The Chow Motive of a Locally Trivial Fibration

Abstract

Guillet and Soulé 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.

Cited by in corpus (2)

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• Position Space Feynman quadrics and their motives