Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds

Eyal Markman

Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds

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Pith's one-line read The Hodge-Weil classes of polarized abelian sixfolds of Weil type with discriminant -1 are algebraic, for every imaginary quadratic field K.

desk verdict A landmark proof of the Hodge conjecture for abelian fourfolds, via algebraicity of Weil classes on sixfolds—long and technical, but with the main delicate step explicitly addressed. read the letter →

arxiv 2502.03415 v2 pith:QDF7UA4Z submitted 2025-02-05 math.AG

Eyal Markman This is my paper · ORCID

classification math.AG MSC 14C2514C3014K22

keywords HodgeconjectureWeilclassesabelianvarietiesoftypesecantsheavesspinorvarietysemiregularityderivedequivalencealgebraiccycles

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 the rational Hodge classes of Weil type on certain abelian sixfolds are algebraic: for every imaginary quadratic field $K$, every polarized abelian sixfold of Weil type with complex multiplication by $K$ and discriminant $-1$ has its two-dimensional space of Hodge-Weil classes spanned by classes of algebraic cycles. The proof constructs, from ideal sheaves of translates of the Abel-Jacobi curve in a genus-3 Jacobian, a reflexive rank-$8d$ sheaf $E$ on $X \times \widehat{X}$ whose characteristic class $\kappa(E)=\exp(-c_1(E)/\mathrm{rank}(E))\,\mathrm{ch}(E)$ is invariant under the subgroup of $\mathrm{Spin}(V)$ fixing a secant plane $P$. Because $\mathrm{Spin}(V)_P$-invariant classes remain of Hodge type in every deformation of the polarized Weil-type structure, $\kappa(E)$ stays Hodge type throughout the 9-dimensional moduli space. A semiregularity theorem for twisted sheaves on abelian varieties then shows the sheaf deforms with the family, and the deformed sheaf's Chern classes realize the Hodge-Weil classes as algebraic classes. A degeneration argument of Schoen then yields the Hodge conjecture for all abelian fourfolds.

What carries the argument

The load-bearing objects are $K$-secant sheaves: coherent sheaves whose Chern characters lie in a rational plane $P$ cutting the even spinor variety in two complex-conjugate pure spinors, which endows $X\times\widehat{X}$ with the structure of a polarized abelian variety of Weil type. The argument runs through Orlov's derived equivalence $\Phi:D^b(X\times X)\to D^b(X\times\widehat{X})$, Chevalley's isomorphism $S\otimes S\cong\wedge^*V$ identifying tensor squares of pure spinors with top exterior powers of maximal isotropic subspaces, and the normalized characteristic class $\kappa(E)=\exp(-c_1(E)/\mathrm{rank}(E))\,\mathrm{ch}(E)$, which remains invariant under the stabilizer $\mathrm{Spin}(V)_P$. The deformation step is carried by a semiregularity theorem for $\mu_r$-twisted sheaves on abelian varieties, proved in Section 7.4 by reducing to the untwisted Buchweitz-Flenner theorem via a projective-bundle construction.

What would settle it

Take the twisted reflexive sheaf $B$ of Section 9.3 and compute its semiregularity map $\sigma_B: \mathrm{Ext}^2(B,B)\to\bigoplus_{q=0}^{2}H^{q+2}(Y,\Omega^q_Y)$. If any non-zero class lies in the kernel, the semiregularity step fails and the deformation argument for Theorem 1.5.1 collapses. Alternatively, exhibit a single polarized abelian sixfold of Weil type of discriminant $-1$ whose Hodge-Weil classes are not algebraic, which would contradict the theorem directly.

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Extended reading notes

Core claim

The central claim is Theorem 1.5.1: the Hodge-Weil classes of polarized abelian sixfolds of Weil type with complex multiplication by $K$ and discriminant $-1$ are algebraic, for every imaginary quadratic field $K$. The proof shows that, starting from a $Q(\sqrt{-d})$-secant plane $P$ spanned by the Chern characters of two ideal sheaves on a genus-3 Jacobian $X$, Orlov's derived equivalence $\Phi:D^b(X\times X)\to D^b(X\times\widehat{X})$ produces an object whose dual $E$ is a simple reflexive sheaf of rank $8d$. The characteristic class $\kappa(E)$ is $\mathrm{Spin}(V)_P$-invariant and therefore remains of Hodge type under every deformation of $X\times\widehat{X}$ as a polarized abelian sixfold of Weil type. The paper proves a semiregularity theorem for $\mu_r$-twisted sheaves on abelian varieties, applies it to a twisted reflexive sheaf descended from $E$, and concludes that $E$ deforms locally over the 9-dimensional moduli space. The deformed sheaves carry the algebraic cycles whose classes span the Hodge-Weil subspace, and Corollary 1.6.1 then states that the Hodge conjecture holds for all abelian fourfolds.

Load-bearing premise

The proof collapses if the semiregularity theorem for $\mu_r$-twisted sheaves on abelian varieties, proved in Section 7.4, has a gap, because without it the reflexive sheaf $E$ cannot be shown to deform with the Weil-type family and no algebraic cycles are produced.

Editorial extensions

If this is right

Theorem 1.5.1 implies that the Hodge-Weil classes are algebraic for every polarized abelian sixfold of Weil type with complex multiplication by any imaginary quadratic field $K$ and discriminant $-1$.
By Schoen's degeneration argument, the same algebraicity then holds for all abelian fourfolds of Weil type, for all imaginary quadratic fields and all discriminants.
The Hodge conjecture for abelian fourfolds follows, since for simple fourfolds the Hodge ring is generated by divisor classes and Hodge-Weil classes, and non-simple fourfolds reduce to known cases.
The deformation theory shows that the normalized characteristic class $\kappa(E)$, rather than the full Chern character, is the quantity that stays of Hodge type under Weil-type deformations and is therefore the correct invariant to track.
The construction yields a concrete reflexive sheaf on $X\times\widehat{X}$ whose deformations parametrize algebraic representatives of the Hodge-Weil classes.

Reading between the lines

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

The equality $\ker(\mathrm{ob}_F)=\ker(\mathrm{ch}(F))$ established for secant sheaves suggests a general principle for abelian varieties: first-order deformability of a secant object is controlled entirely by whether its Chern character stays inside the secant plane, which could simplify deformation arguments in other dimensions.
A proof of Conjecture 7.3.9 in full generality would remove the restriction to families of abelian varieties and could extend the deformation step to other moduli components or to non-abelian base spaces, potentially covering discriminants beyond $-1$.
The normalized characteristic class $\kappa(E)$ being $\mathrm{Spin}(V)_P$-invariant is a weaker condition than full Chern-character invariance; the same normalization may be useful in other settings where a sheaf's Chern character is not fixed along a family.
The paper notes that analogous secant-sheaf constructions exist for Jacobians of higher-genus curves, but semiregularity is proved only in genus 3; testing the same semiregularity computation in higher genus is the natural next step for extending the method beyond sixfolds.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Eyal, the two things to know: this paper proves the Hodge-Weil classes are algebraic for all abelian sixfolds of Weil type of discriminant -1 over every imaginary quadratic field, and derives the Hodge conjecture for all abelian fourfolds. That is a major result, arguably the biggest progress on the Hodge conjecture for abelian varieties in a long time. The reader's take is accurate; I agree with the ACCEPT verdict at moderate confidence.

What is genuinely new: the construction of secant sheaves from ideal sheaves of translates of the Abel-Jacobi curve in a Jacobian of genus 3, Orlov's derived equivalence to produce a sheaf on X × X-hat, the Spin(V)_P-invariance of its characteristic class, and the deformation argument that makes the Hodge-Weil classes algebraic. Prior work covered only Q(i), Q(sqrt(-3)), and special discriminants. This paper handles all imaginary quadratic fields, and the degeneration to fourfolds gives the full Hodge conjecture there. That is not a rehash.

The paper is also honest about its main risk. The general semiregularity conjecture for twisted sheaves (7.3.9) is only asserted to follow from Pridham, with the author saying he cannot reconcile the statements. But the needed abelian case is proved in Section 7.4.2 via a reduction to Buchweitz-Flenner. I looked at that reduction; the diagram chase in Step 4 and the global extension in Lemma 7.4.1 are delicate, but no gap is apparent. The other load-bearing assumption, the general position used to prove reflexivity, is shown to hold generically, so that is not a real concern.

The residual risk is simply the length and depth of the chain—no one can machine-check all of it, and the reader's moderate confidence is fair. But the central argument holds up as far as I can tell, and the paper flags its own limitations rather than hiding them.

Who gets value from this: anyone working on the Hodge conjecture, abelian varieties, derived categories, or semiregularity. It deserves a serious referee—not a desk rejection. I would send it to a specialist who can check the technical sections in detail. My own verdict is that it should go to peer review with the expectation of acceptance after whatever tightening the referee asks for.

Referee Report

0 major / 5 minor

Summary. The paper develops a general framework associating polarized abelian varieties of Weil type to rational K-secant lines in the even cohomology of an abelian n-fold, via the spin representation and Orlov's derived equivalence. The central new results are: (i) a Spin(V)_P-invariance statement for the characteristic class of secant products under deformations of the Weil-type structure; (ii) a proof, for families of abelian varieties, of a twisted version of the Buchweitz-Flenner semiregularity theorem; and (iii) an explicit construction over Jacobians of genus-3 curves of reflexive sheaves whose induced classes deform with the full 9-dimensional moduli of abelian sixfolds of Weil type of discriminant -1. From this the author derives the algebraicity of Hodge-Weil classes for abelian sixfolds of Weil type of discriminant -1 for every imaginary quadratic field, and then, using degeneration and known results, the Hodge conjecture for abelian fourfolds.

Significance. If correct, this is a landmark result: Corollary 1.6.1 would settle the Hodge conjecture for all abelian fourfolds, a problem that has been open in this generality. The paper's strategy is original and substantial: it converts the representation-theoretic Hodge-Weil classes into Chern characters of honest coherent sheaves (up to a twisted descent), proves the needed semiregularity theorem in the abelian case rather than relying on the unverified general conjecture, and gives a very explicit construction for every d and every imaginary quadratic field K. The Spin-invariance argument is essentially parameter-free, and the obstruction-theoretic core (rank-6 obstruction map and 9-dimensional unobstructed subspace) is concrete and checkable. The main limitation is that several long technical verifications are compressed, most notably the semiregularity diagram in Section 7.4.2 and the simultaneous general-position and equivariance requirements in Section 9; these deserve expansion, but I did not find a demonstrated gap.

minor comments (5)

[§7.4.2, Step 4 (diagram p. 48)] The proof that the pullback of a semiregular twisted sheaf to P0 becomes a semiregular untwisted sheaf rests on the displayed commutative diagram, but the commutativity of the right square is asserted to follow from Equation (7.3.5) without the intermediate computation. Since this is the load-bearing step for the twisted semiregularity theorem, please include the explicit Atiyah-class calculation showing how σ_{\tilde E0} is obtained from σ_B after cup product with exp(λ/r), and justify the exponent λ/r rather than λ.
[§1.5 and §7.3 (Conjecture 7.3.9)] The paper honestly states that the general twisted semiregularity conjecture is not fully reconciled with Pridham's theorem and that the author cannot currently compare the two statements. Since the later argument only uses the abelian-variety case proved in Section 7.4, it would be useful to state explicitly at the end of Section 7.4 and again in the proof of Theorem 1.5.1 that Conjecture 7.3.9 itself is not being invoked, only Theorem 7.4.2.
[§9.1–§9.2 (Assumptions 9.1.1 and 9.2.1)] Assumption 9.1.1 is shown to hold generically and, for cyclic orbits, in Lemmas 9.1.2 and 9.1.4. It would be helpful to have an equally explicit statement for Assumption 9.2.1(1)–(2): I could not locate a lemma proving that the fourfold-empty/triple-dimensional intersection conditions and the distinctness of the points tj+si can all be imposed simultaneously with the G1×G2-equivariance needed in Section 9.3. A one-paragraph verification or a pointer to the precise genericity argument would remove this uncertainty.
[§8.1, §8.3 (minor typos)] There are a few typographical errors, e.g., 'polazixed' for 'polarized' at the start of Section 8.1 and 'trhe' for 'the' in Section 8.3; these should be corrected.
[§9.2 (Lemma 9.2.4)] The proof of Lemma 9.2.4 is long and structured into steps; moving the step-by-step verification to an appendix, or at least summarizing the role of each step before the computation, would improve readability without changing the mathematics.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the sixfold algebraicity theorem is established by an independent sheaf construction; the only self-citations are auxiliary and non-load-bearing.

full rationale

The central claim, Theorem 1.5.1, does not reduce by construction to its inputs. Hodge-Weil classes are defined representation-theoretically (Section 1.1; Lemma 2.2.7) and are shown to be algebraic by constructing a reflexive sheaf E on X×X̂ whose characteristic class κ(E) is Spin(V)_P-invariant (Corollary 1.3.2), spans the Hodge-Weil plane together with h^3 (Lemma 8.3.1), and deforms with every polarized abelian sixfold of Weil type of discriminant -1 via the twisted semiregularity theorem proved for abelian families in Section 7.4. That semiregularity result is not borrowed as a black box: the paper proves Conjecture 7.3.9 in the needed abelian case by reduction to the Buchweitz-Flenner theorem, and the author explicitly flags the general conjecture as only expected from Pridham's work ('The author’s ignorance prevents him from comparing the two statements...'), which is a limitation in the exposition but not a circular input to Theorem 1.5.1. The main self-citation is in Lemma 6.2.5, where an abelian-surface Chern-character computation is taken from [M2, Prop. 11.2] to prove the base case of Proposition 6.1.2. This is a prior, independent result about fourfolds of discriminant 1, not an assumption of the sixfold theorem, and the rest of Proposition 6.1.2 is proved by a group-generation argument. The Hodge-conjecture corollary invokes external degeneration and classification results ([S2], [MZ1], [MZ3], [R]) rather than the paper's own conclusions. No fitted parameter is renamed as a prediction, and no definition is shown to encode the target class. The residual risk identified by the reader is technical (the validity of the Step 4 diagram and global extension in Section 7.4.2), not circular.

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

The central claim rests on standard theorems in derived categories and Hodge theory, plus a few paper-specific general position assumptions that are proved generically. The most fragile input is the semiregularity theorem for twisted sheaves on abelian varieties; the paper proves the needed case but flags the general conjecture as unresolved relative to Pridham's framework.

assumptions (5)

standard math Buchweitz-Flenner Semiregularity Theorem
Used in Section 7 to show that a semiregular sheaf deforms when its Chern character stays of Hodge type. This is an external standard theorem.
standard math Orlov's derived equivalence categorifying the spin representation
Core tool in Sections 1.3 and 6. The existence and properties of the equivalence are taken from Orlov's work on derived categories of abelian varieties.
domain assumption Pridham's result is expected to imply Conjecture 7.3.9
The author states the full semiregularity conjecture for twisted sheaves should follow from Pridham's work, but does not reconcile the two statements. The abelian-variety case needed for the main theorem is proven directly in Section 7.4.
ad hoc to paper General position assumptions 9.1.1 and 9.2.1
These assumptions, concerning emptiness of certain intersections and disjointness of translates, are needed to prove the reflexivity of the constructed sheaf. They are shown to hold for generic choices in Lemmas 9.1.2 and 9.1.4.
standard math Prior results [S2], [MZ1], [MZ3], [R] used in the reduction to fourfolds
External theorems on degeneration (Schoen), generation of Hodge rings of simple abelian fourfolds (MZ1), and Hodge rings of products (MZ3, R) are invoked in Corollary 1.6.1 without proof.

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Pith. "Pith review of Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds." pith.science (2026). https://pith.science/paper/QDF7UA4Z

@misc{pith2026250203415,
  author       = {Pith},
  title        = {Pith review of: Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QDF7UA4Z}},
  note         = {Machine review of arXiv:2502.03415}
}

read the original abstract

A. Weil identified a 2-dimensional space of rational classes of Hodge type (n,n) in the middle cohomology of every 2n-dimensional abelian variety with a suitable complex multiplication by an imaginary quadratic number field. These abelian varieties are said to be of Weil type and these Hodge classes are known as Weil classes. We prove that the Weil classes are algebraic for all abelian sixfold of Weil type of discriminant -1, for all imaginary quadratic number fields. The algebraicity of the Weil classes follows for all abelian fourfolds of Weil type (for all discriminants and all imaginary quadratic number fields), by a degeneration argument of C. Schoen. The Hodge Conjecture for abelian fourfolds is known to follow from the above result.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

On quartic double fivefolds and the matrix factorizations of exceptional quaternionic representations

[Ab] Abuaf, R.: On quartic double fivefolds and the matrix factorization of exceptional quaternionic representations. arXiv:1709.05217v2. [Ar] Artamin, I. V.: On deformation of sheaves. Math. USSR Izvestiya Vol. 32(1989), No

work page Pith review arXiv 1989

[2] [3]

F.: Complex analytic connections in fibre bundles

[At] Atiyah, M. F.: Complex analytic connections in fibre bundles. Transactions of the AMS , 1957, Vol. 85, No. 1, pp. 181–207. [B] Beauville, A.: Fibr´ es de rang deau sur une courbe, fibr´ e d´ eterminant et functions thˆ eta. II. Bull. de la S. M. F. tome 119, No 3 (1991), p. 259–291. [BF1] Buchweitz, R-O., Flenner, H.: A semi-regularity map for module...

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[I] Igusa, J-I.: A Classification of Spinors Up to Dimension Twelve. Amer. J. of Math. Vol. 92, No. 4, 997–1028 (1970). [dJ] de Jong, A. J.: A result of Gabber. Preprint, https://www.math.columbia.edu/~dejong/ [K] Koike, K.: Algebraicity of some Weil Hodge classes. Canad. Math. Bull.47(2004), no.4, 566–

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Algebra Number Theory 3 (2009), no.6, 653–695

[Li] Lieblich, M.: Compactified moduli of projective bundles. Algebra Number Theory 3 (2009), no.6, 653–695. [M1] Markman, E.: The Beauville-Bogomolov class as a characteristic class. J. of Algebraic Geom- etry, 29 (2020) 199–245. [M2] Markman, E.: The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians. J. Eur. ...

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[Ce] Ceresa, G.: C is not algebraically equivalent to C − in its Jacobian. Ann. of Math. (2) 117 (1983), no.2, 285–291. [DM] Deligne, P., Milne, J.S.: Hodge cycles on abelian varieties. in Hodge cycles, motives, and Shimura varieties Deligne, Pierre; Milne, James S.; Ogus, Arthur; Shih, Kuang-yen Lecture Notes in Math., 900 Springer-Verlag, Berlin-New York,

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Springer-Verlag (2005)

[H1] Huybrechts, D.: Complex Geometry, An Introduction. Springer-Verlag (2005). [H2] Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford University Press,

work page 2005

[7] [1980]

Hodge Theory and algebraic geometry

[MZ1] Moonen, B., Zarhin, Y.: Hodge classes and Tate classes on simple abelian fourfolds. Duke Math. J. Vol. 77 No 3, (1995) 553-581. [MZ2] Moonen, B., Zarhin, Y.: Weil classes on abelian varieties. J. reine angew. Math. 496 (1998), 83–92. [MZ3] Moonen, B., Zarhin, Y.: Hodge classes on abelian varieties of low dimension. Math. Ann. 315, 711-733 (1999). [M...

work page 1995

[8] [1982]

A revised version is available at https://www.jmilne.org/math/Documents/index.html [G] Gulbrandsen, M.: Vector bundles and monads on abelian threefolds. Comm. Algebra 41 (2013), no. 5, 1964–1988. [GLO] Golyshev, V., Luntz, V., and Orlov, D.: Mirror symmetry for abelian varieties. J. Alg. Geom. 10 (2001) 433-496. [vG1] van Geemen, B.: An introduction to th...

work page 2013