Structural properties of Białynicki-Birula decompositions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918475513987072 |
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| author | Gonzales, Teddy Lowen, Chayim |
| author_facet | Gonzales, Teddy Lowen, Chayim |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27634 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Structural properties of Białynicki-Birula decompositions Gonzales, Teddy Lowen, Chayim Algebraic Geometry 14L30 (Primary) 14M25, 14C15 (Secondary) 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. |
| title | Structural properties of Białynicki-Birula decompositions |
| topic | Algebraic Geometry 14L30 (Primary) 14M25, 14C15 (Secondary) |
| url | https://arxiv.org/abs/2604.27634 |