Motives of quadric bundles

This article is about motives of quadric bundles. In the case of odd dimensional fibers and where the basis is of dimension two we give an explicit relative and absolute Chow-K"unneth decomposition. This shows that the motive of the quadric bundle is isomorphic to the direct sum of the motive of the base and the Prym motive of a double cover of the discriminant. In particular this is a refinement with $\mathbb{Q}$ coefficients of a result of Beauville concerning the cohomology and the Chow groups of an odd dimensional quadric bundle over ${\mathbb{P}}^2$. This Chow-K"unneth decomposition satisfies Murre’s conjectures II and III. This article is a generalization of an article of Nagel and Saito on conic bundles \cite{NS}.