Motivic integration on smooth rigid varieties and invariants of degenerations

Abstract

We develop a theory of motivic integration for smooth rigid varieties. As an application we obtain a motivic analogue for rigid varieties of Serre's invariant for $p$-adic varieties. Our construction provides new geometric birational invariants of degenerations of algebraic varieties. For degenerations of Calabi-Yau varieties, our results take a stronger form.