Resumen de Nearby motives and motivic nearby cycles

Florian Ivorra, Julien Sebag

• We prove that the construction of motivic nearby cycles, introduced by Jan Denef and François Loeser, is compatible with the formalism of nearby motives, developed by Joseph Ayoub. Let kk be an arbitrary field of characteristic zero, and let XX be a smooth quasi-projective kk -scheme. Precisely, we show that, in the Grothendieck group of constructible étale motives, the image of the nearby motive associated with a flat morphism of kk -schemes f:X→A1kf:X→Ak1 , in the sense of Ayoub’s theory, can be identified with the image of Denef and Loeser’s motivic nearby cycles associated with ff . In particular, it provides a realization of the motivic Milnor fiber of ff in the “non-virtual” motivic world.