Rationality questions and motives of cubic fourfolds

In this note we propose an approach to some questions about the birational geometry of smooth cubic fourfolds through the theory of Chow motives. We introduce the transcendental part t(X) of the motive of X and prove that it is isomorphic to the (twisted) transcendental part h tr 2 (F (X)) in a suitable Chow-Künneth decomposition for the motive of the Fano variety of lines F (X). Then we explain the relation between t(X) and the motives of some special surfaces of lines contained in F (X). If X is a special cubic fourfold in the sense of Hodge theory, and F (X) S [2] , with S a K3 surface associated to X, then we show that t(X) t 2 (S)(1). Moreover we relate the existence of an isomorphism between the transcendental motive t(X) and the (twisted) transcendental motive of a K3 surface to conjectures by Hassett and Kuznetsov on the rationality of a special cubic fourfold. Finally we give examples of cubic fourfolds such that the motive t(X) is finite dimensional and of abelian type