Let $X$ be a nice algebraic variety (say smooth, projective) over a field of characteristic 0. Let $G$ be an abelian group acting on $X$. For each subgroup $H$ of $G$, denote by $X^H$ the closed subvariety of points fixed by $H$.

Proposition. $X^H$ is smooth

Of course, the intersection between different $X^H$ could be non-empty. That's why people look at the "inertia stratification"

$X_H=X^H- \bigcup_{H' \subsetneq H} X^{H'}$

My question is: assuming the above proposition, why is each $X_H$ smooth?

Thanks for you help