Intuition for Luna's Étale Slice Theorem

Question

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 $\psi$?

Answer 1

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.