{"id":"6811cd25-1e0c-4f47-a051-4b1f23434c6e","arxiv_id":"2412.04169","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves arithmetic intersection and minimum formulas for toric bundles and computes heights and successive minima of semiabelian compactifications.","lead":"This paper develops an Arakelov-geometric theory of toric bundles, proving formulas for Okounkov bodies, intersection numbers, and successive minima. It uses these to compute heights of compactified semiabelian varieties, generalizing earlier work by Chambert-Loir to arbitrary toric compactifications.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.4.3's non-archimedean continuity argument is sketched and depends on an unverified application of [FL17, Lemma 3.7] to singular/reducible special fibres; if it fails, the left-hand side of Theorem B is not shown to be well-defined.","rationale":"I read the paper as an ambitious extension of the HKM21 bundle BKK formula to Arakelov geometry, with the main novelty being the operational arithmetic Chow theory of Section 2.4 and the adelic polytope machinery. Theorems A, C, D are natural and the computations for semiabelian varieties align with Chambert-Loir; there is no evidence of circularity or fitting. The single most load-bearing assumption is the statement of Proposition 2.4.3 that the arithmetic intersection pairing extends to all integrable adelic line bundles against operational classes. Theorem B explicitly says 'The intersection numbers below are well-defined', and its proof invokes this proposition. In the non-archimedean part of the proof, the reduction to algebraic intersection on the special fibre requires writing the pulled-back operational class as a difference of nef dual cycle classes on each irreducible component. This is cited to [FL17, Lemma 3.7], but the hypotheses of that lemma are not checked for the singular/reducible special fibres that can occur, and the uniform bound needed for continuity is not fully written. If this step fails, the left-hand side of Theorem B has no independent meaning. I agree with the reader's weakest_assumption, and I do not see another issue that precedes this one: Proposition 3.5.2 is important for the proof but would not invalidate the statement if Prop 2.4.3 held. The paper explicitly flags the operational theory as preliminary, which supports this concern. A targeted check of [FL17, Lemma 3.7] and a full expansion of the approximation argument would settle the matter.","tokens_in":37431,"tokens_out":15708,"duration_ms":157726,"concrete_test":"Locate [FL17, Lemma 3.7] and record its exact hypotheses (e.g., normality, Q-factoriality, algebraically closed base field, finite-dimensionality of numerical groups). Then test whether every irreducible component of the special fibre X'_v of a flat projective S-model of a smooth projective variety over a global field satisfies these hypotheses. If not, the proof of Proposition 2.4.3 has a gap for exactly the cycles γ allowed in Theorem B. Additionally, write out the omitted approximation: for two semipositive metrics at sup-norm distance ≤ ε, represent their difference by an effective vertical divisor and verify the bound |deg(c1(L_1)...c1(L_k)φ) difference| ≤ Cε using the nef decomposition; if the bound cannot be established, Theorem B should be restricted to γ in the image of yCH^*(B) or to bases admitting regular models.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem B, which asserts i!ρ(Δ)^{t+i}π^*γ = (t+i)!∫_Δ (c(m)+θ(m)[∞])^i γ dm. For the left-hand side to be meaningful, the paper needs the extension of arithmetic intersection to integrable line bundles against operational classes A^*(X) constructed in Proposition 2.4.3. At a non-archimedean place, the proof bounds intersections of a vertical divisor D by reducing to algebraic intersection on the special fibre and asserting that i^*φ can be written as the difference of nef dual cycle classes, citing [FL17, Lemma 3.7] for full-dimensionality of the nef cone. This is the load-bearing step: special fibres of flat projective models with smooth generic fibre can be singular and reducible, and the paper does not verify that the hypotheses of [FL17, Lemma 3.7] hold for each irreducible component, nor that the decomposition is compatible with the operational pullback i^*. The argument then concludes continuity of the pairing on semipositive metrics from a bound on L = O(D) with D effective vertical, but the passage from that bound to all integrable metrics (difference of uniform limits of semipositive model metrics) is only stated, not proved. The paper itself calls A^*(X) 'preliminary' in Section 2.4 and notes that further compatibility relations may be needed for some applications. Thus, unless the non-archimedean continuity is supplied, Theorem B is not well-defined for the stated generality of γ ∈ A^{g+1-i}(B).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an Arakelov-geometric framework for toric bundles over a projective base, introducing adelic torus bundles, a map rho from toric divisors on the model fibre to divisors on the total space, and the notion of adelic polytopes. It computes Okounkov bodies and Boucksom-Chen transforms for line bundles of the form rho(D)+pi^*L (Theorem 4.0.1 and Theorem 4.1.1), derives formulas for essential and absolute minima (Theorem A), and proves an arithmetic analogue of the Hofscheier-Khovanskii-Monin bundle BKK formula (Theorem B). These results are applied to compactified semiabelian varieties, yielding formulas for heights and successive minima (Theorems C and D) that recover and generalize computations of Chambert-Loir.","tokens_in":37750,"tokens_out":2788,"duration_ms":28293,"significance":"If the proofs are completed, Theorem B is a substantial unification: it extends the arithmetic BKK theorem of Burgos Gil-Philippon-Sombra and the topological bundle BKK theorem of Hofscheier-Khovanskii-Monin to arithmetic intersection numbers on toric bundles with integrable adelic metrics. The framework of adelic polytopes and the rho map provides a coherent language for transferring convex-geometric arguments to arithmetic settings. The applications to semiabelian varieties give clean, explicit height and minima formulas that recover earlier work of Chambert-Loir and go beyond it to arbitrary toric compactifications. The paper is clearly written and contains several original structural contributions, including the category-theoretic treatment of torus bundles and the systematic use of operational arithmetic Chow cohomology.","major_comments":[{"comment":"The extension of the arithmetic intersection pairing to arbitrary integrable line bundles is not adequately justified in the non-archimedean case. The proof bounds intersections with a vertical divisor D by writing i^*phi as a difference of nef dual cycle classes on the special fibre, citing [FL17, Lemma 3.7] for full-dimensionality of the nef cone. However, the special fibre of a flat projective model with smooth generic fibre can be singular and reducible, and the paper does not verify that the hypotheses of [FL17, Lemma 3.7] hold on each irreducible component, nor that the decomposition is compatible with the operational pullback i^*. The subsequent passage from semipositive metrics to all integrable metrics (differences of uniform limits of semipositive model metrics) is stated without proof. Since the left-hand side of Theorem B uses this extension for arbitrary gamma in A^{g+1-i}(B), Theorem B is not shown to be well-defined at the stated level of generality unless this continuity argument is supplied.","section":"Section 2.4, Proposition 2.4.3"},{"comment":"The approximation of an integrable torus bundle by algebraic metrics is only sketched. The proof approximates each pT(m_i) on a basis of M by differences of limits of algebraic semipositive metrics and then extends linearly to all of M, but it does not check that the resulting family pT_k is monoidal, i.e. that the isomorphisms pT_k(m_1+m_2) ~ pT_k(m_1) ⊗ pT_k(m_2) are compatible with the algebraic approximations. The uniform convergence of pρ_{pT_k}(D) to pρ_{pT}(D) is asserted from Lipschitz continuity and convergence of v_k, but the details are not given. This proposition is used in the proof of Lemma 5.4.1 to establish vanishing of higher derivatives of pF_γ in the non-algebraic case; without a complete proof, the induction argument for Theorem B does not cover general integrable torus bundles.","section":"Section 3.5, Proposition 3.5.2"},{"comment":"In the computation for the case where τ_1,...,τ_{t+1} span a maximal cone, the proof states 'We now apply that pρ(pA,θ_v(A)q) = π^*c(pA,θ_v(A)q) and an explicit projection formula.' This identity is not justified in the text: pρ is defined on divisors/polytopes, while c is a map M -> Pic(B), so the meaning of pρ of a point (A,θ_v(A)) and its identification with a pullback from the base requires a nontrivial compatibility statement. Since this step is used to extract the factor pc(A,θ_v(A))^i from the intersection, the argument would benefit from a precise statement and proof of the claimed projection formula in the arithmetic setting.","section":"Section 5.4, Lemma 5.4.1"}],"minor_comments":[{"comment":"The phrase 'Let T be a T-torsor over B, for a split torus T' in the introduction uses the same symbol T for the torus and the torsor, which is confusing; the torsor is later denoted T in Section 3.1 but the double use of T persists.","section":"Introduction, Section 3.2"},{"comment":"The proof of Proposition 3.2.5 says 'This is a combination of Proposition 3.2.5 and [BPS14, Theorem 3.2.4]', which is circular; it should refer to the relevant classification of torus bundle morphisms (e.g. Proposition 3.1.5) rather than to itself.","section":"Proposition 3.2.5"},{"comment":"The global roof function θ is defined as a finite sum Σ n_i θ_i with weights n_i that are 'clear from the context'; for adelic divisors the weights are later specified as 1 for function fields and [K_i:Q_i]/[K:Q] for number fields, but the dependence on the normalization of the arithmetic intersection product is not explained at the point of definition.","section":"Definition 2.1.9"},{"comment":"In the displayed formula of Theorem B the integration variable dm is not explicitly identified (Lebesgue measure on M_R) and the class r8s is introduced only in the preceding paragraph; a short sentence clarifying the normalization of the measure and the class would improve readability.","section":"Theorem B"},{"comment":"The operational arithmetic Chow cohomology A^*(X) is called 'preliminary' and the paper notes that further compatibility relations may be needed for some applications; this caveat should be revisited after the proof of Proposition 2.4.3 is completed, since Theorem B relies essentially on this construction.","section":"Section 2.4"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a serious contribution with a plausible central result, but the two compressed technical points (Proposition 2.4.3 for non-archimedean places and Proposition 3.5.2) are genuinely load-bearing for Theorem B. They are local and likely fixable, but they need to be written out in full or the theorem needs to be restricted to a setting where they are known. I would recommend major revision rather than rejection, with the understanding that the completed proofs must be checked before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read Hultberg's paper. The definite new thing is Theorem B: an arithmetic bundle BKK identity that genuinely contains both the Burgos Gil–Philippon–Sombra arithmetic BKK theorem and the Hofscheier–Khovanskii–Monin bundle BKK theorem. Theorems C and D go beyond Chambert-Loir's semiabelian computations and apply to arbitrary toric compactifications, not just projective space. The adelic polytope language is a clean way to carry the convex geometry through, and the proof strategy—comparing the two sides as polynomials on virtual polytopes—looks sound. I checked the places where it recovers known results, and the cross-checks are consistent.\n\nThe soft spots are exactly where the paper itself is most compressed. Proposition 2.4.3 extends the arithmetic intersection pairing to integrable line bundles against operational classes. The non-archimedean step says it suffices to write the pullback as a difference of nef dual cycle classes and cites [FL17, Lemma 3.7] for full-dimensionality of the nef cone. I don't think the hypotheses are checked for singular or reducible special fibres, and the passage from a bound for semipositive metrics to all integrable metrics is stated rather than proved. Since Theorem B's left-hand side is defined through this pairing, this is load-bearing. Proposition 3.5.2, the approximation of integrable torus bundles by algebraic metrics, is also sketched: linear extension from a basis of M needs a monoidal compatibility argument that is not spelled out. The proof of Theorem D is shorter than I'd like, but it is not the main worry.\n\nThese are fixable gaps, not signs of a wrong central argument. The paper is honest: Section 2.4 is explicitly labelled preliminary, and the reliance on Chambert-Loir's intersection computation is clearly cited and legitimate. My verdict is conditional: I believe the theorems are correct, but I would not use Theorem B at full generality until Section 2.4 is expanded and the approximation argument in 3.5 is fully justified. The paper is for arithmetic geometers working on toric methods and heights; they will get a useful framework and new formulas.\n\nI'd send this to a serious referee. The referee should ask for a revision that proves the non-archimedean continuity properly and expands 3.5, but the paper deserves referee time.","headline":"A genuinely new arithmetic bundle BKK theorem for toric bundles, with solid-looking proofs and two compressed foundational sections that need expansion before the full claims are usable.","tokens_in":38282,"tokens_out":2778,"would_cite":true,"duration_ms":27708,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G40","14M25","52B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves an arithmetic bundle BKK identity in which Arakelov intersections on toric bundles equal an integral over the Newton polytope of base intersection data, and derives from it explicit heights and minima for compactified…","keywords":["toric variety","toric bundle","semiabelian variety","Arakelov geometry","Okounkov body","polytope","adelic line bundle","successive minima"],"falsifier":"Take $B$ an elliptic curve over a number field, a rank-one adelic torus bundle whose single character is a non-torsion class in $\\widehat{\\mathrm{Pic}}(B)$ with a non-algebraic integrable metric, $\\Delta=[0,1]$, $i=1$, and $\\gamma$ the class of a closed point. Compute the model-based left-hand side $\\rho(\\Delta)^{t+1}\\pi^*\\gamma$ independently for a sequence of models and compare it with the integral $\\int_0^1(pc(m)+\\theta(m)[\\infty])\\gamma\\,dm$; a discrepancy would refute the theorem, as would an example in which the linear extension of approximating algebraic metrics in Proposition 3.5.2 fails to be monoidal.","tokens_in":37165,"feed_emoji":"📐","tokens_out":10985,"duration_ms":109788,"temperature":0.7,"pith_summary":"One central claim organizes the paper: on an integrable adelic toric bundle over a smooth projective base, arithmetic intersection numbers of toric divisors against base classes are not transcendental invariants but ordinary integrals of base intersection data over the Newton polytope. The precise statement is the arithmetic bundle BKK theorem, which fixes the mixed number $i!\\rho(\\Delta)^{t+i}\\pi^*\\gamma$ as $(t+i)!\\int_\\Delta (pc(m)+\\theta(m)[\\infty])^i\\gamma\\,dm$. To reach it the paper computes the Okounkov body and Boucksom-Chen transform of a toric-bundle line bundle as a family fibred over the polytope, and derives from this the essential and absolute Zhang minima of such bundles. If the theorem is right, heights and successive minima of toric compactifications of semiabelian varieties are explicit integrals and facewise maxima/minima of ordinary height functions, recovering and generalizing a previous computation for the standard simplex compactification.","feed_headline":"Toric-bundle heights reduce to a single polytope integral","feed_subtitle":"The formula turns Arakelov intersections into base-data integrals, with explicit heights for semiabelian compactifications","key_machinery":"The load-bearing objects are adelic torus bundles—symmetric monoidal functors $T:M\\to\\widehat{\\mathrm{Pic}}(B)$ from the character lattice to the category of adelically metrized line bundles—and the induced maps $\\rho$, $\\rho^{\\mathrm{metr}}$, and $\\hat{\\rho}$ that turn a toric divisor on the model fibre $X_\\Sigma$ into a divisor with metric on the total space of the bundle. The section/eigenspace orthogonality of toric line bundles lets the paper identify global sections of $\\hat{\\rho}(\\Delta)+\\pi^*L$ with sections of $L+T(m)$ on the base. The arithmetic convex chains of Section 5.2, a version of convex-chain algebras over virtual polytopes adapted to roof functions, supply the polynomiality that lets the author compare the two sides of the BKK identity by differentiating along the rays of an adelic fan.","core_discovery":"The discovery is that the arithmetic analogue of the bundle BKK formula holds: for an adelic integrable projective toric bundle $X$ of relative dimension $t$ over a smooth projective base $B$ of dimension $g$, and for any $\\gamma\\in A^{g+1-i}(B)$, the intersections on the left are well-defined and satisfy $i!\\rho(\\Delta)^{t+i}\\pi^*\\gamma=(t+i)!\\int_\\Delta(pc(m)+\\theta(m)[\\infty])^i\\gamma\\,dm$. The data entering the integral are the adelic character map $pc:M\\to\\widehat{\\mathrm{Pic}}(B)$ of the underlying torus bundle, the Newton polytope $\\Delta$ with its roof function $\\theta$, and the archimedean class $[\\infty]$ with constant Green functions. The proof compares two homogeneous polynomial functions on the space of adelic polytopes by differentiating along rays of an adelic fan, using arithmetic convex chains. As a by-product, the Okounkov body of $\\pi^*L+\\rho(\\Delta)$ is the closure of $\\{(m,x):m\\in\\Delta,\\,x\\in\\Delta_{B}(L+T(m))\\}$ and the Boucksom-Chen transform is $\\theta(m)+G_{L+T(m)}(x)$, which yields the stated formulas for Zhang minima and for heights and successive minima of compactified semiabelian varieties.","pith_inferences":["An editorial extension: the computed Boucksom-Chen transform makes toric bundles a natural testbed for equidistribution of small points, since the paper notes the relevant criterion is expressible through such transforms; one could try to verify the equidistribution condition directly from the fibred formula.","The convex-chain method should also yield an arithmetic description of a suitable subring of the arithmetic Chow ring of a toric bundle, replacing the topological arguments used in the geometric bundle BKK theorem; the paper poses this as an open question.","A testable extension is to transfer Theorem B from operational arithmetic Chow cohomology to the homological b-cycle groups defined in Section 2.4; the same derivative comparison would then produce intersection numbers for arbitrary cycles on non-regular models.","A further consequence left implicit is that the identity is polynomial in the adelic polytope, so Minkowski-sum and translation rules for heights on toric bundles should follow the classical BKK additivity pattern."],"forward_implications":["If the theorem is right, the height of any toric compactification of a semiabelian variety with split torus part is $-(d+1)!\\int_\\Delta \\hat{h}(c(m))\\,dm$, so heights become ordinary integrals of the canonical height over the Newton polytope, recovering the standard-simplex computation for every fan.","Zhang minima of toric-bundle line bundles are governed by the maxima and minima over $m\\in\\Delta$ of $\\zeta(L+c(m))+\\theta(m)$, so small-point questions on the bundle reduce to small-point questions on the base twisted by torus characters.","Okounkov bodies of toric bundles fibre over the Newton polytope with fibres equal to Okounkov bodies of base line bundles, giving a geometric picture in which the arithmetic volume of the bundle is an integral of base arithmetic volumes.","The arithmetic bundle BKK identity polarizes: mixed intersection numbers of several toric divisors are symmetric multilinear functions obtainable from the same polytope integral, so all such pairings are computable from base data.","Successive minima of compactified semiabelian varieties are constant up to the dimension of the abelian quotient and are then determined by faces of the polytope, organizing the height filtration entirely by face geometry."],"supporting_citations":[{"why":"The geometric bundle BKK theorem and the convex-chain method that this paper arithmetizes; Theorem B is its arithmetic analogue.","marker":"[HKM21]"},{"why":"The Arakelov dictionary for toric varieties: metrized polytopes, roof functions, and the toric intersection and volume formulas used throughout.","marker":"[BPS14]"},{"why":"Defines the Boucksom-Chen transform and filtered Okounkov bodies used to compute minima and arithmetic volumes.","marker":"[BC11]"},{"why":"Introduces adelic fibre bundles and torsors, the framework in which toric bundles are here redefined.","marker":"[CT01]"},{"why":"Supplies polynomial finitely additive measures of virtual polytopes, the source of the arithmetic convex chains used to prove polynomiality in Theorem B.","marker":"[PK92]"},{"why":"The semiabelian height and minima computations that Theorems C and D extend to arbitrary toric compactifications.","marker":"[Cha00]"},{"why":"Lemma 3.7 is used in Proposition 2.4.3 to extend intersection numbers to integrable metrics at finite places.","marker":"[FL17]"},{"why":"Provides the integrable adelic intersection pairing and Zhang minima in which the statements are formulated.","marker":"[Zha95b]"}],"fun_headline_variants":["Arithmetic BKK: toric bundle heights from polytope integrals","Toric bundle intersections become polytope integrals","Heights of semiabelian compactifications via one polytope","Adelic toric bundles: Arakelov intersections as integrals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The formula is only meaningful if the arithmetic intersection pairing extends from nice model metrics to all integrable metrics on the base; the proof of that extension assumes every base cycle class at a finite place can be written as a difference of nef cycle classes, so an integrable metric for which that decomposition does not control the intersection product would make the left-hand side of the theorem undefined.","fun_headline_variants_meta":{"raw":{"variants":["Arithmetic BKK: toric bundle heights from polytope integrals","Toric bundle intersections become polytope integrals","Heights of semiabelian compactifications via one polytope","Adelic toric bundles: Arakelov intersections as integrals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000236,"raw_usage":{"total_tokens":1499,"prompt_tokens":933,"completion_tokens":566,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":494}},"tokens_in":549,"tokens_out":566,"duration_ms":5707,"temperature":1.0,"reasoning_tokens":494,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:41:11.799620+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $B$ an elliptic curve over a number field, a rank-one adelic torus bundle whose single character is a non-torsion class in $\\widehat{\\mathrm{Pic}}(B)$ with a non-algebraic integrable metric, $\\Delta=[0,1]$, $i=1$, and $\\gamma$ the class of a closed point. Compute the model-based left-hand side $\\rho(\\Delta)^{t+1}\\pi^*\\gamma$ independently for a sequence of models and compare it with the integral $\\int_0^1(pc(m)+\\theta(m)[\\infty])\\gamma\\,dm$; a discrepancy would refute the theorem, as would an example in which the linear extension of approximating algebraic metrics in Proposition 3.5.2 fails to be monoidal.","supporting_citations":[],"review_version":1}