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On the Chow ring of very general abelian varieties and a question of Pirola

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Pith's one-line read The paper proves that on very general abelian varieties of dimension at least 4, every divisor whose square vanishes in the Chow ring is torsion, and uses this to show that all rational sections of the genus-4 Kummer fibration are multiples

desk verdict A serious paper proving a long-open torsion statement for very general abelian varieties and Pirola's conjecture, with a real but standard structural reliance on a companion preprint. read the letter →

arxiv 2607.07053 v2 pith:HHINIX3L submitted 2026-07-08 math.AG

Claire Voisin This is my paper · ORCID
classification math.AG MSC 14C1514D0714H4014K12
keywords Chowringabelianvarietieszero-squaredivisorstorsioninfinitesimalinvariantsnormalfunctionsKummerfibrationgenus4Jacobians
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 tries to settle two related questions in the Chow theory of abelian varieties. First, it argues that on a very general abelian variety of dimension at least four, any divisor whose square vanishes in the Chow ring must be torsion; the same holds for the Jacobian of a very general genus-four curve. Second, it proves that every rational section of the Kummer fibration attached to the universal genus-four Jacobian is a rational multiple of one special section, the Griffiths-Pirola normal function. These results matter because they sharply constrain the ring structure of Chow groups and connect it to transcendence statements of Ax–Schanuel type.

What carries the argument

The central machinery is the Griffiths infinitesimal invariant of a normal function, a first-order derivative measuring how a family of cycles moves in a family of abelian varieties, together with its extension to codimension-2 cycles. The key algebraic step is the square map M_2, which acts as the 2-by-2 minors map on tensors. Proposition 3.1 shows that M_2(ϕ)=0 forces a lift of ϕ to be rank-one. That rank-one property feeds into Proposition 3.16, which uses the Ax–Schanuel theorem to show that nonzero algebraic flat locally decomposable sections of H^1⊗Ω_M do not exist here. For the section theorem, the same square map is used to solve M_2(ϕ)=λ M_2(δγ_P), whose only solutions are multiples

What would settle it

Produce a very general abelian fourfold (or a very general genus-4 Jacobian) with a non-torsion divisor D in Pic^0 satisfying D^2 = 0 in CH^2; this would directly falsify the paper's main theorem. Alternatively, exhibit a rational section of the genus-4 Kummer fibration that is not a multiple of the Griffiths-Pirola section.

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Extended reading notes

Core claim

The paper proves two theorems. First, for a very general abelian variety of dimension at least 4, a divisor D in Pic^0(A) with D^2 = 0 in CH^2(A) must be torsion; the same is true for the Jacobian of a very general genus-4 curve. Second, all rational sections of the Kummer fibration of the universal genus-4 Jacobian are rational multiples of the Griffiths-Pirola normal function, the section defined by the difference of the two trigonal divisors. The proof uses Griffiths infinitesimal invariants of normal functions; the vanishing of the square forces the invariant to have a rank-one representative, and an Ax–Schanuel-type theorem rules out nonzero algebraic flat locally decomposable sections,

Load-bearing premise

The classification of rational sections of the genus-4 Kummer fibration relies on a cited result from the companion paper — that the space of zero-cycles modulo rational equivalence on the generic double symmetric product of a genus-4 curve is one-dimensional, generated by the square of the Griffiths-Pirola divisor — which is not proved in the present paper.

Editorial extensions

If this is right

  • On very general abelian varieties of dimension at least 4, the Chow ring has no exotic zero-square divisors: the set of D in Pic^0 with D^2 = 0 is exactly the torsion subgroup.
  • The same conclusion holds for very general genus-4 Jacobians, and equivalently for divisors on their second symmetric products.
  • The only rational sections of the genus-4 Kummer fibration are multiples of the Griffiths-Pirola section; in particular, no new independent section appears.
  • The paper conjectures that on very general abelian varieties of dimension at least 2k, the only divisors with D^k = 0 are torsion.

Reading between the lines

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

  • If the same infinitesimal-invariant strategy can be extended to higher powers, one would expect that on very general abelian varieties of dimension at least 2k, the condition D^k = 0 in CH^k(A) forces D to be torsion; that would be a natural test of the generality of the rank-one mechanism.
  • The role of Ax–Schanuel suggests a general principle: algebraic flat sections of H^1 ⊗ Ω_M that are locally decomposable are forced to vanish whenever the monodromy group is large; families with small monodromy could provide counterexamples to the zero-square statement.
  • The integer-multiple conclusion for the genus-4 section theorem hints that normal-function multipliers in moduli problems may often be integral; this could be explored in other Franchetta-type settings.
  • The paper explicitly acknowledges that the genus-higher version of the second symmetric product computation is not done here; extending the relevant result to genus greater than 4 is a concrete 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.

Referee Report

3 major / 6 minor

Summary. The paper proves two main theorems. Theorem 1.2 states that on a very general abelian variety of dimension at least 4, any divisor D whose square vanishes in CH^2 is torsion, and the analogous statement holds for a very general genus-4 Jacobian. Theorem 1.8 resolves a conjecture of Pirola: every rational section of the Kummer fibration K=J/±Id over M_4 is a rational multiple of the Griffiths-Pirola section. The proof of Theorem 1.2 combines infinitesimal invariants of normal functions with an algebraic rank-1 reduction (Proposition 3.1), an integrability/horizontality argument (Proposition 3.10), and an Ax-Schanuel input (Proposition 3.16). The proof of Theorem 1.8 uses the one-dimensionality of CH_0 of the generic second symmetric product from the companion paper [20] to compare the square of an arbitrary section with the Griffiths-Pirola square, then applies the infinitesimal invariant analysis of Section 4.1.

Significance. If correct, these are substantial results. Theorem 1.2 gives the first complete description of zero-square divisors on very general abelian varieties of dimension at least 4, and the genus-4 Jacobian case is used for the Pirola conjecture. Theorem 1.8 is a strong Franchetta-type statement for the Kummer fibration in genus 4 and goes beyond previously known results. The paper contains detailed and original algebraic work, especially Proposition 3.1, Claim 3.13, Proposition 4.5, and the monodromy arguments. The main theorems are falsifiable and the structure of the proof is coherent. However, as noted below, several load-bearing ingredients are quoted from the companion preprint [20], and the present manuscript does not include proofs of those ingredients; this makes independent verification difficult and is the main reason the paper needs revision before acceptance.

major comments (3)
  1. [Section 4, Proposition 4.2 and Theorem 4.1] Theorem 1.8 depends crucially on [20, Theorem 1.4], quoted as Theorem 4.1: CH_0(C^(2)_η)_hom⊗Q is 1-dimensional and generated by γ_P,η^2. This is the only input that yields Eq. (65), D^2_{γ,b}=λD^2_{γ_P,b}, and hence Corollary 4.4 and Proposition 4.5. The present paper proves neither the upper bound (from [21]/[12]) nor the nonvanishing (from [20, Thm 2.10]). The companion is not included in this submission, so the referee cannot check whether the generator identification is correct for the generic genus-4 curve over M_4, nor whether the companion secretly uses Theorem 1.2. I ask that the companion proof be included in the submission, or that the dependence on [20, Thm 1.4] be removed or made verifiable.
  2. [Section 3.1, equations (18)–(22) and following text] The identification of the deepest piece L^2H^2(Ω^2_{J_g|J_{g,m}}) with I_{2,W} (resp. I_{2,V}), and the statement that the square map is M_{2,W} (resp. M_{2,V}), are quoted from [20] (including [20, Prop. 5.1]). This identification is load-bearing for Theorem 1.2, because it converts the vanishing δ(D^2)=0 into the algebraic equation M_{2,W}(δD)=0 that is subsequently solved in Proposition 3.1. Without a proof or an exact statement from the companion, the algebraic core of Theorem 1.2 is not self-contained. Please provide the missing proof or include the companion as an appendix.
  3. [Section 3.2, Proposition 3.16] The application of the Ax-Schanuel theorem [6, Theorem A] is very compressed. In particular: (i) the hypotheses of [6, Thm A], which concern special subvarieties and leaves of a connection on a principal bundle, need to be checked for the frame bundle of a generically finite cover M→A_g (resp. M→M_g); (ii) the statement that the images p(F_e∩F_L) are contained in proper ∇-special subvarieties is asserted in one sentence and is not immediate, because F_e∩F_L is an intersection inside the frame bundle and its projection is not obviously special; (iii) the passage from this to algebraic integrability of the distribution α requires explanation. This is a correctness-risk concern. I would recommend expanding this argument, even if the statement is ultimately correct.
minor comments (6)
  1. [Introduction, after Corollary 1.3] Typo: 'neeeded' should be 'needed'.
  2. [Section 2, proof of Lemma 2.1] In the paragraph after Eq. (8), 'η_F vanishes identically' should presumably read 'η^{1,0} vanishes identically'.
  3. [Section 3.1, before Eq. (22)] The notation 'deepest part L^2H^2(Ω^2...)' is used without a precise definition of the filtration degree indexing; this can be confusing because L^* has a decreasing indexing in the preceding subsection. Please clarify whether L^2 denotes the second step of the filtration or its graded piece.
  4. [Section 4.2, Lemma 4.10] The line 'On any irreducible component of this double cover, we have by Theorem 4.9 that γ'=λγ_P' uses a rational λ, while Theorem 4.9 is stated with integers N,N'. The passage from Nγ=N'γ_P to a rational coefficient should be made explicit, including the treatment of components where γ' is torsion.
  5. [References] The spelling of Roitman is inconsistent: [7] uses 'Roitman' and [18] uses 'Rojtman'. Please standardize.
  6. [Remark 1.5] Remark 1.5 says the higher-genus extension needs 'Proposition 3.1(ii)' in higher genus; this should really be a higher-genus analogue of Proposition 3.1(ii), since the current statement is for g=4. The wording is ambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.2 is proved from infinitesimal-invariant machinery and Ax–Schanuel; Theorem 1.8's dependency on the companion result [20, Thm 1.4] is external and not a fitted input or self-citation loop.

full rationale

I followed the derivation chain. Theorem 1.2 is proved in Section 3 by reducing D^2=0 to vanishing of the square of the infinitesimal invariant, applying Proposition 3.1 (a linear-algebra statement proved in the paper) to get a rank-1 lift, then Proposition 3.10 (horizontal lift) and Proposition 3.16 (uses the external Ax–Schanuel theorem [6]). Nothing in this chain assumes that zero-square divisors are torsion; the target conclusion is obtained only at the end via Lemma 2.1 and monodromy. The prior author results [23], [25], [26] are used as standard machinery and are independent of the present theorem. Theorem 1.8 is not proved by fitting: the input Theorem 4.1 from the companion [20] states one-dimensionality of CH_0(C^(2)_η)_hom and generation by γ_P^2; Proposition 4.2 is the direct consequence that any D_γ^2 is a rational multiple of γ_P^2, and the remaining argument compares infinitesimal invariants to upgrade this to D_γ = sqrt(λ)D_{γ_P} (or λ=0, in which case Theorem 1.2(ii) is applied). No parameter is fitted to the claimed output, and the claimed output is not assumed in the inputs. Remark 2.11 explicitly notes that Theorem 2.10 is also implied by Theorem 1.2(ii) but that [20]'s proof is direct, so the paper does not secretly route through its own theorem. The only caveat is that [20, Theorem 1.4] is not proved in this submission and is therefore a genuine structural dependency; a dependency is not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical free parameters are fitted; the number λ in Section 4 is determined by the given section rather than chosen ad hoc. No new particles, forces, dimensions, or entities are postulated. The main external burden is the collection of cited theorems from [20], [21], [6], plus classical Hodge-theoretic results.

assumptions (6)
  • standard math CH_0(C^(2)_η)_hom ⊗ Q is 1-dimensional and generated by γ^2_{P,η} ([20, Theorem 1.4])
    Used in Section 4 to obtain Proposition 4.2; it is quoted from the companion preprint and not proved in this paper.
  • standard math Ax–Schanuel theorem in the form of [6, Theorem A] applies to the non-transverse intersections in Proposition 3.16
    Central to proving that an algebraic flat locally decomposable section of H^1 ⊗ Ω_M must vanish; the hypotheses are stated only briefly.
  • standard math The monodromy/Galois group of the Gauss–Manin connection is Zariski dense in Sp(2g) for the universal families over generically finite covers of A_g or M_g
    Used in Lemma 2.1(iii) and Proposition 3.16 to control monodromy-invariant sections.
  • domain assumption The Griffiths differentials ∇: H^{1,0} → H^{0,1} ⊗ Ω_M and ∇_1: H^{1,0} ⊗ Ω_M → H^{0,1} ⊗ Ω^2_M are injective for the universal families considered
    These are assumptions (i) and (ii) of Lemma 2.1; if they failed, the conclusion that ν is torsion would not follow.
  • standard math Bloch–Roitman and Bloch–Srinivas results on torsion cycles and rational equivalence
    Used in Lemma 3.8 and Theorem 2.8 to connect fiberwise rational equivalence with vanishing of infinitesimal invariants.
  • domain assumption The base curve C is very general of genus 4, in particular non-hyperelliptic with nondegenerate quadric q
    The quadric form q is used throughout Section 3.1 and Section 4; for a very general genus-4 curve this assumption holds.

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Pith. "Pith review of On the Chow ring of very general abelian varieties and a question of Pirola." pith.science (2026). https://pith.science/paper/HHINIX3L

@misc{pith2026260707053,
  author       = {Pith},
  title        = {Pith review of: On the Chow ring of very general abelian varieties and a question of Pirola},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HHINIX3L}},
  note         = {Machine review of arXiv:2607.07053}
}
abstract

We prove that for a very general abelian variety of dimension $\geq 4$, a divisor $D\in {\rm CH}^1(A)$ that satisfies $D^2=0$ in ${\rm CH}^2(A)$ is of torsion. The same result is also established for a very general Jacobian in genus $4$. We use then the second statement in order to prove a conjecture of Pirola, which states that any rational section of the Kummer fibration $K=J/\pm {\rm Id}\rightarrow \mathcal{M}_4$, where $J\rightarrow \mathcal{M}_4 $ is the Jacobian fibration, must be a multiple of the Griffiths-Pirola section given by the difference of the two trigonal divisors.

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