Motivic Zeta Functions on $\mathds{Q}$-Gorenstein Varieties

TL;DR

This paper investigates motivic zeta functions on $\

Contribution

It introduces a method to compute motivic zeta functions on $\

Findings

Derived a closed formula for local motivic zeta functions on abelian quotient singularities.

Showed how weighted blow-ups can reduce candidate poles for certain surface singularities.

Computed motivic invariants for a nonabelian quotient singularity using $\

Abstract

We study motivic zeta functions for $\mathds{Q}$-divisors in a $\mathds{Q}$-Gorenstein variety. By using a toric partial resolution of singularities we reduce this study to the local case of two normal crossing divisors where the ambient space is an abelian quotient singularity. For the latter we provide a closed formula which is worked out directly on the quotient singular variety. As a first application we provide a family of surface singularities where the use of weighted blow-ups reduces the set of candidate poles drastically. We also present an example of a quotient singularity under the action of a nonabelian group, from which we compute some invariants of motivic nature after constructing a $\mathds{Q}$-resolution.