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Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes

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Pith's one-line read The paper proves that for a supercritical collection of free divisor classes on a smooth projective variety, the kernel of the induced Lefschetz operator on (1,1)-classes is spanned exactly by the prime divisors that the collection…

desk verdict The main theorem is likely true and a genuine advance, but the induction hinges on an unproved restriction lemma that needs to be supplied. read the letter →

arxiv 2505.18729 v1 pith:TMTOFDR2 submitted 2025-05-24 math.AG math.MG

Jiajun Hu , Jian Xiao This is my paper · ORCID
classification math.AGmath.MG MSC 14C2014C3014M2552A39
keywords hardLefschetzclassfreedivisorclassessupercriticalcollectionkernelofoperatorAlexandrov-FenchelinequalityKhovanskii-Teissiernulllocustoricvarieties
topics The Hodge Conjecture
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 proves that a natural numerical condition controls the kernel of the Lefschetz-type operators coming from products of divisor classes. On a smooth projective variety of dimension $n$, any collection of $n-2$ free divisor classes that is 'supercritical' defines a multiplication operator on $(1,1)$-classes whose kernel is spanned precisely by the prime divisors annihilated by the product. Equivalently, the collection is a hard Lefschetz class exactly when it kills no prime divisor. This settles the algebraic form of the open problem posed in [SvH23] in the free case, and it gives an algebro-geometric proof of the extremal characterization for the Alexandrov-Fenchel inequality for rational polytopes, along with a new extremal characterization for the Khovanskii-Teissier inequality.

What carries the argument

The load-bearing notion is the supercritical condition $\operatorname{nd}(\sum_{i\in I} L_i)\ge |I|+2$ for every nonempty $I\subseteq[n-2]$, which guarantees that the null locus $\operatorname{Null}_2(\mathbf{L})$ of codimension-two subvarieties killed by $\mathbf{L}$ is a proper Zariski closed set. The induction proceeds by cutting with a general $H\in |L_{n-2}|$; the asserted preservation of supercriticality under restriction lets the induction hypothesis express $\alpha|_H$ as a combination of prime divisors $E_i$ of $H$ killed by the restricted product. A homology-class lemma (Lemma 2.13, from [HHM+25]) identifies each $E_i$, which lies in a codimension-one component $W_i$ of the null locus, as a positive multiple of $L_{n-2}\cdot[W_i]$ in homology, so the combination can be pushed forward to $X$. The remaining class is disposed of by the threefold-based Proposition 3.4 for a single free class of numerical dimension at least three. Throughout, the Hall-Rado criterion for nef classes (Lemma 2.7, from [HX22]) and the Hodge-index/Lorentzian proportionality principle (Lemma 2.5) supply the numerical vanishing steps.

What would settle it

Construct a smooth projective fourfold (for instance toric) with a supercritical pair of free divisor classes $(L_1,L_2)$ and an $(1,1)$-class $\alpha$ killed by $L_1\cdot L_2$ but not representable as a real combination of prime divisors killed by $L_1\cdot L_2$; a toric computation with explicit intersection products would decide this directly, since nef bundles on smooth toric varieties are semiample.

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Extended reading notes

Core claim

Theorem A (Theorem 3.6) states: for a smooth projective variety $X$ of dimension $n$ and a supercritical collection $\mathbf{L} = (L_1,\ldots,L_{n-2})$ of free divisor classes, $\ker \mathbf{L} = \operatorname{span}_{\mathbb{R}}\{[D] : D \in \operatorname{Prime}(X),\ \mathbf{L}\cdot[D]=0\}$. In particular, $\mathbf{L}$ is a hard Lefschetz class if and only if $\mathbf{L}\cdot[D]\neq 0$ for every prime divisor $D$. The same conclusion holds when the classes are only semiample (Remark 1.2). Corollary A derives the equality case of the Alexandrov-Fenchel inequality for a supercritical collection of rational convex polytopes: equality holds exactly when the two polytopes share their supporting hyperplanes in all active normal directions. Corollary B characterizes equality in the Khovanskii-Teissier inequality for two free divisor classes by writing their difference as a combination of prime divisors annihilated by the remaining intersection numbers. The proof is inductive, with a threefold base case and a restriction-to-a-general-hypersurface step.

Load-bearing premise

The induction step assumes, without proof in the text, that restricting a supercritical collection of free divisor classes to a general member of the last linear system produces another supercritical collection; if this restriction property fails, the proof of the general case collapses.

Editorial extensions

If this is right

  • For any supercritical collection of free (or semiample) divisor classes, the kernel of the Lefschetz operator is a subspace of dimension at most the Picard number, and the annihilated prime divisors lie in the augmented base locus of a big class and span extremal rays of the pseudo-effective cone.
  • A supercritical free collection is a hard Lefschetz class precisely when no prime divisor is annihilated, giving a purely divisorial criterion for injectivity of the operator.
  • The Alexandrov-Fenchel equality for a supercritical collection of rational polytopes holds if and only if the two polytopes have identical supporting hyperplanes in every active normal direction, now proved by toric algebraic geometry.
  • The Khovanskii-Teissier equality for intersection numbers $(A^k\cdot B^{n-k})$ of two free divisor classes is characterized by $A-cB$ being a real combination of prime divisors $D_i$ with $A^{k-1}\cdot B^{n-k-1}\cdot[D_i]=0$.
  • When the relevant mixed volume or intersection number vanishes, equality holds automatically and the vanishing itself is characterized by a failure of numerical dimension conditions (Remark 4.9).

Reading between the lines

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

  • The induction step relies on the unproved assertion that supercriticality is preserved under restriction to a general member of the last linear system; verifying this directly for nef (not necessarily free) classes would extend the theorem to the full supercritical nef case and complete the original conjecture.
  • If the same kernel description holds for nef supercritical collections, the toric proof of the AF-extremal theorem would pass from rational polytopes to arbitrary convex bodies by approximation, matching the known polytopal result of [SvH23].
  • The restriction-and-lifting mechanism, especially the use of the general-fiber homology lemma, may be reusable for characterizing kernels of Lefschetz operators on higher-degree cohomology $H^{d,d}(X)$ for collections of $n-2d$ classes, where the kernel description is expected to involve 'degenerate' contributions beyond prime divisors.
  • A natural test is to compute, on a smooth toric fourfold with a supercritical pair of free divisors, whether every class killed by the product is numerically equivalent to a combination of killed torus-invariant divisors; the theorem says yes, and the calculation is purely combinatorial.
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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 the kernel of the Lefschetz-type operator given by cup product with L = L_1 · ... · L_{n-2}, where each L_i is a free divisor class on a smooth projective variety X of dimension n and the collection is supercritical in the sense that nd(L_I) ≥ |I| + 2 for every nonempty I. Theorem A (restated as Theorem 3.6) asserts that ker L is spanned by the classes of prime divisors D with L · [D] = 0, resolving the Shenfeld–van Handel question in this setting. The proof has a 3-fold baby case (Proposition 3.1), an intermediate statement for one free class of numerical dimension at least 3 (Proposition 3.4), and an induction on dimension in which a general member H ∈ |L_{n-2}| is used to reduce to a supercritical collection on H. The paper then derives applications to extremals of the Alexandrov–Fenchel inequality for rational polytopes (Theorem 4.5) and to extremals of the Khovanskii–Teissier inequality (Corollary 4.8).

Significance. If Theorem A is correct, it is a substantial advance: it extends the previous ordered-positivity result [HX23, Theorem A] to arbitrary supercritical collections and gives a complete algebraic characterization of the kernel in the free case, with a clean statement in terms of prime divisors. The toric application reproduces a deep convex-geometric theorem of Shenfeld–van Handel by algebro-geometric means, and the Khovanskii–Teissier application is a useful new equality case. The proof relies on standard tools (Hodge index, Hall–Rado for nef classes, Bertini, and a recent homology lemma of Huang–Huh–Michałek–Wang–Wang) and is mostly self-contained. However, the induction in Theorem 3.6 contains a load-bearing assertion about preservation of supercriticality under restriction that is not proved; this must be supplied before the main theorem can be considered established.

major comments (3)
  1. [§3.3, proof of Theorem 3.6, after Eq. (19)] The proof states: "As L is supercritical, the restriction collection (L_1|_H, ..., L_{n-3}|_H) is supercritical on the lower dimensional variety H." This is the pivot of the induction and is not immediate. Supercriticality on X gives nd_X(L_I) ≥ |I| + 2 for every I ⊂ [n-3], but the induction hypothesis on H needs nd_H(L_I|_H) ≥ |I| + 2, i.e. (L_I)^{|I|+2} · L_{n-2} > 0. The former alone gives only nd_H(L_I|_H) ≥ |I| + 1 in general; the missing unit must come from the mixed condition nd_X(L_I + L_{n-2}) ≥ |I| + 3 together with freeness. No argument connecting these facts is given. Please add a lemma (for instance, using the Hall–Rado criterion of Lemma 2.7 and the monotonicity nd(rL_I + L_{n-2}) ≥ nd(L_I + L_{n-2})) and verify the strict positivity, or restructure the induction so that this restriction property is built in.
  2. [§3.3, proof of Theorem 3.6, Eqs. (21)–(23)] The passage from E_i, a prime divisor of H with L_I|_H · [E_i]_H = 0, to a codimension-one component W_i of Null_2(L) with E_i = H ∩ W_i needs a fuller justification. From L · [E_i]_X = 0 one knows E_i ⊂ Null_2(L). Since H is chosen to meet every component of Null_2(L) transversally at a general point, E_i cannot itself be a component of Null_2(L), but the text does not explain why Null_2(L) must contain a component of dimension n-1 containing E_i, nor why the constant c_i in Eq. (22) is strictly positive for every irreducible component Z of H ∩ W. These facts are used to write [E_i]_X = c_i L_{n-2} · [W_i]_X and hence to transfer the equality from H to X; please make the dimensional and positivity transitions explicit, including the role of the closed union in Definition 2.8.
  3. [§4.2, proof of Corollary 4.8] In the case k ≥ 2, the proof asserts without comment that the collection {A^{(k-1)}, B^{(n-k-1)}} is supercritical once nd(A) ≥ k+1, nd(B) ≥ n-k+1 and nd(A+B) = n. This is true, but it uses the observation that for any subcollection with a' ≥ 1 copies of A and b' ≥ 1 copies of B, the class a'A + b'B = (A+B) + (a'-1)A + (b'-1)B is big, while boundary subcollections are handled by nd(A) ≥ k+1 and nd(B) ≥ n-k+1. Please include this one-line argument; as written, the supercriticality is asserted rather than demonstrated.
minor comments (5)
  1. [Title and abstract] The title contains a typo, "CHARACTERIZA TION" should read "CHARACTERIZATION".
  2. [§2.4, proof of Lemma 2.9] The phrase "there exits I" should be "there exists I".
  3. [§2.5, Lemma 2.13 and §3.3, condition (c)] In Theorem 3.6, condition (c) invokes Lemma 2.13 with W a component of Null_2(L); since the lemma is stated for an arbitrary prime divisor D in an ambient projective variety, the application is legitimate only because each W is a prime divisor of X and L_{n-2} is free. This should be said explicitly, and the constant c should be stated as positive rather than merely real, since the positivity is used later.
  4. [§3.1, proof of Proposition 3.1] The proof of the claim uses the decomposition of Lemma 2.3 and then adds finite movable classes to form a spanning set D_2; the sentence "we can add finite prime divisor classes and finite movable classes" is terse and could be clarified by explaining that every class in H^{1,1}(X) is a numerical combination of effective divisors and movable classes.
  5. [§4.1, proof of Theorem 4.5] When Remark 2.11 is invoked to conclude that each [D] in Eq. (30) spans an extremal ray and is torus-invariant, the text could note explicitly that an extremal ray of the pseudo-effective cone of a toric variety is generated by a torus-invariant divisor class; this is standard but is used for a nontrivial conclusion.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the kernel characterization is derived from Hodge-index induction and Hall-Rado tools, not from its own statement; the main flagged issue is an unproved restriction-supercriticality step, not a circular reduction.

full rationale

The central claim, Theorem A / Theorem 3.6, has the trivial inclusion V_{L,eff} ⊂ ker L and needs the reverse inclusion. The proof obtains the reverse inclusion by induction: the 3-fold baby case (Proposition 3.1) is proved with the Hodge index theorem, divisorial Zariski decomposition, and the intersection-matrix argument; Proposition 3.4 extends this by restriction to an ample hypersurface and the Lefschetz hyperplane theorem; Theorem 3.6 restricts to H ∈ |L_{n-2}| and uses the induction hypothesis plus Proposition 3.4. None of these steps defines ker L as the span of annihilated divisors, nor does it fit a parameter equal to the target statement. The cited [HX22] Hall-Rado criterion in Lemma 2.7 and [HX23, Proposition 5.6] in Remark 2.11 are same-author results, but they are general numerical-dimension and rigidity statements about nef classes; they are not restatements of the kernel characterization, and the paper does not assume the conclusion of Theorem A. The one substantive gap is the sentence in Theorem 3.6: "As L is supercritical, the restriction collection (L_1|_H, ..., L_{n-3}|_H) is supercritical on H." This is asserted without proof; supercriticality on H requires nd_H(L_I|_H) ≥ |I|+2, while the input nd_X(L_I) ≥ |I|+2 alone gives only nd_H(L_I|_H) ≥ |I|+1 in general. The missing increment must come from the mixed supercritical condition nd_X(L_I + L_{n-2}) ≥ |I|+3. This is an omitted verification, not a circular reduction: no equation makes the restricted supercriticality identical to the result being proved. The applications (Corollaries A and B) invoke Theorem A to deduce the extremal characterizations rather than importing the same statement as an input. Overall, the derivation is self-contained in its main logical structure, with no step that reduces to its own conclusion by construction.

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

The proof rests on previously established positivity and Hodge-theoretic tools plus one lemma from a 2025 preprint by other authors. No new free parameters or postulated entities are introduced.

assumptions (5)
  • standard math Hall-Rado theorem for nef classes (nd(L) >= m iff L != 0)
    Invoked in Lemma 2.7 and used in Lemma 2.9 and Theorem 3.6. Proved in the authors' previous work [HX22]; it is a general dimensionality criterion, not assuming the main theorem.
  • standard math Boucksom divisorial Zariski decomposition
    Lemma 2.3, used in Proposition 3.1 to control negative parts of pseudo-effective classes.
  • standard math Hodge index theorem and Hodge-Riemann relations in degree one
    Provides the Lorentzian signature used in Lemma 2.5 and Lemma 3.3, core to the baby case.
  • standard math Lemma 2.12 from [HHM+25]: components of general fibers are algebraically equivalent
    Used to prove Lemma 2.13, which supplies the homology relation for the intersection H cap W_i in the induction step.
  • standard math On smooth toric varieties every nef line bundle is semiample, and the Bernstein-Khovanskii-Kushnirenko theorem
    Used in Section 4 to transfer the kernel theorem to the Alexandrov-Fenchel inequality for rational polytopes.

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Pith. "Pith review of Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes." pith.science (2026). https://pith.science/paper/TMTOFDR2

@misc{pith2026250518729,
  author       = {Pith},
  title        = {Pith review of: Numerical characterization of the hard Lefschetz classes of dimension two, II: supercritical collections of free divisor classes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TMTOFDR2}},
  note         = {Machine review of arXiv:2505.18729}
}
abstract

For $(n-2)$ free divisor classes on a smooth projective variety of dimension $n$, the product of these free divisor classes induces a Lefschetz type operator acting on the N\'{e}ron-Severi space or the cohomology group of $(1,1)$ classes. We give a characterization of this kernel space, when the collection of these free divisor classes is supercritical. This resolves Shenfeld-van Handel's open problem in this setting. As consequences, we provide an algebro-geometric proof of the characterization of the extremals of the Alexandrov-Fenchel inequality for a supercritical collection of rational convex polytopes; we also give a characterization of the extremals of the Khovanskii-Teissier inequality given by the intersection numbers of two arbitrary free divisor classes.

Discussion (0). Continue with ORCID to comment.

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