An introduction to 𝑝-adic and motivic integration, zeta functions and invariants of singularities

Abstract

Motivic integration was introduced by Kontsevich to show that birationally equivalent Calabi-Yau manifolds have the same Hodge numbers. To do so, he constructed a certain motivic measure on the arc space of a complex variety, taking values in a completion of the Grothendieck ring of algebraic varieties. Later, Denef and Loeser, together with the works of Looijenga and Batyrev, developed in a series of articles a more complete theory of the subject, with applications in the study of varieties and singularities. In particular, they developed a motivic zeta function, generalizing the usual ( p p -adic) Igusa zeta function and Denef-Loeser topological zeta function. These notes are a basic introduction to geometric motivic integration, the precedent p p -adic ideas associated with it, and the theory of the above zeta functions related to them. We focus in practical computations and ideas, providing examples and a recent formula obtained by means of partial resolutions.