If a compact Lie group
$K$ acts smoothly on a smooth manifold
$M$, then the set
$M^K$ of fixed points under this action is a smooth submanifold of
$M$. This is proved for example in Duistermaat's book on Lie groups, using the Bochner Linearization Theorem.
I am interested in knowing if some variant of this statement is also true in algebraic geometry. In other words: can one describe a class of algebraic groups where the fixed points for an arbitrary action on a smooth variety is again smooth?