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I am trying to understand the intuition for Luna's Étale Slice Theorem in the affine setting over $\mathbb{C}$.

Here is the setup. Let $X$ be an affine algebraic variety and $G$ a reductive group acting on $X$. Moreover, let $x\in X$ be a closed point and $G_{x}$ its stabilizer under the $G$ action where $G_{x}$ is also a reductive group, and $V\subseteq X$ a $G_{x}$ invariant locally closed affine variety containing $x$.

The group action $G\times V\to X$ by $(g,v)\mapsto g\cdot v$ is $G_{x}$ invariant if the action of $G_{x}$ on $G\times V$ is by $h\cdot (g,v)\mapsto (gh^{-1},hv)$. Indeed, one can check that $gh^{-1}\cdot h\cdot v=gh^{-1}h\cdot v=g\cdot v\in X$ but if $g\in G_{x}$ then $g\cdot v\in V$ since $V$ is $G_{x}$ invariant.

What is the geometric intuition for the quotient $G\times_{G_{x}}V= (G\times V)// G_{x}$?

Formally, it seems we are identifying points $(g,v)\sim (g',v')$ if there exists some $h\in G_{x}$ such that $gh^{-1}\cdot hv$ and $g'\cdot v'$ map to the same point in $X$. Namely $v,v'\in V$ lie in the same $G$ orbit. So $G\times_{G_{x}} V$ is just the set of $G$-orbits in $V$?

This induces a morphism $\psi: G\times_{G_{x}} V\to X$. Say $V$ is an étale slice if $\psi$ is an étale morphism. Luna's theorem shows the existence of such étale slices.

What is the structure of $G\times_{G_{x}} V$ as a variety?

What is the correct way to think about the morphism $\psi$?

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1 Answer 1

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Luna Étale's Slice Theorem is probably the most powerful result we have to understand the local structure of a moduli space.

Moduli spaces (say of semi-stable vector bundles on a smooth projective curve) may be constructed by taking the geometric quotient of a Quot-scheme by a reductive group $G$. The stabilizer of a point $x$ is usually included in $\mathrm{Hom}(E_x, E_x)$, where $E_x$ the vector bundle corresponding to $x$. In case $E_x$ is semi-stable, there are many techniques available to compute $\mathrm{Hom}(E_x, E_x)$. If you are in a favorable situation, the Quot-scheme is smooth at $x$ and Luna Étale's Slice tells you that the singularity of the moduli space at $x$ is locally isomorphic to $\mathbb{C}^n/G_x$.

I would advise reading Seshadri's notes Fibrés vectoriels sur les courbes algébriques (written by Drézet) where all of this is very clearly explained. And with many examples.

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  • $\begingroup$ This post doesn't answer any of the questions of the OP. $\endgroup$ Dec 3, 2022 at 13:34
  • $\begingroup$ @FriedrichKnop : it is indeed correct that my answer does not directly answer to the questions of the OP. However, the chapter on local structures of moduli spaces (and more precisely, the appendix dedicated to Luna's étale slice Theorem) would certainly answer duch questions. $\endgroup$
    – Libli
    Dec 4, 2022 at 16:33
  • $\begingroup$ @FriedrichKnop : In fact, I intended first to rewrite a bit this answer, but I fell short of time and I forgot about it. Please feel free to add your own answer, if you think it may provide additional information. $\endgroup$
    – Libli
    Dec 4, 2022 at 16:37

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