
l�L�X��alwaysexist).Asacorollarywededucethat alwaysbelongs
bto a certain localization of the image of M in M on which theloc
Euler characteristic v naturally extends with rational values. So we
obtain a new invariant of X, the Euler characteristic v�l�L�X���,
which is a rational number and coincides with the usual Euler
characteristic of X when X is smooth. In section 7, we prove that,
��n�1�dwhen X is of pure dimension d, the sequence�p�L�X���Ln
bconvergesto l�L�X��in M.Thisresult,whichisananalogueofa
padic result by J. Oesterle [O], gives, in some sense, a precise meaning
to Nash’s guess one should consider the limit of the p�L�X��’s. Wen
conclude the paper by some remarks on the Greenberg function in
section 8.
For related results concerning motivic Igusa functions, see [D-L].
2. Semi-algebraic sets of power series
(2.1) From now on we will denote by k a fixed algebraicclosure of k,
and by k��t�� the fraction field of k��t��, where t is one variable. Let
x ;...; x be variables running over k��t�� and let ‘ ;...;‘ be vari-1 m 1 r
ables running over Z.A semi-algebraic condition h�x ;...; x ;‘ ;1 m 1
...;‘� is a finite boolean combination of conditions of the formr
(i) ord f�x ;...; x� ord f�x ;...; x�� L�‘ ;...;‘�t 1 1 m t 2 1 m 1 r
(ii) ord f�x ;...; x� L�‘ ;...;‘� mod d;t 1 1 m 1 r
and