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arxiv: 2604.27634 · v1 · submitted 2026-04-30 · 🧮 math.AG

Structural properties of Bia{l}ynicki-Birula decompositions

Teddy Gonzales , Chayim Lowen This is my paper

Pith reviewed 2026-05-07 06:23 UTC · model grok-4.3

classification 🧮 math.AG
keywords Białynicki-Birula decompositionGm-actionequivariant Chow classesfilterable decompositionstratificationtoric varietiesGm-convexityGm-rigidity
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The pith

Białynicki-Birula cell closures are determined by their Gm-equivariant Chow classes when the decomposition is filterable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the Białynicki-Birula decomposition of smooth complete varieties with a Gm-action that fixes only finitely many points. It supplies characterizations of when the decomposition is filterable or forms a stratification and proves that both properties stay the same after reversing the Gm-action. For smooth projective toric varieties the work classifies the cases in which the decomposition may be or must be a stratification. The main result shows that filterability alone forces the cell closures to be recovered from their Gm-equivariant Chow classes, thereby answering questions on Gm-convexity and Gm-rigidity.

Core claim

Assuming only that the Białynicki-Birula decomposition is filterable, the closures of its cells on a smooth complete Gm-variety with finite fixed locus are completely determined by their Gm-equivariant Chow classes. The paper further establishes that filterability and the stratification property are invariant under reversing the Gm-action and gives an explicit classification of smooth projective toric varieties according to whether their Białynicki-Birula decompositions can or must form stratifications.

What carries the argument

The filterable Białynicki-Birula decomposition, whose cell closures are recovered from Gm-equivariant Chow classes.

If this is right

  • Filterability and the property of being a stratification are preserved when the Gm-action is reversed.
  • Smooth projective toric varieties admit a classification according to whether their Białynicki-Birula decompositions may or must be stratifications.
  • Questions on Gm-convexity and Gm-rigidity raised for Richardson varieties receive direct answers under the filterability hypothesis.
  • Equivariant Chow classes serve as complete invariants for the cell closures once filterability holds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Algebraic computation of cell closures becomes possible via Chow classes without constructing the geometric limits explicitly.
  • The invariance under action reversal suggests a natural duality that may extend to other equivariant invariants.
  • The results indicate that equivariant Chow rings carry enough information to reconstruct the poset structure of the decomposition when filterability is present.

Load-bearing premise

The variety is smooth and complete with a Gm-action having only finitely many fixed points, and the Białynicki-Birula decomposition is filterable.

What would settle it

A single counterexample consisting of a smooth complete Gm-variety whose filterable Białynicki-Birula decomposition contains two distinct cell closures with identical Gm-equivariant Chow classes would falsify the determination claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies structural properties of the Białynicki-Birula decomposition on smooth complete Gm-varieties with finite fixed locus. It establishes novel characterizations of filterability and the stratification property, proves that both are invariant under reversal of the Gm-action, classifies smooth projective toric varieties according to whether the decomposition is or can be a stratification, and proves that filterability alone implies the closures of the BB cells are determined by their Gm-equivariant Chow classes. The work resolves several questions posed by Buch-Chaput-Perrin concerning Gm-convexity and Gm-rigidity.

Significance. If the central results hold, the paper supplies useful new criteria for filterability and stratification that are independent of the direction of the Gm-action, together with a determination theorem linking cell closures to equivariant Chow classes under the filterability hypothesis. The toric classification furnishes concrete verification of the general criteria in a setting where explicit computations are feasible. The independent characterizations and the resolution of open questions from Buch-Chaput-Perrin strengthen the literature on equivariant decompositions and rigidity phenomena.

minor comments (3)
  1. [Introduction] In the introduction and abstract, more explicitly separate the new characterizations of filterability/stratification from the extensions of results in Buch-Chaput-Perrin; this would help readers quickly identify the novel contributions.
  2. [Toric classification] The toric classification section would benefit from a short table or enumerated list of low-dimensional examples illustrating the 'may be' versus 'must be' stratification cases, to make the classification more immediately usable.
  3. [Notation and definitions] Notation for BB cells, their closures, and the associated equivariant Chow classes should be checked for consistency across sections, especially when the Gm-action is reversed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript on structural properties of Białynicki-Birula decompositions, including the characterizations of filterability and stratification, invariance under action reversal, the toric classification, and the resolution of questions from Buch-Chaput-Perrin. We appreciate the recommendation for minor revision and the assessment of significance. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper establishes independent characterizations of filterability and stratification for the Białynicki-Birula decomposition (including invariance under Gm-action reversal) and a separate classification for smooth projective toric varieties. The central determination result—that cell closures are determined by their Gm-equivariant Chow classes—holds under the explicit assumption of filterability on a smooth complete variety with finite fixed locus. This assumption is used directly to ensure compatibility with the equivariant Chow ring structure, without reducing the conclusion to a definition, a fitted parameter, or a load-bearing self-citation. The work cites Buch-Chaput-Perrin for context and definitions of Gm-convexity/rigidity but supplies new proofs and answers their questions rather than deriving claims from quantities defined in that prior paper. No self-definitional, fitted-prediction, or uniqueness-imported steps appear; the logical chain remains self-contained against the stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates inside the standard framework of algebraic geometry; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard axioms of algebraic geometry (smooth varieties over algebraically closed fields of characteristic zero)
    Invoked implicitly for the category of Gm-varieties and Chow rings.

pith-pipeline@v0.9.0 · 5461 in / 1321 out tokens · 96539 ms · 2026-05-07T06:23:53.796518+00:00 · methodology

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