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Realization of Cohomology Classes in Grassmannians

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Pith's one-line read A quadratic inequality b²≥ac exactly characterizes which dimension-3 and codimension-3 cohomology classes of a Grassmannian are realizable by irreducible subvarieties, with a short list of boundary exceptions.

desk verdict Main classifications look right, but the Hodge index necessity has a gap in Theorems 7.1 and 8.3 that needs an explicit limit-of-surfaces argument. read the letter →

arxiv 2509.03747 v1 pith:L2XCDRGP submitted 2025-09-03 math.AG

Izzet Coskun , Julius Ross This is my paper · ORCID
classification math.AG MSC 14M1514C2514C17
keywords GrassmannianSchubertclassesirreduciblesubvarietiesrealizableHodgeindextheoremlog-concavesequencescohomologyalgebraiccycles
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

This paper asks which cohomology classes of a Grassmannian G(k,n)—the space of k-dimensional linear subspaces of an n-dimensional vector space—can be represented by a single irreducible algebraic subvariety. In dimensions 2 and 3 and codimensions 2 and 3, it gives a complete answer: apart from a few boundary cases, the only condition beyond nonnegativity of the Schubert coefficients is a quadratic inequality, b²≥ac, between the three coefficients of the class. For the Grassmannians G(2,n), it classifies all classes that are realizable over the rationals: the coefficient sequence must be log-concave with no internal zeros. The result matters because it turns a hard geometric existence question into a checkable algebraic inequality, and it shows exactly where the Hodge index theorem—rather than merely numerical effectiveness—starts to govern which cycles are irreducible.

What carries the argument

The argument is carried by two mechanisms. First, an incidence-correspondence and cone construction (Propositions 4.1–4.6) builds irreducible subvarieties of G(k,n) from irreducible subvarieties of products of projective spaces or smaller Grassmannians, and computes their Schubert classes exactly; this is what realizes every class with b²≥ac. Second, the Hodge index theorem applied to classes like [Y]·x or σ₂,₂⊗1·[f^{-1}Y] forces the determinant inequality ac−b²≤0 on any realizable class, which is what makes the classification sharp. The boundary cases are settled by the known classification of multi-rigid Schubert classes, which says which positive multiples of a single Schubert class can b

What would settle it

An irreducible subvariety of G(3,6) with class σ₃+σ₁,₁,₁ would violate b²≥ac (here b=0, ac=1) and disprove the classification. Conversely, if no such subvariety exists, one can still check the proof's key step directly by computing whether the class σ₂,₂⊗1·[f^{-1}Y] lies in the closure of the effective cone for an arbitrary irreducible sixfold Y—a concrete effective-cone membership calculation in the cohomology of G(2,5)×G(3,6).

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Extended reading notes

Core claim

The central claim is a complete classification in the first nontrivial ranges. Write a dimension-3 or codimension-3 class in G(k,n) as ν=aσ₃+bσ₂,₁+cσ₁,₁,₁, with σλ the Schubert classes forming the standard basis of the cohomology ring. The paper proves that for 3≤k≤n−k, such a class is the class of an irreducible subvariety over Z if and only if a,b,c≥0 and b²≥ac, with precisely listed exceptions: when k=3 and n=6 the only pure classes that survive are σ₃ and σ₁,₁,₁ themselves, and when k=3 and n>6 the class σ₁,₁,₁ must still occur with coefficient 1 while σ₃ may occur with any positive coefficient. The necessity of b²≥ac is derived from the Hodge index theorem applied to limits of surfaces

Load-bearing premise

The load-bearing premise is that the classes obtained by intersecting an irreducible representative with certain divisor classes are still limits of genuine surfaces; the paper proves this in the dimension-3 case but leaves it implicit elsewhere, and if such a class were not a limit of effective surfaces, the numerical inequality that rules out all other classes would not be forced.

Editorial extensions

If this is right

  • For every Grassmannian with 3≤k≤n−k, the dimension-3 and codimension-3 realizability problem is closed: a class either meets the inequality or it does not.
  • Rational realizability in G(2,n) is a convex-geometric condition—log-concavity with no internal zeros—so checking it is an algorithmically simple test.
  • Stabilization reduces all higher-dimension questions to a boundary Grassmannian: a codimension-r class is realizable over Q in G(k,n) iff it is realizable in G(r,2r), so classification problems for fixed r can be solved once and for all.
  • The same Hodge-index obstruction should reappear in every higher dimension; the paper's Question 1.7 makes precise the hope that Hodge-Riemann relations give all obstructions.

Reading between the lines

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

  • One can test the paper's implicit pseudoeffectiveness step directly: in the two cases where it is not proved, compute whether the classes σ₂,₂⊗1·[f^{-1}Y] and x²·[f^{-1}Y] lie in the closure of the effective cone for every realizable Y; if a counterexample appears, the necessity direction would need a different proof even though the theorem may survive.
  • The G(2,n) log-concavity result suggests reading 'realizable over Q' as a convexity phenomenon; one could probe whether the same holds for the other cominuscule Grassmannians, where Schubert classes are indexed by other strict partitions.
  • The boundary rigidities vanish as soon as one passes to a larger Grassmannian: multiples of σ₁,₁,₁ that are not realizable in G(3,6) become realizable in G(3,n) or G(k,n) for k>3. This suggests that rigidity is a small-dimensional phenomenon and that the stabilizing Grassmannian is the right arena for the classification.
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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 which integral cohomology classes of Grassmannians G(k,n) can be represented by irreducible subvarieties. The main results are: (i) a complete classification in dimension 2 and codimension 2 (Theorem 5.1, 5.2); (ii) a classification in dimension 3 and codimension 3 for 3 ≤ k ≤ n−k, giving the condition a,b,c ≥ 0 and b² ≥ ac, with explicit boundary exceptions for G(3,6) and G(2,n) (Theorems 6.1, 7.1, 6.2, 7.2); (iii) a classification of all classes realizable over Q in G(2,n), namely log-concave sequences with no internal zeros (Theorem 9.1); and (iv) additional results for G(3,6) and G(4,8) (Theorems 8.2, 8.3, 9.2). The sufficiency proofs use incidence correspondences and birational cone constructions from products of projective spaces, together with known realizability results of Huh, Hong, and Coskun–Robles. The necessity proofs use the Hodge index theorem to derive the determinant inequality b² ≥ ac. Stabilization results in Section 3 reduce many statements to small Grassmannians.

Significance. If the results are correct, they represent substantial progress on a natural realizability problem for Grassmannian cohomology classes. The paper gives explicit, practically checkable criteria in several nontrivial dimensions and codimensions, and it identifies a clean obstruction from the Hodge-Riemann relations. The stabilization theorems (Corollaries 3.7 and 3.9) are useful and reduce the classification to a finite set of cases. The constructions via birational image of Grassmannian bundles over products of projective spaces are elegant and produce irreducibility in a wide range of cases. The reliance on previously published theorems of Huh, Hong, and Coskun–Robles is appropriate; I see no circularity in the main classification. The main weakness is that several Hodge-index necessity arguments are not fully justified as written, as detailed below.

major comments (3)
  1. [Theorem 7.1 (and Section 7, paragraph after the pullback formulas)] The necessity argument applies the Hodge index theorem to the class S = σ_{2,2} ⊗ 1 · [f^{-1}(Y)]. This is only valid if S is a limit of irreducible surface classes (or otherwise satisfies the weak Lorentzian property). The text does not show this, and effectiveness alone is insufficient. For example, in P² × P², the reducible surface S = ({p}×P²) ∪ (P²×{q}) has D²·S = E²·S = 1 and D·E·S = 0 for D = pr₁*O(1), E = pr₂*O(1), so the Hodge index determinant is positive. The proof in Theorem 6.1 explicitly states that [Z]·x is a limit of irreducible surfaces, but Theorem 7.1 has no such justification for σ_{2,2}⊗1·[f^{-1}Y]. Since this step is the only proof of the inequality b² ≥ ac in the codimension-3 classification, it is load-bearing. Please add a proof, a reference, or an equivalent argument showing that S is a limit of irreducible surfaces (or replace the Hodge-index step with another
  2. [Theorem 8.3] The same issue appears in the proof of Theorem 8.3: after forming x²·[f^{-1}(Y)], the text says 'The Hodge index inequality...' without explaining why this class is a limit of irreducible surfaces. Here the situation is more easily repaired than in Theorem 7.1, because x is the pullback of the hyperplane class from P⁴ and x² is a complete intersection of two basepoint-free divisors; intersecting an irreducible variety with general members of such a linear system yields an irreducible surface by Bertini. As written, however, the necessary justification is omitted. Please state it explicitly.
  3. [Theorem 9.1 (inductive step)] The proof of Theorem 9.1 also applies the Hodge index theorem to the class x^{n−4}·[f^{-1}(Y)] without stating why this is a limit of irreducible surfaces (or, if the dimension is not 2, why the Hodge-Riemann form for the relevant cycle class is weakly Lorentzian). The same concern as in Theorems 7.1 and 8.3 applies. Since Theorem 9.1 is one of the main classification results, this step needs a clear justification. If the dimension count forces this class to be a surface for the relevant m, the argument can likely be repaired by selecting n−4 general hyperplanes and invoking Bertini, but the text should say so.
minor comments (5)
  1. [Theorem 9.2, proof] In the last paragraph, the text reads 'By [Ho05], bσ_{3,1,1} is realizable...' but the class under discussion is bσ_{2,2}. Either the notation should be corrected or the intended Schubert class should be clarified.
  2. [Theorem 6.2(2)] The statement of part (2) does not specify the value of n. From the context it appears to be n = 5; please state this explicitly.
  3. [Theorem 9.1 and its proof] The theorem states that the coefficients are nonnegative rational numbers, but the induction in the proof repeatedly says 'nonnegative integers.' The statements should be harmonized.
  4. [Throughout] There are several typographical and grammatical issues, e.g., 'all the classes that can be realizable' in the abstract, 'Propsition' in Theorem 7.2, and 'prooof' in Theorem 8.3. A careful proofreading pass is recommended.
  5. [Propositions 3.6 and 3.8] The use of the same notation mσ_r and mσ_{1^r} for exceptions in both codimension-r and dimension-r statements is confusing, especially with the convention that σ_λ := σ_{λ^c} in dimension statements. Please clarify the notation in these two propositions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the classifications are derived from independent Hodge-index/Lorentzian obstructions and external realizability theorems; no coefficient is fitted or defined in terms of the target.

full rationale

The derivation chain is not circular. Sufficiency directions construct representatives via incidence correspondences (Prop. 4.3/4.5 and Thm. 4.4/4.6) from realizable classes in products of projective spaces or smaller Grassmannians, using external results [Hu12], [Hu13], [Ho05], and [CR13]. Necessity directions use the Hodge index theorem to force determinant inequalities such as b^2 >= ac (Thm. 6.1, Thm. 7.1, Thm. 8.3, Thm. 9.1). These are genuine obstructions: the target condition b^2 >= ac is proved from Hodge index, not assumed, and no parameter is fitted to the classes being classified. The self-citations [Co09, Co18, CR13] are prior published results with independent standing; [CR13] and [Ho05] classify multi-rigidity/flexibility of Schubert classes, which is an external input used for boundary cases, not an assumption of the main realizability criterion. The skeptic concern about Theorems 7.1 and 8.3—that the classes to which Hodge index is applied are not explicitly shown to be limits of irreducible surfaces—is a possible gap in proof justification, but it is a correctness issue, not circularity. A missing technical lemma does not make the derivation equivalent to its inputs by definition. Therefore no enumerated circularity step is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants and no new entities. It relies on standard Schubert calculus, the Hodge index theorem, and several cited classification theorems. The most fragile input is the implicit application of Hodge-Riemann relations to pulled-back intersection classes that are asserted to be limits of effective cycles.

assumptions (6)
  • standard math Hodge index theorem and Hodge-Riemann relations for pseudoeffective two-dimensional classes
    Used in Theorems 6.1, 7.1, 8.3, and 9.1 to derive the necessary inequality b²≥ac from intersection matrices.
  • standard math Kleiman transversality and the fact that effective cycles in G(k,n) are nonnegative combinations of Schubert classes
    Used at the start of the classification proofs, citing [Co18, Proposition 2.20].
  • standard math Littlewood-Richardson and Pieri rules for Schubert calculus
    Used for pullback and intersection computations in Theorems 6.1, 7.1, 8.3, and 9.1.
  • domain assumption Classification of multiples of Schubert classes, including multi-rigidity from [Ho05] and [CR13]
    Used to handle pure Schubert classes and boundary cases, e.g., mσ1,1,1 realizable over Z if and only if m=1.
  • domain assumption Debarre's connectivity theorem [De96, Theorem 8.1] for irreducible intersections with Schubert varieties
    Load-bearing for the stabilization inclusions in Proposition 3.6.
  • domain assumption Huh's realizability theorems for products of projective spaces, [Hu12, Theorem 21] and [Hu13, Theorem 1]
    External benchmarks used in sufficiency: log-concavity in product spaces and surface classes in P2×P2.

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Pith. "Pith review of Realization of Cohomology Classes in Grassmannians." pith.science (2026). https://pith.science/paper/L2XCDRGP

@misc{pith2026250903747,
  author       = {Pith},
  title        = {Pith review of: Realization of Cohomology Classes in Grassmannians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L2XCDRGP}},
  note         = {Machine review of arXiv:2509.03747}
}
abstract

The study of irreducible subvarieties has recently seen a surge of interest due to connections with convex geometry. In this paper, we study cohomology classes of Grassmannians that are realizable by irreducible subvarieties. We completely classify the cohomology classes that can be realized by irreducible subvarieties in dimensions 2 and 3 and codimensions 2 and 3. We also classify all the classes that can be realizable by an irreducible subvariety for the Grassmannians $G(2,n)$ for $n \leq 6$ and $G(3,6)$. We classify the cohomology classes in all dimensions that are, up to positive multiple, realizable by an irreducible subvariety for the Grassmannians $G(2,n)$.

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