Chai’s conjecture and Fubini properties of motivic integrals.

Let K be a discretely valued field with perfect residue field k. Let G be a semi-abelian variety over K, i.e., an extension of an abelian K-variety A by a K-torus T. The Néron lft-model of G is the minimal extension of G to a smooth group scheme over the value ring of K. We say that G has semi-abelian reduction if the identity component of the special fiber of the Néron lft-model of G is a semi-abelian k-variety. By Grothendieck’s Semi-Stable Reduction Theorem, G acquires semi-abelian reduction over some finite separable extension K’ of K. Chai’s base change conductor c(G) is a positive rational number that measures the defect of semi-abelian reduction of G over K. Chai conjectured that c(G)=c(A)+c(T), and he proved this property if k is finite and also if K has characteristic zero. The proof of the finite residue field case uses Fubini’s theorem for integrals with respect to the Haar measure on the completion of K. The proof of the case where K has characteristic zero uses completely different methods. In this talk, we will first give a general introduction to Néron lft-models and the base change conductor. Then we’ll show how one can use Loeser and Sebag’s motivic integration on rigid varieties to reformulate Chai’s conjecture as a Fubini property for motivic integrals, and how one can use Cluckers and Loeser’s motivic integration on definable sets to prove this Fubini property if K has characteristic zero. This is joint work with Raf Cluckers and François Loeser.