Birational Chow-K\"unneth decompositions

We study the notion of a birational Chow-K\"unneth decomposition, which is essentially a decomposition of the integral birational motive of a variety. The existence of a birational Chow-K\"unneth decomposition is stably birationally invariant and this notion refines the Chow theoretical decomposition of the diagonal. We show that a birational Chow-K\"unneth decompostion exists for the following varieties: (a) Jacobian variety; (b) Hilbert scheme of points on a $K3$ surface and (c) The variety of lines on a stably rational cubic threefold or a stably rational cubic fourfold.