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Project

Dlt zeta functions

Xu (2016) 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 (1998). 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. This dlt motivic zeta function is, together with the dlt topological zeta function, a specialisation of the aforementioned zeta function, the protagonist of this thesis. We show that Xu's dlt motivic zeta function is, unfortunately, not well-defined: it depends on the choice of dlt modification. However, it is possible that the dlt topological zeta function is well-defined. To this end, we provide an explicit construction that yields a dlt modification for non-degenerate polynomials, based on ideas of Ishii (1997,1999). Moreover, we give an explicit formula that computes the dlt motivic zeta function and the dlt topological zeta function on such models. So far, we are not able to convert these ideas into a proof that the dlt topological zeta function is well-defined for non-degenerate polynomials. Using other techniques, we provide some evidence that suggests that the dlt topological zeta function is well-defined in dimension three. In particular, we show that for non-degenerate polynomials in three variables, the local dlt topological zeta function does not depend on the choice of toric dlt modification.

Date:1 Oct 2018 →  19 Dec 2022
Keywords:Zeta functions, Resolution of singularities, Toric geometry
Disciplines:Geometry
Project type:PhD project