Motivic Measures

Abstract

Kontsevich proposed a Haar measure on k[[t]}, k a field of characteristic zero, with values in a universal ring that is obtained in a simple manner from the varieties over k: it assigns to the ideal tn&[[t]] the value double script L sign-n, where double script L sign stands for the affine line over k. This leads to a measure with the same value ring for schemes over k[[t]]. Denef & Loeser and Batyrev have shown that this measure gives rise to invariants with remarkable properties. These are new even in case the scheme is obtained from a k-variety by base change.