Bialynicki-Birula decomposition of a non-singular quasi-projective scheme.

Question

Fix an algebraically closed field $k$, an algebraic one-dimensional torus $G_m$ and a non-singular scheme $X$ of finite type over $k.$

Let us define the following:

Condition 1: $X$ can be covered by $G_m$-invariant quasi-affine open subschemes.

In the paper "Some theorems on actions of algebraic groups" (The Annals of Mathematics, Second Series, Vol. 98, No. 3 (Nov., 1973), pp. 480-497), Bialynicki-Birula constructs, roughly speaking, to any action of $G_m$ on $X$, satisfying Condition 1, two canonical decompositions of $X$ into non-singular $G_m$-invariant locally closed subschemes (Theorem 4.1).

Moreover, Bialynicki-Birula states that if $X$ is projective, Condition 1 is automatically satisfied (he cites Kambayashi, Projective representations of algebraic groups of transformations, Amer. J. Math. 88 (1966), 199-205.)

Question: Assume that $X$ is a non-singular quasi-projective scheme over $k.$ Under what extra assumptions, does $X$ satisfy Condition 1?

Answer 1

It is a theorem of Sumihiro (Equivariant completion, Corollary 2) that a normal variety over an algebraically closed field with an action of a torus is covered by invariant affine open subsets.

Here, the normality hypothesis is necessary : the conclusion does not hold for the action of $\mathbb{G}_m$ on $\mathbb{P}^1$ with $0$ and $\infty$ glued together transversally. The hypothesis that the group is a torus is also necessary. Indeed, the statement already fails for SL(2), even for a proper action. There is an example in Białynicki-Birula and Święcicka's paper "On complete orbit spaces of SL(2) actions II".

Answer 2

Every normal variety with an action of a torus is covered by invariant affine open subsets. This is proved in Hideyasu Sumihiro, Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28.