Beyond twisted maps: applications to motivic integration
Matthew Satriano, Jeremy Usatine
Abstract
We introduce a natural generalization of twisted maps, called \emph{warped maps}. While twisted maps play an important role in the study of Deligne-Mumford stacks, warped maps are better suited for studying Artin stacks. Heuristically, warped maps see the hidden proper-like behavior satisfied by good moduli space maps. Specifically, we show that every arc of a good moduli space admits a \emph{canonical} lift, in a warped sense, thereby proving a valuative criterion for good moduli spaces. Furthermore, we prove that warped maps to an Artin stack $\mathcal{X}$ are given by usual maps to an auxiliary Artin stack $\mathscr{W}(\mathcal{X})$, immediately obtaining a versatile framework for bootstrapping results about usual maps to the setting of warped maps. As an application we obtain a motivic change of variables formula which, given a stacky resolution of singularities $\mathcal{X} \to Y$, canonically expresses any given motivic integral over arcs of $Y$ as a certain motivic integral over warped arcs of $\mathcal{X}$. In particular, this yields a McKay correspondence for linearly reductive groups.