On the rationality and the finite dimensionality of a cubic fourfold

Abstract: Let $X$ be a cubic fourfold in $P^5_{C}$. We prove that, assuming the Hodge conjecture for the product $S \times S$, where $S$ is a complex surface, and the finite dimensionality of the Chow motive $h(S)$, there are at most a countable number of decomposable integral polarized Hodge structures, arising from the fibers of a family of smooth projective surfaces. According to the results in [ABB] this is related to a conjecture proving the irrationality of a very general $X$. If $X$ is special, in the sense of B.Hasset, and $F(X) \simeq S^{[2]}$, with $S$ a K3 surface associated to $X$, then we show that the Chow motive $h(X)$ contains as a direct summand a "transcendental motive" $t(X)$ such that $t(X)\simeq t_2(S)(1)$. The motive of $X$ is finite dimensional if and only if $S$ has a finite dimensional motive, in which case $t(X)$ is indecomposable. Similarly, if $X$ is very general and the motive $h(X)$ is finite dimensional, then $t(X)$ is indecomposable