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Homological rigidity and Schur rigidity of Schubert varieties in rational homogeneous spaces

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Pith's one-line read The paper proves that smooth Schubert varieties in rational homogeneous spaces are homologically rigid whenever all marked Dynkin roots are long roots, covering all ADE types, and supplies a complete rigidity classification for subdiagram-t

desk verdict Solid extension of the rigidity program, but Proposition 3.10 has a real gap in how it treats Picard-one bases, and Example 3.13 is wrong as written. read the letter →

arxiv 2607.18593 v1 pith:YMUUSWNH submitted 2026-07-21 math.AG math.CV

Cong Ding , Qifeng Li This is my paper · ORCID
classification math.AGmath.CV MSC 14M1532G10
keywords SchubertvarietieshomologicalrigiditySchurrationalhomogeneousspacesBilley–PostnikovdecompositionmarkedDynkindiagramsexceptionalpairssubdiagramtype
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Schubert varieties are the basic cycle classes of rational homogeneous spaces—flag varieties and their relatives—and the question is whether a Schubert variety is the only subvariety in its homology class up to the action of the automorphism group ('homological rigidity'), and whether multiples of its class force sums of translates ('Schur rigidity'). The paper proves that in any rational homogeneous space whose marked Dynkin roots are all long roots, every smooth Schubert variety is homologically rigid; because all roots are long in types A, D, and E, this settles the simply-laced cases completely. For Schubert varieties of subdiagram type, it also gives the complete list of non-rigid cases, all of which involve short marked roots in types B, C, or F. In the long-root subdiagram setting, Schur rigidity holds unless the variety is a fiber bundle over projective space. The method is an induction along the Billey-Postnikov decomposition, a tower of fiber bundles reducing the problem to single-marked-root cases.

What carries the argument

The central mechanism is the Billey-Postnikov decomposition: a smooth Schubert variety with Picard number at least two is an iterated Zariski-locally trivial fiber bundle whose fibers are smooth Schubert varieties in single-marked-root rational homogeneous spaces. The proof pushes rigidity down this tower. At each step the key object is an 'exceptional pair'—a single-marked-root Schubert variety admitting a local deformation not coming from the group action—and the induction succeeds exactly when no fiber contains one.

What would settle it

Exhibit a smooth Schubert variety in a type A or D rational homogeneous space of Picard number at least two together with a subvariety in the same homology class that is not a group translate. The paper's Theorem 1.2 predicts that no such pair exists.

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Extended reading notes

Core claim

The paper establishes that on a rational homogeneous space X=G/P, represented by a marked Dynkin diagram, a smooth Schubert variety X_0 is homologically rigid whenever all marked roots are long roots: any subvariety with the same homology class must be g·X_0 for some automorphism g. Since all roots are long in types A, D, and E, every smooth Schubert variety in those spaces is homologically rigid, with no homogeneity assumption on X_0. For Schubert varieties of subdiagram type, the paper gives a complete classification: homological rigidity holds except for explicit configurations in F_4, B_n, and C_n that all involve short marked roots. It further proves Schur rigidity in the long-root subd

Load-bearing premise

The proof rests on a previously compiled list of all exceptional pairs in single-marked-root spaces; if that list is incomplete, the induction over the fiber-bundle tower could miss a non-rigid Schubert variety.

Editorial extensions

If this is right

  • In all type A, D, E rational homogeneous spaces, homological rigidity holds for every smooth Schubert variety without requiring homogeneity.
  • The complete exception list for subdiagram-type Schubert varieties identifies exactly which F_4, B_n, and C_n configurations can deform, so rigidity can be read off the marked Dynkin diagram.
  • In long-root cases, Schur rigidity fails only in the presence of a fiber-bundle structure over projective space; outside that geometric obstruction, multiple homology classes have only the obvious sum-of-translates representatives.
  • The Billey-Postnikov criterion gives a local-to-global test: checking rigidity of any smooth Schubert variety reduces to checking finitely many single-marked-root fibers.
  • The results cover non-homogeneous smooth Schubert varieties, not just group orbits, extending the scope of known rigidity statements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same induction would likely extend to singular Schubert varieties if a Billey-Postnikov tower exists for them; smoothness enters through the exceptional-pair classification rather than through the induction itself.
  • The non-rigid examples in types B_n and C_n are products containing a linear factor with a larger automorphism group; this suggests that in short-root cases the general failure mechanism is again linear-subspace deformations, so a full short-root classification might reduce to tracking maximal linear subspaces.
  • Theorem 1.7's 'not a fiber bundle over projective space' condition is probably sharp in the long-root subdiagram setting: the proof uses it exactly at the Schur step, so any projective-bundle case should admit explicit counterexamples to multiple-class rigidity.
  • Because the proof is combinatorial once the Billey-Postnikov tower is known, the rigidity of any given smooth Schubert variety can in principle be verified by a finite check on Dynkin subdiagrams, independent of the ambient group.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies homological and Schur rigidity for smooth Schubert varieties in rational homogeneous spaces of higher Picard number. The main result, Theorem 1.2, asserts that if all marked roots are long — in particular for G of type ADE — then every smooth Schubert variety is homologically rigid. Theorem 1.3 gives a complete list, for Schubert varieties of subdiagram type, of the cases where homological rigidity fails. Theorem 1.7 extends this to Schur rigidity under a long-root condition and an exclusion of fiber bundles over projective space. The proofs are built on the Billey–Postnikov decomposition of smooth Schubert varieties and on the known Picard-number-one rigidity theorems of Hong–Mok and Hong–Kwon, with an induction on Picard number.

Significance. If the main theorem is correct, it is a substantial extension of the rigidity theory of Schubert varieties from Picard-number-one homogeneous spaces to arbitrary Picard numbers, and the classification in Theorem 1.3 is a valuable complete answer for the subdiagram case. The paper explicitly builds on external classifications and does not appear to be circular; the main ingredients are cited from Hong–Mok, Hong–Kwon, and Richmond–Slofstra. However, the proof of the central induction, Proposition 3.10, has a load-bearing gap concerning Picard-one bases of the Billey–Postnikov tower. The gap is likely repairable in the long-root setting of Theorem 1.2, but as written the proof of the main theorem is incomplete.

major comments (3)
  1. [Section 3.2, Proposition 3.10] The proof is incomplete. In the induction step, after producing the family N_t = π(M_t), the proof asserts: “From the base case ρ=1 of the induction, there exists h(t)·N_t = X_K(v)”. This is a rigidity statement for the Picard-one Schubert variety X_K(v) in G/P_K. The hypothesis of Proposition 3.10 only excludes exceptional pairs among the fibers of the tower; it says nothing about the base X_K(v). The ρ=1 base case is not proved: the argument invokes Theorem 2.6 for the existence of α, but that existence statement is stated only for Picard number at least 2, and a Picard-one Schubert variety appearing as a base need not be rigid. Example 3.13 is a concrete instance: after taking K' = J∪{α_{k-1}}, the base X_{K'}(v) is an exceptional pair of Proposition 3.4(2) (type C_n, marked root α_k), hence is not rigid in the Picard-one sense. Thus the step moving N_t back to X_K(v) is unjustified.
  2. [End of Section 3.2, proof of Theorem 1.2] The proof of Theorem 1.2 checks only that the fibers in the Billey–Postnikov tower are non-exceptional. To run the induction in Proposition 3.10, one must also know that every Picard-one Schubert variety that occurs as a base of the tower is rigid in its ambient space. In the long-root setting this follows from Theorem 1.1, but the paper never states or verifies this base condition. Since Theorem 1.1 does supply the missing rigidity when all marked roots are long, I believe the statement of Theorem 1.2 is likely correct, but the proof as written is not.
  3. [Section 3.2, induction on the fiber X_J(u)] The induction hypothesis is applied to the fiber X_J(u) in P_K/P_J, but the hypotheses of Proposition 3.10 are stated for the original tower associated with X_J(w). One must justify that the restricted tower for X_J(u) again satisfies the no-exceptional-pairs condition. This is probably true by restricting the original tower to the fiber, but it is not stated, and the proof depends on it.
minor comments (5)
  1. [Throughout] There are numerous typographical errors: “homogenous” for “homogeneous”, “subdigram” for “subdiagram”, “clousure”, “veriety”, “denot”, “irreucible”, and “the the” in Example 2.8. These should be corrected.
  2. [Theorem 1.3] The notation R_1 and R_2 is potentially confusing because the root system is also denoted R. Consider using different letters, e.g. A_1 and A_2, or explicitly saying R_1, R_2 are subsets of the set of simple roots.
  3. [Section 2.6, Example 2.8] The notation F• for a flag is informal; define it, e.g., F• = (F_1, F_2).
  4. [Section 3.3] The proofs of Lemmas 3.15–3.17 refer to marked diagrams and product structures; these arguments are very terse. In particular, the transition from “exceptional pairs appear in every Billey–Postnikov decomposition” to “rigidity holds” in Lemma 3.16 needs more detail, even if it is repairable.
  5. [Figure 1 and Section 3.2] Figure 1, “Billey-Postnikov decomposition,” appears to be missing from the text; there is only a caption. Either include the figure or delete the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main rigidity theorems are derived from external classifications and the Billey-Postnikov decomposition theorem, not from their own conclusions.

full rationale

The paper's central results do not reduce to their inputs by construction. Theorem 1.2 is proved from three independent external ingredients: (i) Richmond-Slofstra's Billey-Postnikov fiber-bundle theorem (Theorem 2.6, [18]); (ii) Hong-Mok and Hong-Kwon's Picard-number-one rigidity theorem (Theorem 1.1, [7,10]); and (iii) the exceptional-pair classification for Picard-one subdiagram Schubert varieties (Proposition 3.4, imported from [7,10]). Proposition 3.10 is an induction whose hypothesis (no exceptional pairs in the fibers of a chosen Billey-Postnikov tower) is strictly weaker than the conclusion (homological rigidity of the whole Schubert variety), and the proof works by transporting local deformations down to the base and fibers using Lemmas 3.7-3.9 rather than assuming the target rigidity. The subdiagram classification (Theorem 1.3) is obtained by applying Proposition 3.10 to the imported Picard-one classification, and the Schur-rigidity proof (Theorem 1.7) invokes Theorem 1.2 and Corollary 3.11 only after those are already established. The only author self-citation in the bibliography, [13], is not cited in the actual argument and carries no load; the 'private communication' remark about J. Hong's independent proof is informational, not load-bearing. The skeptical concerns about the rho=1 base case and the Picard-one base X_K(v) in Proposition 3.10 are proof-presentation or correctness issues, not circular reductions: the paper does not define the target result into its hypotheses. Therefore no circularity is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical fitting or data is involved. The free-parameter list is empty because the proofs are deductive. The axioms listed are the principal external theorems on which the central claims rest.

assumptions (5)
  • domain assumption Theorem 1.1 of Hong-Mok and Hong-Kwon: in Picard number one, smooth Schubert varieties are homologically rigid when the marked root is long or the variety is non-linear.
    This is the base case for the induction in Proposition 3.10 and is quoted as Theorem 1.1 in the paper.
  • domain assumption Richmond-Slofstra Billey-Postnikov decomposition theorem: every smooth Schubert variety of Picard number at least two admits a Zariski-locally trivial Mori contraction with fibers that are smooth Schubert varieties of Picard number one.
    This structural input (Theorem 2.6) is needed to build the tower used in the main rigidity induction.
  • domain assumption Proposition 3.4, the complete classification of exceptional pairs, is exhaustive; it is derived from [7,10] and Lemma 3.3.
    The main induction requires that no exceptional pair appears in the fibers; incompleteness of this list would break Theorems 1.2 and 1.3.
  • domain assumption Theorem 1.6 of Hong-Mok: Schur rigidity holds for non-linear smooth Schubert varieties in rational homogeneous spaces of Picard number one.
    Used as the base case for the Schur rigidity proofs in Section 4.
  • standard math Standard facts on homology classes of Schubert varieties and the stabilizer computation in Lemma 2.3.
    These underpin the definitions and the Chow-space arguments; they are standard in Schubert calculus.

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Pith. "Pith review of Homological rigidity and Schur rigidity of Schubert varieties in rational homogeneous spaces." pith.science (2026). https://pith.science/paper/YMUUSWNH

@misc{pith2026260718593,
  author       = {Pith},
  title        = {Pith review of: Homological rigidity and Schur rigidity of Schubert varieties in rational homogeneous spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YMUUSWNH}},
  note         = {Machine review of arXiv:2607.18593}
}
abstract

A Schubert variety $X_0$ on a rational homogenous space $X=G/P$ is said to be homologically rigid, if any subvariety $Z$ on $X$ representing the same homology class with $X_0$ must satisfy $Z=g\cdot X_0$ for some $g\in{\rm Aut_0}(X)$. We say $X_0$ is Schur rigid, if furthermore any subvariety $Z$ on $X$ whose homology class is a multiple $r$ of that of $X_0$ must satisfy $Z=g_1\cdot X_0+\cdots+g_r\cdot X_0$ for some $g_1,\cdots ,g_r\in{\rm Aut_0}(X)$. Homological rigidity and Schur rigidity of Schubert varieties in rational homogeneous spaces of Picard number one have been well studied in extensive literature. In this paper, we study both rigidity problems of Schubert varieties in rational homogeneous spaces of higher Picard numbers. We show that in the long root cases, including all cases when $G$ is of type $ADE$, smooth Schubert varieties have homological rigidity. Besides, we give the complete list of Schubert varieties of subdiagram type with/without homological rigidity. Furthermore, for a Schubert variety $X_0$ of subdiagram type, we show that it has Schur rigidity in long root cases unless $X_0$ admits a fiber bundle structure over the projective space.

Figures

Figures reproduced from arXiv: 2607.18593 by the authors.

Figure 1
Figure 1. Billey-Postnikov decomposition Also, we immediately have Corollary 3.11. Let X0 ⊂ X is a Schubert variety of subdiagram type and L be the set of simple roots in the subdiagram. If I ∩ L contains no short roots, then X0 has homological rigidity in X. Proof. Since I ∩L consists of only long roots, no pairs in the fibers of Billey￾Postnikov decompositions in each step are exceptional, by Proposition 3.4. Then the concl… view at source ↗

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