Geometric Motivic Integration on Artin n-stacks: A Construction and Some Properties

Since its conception by Kontsevich in 1995, the technique of motivic integration has found numerous applications in algebraic geometry and representation theory. The work of Denef, Loeser and Cluckers led to the formulation of different versions of motivic integration ? geometric motivic integration, arithmetic motivic integration and the theory of ?constructible motivic functions?. This book addresses the problem of generalizing the theory geometric motivic integration to Artin n-stacks. We follow the construction of higher Artin stacks as proposed by Toen and Vezzosi. A brief review of this construction along with some of the basic notions of homotopical algebra is also provided. Applications of the theory of motivic integration on varieties have been very fruitful and this work should pave the way for similar results for Artin stacks. Also, some of these ideas may prove useful in generalizing other versions of motivic integration to Artin stacks.