Generalized Bogomolov Inequalities

Julius Ross; Matei Toma; Mihai Pavel

arxiv: 2510.04663 · v2 · submitted 2025-10-06 · 🧮 math.AG

Generalized Bogomolov Inequalities

Mihai Pavel , Julius Ross , Matei Toma This is my paper

Pith reviewed 2026-05-18 09:38 UTC · model grok-4.3

classification 🧮 math.AG

keywords Hodge-Riemann pairsBogomolov pairssemistable sheavesboundednesscohomology classesstabilityalgebraic geometry

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The pith

Hodge-Riemann pairs of cohomology classes are conjectured to satisfy a generalized Bogomolov inequality, with several cases proven.

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

The paper defines Hodge-Riemann pairs of cohomology classes to extend the classical Hodge-Riemann bilinear relations and Bogomolov pairs to extend the Bogomolov inequality for semistable sheaves. It conjectures that every Hodge-Riemann pair is a Bogomolov pair and proves the statement in multiple specific cases. These proofs yield new boundedness results for families of semistable sheaves. A reader would care because the classical versions have long been central to controlling positivity and stability in algebraic geometry, and the generalizations could apply the same controls more broadly. The work focuses on pairs of classes rather than single classes or sheaves alone.

Core claim

We introduce the notion of a Hodge-Riemann pair of cohomology classes that generalizes the classical Hodge-Riemann bilinear relations, and the notion of a Bogomolov pair of cohomology classes that generalizes the Bogomolov inequality for semistable sheaves. We conjecture that every Hodge-Riemann pair is a Bogomolov pair, and prove various cases of this conjecture. As an application we get new results concerning boundedness of semistable sheaves.

What carries the argument

Hodge-Riemann pair and Bogomolov pair, defined as conditions on pairs of cohomology classes that extend classical bilinear relations and inequalities to wider settings.

If this is right

New boundedness results hold for semistable sheaves in the cases where the conjecture is proven.
Explicit instances of the generalized Bogomolov inequality are established for certain pairs of cohomology classes.
The approach produces fresh control on moduli spaces of sheaves through the proven cases.
The definitions allow the classical relations and inequalities to be applied to broader classes of objects.

Where Pith is reading between the lines

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

If the conjecture holds in full generality it could supply a uniform way to derive stability bounds from positivity data in higher-dimensional settings.
The pairs might be checked on concrete examples such as hypersurfaces or toric varieties to locate additional proven cases.
The framework could link to other positivity conditions that appear in the study of Kähler classes or Chern characters.

Load-bearing premise

The newly defined Hodge-Riemann pairs and Bogomolov pairs faithfully capture the essential features of the classical Hodge-Riemann bilinear relations and Bogomolov inequality for semistable sheaves.

What would settle it

Constructing an explicit Hodge-Riemann pair of cohomology classes on a projective variety that fails to satisfy the Bogomolov pair condition would disprove the conjecture.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

This paper defines Hodge-Riemann and Bogomolov pairs of cohomology classes, conjectures one implies the other, proves several cases, and extracts new boundedness results for semistable sheaves.

read the letter

The core contribution is the introduction of Hodge-Riemann pairs and Bogomolov pairs as abstractions of the classical Hodge-Riemann relations and Bogomolov inequality. The authors conjecture that the first kind always gives the second and establish this in multiple cases, which then yields fresh boundedness statements for semistable sheaves on varieties. That is the main takeaway for someone scanning the abstract and setup. They handle the definitions cleanly enough that the classical statements drop out as special cases, and the partial proofs plus the boundedness applications show they have worked through enough details to make the framework operational rather than purely formal. The approach avoids obvious circularity by building directly on standard cohomology and stability background. The main limitation is that the central conjecture stays open outside the cases they treat, so the full strength of the generalization is not yet settled. How closely the new pair notions track the original geometric content will also need checking in further examples, though the paper does not overclaim this. Readers working on moduli spaces of sheaves, stability conditions, or extensions of classical inequalities in algebraic geometry will find the most direct value. The combination of new definitions, partial results, and concrete applications is solid enough to merit a serious referee rather than a desk rejection. I would recommend sending it out for review.

Referee Report

1 major / 2 minor

Summary. The paper introduces the notions of a Hodge-Riemann pair and a Bogomolov pair of cohomology classes, which generalize the classical Hodge-Riemann bilinear relations and the Bogomolov inequality for semistable sheaves, respectively. It conjectures that every Hodge-Riemann pair is a Bogomolov pair, proves various cases of this conjecture, and derives new boundedness results for semistable sheaves as an application.

Significance. If the proved cases of the conjecture are robust and the new boundedness statements are non-trivial, the framework could unify disparate inequalities in algebraic geometry and facilitate further applications to moduli problems. The abstract treatment via pairs offers a potential route to parameter-free or axiomatic derivations in settings where classical Hodge-Riemann relations apply.

major comments (1)

The central conjecture relies on the new definitions faithfully generalizing the classical notions; however, without explicit verification that the Hodge-Riemann pair axioms imply the classical bilinear relations in the standard case (e.g., for the cohomology of a projective manifold), it is unclear whether the proved cases cover the load-bearing instances of the original Bogomolov inequality.

minor comments (2)

Clarify the precise statements of the proved cases of the conjecture, including any restrictions on the underlying variety or sheaf, to make the scope of the boundedness applications explicit.
Add a brief comparison table or diagram contrasting the classical Hodge-Riemann relations with the generalized pair axioms for readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. We address the major comment below.

read point-by-point responses

Referee: The central conjecture relies on the new definitions faithfully generalizing the classical notions; however, without explicit verification that the Hodge-Riemann pair axioms imply the classical bilinear relations in the standard case (e.g., for the cohomology of a projective manifold), it is unclear whether the proved cases cover the load-bearing instances of the original Bogomolov inequality.

Authors: We agree that making the specialization to the classical setting fully explicit will strengthen the paper and clarify that the proved cases include the standard Bogomolov inequality. In the revised manuscript we will add a short subsection (or a detailed remark in Section 2) verifying that the axioms of a Hodge-Riemann pair reduce precisely to the classical Hodge-Riemann bilinear relations when the cohomology classes are taken to be the Kähler class and the Chern classes of a semistable sheaf on a projective manifold. This verification will be carried out in the standard setting of a smooth projective variety over the complex numbers, confirming that our framework recovers the classical statements. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces new definitions of Hodge-Riemann pairs and Bogomolov pairs as generalizations of classical Hodge-Riemann relations and the Bogomolov inequality. It explicitly states a conjecture that every Hodge-Riemann pair is a Bogomolov pair, proves various cases of the conjecture, and derives applications to boundedness of semistable sheaves from those proved cases. No load-bearing step reduces by construction to its inputs via self-definition, fitted parameters renamed as predictions, or self-citation chains. The argument builds on classical background with new notions and partial proofs, remaining self-contained without any quoted reduction of results to prior inputs or definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The paper introduces two new mathematical definitions (Hodge-Riemann pair and Bogomolov pair) and relies on standard background from Hodge theory and sheaf stability. No free parameters or invented physical entities are indicated in the abstract.

axioms (1)

standard math Classical Hodge-Riemann bilinear relations and Bogomolov inequality for semistable sheaves hold in the usual settings.
The generalizations are built directly on these classical statements.

invented entities (2)

Hodge-Riemann pair no independent evidence
purpose: Generalize the classical Hodge-Riemann bilinear relations to pairs of cohomology classes
New definition introduced to extend the classical notion.
Bogomolov pair no independent evidence
purpose: Generalize the Bogomolov inequality for semistable sheaves to pairs of cohomology classes
New definition introduced to extend the classical inequality.

pith-pipeline@v0.9.0 · 5586 in / 1306 out tokens · 27598 ms · 2026-05-18T09:38:10.267001+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1.1 (Hodge-Riemann pairs... Z α·η^{d-1}=0 ⇒ Z α²·η^{d-2}≤0 with equality iff α=0); Conjecture 1.4 (Hodge-Riemann pair implies Bogomolov inequality for η^{d-1}-semistable sheaves)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 3.14 and Theorem 5.1 (Schur polynomials of Kähler classes yield Hodge-Riemann/Bogomolov pairs)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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[Car13] S. A. H. Cardona, Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures, Ann. Global Anal. Geom. 44 (2013), no. 4, 455–469. [Che25] Xuemiao Chen, Admissible Hermitian-Yang-Mills connections over normal varieties, Math. Ann. 392 (2025), no. 1, 487–523. [CRŞ19] Ionuţ Chiose, Rare...

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[DN06] Tien-Cuong Dinh and Viêt-Anh Nguyên, The mixed Hodge-Riemann bilinear relations for compact Kähler manifolds, Geom. Funct. Anal.16 (2006), no. 4, 838–849. [DP04] Jean-Pierre Demailly and Mihai Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2)159 (2004), no. 3, 1247–1274. [FL83] William Fulton and Ro...

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[FL17] Mihai Fulger and Brian Lehmann, Positive cones of dual cycle classes, Algebr. Geom. 4 (2017), no. 1, 1–28. [Gie79] D. Gieseker, On a theorem of Bogomolov on Chern classes of stable bundles, Am. J. Math. 101 (1979), 77–85. [GKP16] Daniel Greb, Stefan Kebekus, and Thomas Peternell,Movable curves and semistable sheaves, Int. Math. Res. Not. IMRN (2016...

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Blaine Lawson, Jr., An intrinsic characterization of Kähler man- ifolds, Invent

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[Joy21] Dominic Joyce, Enumerative invariants and wall-crossing formulae in abelian cate- gories, 2021, arXiv:2111.04694. [Kle66] Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2)84 (1966), 293–344. [Kob87] Shoshichi Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vo...

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[Lam99] A. Lamari, Le cône kählérien d’une surface, J. Math. Pures Appl. (9)78 (1999), no. 3, 249–263. [Lan04] Adrian Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2)159 (2004), 251–276. [Laz04] Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete

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[LY87] Jun Li and Shing Tung Yau, Hermitian-yang-mills connection on non-Kähler mani- folds, Mathematical aspects of string theory, Proc. Conf., San Diego/Calif. 1986, Adv. Ser. Math. Phys. 1, 560-573,

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[MPT25] Damien Mégy, Mihai Pavel, and Matei Toma,Semistability conditions defined by ample classes, Geom Dedicata219 (2025), no

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[NZ18] YanciNieandXiZhang, Semistable Higgs bundles over compact Gauduchon manifolds, J. Geom. Anal.28 (2018), no. 1, 627–642. [Pav24] Mihai Pavel, Moduli spaces of slope-semistable pure sheaves, Ann. Inst. Fourier 74 (2024), no. 5, 2141–2186. [PRT25] Mihai Pavel, Julius Ross, and Matei Toma,Uniform boundedness of semistable pure sheaves on smooth project...

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Stud.54 (1968), no

[Ros68] Hugo Rossi, Picard variety of an isolated singular point, Rice Univ. Stud.54 (1968), no. 4, 63–73. [RT23a] Julius Ross and Matei Toma, Hodge-Riemann bilinear relations for Schur classes of ample vector bundles, Ann. Sci. Éc. Norm. Supér. (4)56 (2023), no. 1, 197–241. [RT23b] , Hodge-Riemann relations for Schur classes in the linear and Kähler case...

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[1] [1]

Math.61 (1987), 181–200

[AA87] Lucia Alessandrini and Marco Andreatta, Closed transverse (p,p)-forms on compact complex manifolds, Compos. Math.61 (1987), 181–200. [Ale18] Lucia Alessandrini, Forms and currents defining generalizedp-Kähler structures, Abh. Math. Semin. Univ. Hamb.88 (2018), no. 1, 217–245. [BDPP13] Sébastien Boucksom, Jean-Pierre Demailly, Mihai Păun, and Thomas...

work page 1987

[2] [2]

[Car13] S. A. H. Cardona, Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures, Ann. Global Anal. Geom. 44 (2013), no. 4, 455–469. [Che25] Xuemiao Chen, Admissible Hermitian-Yang-Mills connections over normal varieties, Math. Ann. 392 (2025), no. 1, 487–523. [CRŞ19] Ionuţ Chiose, Rare...

work page 2013

[3] [3]

[DN06] Tien-Cuong Dinh and Viêt-Anh Nguyên, The mixed Hodge-Riemann bilinear relations for compact Kähler manifolds, Geom. Funct. Anal.16 (2006), no. 4, 838–849. [DP04] Jean-Pierre Demailly and Mihai Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2)159 (2004), no. 3, 1247–1274. [FL83] William Fulton and Ro...

work page 2006

[4] [4]

[FL17] Mihai Fulger and Brian Lehmann, Positive cones of dual cycle classes, Algebr. Geom. 4 (2017), no. 1, 1–28. [Gie79] D. Gieseker, On a theorem of Bogomolov on Chern classes of stable bundles, Am. J. Math. 101 (1979), 77–85. [GKP16] Daniel Greb, Stefan Kebekus, and Thomas Peternell,Movable curves and semistable sheaves, Int. Math. Res. Not. IMRN (2016...

work page 2017

[5] [5]

Blaine Lawson, Jr., An intrinsic characterization of Kähler man- ifolds, Invent

[HL83] Reese Harvey and H. Blaine Lawson, Jr., An intrinsic characterization of Kähler man- ifolds, Invent. Math.74 (1983), no. 2, 169–198. [HL10] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, seconded., CambridgeMathematicalLibrary, CambridgeUniversityPress, Cambridge,

work page 1983

[6] [6]

2021), arXiv: 2111.04694v1

[Joy21] Dominic Joyce, Enumerative invariants and wall-crossing formulae in abelian cate- gories, 2021, arXiv:2111.04694. [Kle66] Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2)84 (1966), 293–344. [Kob87] Shoshichi Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vo...

work page arXiv 2021

[7] [7]

Lamari, Le cône kählérien d’une surface, J

[Lam99] A. Lamari, Le cône kählérien d’une surface, J. Math. Pures Appl. (9)78 (1999), no. 3, 249–263. [Lan04] Adrian Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2)159 (2004), 251–276. [Laz04] Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete

work page 1999

[8] [8]

A Series of Modern Surveys in Mathematics [Re- sults in Mathematics and Related Areas

Folge. A Series of Modern Surveys in Mathematics [Re- sults in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004, Positivity for vector bundles, and multiplier ideals. [LT95] Martin Lübke and Andrei Teleman, The Kobayashi-Hitchin correspondence, World Scientific Publishing Co., In...

work page 2004

[9] [9]

Conf., San Diego/Calif

[LY87] Jun Li and Shing Tung Yau, Hermitian-yang-mills connection on non-Kähler mani- folds, Mathematical aspects of string theory, Proc. Conf., San Diego/Calif. 1986, Adv. Ser. Math. Phys. 1, 560-573,

work page 1986

[10] [10]

[MPT25] Damien Mégy, Mihai Pavel, and Matei Toma,Semistability conditions defined by ample classes, Geom Dedicata219 (2025), no

work page 2025

[11] [11]

[NZ18] YanciNieandXiZhang, Semistable Higgs bundles over compact Gauduchon manifolds, J. Geom. Anal.28 (2018), no. 1, 627–642. [Pav24] Mihai Pavel, Moduli spaces of slope-semistable pure sheaves, Ann. Inst. Fourier 74 (2024), no. 5, 2141–2186. [PRT25] Mihai Pavel, Julius Ross, and Matei Toma,Uniform boundedness of semistable pure sheaves on smooth project...

work page 2018

[12] [12]

Stud.54 (1968), no

[Ros68] Hugo Rossi, Picard variety of an isolated singular point, Rice Univ. Stud.54 (1968), no. 4, 63–73. [RT23a] Julius Ross and Matei Toma, Hodge-Riemann bilinear relations for Schur classes of ample vector bundles, Ann. Sci. Éc. Norm. Supér. (4)56 (2023), no. 1, 197–241. [RT23b] , Hodge-Riemann relations for Schur classes in the linear and Kähler case...

work page 1968