I would like to know to which extent the theory developed for smooth projective varieties in the following articles

A. Białynicki-Birula, Some theorems on actions of algebraic groups. Ann. of Math. (2) , 98:480–497, 1973.

A. Białynicki-Birula, Some properties of the decompositions of algebraic varieties determined by actions of a torus. Bull. Acad. Polon. Sci. S ́er. Sci. Math. Astronom. Phys. , 24(9):667–674, 1976.

extends to the case of smooth non-complete (say, quasi-projective) varieties. Assume the ground field to be $\mathbb{C}$ if you want. There are similar questions on MO, but I haven't found a satisfying answer.