{"id":"935a2924-df54-4d42-9d95-fcd8f67f1ebe","arxiv_id":"2507.10410","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The category of strongly semiample adelic line bundles on a quasi-projective arithmetic variety is equivalent to line bundles on its Berkovich analytification with norm-equivariant continuous semipositive metrics.","lead":"An algebraic geometer shows that two recently developed frameworks for studying heights on quasi-projective arithmetic varieties, adelic line bundles and continuous plurisubharmonic metrics on Berkovich spaces, are equivalent on a natural subcategory. The paper uses this dictionary to define families of Monge-Ampère measures and a new global measure on such varieties.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.2.6 assumes without proof that the approximating Green's functions g_Ei vanish on U^ℶ; Proposition 6.2.4 only proves this for the boundary divisor's Green's function, so the boundary Cauchy estimate is not actually extended to U^an and the inverse map in Theorem A is unproved.","rationale":"The central claim, Theorem A, is an equivalence of categories. The forward direction is reasonably supported by Yuan–Zhang analytification results and the definition of continuous semipositive metrics. The delicate direction is essential surjectivity, and it depends entirely on Proposition 6.2.6. The proof derives the needed inequalities only on δ_D0(U), then extends them to U^an using the claim that both g_Ei and eg0 vanish on U^ℶ. The cited Proposition 6.2.4 proves this only for eg0; no lemma in the manuscript proves it for the approximating Green's functions. Without this, the sequence of model divisors is not shown to satisfy the boundary-norm Cauchy condition on the interior, so the inverse map is not constructed. This is an internal gap in the argument, not a disagreement with external consensus, and it is exactly the weak assumption identified by the Reader. The other issues mentioned in the Reader's verdict—the sketched transfer of Guo's theorem in Theorem 7.3.1 and the apparent parametrization inconsistency in the M(Z) measure of Section 8—are real but ancillary; they do not affect Theorem A. There is no machine-checked verification to offset the gap. The gap may be repairable by a short valuative-criterion argument, but until it is supplied, the proof of the main theorem is incomplete. The appropriate verdict remains CONDITIONAL, matching the Reader's assessment, so no adjustment is needed.","tokens_in":27195,"tokens_out":9519,"duration_ms":116318,"concrete_test":"Verify the missing normalization for the simplest nontrivial case: take U = A^1_Z with boundary divisor D0 = {∞} on the projective model P^1_Z, and let g_Ei arise from the constant section 1 of O(k∞) with k > 0. Prove from the valuative criterion that for every x ∈ U^ℶ the Yuan–Zhang model norm of this section is 1, so g_Ei(x) = 0; if this fails for some k or some x, exhibit the nonzero interior value and show that Proposition 6.2.6's inequality cannot be extended to U^an. Equivalently, re-derive the displayed estimate of Proposition 6.2.6 on all of U^an without invoking the unproved vanishing of g_Ei.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The essential-surjectivity direction of Theorem A rests on Proposition 6.2.6. After obtaining the inequality −ε_i·eg0 ≤ g_Ei − g_Ej ≤ ε_i·eg0 on the compact subset δ_D0(U) and hence on U^b, the proof asserts that the functions g_Ei and eg0 all vanish on U^ℶ by Proposition 6.2.4, and therefore the estimate holds on all of U^an. This is the key step that turns boundary control into the ε_iD0-Cauchy condition of Definition 5.1.4/5.1.5. Proposition 6.2.4 proves vanishing only for eg0, not for g_Ei. Lemma 6.2.3 supplies only that E_i|_U = 0 and that g_Ei is a tropical Fubini–Study Green's function for E_i; it contains no statement about the values of g_Ei on U^ℶ. Without g_Ei vanishing on U^ℶ, the displayed inequality fails at interior points (where eg0 = 0), so the sequence E_i need not be Cauchy in the boundary topology. One cannot simply add a multiple of eg0 to normalize, since that changes the underlying adelic divisor class by a boundary divisor. The gap may be repairable—for E_i supported on the boundary and x ∈ U^ℶ, the valuative criterion plausibly forces the relevant model section to have norm 1—but this argument is absent from the manuscript. Since Proposition 6.2.6 is the only construction of the inverse map, the central equivalence is not established as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a subcategory of Yuan--Zhang adelic line bundles on quasi-projective arithmetic varieties, called strongly semiample adelic line bundles, and a notion of continuous semipositive metric on line bundles over the Berkovich analytification. Its main theorem (Theorem A, restated as Theorem 6.2.7) asserts an equivalence between the category of strongly semiample adelic line bundles on a flat quasi-projective integral scheme U over Spec(Z) and the category of line bundles on U^an equipped with a norm-equivariant continuous semipositive metric. The paper then uses this equivalence to define fiberwise and global Monge--Ampère measures, to identify Yuan--Zhang's invariant adelic line bundle for a polarized dynamical system, and to give a trivially valued-field criterion for non-degeneracy of subvarieties.","tokens_in":27562,"tokens_out":6216,"duration_ms":75015,"significance":"If the main theorem is correct, the paper provides a useful translation between two recently developed frameworks: Yuan--Zhang's adelic line bundles on quasi-projective varieties and Pille-Schneider's global pluripotential theory on Berkovich spaces. The definition of continuous semipositive metrics and the applications to Monge--Ampère measures are natural and likely to be of interest. The proof is largely a synthesis of techniques from Song and Yuan--Zhang and is not fully self-contained, but that is appropriate for the subject. The main shortcoming is a specific missing justification in the proof of the inverse map in Proposition 6.2.6; because this map is essential to Theorem A, the central equivalence is not established as written.","major_comments":[{"comment":"The proof asserts that the functions g_{E_i} and \\tilde{g}_0 'all vanish on U^\\beth by Proposition 6.2.4', but Proposition 6.2.4 concerns only the fixed boundary Green's function \\tilde{g}_{D_0} (denoted \\tilde{g}_0 in the proof). Lemma 6.2.3 guarantees only that E_i|_U = 0 and that g_{E_i} is a tropical Fubini--Study Green's function; it says nothing about the values of g_{E_i} on U^\\beth. Without vanishing of g_{E_i} on the interior, the inequalities -\\varepsilon_i \\tilde{g}_0 \\le g_{E_i} - g_{E_j} \\le \\varepsilon_i \\tilde{g}_0 are established only on \\delta_{D_0}(U) (equivalently on U^b), not on all of U^an, because on U^\\beth the right-hand side vanishes. The Cauchy condition of Definition 5.1.5 is therefore not verified, and the construction of the inverse map does not go through. This is a load-bearing gap: Proposition 6.2.6 is the only construction of the inverse map in Theorem A. The gap may be repairable, for example by proving that for boundary-supported E_i the associated model sections can be normalized to have norm one on U^\\beth, but that argument is not present.","section":"6.2, Proposition 6.2.6"},{"comment":"The theorem claims an integration formula over infinite trivially valued fields by saying that Guo's proof 'carries over mutatis mutandis' to that setting. This is not a proof: the footnote itself notes that one must assume the field is infinite to use Guo Lemma 3.1 and that the argument relies on the theory of forms and currents on Berkovich spaces. Since Proposition 7.3.2 and the non-degeneracy criterion depend on this transfer, the application is conditional on an unproved extension of Guo's result. The authors should either supply the transfer in detail or clearly state Theorem 7.3.1 as a conjecture with the missing steps identified.","section":"7.3, Theorem 7.3.1"}],"minor_comments":[{"comment":"There are several typos and small errors: 'continouous' in the introduction, 'Monge--Am`pere' in the list of applications, 'arithemtic' in Section 2.3, and 'a L + M of strongly semiample' in Definition 6.1.1(2) should read 'a L + M is strongly semiample'.","section":"Various"},{"comment":"In the definition of morphisms of adelic line bundles, the notation 'fdiv(\\ell'_i\\iota\\ell^{-1}_i)' appears to be a typo for 'cdiv'; the symbol fdiv is not defined earlier in the paper.","section":"5.2, Definition 5.2.2"},{"comment":"The definition of the measure \\mu' in (8.1.0.1) uses the length \\ell(\\cdot) of subsets of [0,1] but does not specify that \\ell is Lebesgue measure, and the sets E \\cap I_v for a general Borel set E need not be intervals. The formula should be phrased in terms of the Lebesgue measure of f_v^{-1}(E \\cap I_v) to be unambiguous.","section":"8.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a useful synthesis, but the proof of the central equivalence hinges on Proposition 6.2.6, where the passage from boundary estimates to estimates on all of U^an is unjustified. The author should be asked to supply the missing normalization argument or to restructure the proof. The reliance on unpublished preprints ([YZ24], [Son24], [Guo25]) is heavy but not inappropriate for this subject; however, Theorem 7.3.1 needs more than a 'mutatis mutandis' statement if it is to support the non-degeneracy application."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere is my read on Morrow's paper. The headline: the categorical equivalence between strongly semiample adelic line bundles and continuous semipositive metrics on the Berkovich analytification is a natural and useful statement, and the paper does a good job setting it up. But the proof of the essential surjectivity direction has a gap in Proposition 6.2.6 that is load-bearing. I don't think it kills the result, but it needs to be fixed before the theorem is established.\n\nWhat's genuinely new: the definition of continuous semipositive metrics for quasi-projective varieties (Definition 3.3.6), using nets of tropical Fubini–Study metrics on projective models with compact convergence, is a good addition. It matches the Pille–Schneider notion in the projective case (Lemma 3.3.7) and extends it where there was no clean definition. Theorem A is then a meaningful refinement of Song's bijection: restricting to strongly semiample bundles identifies exactly the line bundles with such metrics. The applications, especially the fiberwise Monge–Ampère measures and the new description of non-degeneracy, are interesting and should be useful.\n\nThe soft spots, in order. Proposition 6.2.6 is the main issue. The proof wants to extend a boundary estimate to all of U^an by saying that g_Ei and eg0 vanish on U^ℶ. But Proposition 6.2.4 only shows this for eg0; nothing in Lemma 6.2.3 or elsewhere proves that the approximating tropical Fubini–Study Green's functions vanish in the interior. Without that, the inequality holds only on the normalized boundary, and the ε_i D0-Cauchy condition is not established. The stress-test note is right that adding a multiple of eg0 would change the divisor class. I suspect the fix is to prove a vanishing statement for the g_Ei directly, perhaps using the valuative criterion for the normalizations of the models, but it is not in the manuscript. This is a real gap, not a nitpick.\n\nLesser issues: Theorem 7.3.1 is explicitly a transfer of Guo's result and is sketched; for a paper that lists this as an application, the referee should ask for the details of the 'carries over mutatis mutandis' claim, especially the trivially valued case. The measure on M(Z) in Section 8 uses a weight 1/(v log v) and a parametrization where the endpoint identifications look inconsistent (|·|_{p,0} and |·|_{p,1} are declared equal in 8.1, contradicting the earlier description). This is minor and probably just typos, but it made me re-read the construction.\n\nWho is this for? People working with adelic line bundles, heights, and Berkovich spaces over Z. They will want to use this dictionary. It deserves a serious referee; the main theorem is plausible and important enough that the gap should be worked out. I'd send it to review, with a referee asked to focus on Proposition 6.2.6.","headline":"The main equivalence is the right dictionary and likely true, but Proposition 6.2.6 has a real gap that needs repair before Theorem A is established.","tokens_in":28084,"tokens_out":4304,"would_cite":true,"duration_ms":44851,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G40","14G22","32U05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For flat quasi-projective schemes over Spec(Z), the paper proves that strongly semiample adelic line bundles form the same category as line bundles on the Berkovich analytification carrying norm-equivariant continuous semipositive metrics.","keywords":["adelic line bundles","Berkovich spaces","pluripotential theory","Monge–Ampère measures","quasi-projective arithmetic varieties","semipositive metrics","tropical Fubini–Study metrics","norm-equivariant metrics"],"falsifier":"Take a non-projective quasi-projective variety such as $\\mathbb{A}^1_{\\mathbb{Z}}$ or $\\mathbb{P}^1_{\\mathbb{Z}}$ minus a point, choose an explicit continuous semipositive metric, and compute whether its approximating tropical Fubini–Study Green's functions can be normalized to vanish on the interior $U^{\\beth}$; if no such normalization exists, the boundary-norm Cauchy condition in Proposition 6.2.6 cannot be verified and the asserted bijection between the two class groups breaks, while a proof that the normalization always exists would complete the missing step.","tokens_in":26964,"feed_emoji":"🔗","tokens_out":16877,"duration_ms":147569,"temperature":0.7,"pith_summary":"The paper aims to show that two currently separate languages for metric data on arithmetic varieties — the algebraic language of adelic line bundles and the analytic language of pluripotential theory on Berkovich spaces — pick out the same objects once each is restricted to the right class. Its main theorem states that for every flat, quasi-projective, integral scheme $U$ of finite type over $\\mathrm{Spec}(\\mathbb{Z})$, the category of strongly semiample adelic line bundles on $U$ (those representable as a limit of semiample model line bundles carrying metrics built from global sections) is equivalent to the category of line bundles on the Berkovich analytification $U^{\\mathrm{an}}$ equipped with a norm-equivariant and continuous semipositive metric. The new ingredient is the definition of a continuous semipositive metric as a compact limit of tropical Fubini–Study metrics coming from projective models of $U$; for projective $U$ it coincides with the existing notion of continuous plurisubharmonic metric. A sympathetic reader cares because the equivalence turns arithmetic objects (heights, intersection numbers, the invariant bundle of a dynamical system) into analytic ones, and yields for the first time Monge–Ampère measures on quasi-projective arithmetic varieties, including on trivially valued fibers, together with a non-degeneracy criterion and a global Monge–Ampère measure.","feed_headline":"Semiample adelic bundles equal semipositive metrics","feed_subtitle":"The equivalence extends Monge–Ampère measures to quasi-projective varieties, including trivially valued fibers.","key_machinery":"The load-bearing new object is the continuous semipositive metric (Definition 3.3.6): a metric on a line bundle over $U^{\\mathrm{an}}$ that is the compactly convergent limit of tropical Fubini–Study metrics, meaning functions of the form $m^{-1}\\max_j(\\log|s_j| + \\lambda_j)$ built from global sections $s_j$ of a semiample multiple of the line bundle on a projective model $X_i$ of $U$. The proof chain runs through arithmetic divisor class groups: strongly semiample adelic line bundles analytify to such metrics (Lemma 6.2.1), and the inverse construction (Proposition 6.2.6) uses the decomposition of $U^{\\mathrm{an}}$ into the compact interior $U^{\\beth}$, where the boundary divisor's Green's function vanishes, and the compact normalized boundary $\\widetilde U^{\\mathrm b}$, where that Green's function is normalized to length one; boundary-norm inequalities proved there are transported to all of $U^{\\mathrm{an}}$ by norm equivariance, exactly matching the Cauchy condition defining an adelic divisor. Full faithfulness of the equivalence is inherited from the Yuan–Zhang analytification of model bundles.","core_discovery":"On its own terms the paper establishes Theorem 6.2.7 (Theorem A): for a flat, quasi-projective, integral scheme $U$ of finite type over $\\mathrm{Spec}(\\mathbb{Z})$, the category $\\underline{\\mathrm{cPic}}(U)^{\\mathrm{s.sa}}$ of strongly semiample adelic line bundles is equivalent to the category $\\underline{\\mathrm{cPic}}(U^{\\mathrm{an}})^{\\mathrm{eqv,sp}}$ of norm-equivariant, continuous semipositive metrized line bundles on $U^{\\mathrm{an}}$. The proof identifies both categories with the same group of arithmetic divisor classes on the Berkovich space, using the interior $U^{\\beth}$ and the normalized boundary $\\widetilde U^{\\mathrm b}$ introduced by Song, together with the boundary-norm topology in which adelic divisors are Cauchy sequences of model divisors. When $U$ is projective, a continuous semipositive metric is exactly a continuous plurisubharmonic metric in the sense of Pille-Schneider, so the theorem extends that global pluripotential theory from projective to quasi-projective varieties. The paper then uses the equivalence to define families of Monge–Ampère measures fiberwise over the Berkovich spectrum of $\\mathbb{Z}$, to show that the invariant adelic line bundle of a polarized dynamical system is strongly semiample under mild hypotheses, to prove a non-degeneracy criterion over trivially valued fields, and to define a global Monge–Ampère measure on $U^{\\mathrm{an}}$.","pith_inferences":["The equivalence suggests that the normalized boundary $\\widetilde U^{\\mathrm b}$ carries exactly the new information that separates quasi-projective from projective varieties, so height and equidistribution statements for quasi-projective $U$ could be reformulated as statements about measures and functions living on that compact boundary alone.","Because the global Monge–Ampère measure depends on the chosen probability measure on $\\mathcal{M}(\\mathbb{Z})$ (weighted by $1/(p\\log p)$ at the finite places), a natural test is whether its total mass in the quasi-projective case is independent of that choice; the paper leaves this open.","The trivially valued non-degeneracy criterion is most useful exactly where classical methods see nothing — namely good-reduction or trivially valued settings where equilibrium measures are supported on Shilov points — so combining it with refined properties of continuous semipositive metrics may yield new non-degeneracy tests for families.","The same compact-limit definition could plausibly define a global psh envelope or a Monge–Ampère operator acting on the boundary, giving quasi-projective analogues of results known for projective varieties over trivially valued fields."],"forward_implications":["Families of Monge–Ampère measures become available for continuous semipositive metrized line bundles on the analytification of every quasi-projective arithmetic variety, fiberwise over the Berkovich spectrum of $\\mathbb{Z}$, including fibers over trivially valued points where no such measures previously existed (Subsection 7.1).","The invariant adelic line bundle of a polarized dynamical system over a quasi-projective arithmetic variety is semiample, and strongly semiample when the base has an affine quasi-projective model, so its analytification carries a norm-equivariant continuous semipositive metric (Corollary 7.2.4).","A closed subvariety $Y$ of a polarized dynamical system is non-degenerate whenever the Monge–Ampère integral of the invariant bundle over the trivially valued fiber of $Y^{\\mathrm{an}}$ is non-zero (Proposition 7.3.2).","For Zariski-dense points of $\\mathcal{M}(\\mathbb{Z})$, the fiberwise Monge–Ampère integral equals an intersection number computed as a limit of classical intersection numbers, extending Guo's integration formula to infinite trivially valued fields (Theorem 7.3.1).","A global Monge–Ampère measure can be defined on $U^{\\mathrm{an}}$ by integrating the fiberwise Monge–Ampère measures against a probability measure on $\\mathcal{M}(\\mathbb{Z})$ (Section 8)."],"supporting_citations":[{"why":"Yuan and Zhang's theory of adelic line bundles on quasi-projective arithmetic varieties supplies the boundary-norm topology, the Cauchy-sequence definition, the intersection pairings, and the analytification map whose full faithfulness closes the proof of Theorem A.","marker":"[YZ24]"},{"why":"Song's bijection between adelic line bundles and norm-equivariant metrized line bundles, together with the interior $U^{\\beth}$ and normalized boundary $\\widetilde U^{\\mathrm b}$ and their Green's-function characterizations, is used to build the inverse map in Proposition 6.2.6.","marker":"[Son24]"},{"why":"Pille-Schneider's global pluripotential theory on Berkovich spaces over general Banach rings provides tropical Fubini–Study metrics, continuous plurisubharmonic metrics, and the family of Monge–Ampère measures that the paper extends to the quasi-projective setting.","marker":"[PS23]"},{"why":"Boucksom–Jonsson's theory of semipositive metrics over trivially valued fields is used to identify pure tropical Fubini–Study metrics with model metrics and to control restrictions of metrics to fibers of the structure map.","marker":"[BJ18]"},{"why":"Guo's integration formula comparing Chern form integration with intersection numbers on quasi-projective varieties is the statement that Theorem 7.3.1 extends to infinite trivially valued fields.","marker":"[Guo25]"},{"why":"Chambert-Loir–Ducros' construction of Monge–Ampère measures on Berkovich spaces by weak convergence is used to define the fiberwise measures in Subsection 7.1.","marker":"[CLD12]"}],"fun_headline_variants":["Semiample adelic bundles equal semipositive metrics","Quasi-projective Monge–Ampère measures from adelic bundles","Extending pluripotential theory to quasi-projective","Adelic bundles meet pluripotential theory"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that every continuous semipositive metric comes from a strongly semiample adelic line bundle relies on the premise that the Green's functions approximating the metric can be normalized to vanish on the interior $U^{\\beth}$, so that estimates proved on the normalized boundary extend to all of $U^{\\mathrm{an}}$; the paper proves this exact vanishing only for the fixed boundary divisor, not for the approximants, and the claimed equivalence of categories fails if the normalization cannot always be made.","fun_headline_variants_meta":{"raw":{"variants":["Semiample adelic bundles equal semipositive metrics","Quasi-projective Monge–Ampère measures from adelic bundles","Extending pluripotential theory to quasi-projective","Adelic bundles meet pluripotential theory"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001266,"raw_usage":{"total_tokens":5222,"prompt_tokens":1022,"completion_tokens":4200,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":638,"completion_tokens_details":{"reasoning_tokens":4132}},"tokens_in":638,"tokens_out":4200,"duration_ms":38943,"temperature":1.0,"reasoning_tokens":4132,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:32:36.012712+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a non-projective quasi-projective variety such as $\\mathbb{A}^1_{\\mathbb{Z}}$ or $\\mathbb{P}^1_{\\mathbb{Z}}$ minus a point, choose an explicit continuous semipositive metric, and compute whether its approximating tropical Fubini–Study Green's functions can be normalized to vanish on the interior $U^{\\beth}$; if no such normalization exists, the boundary-norm Cauchy condition in Proposition 6.2.6 cannot be verified and the asserted bijection between the two class groups breaks, while a proof that the normalization always exists would complete the missing step.","supporting_citations":[],"review_version":1}