On the poles of zeta functions for finite morphisms between normal surfaces

TL;DR

This paper investigates the poles of motivic and topological zeta functions associated with finite morphisms between normal surfaces, establishing inclusion relations and conditions for equality, with applications to quotient maps and log canonical models.

Contribution

It proves pole inclusion results for motivic zeta functions under finite morphisms, identifies cases of equality for quotient maps, and compares log canonical models in relation to zeta functions.

Findings

Poles of motivic zeta functions on the target are contained in those on the source.

Equality of poles occurs in quotient maps induced by finite abelian group actions when the divisor is trivial.

Conditions are provided for when the topological zeta function on the target is a multiple of that on the source.

Abstract

For a divisor representing a function and another divisor representing a differential form on a normal surface singularity, there is a notion of motivic and topological zeta function. In this paper, given a finite morphism between two normal surfaces, we prove that the set of poles of the motivic zeta function associated with the target is contained in the one associated with the source. We illustrate by examples that this inclusion is strict in general, and that on the topological level there are in general no inclusions between the sets of poles on source and target. On the other hand, when the morphism is the quotient map induced by an action of a finite abelian group on $\mathbb{C}^2$, and the divisor associated with the differential form on the source is trivial, we do show equality between the corresponding sets of poles, both on motivic and topological level. In addition, again for the quotient map induced by an action of a finite abelian group on $\mathbb{C}^2$, but now with a general divisor associated with a differential form, we provide a criterion when the topological zeta function on the target is just a multiple of the one on the source. Finally, we compare log canonical models on source and target with a view on zeta functions.