Algebraic cycles of some Fano varieties with Hodge structure of level one

We study Chow groups and étale motivic cohomology groups of smooth complete intersections with Hodge structures of level one, classified by Deligne and Rapoport, with particular attention to fivefolds. We extend these results to an étale motivic context and recover an analogous finite-dimensionality in the sense of Kimura. We further analyse algebraic cycles on other smooth Fano manifolds with Hodge structures of level one and, as an application, we prove the integral Hodge conjecture for smooth quartic double fivefolds by means of the étale motivic approach.