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Concave transforms of compactified S-metrized divisors

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Pith's one-line read For big compactified S-metrized divisors on quasi-projective varieties over adelic curves, concave transforms are constructed and shown to satisfy arithmetic volume and Hilbert-Samuel formulas.

arxiv 2505.14023 v1 pith:3SIB7Z2Q submitted 2025-05-20 math.AG

Debam Biswas , Yulin Cai This is my paper · ORCID
classification math.AG
keywords compactifieds-metrizedconcaveadelicassociatecurvedivisordivisors
verification ladder T0 review T1 audit T2 compute T3 formal

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The reading

Adelic curves are a general setting for Arakelov geometry: a field with absolute values indexed by a measure space, generalizing number fields and function fields. The paper studies compactified S-metrized divisors on quasi-projective varieties, which package a divisor together with a family of Green functions, allowing for singular metrics that occur naturally, for example on moduli spaces. Its first main theorem attaches to each big such divisor a concave function G_D defined on an Okounkov body, a convex region built from valuations of sections. The function records, asymptotically, the position of the Harder-Narasimhan slopes of the spaces of sections. This yields an exact formula for the arithmetic volume as the integral of max{G_D,0}, and an upper bound for the arithmetic χ-volume which becomes an equality when the minimal slope is bounded below.

The second main theorem is an arithmetic Hilbert-Samuel formula: for relatively nef compactified YZ-divisors that are big, the integral of G_D equals the arithmetic self-intersection number (D^{d+1}|U)_S, and the same holds for the χ-volume when the minimal slope condition holds. This extends previously known formulas for projective varieties and continuous metrics to quasi-projective varieties and singular metrics over general adelic curves. The third main theorem uses the concave transform to prove equidistribution of small generic points: Galois orbits distribute according to the positive intersection measure.

The proof strategy is to approximate a compactified divisor by model divisors on projective models, transfer Chen-Moriwaki's concave transform theory to the auxiliary section spaces H_+^0, and then pass to limits using continuity of arithmetic volumes.

Extended reading notes

Core claim

Theorem 5.2.2: If D=(D,g) lies in dDivS,Q(U) ar-nef rel-nef and D is big, then cvol_num_χ(D) = (d+1)! ∫_{Δ(D)°} G_D(λ)dλ = (D^{d+1}|U)_S; in particular cvolχ(D) = (D^{d+1}|U)_S whenever μ_min^asy(D)>−∞. Theorem A also asserts that for big D, cvol(D) = lim_{m→∞} ddeg_+(V_m)/(m^{d+1}/(d+1)!) = (d+1)! ∫_{Δ(D)°} max{G_D(λ),0}dλ.

Load-bearing premise

The proof of the Hilbert-Samuel formula for arithmetically nef divisors rests on Proposition 5.1.3, Step 1, which asserts that [CM22, Theorem 8.8.3] (asymptotic minimal slope positivity for S-ample divisors) remains valid for every adelic curve with the strong tensorial minimal slope property via [CM20, Theorem 7.2.4]. The adaptation is asserted without proof; if it fails, the positivity of μ_min^asy(D) that yields G_D≥0 and cvol(D)=(D^{d+1}|U)_S would break down.

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Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data, and no new physical entities are postulated. The auxiliary sections H_+^0 and the numerical χ-volume cvol_num_χ are definitions, not independent postulates. The load-bearing input consists of technical hypotheses on the adelic curve, the well-definedness of intersection numbers imported from [CG24], and the asserted but unproved adaptation of a theorem from [CM22].

assumptions (4)
  • domain assumption S is a proper adelic curve satisfying ν(Ω∞)<∞, ν(A) not contained in {0,+∞}, and either A is discrete or K is countable, and S has the tensorial minimal slope property of level ≥ C0.
    Assumed in Section 4; needed to run the Harder-Narasimhan filtration and Chen-Moriwaki's measure convergence theorem [CM20, Theorem 6.3.20]. It is known for perfect fields but not for all adelic curves.
  • domain assumption In Section 5, S has the strong tensorial minimal slope property or the Minkowski property of level ≥ C0.
    Used in Proposition 5.1.3 to obtain positivity of the asymptotic minimal slope for S-ample divisors and hence the Hilbert-Samuel equality for arithmetically nef divisors.
  • ad hoc to paper The adaptation of [CM22, Theorem 8.8.3] to all adelic curves with the strong tensorial minimal slope property holds.
    Asserted in Proposition 5.1.3, Step 1, via [CM20, Theorem 7.2.4], but not proved in detail. This is load-bearing for the arithmetically nef Hilbert-Samuel formula.
  • domain assumption The arithmetic intersection number (D^{d+1}|U)_S for dDivS,Q(U) ar-nef rel-nef is well defined as in [CG24, Theorem 11.3].
    Accepted from prior work by the second author and Gubler; underlies the statement of Theorem B.

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Pith. "Pith review of Concave transforms of compactified S-metrized divisors." pith.science (2026). https://pith.science/paper/3SIB7Z2Q

@misc{pith2026250514023,
  author       = {Pith},
  title        = {Pith review of: Concave transforms of compactified S-metrized divisors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SIB7Z2Q}},
  note         = {Machine review of arXiv:2505.14023}
}
read the original abstract

We associate a concave transform to any compactified S-metrized divisor on a quasi-projective variety over an adelic curve. Then we show a Hilbert-Samuel type formula for relatively nef compactified S-metrized YZ-divisors.

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Works this paper leans on

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