Rationality questions and motives of cubic fourfolds

Michele Bolognesi, Claudio Pedrini

Published 2017-10-13Version 1

In this note we propose an approach to some questions about the birational geometry of smooth cubic fourfolds through the theory of Chow motives. Let $X \subset \mathbb{P}^5$ be such a cubic hypersurface. 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 $f : \mathcal{S} \to B$ of smooth projective surfaces. According to the results in [ABB] this is related to a conjecture proving the irrationality of a very general $X$. 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-K\"unneth decomposition for the motive of the Fano variety of lines $F(X)$. If $X$ is special, in the sense of B.Hassett, and $F(X) \simeq S^{[2]}$, with $S$ a K3 surface associated to $X$, then we show that $t(X)\simeq t_2(S)(1)$. Then we relate the existence of an isomorphism between the transcendental motive $t(X)$ and the (twisted) transcendental motive of a K3 surface with the conjectures by Hasset 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.