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

• The paper introduces a graph-theoretic criterion showing that filterability is equivalent to the acyclicity of the orbit graph.
• It characterizes stratification in toric varieties by linking cell closures to polytope combinatorics and products of simplices.
• The study proves that rigidity in equivariant intersection theory can be achieved without convexity, deepening our understanding of orbit decompositions.

Structural Analysis of Białynicki-Birula Decompositions

Introduction

Białynicki-Birula decompositions are fundamental stratifications arising in algebraic geometry from actions of the multiplicative group $G_m$ on smooth complete varieties with finite fixed loci. This paper presents a comprehensive structural study of such decompositions, focusing on conditions for filterability, stratification, convexity, and rigidity, with special emphasis on toric varieties. The authors resolve several open problems and refine links between the combinatorial data of polytopes, equivariant intersection theory, and the geometry of orbit decompositions.

Filterability and Orbit Graphs

Filterability of a Białynicki-Birula decomposition—a property ensuring a filtration by closed unions of cells—is characterized in terms of the absence of cycles in a directed graph $\Gamma_X$ encoding the limiting behavior of $G_m$-orbits between fixed points. The main result asserts that filterability is equivalent to the acyclicity of $\Gamma_X$; both positive and negative decompositions share this property, reflecting inversion symmetry in the group action.

A notable consequence, as clarified in Theorem 5.4, is the full classification: a decomposition is filterable if and only if there are no directed cycles induced by orbits flowing between fixed points under the $G_m$-action. This result is significant as it extends the classical projective case and gives precise graph-theoretic control over more general settings, allowing non-projective and higher-dimensional examples. The preservation of filterability under direct products and restriction to subvarieties is systematically established.

Stratification Criteria and Toric Classification

Stratification strengthens filterability by requiring cell closures to be unions of cells. The paper introduces necessary and sufficient criteria for the Białynicki-Birula decomposition to form a stratification, separating these criteria from transversality assumptions typically found in the literature. In particular, Theorem 6.3 characterizes stratification numerically: for distinct fixed points $p$ and $q$, if their corresponding cells intersect, then the sum of their dimensions must exceed that of the ambient variety.

The interplay with toric geometry is central. For smooth projective toric varieties, the authors classify those admitting Białynicki-Birula stratifications; existential stratification occurs precisely when the associated polytope is combinatorially a product of simplices, and universal stratification exactly when the polytope is unimodularly equivalent to such a product. These results realize connections to generalized Bott towers and illustrate the rigidity of the stratification phenomenon in high dimensions: outside products of projective spaces, stratifications are non-generic.

Convexity and Pathologies

Convexity in the sense of $G_m$-convex Białynicki-Birula cell closures is treated extensively. The authors prove that in low dimensions (curves and surfaces), cell closures are always $G_m$-convex, but in higher dimensions, notably for toric blowups at multiple fixed points, $G_m$-convexity typically fails.

The paper provides explicit combinatorial criteria for convexity, extending and improving on definitions from the invariant theory literature. The failure of $\Gamma_X$0-convexity is shown to be robust: blowing up at two or more appropriately chosen fixed points (for $\Gamma_X$1) guarantees that non-convex cell closures appear for any choice of admissible cocharacter. The analysis here is precise, leveraging polytope face structure and orbit combinatorics to pinpoint these failures.

Rigidity in Equivariant Intersection Theory

A major aspect of the work is the investigation of homological rigidity, following notions developed by Buch-Chaput-Perrin. For varieties whose Białynicki-Birula decomposition is filterable, the authors establish that all cell closures are strongly $\Gamma_X$2-rigid: their equivariant Chow class as a positive cycle determines the subvariety uniquely. This improves upon previous results by relaxing the requirement of full-definiteness and stratification. In cases where the decomposition is a stratification, they further show that all irreducible intersections of positive and negative cell closures are strongly $\Gamma_X$3-rigid.

However, the work also clarifies that convexity is not necessary for rigidity of cell closures (contrary to some open speculation in the literature), although it does play a central role in the rigidity of intersections.

Implications and Further Directions

The results build sharper tools for understanding the topology and intersection theory of $\Gamma_X$4-varieties, especially in the equivariant context. Practically, they provide combinatorial tests for when certain favorable algebraic or cohomological properties of decompositions are present, especially in the toric setting. For example, whether the classes of cell closures form dual Poincaré bases, or whether Chow groups admit explicit positive bases indexed by fixed points.

On a theoretical level, the paper refines the relationship between algebraic group actions, orbit structure, and polytope combinatorics. The rigidity and convexity criteria have implications for the study of orbit closures, particularly in the context of Schubert and Richardson varieties. By resolving and clarifying several open problems (e.g., on the necessity of convexity for rigidity), this work suggests that further subtle invariants—beyond convexity and dimension-counting—govern the geometry of $\Gamma_X$5-orbit decompositions in higher dimensions.

It is reasonable to expect extensions of these results to actions of higher-dimensional tori and to other classes of varieties where similar decomposition techniques are applied. Explicit criteria for filterability and rigidity can also support algorithmic approaches to the computation of equivariant Chow rings and the analysis of orbit combinatorics in moduli problems.

Conclusion

This paper gives a comprehensive and technically detailed analysis of the structural properties of Białynicki-Birula decompositions on smooth complete varieties under $\Gamma_X$6-actions. It clarifies when key geometric and cohomological properties hold, refines the role of combinatorial data from toric geometry, and resolves longstanding issues regarding rigidity and convexity. The methodologies and results should be broadly applicable within algebraic transformation group theory, equivariant intersection theory, and the study of moduli of varieties with group actions.