Question

Hello!

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?

Thank you!

Hanno

Answer 1

Linearly reductive groups, that is, linear algebraic groups, say over a field, in which the functor of invariants from finite dimensional representations to vector spaces is exact. I am not sure this is in the literature in this generality, but it is not so hard to prove with a formal scheme argument.

Also, I would conjecture that this is optimal, that is, given a non linearly reductive algebraic group, one can find a smooth variety on which this acts, such that the fixed point locus is not smooth.