Motivic integration on Berkovich spaces

Abstract

We define a motivic measure on the Berkovich analytification of an algebraic variety defined over a trivially valued field, and introduce motivic integration in this setting. The construction is geometric with a similar spirit as Kontsevich's original definition, and leads to the formulation of a functorial theory which mirrors, in this aspect, the approach of Cluckers and Loeser via constructibe motivic functions. A version of the integral over nontrivially valued fields and its relation to Hrushovski and Kazhdan's integration are also discussed.

v4: 44 pages, final version, to appear in a volume dedicated to Vyacheslav Shokurov on the occasion of his 70th birthday