# Does Borel fixed-point theorem hold for Deligne-Mumford stacks?

Let $$X$$ be a proper Deligne-Mumford stack over $$\mathbb{C}$$ with an action by a complex torus $$T$$. Let $$X^T$$ denote the fixed locus.

Question: Is the following statement true?

If every point of $$X^T$$ is a smooth point of $$X$$, then $$X$$ is smooth.


If $$X$$ is a scheme, I know the answer is yes and it is proved by applying the Borel fixed-point theorem to the singular locus of $$X$$. So the question will be solved if the fixed-point theorem holds for DM stacks. But I could not find a reference for this.

EDIT 1: The previous version is a bit misleading: the assumption for the statement is that for any $$x\in X^T$$, the local ring $$\mathcal{O}_{x,X}$$ is regular (but not $$\mathcal{O}_{x,X^T}$$).

EDIT 2: Following Ariyan's comment, I present my attempted proof for the case of schemes: Let $$X'$$ be the singular locus of $$X$$. Then $$T$$ acts on $$X'$$ and leaves each of its irreducible components invariant. For each component, we apply Borel fixed-point theorem to get a fixed point $$x$$. By our assumption, it is a smooth point of $$X$$, and hence $$x\not\in X'$$, a contradiction.

EDIT 3: The following result appears in Graber-Pandharipande's paper on virtual localization formula. Hope it is useful:

If $$V$$ is a DM stack with a $$\mathbb{C}^{\times}$$-action which has no fixed points, then the equivariant Chow group $$A_*^{\mathbb{C}^{\times}}(V)$$ vanishes after localizing the equivariant parameter.

Back to the original question. Suppose the singular locus $$X'$$ of $$X$$ has no fixed-points. (If it has, we win.) We apply their result to $$X'$$ to get the vanishing property of the equivariant Chow of $$X'$$. A potential contrdictation is that the fundamental class $$[X']$$ cannot be killed by the equivariant parameter and the reason should have something to do with the properness of $$X'$$. But I don't know if it is true.

• Welcome Chi Hong Chow. Minor comment: "proper DM stack of finite type over $\mathbb{C}$" is a bit redundant. It suffices to say "proper DM stack over $\mathbb{C}$". Also, could you explain how the argument using Borel's fixed-point theorem goes for schemes? Commented Sep 17, 2021 at 15:58
• @AriyanJavanpeykar Thanks! Edited based on your comments. Also clarified the assumption for the statement which had potential to cause confusion. Commented Sep 18, 2021 at 8:32