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Incidence equivalence, a survey

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Pith's one-line read This survey argues that a single Hodge-theoretic condition — all primitive Lefschetz components of algebraic odd cohomology being algebraic — makes incidence equivalence coincide with Abel-Jacobi equivalence for codimension-i cycles algebra

desk verdict Useful survey, but the advertised new criterion is built on a misindexed definition and an unproved star-operator step. read the letter →

arxiv 2607.22233 v1 pith:3O5ZZUNP submitted 2026-07-24 math.AG

Chris Peters This is my paper · ORCID
classification math.AG MSC 14C2514C3014K30
keywords incidenceequivalenceAbel-JacobiintermediatejacobiansLefschetzdecompositiongeneralizedHodgeconjecturealgebraiccyclesstaroperatorheightpairing
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 survey revisits a question from the early 1970s: for cycles algebraically equivalent to zero, is the geometrically defined incidence equivalence the same as the transcendentally defined Abel-Jacobi equivalence? Its main new claim is that a positive answer in codimension i follows from a Hodge-theoretic condition: every primitive component in the Lefschetz decomposition of every algebraic class in H^{2i-1} must itself be algebraic. The paper proves this implication in Corollary 7.8 using the Hodge star operator and a positivity statement, and it shows the condition holds in important cases: codimensions 1, 2, and d, abelian varieties, and odd-dimensional complete intersections whose odd cohomology is concentrated in the middle degree. If the two equivalences coincide, Abel's theorem for divisors gains a higher-codimension analogue, and the asymptotic invariant of the archimedean height pairing becomes a purely geometric quantity. The survey also gathers known results and open connections to the generalized Hodge conjecture.

What carries the argument

The argument runs through the Hodge star operator and the Lefschetz decomposition. A classical formula from the theory of Kähler varieties expresses the Hodge star of a primitive class via the Weil operator and the Lefschetz operator; under condition (Λ, X, i) this formula implies that the star of an algebraic class in H^{2i-1} belongs to the algebraic part of H^{2d-2i+1} and has positive self-intersection unless the class is zero. That makes the cup-product pairing between H^{2i-1}_alg and H^{2d-2i+1}_alg non-degenerate on the first factor, and a known criterion — based on biextensions — turns this non-degeneracy into the equality of incidence and Abel-Jacobi equivalence.

What would settle it

For a smooth projective variety X and an index i satisfying condition (Λ, X, i), compute the cup-product pairing H^{2i-1}_alg(X) × H^{2d-2i+1}_alg(X) and check non-degeneracy; finding a nonzero class v in H^{2i-1}_alg(X) whose Hodge star is not in H^{2d-2i+1}_alg(X), or whose self-intersection v·(∗v) is zero, would refute equation (14) and invalidate the proof of the corollary.

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Extended reading notes

Core claim

The central new result, Corollary 7.8, states: if every primitive component in the Lefschetz decomposition of every class in H^{2i-1}_alg(X, Q) is itself algebraic — the condition the author names (Λ, X, i) — then for codimension-i cycles algebraically equivalent to zero, incidence equivalence coincides with Abel-Jacobi equivalence after tensoring with Q. The condition is shown to hold for codimensions 1, 2, and d, for abelian varieties because the Λ-operator is algebraic there, and for odd-dimensional complete intersections whose odd cohomology is concentrated in the middle degree. In the codimension-2 case the paper repairs a gap in an earlier published proof.

Load-bearing premise

The proof of Corollary 7.8 rests on equation (14), which asserts that the Hodge star of an algebraic class in H^{2i-1} is again algebraic and has positive self-intersection; this is stated in one sentence from a classical formula, and if the star fails to preserve algebraicity the non-degeneracy of the cup-product pairing is not established.

Editorial extensions

If this is right

  • A uniform proof of the codimension-2 case replaces a previously incomplete argument, and the same Hodge-theoretic criterion covers all codimensions.
  • For abelian varieties, incidence and Abel-Jacobi equivalence coincide for cycles of every codimension.
  • If the generalized Hodge conjecture holds in the relevant odd degree, the coincidence follows for any smooth projective variety.
  • On odd-dimensional complete intersections of dimension 2m+1 with no odd cohomology outside the middle, the two equivalences agree for codimension m+1 cycles.
  • When the coincidence holds, the leading asymptotic coefficient of the archimedean height pairing is the local geometric height pairing, a genuine invariant of the cycles.

Reading between the lines

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

  • The condition (Λ, X, i) is formulated as an assumption on primitive components; testing it on examples beyond complete intersections and abelian varieties — for instance, low-degree hypersurfaces with richer odd cohomology — would map the true boundary of the theorem.
  • The positivity assertion underlying (14) is reminiscent of the Hodge-Riemann relations; if it could be derived from those relations directly, the dependence on the explicit star-operator formula might be removed, and the method might extend to variations of Hodge structure.
  • A single algebraic class whose Hodge star is not algebraic would pinpoint exactly where the geometric-transcendental bridge breaks; hunting for such a class may be more tractable than attacking the generalized Hodge conjecture head-on.
  • The same correspondence-based strategy could be applied to other cycle-equivalence relations defined by kernels of correspondences, suggesting a general principle: Hodge-theoretic non-degeneracy equals geometric equivalence.
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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 / 4 minor

Summary. The paper surveys Griffiths' incidence equivalence for cycles algebraically equivalent to zero on smooth complex projective varieties, and its relation to Abel-Jacobi equivalence, denoted GC(X,i). It reviews Chow groups, correspondences, standard conjectures, intermediate Jacobians, and Müller-Stach's biextension approach. The main new assertion is Corollary 7.8: a condition (Λ,X,i) on the Lefschetz components of odd-degree algebraic cohomology classes implies GC(X,i); the proof rests on a Hodge-star computation recorded as equation (14). The paper also claims consequences for codimension two cycles, for abelian varieties, and for odd-dimensional varieties with only one middle odd Betti number.

Significance. If the new criterion is correct, it gives a clean structural explanation of Murre's codimension-two theorem and a useful bridge between incidence equivalence, the generalized Hodge conjecture, and the standard conjectures. The survey is well organized and collects relevant older literature, which is valuable. However, the proof of the central new result is currently a sketch with a definitional inconsistency and an unproved Hodge-star step; as written, Corollary 7.8 is not established. The underlying strategy is plausible and may be repairable, but the manuscript in its present form needs substantial corrections before the main claim can be accepted.

major comments (3)
  1. [§6, Eq. (12)] The definition H_alg^{2i-1}:=N^i H^{2i-1} is incompatible with the coniveau convention stated earlier in §6. For Z⊂X of codimension at least i, the real codimension is at least 2i, so the local cohomology H^m_Z(X) vanishes for m<2i; in particular H^{2i-1}_Z(X)=0. Hence H_alg^{2i-1}=0 for all i, making Condition (Λ,X,i) vacuous and Lemma 6.3 false (it would imply J^i_alg=0). The definition must be reindexed, e.g. H_alg^{2i-1}=N^{i-1}H^{2i-1}, and then Corollary 6.2, Criterion 7.3, and Condition (Λ,X,i) must be rechecked under the corrected convention.
  2. [§7, Eq. (14)] The assertion 'if v∈H_alg^{2i-1} then *v∈H_alg^{2d-2i+1}' is not derived. Weil's p.76 formula writes *v as a combination of L^{d-2i+1+r} C(v_{2i-1-2r}); the proof does not show that the Weil operator C preserves the relevant algebraic subspace, nor that L sends the corrected coniveau level into the required one. With the natural correction H_alg^{2i-1}=N^{i-1}, a Gysin class supported on codimension i-1 has star supported on the same subvariety, so *v∈N^{i-1}H^{2d-2i+1}, while H_alg^{2d-2i+1}=N^{d-i}H^{2d-2i+1}; these differ unless i=j. Example: X a 4-fold, i=2, v=i_*w with w∈H^1(D) for a divisor D; then *v is supported on D, hence lies in N^1H^5, not in N^2H^5. Thus (14) is unsupported and the non-degeneracy argument in Corollary 7.8 collapses.
  3. [Summary 7.11(4)] The stated equivalences Λ(X,i)+Λ(X,j) ⇔ GC(X,i)+GC(X,j) and Λ(X,i)⇔GC(X,i) when d+1=2i are not proved in §7. The text proves only the implication Λ⇒GC, and that proof depends on the unproved equation (14). Remark 7.10 in fact says that Condition (Λ,X,i) gives only one inclusion J^j_alg⊂J^i_alg^*, and that the reverse inclusion requires Condition (Λ,X,j). Either supply the missing converse argument or weaken the summary to state only the implication proved.
minor comments (4)
  1. [Title and abstract] The title contains spacing/typo artifacts ('INCIDENCE EQUIV ALENCE, A SUR VEY.') that should be cleaned; same for the abstract.
  2. [§5.2] Typo 'whch' should be 'which'.
  3. [Summary 7.11(3)] The expression 'G(X,2)' should be 'GC(X,2)'.
  4. [Remark 7.6] The verification of Λ(X,2) is very terse: the assertion 'H^1_prim,alg=H^1(X)_Q by Abel's theorem' is not a standard formulation of Abel's theorem and needs a precise reference or proof. Once the definition of H_alg is corrected, this point becomes non-obvious and should be expanded.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Corollary 7.8 is a conditional implication, not an input-output fit; self-citations are background references.

full rationale

The derivation chain is not circular in the sense targeted by this pass. The central new assertion, Corollary 7.8, says the geometric Condition (Λ,X,i) implies GC(X,i). The proof reduces GC to a non-degeneracy statement via Criterion 7.3 and then invokes equation (14), which is an auxiliary positivity/Weil-operator assertion. Condition (Λ,X,i) is not defined in terms of GC(X,i), and the conclusion is not a renamed version of the hypothesis: the condition only constrains primitive Lefschetz components of algebraic odd cohomology classes, while GC compares two equivalence relations on cycles. No parameter is fitted to data and no quantity is renamed as a prediction. The paper's self-citations ([3], [20], [22]) are standard background references for Hodge theory, motives and period mappings; they are not the load-bearing step for Corollary 7.8, which instead invokes Murre, Müller-Stach, Weil and Griffiths-Schmid. The sections do contain a serious technical concern that the reviewer's note raises: equation (12) defines H_alg^{2i-1}=N^i H^{2i-1}, which under the coniveau convention of §6 would be zero for degree 2i-1; and (14) is asserted in one sentence without fully proving that the Weil operator and the relevant L-powers preserve the algebraic subspace. But these are correctness/consistency gaps, not circular reductions: (14) is stronger than the hypothesis and is not a restatement of the target result. The manuscript even flags an earlier proof (Murre's Lemma 5.2) as incomplete, which is an explicit limitation rather than a circular appeal. Accordingly, no circular step is exhibited and the score is set at 1 rather than 0 only to acknowledge the presence of minor background self-citations.

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

The proof of Corollary 7.8 rests on cited Hodge-theoretic machinery—Weil's star formula, Müller-Stach's biextension criterion, Murre's description of H_alg—rather than on new postulates. No empirical free parameters appear.

assumptions (6)
  • domain assumption Weil's formula for the Hodge star operator on primitive Lefschetz components: ∗(L^r v_{k−2r}) = (−1)^{...} a_{d,r} L^{d−k+r} C(v_{k−2r}), from Weil [24, p. 76].
    Used in §7 to derive equation (14) and hence the non-degeneracy of the cup product; not proved in the survey.
  • domain assumption Müller-Stach's Theorem 7.1: the Bloch biextension E_eta^alg splits iff η ~_inc 0, together with the corrections in [17, Satz 3.2.1] and [21, Lemma 5.1, 5.2].
    Basis of Criterion 7.3 and Corollary 7.2; the survey does not prove the theorem itself.
  • standard math Abel's theorem and the Lieberman-Saito results: incidence equivalence equals Abel-Jacobi equivalence for divisors and for 0-cycles, and H_prim,alg^1(X) = H^1_Q(X).
    Used for GC(X,1), GC(X,d), and for verifying Condition (Λ,X,2).
  • domain assumption Murre's Lemma 6.3: H_alg^{2i-1}(X)_C = T(J_alg^i(X)) ⊕ ̄T(J_alg^i(X)), so H_alg^{2i-1}(X) is a level ≤ 1 Hodge substructure.
    Used to identify dual tori and to justify Criterion 7.3.
  • standard math Hodge-Riemann bilinear relations and the positive-definiteness of v·∗v for real harmonic forms.
    Used to conclude v·(∗v) > 0 for nonzero v in (14).
  • domain assumption The Weil operator C preserves H_alg and the Lefschetz operator L preserves H_alg.
    Needed for (14) but not explicitly stated by the survey; it is a key compatibility condition for the new proof.

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Pith. "Pith review of Incidence equivalence, a survey." pith.science (2026). https://pith.science/paper/3O5ZZUNP

@misc{pith2026260722233,
  author       = {Pith},
  title        = {Pith review of: Incidence equivalence, a survey},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3O5ZZUNP}},
  note         = {Machine review of arXiv:2607.22233}
}
abstract

This is a survey of results on incidence equivalence, a notion introduced by P$.$Griffiths around 1970 when trying to extend the classical properties of the Abel-Jacobi map for curves. Using the intermediate jacobians and the associated Abel-Jacobi maps in higher dimension, a natural question came up: is the geometrically defined incidence equivalence relation the same as Abel-Jacobi equivalence, which is of transcendental nature? I give an overview of results related to this question and to several classical conjectures that are far from resolved, such as Grothendieck's generalized Hodge conjecture. The motivation for writing this survey came from a recently observed unexpected connection of Griffiths' question to the asymptotic behaviour of the archimedean height pairing in a geometric setting.

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