In section 6 of Kawamata's paper https://arxiv.org/abs/math/0205287, he defines the canonical covering stack. In the proof of theorem 6.5, he considered a flop $$X\xrightarrow{\phi}W\xleftarrow{\psi}Y$$ and considered the canonical covering stacks of them, denoted by $\mathcal{X},\mathcal{Y},\mathcal{W}$. Here $X$ is a$\mathbb{Q}$-factorial threefold with terminal singularity. I want to ask why we can define a canonical covering stack for $W$? If my understanding is correct, for a small contraction, the canonical divisor after the contraction can never be $\mathbb{Q}$-Cartier. And I think $\phi$ is one of this case.
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$\begingroup$ The canonical divisor is never $\mathbb{Q}$-Cartier after a flipping contraction, but here you have a flop not a flip. The point is that here $K_X$ is $\phi$-trivial so there's no obstruction to $K_X$ being pulled back from $W$. Here $W$ should have $\mathbb{Q}$-Gorenstein canonical singularities. An explicit example where $W$ is even Gorenstein is the Atiyah flop where $W$ is the cone over a quadric surface. $\endgroup$– Dori BejleriCommented Jul 2, 2021 at 19:23
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$\begingroup$ @DoriBejleri Thank you very much. Do you know somewhere that I can find the proof of the $\mathbb{Q}$-Gorenstein property after a flop contraction? $\endgroup$– Peter LiuCommented Jul 3, 2021 at 4:36
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