This is a construction [Definition 6.1] given in the paper D-equivalence and K-equivalence by Kawamata.



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> > Let $X$ be a normal quasiprojective variety such that the canonical divisor $K\_X$ is a $\mathbb{Q}-$Cartier divisor. Each point $x \in X$ has an open neighborhood $U\_x$ such that $m\_xK\_X$ is a 
> >  principal Cartier divisor on $U\_x$ for a minimum positive integer $m\_x$.
> >  The canonical covering $\pi\_x: \tilde U\_x \to U\_x$ is a finite 
> >  morphism of degree $m\_x$ from a normal variety 
> >  which is etale in codimension $1$ and such that $K\_{\tilde U\_x}$ is a Cartier 
> >  divisor. 
> >  The canonical coverings are etale locally uniquely determined,
> >  thus we can define the canonical covering stack $\mathcal{X}$ as 
> >  the stack above $X$ given by the collection of canonical coverings 
> >  $\pi\_x: \tilde U\_x \to U\_x$.
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I have two question about this construction:


(1) Why "The canonical coverings are etale locally uniquely determined"?


(2)Should I think the canonical covering stack $\mathcal{X}$ as a gluing of stacks $Sch / U\_x$ or as a gluing of quotient stack $[U\_x / \mu\_x]$(I only vaguely thought $\mu\_x$ should be some group in the construction in the etale covering).



Uniqueness of the canonical etale coverings
I recommend reading about "root stacks" as studied by Charles Cadman. Given an invertible sheaf, e.g., $\mathcal{O}\_X(mK\_X)$, and given an integer $m$, there is an Artin stack $\mathcal{R}\to X$ parameterizing $m^{\text{th}}$ roots of the pullback of this invertible sheaf. This stack is a $\mu\_m$-gerbe over $S$, hence it is Deligne-Mumford in characteristic $0$. Over the smooth locus $U$ of $X$, there is a section $s\_U:U \to \mathcal{R}$, since $\omega\_U$ is an $m^{\text{th}}$ root of $\mathcal{O}\_X(mK\_X)|\_U$. The claim is that there is a unique triple $(\mathcal{X}\_m,s,i\_m)$ of a normal, separated Artin stack $\mathcal{X}\_m$, a finite morphism $s:\mathcal{X}\_m \to \mathcal{R}$, and a morphism over $\mathcal{R}$, $i\_m:U\to \mathcal{X}\_m$ such that $i\_m$ is representable by dense open immersions. In particular, this means that $s$ is the "integral closure of $\mathcal{R}$ in the field of fractions of $U$". Since that construction is compatible with smooth base change, you can construct the triple étale locally over $X$, which is what Kawamata describes. Finally, you can check that the natural induced morphism $\mathcal{X}\_m \to \mathcal{X}\_{mn}$ is an isomorphism. So the construction (locally) only depends on the "local index" of $K\_X$. 


