Article Dans Une Revue International Mathematics Research Notices Année : 2023

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

Résumé

Using 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.

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Dates et versions

hal-03014185 , version 1 (20-11-2020)
hal-03014185 , version 2 (04-01-2022)

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Mattia Cavicchi, Frédéric Déglise, Johannes Nagel. Motivic decompositions of families with Tate fibers: smooth and singular cases.. International Mathematics Research Notices, 2023, 2023 (16), pp.14239-14289. ⟨10.1093/imrn/rnac223⟩. ⟨hal-03014185v2⟩
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