A short course on geometric motivic integration
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].
scientific article; zbMATH DE number 6059496