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.



varieties whose canonical bundle has finite order in Pic?
(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](http://arxiv.org/abs/1001.4912)) that the degree of the cover $Y \rightarrow X$ must divide $d+1$.


