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 Chow-Kunneth decomposition. This relative Chow-Kunneth decomposition shows that the motive of the quadric bundle is isomorphic to the direct sum of Tate twists of the motive of the base and of the Prym motive of a double cover of the discriminant. In particular this is a refinement with -coefficients of a result of Beauville concerning the cohomology and the Chow groups of an odd dimensional quadric bundle over . This relative Chow-Kunneth decomposition induces an absolute Chow-Kunneth decomposition which satisfies parts of Murre's conjectures. This article is a generalization of an article of Nagel and Saito on conic bundles (Nagel and Saito in Int Math Res Not IMRN 16, 2978-3001, 2009).