# Does the quotient of a variety with log terminal singularities also have log terminal singularities?

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?

• 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. Commented Aug 7, 2021 at 21:14
• @JoaquínMoraga I do not understand : this is a hypersurface singularity, and as such it is Gorenstein, so the canonical divisor is actually Cartier! Commented Aug 7, 2021 at 22:30
• @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. Commented Aug 7, 2021 at 22:51

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