Motivic apsects of a remarkable class of Calabi-Yau threefolds

In this note we consider the motivic aspect of the middle cohomology of more than 200 classes of quasi-smooth Calabi–Yau threefolds inside weighted projective 4-space which come with an action of a cyclic group of even order. The action induces a self-dual Chow–Künneth decomposition. All but one component correspond to Fano threefolds. For these the generalized Hodge conjecture is known, but thanks to the nature of the decomposition we can give a direct proof for one of the components.