Motivic zeta functions on Q-Gorenstein varieties

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.

Monodromy conjecture for semi‐quasihomogeneous hypersurfaces

“…Theorem 1.1 follows from the next one, which we prove using the main result of [12] allowing computations of motivic zeta functions from partial embedded resolutions: Theorem 1.2. Let 𝑤 0 , … , 𝑤 𝑛 ∈ ℤ 𝑛+1 >0 be a weight vector.…”

“…Theorem 1.1 follows from the next one, which we prove using the main result of [12] allowing computations of motivic zeta functions from partial embedded resolutions: Theorem Let $$w_0,\ldots ,w_n\in \mathbb {Z}_{>0}^{n+1}$$ be a weight vector. Let $$f\in \mathbb {C}[x_0,\ldots ,x_n]$$ be a semi‐weighted homogeneous polynomial of initial degree d with respect to these weights.…”

“…Locally around a generic point of 𝑆 𝑘 , 𝑌 is analytically isomorphic to ℂ 𝑛+1 ∕𝐺 𝑘 for some finite abelian group 𝐺 𝑘 , acting diagonally on the coordinates 𝑡 0 , … , 𝑡 𝑛 of ℂ 𝑛+1 and small (i.e., not containing rotations around the hyperplanes other than the identity); 𝑓•𝜇 is given by 𝑡 . Then, by [12,Theorem 4],…”

“…By [12,Theorem 4], the zeta function 𝑍 𝑚𝑜𝑡 𝑓,𝑥 𝛽 (𝑠) is as in (2.2) but with the relative canonical divisor replaced by the above form. Running the proof of Theorem 1.2, we note that everything works similarly.…”

“…Let E be the exceptional divisor and H the strict transform of $$\lbrace f=0\rbrace$$. We show first that μ is an embedded $$\mathbb {Q}$$‐resolution of f , see [12, section 1.4]. By definition, this means that $$E\cup H$$ has $$\mathbb {Q}$$‐normal crossings, that is, it is locally analytically isomorphic to the quotient of a union of coordinate hyperplanes among those given by a local system of coordinates $$t_0,\ldots , t_n$$ with a diagonal action of a finite abelian subgroup of G of $$GL_{n+1}(\mathbb {C})$$, that is, locally $$$$\begin{equation*} f\circ \mu = t_0^{N_0}\ldots t_n^{N_n}:\mathbb {C}^{n+1}/G\rightarrow \mathbb {C} \end{equation*}$$$$for some $$N_i\in \mathbb {N}$$.…”