  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 $CH_0(F)$ which
canonically splits under the action of $\phi$, 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.

Algebraic Geometry