Let us have a (possibly singular) irreducible projective variety $X$ over $\mathbb{C}$, with an algebraic $\mathbb{C}^\*$-action that has finitely many fixed points $\{x\_1,\dotsc,x\_n\}$. One can define the attracting sets 
$$U\_k = \{x \in X \mathrel| \lim\_{t\rightarrow \infty}t\cdot x =x\_k\}$$ that decompose $X$ into a disjoint union. When $X$ is smooth, Białynicki-Birula's theorem is that these are affine bundles over $x\_k$, hence affine spaces. That gives us an affine cell decomposition of $X$.



> 
> Is this still true when $X$ is singular?
> 
> 
> 



Białynicki-Birula decomposition for singular projective variety
No, not at all. Take any Schubert variety, $X\_w \subset G/B$. Then for one choice of $\mathbb{C}^\* \subset T$ the BB decomposition is a cell decomposition, but for others (e.g., when the Schubert variety is singular and the torus is chosen to be attractive at the "base point" $B/B$) it will not be.


The simplest example is probably given by a singular quadric cone in $\mathbb{P}^3$ with unique singular point $x$. This has many $\mathbb{C}^\*$-actions which are attractive near $x$ with finitely many fixed points, and hence the attractive set is a singular affine quadric.


My guess is that it is a miracle when it is, and studying singular examples shows how amazing it is that it holds in the smooth case.


PS: I just did the classic "read the comments after drafting an answer" and I see that Piotr ([1](https://mathoverflow.net/questions/356995/bialynicki-birula-decomposition-for-singular-projective-variety#comment897317_356995)) and Nicolas ([1](https://mathoverflow.net/questions/356995/bialynicki-birula-decomposition-for-singular-projective-variety#comment897251_356995) [2](https://mathoverflow.net/questions/356995/bialynicki-birula-decomposition-for-singular-projective-variety#comment897319_356995) [3](https://mathoverflow.net/questions/356995/bialynicki-birula-decomposition-for-singular-projective-variety#comment897591_356995)) have already answered the question. Posting anyway in case it helps someone someday ….


