Global pluripotential theory for adelic line bundles

Jackson S. Morrow

REVIEW 2 major objections 3 minor 41 references

Global pluripotential theory for adelic line bundles

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2507.10410 v1 pith:MTDVPVLG submitted 2025-07-14 math.AG math.NT

Jackson S. Morrow This is my paper · ORCID

classification math.AGmath.NT MSC 14G4014G2232U05

keywords adeliclinebundlesBerkovichspacespluripotentialtheoryMonge–Ampèremeasuresquasi-projectivearithmeticvarietiessemipositivemetricstropicalFubini–Studynorm-equivariant

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $U^{\mathrm{an}}$.

Load-bearing premise

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.

Editorial extensions

If this is right

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).

Reading between the lines

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

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.

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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

Colleague,

Here 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.

What'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.

The 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.

Lesser 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.

Who 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.

Referee Report

2 major / 3 minor

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.

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 (2)

[6.2, Proposition 6.2.6] 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.
[7.3, Theorem 7.3.1] 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.

minor comments (3)

[Various] 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'.
[5.2, Definition 5.2.2] 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.
[8.1] 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.

Circularity Check

1 steps flagged · score 2.0 of 10

Theorem A is largely a definitional translation: both categories are built from the same tropical Fubini–Study model data, and the forward direction follows by definition; the inverse proof has an unproved normalization step that is a correctness gap rather than circularity.

self definitional [Definition 3.3.6; Definition 6.1.1; Lemma 6.2.1 (Section 6.2)]
"The first statement now follows from our definition of continuous semipositive metric (Definition 3.3.6)."

The forward direction of Theorem A is asserted to follow directly from the definition of continuous semipositive. Definition 3.3.6 defines a continuous semipositive metric as the compact limit of tropical Fubini–Study metrics on projective models, while Definition 6.1.1 defines strongly semiample adelic line bundles as Cauchy sequences of the same global tropical Fubini–Study model data. Thus the two categories are constructed from identical building blocks, and the easy direction of the claimed equivalence is true by construction. The residual content is confined to the boundary-topology Cauchy condition in Proposition 6.2.6, so the circularity is partial rather than total.

full rationale

This paper contains no fitted parameters, no empirical predictions, and no load-bearing self-citations: the cited prior work (Yuan–Zhang, Song, Pille-Schneider) is external and supplies the dictionary between adelic line bundles and metrized Berkovich bundles. The main concern is definitional overlap: Definition 3.3.6 defines 'continuous semipositive' as a compact limit of tropical Fubini–Study metrics on projective models, while Definition 6.1.1 defines 'strongly semiample' as a Cauchy sequence of the same global tropical Fubini–Study model data; Lemma 6.2.1 therefore obtains the forward direction immediately from the definition. That makes Theorem A largely a translation between two formulations of the same approximations, though the inverse direction (Proposition 6.2.6) is not tautological: it must prove the boundary-topology Cauchy condition, and it does so using Song's description of the normalized boundary. A separate correctness gap, not a circularity, is that Proposition 6.2.6 asserts 'the functions g_{E_i} and tilde g_0 all vanishing on U^ℶ by Proposition 6.2.4', whereas Proposition 6.2.4 proves vanishing only for tilde g_0; the vanishing of the approximating g_{E_i} on U^ℶ is not established. This gap affects the validity of the inverse map but does not reduce the argument to its own inputs. Score 2 reflects the partial definitional character of the main equivalence, without treating the proof gap as circular.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

The central claim rests on the established theories of Yuan-Zhang, Song, and Pille-Schneider, plus a transfer of Guo's result that is asserted rather than fully proved. One hand-chosen measure on M(Z) is introduced for the global Monge-Ampère construction.

free parameters (1)

Weights in base measure μ on M(Z) = 1/(p log p) for finite primes p; 1/e for Archimedean; normalized to total mass 1
Chosen by hand to make μ' finite; the global Monge-Ampère measure in Definition 8.2.1 depends on this arbitrary choice, as acknowledged in Remark 8.2.3(1).

assumptions (4)

domain assumption Yuan-Zhang's construction of adelic line bundles on quasi-projective arithmetic varieties, including the boundary topology and the isomorphism CaCl(U) ≅ cPic(U) (YZ24)
Used throughout Section 5 and in proofs of Lemmas 6.2.1 and 6.2.3 and Proposition 6.2.6.
domain assumption Song's results: compactness of U^ℶ and ilde U^b, and the bijection between adelic line bundles and continuous norm-equivariant metrized bundles
Theorem 2.4.4 and the starting point cited in the introduction; Propositions 6.2.4 and 6.2.5 rely on it.
domain assumption Pille-Schneider's global pluripotential theory, particularly the characterization of continuous psh metrics as uniform limits of tropical Fubini-Study metrics (PS23 Prop 2.21)
Used in Lemma 3.3.7, Lemma 3.3.8, and Section 7.1.
ad hoc to paper Guo's integration formula transfers to infinite trivially valued fields
Theorem 7.3.1 is proved by asserting this transfer; no detailed verification is provided.

invented entities (1)

Measure μ on M(Z) with weights 1/(v log v)
purpose: Base measure used to define the global Monge-Ampère measure by integrating fiberwise measures over M(Z)
Introduced in Section 8.1; the choice of weights is arbitrary (Remark 8.2.3) and there is no independent falsifiable handle outside the construction.

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Cite this review

Pith. "Pith review of Global pluripotential theory for adelic line bundles." pith.science (2026). https://pith.science/paper/MTDVPVLG

@misc{pith2026250710410,
  author       = {Pith},
  title        = {Pith review of: Global pluripotential theory for adelic line bundles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MTDVPVLG}},
  note         = {Machine review of arXiv:2507.10410}
}

read the original abstract

In this work, we relate recent work of Yuan--Zhang and Song on adelic line bundles over quasi-projective arithmetic varieties to recent advances in pluripotential theory on global Berkovich spaces from Pille-Schneider. In particular, we establish an equivalence between subcategories of adelic line bundles on quasi-projective varieties and line bundles on their Berkovich analytifications equipped with a continuous plurisubharmonic metric. We also provide several applications of this equivalence. For example, we generalize a construction of Pille-Schneider concerning families of Monge--Amp\`ere measures on analytifications of projective arithmetic varieties to the quasi-projective setting. With this construction, we offer a new description of non-degenerate subvarieties which involves Monge--Amp\`ere measures over trivially valued fields. Finally, we define a Monge--Amp\`ere measure on the analytification of a quasi-projective arithmetic variety.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Marc Abbound, Unlikely intersections problem for automorphisms of M arkov surfaces , Preprint, arXiv:2401.05762v3 (March 19, 2024)

work page Pith review arXiv 2024

[2] [2]

S\'ebastien Boucksom and Dennis Eriksson, Spaces of norms, determinant of cohomology and F ekete points in non- A rchimedean geometry , Adv. Math. 378 (2021), Paper No. 107501, 124. 4192993

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