No abstract

We study motivic zeta functions for Q-divisors in a 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 Q-resolution.

Let f = (f1, . . . , f l ) : U → K l , with K = R or C, be a K-analytic mapping defined on an open set U ⊂ K n , and let Φ be a smooth function on U with compact support. In this paper, we give a description of the possible poles of the local zeta function attached to (f , Φ) in terms of a log-principalization of the ideal I f = (f1, . . . , f l ). When f is a non-degenerate mapping, we give an explicit list for the possible poles of Z Φ(s, f ) in terms of the normal vectors to the supporting hyperplanes of a Newton polyhedron attached to f , and some additional vectors (or rays) that appear in the construction of a simplicial conical subdivision of the first orthant. These results extend the corresponding results of Varchenko to the case l 1, and K = R or C. In the case l = 1 and K = R, Denef and Sargos proved that the candidate poles induced by the extra rays required in the construction of a simplicial conical subdivision can be discarded from the list of candidate poles. We extend the Denef-Sargos result to arbitrary l 1. This yields, in general, a much shorter list of candidate poles, which can, moreover, be read off immediately from Γ(f ).wheref (y) := (f 1 (y), . . . ,f l (y)), in V .We can choose a small neighbourhood V b of b over which (2.3)-(2.5) are valid, and |ε(y)| K , |f (y)| K , |η(y)| K are R-analytic. Then |ε(y)| s K , |f (y)| s K are R-analytic in y for any s ∈ C, and holomorphic in s ∈ C for any y ∈ V b .Since h −1 (support Φ) is compact, we can take a finite covering of the form {V b }, where the V b are homeomorphic under φ V to the polydisc P (0) in K n defined by |y i | K < , with sufficiently small and for 1 i n. By picking a smooth partition of the unity subordinate to {V b }, and using the previous discussion,becomes a finite sum of integrals of the following two types:where Ψ is a C ∞ 0 function with support contained in a polydisc P (0) and e s ln |f * (y)|K is R-analytic for y ∈ V b and holomorphic in s ∈ C, or J(s) := K n Θ(y, s)