Adelic Line Bundles: Intersection & Heights

• Adelic line bundles are line bundles equipped with compatible metric data from all places or models, unifying arithmetic and geometric perspectives.
• They extend classical theories by incorporating intersection, positivity, and volume measures through coherent local and global data.
• Their framework underpins applications in p-adic Arakelov theory, algebraic dynamics, and analytic realizations on Berkovich spaces, enabling new equidistribution and height results.

An adelic line bundle is, in the arithmetic-geometric sense, a line bundle equipped with compatible metric or model data that records contributions from all places or from all projective models of a variety, thereby supporting intersection theory, positivity, volume theory, and height theory in a single framework. In the Yuan–Zhang formalism for quasi-projective varieties, adelic line bundles are limit objects of Hermitian line bundles on projective models under a boundary topology; over adelic curves they are line bundles endowed with measurable, dominated families of continuous metrics on analytifications; and in a distinct holomorphic one-dimensional literature they appear on adelic projective lines and solenoids, where classification is by rational degree rather than integral degree (Yuan et al., 2021, Sédillot, 2023, Burgos et al., 2016).

1. Foundational definitions and ambient categories

For a quasi-projective variety $U$ over $k$, Yuan–Zhang define adelic objects by varying projective models of $U$. Fixing a compactification $U\subset X_0$ with boundary given by an effective divisor $E_0$, they introduce the boundary norm

$$D_{E_0}:=\inf\{\varepsilon>0:\,-\varepsilon E_0\le D\le \varepsilon E_0\},$$

which induces the boundary topology. An adelic line bundle on $U$ is represented by data

$(L,(X_i,L_i,\ell_i)_{i\ge1}),$

where $L$ is a line bundle on $U$, the $k$0 are projective models, the $k$1 are model line bundles, and the identifications $k$2 satisfy a Cauchy condition in the boundary topology. The isomorphism classes form $k$3, and there is a canonical identification $k$4 (Yuan et al., 2021).

A parallel formulation is used over adelic curves. For a proper adelic curve

$k$5

or equivalently in the notation

$k$6

the product formula

$k$7

is part of the structure. If $k$8 is a projective $k$9-scheme and $U$0 a line bundle on $U$1, then an adelic line bundle is a pair

$U$2

where $U$3 is a dominated and measurable family of continuous metrics on the analytic line bundles $U$4 (Chen et al., 2022, Sédillot, 2023).

For projective varieties over number fields, a classical adelic description already appears as

$U$5

where $U$6 is a line bundle and the $U$7 are continuous $U$8-invariant metrics coherent with an integral model at all but finitely many places. This is the setting of the arithmetic Hodge index theorem for adelic line bundles over number fields (Yuan et al., 2013).

These formulations are not identical, but they are compatible in spirit: each replaces a single global model by a coherent collection of local or asymptotic data. This suggests that the adjective “adelic” is functioning as a structural principle rather than a single rigid definition.

2. Positivity, intersection theory, and asymptotic volume

Yuan–Zhang extend the standard positivity hierarchy to adelic line bundles on quasi-projective varieties. An adelic line bundle is strongly nef if it is represented by a Cauchy sequence of model line bundles each of which is nef. It is nef if there exists a strongly nef adelic line bundle $U$9 such that

$U\subset X_0$0

is strongly nef for every positive integer $U\subset X_0$1. It is integrable if it is a difference of two strongly nef adelic line bundles. These implications hold: $U\subset X_0$2 The associated subcategories are denoted $U\subset X_0$3, $U\subset X_0$4, and $U\subset X_0$5 (Yuan et al., 2021).

Intersection theory is then defined by limiting from models. For a flat essentially quasi-projective integral scheme $U\subset X_0$6 of absolute dimension $U\subset X_0$7, there is a multilinear pairing

$U\subset X_0$8

If $U\subset X_0$9 are nef, then

$E_0$0

A relative intersection theory via the Deligne pairing is also available for projective flat morphisms $E_0$1 (Yuan et al., 2021).

Volume theory is governed by effective sections and asymptotic growth. For a quasi-projective $E_0$2,

$E_0$3

the limit exists, and if $E_0$4 is nef then

$E_0$5

For nef $E_0$6,

$E_0$7

These statements extend Hilbert–Samuel-, Fujita-, and Siu-type results to the adelic setting (Yuan et al., 2021).

Over adelic curves, the central asymptotic invariant is the $E_0$8-volume. For an ample adelic line bundle $E_0$9 on an integral projective $$D_{E_0}:=\inf\{\varepsilon>0:\,-\varepsilon E_0\le D\le \varepsilon E_0\},$$0-scheme of dimension $$D_{E_0}:=\inf\{\varepsilon>0:\,-\varepsilon E_0\le D\le \varepsilon E_0\},$$1,

$$D_{E_0}:=\inf\{\varepsilon>0:\,-\varepsilon E_0\le D\le \varepsilon E_0\},$$2

The arithmetic Hilbert–Samuel theorem identifies this with arithmetic self-intersection: $$D_{E_0}:=\inf\{\varepsilon>0:\,-\varepsilon E_0\le D\le \varepsilon E_0\},$$3 for semi-positive metrics on ample line bundles, and in a semiample semipositive form as well (Chen et al., 2022, Luo, 2023).

Arithmetic Hodge index theorems place sign constraints on intersection pairings. Over number fields, if $$D_{E_0}:=\inf\{\varepsilon>0:\,-\varepsilon E_0\le D\le \varepsilon E_0\},$$4 is integrable and $$D_{E_0}:=\inf\{\varepsilon>0:\,-\varepsilon E_0\le D\le \varepsilon E_0\},$$5 are nef and big with

$$D_{E_0}:=\inf\{\varepsilon>0:\,-\varepsilon E_0\le D\le \varepsilon E_0\},$$6

then

$$D_{E_0}:=\inf\{\varepsilon>0:\,-\varepsilon E_0\le D\le \varepsilon E_0\},$$7

with equality characterized by descent from the base under additional boundedness hypotheses (Yuan et al., 2013). Over finitely generated fields, the corresponding conclusion is that the pushforward intersection class is pseudo-effective, and numerical triviality again characterizes classes pulled back from the base (Yuan et al., 2013).

3. Heights, slopes, and equidistribution

The height theory of adelic line bundles is built directly from their intersection theory. For an integrable adelic line bundle $$D_{E_0}:=\inf\{\varepsilon>0:\,-\varepsilon E_0\le D\le \varepsilon E_0\},$$8 on a quasi-projective variety over a finitely generated field, the height of a point $$D_{E_0}:=\inf\{\varepsilon>0:\,-\varepsilon E_0\le D\le \varepsilon E_0\},$$9 is

$U$0

and for a closed subvariety $U$1,

$U$2

when the denominator is nonzero. In the projective case these recover Moriwaki-type heights after choosing a polarization (Yuan et al., 2021).

Over adelic curves, the section spaces $U$3 are adelic vector bundles, so slope theory enters. The asymptotic minimal slope is defined by

$U$4

For function fields of characteristic $U$5, boundedness of minimal slopes is automatic for adelic line bundles, and this is a technical engine for equidistribution (Luo, 2022).

If $U$6 is big and semipositive on a variety over an adelic curve, and $U$7 is a generic net satisfying

$U$8

then the associated measures $U$9 converge to $(L,(X_i,L_i,\ell_i)_{i\ge1}),$0 (Luo, 2022). In the quasi-projective arithmetic setting, differentiability of geometric and arithmetic adelic volume on the big cone is expressed by

$(L,(X_i,L_i,\ell_i)_{i\ge1}),$1

and this differentiability underlies a variational proof of equidistribution for small generic sequences (Biswas, 2023).

A related differentiability statement over an adelic curve is

$(L,(X_i,L_i,\ell_i)_{i\ge1}),$2

This is used to obtain logarithmic equidistribution for small generic points (Sédillot, 2023).

The common mechanism is that adelic line bundles carry both global arithmetic growth and local metric data. This makes them the natural input for variational arguments in equidistribution.

4. Abelian, $(L,(X_i,L_i,\ell_i)_{i\ge1}),$3-adic, Néron-theoretic, and dynamical constructions

For abelian varieties, line bundles give rise to adelic theta groups. If $(L,(X_i,L_i,\ell_i)_{i\ge1}),$4 is an abelian variety over an algebraically closed field $(L,(X_i,L_i,\ell_i)_{i\ge1}),$5, Grieve defines

$(L,(X_i,L_i,\ell_i)_{i\ge1}),$6

using prime-to-$(L,(X_i,L_i,\ell_i)_{i\ge1}),$7 torsion. For a line bundle $(L,(X_i,L_i,\ell_i)_{i\ge1}),$8 on $(L,(X_i,L_i,\ell_i)_{i\ge1}),$9, the adelic theta group $L$0 is a central extension

$L$1

Its commutator pairing yields a class in $L$2, and the map

$L$3

is an injective group homomorphism, functorial in $L$4 (Grieve, 2013).

A different adelic refinement appears in $L$5-adic Arakelov theory. Fixing a continuous idele class character

$L$6

a $L$7-adic adelic metric on a line bundle $L$8 consists of log functions at places above $L$9 and $U$0-valuations away from $U$1. The associated $U$2-adic height is

$U$3

On abelian varieties, canonical $U$4-adic metrics are constructed from the Poincaré bundle, and the resulting heights recover the canonical Mazur–Tate height and, for Jacobians, the Coleman–Gross height (Besser et al., 2021).

On an abelian scheme over a smooth curve, the Poincaré bundle extends to the identity components of the Néron models: $U$5 The extended rigidified Poincaré bundle admits an integrable adelic extension $U$6, uniquely characterized by

$U$7

Pulling this bundle back along sections produces an adelic line bundle on the base whose height function satisfies

$U$8

so variation of the Néron–Tate pairing is encoded exactly by an adelic line bundle on the base curve (Chen, 11 Jun 2026).

Adelic line bundles are also canonical in algebraic dynamics. For a polarized system $U$9 with

$k$00

there is a unique $k$01-admissible or $k$02-invariant adelic line bundle lifting $k$03 (Yuan et al., 2013). In the finitely generated-field setting, the arithmetic Hodge index theorem implies rigidity of preperiodic sets: equality or Zariski density of common preperiodic points forces equality of the corresponding adelic splittings on $k$04 (Yuan et al., 2013). In families of curves, the admissible canonical bundle becomes a big adelic line bundle, and this bigness feeds into a uniform Bogomolov-type theorem over global fields of all characteristics (Yuan, 2021).

5. Analytification, Berkovich metrics, and global pluripotential theory

Adelic line bundles admit analytic realizations on Berkovich spaces. Yuan–Zhang construct a fully faithful functor

$k$05

so an adelic line bundle can be viewed as a metrized line bundle on a global analytic space (Yuan et al., 2021).

This analytic viewpoint is sharpened in global pluripotential theory. For a quasi-projective arithmetic variety $k$06, a metric on $k$07 is called continuous semipositive if it is a compact uniform limit on compact subsets of tropical Fubini–Study metrics coming from projective models. In the projective semiample case, continuous semipositive and continuous plurisubharmonic metrics coincide (Morrow, 14 Jul 2025).

The central equivalence is

$k$08

where $k$09 denotes strongly semiample adelic line bundles and $k$10 denotes norm-equivariant continuous semipositive metrized line bundles on $k$11 (Morrow, 14 Jul 2025). This identifies a model-theoretic subcategory of adelic line bundles with an analytic category defined by semipositive metrics.

The same paper defines fiberwise Monge–Ampère measures

$k$12

on the fibers $k$13, and then globalizes by integrating against a fixed probability measure on the Berkovich spectrum of $k$14. For strongly semiample adelic line bundles, the total mass on each fiber satisfies

$k$15

a quasi-projective and trivially valued extension of Guo’s theorem (Morrow, 14 Jul 2025).

In the toric-bundle setting, adelic line bundles become explicitly polyhedral. Chambert-Loir–Tschinkel’s functorial philosophy is encoded by monoidal functors

$k$16

and the arithmetic of toric bundle line bundles is expressed by roof functions, Okounkov bodies, and Boucksom–Chen transforms. Heights, minima, and arithmetic intersection numbers become integrals over polytopes, culminating in an arithmetic analogue of the Hofscheier–Khovanskii–Monin formula for toric bundles (Hultberg, 2024).

6. Solenoidal and proalgebraic one-dimensional adelic geometries

A distinct but important use of the phrase appears on the adelic Riemann sphere and the adelic projective line. Here the geometry is built from inverse limits of branched covers rather than from all places of a global field.

For the adelic Riemann sphere $k$17, which is the topological suspension of the adelic solenoid $k$18, holomorphic line bundles are defined by clutching maps on the algebraic solenoid. For each $k$19, the line bundle $k$20 is defined by the transition function $k$21. The adelic Birkhoff–Grothendieck theorem states that every holomorphic vector bundle splits as

$k$22

Line bundles are classified by the rational Chern character

$k$23

and

$k$24

The rationality comes from the fact that self-maps of the adelic solenoid have rational degree (Burgos et al., 2016).

An analogous result holds for the adelic projective line $k$25 viewed as a proalgebraic inverse limit of branched covers. Holomorphic functions on adelic disks and annuli are given by Laurent–Puiseux series with rational exponents, and holomorphic line bundles are determined by clutching functions on adelic annuli. The holomorphic Picard theorem identifies

$k$26

with

$k$27

and tensor product corresponds to addition of rational degrees: $k$28 The Birkhoff–Grothendieck splitting theorem again holds with rational summands (Burgos et al., 2020).

These one-dimensional theories are not Arakelov theories of metrized line bundles on varieties over global fields. They are nevertheless genuinely adelic in the sense that the inverse-limit geometry simultaneously encodes all finite coverings. A common source of confusion is the reuse of the same phrase for these different settings; the literature makes clear that the shared feature is simultaneous control of all levels, places, or covers, while the specific objects and invariants differ substantially.