A short course on geometric motivic integration
type: Item
instance of: scholarly article
zbMATH Open document ID: 1279.14021
author: Manuel Blickle
publication date: 2012-07-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0507404
Mathematics Subject Classification ID: 14E18
zbMATH DE Number: 6059496
zbMATH Keywords: motivic integration
zbMATH Keywords: birational geometry
MaRDI profile type: Publication
arXiv ID: math/0507404
title: A short course on geometric motivic integration
review text: Motivic integration was introduced by Kontsevich in analogy to \(p\)-adic integration, to prove that two birationally equivalent smooth Calabi-Yau varieties have the same Hodge numbers (conjectured by Batyrev). It was then extended by Denef and Loeser to singular varieties over a field of characteristic zero and they used this theory to introduce new invariants of singularities.NEWLINENEWLINE This course gives an accessible introduction to this theory in the case of smooth varieties. In particular it introduces and studies briefly:NEWLINENEWLINEThe Arc space of a variety \(X\): this is the scheme that parametrizes formal curves traced on \(X\). It also is the domain of a motivic integral.NEWLINENEWLINEThe Grothendieck ring of algebraic varieties which is the value ring of a motivic integral.NEWLINENEWLINE A formula for a motivic integral after a change of variables given by a birational map.NEWLINENEWLINE The course ends with an application of motivic integration, which is Mustata's formula of the log canonical threshold of a pair in terms of dimensions of some irreducible components of jet schemes.NEWLINENEWLINEFor the entire collection see [Zbl 1241.14001].
reviewed by: Hussein Mourtada
description: scientific article; zbMATH DE number 6059496