Motivic decompositions of families with Tate fibers: smooth and singular cases.

36 pages, version 4: a new condition of tamely ramified Artin-Tate motive has been introduced which allows to extend the main result. New introduction and link with WIldeshaus' work explained.International audienceUsing the homotopy t-structure, we build Chow-Künneth and refined Chow-Künneth decomposition of relative Artin-Tate motives.We apply Wildeshaus’s theory of motivic intermediate extensions to the motivic decomposition conjecture, formulated by Deninger–Murre and Corti–Hanamura. We first obtain a general motivic decomposition for the Chow motive of an arbitrary smooth projective family $f:X \rightarrow S$ whose geometric fibers are Tate. Using Voevodsky’s motives with rational coefficients, the formula is valid for an arbitrary regular base $S$, without assuming the existence of a base field or even of a prime integer $\ell $ invertible on $S$. This result, and some of Bondarko’s ideas, lead us to a generalized formulation of Corti–Hanamura’s conjecture. Secondly we establish the existence of the motivic decomposition when $f:X \rightarrow S$ is a projective quadric bundle over a characteristic $0$ base, which is either sufficiently general or whose discriminant locus is a normal crossing divisor. This provides a motivic lift of the Bernstein–Beilinson–Deligne decomposition in this setting