On the Chow groups of the variety of lines of a cubic fourfold

Let $X$ be a smooth complex cubic fourfold and let $F$ be the variety of lines of $X$. The variety $F$ is known to be a smooth projective hyperkaehler fourfold, which is moreover endowed with a self rational map $\phi : F -\rightarrow F$ first constructed by C. Voisin. Here we define a filtration of Bloch–Beilinson type on the Chow group of zero-cycles $C{H}_0(F)$ which canonically splits under the action of $\varphi$, thereby answering in this case a question of A. Beauville. Moreover, we show that this filtration is of motivic origin, in the sense that it arises from a Chow–Kuenneth decomposition of the diagonal.