Hyper-Kähler Varieties with a Motive of Abelian Type

Abstract

According to a well-known Conjecture the Chow motive of a hyper-Kähler variety should be of abelian type. In this paper we consider the case of the hyper-Kähler varieties associated to a cubic fourfold $$X\subset \textbf{P}^5$$ : the Fano variety of lines F(X), the eightfold Z constructed in [13] and the 10-dimensional projective compactification $$\bar{\mathcal {J}}$$ of the Jacobian fibration constructed in [12]. We show that the Chow motives of F(X) and Z are of abelian type if X has a motive of abelian type. This in fact the case for a one dimensional family of cubic fourfolds inside the Hassett divisor $$\mathcal {C}_d$$ . If X is a general cubic fourfold than the motive $$h(\bar{\mathcal {J}})$$ is of abelian type if the surface $$\Sigma _2$$ of lines of second type has a motive of abelian type.