{"id":"8cfdbe9d-13f2-4a06-9af7-48ccca025f3f","arxiv_id":"2504.14415","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A functorial Abel-Jacobi map is constructed for all compact tropical varieties, and the tropical Ceresa class of a curve is computed explicitly from the graph and edge lengths.","lead":"This paper builds a tropical version of Abel-Jacobi theory, attaching intermediate Jacobians and Abel-Jacobi maps to tropical varieties of any dimension. It also gives an explicit combinatorial formula for the tropical Ceresa class of a curve, useful for detecting when a cycle is algebraically trivial.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.3(3) proof assumes ∂γ is exactly an algebraic fundamental chain, not merely equal modulo boundaries; the missing θ term makes the proof incomplete, though the gap is fixable and the exact case used by AJ is proven.","rationale":"The reader correctly identified Theorem 2.3(3) as the load-bearing step for the Abel–Jacobi construction. My reading agrees that this is the place where the construction is most delicate, but the precise issue is different from what the reader stated: it is not a suspected failure of marking independence for chains with algebraic boundaries, but an expository gap in the proof of the modulo-boundary version of the theorem. The proof as printed equates a chain-level coefficient of ∂γ with the coefficient of an algebraic fundamental chain, which is legitimate only when the boundary is exactly algebraic. The hypothesis of Theorem 2.3(3) allows a boundary-term ambiguity, and the missing θ term is not discussed. However, the gap is easily closed: the ambiguous term is itself a boundary, so the conclusion of the theorem remains true. Moreover, the central applications of the paper, including the definition of AJ and the Ceresa class computations, only need the exact-boundary case, and that case is fully justified by the written proof. I found no other load-bearing concern in the construction: the chain-level monodromy is a chain map, the quotient by L gives a well-defined Abel–Jacobi map for homologically trivial cycles, functoriality follows from Theorem 2.3(4), and the vanishing on rationally equivalent cycles is handled by an explicit bounding chain computation. The Ceresa formulas are consistent with the known examples and with [CEL24] and [Rit24] in the cases checked. The conjectural part of Appendix A is clearly labeled and does not affect the main theorems. For these reasons the appropriate recommendation is to keep the reader's ACCEPT verdict, with a minor revision requested only for the proof of Theorem 2.3(3).","tokens_in":38928,"tokens_out":33795,"duration_ms":299384,"concrete_test":"Re-derive the marking-change calculation in the proof of Theorem 2.3(3) with ∂γ written as α + ∂θ, where α = Σ_R a_R[R,v_R] is the fundamental chain of an algebraic p-cycle and θ ∈ C_{p,p+1}(X). After changing the marked point of a p-face Q, compute the difference N(γ) − N′(γ); verify that the extra term [Q, u∧(∂θ)_Q] equals ∂(Σ_P [P,u∧w_P]) for θ = Σ_P[P,w_P], so the total difference is a boundary in B_{p+1,p}(X). If this identity fails, Theorem 2.3(3) is false; if it holds, the stated theorem is correct and the written proof needs one additional sentence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The well-definedness of AJ and the Ceresa formulas all rest on Theorem 2.3(3): for a (p,p+1)-chain γ whose boundary represents an algebraic p-cycle, N(γ) modulo B_{p+1,p} is independent of the choices of markings. In the proof, when the marking of a p-face Q is changed, the difference is written as Σ_{P·≻Q} sgn(P,Q)[Q,u∧π_{P,Q}(z_P)] = a_Q[Q,u∧v_Q] = 0. The middle equality identifies the Q-coefficient of ∂γ with the coefficient a_Q v_Q of the fundamental chain of the algebraic cycle. This is justified only if ∂γ equals that fundamental chain exactly. The theorem's hypothesis is weaker: ∂γ is represented by an algebraic cycle modulo B_{p,p}, i.e., ∂γ = α + ∂θ for some (p,p+1)-chain θ. Under this weaker hypothesis the difference gains the term [Q, u∧(∂θ)_Q], which is not obviously zero. The written proof does not mention θ or this extra term. The gap is fixable: writing θ = Σ_P[P,w_P], one checks that [Q,u∧(∂θ)_Q] = ∂(Σ_P [P,u∧w_P]), so the total marking-change difference is still a boundary in B_{p+1,p}(X). But this argument is absent, so the text proves only the exact-boundary case, not the full statement of Theorem 2.3(3). Every application in the paper (Prop 3.6, Thm 3.10, Thm 6.4) uses the exact-boundary case, so the central construction of AJ is not threatened; nevertheless the foundational lemma needs a corrected proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":39323,"tokens_out":11575,"duration_ms":100508,"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":[{"comment":"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":"Section 2.11, Theorem 2.3(3)"},{"comment":"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.","section":"Section 3.5, Proposition 3.13"}],"minor_comments":[{"comment":"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":"Section 2.11, Theorem 2.3(1)"},{"comment":"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":"Section 2.8"},{"comment":"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":"Section 3.1"},{"comment":"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 ♭.","section":"Section 6.2, proof of Theorem 6.6"},{"comment":"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.","section":"Appendix A, Proposition A.2"}],"recommendation":"major_revision","confidential_remarks":"Both major issues are local and fixable; the main construction of AJ and the Ceresa formulas remain defensible after the correction to Theorem 2.3(3) and the added basepoint hypothesis in Proposition 3.13. The conditional comparison in Appendix A is clearly labeled and does not affect the main theorems. I see no circularity or novelty disclosure concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThis 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.\n\nThe 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.\n\nOther 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.\n\nThis 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.\n\nBest,\n\n[Your name]","headline":"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.","tokens_in":39790,"tokens_out":2359,"would_cite":true,"duration_ms":19522,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14T10","14T20","05C25","05E14","14C25","14H40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Abel-Jacobi map","Albanese","Algebraic cycles","Ceresa cycle","Intermediate Jacobian","Tropical variety","Tropical homology","Monodromy"],"falsifier":"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.","tokens_in":38724,"feed_emoji":"🌴","tokens_out":10181,"duration_ms":86723,"temperature":0.7,"pith_summary":"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.","feed_headline":"Tropical varieties now have Abel-Jacobi maps in every dimension","feed_subtitle":"One monodromy construction recovers curve Jacobians and turns the Ceresa class into a graph sum.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the tropical homology groups $H^{p,q}$ that form the coefficient spaces for the intermediate Jacobians.","marker":"[IKMZ19]"},{"why":"Defines the eigenwave or monodromy operator and intermediate Jacobians for tropical manifolds, which the paper extends to all dimensions.","marker":"[MZ14]"},{"why":"Gives the Abel-Jacobi map and Jacobian for tropical curves that Theorem 5.1 recovers from the new construction.","marker":"[MZ08]"},{"why":"Supplies the weight-monodromy and Kähler package under which the intermediate Jacobians are genuine real tori.","marker":"[AP20]"},{"why":"Provides chain-level tropical homology and Lefschetz properties used in proving $N\\circ cl=0$ and functoriality.","marker":"[JRS18]"},{"why":"Introduces the classical Ceresa cycle whose tropical analogue motivates the Ceresa class computations.","marker":"[Cer83]"},{"why":"Defines the related Morita or tropical Ceresa class for integer edge lengths that Appendix A compares with $v(\\Gamma)$.","marker":"[CEL24]"},{"why":"Gives a tropical obstruction to algebraic equivalence of Ceresa cycles that is recovered and generalized by the monodromy obstruction.","marker":"[Zha15]"},{"why":"Provides an alternate bounding chain for the tropical Ceresa cycle and a Ceresa period computation compared in Section 6.4.","marker":"[Rit24]"},{"why":"Contains the classical basepoint-dependence formula whose tropical analog is Theorem 6.6.","marker":"[HR04]"}],"fun_headline_variants":["Abel-Jacobi maps now exist for every compact tropical variety","Tropical Abel-Jacobi theory: from curves to all dimensions","Explicit tropical Ceresa class via graph combinatorics","Functorial Abel-Jacobi maps on all compact tropical varieties","Tropical intermediate Jacobians tame Abel-Jacobi and Ceresa"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Abel-Jacobi maps now exist for every compact tropical variety","Tropical Abel-Jacobi theory: from curves to all dimensions","Explicit tropical Ceresa class via graph combinatorics","Functorial Abel-Jacobi maps on all compact tropical varieties","Tropical intermediate Jacobians tame Abel-Jacobi and Ceresa"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000268,"raw_usage":{"total_tokens":1595,"prompt_tokens":900,"completion_tokens":695,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":611}},"tokens_in":516,"tokens_out":695,"duration_ms":6308,"temperature":1.0,"reasoning_tokens":611,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:49:00.972140+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}