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