TITLE: Smoothness of fix point components of finite group action on smooth variety
QUESTION [5 upvotes]: Let $X$ be a smooth complex algebraic variety, and $\varphi: \Gamma\curvearrowright X$ an action (by automorphisms) of a finite group $\Gamma$ on $X$.

Can we say that each irreducible component of the fix point set $X^{\Gamma}$ is smooth?

Here $X^{\Gamma}:=\{ x\in X\; | \; g(x):=\varphi_g (x)=x\;\forall g\in\Gamma\; \}$.
Remark: without loss of generality we can assume $X$ affine, since any $\Gamma$-variety can be covered by affine $\Gamma$-invariant open subsets.

REPLY [9 votes]: Yes. The reason is that locally around a fixed point, the action linearizes, i.e. is analytically equivalent to a linear action of $\Gamma $ onto a vector space -- so the fixed locus is locally isomorphic to a linear space. This fact, which follows easily from an averaging process, goes back (at least) to H. Cartan, in Algebraic geometry and topology (a symposium in honor of S. Lefschetz), pp. 90-102, Princeton University Press, 1957.