Structural properties of Białynicki-Birula decompositions

We investigate several aspects of the Bialynicki-Birula decomposition of a smooth complete ${\mathbb{G}}_m$-variety with finite fixed locus. Our results include novel characterizations of when the Bialynicki-Birula decomposition is filterable or forms a stratification, showing that these properties are invariant under reversing the ${\mathbb{G}}_m$-action. We additionally classify the smooth projective toric varieties for which the Bialynicki-Birula decomposition either may or must be a stratification. Our study of ${\mathbb{G}}_m$-convexity and ${\mathbb{G}}_m$-rigidity – properties recently introduced by Buch–Chaput–Perrin – answers several questions posed in their $\mathit{\text{Equivariant}}\mathit{\text{rigidity}}\mathit{\text{of}}\mathit{\text{Richardson}}\mathit{\text{varieties}}$. In particular, assuming only filterability of the decomposition, we show that the Bialynicki-Birula cell closures are determined by their ${\mathbb{G}}_m$-equivariant Chow classes.