Tropical Abel-Jacobi theory

Daniel Corey; Leonid Monin; Omid Amini

REVIEW 2 major objections 5 minor 62 references

Tropical Abel-Jacobi theory

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Every compact tropical variety carries a functorial Abel-Jacobi map into an intermediate Jacobian, and for tropical curves the Ceresa class is computed by an explicit graph formula.

desk verdict A substantial higher-dimensional tropical Abel-Jacobi theory with explicit Ceresa formulas; a fixable gap in the written proof of Theorem 2.3(3) does not threaten the main results. read the letter →

arxiv 2504.14415 v1 pith:Y4OMC5AP submitted 2025-04-19 math.AG math.COmath.NT

Omid Amini , Daniel Corey , Leonid Monin This is my paper · ORCID

classification math.AGmath.COmath.NT MSC 14T1014T2005C2505E1414C2514H40

keywords Abel-JacobimapAlbaneseAlgebraiccyclesCeresacycleIntermediateJacobianTropicalvarietyhomologyMonodromy

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

The paper builds an Abel-Jacobi theory for compact tropical varieties of any dimension. To a tropical variety $X$ it attaches intermediate Jacobians $JH^{p,q}(X)=H^{p,q}(X,\mathbb{R})/L^{p,q}(X)$, where $L^{p,q}(X)$ is the image of tropical monodromy on integral homology, and constructs a functorial Abel-Jacobi map $AJ:A^\circ_p(X)\to JH^{p+1,p}(X)$ from homologically trivial cycles modulo rational equivalence. The map is built by choosing a chain that bounds the cycle and applying the monodromy operator. In dimension one the construction reproduces the existing Abel-Jacobi map for tropical curves. As a concrete application, the tropical Ceresa class of a curve is expressed purely in terms of a spanning tree, a basepoint, and edge lengths, and rational edge lengths force the class to be torsion. A nonzero monodromy image of the Ceresa class obstructs the Ceresa cycle from being algebraically equivalent to zero.

What carries the argument

The load-bearing object is the tropical monodromy operator $N$, defined on chains by $N([P,v])=\sum_{Q\prec P}\operatorname{sgn}(P,Q)[Q,w_{P,Q}\wedge\pi_{P,Q}(v)]$, where $w_{P,Q}$ is the displacement between chosen marking points on a polytope and its codimension-one face. The quotient $JH^{p,q}(X)=H^{p,q}(X,\mathbb{R})/L^{p,q}(X)$, with $L^{p,q}(X)$ the image of $N^{p-q}$ from integral homology, is the tropical intermediate Jacobian. The operator converts a bounding chain into a linear functional on tropical cocycles; its chain-level properties, especially independence of markings for chains whose boundary is algebraic, are what make the Abel-Jacobi map well-defined and functorial.

What would settle it

On a projective tropical manifold, choose one homologically trivial $p$-cycle and two different bounding chains with different markings; compute the two Abel-Jacobi classes as in Section 3. If the difference is not contained in $L^{p+1,p}(X)$, then Proposition 3.5 and Theorem 1.1 are false. The $K_4$ curve example in Section 6.5 offers a concrete place to run this check for $p=1$ on its Jacobian.

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Extended reading notes

Core claim

The central claim is that every compact tropical variety has a well-defined, functorial Abel-Jacobi map $AJ:A^\circ_p(X)\to JH^{p+1,p}(X)$ landing in a tropical intermediate Jacobian. The paper proves this by showing that applying the tropical monodromy operator $N$ to any bounding chain of a homologically trivial cycle gives a period that is independent of the auxiliary choices, modulo the lattice $L^{p+1,p}(X)$. It then verifies that rationally equivalent cycles have vanishing image, so the map descends to the Chow group of homologically trivial cycles. For a tropical curve the target $JH^{1,0}(\Gamma)$ is the usual Jacobian and the map is the known Abel-Jacobi map. The paper also computes the Ceresa class of a tropical curve explicitly as $v_\flat(\Gamma)=\sum_{e,\varepsilon} \operatorname{sgn}^\flat_T(e,\varepsilon)\ell(e)\,a_\varepsilon\otimes(b_\varepsilon\wedge b_e)$ in $JH^{2,1}(\mathrm{Jac}(\Gamma))$, and derives a basepoint-free quotient class with a similar graph-theoretic formula.

Load-bearing premise

The load-bearing premise is the marking-independence assertion proved in Section 2.11: a chain whose boundary is an algebraic cycle has a monodromy image that does not change when the auxiliary marking points on its faces are changed. If that independence failed, the Abel-Jacobi map would not be well-defined.

Editorial extensions

If this is right

Any compact tropical variety satisfying the weight-monodromy property carries real-torus intermediate Jacobians, so homologically trivial cycles acquire period invariants in every dimension.
For tropical curves the new Abel-Jacobi map coincides with the classical one, so the one-dimensional theory is exactly a special case of the general construction.
Algebraically trivial tropical $p$-cycles satisfy $N(AJ(\alpha))=0$ in $Q^{p+2,p-1}(X)$, giving a computable obstruction to algebraic equivalence.
The pointed and unpointed Ceresa classes of a tropical curve are determined only by a spanning tree, a basepoint, and the edge lengths; the formulas reproduce the known computations for the graphs $K_4$ and $TL_3$.
Tropical curves with rational edge lengths have torsion Ceresa classes, and such curves are dense in tropical moduli space.

Reading between the lines

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

The same chain-level construction should extend to tropical varieties with boundary or mild singularities, since the balancing condition needed for marking independence is already local; the paper does not pursue this extension.
The explicit graph formula makes a computer search feasible: one can sample random metric graphs and test whether a nonhyperelliptic curve has vanishing pointed or unpointed Ceresa class, which would settle the paper's Question 7.2.
If the conjectural Johnson-homomorphism formula in Appendix A holds, the tropical comparison theorem would give an effective way to compute the Johnson homomorphism on mapping classes of the form $T_\gamma T_{\gamma'}^{-1}$ beyond the classical bounding-pair case.

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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,

This paper is the first systematic higher-dimensional tropical Abel-Jacobi theory. The definitions of intermediate Jacobians via images of monodromy, the functorial AJ map, and the explicit tropical Ceresa class formulas are genuinely new. The curve case recovers Mikhalkin-Zharkov, and the examples in Section 6 match the computations of CEL24 and CL24/Ritter, which is a good sign. The proof of the Ceresa formula in Theorem 6.4 is worked out in detail with explicit bounding chains; that is real, reproducible mathematics.

The soft spot I see is in Theorem 2.3(3), the foundational lemma that AJ is well-defined. The statement says that if ∂γ is represented by an algebraic cycle modulo boundaries, then N(γ) is marking-independent. In the proof, when changing a marking on a p-face Q, the difference is identified with an expression that uses the coefficient of Q in ∂γ, and the text writes ∂γ as exactly the fundamental chain of the algebraic cycle. But the hypothesis only gives ∂γ = α + ∂θ. The extra ∂θ term contributes to the difference and the written proof does not address it. This is a genuine gap in the written argument. It looks fixable—one should be able to absorb the θ-term as a boundary—and all the applications (Prop 3.6, Thm 3.10, Thm 6.4) use the exact-boundary case, so the central construction is not threatened. Still, the lemma needs a corrected proof.

Other than that, the paper is dense but careful. Many foundational properties are cited from earlier papers rather than proven, which is normal for a theory-building paper. The comparison with the Morita class in Appendix A is explicitly conditional on Conjecture A.4, and the main results do not depend on it.

This paper is for tropical geometers and anyone working on Ceresa classes or Abel-Jacobi-type obstructions. It deserves a serious referee: the referee should double-check Theorem 2.3(3) and ask the authors to fix the proof, but the overall framework and the explicit formulas are solid. I would want to see a revision, not a rejection.

Best,

[Your name]

Referee Report

2 major / 5 minor

Summary. The paper develops a tropical analogue of Abel-Jacobi theory for compact tropical varieties of arbitrary dimension. It defines intermediate Jacobians JH^{p,q}(X) as quotients of tropical cohomology by the image of the monodromy operator, constructs an Abel-Jacobi map AJ: A^o_p(X) -> JH^{p+1,p}(X), proves functoriality and an obstruction to algebraic equivalence, and introduces a tropical Albanese variety. For tropical curves, the paper identifies JH^{1,0}(Gamma) with the Mikhalkin-Zharkov Jacobian and recovers the classical Abel-Jacobi map of [MZ08]. It then defines pointed and unpointed tropical Ceresa classes and gives explicit combinatorial formulas (Theorems 6.4 and 6.8) in terms of a spanning tree and edge lengths, with worked examples for K4 and TL3. Appendix A compares the unpointed class with the Morita class of [CEL24], conditional on a new conjecture on the Johnson homomorphism, and this conditional status is clearly stated.

Significance. If the central construction is correct, this is a substantial step: it provides a systematic and functorial tropical intermediate Jacobian theory that goes beyond the curve case, gives a purely combinatorial Ceresa class formula, and connects the tropical class to the Morita and Johnson-homomorphism literature. The paper's strengths include explicit chain-level computations, checkable formulas that reproduce prior computations in [Zha15, CEL24, CL24, Rit24], and a transparent separation of the conditional comparison in Appendix A from the main results. The two issues I raise are local and fixable, but one of them occurs in the foundational monodromy lemma on which the well-definedness of AJ relies, so the manuscript needs a revision rather than a simple accept.

major comments (2)

[Section 2.11, Theorem 2.3(3)] The proof of Theorem 2.3(3) proves only the case in which the boundary of gamma is exactly equal to the fundamental chain of an algebraic cycle, not the stated hypothesis that d(gamma) is represented by an algebraic cycle modulo B_{p,p}(X). Under the stated hypothesis one has d(gamma) = alpha + d(theta), and the marking-change difference acquires the extra term [Q, u ^ (d(theta))_Q]. The printed computation identifies the Q-coefficient of d(gamma) with a_Q v_Q, which is valid only when theta = 0. The statement is nevertheless true: writing theta = sum_P [P, w_P] gives [Q, u ^ (d(theta))_Q] = (d( sum_P [P, u ^ w_P] ))_Q, so the extra term is still a boundary. Because Theorem 2.3(4) invokes (3) and Proposition 3.6 uses Theorem 2.3(4), this gap is in the logical chain supporting the functoriality of AJ, although the exact-boundary case used in the main construction of AJ is proved. Please repair the proof, or else state the exact-boundary version as a separate lemma and prove the general modulo-boundaries version with the missing theta term.
[Section 3.5, Proposition 3.13] The universal property of the tropical Albanese is stated without a basepoint hypothesis. The Abel-Jacobi map AJ: X -> Alb(X) used in the diagram is defined from a basepoint and sends that basepoint to the identity of Alb(X). If the given morphism X -> A does not send that basepoint to the identity of A, the required factorization cannot exist, so the proposition as stated is false. The proof also does not address uniqueness. The statement should require that f sends the chosen basepoint to 0_A, or should be reformulated up to translation, and uniqueness should be justified by the induced map on H_{1,0} or by the universal property of the quotient torus.

minor comments (5)

[Section 2.11, Theorem 2.3(1)] The target of the induced map is printed as H_{p-1,q+1}(X), which contradicts the definition N: H_{p,q}(X) -> H_{p+1,q-1}(X) at the beginning of the section; the indices should be corrected.
[Section 2.8] In the sentence defining the sheaf versions, the left-hand side is written as F_p(U) = F_p(U)^* and F_p^Z(U) = (F_p^Z(U))^*, but the left-hand sides should use the sheaf notation F^p or F^p_Z to avoid identifying the cosheaf with its dual.
[Section 3.1] In the sentence 'Given a pair of integers q <= p, the (p,q)-th define L_{p,q}(X) := ...', the word 'intermediate Jacobian' or 'lattice' is missing; the sentence should be completed.
[Section 6.2, proof of Theorem 6.6] In the displayed representatives of the cycles, the second cycle is written as Y(Gamma_♭) = sum [J s_e + ♭, t_e + ♭ K, b_e], but the second basepoint should be denoted by a different symbol, say ♭', to avoid confusion with the first ♭.
[Appendix A, Proposition A.2] The notation B(delta_Gamma) is reused for two different groups: the earlier quotient W_{-4}/((delta_Gamma - I)W_{-2} + W_{-6}) and the displayed quotient X⊗Y∧Y/((delta_Gamma - I)(X∧X⊗Y)). The sentence 'B(delta_Gamma) ~= B(delta_Gamma)/omega ∧ Y' should be rewritten with distinct symbols for these two quotients.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Abel-Jacobi map and Ceresa formulas are derived from the chain-level monodromy N with no fitted parameters; Appendix A is explicitly conjectural.

full rationale

The construction is self-contained: L^{p,q}(X)=im(N^{p-q}:H^{q,p}(X,Z)->H^{p,q}(X,R)), JH^{p,q}=H^{p,q}/L^{p,q}, and AJ is defined by choosing a bounding chain gamma with d(gamma)=alpha and setting AJ(alpha)=Psi_gamma. Well-definedness is proven in Props. 3.3-3.5 and Theorem 3.10 from the chain-level identities for N (Theorem 2.3), not by importing a fitted target. In the curve case, Theorem 5.1 identifies JH^{1,0} with Jac(Gamma) and the two Abel-Jacobi maps via the proved equality N=I (Prop. 5.2); this is a comparison, not a definitional renaming. The Ceresa formulas (Thms. 6.4 and 6.8) are computed from an explicit bounding chain xi_flat(Gamma) and the torus monodromy computation in Prop. 4.1; no parameter is fitted from the results being 'predicted'. The agreement with [CEL24], [CL24], and [Rit24] in Sections 6.5-6.6 is offered as a consistency check. The only result that depends on a genuinely new input is Theorem A.3, and the paper states it conditionally: 'If Conjecture A.4 holds, then we have Phi_Gamma(n(Gamma)) = v(Gamma)'; the conjecture is labeled as conjectural and is not needed for the main claims. Self-citations such as [AP20] for the weight-monodromy property and [MZ14] for the eigenwave operator provide background or context; the core well-definedness of AJ does not rely on them for the exact-boundary chains used in the paper. I find no circular step. (A separate, non-circular proof gap in Theorem 2.3(3): the line 'Because d(gamma) is the fundamental class of an algebraic cycle' assumes exact equality with the fundamental chain, whereas the theorem's hypothesis allows d(gamma)=alpha+d(theta) modulo boundaries; the missing theta term is absent from the displayed computation. This is a correctness issue, not circularity, and the applications use the exact-boundary case alpha=d(gamma).)

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted constants or hand-adjusted parameters appear in the paper; edge lengths and graph data are arbitrary inputs to the formulas. All objects (intermediate Jacobians, Albanese, Ceresa class) are constructions from existing tropical homology and monodromy; no new physical or geometric primitives are postulated, and the explicit formulas give falsifiable consistency checks against prior computations.

assumptions (4)

domain assumption Tropical homology groups H^{p,q}(X,A) and their cellular chain complex description exist as reviewed in Section 2.8, with Poincare duality for Kaehler tropical varieties and projective tropical manifolds.
The definitions and properties are cited to [IKMZ19, MZ14, JSS19, AP20]; the paper builds on them without reproving them.
domain assumption The weight-monodromy property (WMP) holds for Kaehler tropical varieties, so N^{p-q}: H^{q,p}(X,R) to H^{p,q}(X,R) is an isomorphism and the intermediate Jacobian is a real torus.
Invoked in Section 2.12 and used for the torus structure of JH and for Theorem 3.12; proved in [AP20].
domain assumption The classical tropical Jacobian and Abel-Jacobi map for curves of [MZ08] are taken as the benchmark; Theorem 5.1 compares the new construction to them.
Used in Sections 5 and 6 to identify JH^{1,0}(Gamma) with Jac(Gamma).
ad hoc to paper Conjecture A.4 gives a formula for the Johnson homomorphism on T_gamma T_{gamma'}^{-1}; Theorem A.3 comparing the unpointed Ceresa class to the Morita class is conditional on it.
The paper introduces this as a new conjecture, not a proven result; it is clearly flagged and not needed for the main theorems.

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Pith. "Pith review of Tropical Abel-Jacobi theory." pith.science (2026). https://pith.science/paper/Y4OMC5AP

@misc{pith2026250414415,
  author       = {Pith},
  title        = {Pith review of: Tropical Abel-Jacobi theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y4OMC5AP}},
  note         = {Machine review of arXiv:2504.14415}
}

read the original abstract

To a compact tropical variety of arbitrary dimension, we associate a collection of intermediate Jacobians defined in terms of tropical homology and tropical monodromy. We then develop an Abel-Jacobi theory in the tropical setting by defining functorial Abel-Jacobi maps. We introduce, in particular, tropical Albanese varieties and formulate obstructions to algebraic equivalence of tropical cycles. In dimension 1, we show that this recovers the existing Abel-Jacobi theory for tropical curves. As an application, we consider the Ceresa class of a tropical curve which is defined as the image of the Ceresa cycle in an appropriate intermediate Jacobian under the Abel-Jacobi map. We give an explicit formula for this class entirely in terms of the combinatorics of the tropical curve.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

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