Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an orbifold? I.e., does $X/\!/G$ have, at worst, quotient singularities?
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1$\begingroup$ If $X$ is singular then this is certainly not true $\endgroup$– HenriCommented Apr 10 at 9:20
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$\begingroup$ Sorry I meant to include the word smooth. It is corrected now. $\endgroup$– Dr. EvilCommented Apr 11 at 7:10
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$\begingroup$ How about Luna's slice theorem? This post. $\endgroup$– Display NameCommented Apr 12 at 7:19
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$\begingroup$ The Luna slice theorem does indeed seem to be relevant. Although I found the answer to this post better suited for the question. If somebody can expand this into an answer, I would be grateful. mathoverflow.net/questions/435428/… $\endgroup$– Dr. EvilCommented Apr 13 at 19:41
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