Motivic Integration Quotient Singularities and the McKay Correspondence

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Motivic Integration, Quotient Singularities and the McKay Correspondence JAN DENEF1 and FRANC! OIS LOESER2 1University of Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium. e-mail: 2De? partement de mathe? matiques et applications, E? cole Normale Supe? rieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France (UMR 8553 du CNRS). e-mail: Franc8 (Received: 23 March 1999; accepted in ¢nal form: 29 December 2000) Abstract. The present work is devoted to the study of motivic integration on quotient singularities.We give a new proof of a form of the McKay correspondence previously proved by Batyrev. The paper contains also some general results on motivic integration on arbitrary singular spaces. Mathematics Subject Classi¢cation (2000). 14A15; 14A20; 14B05; 32S45; 32S05; 32S35. Key words. Motivic integration, McKay correspondence, quotient singularities, orbifold Introduction Let X be an algebraic variety, not necessarily smooth, over a ¢eld k of characteristic zero. We denote by L?X ? the k-scheme of formal arcs on X : K-points of L?X ? correspond to formal arcs SpecK ??t ! X , for K any ¢eld containing k.

  • variety over

  • fraction ¢eld

  • l?x ?

  • arc

  • l?u? ?

  • k?t-semi-algebraic

  • closed ¢eld containing


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Compositio Mathematica131:267^290, 2002. #2002Kluwer Academic Publishers. Printed in the Netherlands.
Motivic Integration, Quotient Singularities and the McKay Correspondence
267
JAN DENEF1and FRAN !COIS LOESER2 1University of Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium. e-mail: jan.denef@wis.kuleuven.ac.be 2 45 rue d’Ulm,De¤ partement matiques de mathe¤ rieure, et applications, EŁ cole Normale Supe¤ 75230 Paris Cedex 05, France (UMR 8553 du CNRS). e-mail: Franc8 ois.Loeser@ens.fr
(Received: 23 March 1999; accepted in ¢nal form: 29 December 2000)
Abstract.The present work is devoted to the study of motivic integration on quotient singularities. We give a new proof of a form of the McKay correspondence previously proved by Batyrev. The paper contains also some general results on motivic integration on arbitrary singular spaces.
Mathematics Subject Classi¢cation (20 0 0).14A15; 14A20; 14B05; 32S45; 32S05; 32S35.
Key words.Motivic integration, McKay correspondence, quotient singularities, orbifold
Introduction
LetXbe an algebraic variety, not necessarily smooth, over a ¢eldkof characteristic zero. We denote byXÞthek-scheme of formal arcs onX:K-points ofXÞ correspond to formal arcsSpecK½½t !X, forKany ¢eld containingk. In a recent paper [8], we developed an integration theory on the spaceXÞwith values in c M, a certain ring completion of the Grothendieck ringMof algebraic varieties overk(the de¢nition of these rings is recalled in Section 1.9), based on ideas of Kontsevich [12]. In the most interesting cases, the integrals we consider belong to a much smaller ringMloc½ððL1Þ=ðLi1ÞÞiX1, on which the usual Euler characteristic and Hodge polynomial may be extended in a natural way to an Euler characteristicEuand a Hodge polynomialHbelonging respectively toQand the ring
Z½u;v½ðuvÞ124ðuvuvÞi11iX135:
WhenXis smooth and one considers the total measure ofXÞ, these invariants reduce to the usual Euler characteristic and Hodge polynomial, but in general one obtains interesting new invariants (see [2,3,5,8,18]).
268
JAN DENEF AND FRANC! OIS LOESER
WhenXis a normal variety with at most Gorenstein canonical singularities, one can use the canonical class to de¢ne a measuremGorðAÞfor certain subsetsAof XÞ. Now assumeXis the quotient of the af¢ne spaceAknby a ¢nite subgroup Gof orderdofSLnðkÞ. We make the assumptionkcontains alldth roots of unity. We denote byXÞ0the set of arcs whose origin is the point 0 inX. One of the main results of the present paper is Theorem 3.6, which expressesmGorðLðXÞ0Þin terms of representation theoretic weightswðgÞof the conjugacy classes of elements gofG, de¢ned as wðgÞ:¼Xeg;i=d; 1WiWn with 1Weg;iWdandxeg;ithe eigenvalues ofgfori¼1;. . .;n,xbeing a ¢xed primi-tivedth root of unity ink. More precisely, the image ofmGorðLðXÞ0Þis equal to that P½g2ConjðGÞLwientM=ofM, withLthe class of the af¢ne ofðgÞin a certain quotc c c line. The quotientM=is de¢ned by requiring that the class of a quotient of a vector spaceVby a ¢nite group acting linearly should be that ofV. This condition is mild enobugh to guarantee thatmGorðLðXÞ0ÞandP½g2ConjðGÞLwðgÞhave the same image inK0ðCHMkÞ, an appropriate completion ofK0ðCHMkÞ, the Grothendieck group of the pseudo-Abelian category of Chow motives overk, and in particular have the same Hodge polynomial and Euler characteristic. This result ^ at least for the Hodge realization ^ is due to Batyrev [6] and implies, whenXhas a crepant resolution, a form of the McKay correspondence which has been conjectured by Reid [16] and proved by Batyrev [6]. The aim of the present paper is to present an alternative proof of Batyrev’s result and also to develop further, some basic properties of motivic integration which were not covered in [8]. Though Batyrev also uses integration on spaces of arcs, the approach we follow here, which was inspired to us by Maxim Kontsevich, is some-what different. One of the main differences is that we are able to work directly on the singular spaceXinstead of going to desingularizations. This allows us to have a more local approach, in the sense that we can directly calculate the part of the motivic integral coming from each conjugacy class. More precisely, for each elementgin the groupG, we considerXÞg0;g, the set of arcsjinXÞ0, which are not contained in the discriminant and may be lifted inAknÞto a fractional arc~jðt1=dÞsuch that~jðxt1=dÞ ¼g~jðt1=dÞ. We prove that the image of mGorðLðXÞg0;gÞincM=is equal to that ofLwðgÞ. Let us now brie£y review the content of the paper. In Section 1, we recall some material on semi-algebraic geometry overkððtÞÞfrom [8]. In fact, we need to generalize slightly semi-algebraic geometry as developed in [8] to ‘k½t-semi-algebraic geometry’ which allows expressions involvingt, sincek½t-morphisms naturally appear in Section 2. Fortunately, this is quite harmless, since most proofs remain the same. This material onk½t-semi-algebraic geometry might be useful elsewhere. One of the main technical dif¢culties of the section is Theorem 1.16 were we extend the crucial change of variables formula [8] to certain maps which are not birational.
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