# Uniqueness of the canonical etale coverings

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

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 $\pi_x: \tilde U_x \to U_x$.

(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).
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 $K_X$.
• @Jason: According to paper of Cadman you're referring to, he indicates in Example 2.4.1 that his root stack, which I'm guessing is your $\mathcal R$, is covered by the quotient stacks $[\tilde{U}_x/\mu_{m_x}]$, as the OP asked about. Is your $\mathcal{X}_m$ then more closely associated to the $\tilde{U}_x$? I'm just trying to see how your description fits in with Kawamata's. For example can you describe an atlas for each of these? – HNuer May 7 '13 at 21:32
• @Jason: Also it seems from later on in Kawamata's paper, namely at the end of the proof of Theorem 6.5 on p. 22 (of the version on the Arxiv) where he says explicitly that the canonical covering stack restricted to one of the $U_x$ is the quotient stack $[\tilde{U}_x/\mu_{m_x}]$. So is his stack really the same as your $\mathcal X_m$? I admit that maybe your $\mathcal X_m$ (and not $\mathcal R$) is actually just Cadman's $X_{L,s,r}$. Is this the case? – HNuer May 7 '13 at 22:01