The dlt motivic zeta function is not well-defined

TL;DR

This paper demonstrates that the dlt motivic zeta function, introduced as an adaptation of the classical motivic zeta function, is not well-defined because it depends on the choice of dlt modification, contrary to previous claims.

Contribution

The paper provides explicit counterexamples showing the dlt motivic zeta function's dependence on the modification choice, challenging prior assumptions of its well-definedness.

Findings

The dlt motivic zeta function varies with different dlt modifications.

Explicit examples demonstrate the non-uniqueness of the zeta function.

The claim of the zeta function being well-defined is incorrect.

Abstract

In arXiv:1408.4708, Xu defines the dlt motivic zeta function associated to a regular function $f$ on a smooth variety $X$ over a field of characteristic zero. This is an adaptation of the classical motivic zeta function that was introduced by Denef and Loeser. The dlt motivic zeta function is defined on a dlt modification via a Denef-Loeser-type formula, replacing classes of strata in the Grothendieck ring of varieties by stringy motives. We provide explicit examples that show that the dlt motivic zeta function depends on the choice of dlt modification, contrary to what is claimed in arXiv:1408.4708, and that it is therefore not well-defined.