Nearby motives and motivic nearby cycles

Abstract

We prove that the construction of motivic nearby cycles, introduced by Jan Denef and Francois Loeser, is compatible with the formalism of nearby motives, developed by Joseph Ayoub. Let $$k$$ be an arbitrary field of characteristic zero, and let $$X$$ be a smooth quasi-projective $$k$$ -scheme. Precisely, we show that, in the Grothendieck group of constructible etale motives, the image of the nearby motive associated with a flat morphism of $$k$$ -schemes $$f:X\rightarrow \mathbb A ^1_k$$ , in the sense of Ayoub’s theory, can be identified with the image of Denef and Loeser’s motivic nearby cycles associated with $$f$$ . In particular, it provides a realization of the motivic Milnor fiber of $$f$$ in the “non-virtual” motivic world.