A note on varieties of weak CM-type

Abstract: CM-type projective varieties X of complex dimension n are characterized by their CM-type rational Hodge structures on the cohomology groups. One may impose such a condition in a weakest form when the canonical bundle of X is trivial; the rational Hodge structure on the level-n subspace of $H^n(X;Q)$ is required to be of CM-type. This brief note addresses the question whether this weak condition implies that the Hodge structure on the entire $H^\ast(X;Q)$ is of CM-type. We study in particular abelian varieties when the dimension of the level-n subspace is two or four, and K3 $\times T^2$. It turns out that the answer is affirmative. Moreover, such an abelian variety is always isogenous to a product of CM-type elliptic curves or abelian surfaces. This extends a result of Shioda and Mitani in 1974.