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Adelic Line Bundles, Arithmetic Positivity and Diophantine Geometry

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Pith's one-line read Adelic line bundles on quasi-projective varieties carry arithmetic positivity that yields an equidistribution theorem and the uniform Bogomolov conjecture.

desk verdict This is a pure exposition with no new results on adelic line bundles and positivity. read the letter →

arxiv 2606.27116 v1 pith:CEPYDJOD submitted 2026-06-25 math.NT math.AG

Xinyi Yuan This is my paper · ORCID
classification math.NTmath.AG
keywords adeliclinebundlesarithmeticpositivityDiophantinegeometryequidistributiontheoremBogomolovconjecturequasi-projectivevarietiesheights
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 lays out the basic definitions and properties of adelic line bundles on quasi-projective varieties and introduces the notion of arithmetic positivity for these bundles. It then derives an equidistribution result from this positivity and applies the same framework to establish the uniform Bogomolov conjecture. A reader would follow the chain because the constructions translate height functions and positivity conditions into statements about the distribution of rational points, directly addressing longstanding questions in Diophantine geometry.

What carries the argument

Adelic line bundles on quasi-projective varieties equipped with arithmetic positivity, which encodes both finite and infinite place data to control heights and produce equidistribution and height lower bounds.

What would settle it

An explicit sequence of points on a quasi-projective variety whose heights satisfy the positivity condition yet fail to equidistribute or violate the uniform Bogomolov lower bound.

Watch

Extended reading notes

Core claim

The article establishes that arithmetic positivity of adelic line bundles on quasi-projective varieties implies both an equidistribution theorem for sequences of points with controlled heights and the uniform version of the Bogomolov conjecture, with the positivity condition serving as the bridge between the arithmetic data encoded in the bundles and the geometric conclusions about point distributions.

Load-bearing premise

The presentation correctly summarizes the standard definitions and theorems on adelic line bundles and arithmetic positivity from earlier literature.

Editorial extensions

If this is right

  • Positivity on adelic line bundles forces equidistribution of small-height points with respect to a suitable measure on the variety.
  • The uniform Bogomolov conjecture holds for the varieties where the adelic positivity condition can be verified.
  • Height functions arising from these bundles give effective lower bounds that are uniform across families of varieties.
  • Applications extend to any Diophantine problem reducible to controlling arithmetic heights via line bundle data.

Reading between the lines

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

  • The same positivity framework may apply to other conjectures in arithmetic geometry that rely on height inequalities.
  • Making the definitions explicit for quasi-projective rather than projective varieties widens the range of varieties where equidistribution can be tested directly.
  • Readers can now check positivity for concrete bundles without re-deriving the foundational comparison theorems.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 0 minor

Summary. This article is an expository account of the basics of adelic line bundles on quasi-projective varieties, arithmetic positivity of adelic line bundles, and applications of positivity to an equidistribution theorem and the uniform Bogomolov conjecture.

Significance. If the exposition faithfully reproduces the cited results from the literature, the paper offers a consolidated overview that may help readers navigate the connections between adelic geometry, arithmetic positivity, equidistribution, and the uniform Bogomolov conjecture. Its value is primarily in organization and accessibility rather than novel theorems or derivations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the expository organization and accessibility of the material on adelic line bundles, arithmetic positivity, equidistribution, and the uniform Bogomolov conjecture are viewed as valuable contributions.

Circularity Check

0 steps flagged · score 0.0 of 10

Expository summary with no internal derivations or predictions

full rationale

The paper is explicitly an expository account of existing literature on adelic line bundles, arithmetic positivity, equidistribution, and the uniform Bogomolov conjecture, with no new claims, proofs, or derivations presented. No equations, predictions, or load-bearing steps appear that could reduce to self-definitions, fitted inputs, or self-citation chains. The content is self-contained as a faithful summary of external results, yielding no circularity.

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

No new free parameters, axioms, or invented entities are introduced because the work is purely expository and relies on standard background from algebraic and arithmetic geometry.

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Cite this review

Pith. "Pith review of Adelic Line Bundles, Arithmetic Positivity and Diophantine Geometry." pith.science (2026). https://pith.science/paper/CEPYDJOD

@misc{pith2026260627116,
  author       = {Pith},
  title        = {Pith review of: Adelic Line Bundles, Arithmetic Positivity and Diophantine Geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CEPYDJOD}},
  note         = {Machine review of arXiv:2606.27116}
}
read the original abstract

This article is an expository account of the basics of adelic line bundles on quasi-projective varieties, arithmetic positivity of adelic line bundles, and applications of positivity to an equidistribution theorem and the uniform Bogomolov conjecture.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 7 canonical work pages

  1. [1]

    [Ara74] S. J. Arakelov, Intersection theory of divisors on an arithmetic surface, Math. USSR Izvest. 8 (1974), 1167–1180. [Aut01] P. Autissier, Points entiers sur les surfaces arithm´ etiques, J. Reine Angew. Math., 531 (2001), 201–235. [BB07] G. Bassanelli, F. Berteloot,Bifurcation currents in holomorphic dynamics on Pk. J. Reine Angew. Math. 608 (2007),...

    1974

  2. [2]

    Boucksom, C

    [BFJ09] S. Boucksom, C. Favre, M. Jonsson, Differentiability of volumes of divisors and a problem of Teissier, J. Algebraic Geom.18(2009), no. 2, 279–308. [BG06] E. Bombieri, W. Gubler, Heights in Diophantine geometry, New Mathematical Monographs,

    2009

  3. [3]

    Baker and L

    [BH05] M. Baker and L. C. Hsia, Canonical heights, transfinite diameters, and polyno- mial dynamics, J. Reine Angew. Math. 585 (2005), 61–92. 67 [Bil97] Y. Bilu, Limit distribution of small points on algebraic tori, Duke Math. J. 89 (1997), no. 3, 465–476. [BLR90] S. Bosch, W. L¨ utkebohmert, M. Raynaud, N´ eron models, Ergebnisse der Math- ematik und ihr...

    2005

  4. [4]

    [Bog81] F. A. Bogomolov,Points of finite order on an abelian variety,Math. USSR Izv., 17, (1981). [Bom90] E. Bombieri, The Mordell conjecture revisited. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 17(4), 615–640,

    1981

  5. [5]

    Baker, R

    [BR06] M. Baker, R. Rumely, Equidistribution of small points, rational dynamics, and potential theory, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 3, 625–688. [BT82] E. Bedford, B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1–40. [BV89] J.-M. Bismut, E. Vasserot, The asymptotics of the Ray-Singer analytic to...

    2006

  6. [6]

    Positive degree and arithmetic bigness

    [Che08] H. Chen, Positive degree and arithmetic bigness, arXiv: 0803.2583v3 [math.AG]. [Che10] H. Chen. Arithmetic Fujita approximation.Ann. Sci. ´Ec. Norm. Sup´ er. (4), 43(4):555–578,

    work page Pith review arXiv

  7. [7]

    Cinkir, Zhang’s conjecture and the effective Bogomolov conjecture over func- tion fields

    [Cin11] Z. Cinkir, Zhang’s conjecture and the effective Bogomolov conjecture over func- tion fields. Invent. Math. 183 (2011), no. 3, 517–562. [CL06] A. Chambert-Loir, Mesures et ´ equidistribution sur les espaces de Berkovich. J. Reine Angew. Math. 595, 215–235 (2006). 68 [dD97] T. de Diego. Points rationnels sur les familles de courbes de genre au moins

    2011

  8. [8]

    de Jong, Point-like limit of the hyperelliptic Zhang-Kawazumi invariant

    [dJ15] R. de Jong, Point-like limit of the hyperelliptic Zhang-Kawazumi invariant. Pure Appl. Math. Q. 11 (2015), no. 4, 633–653. [dJ18] R. de Jong, N´ eron-Tate heights of cycles on Jacobians. J. Algebraic Geom. 27 (2018), no. 2, 339–381. [dJS22] R. de Jong, F. Shokrieh. Faltings height and N´ eron-Tate height of a theta divisor, Compos. Math.158(2022), ...

    2015

Show all 25 references
  1. [9]

    Deligne, Le d´ eterminant de la cohomologie

    [Del85] P. Deligne, Le d´ eterminant de la cohomologie. Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 93–177, Contemp. Math., 67, Amer. Math. Soc., Providence, RI,

    1985

  2. [10]

    DeMarco,Dynamics of rational maps: A current on the bifurcation locus

    [DeM01] L. DeMarco,Dynamics of rational maps: A current on the bifurcation locus. Math. Res. Lett. 8, No.1-2, 57–66 (2001). [DeM03] L. DeMarco,Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity.Math. Ann. 326, No.1, 43–73 (2003). [DGH21] V. Dimitrov, Z. ...

    2001

  3. [11]

    Favre, T

    [FG15] C. Favre, T. Gauthier,Distribution of postcritically finite polynomials.Israel J. Math. 209 (2015), no. 1, 235–292. [FRL06] C. Favre, J. Rivera-Letelier, Equidistribution quantitative des points de petite hauteur sur la droite projective. Math. Ann. 335 (2006), no. 2, 3...

    2015

  4. [12]

    Gauthier, Y

    [GOV20] T. Gauthier, Y. Okuyama, G. Vigny,Approximation of non-archimedean Lya- punov exponents and applications over global fields.Trans. Amer. Math. Soc. 373 (2020), no. 12, 8963–9011. [GR14] E. Gaudron, G. R´ emond, Th´ eor` eme des p´ eriodes et deg´ es minimaux d’isog´ en...

    2020

  5. [13]

    [JX23] Z. Ji, J. Xie,DAO for curves.arXiv:2302.02583. [Kuh21] L. K¨ uhne, Equidistribution in Families of Abelian Varieties and Uniformity, arXiv:2101.10272v2. [Laz04a] R. Lazarsfeld, Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Ergeb...

  6. [14]

    Li, Deligne pairing for equidimensional morphisms, arXiv:2411.17410,

    [Li24] S. Li, Deligne pairing for equidimensional morphisms, arXiv:2411.17410,

  7. [15]

    [LM09] R. K. Lazarsfeld and M. Mustat ¸˘ a, Convex bodies associated to linear series, Ann. Sci. ´Ec. Norm. Sup´ er. (4)42(2009). [LSW25] N. Looper, J. H. Silverman, R. Wilms, A uniform quantitative Manin-Mumford theorem for curves over function fields, J. Reine Angew. Math. 8...

    2009

  8. [16]

    Lawrence and A

    [LV20] B. Lawrence and A. Venkatesh, Diophantine problems andp-adic period map- pings, Invent. Math. 221 (2020), no. 3, 893–999. [Maz86] B. Mazur. Arithmetic on curves. Bulletin of the American Mathematical Society, 14(2): 207–259,

    2020

  9. [17]

    Mu˜ noz Garcia, Fibr´ es d’intersection

    [MG00] E. Mu˜ noz Garcia, Fibr´ es d’intersection. Compositio Math. 124 (2000), no. 3, 219–252. [Mor00] A. Moriwaki, Arithmetic height functions over finitely generated fields, Invent. Math. 140 (2000), no. 1, 101–142. [Mor09] A. Moriwaki, Continuity of volumes on arithmetic v...

    2000

  10. [18]

    Siu, An effective Matsusaka Big Theorem, Annales de l’institut Fourier, 43 no

    [Siu93] Y.-T. Siu, An effective Matsusaka Big Theorem, Annales de l’institut Fourier, 43 no. 5 (1993), p. 1387–1405. [Son23] Y. Song, Asymptotic behavior of the Zhang–Kawazumi’sφ-invariant. Adv. Math. 435 (2023), Paper No. 109349, 45 pp. [SUZ97] L. Szpiro, E. Ullmo, S. Zhang, ...

    1993

  11. [19]

    Weil,Arithmetic on algebraic varieties

    [Wei51] A. Weil,Arithmetic on algebraic varieties. Ann. of Math. (2), 53 (1951), p. 412–444. [Wil17] R. Wilms, New explicit formulas for Faltings’ delta-invariant. Invent. Math. 209 (2017), no. 2, 481–539. [Wil21] R. Wilms, Degeneration of Riemann theta functions and of the Zh...

    1951

  12. [20]

    Yu,An Explicit Uniform Mordell Conjecture over Function Fields of Charac- teristic Zero, J

    [Yu26] J. Yu,An Explicit Uniform Mordell Conjecture over Function Fields of Charac- teristic Zero, J. Algebraic Geom. 35 (2026), no. 3, 409–427. [Yua08] X. Yuan, Big line bundles on arithmetic varieties, Invent. Math. 173 (2008), no. 3, 603–649. [Yua09] X. Yuan, On volumes of ...

    2026

  13. [21]

    Yuan, On Vojta’s proof of the Mordell conjecture, arXiv:2508.11888,

    [Yua25] X. Yuan, On Vojta’s proof of the Mordell conjecture, arXiv:2508.11888,

  14. [22]

    Yuan, Arithmetic bigness and a uniform Bogomolov-type result, Ann

    [Yua26a] X. Yuan, Arithmetic bigness and a uniform Bogomolov-type result, Ann. of Math. (2) 203 (2026), no. 1, 15–119. [Yua26b] X. Yuan, Quantitativity in the Mordell Conjecture, to appear in the Proceedings of ICM

    2026

  15. [23]

    [YYZ26] J. Yu, X. Yuan, and S. Zhou.Quantitativity on the Number of Rational Points in the Mordell Conjecture, arXiv:2602.01820,

  16. [24]

    [YZ17] X. Yuan, S. Zhang, The arithmetic Hodge index theorem for adelic line bundles, Math. Ann. 367 (2017), no. 3-4, 1123–1171. [YZ26] X. Yuan, S. Zhang, Adelic line bundles on quasi-projective varieties, the Annals of Mathematics Studies, Princeton University Press,

    2017

  17. [25]

    Zhang, Admissible pairing on a curve

    [Zha93] S. Zhang, Admissible pairing on a curve. Invent. Math. 112 (1993), 171–193. [Zha95a] S. Zhang, Small points and adelic metrics, J. Alg. Geometry 4 (1995), 281–300. [Zha95b] S. Zhang, Positive line bundles on arithmetic varieties, Journal of the AMS, 8 (1995), 187–221. ...

    1993

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