Motivic Integration and Logarithmic Geometry

TL;DR

This thesis employs logarithmic geometry to develop new constructions and formulas for motivic invariants, enhancing understanding of motivic zeta functions, nearby cycles, and fixed points without relying on characteristic zero assumptions.

Contribution

It introduces a characteristic-zero-free construction of the motivic Serre invariant and a new formula for the motivic zeta function using log geometry, impacting the study of monodromy and mirror symmetry.

Findings

A new construction of the motivic Serre invariant relying only on resolution of singularities.

A formula for the motivic zeta function that reduces candidate poles and aids in the monodromy conjecture.

Insights into motivic nearby cycles and fixed points of G-actions.

Abstract

In this thesis, we use logarithmic methods to study motivic objects. Let R be a complete discrete valuation ring with perfect residue field k, and denote by K its fraction field. We give in chapter 2 a new construction of the motivic Serre invariant of a smooth K-variety and extend it additively to arbitrary K-varieties. The main advantage of this construction is to rely only on resolution of singularities and not on a characteristic zero assumption, as did previous results. As an application, we give a conditional positive answer to Serre's question on the existence of rational fixed points of a G-action on the affine space, for G a finite l-group. We end the chapter by showing how the logarithmic point of view that we use in our construction leads to a new understanding of the motivic nearby cycles with support of Guibert, Loeser and Merle as a motivic volume. In chapter 4 we use the theory of logarithmic geometry to derive a new formula for the motivic zeta function via the volume Poincar\'e series. More precisely, we show how to compute the volume Poincar\'e series associated to a generically smooth log smooth R-scheme in terms of its log geometry, more specifically in terms of its associated fan in the sense of Kato. This formula yields a much smaller set of candidate poles for the motivic zeta function and seems especially well suited to tackle the monodromy conjecture of Halle and Nicaise for Calabi-Yau K-varieties, for which log smooth models appear naturally through the Gross-Siebert programme on mirror symmetry. We end the chapter by showing how this formula sheds new light on previous results regarding the motivic zeta function of a polynomial nondegenerate with respect to its Newton polyhedron, and of a polynomial in two variables.