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 \begin{equation} X=\bigsqcup_{i=1\ldots k} X^+(F_k)\tag{*}\label{star} \end{equation} (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!

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