A short course on geometric motivic integration

Abstract

ABSTRACT. These notes grew out of the authors effort to understand the theory of motivic integration. They give a short but thorough introduction to the flavor of motivic integration which nowadays goes by the name of geometric motivic integration. Motivic inegration was introduced by Kontsevich and the foundations were worked out by Denef, Loeser, Batyrev and Looijenga. We focus on the smooth complex case and present the theory as self contained as possible. As an illustration we give some applications to birational geometry which originated in the work of Mustaţǎ. CONTENTS

Prolongations in differential algebra

“…The following propositions are versions of standard facts about jet spaces, and can be proved in the same way (for example, see [Bli05]). …”

Deforming spaces of m-jets of hypersurfaces singularities

“…We proved in [24] the following result: 24]). Let f := x a 1 1 + x a 2 2 + · · · + x a k 2 + · · · + x an n , a k > 1, and we suppose that the polynomials g i , i ≥ 1, belong to the integral closure of the ideal generated by x a 1 1 , x a 2 2 , · · · , x an n . Then the morphism p m : (W(S) m ) red → S := Spec K[[s]] is flat for all 0 ≤ m ≤ ∞.…”

“…In the year 1995, Kontsevich, using these spaces, introduced the motivic integration to resolve the Batyrev conjecture on the Calabi-Yau varieties (see [21]). For more references on motivic integration see [1], [5], [6] and [28].…”

Generalised Poincaré series and embedded resolution of curves

“…We define the class [k * ] of the group of units k * of k as the class [G m ] ∈ K 0 (ν k ). Since G m ∼ = Spec k x, 1 x , we get [G m ] = Spec k x, 1…”