Suppose $\mathsf{k}$ is an algebraically closed field of zero characteristic. Chevalley-Shephard-Todd (C-S-T) Theorem in one of its equivalent versions is the following result:

(C-S-T): Let $G$ be a finite subgroup of $GL(V)$, where $V$ is a finite dimensional vector space. Then the quotient variety $V/G$ is smooth if and only if $G$ is a group generated by pseudoreflections.

My first question is the following:

(Question 1) Let $G$ be a finite group of automorphisms of an irreducible affine smooth variety $X$. When is the quotient variety $X/G$ smooth?

Of course this holds if the action of $G$ is free, but I'm looking for a condition that is similar in nature to that of C-S-T Theorem. I've heard something among these lines: $G$ is generated by elements $g$ such that $X^g = \{ x \in X| g.x=x \}$ is a divisor, etc; but I could not find an exact statement or reference.

I would greatly appreciate an answer, either in form of an adequate reference, or a proof of an exact statement.

My second question is related to the first one as follows. We have a result by Kantor (1977) and Levasseur (1981) that is:

(K-L): Let $V$ be a finite dimensional vector space, $G$ a finite subgroup of $GL(V)$. Then the quotient map $V \mapsto V/G$ induces an isomorphism $\mathcal{D}(V)^G \simeq \mathcal{D}(V/G)$ if and only if $G$ contains no pseudoreflections. (cf. Levasseur, Thm 5, MR633520)

Here of course $\mathcal{D}(.)$ is the ring of differential operators, and the action of $G$ on this ring is by conjugation.

(Question 2) Let $X$ be an irreducible affine smooth variety, and $G$ a finite group of automorphisms of it. The quotient map $X \mapsto X/G$ induces an injective map $\mathcal{D}(X)^G \rightarrow \mathcal{D}(X/G)$. When is this map an isomorphism?

Again there are sufficient conditions for this to hold, such as the action of $G$ being free (cf. Cannings, Holland, 3.7, MR1305263, 1994). I wonder if there is a condition for this to hold analogue to the one considered by Kantor and Levasseur, i.e., $G$ having no "pseudoreflections", in the sense I suggested in Question (1).

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    $\begingroup$ Should $X$ be affine in Question 1 (judging by title looks like that's what you intended)? $\endgroup$ – Sam Hopkins Jun 29 at 18:24
  • $\begingroup$ Yes, it is, thank you for pointing out. I will edit it. $\endgroup$ – jg1896 Jun 29 at 23:14
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    $\begingroup$ I do not know about the general case, but over the complex numbers, $X/G$ is smooth if and only if the action on $T_xX$ of the stabilizer $G_x$ of each $x\in X$ satisfies the CST condition. The key is Cartan's theorem on local holomorphic linearization, see the reference I gave here. $\endgroup$ – Moishe Kohan Jul 15 at 16:34
  • $\begingroup$ This is extremely helpful! I will look Cartan's work. Thank you very much @MoisheKohan $\endgroup$ – jg1896 Jul 15 at 16:39

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