2020/07/22 by Cavicchi, Mattia, Déglise, Frédéric, Nagel, Jan
#Algebraic Geometry (math.AG) #FOS: Mathematics #Primary: 19E15 14F42 14C15 #Secondary: 14F08 18G80
paper · doi:10.48550/arxiv.2007.11447
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 → 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 ℓ invertible on S. This result, and some of Bondarko' ideas, lead us to a generalized formulation of Corti-Hanamura's conjecture. Secondly we establish the existence of the motivic decomposition when f:X → 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.