Points of Small Height on Semiabelian Varieties

The Equidistribution Conjecture is proved for general semiabelian varieties over number fields. Previously, this conjecture was only known in the special case of almost split semiabelian varieties through work of Chambert-Loir. The general case has remained intractable so far because the height of a semiabelian variety is negative unless it is almost split. In fact, this places the conjecture outside the scope of Yuan's equidistribution theorem on algebraic dynamical systems. To overcome this, an asymptotic adaption of the equidistribution technique invented by Szpiro, Ullmo, and Zhang is used here. It also allows a new proof of the Bogomolov Conjecture and hence a self-contained proof of the Strong Equidistribution Conjecture in the same general setting.

Throughout this article, G denotes an arbitrary semiabelian variety over a number field K ⊂ Q with maximal subtorus T of dimension t and maximal abelian quotient π : G → A of dimension g. For a place ν of K, we denote by K ν the associated completion of K, by K ν the algebraic closure of K ν , and by C ν the completion of the algebraic closure K ν . For a quasi-projective algebraic variety X over a number field K and a place ν of K, we denote by X an Cν the C ν -analytic space associated with X Cν . If ν is archimedean, this means that X an Cν is a complex (analytic) space (see [21] for this notion). If ν is non-archimedean, X an Cν is a Berkovich C ν -analytic space (see [3,Section 3.4]).
In order to state our main results, we need a canonical height h on G. For details, the reader is referred to Section 2 and [41, Sections 2 and 3]. To simplify our exposition, we enlarge K if necessary so that we can assume that T = G t m . Then (P 1 ) t is naturally a G t m -equivariant compactification of G t m , and the multiplication-by-n map [n] : G t m → G t m extends to a map [n] : (P 1 ) t → (P 1 ) t (n ≥ 2 an integer). This yields a compactification G = G× G t m (P 1 ) t of G, and π extends to a map π : G → A, whose fibers are isomorphic to (P 1 ) t . The boundary (P 1 ) t \ G t m gives rise to a Weil divisor G = A × G t m ((P 1 ) t \ G t m ) on G. Letting M G denote the line bundle associated with this divisor, we have [n] * M G = M ⊗n G . In addition, we fix an ample symmetric line bundle N on A and set L = M G ⊗ π * N. Tate's limit argument allows us to define a unique canonical height h L (x) for each closed point x ∈ G, starting from the Weil heights of M G and π * N.
This already suffices to define the main object of our study. We say that a sequence (x i ) ∈ G N of closed points is generic (resp. strict) if none of its subsequences is contained in a proper algebraic subvariety (resp. a proper algebraic subgroup) of G. Furthermore, we say that a sequence (x i ) ∈ G N of closed points is a sequence of small points if h L (x i ) → 0.
As in the case of abelian varieties, the following two conjectures convey significant information about the diophantine geometry of semiabelian varieties: For each place ν ∈ Σ(K), a closed point x ∈ G yields a 0-cycle O ν (x) = (x ⊗ K C ν ) an on the C ν -analytic group G an Cν associated with G. We write δ y for the Dirac measure associated with a point y ∈ G an Cν . Equidistribution Conjecture (EC). For every generic sequence (x i ) ∈ G N of small points, the measures y∈Oν (x i ) δ y converge weakly to the measure c 1 (L ν ) ∧g+t / deg L (G). More explicitly, (EC) asserts that for every compactly supported f ∈ C 0 (G an Cν ). The measure c 1 (L ν ) ∧g+t arises naturally in a refined approach to the canonical height h L introduced above (see Sections 1 and 2 for details). In fact, the line bundle L can be endowed with a canonical ν-metric. This yields a ν-metrized line bundle L ν = (L, · ν ). Letting ν vary over all places of K, these canonical metrics combine to an (adelically) metrized line bundle L = (L, { · ν }). The (adelic) height function h L associated with L coincides with the Néron-Tate height h L from above. Additionally, it enables us to assign a height h L (X) with every algebraic subvariety X ⊆ G. Most importantly, each ν-metrized line bundle L ν supplies us with a regular Borel measure c 1 (L ν ) ∧g+t on the analytic spaces G an Cν (see Section 1.3). For an archimedean place ν, it is well-known that c 1 (L ν ) ∧g+t is the Haar measure on the maximal compact subgroup of G an Cν . In fact, this is a special case of our Lemma 20 below. For a non-archimedean place ν, the determination of c 1 (L ν ) ∧g+t is more intricate since even for abelian varieties the reduction of G with respect to ν plays a role. A complete description of the abelian case is given by Gubler in [29,Example 7.2]. It seems very likely that his techniques can be also used for general semiabelian varieties. Since this seems, unfortunately, a lengthy distraction from our main investigation, we leave it nevertheless to the interested reader.
Bogomolov Conjecture (BC). Let X be a geometrically irreducible algebraic subvariety of G. Then, either X is an irreducible component of an algebraic subgroup of G or there exists some ε = ε(X) > 0 such that is not Zariski-dense in X.
Closely related to these two conjectures is a formal strengthening of the first one. Strong Equidistribution Conjecture (SEC). For every strict sequence (x i ) ∈ G(Q) N of small points, the measures y∈Oν (x i ) δ y converge weakly to c 1 (L ν ) ∧g+t / deg L (G). In fact, it is easy to prove the equivalence (EC) ∧ (BC) ⇔ (SEC). In the nineties of the last century, considerable efforts were dedicated to prove the above conjectures in various settings. An important special case of (BC) is the embedding X = C ֒→ G = Jac(C) of an algebraic curve C over K into its Jacobian variety Jac(C). Before being completely settled by Ullmo [63], this case of (BC) was proven by Szpiro [61] and Zhang [68,70] in numerous cases. In the meantime, Zhang proved (BC) for algebraic tori in [69]. A complete proof of (BC) for abelian varieties was given by Zhang [71]. (The reader may also consult the surveys [1,72].) For abelian varieties, (EC) was proven in a joint work of Szpiro, Ullmo, and Zhang [62]. Bilu [4] proved directly (SEC) for algebraic tori and gave a deduction (SEC) ⇒ (BC). A further advancement was made by Chambert-Loir [11], who gave a proof of (SEC) in the case of almost split semiabelian varieties. Up to the present work, his work has contained the best result in the direction of (SEC). Indeed, the canonical height h L (G) is strictly negative unless it is almost split ( [11,Corollaire 4.3]). In addition, all points of negative height lie on the boundary G \ G ( [11,Lemma 3.9]). This means that no generic sequence (x i ) ∈ G N of closed points can satisfy (2) h unless G is almost split. Yuan's general equidistribution theorem for algebraic dynamical systems ([67, Theorem 10.2]) is hence empty in this situation as the equidistribution method developed by Szpiro, Ullmo, and Zhang [62] does generally only apply to sequences satisfying (2). The reader is referred to [11] for details. Using different methods, David and Philippon [15] 1 proved (BC) for general semiabelian varieties. However, their method seems completely incapable to approach (EC). In this article, we tackle both (EC) and (BC) for general semiabelian varieties with arithmetic intersection theory. We proceed in a way that is surprisingly close to the method of Szpiro, Ullmo, and Zhang in spite of the above-mentioned obstacle. Our main result is as follows. Theorem 1. (SEC) is true for every semiabelian variety G over Q and every strict sequence (x i ) ∈ G N of small points.
As already mentioned, (SEC) is a direct consequence of (EC) and (BC); we refer to Section 6 for the deduction of Theorem 1 from (EC) and (BC). The fastest way to prove Theorem 1 is hence to prove merely (EC) and to rely on [15] for (BC). However, our technique to prove (EC) can be also used to give a new proof of (BC), which is remarkably close to Zhang's proof of (BC) in the case of abelian varieties [71]. Consequently, we can give a self-contained and genuinely Arakelov-theoretic proof of Theorem 1. This seems worth to afford the detour of proving (BC) anew, and we do so in Section 5 after some preparation in Section 4.
The centerpiece of our argument is Proposition 13, which includes (EC) as a special case. Section 3 is completely devoted to its proof. The main idea is rather simple and we describe it here for semiabelian varieties of toric rank 1. As already mentioned, a semiabelian variety G with maximal torus T = G m has canonical height zero if and only if it is almost split, which means that the associated extension class η ∈ Ext 1 Q (A, G m ) = A ∨ (Q) is a torsion point. For an ample symmetric line bundle N on A ∨ , this is equivalent to the Néron-Tate height h N (η) being zero. One can hence suspect that h N (η) quantifies the obstruction to proving (EC) by means of the standard equidistribution arguments.
The Q-isogeny class of G contains semiabelian varieties associated with extension classes is arbitrary small. In fact, if η ′ is such that nη ′ = η for some positive integer n, then h N (η ′ ) = n −2 h N (η). Writing G n for the semiabelian variety described by η ′ , there is an isogeny ϕ n : G n → G of degree n (see Section 3). Additionally, it is not hard to see that we only need to prove (EC) for a single element in the isogeny class of G. It is hence reasonable to replace our original G with some G n , n ≫ 1, and to hope that this facilitates the proof of (EC) with the traditional procedure.
From [11,Théorème 4.2], we deduce that for a certain compactification G n of G n if G is not almost split. Thus, merely replacing G with a fixed G n , n ≫ 1, is not sufficient, but working asympotically as n → ∞ seems prospective.
In other words, one should try to carry out the argument of Szpiro, Ullmo, and Zhang [62] for each level G n and observe what happens as n → ∞. The integer n is not the only parameter that appears here. As in previous proofs of (EC), a rational 2 scaling factor λ → 0 comes up. It turns out that there is some interplay between these both parameters: Up to suppressing some easily controllable terms, written (. . . ), we obtain an upper bound (compare (32) below) for every integer n and every real λ ∈ (0, n −1 ]. Choosing n a square number and λ = n −3/2 yields an upper bound ≪ G,f n −1/2 so that (EC) follows with n → ∞.
To conclude this sketch of our argument, let us briefly discuss the provenience of the two terms on the right-hand side of (4). The first term stems from the error term in the expansion (compare [71, p. 162] and Lemma 16 below) for some integer n ≫ 1. The integer n appears in the implicit constant of O G,f,n (|λ| 2 ), and we have to render the dependency more precisely. This is done by applying the projection formula to (a compactification of) the isogeny ϕ n : G n → G of degree n. The second term is more or less |λ −1 h L (G n )|, which is majorized by (3). That there is a suitable choice of n and λ relies utimately on the fact that |λ −1 h L (G n )| decreases faster than deg(ϕ n ) increases. To make a long story short, the quadraticity of the Néron-Tate height on the dual abelian variety A ∨ is played off against the linearity in the toric part of G, and quadratic decay prevails over linear growth.
As is well-known (cf. [71,Corollary 4]), (SEC) implies directly Lang's conjecture, which was proven by Hindry [32] in our setting. Furthermore, Poonen [54] and Zhang [73,Remark (3) on p. 41] pointed out that (SEC) and the Mordell-Lang Conjecture (MLC) imply a common generalization. For a finitely generated subgroup Γ of G(Q), we set Mordell-Lang plus Bogomolov Conjecture (MLBC). For any subvariety X ⊂ G that is not a translate of a semiabelian subvariety of G by a point in Γ ′ , there exists some ε > 0 By the time [54] and [73] were written, McQuillan [49] had already proven (MLC) for general semiabelian varieties so that their arguments were only conditional on the then-missing (SEC) for an archimedean place ν. In [55], Rémond gave a proof of (MLBC) that avoids (SEC) and uses instead his version [56] of Vojta's inequality for semiabelian varieties as well as (BC) for semiabelian varieties [15]. Our Theorem 1 renders the original proofs of Poonen [54] and Zhang [73] unconditional, giving an alternative to Rémond's approach.
Finally, let us remark that the availability of (EC) for general semiabelian varieties also allows to extend Zhang's equidistribution result on almost division points [73, Theorem 1.1] (cf. Remark (3) on p. 41 in loc.cit.).
Notations and conventions. General. For two terms a and b, we write a ≪ b if there exists a positive real number c such that a ≤ c · b. If c depends on some data, say an algebraic variety X, we write a ≪ X b etc. If there is no subscript, the implied constant c is absolute. We use ≫ similarly.
Number fields. Throughout this article, we let K ⊂ Q denote a number field with integer ring S = Spec(O K ). In addition, Σ f (K) (resp. Σ ∞ (K)) is the set of non-archimedean (resp. archimedean) places, and we set Σ(K) = Σ f (K)∪Σ ∞ (K). For each ν ∈ Σ(K), we let K ν denote the ν-adic completion of K. By C ν is denoted the completion of the algebraic closure K ν of K ν and by O ν its ring of integers. Additionally, k ν is the residue field of K ν and p ν its characteristic. For all ν ∈ Σ f (K), the absolute value | · | ν on C ν is normalized such that . We use the standard values of R and C for archimedean places. This normalization leads to an additional factor δ ν = 2 if ν is complex archimedean, 1 otherwise, in some identities. For an algebraic variety X over S, we denote by X ν its completion at ν. Algebraic Geometry (General). Denote by k an arbitrary field. A k-variety is a reduced scheme of finite type over k. In particular, a k-subvariety is determined by its underlying topological space and we frequently identify both. The tangent bundle of a k-variety X is written T X and its fiber over a point x ∈ X is denoted by T x X. Furthermore, X sm denotes the smooth locus of X. If X is an irreducible k-variety, we write η X for its generic point. For a non-negative integer d and a k-variety X, a d-cycle on X is a finite formal sum r i=1 n i Z i where each n i is an integer and each Z i is a k-irreducible subvariety of X having dimension d. The unity of a k-algebraic group G is written e G .
For a line bundle L over a general scheme, we denote by F (L) its sheaf of sections.
Line bundles and intersection theory. For line bundles L 1 , L 2 , . . . , L d on a proper algebraic variety X of dimension d over a field k, we use the intersection numbers L 1 · L 2 · · · L d ∈ Z defined by Kleiman [38] and Snapper [59] (see [39,Section VI.2] for a good introduction). These coincide with the numbers in the terminology of [18]. If {M 1 , M 2 , . . . , M r } = {L 1 , L 2 , . . . , L d } and each M i occurs n i -times among L 1 , L 2 , . . . , L d , we set M n 1 1 · M n 2 2 · · · M nr r := L 1 · L 2 · · · L d ; a similar notation is used for the Borel measures defined in Section 1.3 and the arithmetic intersection numbers defined in Section 1.6. Furthermore, we write deg L (X) for L d . We define The group law of Picard groups of line bundles, as well as of their various metrized versions introduced in Section 1, is written additively. For a non-zero rational section s : X L of a line bundle L, its divisor is denoted by div(s). The support of a divisor D (resp. a cycle Z) is written |D| (resp. |Z|).
Continuity and smoothness. We use C 0 (resp. C ∞ ) as an abbreviation for continuous (resp. smooth). For any topological space X, C 0 c (X) denotes the continuous functions on X. Complex spaces, differential forms, and currents. Let S be a reduced complex (analytic) space. Recall that this means that S is locally biholomorphic to a closed analytic subvariety V in a complex domain U ⊂ C n . A function f on S is smooth if, for each such sufficiently small local chart, it is the restriction of a smooth function on U. We write C ∞ (S) for the smooth real-valued functions on S. In the same way, we use local charts to define plurisubharmonic functions on S as restrictions. A differential form ω on S is simply a differential form on the smooth locus S sm of S. We say that ω has C ∞ -coefficients (or is C ∞ ) if S can be covered by local charts V ⊂ U ⊂ C n as above such that for each such chart the differential form ω| V sm is the restriction of a C ∞ -differential form on U. There are also well-defined linear operators d and d c = i/4π(∂ − ∂) on the C ∞ -differential forms on S. For each local chart V ⊂ U ⊂ C n , these are simply the restrictions of the operators of the same name on C n .
Having defined differential forms on S, we can define currents by duality. A current on S is called semipositive here if it is called (weakly) positive in the standard terminology (e.g., as in [30, Definition 2.9.3], which extends verbatim to complex spaces). For two currents T 1 and T 2 on S, we use T 1 ≥ T 2 as a shorthand for the statement that T 1 − T 2 is semipositive. For each C 0 -hermitian metric · on a holomorphic line bundle L over S, we can define a Chern current c 1 (L, · ) in the usual way; if s : U → L is a non-zero section over some open subset U ⊂ S, we set c 1 (L, · )| U = dd c (− log s ). This is a C ∞ -differential form on S if · is C ∞ .
Tangent spaces. Let Y be a complex manifold (e.g., (X sm ) an Cν for an algebraic variety X over K and some ν ∈ Σ ∞ (K)). To Y is associated its real tangent bundle T R Y and its holomorphic tangent bundle T 1,0 C Y (e.g., (T X) an Cν for a smooth complex algebraic variety X and some ν ∈ Σ ∞ (K)). The reader is referred to [ Admissible formal schemes, special and generic fibers. For each ν ∈ Σ f (K), we define admissible formal schemes over Spf(O ν ) as in [27, 2.6]. As there, we assign with an admissible formal scheme X ν over Spf(O ν ) a Berkovich C ν -analytic space X ν,η , its generic fiber, and a k ν -scheme X ν,s of locally finite type, its special fiber.
Metrized pseudo-divisors. We say that D is a ν-metrized pseudo-divisor on a K-variety X if it is a metrized pseudo-divisor on X × K K ν as defined in [27, 3.4]. In the terminology of [26], a ν-metrized pseudo-divisor on X for some ν ∈ Σ ∞ (K) is the same as a hermitian pseudo-divisor on X × K K ν .

Arithmetic Intersection Theory
In this section, we recall the basics of arithmetic intersection theory for (adelically) metrized line bundles. Basic references are [11,12,13,24,25,26,27,48,52,67,70]. Since we only need to associate intersection numbers with integrable (adelically) metrized line bundles on projective varieties over a number field, we can work rather economically, avoiding some intricacies of the original arithmetic intersection theory of Gillet and Soulé [7,17,20,60] such as arithmetic Chow rings and so forth.
1.1. Setup. Throughout this section, we consider an irreducible, projective K-variety X of dimension d. Let ν ∈ Σ f (K) and e a positive integer. Since X is projective, the analytic space X an Cν is Hausdorff and compact for every ν ∈ Σ(K) ([3, Theorem 3.4.8 (ii)]). Let L 1 , L 2 , . . . , L k be line bundles on X. (The case k = 0 is allowed here.) A formal O νmodel (X ν , L 1,ν , . . . , L k,ν ) of (X, L ⊗e 1 1 , . . . , L ⊗e k k ) consists of an admissible formal scheme X ν over Spf(O ν ) and line bundles L i,ν on X ν such that X an Cν = X ν,η and (L ⊗e i i ) an Cν = L i,ν,η . Similarly, an S-model (X , L 1 , . . . , L k ) of (X, L ⊗e 1 1 , . . . , L ⊗e k k ) is a flat, integral, projective S-scheme X and a collection of line bundles L i on X such that X = X × S K and L ⊗e i gives naturally rise to a formal O ν -model (X ν , L 1,ν , . . . , L k,ν ) of (X, L ⊗e 1 1 , . . . , L ⊗e k k ) by taking formal completions over ν.
If ν ∈ Σ f (K), the first two conditions imply automatically v + w ≤ max{ v , w } for all x ∈ X an Cν and v, w ∈ L an Cν | x . For a sequence of ν-metrics · (n) on L, we say that · (n) converges uniformly to a ν-metric · if the C 0 -functions · (n) / · converge uniformly to 1 on X an Cν . A ν-metrized line bundle L = (L, · ) consists of a line bundle L on X and a ν-metric on L. An isometry f : (L, · ) → (M, · ′ ) between two ν-metrized line bundles is an isomorphism f : L → M of line bundles that transports · to · ′ . The set of isometry classes of ν-metrized line bundles on X is denoted by Pic ν (X). For L, M ∈ Pic ν (X), N ∈ Pic ν (Y ), and every algebraic map f : X → Y , we define L + M , −L, and f * N as elements of Pic ν (X) in the obvious way. If M = (L ⊗e , · ), e a non-zero integer, is a ν-metrized line bundle, there is a unique ν-metrized line bundle L = (L, · 1/e ) such that eL ≈ M . For later applications we also set Pic ν (X) Q = Pic ν (X) ⊗ Z Q.

Lν on L an
Cν is called formally semipositive if L ν,s can be chosen to be a nef line bundle on the special fiber X ν,s . A general ν-metrized line bundle is called semipositive if its ν-metric is the uniform limit of formally semipositive ν-metrics.
For every g ∈ C 0 (X an Cν ), we define the ν-metrized line bundle O X (g) = (O X , · ) by setting 1 x ν = e −g(x) for all x ∈ X an Cν . For a ν-metrized line bundle L, we write L(g) instead of L ⊗ O X (g).

Borel measures.
For both archimedean and non-archimedean places ν ∈ Σ(K), a collection of semipositive ν-metrized line bundles L 1 , L 2 , . . . , L d ∈ Pic ν (X) gives rise to a finite regular Borel measure Cν . If X is smooth and the metrics of L 1 , L 2 , . . . , L d are C ∞ , we just take the Borel measure given by integrating with the wedge product of the Chern forms c 1 (L i ) (i ∈ {1, . . . , d}). If the metrics of L 1 , . . . , L d are only C 0 or ν ∈ Σ f (K), the definition of c 1 (L 1 ) ∧ c 1 (L 2 ) ∧ · · · ∧ c 1 (L d ) is more involved, but we nevertheless retain the notation from the smooth archimedean case for ease of notation.
We start with defining c 1 (L 1 )∧c 1 (L 2 )∧· · ·∧c 1 (L d ) for archimedean ν ∈ Σ ∞ (K). Let U ⊆ X an on U. This current depends only on c 1 (L i | U ) and not on the local potentials (− log s i ). Consequently, the currents T U 1 and T U 2 agree on U 1 ∩ U 2 for any two open sets U 1 and U 2 as above. A partition of unity argument (cf. [ Cν . It is a consequence of the Chern-Levine-Nirenberg (CLN) inequalities [14] that c 1 (L 1 ) ∧ c 1 (L 2 ) ∧ · · · ∧ c 1 (L d ) does not charge locally pluripolar sets and has finite mass (cf. [30, Theorem 3.14] 4 ). 5 For non-archimedean ν ∈ Σ f (K), we use the measures introduced by Chambert-Loir (cf. [12, Section 2]) and define c 1 (L 1 ) ∧ · · · ∧ c 1 (L d ) as in [27, (3.8)]. For this, we choose nonzero sections s i : as the local height of X in the sense of Gubler [26, Section 9], we obtain a functional which induces a Borel measure on X an Cν by Riesz representation theorem [58, Theorem 2.14]. The ensuing measure is independent from the choice of s i by [27, Theorem 3.5 (c)], and we 4 Literally, this theorem only applies if X an Cν is smooth. By Hironaka's resolution theorem [33] (see also [40]), there always exists a smooth variety X and a birational, projective morphism f : X → X. One can then use [30,Theorem 3.14] to prove that c 1 (f * L 1 ) ∧ c 1 (f * L 2 ) ∧ · · · ∧ c 1 (f * L d ) does not charge locally pluripolar sets in X. In particular, no mass is attached to the ramification locus E of f . We can then obtain the same assertion for c 1 (L 1 ) ∧ c 1 (L 2 ) ∧ · · · ∧ c 1 (L d ) by means of Lemma 3 (c). 5 Finiteness also implies the claimed regularity by [ denote it by c 1 (L 1 ) ∧ c 1 (L 2 ) ∧ · · · ∧ c 1 (L d ) in the sequel. Moreover, the measure is finite by [27, Corollary 3.9 (c)] and hence regular by [58,Theorem 2.18].
In spite of the different definitions, the following two lemmas allow to treat both cases ν ∈ Σ f (K) and ν ∈ Σ ∞ (K) in a uniform manner.

Lemma 2.
For each i ∈ {1, . . . , d}, let L i be a line bundle on X and · (n) i a sequence of semipositive ν-metrics converging uniformly to · i . Writing L (c) (Projection Formula) Let Y be an irreducible, projective K-variety, and let f : Y → X be a generically finite map of degree deg(f ). Then, the push-forward measure Proof. In case of ν ∈ Σ ∞ (K), the first three assertions are evident if all metrics are C ∞ . The local nature of the first three statements allows as above to use plurisubharmonic smoothings (cf. [30,Proposition 1.42]) and Lemma 2. For the fourth statement, which is stated as [16,Corollary 9.3], one needs a global C ∞ -regularization of the C 0 -metrics (compare our Appendix A).
The non-archimedean case ν ∈ Σ f (K) is just [27, Corollary 3.9] for the first three assertions and [27, Proposition 3.12] for the fourth one.
We say that L ∈ Pic ν (X) Q is integrable if there exists a non-zero integer n and semipositive L 1 , L 2 ∈ Pic ν (X) such that nL = L 1 − L 2 . By (a) of the above lemma, we can define signed Borel measures 1.4. Hermitian line bundles on arithmetic varieties. Let X be a flat, integral, projective S-scheme of relative dimension d + 1. A hermitian line bundle L on X is a collection (L, { · ν } ν∈Σ∞(K) ) consisting of a line bundle L on X and a ν-metric · ν on L K for each archimedean place ν ∈ Σ ∞ (K). We say that the hermitian line bundles L and M are isometric if there is an isomorphism L ≈ M preserving the metrics at all archimedean places. The arithmetic Picard group Pic(X ) is the set of isometry classes of hermitian line bundles on X .
Again, there is a natural notion of isometry between metrized line bundles and the isometry classes of metrized line bundles form a Picard group Pic(X). If L and M are metrized line bundles with underlying line bundles L and M, there is an obvious way to endow L ⊗ M and L −1 with the structure of metrized line bundles. We write L ⊗ M and L −1 , respectively, for these metrized line bundles. If f : Y → X is an algebraic map between irreducible, projective K-varieties and L is a metrized line bundle, we can endow the pull-back f * L with a canonical adelic metric, obtaining a metrized line bundle f * L. For an inclusion f : Metrized line bundles are induced by hermitian line bundles on S-models. In fact, let (X , L) be an S-model of (X, L ⊗e ) and let L = (L, { · ν } ν∈Σ∞(K) ) ∈ Pic(X ) be a hermitian line bundle.
is an adelic metric on L, and we call adelic metrics of this type algebraic. The corresponding metrized line bundles are called algebraically metrized.
All other adelic metrics of interest for us arise from algebraic adelic metrics by means of a limit process. Let { · i,ν }, i ∈ N, be a sequence of adelic metrics on L. We say that these metrics converge uniformly to an adelic metric { · ν } on L if there exists a finite set of Finally, an element L ∈ Pic(X) is called vertically integrable if each L ν ∈ Pic ν (X), ν ∈ Σ(K), is integrable. Again, we can extend this terminology to Pic(X) Q = Pic(X) ⊗ Z Q.

Arithmetic intersection pairings. Given vertically integrable
We rely on Gubler's theory of local heights [24,25,26] for this task. We start by choosing non-zero rational sections s i : X L i such that There always exist rational sections s i meeting this condition. Each section s i defines a ν- By our assumption on vertical integrability, we can use [26,Theorem 10.6] to obtain a collection of (unique) local heights A similar definition for ν ∈ Σ ∞ (K) is given in [26,Theorem 10.6] under the assumption that the hermitian metrics · i,ν are C ∞ . This assumption can be lifted by using the induction formula of [26, Proposition 3.5] as a recursive definition for the integrals encountered are finite by the CLN inequalities [30,Theorem 3.14]. Using a regularization lemma [48, Théorème 4.6.1], which is essentially due to Richberg [57], one can deduce the standard properties for the local heights thus defined in the C 0 -case from the C ∞ -case. This is straightforward, but requires some checking. The sceptic reader is referred to Appendix A for further indications. Arithmetic intersection numbers can be simply defined as sums of local heights, by setting For this to be a valid definition, the right-hand side has to be independent of the chosen sections s i and all except finitely many summands have to be zero. The former fact follows from the product formula by [26,Propositions 3.7 and 9.4], and the latter one follows from the compatibility with ordinary intersection theory [25,Section 6]. This compatibility also shows that the intersection numbers in (7) generalize those defined by Gillet and Soulé [20,60] and their extension by Zhang [70].
To simplify notation, we write . From [26, Theorem 10.6 (b)], we know that (c) (Projection Formula) Let Y be an irreducible, projective K-variety, and let f : Y → X be a generically finite map of degree deg(f ). Then, Each of the first four statements follows from the respective property of Gubler's local heights ([26, Theorem 10.6]), which extends to archimedean C 0 -metrics by Appendix A.
The final statement is a consequence of [26, Propositions 3.7 and 9.4] and the product formula.
We need also an extension of the projection formula mentioned in the above lemma, which is a slight generalization of [51, Proposition 1.3] to our setting.
Lemma 5. Let Y be an irreducible, projective K-variety, and let f : Y → X be a proper surjective map. Set d = dim(X) as well as d ′ = dim(Y ). Then, Proof. The proof of [9, Proposition 2.3] can be straightforwardly adapted to our situation, starting from Lemma 4 (c) and using the induction formula [26, Proposition 3.5 and Remark 9.5].
We conclude with two further direct consequences of the induction formula [26, Proposition 3.5 and Remark 9.5]. Fix some ν ∈ Σ(K) and f ∈ C 0 (X an If ν ∈ Σ ∞ (K) and we consider κ ∈ R as a constant function on X an Cν , we hence have If L is nef, this means by the algebraic Riemann-Roch Theorem ([46, Corollary 1.4.41]).
As for the Borel measures defined in Section 1.3, Lemma 4 (a) allows to extend the definition of the arithmetic intersection number to integrable elements of Pic(X) Q . The above results evidently remain valid in this generality.

1.7.
Heights. Using the intersection numbers defined above, we can define the height h L (Y ) of a subvariety Y ⊆ X with respect to a metrized line bundle L = (L, { · ν } ν∈Σ(K) ) such that L is ample. In fact, we set We give an analogous definition of the height for a closed point x ∈ X. Writing K(x) for the residue field of x, we set h L (x) = ( L| x )/[K(x) : K]. Combining (7) with the induction formula [26, Proposition 3.5 and Remark 9.5], we obtain for any non-zero section s ∈ H 0 (X, L) such that x / ∈ |div(s)|.
1.8. Positivity. Having arithmetic intersection numbers and heights at our disposal, we collect here various notions of positivity for metrized line bundles. An algebraic adelic metric { · ν } ν∈Σ(K) on L that arises from an S-model (X , L) of (X, L ⊗e ) and a hermitian line bundle (c) semipositive if it is both vertically and horizontally semipositive. A metrized line bundle is called semipositive if its adelic metric is the uniform limit of semipositive algebraic adelic metrics. It is called integrable if it is the difference of two semipositive metrized line bundles.
For a vertically semipositive L ∈ Pic(X), each ν-metrized line bundle L ν ∈ Pic ν (X), ν ∈ Σ(K), is semipositive. Consequently, an integrable metrized line bundle is also vertical integrable in the sense of Section 1.5. An element L ∈ Pic(X) Q is called semipositive (resp. integrable) if there exists some integer n ≥ 1 such that n L is contained in the image of Pic(X) and semipositive (resp. integrable) according to the above definition.
1.9. Arithmetic volumes. Let again L = (L, { · ν } ν∈Σ(K) ) be a metrized line bundle on X and write L ⊗N = (L ⊗N , { · ⊗N ν }). For each integer N ≥ 1, we consider the global sections V N = H 0 (X, L ⊗N ) as a r N -dimensional K-vector space and form the tensor product V N,A = V N ⊗ K A K with the adeles A K of K. For each ν ∈ Σ(K), we can additionally endow V N with a sup-norm there is a unique invariant Haar measure vol N (·) on V N,A that is normalized such that the induced quotient measure on V N,A /V N has total mass 1. We then define the adelic unit ball Let s 1 , . . . , s r N : X → L ⊗N be a basis of V N . By the coherence condition in the definition of metrized line bundles, we have s i for almost all ν ∈ Σ(K). This shows that vol N (B N ) is a non-zero real so that we can set The arithmetic volume of a metrized line bundle L ∈ Pic(X) is defined by If the adelic metric of L is induced by a hermitian line bundle L on some S-model of X, the volume vol χ ( L) agrees with the volume denoted vol χ (L) in [37]. We collect some standard results on vol χ (·) in the following lemma. Lemma 6. Let L = (L, { · ν }) be a metrized line bundle. Then, (a) for any ν ∈ Σ ∞ (K) and any real κ ∈ R, which is considered as a real-valued constant function on X an Cν , we have vol χ ( L(κ)) = vol χ ( L) + δ ν κ(d + 1)vol(L).
(b) Let L i = (L, { · ν }) be a sequence of metrized line bundles such that the adelic metric of L i converges to L. If L is big, we have Otherwise, we have vol χ ( L i ) = vol χ ( L) for all i. (c) Assume that the adelic metric of L is a uniform limit of algebraic adelic metrics. For each integer k ≥ 1, we have vol χ ( L ⊗k ) = k d+1 vol χ ( L).
Proof. The first assertion follows from χ sup ( L ⊗N (κ)) = χ sup ( L ⊗N ) + κ · r N N for real ν and similarly for complex ν. 6 For the second one, we define a C 0 -function e −φ i,ν = · i,ν / · ν on X an Cν for each i and ν ∈ Σ(K). There exists a finite subset Σ ⊂ Σ(K) such that φ i,ν = 0 for all ν ∈ Σ(K) \ Σ. For each ν ∈ Σ, we have φ i,ν → 0 (i → ∞) uniformly on X an Cν . The claim then follows from Part (c) is [37,Theorem 3.3.2] in the case where the adelic metric of L is algebraic. Using part (b) of the lemma, we can extend this result.
Lemma 6 (c) allows us to extend the definition of vol χ (·) to Pic(X) Q . The next lemma is a straightforward consequence of Ikoma's version of Yuan's bigness theorem [66].
Proof. Because of homogenity (Lemma 4 (a) and Lemma 6 (c)), we can assume that L, M ∈ Pic(X). By assumption, there exists a sequence ( L i ) (resp. ( M i )) of semipositive algebraically metrized line bundles whose adelic metrics converge uniformly towards the metric of L (resp. M ). We can assume that both L i and M i are given by hermitian line bundles on the same S-model X of X. 7 Let ε > 0 be a real number. By Lemma 6 (b), we have Taking the limit ε → 0 finishes the proof.

1.10.
Minkowski's Theorem. The following lemma is typical in applications of Arakelov theory to diophantine geometry. Lemma 8. Let ν ∈ Σ(K) and a real ε > 0 be given. For each L = (L, { · ν }) ∈ Pic(X) with big L, there exists a non-zero section s ∈ H 0 (X, L ⊗N 0 ), N 0 a positive integer, such that Proof. In this proof, we use the notations V, V N , r N , V N,ν , B N , vol N as in Section 1.9. The lemma is implied by an adelic version of Minkowski's Second Theorem [6,Theorem C.2.11]. To use the theorem, fix an identification V N ≈ K r N and note that the Haar measure used there agrees with vol N ([6, Proposition C.1.10]). Setting 7 In fact, if L i and M i are induced from hermitian line bundles L i ∈ Pic(X i ) and M i ∈ Pic(X ′ i ), then we can replace X i and X ′ i with the Zariski closure X of the diagonally embedded copy of X in X i × X ′ i and the line bundles L i and M i with their pullbacks to X . The theorem yields hence a non-zero section s ∈ S, which means that and log s µ ≤ 0. Using the algebraic Riemann-Roch Theorem ([46, Corollary 1.4.41]), we infer the assertion of the lemma.

Semiabelian Varieties
We start by summarizing the basic facts on homomorphisms and compactifications of semiabelian varieties that are essential for our main proof. The reader may also compare with [41, Sections 1 and 2].
2.1. Basics. A semiabelian variety G over a field k is a connected smooth algebraic k-group that is the extension of an abelian variety A over k by a k-torus T . This means that there exists an exact sequence in the abelian category of commutative k-algebraic groups of finite type ([19, Théorème VI A .3.2]). Both T and A are uniquely determined by G so that we may call T the toric part of G and G → G/T = A (or just A) the abelian quotient of G in the following. Furthermore, the exact sequence (12) describes a Yoneda extension class in Ext 1 k (A, T ). In the sequel, we write η G for the extension class associated to a semiabelian variety G in this way. Each homomorphism ϕ : B → A (resp. ϕ : T → S) of abelian varieties (resp. tori) induces a pullback ϕ * : Ext 1 k (A, T ) → Ext 1 k (B, T ) (resp. a pushforward ϕ * : Ext 1 k (A, T ) → Ext 1 k (A, S)). The Weil-Barsotti formula (see [53,Section III.18] or the appendix to [50]) gives a canonical identification Ext 1 k (A, G m ) = A ∨ (k). If T is split (i.e., T = G t m ), we make frequent use of the identity Ext 1 k (A, G t m ) = Ext 1 k (A, G m ) t = (A ∨ ) t (k). With this identification, it is easy to describe pushforwards. If ϕ : G t m → G t ′ m is the homomorphism described by ϕ * (Y v ) = t u=1 X auv u in standard coordinates X 1 , . . . , X t (resp. Y 1 , . . . , Y t ′ ) on G t m (resp. G t ′ m ), then the pushforward Homomorphisms and compactifications. Let G (resp. G ′ ) be a semiabelian variety over a field k with maximal abelian quotient A (resp. A ′ ) and split maximal subtorus G t m (resp. G t ′ m ). Recall from Lemma [41, Lemma 1] that homomorphisms ϕ : G → G ′ give rise to commutative diagrams with homomorphisms ϕ tor : G t m → G t ′ m and ϕ ab : A → A ′ . Conversely, a (unique) homomorphism ϕ exists for a pair (ϕ tor , ϕ ab ) if and only if (ϕ tor ) * η G = ϕ * ab η G ′ ∈ Ext 1 (G t ′ m , A). We use the compactification G of G given in [41,Construction 5], which differs from the one used in [11,Section 4]. This means that we consider the inclusion G t m ֒→ (P 1 ) t and form the contraction product G × G t m (P 1 ) t = (G × (P 1 ) t )/G t m , which is a smooth k-variety G into which G embeds. The abelian quotient π : G → A extends to an algebraic map π : G → A.
Writing η G = (Q 1 , . . . , Q t ) ∈ A ∨ (k) t = Pic(A) t , our compactification can be also described as the embedding (14) G ). Replacing the factor Proj(Sym(O A ⊕ F (Q i ) ∨ )) in (14) with this divisor, we obtain a Weil divisor D G,i ⊗ π * N is ample. In the sequel, G,i ). We also have to compactify (the graphs of) homomorphisms. A general procedure for this is outlined in [41,Construction 7], but we only need some special cases and we describe these in detail here. Assume that ϕ : G → G ′ is a homomorphism of semiabelian varieties such that ϕ tor = [n] G t m in the notation of (13). The homomorphism [n] G t m : G t m → G t m extends to a map [n] G t m : (P 1 ) t → (P 1 ) t so that there is a map It is easy to see that this map descends to a map

Canonical metrics and heights.
Our references for this subsection are [10,11,69]. We let G be a semiabelian variety over a number field K. Denote by π : G → A its abelian quotient and by T its toric part. Enlarging K if necessary, we may assume that T = G t m . In this situation, the last subsection yields a compactification G and line bundles M We next aim to decorate M (κ) G,i and N with adelic metrics, following Zhang [70]. The technical result that we need is summarized in the following lemma.
Lemma 9. Let X be a projective variety, L a line bundle on X, f : X → X a surjective algebraic map, d > 1 an integer, and φ : L ⊗d → f * L an isomorphism of line bundles over X. Then, (a) for each ν ∈ Σ(K), there exists a unique ν-metrized line bundle L ν = (L, · ν ) such that φ is an isometry L ⊗d ν → f * L ν , (b) on replacing φ with cφ, the metric · ν changes to |c| 1/(d−1) ν · ν , (c) the metrics · ν thus obtained combine to an adelic metric { · ν } ν∈Σ(K) on L, . Let L 0 be a very ample line bundle on X such that (L ⊗ L 0 ) ⊗e is also very ample. In other words, the global sections yield projective embeddings ι L 0 : X ֒→ P k 1 K and ι (L⊗L 0 ) ⊗e : X ֒→ P k 2 K . Composing with the diagonal map, we thus obtain an embedding ι : (1) where pr i (i = 1, 2) denotes the projection to the i-th factor.
Let X be the Zariski closure of X in P k 1 S × P k 2 S . We set L = (pr * 2 O P k 2
For each integer n ≥ 1, we consider the graph embedding ι Γ = (id X , f •n ) : X ֒→ Γ(f •n ) ⊂ X × X and the Zariski closure Γ(f •n ) of Γ(f •n ) in X × X . Writing pr 2 : X × X → X for the projection to the second factor, we define the line bundle L n = (pr * 2 L)| Γ(f •n ) . The isomorphism φ induces an isomorphism φ n : L ⊗d n → (f •n ) * L over X. Through ι Γ and φ n , the tuple (Γ(f •n ), L n ) is an S-model of (X, L ⊗d n e ). For each ν ∈ Σ f (K), we have induced formal ν-metrics · 1/d n e Ln,ν on X. An inspection of the argument of [70, Theorem 2.2 (a,b)] shows that the ν-metric · 1/d n e Ln,ν converges uniformly to the ν-metric · ν from (a) as n → ∞. Furthermore, there exists a non-empty open U ⊂ S such that each iterate f •n exists to a map f •n : X | U → X | U and each φ n extends to an isomorphism φ n : L ⊗d | U → f * L| U over X | U . For each ν ∈ U, this implies · ν = · 1/d n e pr * 2 Ln,ν = · 1/e pr * 2 Lν , which shows that { · ν } ν∈Σ(K) is an adelic metric.

Using [n]
* M G,i is not as easy to establish as for N . In [11,Proposition 3.6], specific regular models of abelian varieties constructed by Künnemann [42] are used for this. We give a more elementary proof here.
In this way, we obtain a multiprojective embedding g 0 × g 1 × · · · × g t : A ֒→ P k 0 K × P k 1 K × · · · P kt K and we denote by A the Zariski closure of A in P k 0 S × P k 1 S × · · · P kt S . Writing pr i : P k 0 S × P k 1 S × · · · × P kt S → P k i S for the various projections, we set N i = pr * i O P k i S (1)| A (0 ≤ i ≤ t) and Q i = N i ⊗N −1 0 . By construction, (A, N 0 , Q 1 , . . . , Q t ) is an S-model of (A, N ⊗3 , Q 1 , . . . , Q t ).
We next define whose generic fiber is evidently G → A. As in Subsection 2.2, the ideal Sym(0 ⊕ F (Q i ) ∨ ) (resp. Sym(O A ⊕ 0)) describes a Cartier divisor on Proj(Sym(O A ⊕ F (Q i ) ∨ )). Replacing the factor Proj(Sym(O A ⊕ Q ∨ i )) in (15) with this divisor, we obtain a Cartier divisor D G,i for any ν ∈ Σ ∞ (K) (see [41,Lemma 14]) so that we obtain an adelic metric { · M (κ) G,i × S k ν is relatively ample with respect to π × S k ν . For each integer n ≥ 1, an application of [23, Théorème 4.6.13 (ii)] yields that the line bundle (M for every proper curve C ⊆ G × S k ν . Taking the limit n → ∞, this implies (M (κ) 2.4. Heights and Homomorphisms. We next recall a lemma that controls the behavior of our canonical height under homomorphisms. Lemma 11. Let G i be a semiabelian variety with maximal torus T i = G t i m and maximal abelian quotient π : for all closed points x ∈ G 1 .
Proof. The first inequality is equivalent to h N 2 (ϕ ab (x)) ≪ N i ,ϕ h N 1 (x) for all closed points x ∈ A 1 . It follows hence from an application of [65,Proposition 2.3] to the map ϕ : A 1 A 2 . As usual, the term O(1) in loc. cit. disappears by Tate's limit argument. The second inequality can be deduced from [41,Lemma 10], letting the second homomorphism ϕ ′ in this lemma be the zero homomorphism. G,i to ease notation. Lemma 12. There exists a finite collection {(U j , ψ j )} j∈J where U j , j ∈ J, are open sets covering A an Cν and ψ j , j ∈ J, are holomorphic maps For each j, j ′ ∈ J and each i ∈ {1, . . . , t}, the quotient : (π an Cν ) −1 (U j ∩ U j ′ ) → C is a locally constant function with values in S 1 = {z ∈ C | |z| = 1}. Furthermore, we have (17) ker(dψ i ) y ∩ ker(dπ) y , i ∈ {1, . . . , t}, for any j, j ′ ∈ J and all y ∈ (π an Cν ) −1 (U j ∩ U j ′ ). In the following, Weil functions are used to describe archimedean metrics. The reader is referred to [44,Chapter 10] Let U be an open, simply connected subset of A an Cν , and let us assume additionally that U and div(s i ) an Cν are disjoint. We can then lift the inclusion π an Cν | U : U ֒→ A an Cν to a map π : U → C g where C g is interpreted as the universal covering of A an Cν . By [43,Chapter X], there exists a normalized theta function ϑ i on C g whose (analytic) divisor div(ϑ i ) is the pullback of div(s i ) an Cν along the universal covering C g ։ A an Cν . Rescaling if necessary, we may assume that f i (e G ) = 1 and ϑ i (0, . . . , 0) = 1. We define a map ψ : G an Cν | U → U × ((P 1 Cν ) t ) an by setting ψ(y) = (π(y), ψ 1 (y), . . . , ψ t (y)) , ψ i (y) = f i (y)/ϑ i ( π(y)), for y ∈ G an Cν | U ; this extends uniquely to a biholomorphism G is a meromorphic function on U j ∩ U j ′ , this implies (16), whence also (17). We can also define C ∞ -functions λ i : G an i (y)| for all y ∈ U j . The mentioned proposition in [64] states then that λ i is the unique Weil function for D Lemma 9 (a) implies that · G,i = e −|λ i | | · | where | · | is the ordinary absolute value on rational functions. Similarly, we can argue with φ (n) Gm and obtain that · Gm = e −| log |z|| | · |. Combining these two identities and using |log(·)| • ψ

The Equidistribution Conjecture
As in the introduction, we let K be a number field, ν ∈ Σ(K) an arbitrary place, G a semiabelian variety over K, T = G t m the torus of G, and π : G → A its abelian quotient. Thus, G is given by an exact sequence . We may then use the compactification G and the map π : G → A described in Section 2.2. Write M (resp. N) for the vertically semipositive metrized line bundle M G ∈ Pic(G) (resp. N ∈ Pic(A)) defined in Section 2.3 and set L = M + π * N.
Our aim in this section is to prove (EC) in a slighter stronger form, namely for arbitrary subvarieties X ⊆ G. This extra strength is needed in Section 5 for the proof of (BC). The case X = G corresponds to (EC).
Proposition 13. Let X ⊆ G be an irreducible algebraic subvariety. Set d = dim(X), d ′ = dim(π(X)), and define the Borel measure Cν . Furthermore, let (x i ) ∈ G N be a generic sequence of small points contained in X. For any f ∈ C 0 (X an Cν ), we have then The proposition is proven at the end of this section, after a series of preparatory lemmas. Before starting with them, we have to introduce some further objects. For each integer n ≥ 1, we choose (arbitrary) η (n) i ∈ A ∨ (Q), 1 ≤ i ≤ t, such that n·η (n) i = η i . Let G n be the semiabelian variety given by the extension class (η . From Section 2.2, we know that there exists an isogeny ϕ n : G n → G such that, in the notation of (13), we have ϕ n,tor = [n] and ϕ n,ab = id A . We have again a standard compactification G n of G n and a map π n : G n → A from Section 2.2. In addition, the homomorphism ϕ n extends to a map ϕ n : G n → A. Let M n ∈ Pic(G n ) denote the vertically semipositive metrized line bundle M Gn defined in Section 2.3. The construction in Section 2.2 shows that ϕ * n M ≈ M n , which we can use to ensure ϕ * n M = M ⊗n n by Lemma 9 (b). Writing L n = M n + π * n N , we have an identity L n = ϕ * n (n −1 M + π * N ), which we invoke frequently. We use X n as a shorthand for the preimage ϕ −1 n (X) ⊆ G n . Our next lemma controls the growth of geometric degrees as n → ∞. In its proof and the one of Lemma 15 below, we use the standard notation from Fulton's book [18] freely.
because deg(ϕ n ) = n t , ϕ * n M = nM n , and ϕ * n π * N = π * n N. Note that M is nef by [41,Lemma 3] and that π * N is nef because N is ample. Using [38, Theorem III.2.1], we infer The lemma is proven if we can show that the degree on the right-hand side of (21) is positive. Set η = η π(X) . By an ascending induction on the fiber dimension d − d ′ , we can deduce from the projection formula. Since M η is ample on X η and N is ample on π(X), the two factors on the right-hand side of this identity are strictly positive by [18,Lemma 12.1].
The next lemma justifies the choice of the measure µ ν in Proposition 13.
Proof. Another use of the projection formula reveals that which is clearly zero whenever i > d ′ . With (20), we infer that Recall from the proof of Lemma 14 above that (M| X ) d−d ′ · (π * N| X ) d ′ > 0. Invoking Lemma 3 (a) and (c), we obtain similarly The lemma follows by combining these two asymptotic estimates.

Proof. We have
h Ln(λfn) (X n ) = ( L n (λf n )| Xn ) d+1 (d + 1)(L n | Xn ) d by definition and by Lemma 4 (a) and (b). Since Cν f n µ n,ν by (8), the lemma boils down to As ϕ * n M = n M n , ϕ * n π * N = π * n N and deg(ϕ) = n t , Lemma 4 (c) implies that By Lemma 4 (a) and (8), we have for any i ≥ 2. By the integrability assumption, there exist semipositive P 1 , P 2 ∈ Pic ν (X) Q such that O X (f ) = P 1 − P 2 . The above integral is then bounded from above by . By the projection formula, this is zero if j > d ′ . Consequently, the term in (23) is ≪ X,f |λ| i n t−d+d ′ +i−1 . In combination with Lemma 14, this gives that each term on the left-hand side of (22) is ≪ X,f |λ| i n i−1 . Since |λ| i n i−1 ≤ |λ| 2 n, we obtain (22).
Starting with the following lemma, we fix some place ν 0 ∈ Σ ∞ (K) so that we can regard a real number κ as a constant function on X an Cν 0 and define L n (κ) as in Section 1.5. For our purposes, it is immaterial which place ν 0 we choose, even whether ν = ν 0 or ν = ν 0 , so that we omit any further reference to the place ν 0 in the following.
Lemma 17. There exists some positive real κ ≪ G n −2 such that L n (κ) is horizontally semipositive. Consequently, we have for every irreducible algebraic subvariety X ⊂ G containing a generic sequence of small points.
For the compactication G * used by Chambert-Loir, an explicit height formula [11,Théorème 4.2] implies the same asymptotics in the case X = G * . By Zhang's proof of (BC), (25) implies that π(X n ) is a translate of an abelian subvariety of A by a torsion point. However, we have no use for this information, and it does not simplify the arguments below.
Proof. By definition, L n (κ) is horizontally semipositive if and only if Our first assertion hence follows directly from the statement of [11,Lemme 4.5] if t = 1. In general, the compactification used there differs from ours and the argument has to be slightly adjusted. But this is straightforward and hence left to the reader. By Zhang's ampleness theory in the incarnation of [11, Théorème 1.5], we have for any subvariety Y ⊆ G n , which proves the lower bound in (24). The upper bound in (24) is another direct consequence of Zhang's inequalities because X contains a generic sequence of small points. Finally, π n (X n ) ⊆ A also contains a generic sequence of small points in this situation, whence (25).
We next give an asymptotic lower bound on the arithmetic volume of L n (λf n )| Xn .
Lemma 18. Assume that O X (f ) ∈ Pic(X) Q is integrable and that L n (κ), κ > 0, is horizontally semipositive. For any positive integer n and any real number λ ∈ [−n −1 , n −1 ], we have Note that the two terms on the left-hand side are equal if L n (λf n )| Xn is vertically semipositive (see [37, Theorem 3.5.1 and Remark 3.5.4]). In our situation, there are however two obstructions to this line of reasoning. First, L n (λf n )| Xn may not be semipositive for small λ even if L n is so. This problem has to be dealt with already in the almost split case (cf. [11]). The second problem is that L n is not horizontally semipositive, which is a new problem for general semiabelian varieties. To work around this, we follow the argument given in [66, Section 3.2] but have to pay additional attention to the errors terms suppressed therein.
By (9) and Lemma 6 (a), the left-hand side of the above inequality equals We can now reap the proceeds of the above lemmas.
Proof of Proposition 13. It is well- then there exist g n ∈ C 0 (G an Cν ) converging uniformly to f such that each O G (g n ) ∈ Pic(G) Q is integrable; in fact, this follows from [25,Theorem 7.12] and [28,Proposition 3.4] (see also [66,Lemma 3.5] and [67,Section 10.4]). For the proof of the proposition, we can hence always assume that O G (f ) is integrable.
From now on, let n be a fixed integer. We choose also a non-zero rational λ ∈ [−n −1 , n −1 ] and a real ε > 0; an explicit choice of λ is given below. By Lemma 17, we can find some positive real κ ≪ G n −2 such that L n (κ) is horizontally semipositive.

Equilibrium Measures
In preparation for the proof of (BC) in Section 5, we investigate here the measures from Proposition 13 in more detail. We continue with the notation of Section 3 but restrict to an archimedean place ν ∈ Σ ∞ (K) throughout this section. Choose a local trivialization {(U j , ψ j )} j∈J , J finite, of G an Cν as in Lemma 12 and write ψ j = (ψ G,i . Let X ⊆ G be a geometrically irreducible algebraic subvariety of positive dimension. Set d = dim(X), d ′ = dim(π(X)), and t ′ = d − d ′ . We let I X be the set of all t ′ -tuples (i 1 , i 2 , . . . , i t ′ ) such that where η π(X) is the generic point of π(X). For each i = (i 1 , i 2 , . . . , i t ′ ) ∈ I X , we define the subset Using (16), we see that X i is a closed and hence compact real-analytic subset of X an Cν . We next define complex-analytic subsets E i ⊂ X an Cν , i ∈ I X , such that each X i has a simple structure away from E i . For this, we first set and note that by (17) we have E is a closed complex-analytic subset of X an Cν . We set We collect the main properties of X i and E i in the following lemma.
Lemma 19. Let j ∈ J and i ∈ I X . Then, (a) E i is a closed complex-analytic subset of X an Cν having dimension < d, restricts to a local biholomorphism with codomain (π(X) sm × G t ′ m ) an Cν , (c) X i \ E i is a union of finitely many real-analytic manifolds, each having dimension d + d ′ , and restricts to a real-analytic local isomorphism with codomain (π(X) sm ) an Proof. (a): It is enough to show that the closed complex-analytic subset j∈J E Cν is irreducible as a complex-analytic set, we only have to find a point y ∈ X an Cν not contained in j∈J E (j) i . By assumption (33), there exists a closed point z ∈ π(X) sm Cν such that the fiber X| z is of dimension t ′ and Let U j , j ∈ J, be such that z an ∈ U j . Note that ψ (j) i | (X|z) an is the analytification of an algebraic map f : X| z → (P 1 ) t ′ (either by Chow's theorem [31,Theorem M.3] or by inspecting the proof of Lemma 12) such that M G,i | z ≈ f * pr * i M Gm where pr i : (P 1 ) t ′ → P 1 is the projection to the i-th factor. It is easy to see that dim(f (X| z )) = t ′ ; for the projection formula ([18, Proposition 2.5 (c)]) would else imply that the intersection number in (34) is zero. By [31, Lemma L.6 and Theorem N.1], this implies that there exists some smooth point y ∈ (X| z ) an such that the rank of (dψ (j) i | (X|z) an ) y is t ′ = dim(X| y ). This means nothing else but ker(dψ Any real-analytic set is locally a union of finitely many real-analytic manifolds (e.g., by [47,Theorem 2]). By compactness, we can hence write X i as a union of finitely many realanalytic manifolds. It only remains to show that X i \ E i has local dimension d + d ′ everywhere. Using Cν \ E i ) and (b), the two assertions follows from the standard fact that (S 1 ) t ′ ⊂ (G t ′ m ) an Cν is a real-analytic submanifold of dimension t ′ .
With the information of Lemma 19 at our disposal, we can conclude this section with an explicit description of the measures introduced in Proposition 13. Let ω t ′ be the unique (S 1 ) t ′invariant t ′ -form on the compact real Lie group (S 1 ) t ′ such that (S 1 ) t ′ ω t ′ = 1. For each j ∈ J, the pullback (ψ (j) i ) * ω t ′ is a t ′ -form on (X i ∩ π −1 (U j )) \ E i . By (16) and (S 1 ) t ′ -invariance, these forms glue together to a t ′ -form ω i on X i \ E i .
Proof. By linearity, it suffices to prove that for each i ∈ I X and all f ∈ C 0 c (X an Cν ). Since the subsets E i ⊂ X an Cν are locally pluripolar by Lemma 19 (a), we can further restrict to f ∈ C 0 c (X an Cν \ E i ). Using a partition of unity and Lemma 19 (b), we can even restrict to the case where f ∈ C 0 c (U) with U ⊆ X an Cν \ E i a relatively compact, open subset such that, for some j ∈ J, the map π an Cν × ψ Cν is an open subset and i , the left-hand side of (35) becomes i )| −1 U and z 1 , . . . , z t ′ the standard coordinates on (G t ′ m ) an Cν . The Borel measure dd c log |z 1 | ∧ dd c log |z 2 | ∧ · · · ∧ dd c log |z t ′ | ∧ c 1 (pr * 0 N ν | U ′ 0 ) ∧d ′ is the product of the measures induced by dd c log |z 1 |, dd c log |z 2 |, . . . , dd c log |z t ′ | on (G 1 m ) an Cν and c 1 (pr * 0 N ν | U ′ 0 ) ∧d ′ on U ′ 0 ; indeed, this follows from the corresponding fact for C ∞ -forms by plurisubharmonic smoothings (combine [ It is an elementary exercise to compute that h(e iφ ) dφ 2π for every h ∈ C 0 c ((G m ) an Cν ). Using Fubini's Theorem once again, we see that (37) equals By the substitution formula and Lemma 19 (d), this equals U f ω i as claimed.

The Bogomolov Conjecture
Our argument for deducing (BC) from the archimedean case of Proposition 13 follows Zhang's argument [71], which itself is a generalization of an argument due to Ullmo [63]. The new difficulty is that the measures µ ν , ν ∈ Σ ∞ (K), from Proposition 13 are not described by smooth differential forms, which is why we need Lemma 20. For an algebraic torus G = G t m or a split semiabelian variety G = G t m × A, our deduction of (BC) from (EC) is genuinely different from the one given, respectively, by Zhang [69,Theorem 6.2] or Chambert-Loir [10,Section 7]. In fact, these proofs do not seem to extend well to general semiabelian varieties.
Proposition 21. (BC) is true for every semiabelian variety G over Q.
Before proving the proposition, we start with a lemma.
Lemma 22. Let X be a geometrically irreducible subvariety of G. Assume that the stabilizer group is trivial (i.e., equal to {e G Q }). For all integers m ≫ X 1, the algebraic map is then quasi-finite of generic degree 1.
Proof. This can be proven in the same way as [71, Lemma 3].
With this lemma, we can start the main proof of this section.
Proof of Proposition 21. We may and do assume that X is of positive dimension. Let G t m be the maximal torus of G. We first reduce to the case Stab G Q (X Q ) = {e G Q }. By enlarging K, we can assume that Stab G Q (X Q ) is H Q for some algebraic subgroup H ⊆ G. Consider the quotient ϕ : G ։ G/H =: G ′ . It is well-known that G ′ is a semiabelian variety ([8, Corollary 5.4.6]). Denote its maximal torus by G t ′ m and its maximal abelian quotient by π ′ : G ′ → A ′ . The map ϕ is a homomorphism ([36, Theorem 2]), and the image X ′ = ϕ(X) is an irreducible subvariety of Evidently, X ′ is not an algebraic subgroup of G ′ unless X is an algebraic subgroup of G.
We can then reduce (BC) for X to (BC) for X ′ . Let G = G × G t m (P 1 ) t and G ′ = G ′ × G t ′ m (P 1 ) t ′ be the standard compactifications from Section 2.2 and let π : G → A and π ′ : G ′ → A ′ be the associated projections. We also use the nef line bundles M G and M G ′ as defined there, and we fix an ample symmetric line bundle N (resp. N ′ ) on A (resp. A ′ ). Set L = M G ⊗ N (resp. L ′ = M G ′ ⊗ N ′ ) and endow all line bundles with the adelic metrics from Section 2.3. By Lemma 11 applied to ϕ, we have for every closed point x ∈ G. If there exists some ε > 0 such that for some sufficiently small ε ′ > 0. We may hence assume that Stab G (X) = {e G } in the following. We argue by contradiction and assume that (BC) is wrong for X. This means that there exists a Zariski-dense sequence (x i ) ∈ X N of small points. By Lemma 22, we can fix some integer m such that α m is quasi-finite and of generic degree 1. Pick a bijection and define the new sequence y i = (x φ 1 (i) , . . . , x φm(i) ), i ∈ N, which is clearly Zariski-dense in X m . Using [71,Lemma 4.1], we can even assume that (y i ) is generic by passing to a subsequence. From their construction, both (y i ) ∈ X m (Q) and (α m (y i )) ∈ G m−1 (Q) are sequences of small points in G m and G m−1 , respectively. Let U ⊆ X m be a dense open subset such that α m | U : U → α m (U) is an isomorphism. For sufficiently large i, we have y i ∈ U and α m (y i ) ∈ α m (U).
For the sequel, fix an arbitrary archimedean place ν ∈ Σ ∞ (K). Proposition 13 yields Borel measures µ 1 and µ 2 on (X m ) an Cν and α m (X m ) an Cν , respectively, such that the following two assertions are true: (a) For every f ∈ C 0 c ((X m ) an Cν ), we have 1 #O ν (y i )  for every g ∈ C 0 c (α m (X m ) an Cν ). By Lemma 20, there exist finitely many real-analytic submanifolds {N k } 1≤k≤K of α m (X m ), each endowed with a positive C ∞ -volume form ω k , such that (40) αm(X m ) an Cν gµ 2 = K k=1 N k gω k .
Let f ∈ C 0 c (V m ) with f (x, . . . , x) = 0. Considering the Taylor expansion of dα m at (x, . . . , x), we infer that since ω ′ k is positive at x by construction. For ε → 0, we obtain a contradiction.
Proof of Theorem 1. Let (x i ) ∈ G N be a strict sequence of small height. If (x i ) were not generic, there would exist a proper algebraic subvariety X and a Zariski-dense subsequence (x n i ) ∈ X N of small points. Proposition 21 implies that X is contained in a finite union of proper algebraic subgroups of G. This contradicts the strictness of (x i ) and hence (x i ) must be generic. This allows us to apply Proposition 13, concluding the proof of (SEC).

Appendix A. Global regularization and archimedean local heights
In this appendix, we indicate how to extend the archimedean local heights defined by Gubler [26] to semipositive C 0 -metrics through a global regularization.
If |Z| ⊆ |div(s d ′ +1 )|, we have to replace s d ′ +1 with a non-zero rational section s ′ d ′ +1 : X L d ′ +1 such that |Z| div(s ′ d ′ +1 ); again, the above argument then shows that both the inner and outer limit exist. In summary, the quantity in (43) is well-defined and we can take it as a definition of λ D 1 ,..., D d ′ (Z). By construction, this extends indeed Gubler's local heights.
Finally, our definition of λ D 1 ,..., D d ′ (Z) by a limit process also allows us to observe that [26, Propositions 3.4, 3.5, 3.6, 3.7, 3.8 and Theorem 10.6] remain true in our slightly generalized setting. In particular, we realize retrospectively that the induction formula of [26,Proposition 3.5] can be also used as a straightforward definition of the local heights λ D 1 ,..., D d ′ (Z) for general semipositive ν-metrized line bundles on X.