Structural properties of Bia{\l}ynicki-Birula decompositions

TL;DR

This paper studies the properties of Bialynicki-Birula decompositions in smooth complete G_m-varieties, providing new characterizations, classifications, and insights into their invariance and relation to Chow classes.

Contribution

It offers novel characterizations of when the decomposition is a stratification, classifies certain toric varieties, and answers questions on G_m-convexity and G_m-rigidity.

Findings

Decomposition properties are invariant under reversing the G_m-action.

Classified smooth projective toric varieties based on stratification properties.

Established that cell closures are determined by G_m-equivariant Chow classes under filterability.

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