MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top


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!


share|cite|improve this question
up vote 5 down vote accepted

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.

share|cite|improve this answer
@Angelo: Thank you very much! – Hanno Becker Apr 12 '11 at 14:11
A reference is Birger Iversen, A fixed point formula for action of tori on algebraic varieties, Invent. Math., 1972. – Dan Petersen Mar 2 '12 at 16:15

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.