Twisted jets, motivic measures and orbifold cohomology

We introduce the notion of twisted jets. For a Deligne-Mumford stack X of finite type over ℂ, a twisted ∞-jet on X is a representable morphism D → X such that D is a smooth Deligne-Mumford stack with the coarse moduli space Spec ℂ[[t]]. We study a motivic measure on the space of the twisted ∞-jets on a smooth Deligne-Mumford stack. As an application, we prove that two birational minimal models with Gorenstein quotient singularities have the same orbifold cohomology as a Hodge structure.