The chow motive of semismall resolutions

We show that the topological Decomposition Theorem for a proper semismall map f : X → Y implies a "motivic" decomposition theorem for the rational algebraic cycles of X and, in the case X is compact, for the Chow motive of X. We work in the category of pure Chow motives over a base. Under suitable assumptions on the stratification, we also prove an explicit version of the motivic decomposition theorem and compute the Chow motives and groups in some examples, e.g. the nested Hilbert schemes of points of a surface. In an appendix with T. Mochizuki, we do the same for the parabolic Hilbert scheme of points on a surface.