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.