Motivic integration over Deligne–Mumford stacks

Abstract

The aim of this article is to develop the theory of motivic integration over Deligne-Mumford stacks and to apply it to the birational geometry of Deligne-Mumford stacks.

Mass Formulas for Local Galois Representations and Quotient Singularities. I: A Comparison of Counting Functions

“…A slightly different version was obtained by Denef-Loeser [DL02] over a field of characteristic zero containing all ♯Γ-th roots of unity with Γ the given finite group. Yasuda [Yas06] then generalized it to an arbitrary perfect field of characteristic prime to ♯Γ. Now we recall these results.…”

“…To deduce the general case of the theorem from a result in [Yas06], we need to first prove Theorem 5.9 below.…”

“…To generalize the result to the tame case over an arbitrary perfect base field, we need the notion of twisted 0-jets from the paper [Yas06], and in particular, need to use Deligne-Mumford stacks.…”

“…The following is the McKay correspondence in the simplest case: Yas06]). Suppose that Γ ⊂ GL n (κ) has no pseudo-reflection.…”

Weighted Ehrhart theory and orbifold cohomology

“…When P is reflexive, Batyrev and Dais [3] showed that δ i is the 2ith stringy Betti number of a toric variety X associated to P . Furthermore, in this case, results of Yasuda [31] imply that δ i is equal to the dimension of the 2ith orbifold cohomology group of the canonical orbifold associated to X. This interpretation of the Ehrhart δ-polynomial of P was used by Mustaţǎ and Payne in [23] and Karu in [18].…”