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: as Dori Bejleri comments below, my question is about "the decomposition into locally closed affine cells and the implications that has about the cohomology"; the question linked in the comment by Vivek Shende, on the other hand, "asks about the existence of an open cover by torus invariant affines, which while important to the proof of the BB decomposition, is not the same thing".

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: as Dori Bejleri comments below, my question is about "the decomposition into locally closed affine cells and the implications that has about the cohomology"; the question linked in the comment by Vivek Shende, on the other hand, "asks about the existence of an open cover by torus invariant affines, which while important to the proof of the BB decomposition, is not the same thing".

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.

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: as Dori Bejleri comments below, my question is about "the decomposition into locally closed affine cells and the implications that has about the cohomology"; the question linked in the comment by Vivek Shende, on the other hand, "asks about the existence of an open cover by torus invariant affines, which while important to the proof of the BB decomposition, is not the same thing".