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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.

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    $\begingroup$ See math.stackexchange.com/questions/348105/… $\endgroup$
    – dhy
    Aug 29, 2014 at 0:29
  • $\begingroup$ @dhy: thanks. I'll leave the question open but that may be as good as it gets. $\endgroup$ Aug 29, 2014 at 1:34
  • $\begingroup$ See the article by Beauville "Variétés Kähleriennes dont la première classe de Chern est nulle". J. Differential Geom. 18 (1983), no. 4, 755–782 (1984). $\endgroup$
    – naf
    Aug 29, 2014 at 8:41
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    $\begingroup$ I'm not sure the problem fully reduces to understanding varieties with trivial canonical bundle, because you also need to understand what kind of quotients are possible. For example, suppose $Y$ is hyperkähler of dimension $2d$. Then Oguiso and Schröer showed (arxiv.org/abs/1001.4912) that $n$ must divide $d+1$. $\endgroup$
    – user5117
    Aug 29, 2014 at 9:27
  • $\begingroup$ @ArtiePrendergast-Smith thanks, if you wrote even just that$+\epsilon$ as an answer I'd be happy to accept it. $\endgroup$ Sep 9, 2014 at 14:19

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(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 $d+1$.

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