A refinement of the Grothendieck ring of varieties, and specialization of birational types

It is an open problem whether the Grothendieck ring of varieties detects the birational type of a smooth and proper variety. In this talk, I will present a refinement of the Grothendieck ring that detects birational types for trivial reasons, and show that the motivic nearby fiber of Denef and Loeser can be defined at this finer level. This construction is heavily influenced by the theory of motivic integration developed by Hrushovski and Kazhdan. It leads to a unified treatment of the specialization results for stable birational types obtained with Evgeny Shinder and the specialization results for birational types due to Kontsevich and Tschinkel. These results are part of a joint project with John Christian Ottem.