arxiv: 2605.02129 · v1 · submitted 2026-05-04 · 🧮 math.AG · math.AT

Recognition: 3 theorem links

· Lean Theorem

From Diaz's Enriques Product to an n-Fold Cup-Product Bockstein Family of Integral Hodge Counterexamples

Abdul Rahman

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:51 UTC · model grok-4.3

classification 🧮 math.AG math.AT

keywords integral Hodge conjectureEnriques surfacesBockstein homomorphismcup productintegral Hodge classesalgebraic cyclesMacPherson-Vilonen obstructionfinite coefficients

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The pith

An n-fold cup-product Bockstein family produces non-algebraic 2-torsion integral Hodge classes on products of Enriques surfaces under a Brauer-separation hypothesis.

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

The paper reinterprets Diaz's four-dimensional counterexample to the integral Hodge conjecture as the n=2 case of a general construction. It forms the class Θ_n by cupping the K3 double-cover class from one Enriques factor with Brauer-detecting classes from the remaining factors on the product X_n of n Enriques surfaces, then takes the Bockstein to obtain the integral class Δ_n. The argument shows unconditionally that Δ_n has nonzero image in a distinguished Enriques-Brauer component of the MacPherson-Vilonen obstruction channel. Under the additional hypothesis that algebraic codimension-n cycles map to zero in that component, Δ_n supplies a non-algebraic 2-torsion integral Hodge class in every even dimension 2n.

Core claim

The central claim is that the Bockstein Δ_n of the cup-product class Θ_n = π_1^* α_1 ∪ ⋯ ∪ π_n^* β_n lies in H^{2n}(X_n, Z(n)) and maps nontrivially into the distinguished Enriques-Brauer component of the MV obstruction channel; once the Brauer-separation hypothesis is assumed, this implies that Δ_n is a non-algebraic 2-torsion integral Hodge class on the product of n Enriques surfaces.

What carries the argument

The n-fold cup-product Θ_n in finite-coefficient cohomology whose Bockstein Δ_n is detected by its nonzero image in the Enriques-Brauer component of the MacPherson-Vilonen obstruction channel.

If this is right

If the Brauer-separation hypothesis holds, then each product X_n carries a non-algebraic 2-torsion integral Hodge class in codimension n.
The nonzero image of Δ_n in the obstruction channel is proved unconditionally via external products of perverse sheaves, categorical Bockstein compatibility, and a Leibniz rule for the MV boundary.
The non-decomposable case of the separation hypothesis reduces via integral even Chow-Kunneth projectors to a coefficient-level algebraic-control problem on H^1(S_1, Z/2(1)).
A motivic finite-coefficient lift of the tower exists via the cone 1_X(n)/2.

Where Pith is reading between the lines

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

Similar iterated cup-product constructions may produce torsion Hodge counterexamples on other varieties whose Brauer groups or finite-coefficient cohomology carry suitable detecting classes.
Full verification of the separation hypothesis for non-decomposable cycles would complete the proof that these classes are genuine counterexamples in every dimension.
The recorded motivic lift suggests that the examples may interact with questions about the motivic Hodge conjecture or the algebraicity of classes in motivic cohomology.

Load-bearing premise

Algebraic codimension-n cycle classes have zero image in the distinguished Enriques-Brauer component of the MacPherson-Vilonen obstruction channel.

What would settle it

An explicit algebraic cycle on X_n whose image in the Enriques-Brauer component of the MV obstruction channel equals the image of Δ_n would show that Δ_n is algebraic.

read the original abstract

We reinterpret Diaz's construction of Chow-trivial smooth projective varieties violating the integral Hodge conjecture as the level-two case of an $n$-fold cup-product Bockstein mechanism. Diaz's dimension-four example is $V=S_1\times S_2$, where $S_1,S_2$ are Enriques surfaces, and its obstruction is the Bockstein of $\pi_1^*\alpha_1\cup\pi_2^*\beta_2\in H^3(V,\mathbb Z/2(2))$. Here $\alpha_1$ is the K3 double-cover class and $\beta_2$ is an Enriques Brauer-detecting class. We extend the finite-coefficient source construction to $X_n=S_1\times\cdots\times S_n$ by forming $\Theta_n=\pi_1^*\alpha_1\cup\pi_2^*\beta_2\cup\cdots\cup\pi_n^*\beta_n \in H^{2n-1}(X_n,\mathbb Z/2(n))$,with Bockstein $\Delta_n=\delta(\Theta_n)\in H^{2n}(X_n,\mathbb Z(n))$. Using external products of perverse sheaves, categorical Bockstein compatibility, and a Leibniz rule for the MacPherson--Vilonen boundary, we prove unconditionally that $\Delta_n$ has nonzero image in a distinguished Enriques--Brauer component of the MV obstruction channel. Under the Brauer-separation hypothesis, which asserts that algebraic codimension-$n$ cycle classes have zero image in this same component, the class $\Delta_n$ is a non-algebraic $2$-torsion integral Hodge class. We verify this separation for decomposable algebraic cycles and reduce the remaining non-decomposable case, via integral even Chow--K\"unneth projectors on the Enriques factors, to a single coefficient-level algebraic-control problem involving the $H^1(S_1,\mathbb Z/2(1))$ Enriques double-cover direction. We also record a motivic finite-coefficient lift of the tower via the finite-coefficient cone $\mathbf 1_X(n)/2:=\operatorname{Cone}(\mathbf 1_X(n)\xrightarrow{\times 2} \mathbf 1_X(n))$, and explain which formal part of the MacPherson--Vilonen zig-zag construction lifts motivically under Betti realization.

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

The n-fold cup-product extension of Diaz's example is cleanly executed with solid unconditional results on the obstruction image, but the non-algebraicity of Δ_n still rests on an unproven Brauer-separation hypothesis after reduction.

read the letter

The paper extends Diaz's construction of integral Hodge counterexamples from dimension four to arbitrary dimension by forming an n-fold product of Enriques surfaces and cupping the relevant classes. The new object is the class Δ_n, the Bockstein of Θ_n, and the claim is that it gives a 2-torsion non-algebraic integral Hodge class under a separation hypothesis. What the paper does well is the unconditional verification that Δ_n has nonzero image in the distinguished Enriques-Brauer component of the MV obstruction channel. This follows from external products of perverse sheaves, categorical Bockstein compatibility, and the Leibniz rule for the MacPherson-Vilonen boundary. They also check the property for decomposable algebraic cycles and record a motivic finite-coefficient lift using the cone of multiplication by two on the unit object. The soft spot is the treatment of non-decomposable cycles. The separation hypothesis is reduced, using Chow-Künneth projectors on the Enriques factors, to a coefficient-level problem on H^1(S_1, Z/2(1)). This reduced problem is stated but left open, so the main conclusion that Δ_n is non-algebraic remains conditional rather than proved. This paper is for algebraic geometers working on the integral Hodge conjecture and on motivic methods. The n-fold family organizes earlier examples into a tower and the motivic lift is a useful formal addition. The unconditional parts are solid and the reduction is clearly laid out, so the paper deserves peer review. Referees can assess whether the coefficient control problem can be solved or if the hypothesis needs to remain explicit.

Referee Report

2 major / 0 minor

Summary. The manuscript reinterprets Diaz's Enriques product construction as the n=2 case of an n-fold cup-product Bockstein family. On the product X_n of n Enriques surfaces, it defines Θ_n as the cup product of pullbacks of the K3 double-cover class and Enriques Brauer classes, takes its Bockstein Δ_n in integral cohomology, and shows unconditionally that Δ_n maps nontrivially to a distinguished Enriques-Brauer component of the MacPherson-Vilonen obstruction channel. It verifies that decomposable algebraic cycles map to zero in this component and reduces the separation question for non-decomposable cycles to a coefficient problem in H^1(S_1, Z/2(1)) using Chow-Künneth projectors. Under the resulting Brauer-separation hypothesis, Δ_n provides a non-algebraic 2-torsion integral Hodge class, yielding counterexamples to the integral Hodge conjecture in all dimensions.

Significance. If the Brauer-separation hypothesis is established, the work supplies an explicit infinite family of counterexamples to the integral Hodge conjecture in arbitrarily high dimensions, extending the dimension-4 example of Diaz. The unconditional proof that the image of Δ_n is nonzero in the obstruction channel, relying on external products of perverse sheaves, categorical Bockstein compatibility, and a Leibniz rule for the MacPherson-Vilonen boundary, constitutes a substantial technical advance. The motivic lift via the finite-coefficient cone and the partial motivic compatibility under Betti realization are also positive features. The conditional status of the main claim, however, limits the immediate impact until the separation hypothesis is resolved.

major comments (2)

Abstract and the section on the non-decomposable case: The Brauer-separation hypothesis is invoked to conclude that Δ_n is non-algebraic. Separation is verified for decomposable algebraic codimension-n cycles, but the reduction of the non-decomposable case via integral even Chow-Künneth projectors on the Enriques factors is stated as reducing the question to a single coefficient-level algebraic-control problem on the image in H^1(S_1, Z/2(1)) along the double-cover direction; this reduced problem is formulated but not solved or verified in the manuscript, leaving the hypothesis as an assumption rather than a theorem.
The unconditional proof of nonzero image (via external products of perverse sheaves, categorical Bockstein compatibility, and Leibniz rule for the MacPherson-Vilonen boundary) is load-bearing for the construction but is presented as holding independently of the separation hypothesis; the manuscript should explicitly cross-reference the precise statement of the distinguished Enriques-Brauer component in the MV obstruction channel to the relevant diagram or equation defining the obstruction map.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation of the technical contributions, and constructive suggestions. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: Abstract and the section on the non-decomposable case: The Brauer-separation hypothesis is invoked to conclude that Δ_n is non-algebraic. Separation is verified for decomposable algebraic codimension-n cycles, but the reduction of the non-decomposable case via integral even Chow-Künneth projectors on the Enriques factors is stated as reducing the question to a single coefficient-level algebraic-control problem on the image in H^1(S_1, Z/2(1)) along the double-cover direction; this reduced problem is formulated but not solved or verified in the manuscript, leaving the hypothesis as an assumption rather than a theorem.

Authors: The referee is correct that the manuscript presents the Brauer-separation hypothesis as an assumption rather than a fully resolved theorem. We verify separation explicitly for decomposable cycles and reduce the non-decomposable case, via the integral even Chow-Künneth projectors, to a coefficient-level question in H^1(S_1, Z/2(1)). This reduction is itself a contribution, as it isolates the remaining obstruction to a single, concrete algebraic-control problem. We will revise the abstract and the relevant section to state more explicitly that the reduced coefficient problem remains open and that the main theorem is therefore conditional on the hypothesis. No claim is made that the hypothesis has been proved in full. revision: partial
Referee: The unconditional proof of nonzero image (via external products of perverse sheaves, categorical Bockstein compatibility, and Leibniz rule for the MacPherson-Vilonen boundary) is load-bearing for the construction but is presented as holding independently of the separation hypothesis; the manuscript should explicitly cross-reference the precise statement of the distinguished Enriques-Brauer component in the MV obstruction channel to the relevant diagram or equation defining the obstruction map.

Authors: We agree that an explicit cross-reference will improve readability. The proof that the image of Δ_n is nonzero in the distinguished Enriques-Brauer component of the MacPherson-Vilonen obstruction channel is indeed unconditional and independent of the separation hypothesis; it relies on the external-product construction of perverse sheaves, categorical compatibility of the Bockstein, and the Leibniz rule for the boundary map. We will insert a direct cross-reference from the statement of the nonzero image to the diagram or equation that defines the obstruction map and identifies the Enriques-Brauer component. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external tools and an explicit open assumption

full rationale

The paper proves unconditionally (via external products of perverse sheaves, categorical Bockstein compatibility, and Leibniz rule for the MacPherson-Vilonen boundary) that Δ_n has nonzero image in the distinguished Enriques-Brauer component. It verifies separation for decomposable algebraic cycles and reduces the non-decomposable case, via integral even Chow-Künneth projectors, to an explicit coefficient-level control problem on H^1(S_1, Z/2(1)). The Brauer-separation hypothesis is stated as an assumption rather than derived or fitted; no equation or claim reduces the target class to a self-definition, a renamed input, or a load-bearing self-citation. The central result is therefore conditional on an independent open statement and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard cohomology and sheaf axioms plus one domain assumption about Chow-Kunneth projectors; the Brauer-separation hypothesis is treated as an external assumption rather than an axiom internal to the derivation.

axioms (2)

standard math Standard properties of cup products, Bockstein homomorphisms, external products of perverse sheaves, and the Leibniz rule for the MacPherson-Vilonen boundary.
Invoked to construct Θ_n and prove that its Bockstein has nonzero image in the Enriques-Brauer component.
domain assumption Existence of integral even Chow-Kunneth projectors on the Enriques factors.
Used to reduce the non-decomposable algebraic cycle case to a coefficient-level problem.

pith-pipeline@v0.9.0 · 5753 in / 1507 out tokens · 42831 ms · 2026-05-08T18:51:03.066057+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.

Foundation/AlexanderDuality.lean (D=3 forcing) alexander_duality_circle_linking unclear

?

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

Δ_n = δ(Θ_n) ∈ H^{2n}(X_n, Z(n)) ... non-algebraic 2-torsion integral Hodge class
Cost / FunctionalEquation (J-cost uniqueness) washburn_uniqueness_aczel unclear

?

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

coefficient sequence 0 → Z(p) →×2 Z(p) → Z/2(p) → 0 ... Bockstein δ

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

53 extracted references · 6 canonical work pages · 1 internal anchor

[1]

Beilinson, Alexander and Bernstein, Joseph and Deligne, Pierre , title =
[2]

Mathematische Annalen , volume =

Brieskorn, Egbert , title =. Mathematische Annalen , volume =. 1968 , pages =

1968
[3]

Inventiones Mathematicae , volume =

Goresky, Mark and MacPherson, Robert , title =. Inventiones Mathematicae , volume =. 1983 , pages =

1983
[4]

Grothendieck, Alexander , title =
[5]

Hamm, Helmut A. and L. Rectified homotopical depth and Grothendieck conjectures , booktitle =. 1991 , pages =

1991
[6]

Topology , volume =

Lamotke, Klaus , title =. Topology , volume =. 1981 , pages =

1981
[7]

Milnor, John , title =
[8]

Publications Math

Mumford, David , title =. Publications Math. 1961 , pages =

1961
[9]

2020 , eprint =

Saito, Morihiko , title =. 2020 , eprint =

2020
[10]

Publications of the Research Institute for Mathematical Sciences , volume =

Saito, Morihiko , title =. Publications of the Research Institute for Mathematical Sciences , volume =. 1990 , pages =

1990
[11]

Topology of Singular Spaces and Constructible Sheaves , series =

Sch. Topology of Singular Spaces and Constructible Sheaves , series =
[12]

arXiv preprint , year =

Jung, Seung-Jo and Saito, Morihiko , title =. arXiv preprint , year =. 2601.13151 , archivePrefix =

work page arXiv
[13]

Generalizations of intersection homology and perverse sheaves with duality over the integers

Greg Friedman , title =. Michigan Mathematical Journal , volume =. 2019 , pages =. 1612.09353 , archivePrefix =

work page arXiv 2019
[14]

Topological invariance of torsion-sensitive intersection homology

Greg Friedman , title =. Geometriae Dedicata , volume =. 2023 , pages =. 1907.07790 , archivePrefix =

work page arXiv 2023
[15]

Cohomologie non ramifi

Colliot-Th. Cohomologie non ramifi. Duke Mathematical Journal , volume =. 2012 , eprint =

2012
[16]

Commentarii Mathematici Helvetici , volume =

Benoist, Olivier and Ottem, John Christian , title =. Commentarii Mathematici Helvetici , volume =. 2020 , pages =

2020
[17]

Annales scientifiques de l'

Bloch, Spencer and Ogus, Arthur , title =. Annales scientifiques de l'. 1974 , pages =

1974
[18]

Commentarii Mathematici Helvetici , volume =

Goresky, Mark and Siegel, Paul , title =. Commentarii Mathematici Helvetici , volume =. 1983 , pages =

1983
[19]

Nikulin, V. V. , title =. Mathematics of the USSR-Izvestiya , volume =. 1980 , pages =

1980
[20]

Atiyah, M. F. and Hirzebruch, F. , title =. Topology , volume =. 1962 , pages =

1962
[21]

Journal of the American Mathematical Society , volume =

Totaro, Burt , title =. Journal of the American Mathematical Society , volume =. 1997 , pages =

1997
[22]

Torsion cohomology classes and algebraic cycles on complex projective manifolds , journal =

Soul. Torsion cohomology classes and algebraic cycles on complex projective manifolds , journal =. 2005 , pages =

2005
[23]

Moduli Spaces and Arithmetic Geometry , series =

Voisin, Claire , title =. Moduli Spaces and Arithmetic Geometry , series =. 2006 , pages =

2006
[24]

Journal of the Institute of Mathematics of Jussieu , volume =

Totaro, Burt , title =. Journal of the Institute of Mathematics of Jussieu , volume =. 2021 , pages =

2021
[25]

and Migliorini, Luca , title =

de Cataldo, Mark Andrea A. and Migliorini, Luca , title =. Annales scientifiques de l'. 2005 , pages =

2005
[26]

and Migliorini, Luca , title =

de Cataldo, Mark Andrea A. and Migliorini, Luca , title =. Bulletin of the American Mathematical Society , volume =. 2009 , pages =

2009
[27]

Journal of Algebra , volume =

Williamson, Geordie , title =. Journal of Algebra , volume =. 2017 , pages =

2017
[28]

and Shaneson, Julius L

Cappell, Sylvain E. and Shaneson, Julius L. , title =. Annals of Mathematics , volume =. 1991 , pages =

1991
[29]

Friedman, Greg , title =
[30]

Voisin, Claire , title =
[31]

Fulton, William , title =
[32]

2026 , eprint =

Abdul Rahman , title =. 2026 , eprint =

2026
[33]

Mathematische Zeitschrift , volume =

Kloosterman, Remke , title =. Mathematische Zeitschrift , volume =. 2022 , pages =

2022
[34]

American Journal of Mathematics , volume =

Artin, Michael , title =. American Journal of Mathematics , volume =. 1966 , pages =

1966
[35]

Publications Math

Lipman, Joseph , title =. Publications Math. 1969 , pages =

1969
[36]

, title =

Bredon, Glen E. , title =
[37]

Hatcher, Allen , title =
[38]

Herbert , title =

Clemens, C. Herbert , title =. Advances in Mathematics , volume =. 1983 , pages =

1983
[39]

Defect of a nodal hypersurface , journal =

Cynk, S. Defect of a nodal hypersurface , journal =. 2001 , pages =

2001
[40]

, title =

Milne, James S. , title =
[41]

Dix Expos

Grothendieck, Alexander , title =. Dix Expos. 1968 , pages =

1968
[42]

Inventiones Mathematicae , volume =

Brieskorn, Egbert , title =. Inventiones Mathematicae , volume =. 1966 , pages =

1966
[43]

and Raymond, Frank , title =

Neumann, Walter D. and Raymond, Frank , title =. Algebraic and Geometric Topology , series =. 1978 , pages =

1978
[44]

and Hulek, Klaus and Peters, Chris A

Barth, Wolf P. and Hulek, Klaus and Peters, Chris A. M. and Van de Ven, Antonius , title =
[45]

Beauville, Arnaud , title =
[46]

Zack and Schaffler, Luca , title =

Alexeev, Valery and Engel, Philip and Garza, D. Zack and Schaffler, Luca , title =. Nagoya Mathematical Journal , volume =. 2025 , pages =

2025
[47]

Torsion Trajectories from Local Discriminants to Global Obstructions

Abdul Rahman , title =. arXiv , primaryClass =:2605.00355 , year =

work page internal anchor Pith review Pith/arXiv arXiv
[48]

arXiv preprint , eprint =

Tubach, Swann , title =. arXiv preprint , eprint =
[49]

arXiv preprint , eprint =

Ruimy, Rapha. arXiv preprint , eprint =
[50]

Inventiones Mathematicae , volume =

MacPherson, Robert and Vilonen, Kari , title =. Inventiones Mathematicae , volume =
[51]

Ukrainian Mathematical Journal , volume =

Lyubashenko, Volodymyr , title =. Ukrainian Mathematical Journal , volume =. 2001 , eprint =

2001
[52]

Applied Categorical Structures , volume =

Positselski, Leonid , title =. Applied Categorical Structures , volume =. arXiv:1404.5011v6 , archivePrefix =

work page arXiv
[53]

, title =

Diaz, Humberto A. , title =. Mathematical Proceedings of the Cambridge Philosophical Society , volume =. 2204.13818 , archivePrefix =

work page arXiv