The Chow Motive of a Locally Trivial Fibration

TL;DR

This paper proves a motivic decomposition for locally trivial fibrations, explicitly constructing an isomorphism of Chow motives under certain conditions, and verifies related conjectures in algebraic geometry.

Contribution

It provides an explicit isomorphism of Chow motives for fibrations with special fibers, extending previous decomposition results and confirming conjectures by Murre.

Findings

Established an explicit isomorphism of motives for certain fibrations.

Verified Murre's conjectures for these fibrations.

Extended the understanding of motivic decompositions in algebraic geometry.

Abstract

Guillet and Soul\'e have shown that, for a fibration $\pi: Y \to X$ with fibre $Z$, locally trivial in the Zariski topology, we have a decomposition \[ [Y] = [X] \cdot [Z], \] where $[\cdot]$ denotes a class in the Grothendieck group $K_{0}(\mathcal{M}_{Rat}(k))$ associated to the category of (pure effective) Chow motives $\mathcal{M}_{Rat}(k)$ for a field $k$. By assuming some additional properties for the fibre $Z$, we construct an explicit isomorphism $h(Y) \cong h(X) \otimes h(Z)$ in the category $\mathcal{M}_{Rat}(k)$, and we use it to prove, for this type of fibrations, some conjectures disscussed by Murre.