Structural properties of Białynicki-Birula decompositions

Abstract

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 $\text{Equivariant rigidity of Richardson varieties}$. 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.

42 pages, 10 figures