Let $X$ be a *smooth* complex algebraic variety, and $\varphi: \Gamma\curvearrowright X$ an action (by automorphisms) of a *finite* group $\Gamma$ on $X$.



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



Smoothness of fix point components of finite group action on smooth variety
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. 


