Q:

varieties whose canonical bundle has finite order in Pic?

Is there a structure theorem for such varieties?
If X is a smooth and proper/projective variety whose canonical bundle $\omega_X$ has finite order in the Picard group, do we know anything about X?
EDIT: As was pointed out in the comments, if $\omega_X^n = O_X$ then one can find a cycling covering of order n, $Y \to X$, with $\omega_Y = O_Y$. So the problem reduces to understanding varieties with trivial canonical bundle. I'll leave the question as is for a few days more, in case someone else wants to contribute. If nothing happens I will delete it.

A:

(Comment copied here at user125763's request.)
There seem to be two parts to the problem.
The first part is the Bogomolov–Beauville theorem that any such variety $X$ has a finite étale cover $Y \rightarrow X$ such that $Y$ is a product of 1) "strict" Calabi–Yaus, 2) abelian varieties, and 3) hyperkähler varieties.
The second part is understanding what quotients are possible. For example, suppose Y is hyperkähler of dimension $2d$. Then Oguiso and Schröer showed (arxiv.org/abs/1001.4912) that the degree of the cover $Y \rightarrow X$ must divide $d+1$.

