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Let $X$ be a smooth complex affine variety, let $G$ be a complex reductive group acting on $X$. Suppose that the stabilizer $G_x$ of a point $x\in X$ is reductive and connected. Let $\varphi: X\to X//G$ be the GIT quotient. I would like to understand the germ of $X//G$ at the point $\varphi(x)$, and in particular understand if the following is correct:

Guess. The analytic germ of $X//G$ at $\varphi(x)$ is isomorphic to the analytic germ at $0$ of the GIT quotient of $T_X(x)/T_{O_x}(x)$ by the linear action of $G_x$.

Question. Is the above guess correct? If yes, is there a reference for this statement? In particular, how to prove this statement in the case when $x$ is fixed by $G$?

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    $\begingroup$ If by "germ" you mean "etale germ", then that follows from Luna's etale slice theorem. $\endgroup$ – Jason Starr Apr 10 '17 at 20:21
  • $\begingroup$ Jason, thanks for your comment and for evoking. It will take me some time to learn what is etale germ (I'll try)... What if I just put analytic germ, would this then work? $\endgroup$ – aglearner Apr 10 '17 at 21:39
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    $\begingroup$ Yes. If $(X,p)$ and $(Y,q)$ are complex algebraic varieties that are etale equivalent, then the underlying complex analytic spaces have biholomorphic (analytic) open neighborhoods. $\endgroup$ – Jason Starr Apr 10 '17 at 21:53
  • $\begingroup$ Jason, one more question. Suppose that $x$ is a fixed point of $G$-action. How then one proves that the guess is correct? (I think I understood the formulation of Luna etale slice, but one also needs this) $\endgroup$ – aglearner Apr 10 '17 at 22:43
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For a fixed point your guess is right and one doesn't need Luna's slice theorem to prove it: Let $T$ be the tangent space in $x$ and let $\mathfrak m_x\subset\mathbb C[X]$ be the maximal ideal. Then the canonical surjective linear map $\mathfrak m_x\to\mathfrak m_x/\mathfrak m_x^2\cong T^*$ has a $G$-equivariant section giving rise to a $G$-linear map $T^*\hookrightarrow\mathbb C[X]$ and therefore to a morphism $X\to T$ which maps $x$ to $0$ and is étale in $x$. Now use the fact that invariants commute with completions (by complete reducibility) to see that $X//G\to T//G$ is étale in the image of $x$. As Jason indicated this implies an isomorphism of analytic neighborhoods.

It is true that the above argument is part of Luna's slice theorem but it is actually only the "easy" part. In fact, Luna's slice theorem is not just a statement about étale or formal neighborhoods but has a global aspect, as well. One consequence is, e.g., that the unstable set with respect to $x$ (i.e., the set of points having $x$ in their orbit closure) is globally isomorphic to the unstable set of $0$ in $T$.

Concerning your more general guess: One has to be very careful with Luna's slice theorem: it requires that the orbit of $x$ is closed. This is much stronger than $G_x$ being reductive. So, your guess is ok if $Gx$ is closed in $X$ (which is trivial if $x$ is a fixed point).

Otherwise, there is the following counterexample. Let $G=SL(2,\mathbb C)$, let $X$ be the space of $5$-forms, and $x=\xi\eta(\xi+\eta)^3\in X$. It is easily seen that $G_x=1$. So $V:=T_x(X)/T_{\mathcal O_x}(x)=\mathbb C^3$.

Let $N\subset X$ be the set of unstable $5$-forms. Hilbert has proved that these are those $5$-forms having a linear factor of multiplicity $\ge3$. So $x\in N$. Now, it is well known that $X//G$ is singular in the image of $0$. In particular, it is not isomorphic to $V//G_x=V$.

The problem of this example is that the codimension of $N$ in $X$ is $2$. Thus, $S\cap N$ is nontrivial and maps to $0$ where $S\cong V$ is a slice to the orbit of $x$.

Edit: Concerning the equivalence of étale and analytic singularities see this MO thread.

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  • $\begingroup$ Thanks a lot for this answer! I wonder if there is some textbook or lecture notes that I might consult to fill in some details? For example, the fact that "invariants commute with completions (by complete reducibility) "? (I am aware of the book of Mumford Fograty, but find it a bit tough) $\endgroup$ – aglearner Apr 11 '17 at 8:20
  • $\begingroup$ I just realized that "invariants commute with completions" is not a simple consequence of complete reducibility. In fact, its proof is more or less half of the proof of Luna's slice theorem. You can find a proof (in german, available <a href="algeo.math.fau.de/fileadmin/algeo/users/knop/papers/…) as an appendix to an article of Slodowy on the LST. See the proof of Lemma 3 with $M=$ trivial module. $\endgroup$ – Friedrich Knop Apr 11 '17 at 10:15
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    $\begingroup$ I screwed up the link: algeo.math.fau.de/fileadmin/algeo/users/knop/papers/… $\endgroup$ – Friedrich Knop Apr 11 '17 at 10:23

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