  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.
