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](https://mathoverflow.net/questions/356995/bia%C5%82ynicki-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!



Plus and minus Białynicki-Birula decomposition for normal variety
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](https://doi.org/10.1215/kjm/1250523277). J. Math. Kyoto Univ. 14 (1974), 1–28) according to which $X$ can be embedded equivariantly into a projective space.


The assertions are more generally valid for any normal complete variety. The paper "Konarski, Jerzy: Decompositions of normal algebraic varieties determined by an action of a one-dimensional torus.
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 295–300" seems to be a good reference.


