Let $X$ be an algebraic variety with a $\mathbb C^*$ action such that the fixpoints set is finite. By theorem 4.3 in the paper of Bialynicki-Birula "Some theorems on actions of algebraic groups", there exist the ($+$) and the ($-$)-decomposition if $X$ is complete.
Is there such a decomposition in the case the variety is not complete? Or more precisely, having an algebraic variety $X$ (not necessarily complete) with a $\mathbb C^*$ action such that the fixpoints set is finite, is $X$ known to have a Bialynicki-Birula cell decomposition as well? If not, is there any other cell decomposition that is known for this case? Any suggestions for references? Thanks.