In this paper, M. Romagny defines for an action of a group scheme $G$ on a stack $X$ the fixed point stacks $X^G$ associated to the group action on a stack and in Theorem 3.3 he proves that if

  1. the group $G$ is proper and flat of finite representation
  2. $X$ is a Deligne-Mumford stack

then $X^G$ is algebraic. Later in this note, he proves that condition 2. can be relaxed to $X$ being algebraic with the diagonal being locally of finite presentation.

I am mostly interested in actions by complex tori on algebraic stacks locally of finite type. In this case, one doesn't have the properness from condition 1. Is it still true that the stack of fixed points $X^G$ is algebraic?

I am aware that it is common to take fixed points of Deligne-Mumford stacks as in Graber--Pandharipand with respect to the torus action, but the same approach doesn't seem to work in the completely general case.


1 Answer 1


I hadn't seen this question until Arkadij contacts me directly. The answer is yes: if $G$ is a group scheme of multiplicative type then the fixed point stack is algebraic. This is now here : https://arxiv.org/abs/2101.02450.


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