# Plus and minus Białynicki-Birula decomposition for normal variety

We work over $$\mathbb{C}$$. Let $$X$$ be a normal projective irreducible variety, and let $$\mathbb{C}^*$$ act nontrivially on $$X$$. The fixed point locus of $$X$$, namely $$X^{\mathbb{C}^*}$$, can be decomposed into a disjoint union of connected fixed point components, let us call them $$F_1,\ldots,F_s$$. Moreover, for any $$k=1,\ldots,s$$, we can define the plus cell as

$$X^+(F_k)=\{p\in X\mid \lim_{t\to 0} t\cdot p \in F_k\}$$

and similarly also the minus cell, by considering the limit at $$\infty$$.

If $$X$$ is smooth, then the celebrated theorem of Bialynicki-Birula tell us that $$$$X=\bigsqcup_{i=1\ldots k} X^+(F_k)\tag{*}\label{star}$$$$ (and similarly for the minus cell), and moreover this is an affince cell decomposition.

Question: If $$X$$ is only normal, do we still have a decomposition as in \eqref{star} (but in this case the cell will not be afine, as explained in Białynicki-Birula decomposition for singular projective variety)?

I've tried to search on the literature, but what I've found is usually much more deep of what I'm asking and I have a hard time trying to translate that result in my specific case. Any help, or reference, would be much appreciated. Thanks in advance!

• In your question, did you intend to write, "If $X$ is only normal, ..."? Commented Jun 7, 2022 at 10:11
• Dear @JasonStarr, sure, thanks for spotting the typo! Commented Jun 7, 2022 at 10:29
• Have you looked at Hausel-Hitchin arXiv:2101.08583, Section 2.1? Commented Jun 7, 2022 at 10:35
• Such a generalization is I think due to Drinfeld and employed in several papers by Jelisiejew. Commented Jun 7, 2022 at 10:46
• The references: arXiv:1308.2604 (Drinfeld), arXiv:1710.06124 (Jelisiejew, see Section 3), arXiv:1805.11558 (generalization to reductive groups, Jelisiejew-Sienkiewicz) Commented Jun 7, 2022 at 10:50

Yes, \eqref{star} is always a disjoint union (that's obvious). Moreover, each set $$X^+(F_k)$$ is locally closed and the map $$x\mapsto\lim_{t\to0}t\cdot x$$ induces an affine morphism $$\pi_k:X^+(F_k)\to F_k$$. In general, the morphism $$\pi_k$$ is not a fiber bundle anymore.
These assertions can be easily reduced to the smooth case by using a theorem of Sumihiro (Sumihiro, Hideyasu: Equivariant completion. J. Math. Kyoto Univ. 14 (1974), 1–28) according to which $$X$$ can be embedded equivariantly into a projective space.
• Dear Prof. Knop, thank you very much for the answer and the references. Before accepting it though, I have few questions. You say at the beginning "it's obvious": are you referring to the fact that the union is disjoint (and I agree with you on this), or are you referring to the fact that $X$ coincides with the union? I was precisely asking about the second point, but with your suggestion I think I know how to prove the second question (see the comment below): Commented Jun 8, 2022 at 6:27
• Using Sumihiro's theorem, we embed $X$ $\mathbb{C}^*$-equivariantly into a $\mathbb{P}^n$. Now we use the BB-decomposition of the projective space, obtaining $$\mathbb{P}^n=\bigsqcup_{i=1}^k (\mathbb{P}^n)^+(F_i),$$ where $F_i$ are the connected fixed point components of $\mathbb{P}^n$ under the $\mathbb{C}^*$-action. Then since $X$ is $\mathbb{C}^*$-invariant we consider the intersection of the BB-decomposition with $X$, obtaining $$\mathbb{P}^n\cap X=X=\bigsqcup_{i=1}^k X^+(F_i\cap X),$$ where maybe some intersection will be empty, but we still obtain a $+$-decomposition of $X$. Commented Jun 8, 2022 at 6:28
• @Mago: That's correct. That $(*)$ is a disjoint union follows from the fact that for every $x$ the limit $\lim_{t\to0} t\cdot x$ exists (since $X$ is complete) and is unique (since $X$ is separated) and that the limit point is a $\mathbb C^*$-fixed point. But all this follows also from the Sumihiro argument once you accept the assertions for the projective space. Commented Jun 8, 2022 at 14:44
• @Mako: One more thing: Observe that $F_i\cap X$ might have several connected components. Then $X^+(F_i\cap X)$ decoposes accordingly. Commented Jun 8, 2022 at 14:54