Algebraicity of Hodge classes on some generalized Prym Varieties

Deepam Patel; Yilong Zhang

REVIEW 3 major objections 4 minor 15 references

Algebraicity of Hodge classes on some generalized Prym Varieties

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that all top-wedge Hodge classes on Prym varieties attached to abelian covers of curves are algebraic cycles.

desk verdict A serious generalization of Schoen's theorem with a genuine new framing, but Lemma 4.2's proof is wrong as written and it is the load-bearing step. read the letter →

arxiv 2506.13729 v2 pith:24Q5BSZU submitted 2025-06-16 math.AG

Deepam Patel , Yilong Zhang This is my paper · ORCID

classification math.AG MSC 14C3014K2214H40

keywords HodgeconjecturealgebraiccyclesPrymvarietiesabeliancoversgeometricclassfieldtheoryWeilclassessymmetricproductsAbel-Jacobimaps

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 certain Hodge classes on Prym varieties are algebraic, meaning they are the cohomology classes of actual algebraic cycles. Starting from an étale cover $C \to C'$ of smooth projective curves with finite abelian Galois group $G$, the Prym variety $B$ is the part of the Jacobian of $C$ on which $G$ acts through non-trivial characters. The paper shows that the full top exterior power of $H^1(B,\mathbb{Q})$ over the non-trivial part of the group ring, a Hodge substructure of dimension $|G|-1$, is generated by algebraic cycles. This removes the cyclic and primitive-character restrictions from an earlier result in the literature. If correct, it supplies algebraic representatives for a natural family of Hodge classes that are generally not divisor classes.

What carries the argument

The machine that drives the argument is the base-change diagram from unramified geometric class field theory: because $C \to C'$ is abelian, it is pulled back from an isogeny $A \to J(C')$, so there is a commutative square in which the cover $W \to \mathrm{Sym}^h(C')$ sits over $A \to J(C')$. The distinguished object is the special fiber $\mathbb{P}^{g(C')-1} = |K_{C'}|$ of $AJ_h$ over the canonical point of $J(C')$; in $W$ its inverse image is $|G|$ disjoint projective spaces indexed by the group. Their classes, after quotienting by the diagonal class, form the summand $U$ in the computed decomposition $H^*(W,\mathbb{Q}) \cong H^*(\mathrm{Sym}^h(C'),\mathbb{Q}) \oplus U(-h/2)$. Lemma 4.2 then uses the diagonal subgroup $G_0 \subset G^h$ acting on $C^h$, together with Schur's lemma, to show that $\phi^*U$ sits inside the top Künneth factor $\wedge^h H^1(C,\mathbb{Q})$ of $H^h(\mathrm{Sym}^h C,\mathbb{Q})$; Macdonald's symmetric-product formula, the Abel-Jacobi map, and the Lefschetz operator complete the transfer of algebraicity from $U$ to $U_{\mathrm{Weil}}$.

What would settle it

Compute, for a small abelian cover such as a $\mathbb{Z}/3$-cover of a genus-3 curve, the decomposition of $H^4(\mathrm{Sym}^4 C,\mathbb{Q})$ by the diagonal subgroup $G_0$; if the projection of $\phi^*U$ to any summand $\wedge^{4-2i}H^1(C,\mathbb{Q})$ with $i>0$ is nonzero, Lemma 4.2 and the identification with $U_{\mathrm{Weil}}$ fail. A direct dimension count of $H^4(W,\mathbb{Q})$ for the same cover would also test the predicted decomposition of Theorem 1.1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: for every étale abelian cover $C \to C'$ with $g(C') \ge 2$ and $h = 2g(C') - 2$, the subspace $U_{\mathrm{Weil}} = \Lambda^h_{\mathbb{Q}[G]^{\mathrm{nt}}} H^1(B,\mathbb{Q}) \subset H^h(B,\mathbb{Q})$ consists of Hodge classes and is represented by algebraic cycles. Over $\mathbb{C}$ this subspace is spanned, one vector per non-trivial character $\chi$, by the wedge of the $h$ copies of the $\chi$-isotypic component; it is a Hodge substructure by a pairing of conjugate Hodge types, and it has dimension $|G|-1$. The proof constructs the cycles on an auxiliary space: the Abel-Jacobi map from $\mathrm{Sym}^h(C')$ to $J(C')$ has a special fiber $\mathbb{P}^{g(C')-1}$ over the canonical class, and its preimage in the abelian cover $W$ of $\mathrm{Sym}^h(C')$ is a union of $|G|$ projective spaces whose classes, modulo their $G$-invariant sum, span exactly the summand $U$ appearing in Theorem 1.1. Pulling $U$ back to $\mathrm{Sym}^h(C)$, pushing forward by the Abel-Jacobi map to $J(C)$, and applying powers of the Lefschetz operator identifies $U$ with $U_{\mathrm{Weil}}$ on the Prym variety; the algebraicity of the Lefschetz inverse on abelian varieties then makes the classes algebraic.

Load-bearing premise

The load-bearing premise is that pulling the algebraic classes $U$ back to the symmetric product of the covering curve places them in the very top cohomology slice and nowhere else; the proof infers this from how the deck-transformation group acts on the lower slices, and if any lower slice shared the same group characters the identification with the Prym's Weil classes would break.

Editorial extensions

If this is right

For every étale abelian cover with base genus at least 2, the full nontrivial-character Weil class space $U_{\mathrm{Weil}}$ is algebraic, so the Hodge conjecture holds on the Prym variety for this entire subspace.
The result covers all nontrivial characters at once; in cyclic covers whose order is not prime these include characters that the earlier primitive-character theorem did not reach.
The same proof, as the paper notes for an arbitrary rational representation $V$ of $G$, produces algebraic cycles for the corresponding top-wedge classes on each isotypic abelian subvariety $J(C)_V$.
Because the Lefschetz standard conjecture for abelian varieties is already known, the algebraicity conclusion transfers through the Lefschetz operator without needing the full Hodge conjecture.

Reading between the lines

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

The paper does not spell this out, but the same class-field-theory base change should carry the construction into the ramified setting with Jacobians with modulus, where the special fiber would be replaced by a corresponding Brill-Noether-like locus; this would give algebraic representatives for Prym-type classes attached to ramified abelian covers.
A direct consequence of the argument is that algebraicity of these classes is a formal consequence of the base-change diagram plus the Lefschetz standard conjecture on abelian varieties; one would not expect it to depend on special properties of cyclotomic fields, unlike the cyclic examples that motivated it.
One testable extension is to realize the same cycles in $\ell$-adic cohomology and verify that they are Tate classes under a finite-field analogue of the abelian cover; the paper notes the proofs adapt to étale cohomology, so such a check would be a natural next step.

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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

Dear [colleague],

Read the Patel–Zhang paper. It extends Schoen's theorem from cyclic primitive-character covers to all finite abelian covers and all nontrivial characters. That is a real mathematical step, and the geometric class field theory framing is a genuinely useful way to see why the cycles exist. Theorem 1.1's Leray spectral sequence computation looks coherent, and the construction of the algebraic cycles Q_t is explicit and checkable.

The soft spot is Lemma 4.2, and it is load-bearing. The subgroup G0 = {(g,-g,0,...,0)} lies in N, hence in H, so it acts trivially on H^h(W,Q). It cannot induce the standard G-action on U. That looks like a typo—possibly they meant (g,0,...,0)—but the Schur's lemma argument is not right even after that correction. U is reducible, and a non-faithful action on a lower Kunneth summand does not imply the summand has no characters in common with U. For G = Z/3 and h = 4, the ^2 H^1(C) summand contains nontrivial characters, so the projection of phi*U to that summand is not shown to vanish. The containment phi*U ⊂ ^h H^1(C) is therefore not established.

This is not a cosmetic complaint. The identification of U with U_Weil in Theorem 1.2 passes through that containment via diagram (4.3.1). If Lemma 4.2 fails, the chain of maps does not identify the two Hodge structures. I don't think the final theorem is false—Schoen's result and the overall framework make it plausible—but the proof as written has a real gap at exactly the point where the main theorem needs support.

The rest of the paper is solid. The representation theory setup is standard, the comparison with Schoen's setting in Section 5 is helpful, and the references look appropriate. The paper is worth serious referee time. A referee should ask for a corrected Lemma 4.2, likely via a direct character computation on the Macdonald summands, and then re-check the isomorphism in (4.3.1).

Recommendation: send to peer review, with the lemma as the main issue.

Referee Report

3 major / 4 minor

Summary. This paper studies algebraic cycles on Prym varieties associated with etale abelian covers C -> C1 of smooth projective curves over C. The authors use geometric class field theory to construct an abelian cover W of the symmetric product Sym^h(C1), with h = 2g(C1) - 2, and prove (Theorem 1.1) a description of the cohomology of W: it agrees with that of Sym^h(C1) except in middle degree, where an additional |G|-1 dimensional Hodge substructure U generated by algebraic cycles appears. They then claim (Theorem 1.2) that the Weil-type Hodge classes U_Weil = Λ^h_{Q[G]^nt} H^1(B,Q) inside H^h(B,Q) of the associated Prym variety B are algebraic, by identifying U with U_Weil through Abel-Jacobi maps and Lefschetz operators. If correct, this extends Schoen's theorem for cyclic covers and primitive characters to arbitrary abelian covers and all nontrivial characters.

Significance. The potential significance is substantial: Theorem 1.2 would establish algebraicity of the full non-trivial-character part of the cohomology of Prym varieties attached to abelian covers, generalizing Schoen's cyclic primitive-character result. The paper also offers a conceptual reinterpretation via unramified geometric class field theory, and the proof of Theorem 1.1 via the Leray spectral sequence with explicit cycle classes Q_t is concrete and appears sound. The main result, however, rests on Lemma 4.2, whose proof contains a false group-action assertion and an invalid Schur's lemma argument. Since the identification of U with U_Weil is not established, the central algebraicity theorem is currently unsupported.

major comments (3)

[§4.2 (diagram 4.1.2 and following paragraph)] The subgroup G0 = {(g,-g,0,...,0)} is contained in N = {Σ t_k = 0}, hence in H = N S_h, so it acts trivially on H^h(W,Q) = H^h(C^h,Q)^H. The sentence claiming that the composition G0 -> G^h S_h -> \tilde H/H ≅ G is an isomorphism and that the induced G0-action on U is the standard G-action is therefore false: the image of G0 in \tilde H/H is zero. This invalidates the equivariance framework used in the proof of Lemma 4.2.
[§4.3, Lemma 4.2] Even if G0 is replaced by a subgroup that does induce the G-action, the Schur's lemma step is invalid. U is not an irreducible G-representation in general, and a non-faithful action on a Kunneth summand does not preclude common irreducible constituents with U. For instance, when G = Z/3 and g(C1) = 3 (so h = 4), the summand Λ^2 H^1(C,Q)(-1) of H^4(Sym^4 C,Q) contains a nontrivial character: for v in the trivial part H^1(C1,Q) and w in the χ-isotypic part of H^1(C,Q), the class v∧w transforms by χ under the diagonal action and by χ^{-1} under the stated G0-action, and such characters also occur in U. Hence the projection of φ^*U to a lower Kunneth factor is not shown to vanish, and the containment φ^*U ⊆ Λ^h H^1(C,Q) is not proved.
[§4.4, Theorem 4.3 and diagram (4.3.1)] The identification of φ^*U with Φ^*U_h and the final conclusion of Theorem 1.2 depend on Lemma 4.2. Since Lemma 4.2 is not established, the chain of isomorphisms in diagram (4.3.1) does not prove that the Weil Hodge classes U_Weil are algebraic. A corrected proof of Lemma 4.2, or a different argument for containment of φ^*U in the top Kunneth factor, is required.

minor comments (4)

[§3.2, Proposition 3.2] The notation 'Q' for the special fiber P^{g-1} is confusing because Q is also the coefficient field; please use a different symbol such as P.
[§4 heading] The section heading contains a typo: 'Prym V ariety' should be 'Prym Variety'.
[§2.3, Lemma 2.6] The notation \bar σ is not defined; please specify the complex conjugation action on Hom_Q(F_i,C).
[§4.1, Lemma 4.1] The construction of the morphism φ via the universal property of the fiber product is only sketched in one sentence; a slightly more detailed explanation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: U_Weil is an independently defined exterior-power Hodge structure, and the algebraicity claim is established by explicit maps (Abel–Jacobi and Lefschetz) rather than by fitting or by definition.

full rationale

The paper's derivation chain is not circular. The Hodge substructure U in Theorem 1.1 is algebraic by construction: it is generated by inverse images of the special fiber P^{g-1} of the Abel–Jacobi map over the canonical class (Proposition 3.2), and its algebraic nature does not presuppose the algebraicity of U_Weil. The target object U_Weil is defined independently in (1.1.2) and (2.11.2) as the top exterior power over Q[G]^nt of H^1(B,Q), and the fact that it consists of Hodge classes is checked separately via the balanced-Hodge-type criterion in Lemma 2.6. The identification of U with U_Weil is the actual content of Theorem 1.2, proved through Lemma 4.2, the commutative diagram (4.3.1), and the algebraicity of the Lefschetz operator on abelian varieties. No parameter is fitted to the target class, and no definition of U_Weil is phrased in terms of U or of the cycles that represent U. The cited external results, including Macdonald's symmetric-product formula and Kleiman's standard conjecture for abelian varieties, are independent inputs and are not self-citations of the present authors. The skeptical concern about Lemma 4.2 is a possible mathematical gap in the Schur-lemma argument, not a circularity: even if that lemma failed, the theorem would be unsupported rather than equivalent to its assumptions by construction. Under the standard for this pass, that is a correctness issue, not a circularity finding.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters are fitted and no new entities are postulated. The construction fixes h = 2g(C') - 2 from the base curve, and no constants are tuned to data. The proof depends on standard theorems from geometric class field theory, the projective bundle description of Abel-Jacobi maps, Macdonald's symmetric product formula, the Weil class criterion, and the known Lefschetz standard conjecture for abelian varieties, all cited. These are accepted background results, not targets of the paper, so they do not create circularity.

assumptions (8)

standard math Geometric class field theory (Theorem 2.1): Abel-Jacobi maps identify abelianized fundamental groups of Sym^d(C') and Pic^0(C'), so every abelian etale cover of Sym^d(C') is a pullback from a cover of the Jacobian.
Invoked in the Cartesian diagram (2.1.1) and in the definition of the helper cover W; dependence on Serre [14] and Kallel-Taamallah [3].
standard math Projective bundle theorem for Abel-Jacobi maps (Theorem 2.2): for d = 2g - 2, AJ_d is a P^{g-2}-bundle over the complement of a point, with special fiber P^{g-1} over the canonical class.
Used in the Leray spectral sequence computation in Proposition 3.1 and to produce the algebraic cycles Q_t.
standard math Decomposition of abelian varieties by rational G-representations (Theorem 2.4): an abelian variety with G-action splits into pairwise orthogonal G-stable subvarieties.
Defines the Prym variety B = J(C)_{V_nt} and the isotypic decomposition in Section 2.2.
standard math Weil class criterion (Lemma 2.6): if the multiplicities n_sigma and n_sigmabar agree, the top exterior power over the endomorphism ring consists of Hodge classes.
Shows that U_Weil consists of Hodge classes in Section 2.6.
standard math Macdonald's formula (4.1.3): H^h(Sym^h C,Q) decomposes as a direct sum of wedge powers of H^1(C,Q) with Tate twists.
Used to locate the pullback of U inside the cohomology of Sym^h C in Section 4.3.
standard math Nontrivial finite-order rank-one local systems on a complex torus have vanishing cohomology, asserted in (3.1.4).
Used in the spectral sequence to eliminate all E_2 terms except the one producing U; stated without proof in Section 3.2.
standard math Lefschetz standard conjecture for abelian varieties (Kleiman [4]): the inverse Lefschetz operator is algebraic.
Used in Theorem 4.3 to move algebraicity from the complementary dimension back to U_Weil.
standard math Balanced Hodge type: each nontrivial character occurs with multiplicity h = 2g(C') - 2 in H^1(C,C), evenly split between H^{1,0} and H^{0,1}.
Used to know that U_Weil consists of Hodge classes; proved in Lemma 2.9 via Euler characteristic of local systems.

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Pith. "Pith review of Algebraicity of Hodge classes on some generalized Prym Varieties." pith.science (2026). https://pith.science/paper/24Q5BSZU

@misc{pith2026250613729,
  author       = {Pith},
  title        = {Pith review of: Algebraicity of Hodge classes on some generalized Prym Varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24Q5BSZU}},
  note         = {Machine review of arXiv:2506.13729}
}

read the original abstract

In this article, we revisit the construction of some algebraic cycles due to Chad Schoen on certain Prym Varieties. More precisely, we show that these cycles arise naturally from (unramified) geometric class field theory, and apply it to prove the algebraicity of certain Hodge classes on some generalized Prym Varieties.

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Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Arbarello, M

E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris.Geometry of algebraic curves. Vol. I, volume 267 ofGrundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1985

work page 1985

[2] [2]

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Carel Faber. Prym varieties of triple cyclic covers.Math. Z., 199(1):61–79, 1988

work page 1988

[3] [3]

The geometry and fundamental group of permutation products and fat diagonals

Sadok Kallel and Walid Taamallah. The geometry and fundamental group of permutation products and fat diagonals. Canad. J. Math., 65(3):575–599, 2013

work page 2013

[4] [4]

S. L. Kleiman. Algebraic cycles and the Weil conjectures. InDix expos´ es sur la cohomologie des sch´ emas, volume 3 ofAdv. Stud. Pure Math., pages 359–386. North-Holland, Amsterdam, 1968

work page 1968

[5] [5]

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Kenji Koike. Algebraicity of some Weil Hodge classes.Canad. Math. Bull., 47(4):566–572, 2004

work page 2004

[6] [6]

Rodr´ ıguez.Decomposition of Jacobians by Prym varieties, volume 2310 ofLect

Herbert Lange and Rub´ ı E. Rodr´ ıguez.Decomposition of Jacobians by Prym varieties, volume 2310 ofLect. Notes Math. Cham: Springer, 2022

work page 2022

[7] [7]

I. G. Macdonald. Symmetric products of an algebraic curve.Topology, 1:319–343, 1962

work page 1962

[8] [8]

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Eyal Markman. The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians.J. Eur. Math. Soc. (JEMS), 25(1):231–321, 2023

work page 2023