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Boutot's theorem says that if $X$ is a variety over a field of characteristic 0 with rational singularities, and if $G$ is a reductive group acting on $X$, then the quotient $X/G$ has rational singularities as well.

Is it known whether an analog of this result is true for log terminal singularities?

Namely suppose $X$ is a variety over a field of characteristic 0 with log terminal singularities and an action of a reductive group $G$. Then must $X/G$ also have log terminal singularities?

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    $\begingroup$ No, the singularity $\{xy-zw=0\}\subset \mathbb{A}^4$ is the quotient of $\mathbb{A}^4$ by a torus, but its canonical divisor is not $\mathbb{Q}$-Cartier, so is not log terminal. $\endgroup$ Commented Aug 7, 2021 at 21:14
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    $\begingroup$ @JoaquínMoraga I do not understand : this is a hypersurface singularity, and as such it is Gorenstein, so the canonical divisor is actually Cartier! $\endgroup$ Commented Aug 7, 2021 at 22:30
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    $\begingroup$ @OlivierBenoist Oops. I meant to quotient the hypersurface by a finite group afterwards. It seems I can't edit anymore. Anyways, any 3-fold toric singularity which is not $\mathbb{Q}$-Gorenstein will make it. $\endgroup$ Commented Aug 7, 2021 at 22:51

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Now I can give you a definite answer. In general, the quotient of a klt singularity by a reductive group is not klt, because for instance, the canonical divisor of the quotient may not be $\mathbb{Q}$-Cartier.

However, one can define a broader notion: klt type. A singularity $(X;x)$ is said to be of klt type if there exists a boundary $B$ through $x$ for which $(X,B;x)$ is klt. Then, one gets the following theorem:

Theorem: Let $X$ be an affine variety with klt type singularities over an algebraically closed field of characteristic zero and $G$ be a reductive group acting on $X$. Then $X/\!/G$ is of klt type again.

The previous is Theorem 1 in https://arxiv.org/abs/2111.02812. I should also mention that the klt type property is an etale property, i.e., if you can check it in an etale cover of $X$ then it holds for $X$. This is Proposition 4.1. in https://arxiv.org/abs/2111.02812. Furthermore, the klt type condition

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