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

TL;DR

This paper develops a broad motivic decomposition framework for families with Tate fibers, extending previous conjectures and applying to smooth and singular cases, including quadric bundles, using advanced motivic theories.

Contribution

It introduces a general motivic decomposition formula for families with Tate fibers and proves its existence for certain quadric bundles, extending prior conjectures.

Findings

General motivic decomposition for smooth projective families with Tate fibers.

Motivic lift of Bernstein-Beilinson-Deligne decomposition for quadric bundles.

Applicability to arbitrary regular bases without field or prime assumptions.

Abstract

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' 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.