The distribution of Galois orbits of points of small height in toric varieties

We address the distribution properties of points of small height on proper toric varieties and applications to the related Bogomolov property. We introduce the notion of monocritical toric metrized divisor and we prove that equidistribution occurs for every generic, small sequence with respect to a toric metrized divisor, for every place if and only if the divisor is monocritical. Furthermore, when this is the case, the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of the ambient toric variety. We also study the $v$-adic modulus distribution of Galois orbits of small points. We characterize, in terms of the given toric semipositive metrized divisor, the cluster measures of $v$-adic valuations of Galois orbits of generic small sequences. The Bogomolov property now says that a subvariety of the principal orbit of a proper toric variety that has the same essential minimum than the toric variety with respect to a monocritical toric metrized divisor, must be a translate of a subtorus. We also give several examples, including a non-monocritical divisor for which the Bogomolov property does not hold.


Introduction
The study of the limit distribution of Galois orbits of points of small height was initiated by Szpiro, Ullmo and Zhang in their seminal paper [SUZ97]. They proved the equidistribution of the Galois orbits of sequences of points in an abelian variety over a number field, with Néron-Tate height converging to zero, over the Archimedean places. It was motivated by the Bogomolov conjecture on abelian varieties, and eventually applied in the affirmative answer to this conjecture by Ullmo [Ull98] and Zhang [Zha98], see also [Cin11,Ghi09,Gub07,Yam13] for similar results in the function field case.
This equidistribution result was widely generalized, in particular to other varieties and other height functions and, with the introduction of Berkovich spaces, to the equidistribution over non-Archimedean places [Bil97,Cha00,FR06,Cha06,BR06,Yua08,BB10,Che11]. We next introduce the necessary background to explain this generalization.
Let K be a global field, that is, a field which is either a number field or the function field of a regular projective curve over an arbitrary field, and M K its set of places. Let X be a proper algebraic variety over K of dimension n, and D = (D, ( · v ) v∈M K ) a semipositive metrized (Cartier) divisor with D big. Let h D : X(K) −→ R be the associated height function on the set of algebraic points of X, see §2 for details. It is a generalization of the notion of height of algebraic points considered by Weil, Northcott and others.
The essential minimum of X with respect to D, denoted by µ ess D (X), is the smallest possible limit value of the height of a generic net of algebraic points of X. Consequently, we say that a net (p l ) l∈I is D-small if lim l h D (p l ) = µ ess D (X).
A fundamental inequality by Zhang [Zha95] shows that the essential minimum can be bounded below in terms of the height and the degree of D: . (1.1) We say that D is quasi-canonical if this lower bound for the essential minimum is an equality (Definition 2.7). Examples of quasi-canonical metrized divisors are given by the canonical metrics on divisors of toric and abelian varieties and, more generally, by the metrics coming from algebraic dynamical systems. For a place v ∈ M K , we denote by X an v the v-adic analytification of X. If v is Archimedean, it is a complex space whereas, if v is non-Archimedean, it is a Berkovich space over C v , the completion of the algebraic closure of the local field K v . We endow the space of probability measures on X an v with the weak- * topology with respect to the space of continuous functions on X an v . For an algebraic point p of X, we denote by Gal(p) v its v-adic Galois orbit, that is, the orbit of p in X an v under the action of the absolute Galois group of K. We set for the uniform probability measure on Gal(p) v . We also denote by c 1 (D, · v ) ∧n the v-adic Monge-Ampère measure of D, see for instance [BPS14a,§1.4]. It is a measure on X an v of total mass deg D (X). The following statement is representative of several equidistribution theorems for Galois orbits of small points in the literature. In this form, it is due to Yuan [Yua08, Theorem 3.1] for number fields and to Gubler [Gub08, Theorem 1.1] for function fields.
Theorem 1.1. Let X a projective variety over K of dimension n, and D a quasicanonical semipositive metrized divisor on X with D ample. Let (p l ) l∈I be a generic D-small net of algebraic points of X. Then, for every v ∈ M K , the net of probability measures (µ p l ,v ) l∈I converges to A common feature of this result and its variants and generalizations, is the assumption that the lower bound (1.1) is an equality or, in other words, that the metrized divisor D is quasi-canonical. This severely restricts their range of application. Nonetheless, these results do apply to the important case of metrics arising from algebraic dynamical systems and moreover, they have a very strong thesis: not only the Galois orbits of points of small height do converge, but the limit measure is given by the normalized v-adic Monge-Ampère measure.
The motivation of this paper is to start the study what happens when we remove the hypothesis that D is quasi-canonical. Some of our typical questions are: is there always an equidistribution phenomenon for Galois orbits of D-small points? If not, can we give conditions on D, beyond being quasi-canonical, under which such a phenomenon occurs? When equidistribution occurs, can we describe the limit measure?
We address these questions and some of its continuations in the toric setting. Our approach is based on the techniques developed in the series [BPS14a,BMPS12,BPS14b]. These techniques are well-suited for the study of toric metrics and their associated height functions. In the sequel, we recall the basic constructions.
Let X be a proper toric variety over K of dimension n, given by a complete fan Σ on a vector space N R R n , and a big toric divisor D on X, given by a virtual support function Ψ D : N R → R. This toric divisor also defines an n-dimensional polytope ∆ D in the dual space M R := N ∨ R . Let D = (D, ( · v ) v∈M K ) be a semipositive toric metrized divisor on X. To it we associate an adelic family of concave functions ψ D,v : N R → R, v ∈ M K , called the metric functions of D. They satisfy that |ψ D,v − Ψ D | is bounded on N R for all v, and that ψ D,v = Ψ D for all v except for a finite number. We also associate to D an adelic family of continuous concave functions on the polytope ϑ D,v : ∆ D → R, v ∈ M K , called the local roof functions of D. They verify that ϑ D,v is the zero function for all v except for a finite number. The global roof function is a concave function ϑ D : ∆ D → R defined as a weighted sum of the local roof functions.
The metric functions and the roof functions convey lot of information of the pair (X, D). For instance, the essential minimum of X with respect to D can be computed as the maximum of the global roof function [BPS14b, Theorem 1.1]: (1.2) In the toric setting, the condition that the metrized divisor D is quasi-canonical is very restrictive, since it is equivalent to the condition that its global roof function is constant (Proposition 5.3). Thus, the only toric metrics to which the equidistribution theorem 1.1 applies are those whose global roof function is constant.
To identify the toric metrics having good equidistribution properties, we introduce the notion of monocritical toric metrized divisor. The semipositive toric metrized divisor D is monocritical if a certain function on a space of measures attains its infimum at a unique measure (Definition 4.14). When this is the case, the minimizing measure determines a point The condition for D of being monocritical can be characterized in terms of its global roof function: given a point x max ∈ ∆ D maximizing ϑ D , the sup-differential ∂ϑ D (x max ) is a convex subset of N R containing the point 0. Then D is monocritical if and only if 0 is a vertex of this convex subset and, when this is the case, the critical point of D can be computed from the sup-differential of the local roof functions at x max (Proposition 4.16).
Let T G n m,K be the torus of X, which can be identified with X 0 , the principal open subset of X. For each v ∈ M K , we denote by S v the compact subtorus of T an v . To a monocritical toric metrized divisor D with critical point u ∈ v∈M K N R , we associate a probability measure λ Sv,uv on X an v (Definition 5.1). When v is Archimedean, it is the uniform measure on a translate of S v (S 1 ) n whereas, when v is non-Archimedean, it is the Dirac measure at a translate of the Gauss point of T an v . The following is the main result of this paper (Theorem 5.2).
Theorem 1.2. Let X be a proper toric variety over K and D a semipositive toric metrized divisor on X with D big. Then D is monocritical if and only if for every place v ∈ M K and every generic D-small net (p l ) l∈I of algebraic points of X 0 , the net of probability measures (µ p l ,v ) l∈I on X an v converges. When this is the case, the limit measure agrees with λ Sv,uv , where u v ∈ N R is the v-adic component of the critical point of D.
Quasi-canonical toric metrized divisors are monocritical, and Theorem 1.2 reduces to Theorem 1.1 in this case. Our result produces a wealth of new examples of metric satisfying the equidistribution property, that were not covered by the previous results. In particular, this is the case for toric divisors over a number field K with positive smooth metrics at the Archimedean places and canonical metrics at the non-Archimedean ones (Theorem 6.4). Here we state a simplified version for the case when K = Q. Corollary 1.3. Let X be a proper toric variety over Q and D a semipositive toric metrized R-divisor with D big. We assume that the v-adic metric of D is, when v is the Archimedean place, smooth and positive and, when v is non-Archimedean, equal to the v-adic canonical metric of D. Then, for every generic D-small sequence (p l ) l≥1 of algebraic points of X 0 and every place v ∈ M Q , the sequence (µ p l ,v ) l≥1 on X an v converges to the probability measure λ Sv,0 .
This corollary covers many typical examples of metrics on toric varieties like weighted projective spaces and toric bundles, see § 6.2. For instance, let X = P 1 Q and D the divisor of the point at infinity equipped with the Fubini-Study metric at the Archimedean place and the canonical metric at the non-Archimedean places. Its essential minimum is µ ess D (X) = log(2) 2 and, for every generic sequence of algebraic points of P 1 Q with height converging to this quantity, its ∞-adic Galois orbits converge to the Haar probability measure on S 1 , the unit circle of the Riemann sphere (Example 6.5). This an example where equidistribution does occur, but the limit measure is not given by the v-adic Monge-Ampère measure as in Theorem 1.1.
In the other extreme, classical examples of translates of subtori with the canonical metric can behave badly with respect to equidistribution. For instance let X be the line of P 2 Q of equation 2z 1 − z 2 = 0 and D the metrized divisor on X given by the restriction of the canonical metrized divisor at infinity of P 2 Q . As explained in Example 6.1, Theorem 1.2 implies that D does not satisfy the equidistribution property in the sense of Definition 2.9.
We also study the modulus distribution of the v-adic Galois orbits of D-small nets of algebraic points in the general, non necessarily monocritical, toric case.
Hence, for an algebraic point p of X 0 = T, the direct image measure ν p,v := (val v ) * µ p,v is a probability measure on N R that gives the modulus distribution of its v-adic Galois orbit.
To each semipositive toric metrized divisor D with D big, we associate an adelic family of nonempty subsets of N R . We endow the space of probability measures on N R with the weak- * topology with respect to the bounded continuous functions on N R . For a probability measure ν on N R , we denote by supp(ν) ⊂ N R its support and, if ν has finite first moment, we denote by E[ν] its expected value. The next result characterizes the limit behavior of the modulus distribution for D-small nets (Theorem 4.3 and Corollary 4.12).
Theorem 1.4. Let X be a proper toric variety over K, D a semipositive toric metrized divisor on X with D big, and v ∈ M K . For every D-small net (p l ) l∈I of algebraic points in X 0 , the net of probability measures (ν p l ,v ) l∈I has at least one cluster point. Every such cluster point is a measure ν v with finite first moment that satisfies Conversely, for every probability measure ν v on N R which has finite first moment and satisfies (1.4), there is a D-small net (p l ) l∈I of algebraic points of X 0 such that ν v is the limit of the net (ν p l ,v ) l∈I .
In the situation of Theorem 1.4, when F v consist of only one point u v , the net (ν p l ,v ) l∈I converges to the measure δ uv . In this case we say that D satisfies the modulus concentration property at the place v.
In fact, D is monocritical if and only if for every place v, the set F v consists of only one point. Hence, our results imply that D satisfies the equidistribution property at every place if and only if it satisfies the modulus concentration property at every place. Observe also that, in contrast with Theorem 1.2, the net of algebraic points in Theorem 1.4 does not need to be generic.
In the absence of modulus concentration, there is a wealth of limit measures of v-adic Galois orbits of D-small nets of algebraic points. For instance, consider the projective line over a number field K and any adelic set E = (E v ) v∈M K of global capacity 1, whose associated equilibrium measures are compatible with the collection of sets in (1.3) (see Theorem 7.2 for the precise condition). Using Rumely's Fekete-Szegő theorem [Rum02], we show that, for all v, the equilibrium measure of E v can be realized as the limit measure of a sequence of v-adic Galois orbits of D-small points (Theorem 7.2).
As we already mentioned, the original motivation in [SUZ97] to search for equidistribution results of Galois orbits of small points was to prove the Bogomolov conjecture. The Bogomolov conjecture for toric varieties can be stated as follows: let X be a toric variety over K and D can an ample toric divisor on X equipped with the canonical metric. Let V ⊂ X 0,K be a subvariety which is not a translate of a subtorus by a torsion point. Then there exists ε > 0 such that the subset of algebraic points of V of canonical height bounded above by ε, is not dense in V .
Equivalently, if V ⊂ X 0,K is a subvariety with µ ess D can (V ) = 0, then V is a translate of a subtorus by a torsion point. It is the toric counterpart of the Bogomolov conjecture for abelian varieties proved by Ullmo and Zhang. This conjecture was proved by Zhang [Zha95] for number fields, and later Bilu gave a proof using his equidistribution theorem [Bil97]. Here we extend this result to an arbitrary monocritical metric on a toric variety over a number field (Theorem 5.12).
Theorem 1.5. Let X be a proper toric variety over a number field K and D a monocritical toric metrized divisor on X with critical point is called a D-special subvariety. We say that a given toric metrized divisor D satisfies the Bogomolov property 1 if every D-special subvariety is a translate of a subtorus (Definition 5.11). This property is intimately related with the equidistribution property. Indeed, we give an example of a metrized divisor D on P 2 Q , such that the line of equation z 0 + z 1 + z 2 = 0 is D-special (Example 6.6). This line is certainly not a translate of a subtorus, and so D does not satisfy the Bogomolov property. This metrized divisor is a variant of the one in Example 6.1, and does not verify modulus concentration nor equidistribution for any place of Q.
We comment briefly on the techniques of proof. To prove Theorem 1.4, for each place v we construct an upper-semicontinuous concave functional Φ v on the space of probability measures with finite first moment on N R . Using the toric dictionary from [BPS14a, BMPS12, BPS14b], we reduce the study of the modulus distribution of v-adic Galois orbits to the optimization of this functional, which we treat by applying Prokorov's compactness theorem.
Once we know that a monocritical divisor D satisfies modulus concentration at every place, we can construct another toric metric on D that is quasi-canonical and such that the D-small points are also small with respect to this new metric. With this trick, the toric equidistribution theorem 1.2 is derived from Theorem 1.1.
The Bogomolov property for monocritical metrics (Theorem 1.5) follows from Theorem 5.7, a variant of the toric equidistribution theorem for D-small nets that are not necessarily generic but only strict, in the sense that they eventually avoid any fixed translate of subtorus.
These results arise several interesting questions. For instance: is it possible that a given semipositive toric metrized divisor D satisfies the equidistribution property at one place and not at another? We study this for the projective line showing that, under a natural rationality hypothesis, the equidistribution property holds at a given place if and only if it holds at every place (Proposition 7.5). However, this conclusion is not true without this rationality hypothesis (Remark 7.7) and we have neither settled this question for the projective line in full generality, nor treated toric varieties of higher dimension.
It would also be interesting to see if the converse of Theorem 1.5 holds: Let X be proper toric variety with dim X ≥ 2. Given a semipositive toric metrized divisor D on X, with D big satisfying the Bogomolov property, is D necessarily monocritical? In Proposition 6.7 we show that this is true in a very particular case. Extending this to the general case would reinforce the link between the equidistribution and the Bogomolov properties.
The results of this paper also inspire questions for general varieties and metrized divisors. For instance, from Corollary 1.3, it is plausible to conjecture that a toric divisor equipped with a positive smooth, but not necessarily toric, Archimedean metric and canonical non-Archimedean metrics, does satisfy the equidistribution property. A puzzling question is that of computing the essential minimum, with a formula generalizing (1.2) to the general, non-toric, case. Even more challenging seems the problem of generalizing the crucial notion of monocritical metrized divisor.
Several of the results presented in this introduction hold in greater generality and their thesis are stronger. We refer to the body of the paper for these versions. The structure of the paper is as follows. In §2 we give the preliminaries on Galois orbits and height of points. In §3 we introduce the upper semi-continuous concave functional Φ v and study its properties. In §4 we study the modulus distribution of v-adic Galois orbits of D-small nets of points in toric varieties. In §5 we prove the toric equidistribution theorem 1.2 and its variants, together with the Bogomolov property for monocritical toric metrized divisors. In §6 we give examples illustrating a number of phenomena, including a non-monocritical toric metrized divisor not verifying the Bogomolov property. Finally, in §7 we use potential theory to study the limit measures that appear in the absence of modulus concentration.
By a global field K we mean either a number field or the function field of a regular projective curve over an arbitrary field, equipped with a certain set of places, denoted by M K . Each place v ∈ M K is a pair consisting of an absolute value | · | v on K and a positive weight n v ∈ Q >0 , defined as follows.
The places of the field of rational numbers Q consist of the Archimedean and the p-adic absolutes values, normalized in the standard way, and with all weights equal to 1. For the function field K(C) of a regular projective curve C over a field k, the set of places is indexed by the closed points of C. For each closed point v 0 ∈ C, we consider the absolute value and weight given, for α ∈ K(C) × , by with c k = #k if the base field k is finite and c k = e otherwise, and where ord v0 (α) denotes the order of α in the discrete valuation ring O C,v0 .
Let K 0 denote either Q or K(C). In the general case when K is a finite extension of K 0 , the set of places of K is formed by the pairs v = (| · | v , n v ) where | · | v is an absolute value on K extending an absolute value | · | v0 with v 0 ∈ M K0 and where K v denotes the completion of K with respect to | · | v , and similarly for K 0,v0 . This set of places satisfies the following basic properties.
Proposition 2.1. Let K 0 denote either Q or K(C), the function field of a regular projective curve C over a field k. Let K be a finite extension of K 0 and M K the associated set of places as above. Then (1) for all v 0 ∈ M K0 , we have v|v0 n v = n v0 ; (2) for all α ∈ K × , we have v∈M K n v log |α| v = 0 (product formula).
In the function field case there is a subtlety, due to the fact that a given field may have different structures of global field depending on the choice of base curve.
Let C be a regular projective curve over k and K(C) → K a finite extension of fields. Then there is a regular projective curve B over k and a finite morphism π : B → C such that K K(B) and the previous extension can be identified with π * : K(C) → K(B), see for instance [Liu02,Proposition 7.3.13 and Lemma 7.3.10].
We could give to K the structure of global field defined directly by the curve B, but the obtained absolute values of K would not be extensions of those of K 0 . To remedy this, we renormalize these absolute values of K and, to preserve the product formula, we also change the weights.
From the valuative criterium of properness, for each closed point v 0 ∈ C, the absolute values of K extending |·| v0 are in bijection with the closed points of the fiber above v 0 . Moreover, since the map π is finite, for each closed point v ∈ π −1 (v 0 ), the ring O B,v is a finite module over O C,v0 . It follows from [Bou85, Chapitre 6, Proposition 2 in §8.2 and Theorem 2 in §8.5] that the absolute value and weight corresponding to v are given, for α ∈ K(B) × , by with e v/v0 the ramification index of v over v 0 . The same results in loc. cit. give the formula in (1). For the product formula in (2), we obtain from (2. deg(div(α)) = 0, because the degree of a principal divisor on B is zero, which concludes the proof.
For v ∈ M K , we choose an algebraic closure K v ⊂ K v of K v . The absolute value | · | v on K v has a unique extension to K v . We denote by C v the completion of K v with respect to this extended absolute value. We also choose an algebraic closure K of K and an embedding  v : Let X be a variety over K, that is, a reduced and irreducible separated scheme of finite type over K. The elements of X(K) are called the algebraic points of X. For p ∈ X(K), its Galois orbit is Gal(p) := Gal(K/K) · p ⊂ X(K), that is, the orbit of p under the action of the absolute Galois group of K.
For v ∈ M K , we denote by X an Kv the v-adic analytifications of X over K v and by X an v the v-adic analytifications of X over C v . If v is Archimedean, they both coincide with a complex space (X Kv is equipped with an anti-linear involution if K v R). If v is non-Archimedean, they are Berkovich spaces over K v and C v , respectively. These spaces are related by ([Ber90, Corollary 1.3.6]) There is a map X(C v ) −→X an v . Using the chosen inclusion  v : K → C v , we obtain a map X(K) → X(C v ) and, by composition the previous map, an inclusion The v-adic Galois orbit of an algebraic point p ∈ X(K), denoted by Gal(p) v , is defined as the image of Gal(K/K) · p under ι v . It is a finite subset which does not depend on the choice of the inclusion  v . We also denote by µ p,v the uniform discrete probability measure on X an v supported on Gal(p) v , that is where δ q is the Dirac measure at the point q ∈ X an v . Hence, for a continuous function f : An R-divisor on X is a linear combination of Cartier divisors on X with real coefficients. A metrized R-divisor D on X is an R-divisor D on X equipped with a quasi-algebraic family of v-adic metrics ( · v ) v∈M K , see [BMPS12,§3] for details. In loc. cit., for each v ∈ M K the metric · v is defined over the analytic space X an Kv . Note that this space was denoted "X an v " in loc. cit. but since we will study equidistribution problems of Galois orbits of points that are defined over varying extensions of K of arbitrary large degree it is more convenient to work on the space X an v instead that in the space X an Kv . Hence we have changed the notation accordingly. With this point of view, every object on X an Kv will be seen as an object on X an v by taking its inverse image under the projection π v . For instance let D be a metrized R-divisor on X and s a rational R-section of D [BMPS12,§3]. In loc. cit., the v-adic metric · v is described by a continuous function s v : X an Kv \ | div(s)| → R >0 . In the current paper we denote by s v the function on X an v \ | div(s)| given by the composition Clearly this function is invariant under the action of Gal( To a metrized R-divisor D on X we can associate a height function Given p ∈ X(K), choose a rational R-section s of D such that p ∈ | div(s)|. Choose a finite extension F of K such that p ∈ X(F). For each w ∈ M F over a place v ∈ M K , we can choose an embedding σ w : F → C v such that the restriction of the absolute value | · | v of C v agrees with | · | w . We denote also by σ w the induced map X(F) → X an v .
Definition 2.2. Let X be a variety over K, D a metrized R-divisor on X, and p ∈ X(K). With the above notation, the height of p with respect to D is defined as The height is independent of the choice of the rational R-section s, the extension F and the embeddings σ w .
Instead of choosing a finite extension where the point p is defined, it is possible to express the height of an algebraic point in terms of its Galois orbit.
Proposition 2.3. With the previous hypothesis and notation, the height of p with respect to D is given by Proof. Choose a finite normal extension F ⊂ K of K such that p ∈ X(F). For each v ∈ M K we denote M F,v the set of places of F above v. Write G = Gal(F, K) and let F G be the fixed field. Then F/F G is a Galois extension with Galois group G and F G /K is purely inseparable. Hence, for v ∈ M K , Then, from the definition of the height of p in Definition 2.2 and Proposition 2.1(1), it follows that The group G acts on M F,v and, since p is defined over F, also on Gal(p) v . Both actions are transitive. Therefore, choosing w 0 ∈ M F,v , The statement follows from this together with (2.5).
The essential minimum of X with respect to D is defined as (2.6) Roughly speaking, the essential minimum is the generic infimum of the height function.
Definition 2.4. Let X be a variety over K and D a metrized R-divisor on X. A net (p l ) l∈I of algebraic points of X is D-small if The net (p l ) l∈I is generic if, for every closed subset Y X, there is l 0 ∈ I such that p l ∈ Y (K) for l ≥ l 0 .
Proposition 2.5. Given a variety X over K and D a metrized R-divisor on X, there exists a generic D-small net of algebraic points of X. Moreover, every generic net (p l ) l≥1 of algebraic points of X satisfies Proof. The second statement is clear from the definition of the essential minimum. For the first statement, let I be the set of hypersurfaces of X, ordered by inclusion. This is a directed set. For each Y ∈ I, denote by c(Y ) its number of irreducible components and choose a point Clearly, the net (p Y ) Y ∈I is generic and D-small.
Remark 2.6. When K is a number field, the collection of subvarieties of X is countable. Using this fact, we can strengthen Proposition 2.5 to show the existence of a generic D-small sequence of algebraic points.
Suppose now that the variety X is proper over K and of dimension n. A metrized R-divisor D on X is semipositive if it can be written as with D i a semipositive metrized divisor and α i ∈ R ≥0 , i = 1, . . . , r. Recall that D i is semipositive if each of its v-adic metrics is a uniform limit of a sequence of semipositive smooth (respectively, algebraic) metrics in the Archimedean (respectively, non-Archimedean) case.
Given a semipositive metrized R-divisor D, we can extend the notion of height of points to subvarieties of higher dimension. In particular, the height of X, denoted by h D (X), is defined. Moreover, for each v ∈ M K we can consider the associated v-adic Monge-Ampère measure, denoted by c 1 (D, · v ) ∧n . It is a measure on X an v of total mass deg D (X), see for instance [BPS14a,§1.4] for the case when D is a divisor. The v-adic Monge-Ampère measure of an R-divisor is defined from that of divisors by polarization and multilinearity.
A theorem of Zhang shows that, when K is a number field, D is an ample divisor and D is semipositive, the essential minimum can be bounded below in terms of the height of X and the degree of D [Zha95, Theorem 5.2]: . (2.7) This inequality can be generalized to global fields and semiample big divisors, see for instance [Gub08, Proposition 5.10].
In some cases, the inequality (2.7) is an equality. For instance, this happens for the canonical metric on divisors of toric and abelian varieties, and for the canonical metrics coming from dynamical systems. This motivates the following definition.
Definition 2.7. Let X be a proper variety over K of dimension n, and D a semipositive metrized R-divisor on X with D big. Then D is quasi-canonical if .
In other words, quasi-canonical metrized R-divisors are those for which Zhang's lower bound for the essential minimum is attained.
As we will see in §5, the condition for a toric metric of being quasi-canonical is very restrictive. The following observation is a direct consequence of Proposition 2.5 and of the inequality (2.7).
Proposition 2.8. Let X be a proper variety over K of dimension n and D a semipositive metrized divisor on X with D big and semiample. Then there exists a generic net (p l ) l∈I of algebraic points of X with if and only if D is quasi-canonical.
We discuss now the equidistribution of Galois orbits of points of small height. Let X be a proper variety over K and v ∈ M K . We endow the space of probability measures on X an v with the weak- * topology with respect to the space of continuous functions on X an v . In particular, a net of probability measures (µ l ) l∈I converges to a probability measure µ if, for every continuous function f : Definition 2.9. Let D be a metrized R-divisor on X. A probability measure µ on X an v is a v-adic limit measure for D if there exists a generic D-small net (p l ) l∈I of algebraic points of X such that the net of probability measures (µ p l ,v ) l∈I converges to µ. We say that D satisfies the v-adic equidistribution property if, for every generic D-small net (p l ) l∈I as above, the net of measures (ν p l , ) l∈I converges.
Clearly, when the v-adic equidistribution property holds, there exists a unique limit measure.
Remark 2.10. When K is a number field, the analytic space X an v is metrizable and so is the space of probability measures on it. Using this together with Remark 2.6 and standard arguments, one can reduce to sequences, instead of nets, when studying equidistribution properties over number fields.
In the literature there are many equidistribution theorems of Galois orbits of points of small height. All these equidistribution results deal with generic nets (or sequences when K is a number field) of algebraic points satisfying the equality (2.8). In view of Proposition 2.8, the existence of such a net implies that the metric is quasi-canonical. Moreover, the condition (2.8) for this net is equivalent of being D-small. Thus we can reformulate a general equidistribution result in the following form.
Theorem 2.11. Let K be a global field and X a projective variety over K of dimension n. Let D be a semipositive metrized divisor on X such that D is ample. If D is quasi-canonical then, for every place v ∈ M K , (1) D satisfies the v-adic equidistribution property; (2) the limit measure is the normalized Monge-Ampère measure This result is due to Yuan [Yua08, Theorem 3.1] in the number field case and, with more general hypotheses, to Gubler [Gub08, Theorem 1.1] in the function field case.
Written in this form, it is clear that this equidistribution theorem has a very restrictive hypothesis, that the metrized divisor D is quasi-canonical. But it also has a very strong thesis: not only the Galois orbits of points of small height converge to a measure, but this limit measure can be identified with the normalized Monge-Ampère measure of the metrized divisor.
The main objective of this paper is to start the study of what happens when the hypothesis of D being quasi-canonical is removed. We will work with toric varieties and toric metrics because, in this case, the tools developed previously allow us to work very explicitly. In this setting, we will see that the first statement in Theorem 2.11 holds in much great generality, but, if the metric is not quasicanonical, the limit measure does not need to agree with the normalized Monge-Ampère measure.

Auxiliary results on convex analysis
In this section we gather several definitions and results on convex analysis that we will use in our study of toric height functions. For a background in convex analysis, see for instance [BPS14a,§2].
Let N R R n be a real vector space of dimension n and M R = Hom(N R , R) = N ∨ R its dual. The pairing between x ∈ M R and u ∈ N R will be denoted either by x, u or u, x . Following [BPS14a, §2], a convex subset C is nonempty. The relative interior of C, denoted by ri(C), is the interior C relative to the minimal affine subspace containing it.
Let C ⊂ M R be a convex subset and g : C → R a concave function. The supdifferential of g at a point x ∈ C is It is a convex subset of N R . The stability set of g is the subset of N R defined by (3.1) Let E ⊂ N R be a convex subset. A nonempty subset F ⊂ E is a face of E if every closed segment S ⊂ E whose relative interior has nonempty intersection with F , is contained in F .
Lemma 3.1. Let C ⊂ M R be a compact convex subset and g 1 , g 2 : C → R two continuous concave functions. Denote by C max the convex subset of C of the points where g 1 + g 2 attains its maximum value and choose x ∈ C max . For i = 1, 2, consider the concave function φ i : ) is nonempty and does not depend on the choice of x ∈ C max ; (3) the minimal face of ∂g 1 (x) containing ∂g 1 (x) ∩ (−∂g 2 (x)) does not depend on the choice of x ∈ C max ; (4) the function φ i is nonpositive and vanishes precisely on ∂g i (x).
Proof. The restriction to C max of the sum g 1 +g 2 is constant, and so the restrictions to this set of g 1 and g 2 are affine and with opposite slopes. In other words, there is u 0 ∈ N R such that, for all x 1 , x 2 ∈ C max , For the statement (1), let i = 1, 2 and u ∈ ∂g i (x ). Since x is in the relative interior of C max , there exists ε > 0 such that x −ε(x−x ) ∈ C max . By the definition of the sup-differential, for all z ∈ C, (3.4) To prove the statement (2) note that, since g 1 + g 2 attains its maximum value at x, by [BPS14a, Proposition 2.3.6(2)] Hence ∂g 1 (x) ∩ (−∂g 2 (x)) = ∅, as stated. Now let u be a point in this intersection. Then Using this together with (3.4)and (3.5), we obtain Hence u ∈ ∂g 1 (x ) ∩ (−∂g 2 (x )), as stated.
For the next statement, consider the convex set B = ∂g 1 (x) ∩ (−∂g 2 (x)) which, thanks to (2), does not depend on the choice of x ∈ C max . Denote by F x the minimal face of ∂g 1 (x) containing it. By (1), it is enough to consider the case when x ∈ ri(C max ). By the same statement, the set ∂g 2 (x) does not depend on the choice of x ∈ ri(C max ), proving (3).
Definition 3.2. Let C ⊂ M R be a compact convex subset and g 1 , g 2 : C → R two continuous concave functions. Let C max be the convex subset of C of the points where g 1 + g 2 attains its maximum value. Given x ∈ C max , we define the convex subset of N R B(g 1 , g 2 ) = ∂g 1 (x) ∩ (−∂g 2 (x)) and the convex subset F (g 1 , g 2 ) ⊂ ∂g 1 (x) as the minimal face of ∂g 1 (x) that contains B(g 1 , g 2 ). By Lemma 3.1, these convex subsets do not depend on the choice of x.
We also define the convex subset A i (g 1 , g 2 ) = ∂g i (x) ⊂ N R , i = 1, 2. By the same lemma, this convex subset does not depend on the choice of the point x.
Remark 3.3. In the setting of Lemma 3.1, it is not always true that ∂g i (x) = ∂g i (x ) for x, x ∈ C max . An example of this situation is when g i is the zero function on the interval [0, 1] ⊂ R and x = 0, x = 1/2. Hence, in Definition 3.2 we cannot define the sets A i (g 1 , g 2 ) using an arbitrary point x ∈ C max .
Lemma 3.4. Let C ⊂ M R be a compact convex subset with nonempty interior and g 1 , g 2 : C → R two concave functions. Then B(g 1 , g 2 ) is bounded and F (g 1 , g 2 ) contains no lines.
Proof. The convex set B(g 1 , g 2 ) is not bounded if and only if it contains a ray, that is, a subset of the form R ≥0 u 1 + u 2 with u i ∈ N R , i = 1, 2, and u 1 = 0. Suppose that this is the case. This implies that, for x ∈ C max and all t ≥ 0, Hence, for all z ∈ C and t ≥ 0, Letting t → ∞, this implies C ⊂ {z | u 1 , z − x = 0}, contradicting the hypothesis that C has nonempty interior. Hence B(g 1 , g 2 ) is bounded. Similarly, if F (g 1 , g 2 ) contains a line Ru 1 + u 2 , then, for x ∈ C max and t ∈ R, This also implies that C is contained in the affine hyperplane {z | u 1 , z − x = 0} and contradicts the hypothesis that C has nonempty interior. Hence F (g 1 , g 2 ) contains no lines.
Let C b (N R ) be the space of bounded continuous functions on N R , and let · be an auxiliary norm on N R that we fix.
Definition 3.5. We denote by P the space of Borel probability measures on N R endowed with the weak- * topology with respect to C b (N R ). This is the smallest topology on P such that, for all ϕ ∈ C b (N R ), the function µ → ϕ dµ is continuous.
We denote by E ⊂ P the topological subspace of probability measures with finite first moment, that is, the probability measures on N R satisfying For µ ∈ E, the expected value is The weak- * topology of P with respect to C b (N R ) is called the "topologieétroite" in [Bou69,§5]. By Proposition 5.4.10 in loc. cit., the topological space P is complete, metrizable and separable. Latter we will consider other topologies on the underlying spaces of P and E. When this is the case, we will state explicitly the used topology.
For µ ∈ P, its support, denoted by supp(µ), is the set of all points in N R such that all its neighborhoods have positive measure. The set of probability measures on N R with finite support is contained in E, and is dense therein.
For the rest of this section, we fix a compact convex subset C ⊂ M R with nonempty interior and two continuous concave functions g 1 , g 2 : C → R. The Legendre-Fenchel dual g ∨ i is a concave function on N R with stability set C. We introduce the function Φ : E → R given, for µ ∈ E, by This function will play a central role in the proof of the main results in the next section.
It follows easily from its definition that Φ is concave. In general, this function is not continuous, as the following example shows.
Nevertheless, we have the following result.
Proposition 3.7. The function Φ is upper semicontinuous.
To prove this proposition, we need the following lemma.
Lemma 3.8. Let φ : N R → R be a continuous function. If φ is bounded above (respectively below), then the map is upper semicontinuous (respectively lower semicontinuous).
Proof. We prove the case of a function bounded above, the other case being analogous. Let µ ∈ P and ε > 0 be given and, for l ≥ 1, put The sequence of functions (φ l ) l≥1 is monotone and converges pointwise to φ. So Lebesgue's monotone convergence theorem implies that there is l 0 ≥ 1 such that Since ε is arbitrary, lim sup l→∞ φ dµ l ≤ φ dµ, proving the lemma.
Proof. Let notation be as in Lemma 3.1 and for short put Fix a point x ∈ ri(C max ) and let φ i be as in (3.2). For every µ ∈ E we can write Φ(µ) in terms of the functions φ i as (3.9) By Lemma 3.1(4), the functions φ i are nonpositive and vanish precisely on the sets A i . It follows from (3.9) that Φ is nonpositive and vanishes for every µ ∈ E satisfying (3.8).
We next prove that the first condition in If θ < 1, we put If θ = 0, then E[µ] = u 2 and so E[µ] ∈ A 1 \ F , contradicting the fact that E[µ] ∈ B ⊂ F . Suppose that 0 < θ < 1 and set Therefore E[µ] = θu 1 + (1 − θ)u 2 ∈ ri(u 1 u 2 ), the relative interior of the segment u 1 u 2 . Since E[µ] is in B and hence in F , we have ri(u 1 u 2 ) ∩ F = ∅. Moreover, the whole segment is contained in A 1 , and F is a face of A 1 . We deduce that this segment is contained in F . Therefore u 2 ∈ F , contradicting the fact that u 2 ∈ A 1 \F . We conclude that θ = 1 and so supp(µ) ⊂ F . This proves the first condition and completes the proof.
The function Φ satisfies also the following property.
Lemma 3.10. There are constants c 1 ≥ 0 and c 2 > 0 such that, for all µ ∈ E, Proof. Let Ψ be the support function of C, which is the function on N R given by Put c 1 = 4 max y∈C (|g 1 (y)|, |g 2 (y)|). It follows from their definition that the func- (3.10) Let x be a point in the interior of C. On M R , we consider the norm induced by the chosen norm · in N R . Since x is interior, we can find a constant c 2 > 0 such that B(x, c 2 ), the closed ball of center x and radius c 2 , is contained in C. Then By (3.10) and (3.11), as stated.
Proposition 3.11. Let (µ l ) l∈I be a net of measures in E such that Then (µ l ) l∈I has at least one cluster point in P, and every such cluster point µ lies in E and satisfies Proof. Replacing (µ l ) l∈I by a subnet if necessary, we assume that Φ(µ l ) ≥ −1 for all l ∈ I. Let c 1 , c 2 be the constants of Lemma 3.10 and set K = (c 1 + 1)/c 2 > 0. This lemma implies that each µ l is in the subset of E given by This subset is compact thanks to Prokhorov's theorem [Bou69, Théorème 5.5.1], and it is metrizable because P is. Hence, the net (µ l ) l∈I has at least one cluster point, and every such cluster point µ lies in E, proving the first statement.
To prove the last statement, let (µ k ) k∈I be a subnet converging to µ. By Proposition 3.7, the function Φ is upper-semicontinuous and so Hence Φ(µ) = 0, and the statement follows from Proposition 3.9.
As we have seen in Example 3.6, the function Φ is not continuous. We now consider another topology on E with respect to which the function Φ is continuous.
Given µ, µ ∈ P, denote by Γ(µ, µ ) the set of probability measures on N R × N R with marginals µ and µ . That is, a probability measure ν on where p i is the projection of N R ×N R onto its i-th factor, and (p i ) * the direct image of measures.
Definition 3.12. The Kantorovich-Rubinstein distance (or Wasserstein distance of order 1 ) on E is defined, for µ, µ ∈ E, by The Kantorovich-Rubinstein topology (or KR-topology for short) of E is the topology induced by this distance.
For a Lipschitz continuous function ψ : N R → R, denote by Lip(ψ) its Lipschitz constant, given by Lipschitz constants and the Kantorovich-Rubinstein distance are related by the duality formula: for µ, µ ∈ E and a Lipschitz continuous function ψ : Remark 3.13. By [Vil09, Theorem 6.9], the KR-topology agrees with the weak- * topology on E with respect to the space of continuous functions ϕ : for a c ∈ R and all u ∈ N R . In particular, the KR-topology is stronger than the topology of E induced by that of P as in Definition 3.5.
Proposition 3.14. The function Φ is continuous with respect to the KR-topology.
In particular, if (µ l ) l∈I is a net of measures in E that converges to a measure µ ∈ E with respect to the KR-topology and then lim l Φ(µ l ) = 0.
Proof. Let (µ l ) l∈I be a net on E that converges to a measure µ ∈ E with respect to the KR-topology. By Remark 3.13, Therefore lim l Φ(µ l ) = Φ(µ) and so Φ is continuous, proving the first statement. The second statement follows from the first one and Proposition 3.9.
We also need the following easy result, that we include here for the lack of a suitable reference.
. . , r, be convex subsets and E = E 1 + · · · + E r their Minkowski sum. For a point u 0 ∈ E, the following conditions are equivalent: (1) the point u 0 is a vertex of E; (2) the equation u 0 = i z i with z i ∈ E i has a unique solution and, for i = 1, . . . , r, the point z i in this solution is a vertex of E i .
Proof. First assume that u 0 is a vertex of E. Suppose that the equation belong to E, they are different and satisfy u 0 = 1 2 (u 1 + u 2 ), contradicting the fact that u 0 is a vertex of E. Hence the equation u 0 = i z i has a unique solution with are different, belong to E and u 0 = 1 2 (u 1 + u 2 ), contradicting the assumption that u 0 is a vertex of E. Thus we have proved that (1) implies (2).
Assume now that the statement (2) is true but u 0 is not a vertex of E. Then there are two different points u 1 , u 2 ∈ E with u 0 = 1 2 (u 1 + u 2 ). Since E is the Minkowski sum of the sets E i , we can write The equation u 0 = i z i has a unique solution and so z i = 1 2 (z i + z i ) for all i. Since z i is a vertex of E i , this implies z i = z i . Therefore u 1 = u 2 , contradicting the assumptions and thus proving that (2) implies (1).

Modulus distribution
In this section, we study the asymptotic modulus distribution of the Galois orbits of nets of points of small height in toric varieties. Our approach is based on the techniques developed in the series of papers [BPS14a,BMPS12,BPS14b]. These techniques are well-suited for the study of toric metrics and their associated height functions. In the sequel, we recall the basic constructions and results.
Let K be a global field and T G n m,K a split torus of dimension n over K. Let N = Hom(G m,K , T) and M = Hom(T, G m,K ) = N ∨ be the lattices of cocharacters and of characters of T, respectively, and write N R = N ⊗ R and M R = M ⊗ R. We also fix an auxiliary norm · on N R .
Let v ∈ M K . We denote by T an v the v-adic analytification of T and by S v its compact subtorus. In the Archimedean case, S v is isomorphic to (S 1 ) n whereas, in the non-Archimedean case, it is a compact analytic group, see [BPS14a, § 4.2] for a description. Moreover, there is a map val v : T an v → N R , defined, in a given splitting, by val (4.1) This map does not depend on the choice of the splitting, and the compact torus S v coincides with its fiber over the point 0 ∈ N R . Let X be a proper toric variety over K with torus T, described by a complete fan Σ on N R . To each cone σ ∈ Σ corresponds an affine toric variety X σ , which is an open subset of X, and an orbit O(σ) of the action of T on X. The affine toric variety corresponding to the cone σ = {0} is the principal open subset X 0 . It coincides with the orbit O(0) and is canonically isomorphic to the torus T.
An R-divisor D on X is toric if it is invariant under the action of T. Such an R-divisor defines a virtual support function on Σ, that is a function Ψ D : N R −→ R whose restriction to each cone of the fan Σ is linear. We also associate to D the subset of M R given by If D is pseudo-effective, then ∆ D is a polytope and, otherwise, it is the empty set. Properties of the R-divisor D can be read off from its associated virtual support function and polytope. In particular, D is nef if and only if Ψ D is concave, and D is big if and only if ∆ D has nonempty interior.
A quasi-algebraic metrized divisor D = (D, ( · v ) v∈M K ) on X is toric if and only if the v-adic metric · v is invariant with respect to the action of S v , for all v. Such a toric metrized R-divisor on X defines a family of continuous functions ψ D,v : N R → R indexed by the places of K. For each v ∈ M K , this function is given, where s D is the canonical rational R-section of D as in [BMPS12,§ 3]. This adelic family of functions satisfies that |ψ D,v − Ψ D | is bounded for all v, and that ψ D,v = Ψ D for all v except for a finite number. In particular, the stability set of each ψ D,v coincides with ∆ D . For each v ∈ M K , we also consider the v-adic roof function ϑ D,v : ∆ D → R, that is given by This is an adelic family of continuous concave functions on ∆ D which are zero except for a finite number of places. The global roof function ϑ D : The essential minimum of X with respect to D defined in (2.6) can be computed as the maximum of its roof function [BPS14b, Theorem 1.1], that is is equivalent to the condition that D is nef. For the rest of this section, we suppose that X is a proper toric variety over the global field K with torus T, and that D is a semipositive toric metrized R-divisor with D big. We also fix the notation below. Recall from §3 that P denotes the space of probability measures on N R endowed with the weak- * topology with respect to the space C b (N R ), and that E denotes the subspace of probability measures with finite first moment.
Notation 4.2. Let v ∈ M K . We denote by g i,v , i = 1, 2, the concave functions on ∆ D given by Thus ϑ D = n v (g 1,v + g 2,v ). We consider the convex subsets of N R given by Definition 3.2 We also denote by Φ v the function on E obtained by applying Definition (3.6) to the set C = ∆ D and the functions g i,v , i = 1, 2.
Given v ∈ M K and a point p ∈ X(K), we consider the discrete probability measure on N R defined by where µ p,v is the uniform discrete probability measure on X an v supported on the vadic Galois orbit of p as in (2.4). This probability measure on N R gives the modulus distribution of the v-adic Galois orbit of the point p. The next result characterizes the limit behavior of this modulus distribution for nets of points of small height.
Theorem 4.3. Let notation and hypothesis be as above. For each v ∈ M K and every D-small net (p l ) l∈I of algebraic points in the principal open subset X 0 , the net (ν p l ,v ) l∈I of measures in P has at least one cluster point. Every such cluster point ν v lies in E and satisfies The proof of Theorem 4.3 is given below, after a definition and an auxiliary result.
Definition 4.4. A centered adelic measure ν on N R is a collection of measures ν v ∈ E, v ∈ M K , such that ν v = δ 0 , the Dirac measure at the point 0 ∈ N R , for all but a finite number of places v, and such that (4.6) We denote by H K the set of all centered adelic measures on N R .
We introduce the function η D : This function extends the notion of height of points to the space H K . Indeed, for p ∈ X 0 (K), the collection is a centered adelic measure on N R , because of the product formula in Proposition 2.1(2). Moreover, the canonical R-section s D does not vanish at p and, by Proposition 2.3 and (4.2), Lemma 4.5. For every centered adelic measure ν = (ν v ) v∈M K , (4.10) In particular, for p ∈ X 0 (K), Proof. Let ∆ D,max be the set of points maximizing the roof function ϑ D and choose , where g i,v denotes the concave function on ∆ D in Notation 4.2 and g ∨ i,v its Legendre dual as in (3.1).
To prove the first inequality in (4.10), fix v ∈ M K . By [BPS14a, Propositions 2.3.1(1) and 2.3.3(3)], (4.14) where w runs over the places of K different from v, the symbol denotes the sup-convolution and, for a concave function ψ and a nonzero constant λ, the expression ψλ denotes the right multiplication as in [BPS14a,§2.3].
By the equality (4.14), the definitions of the sup-convolution and the right multiplication, and condition (4.6), we deduce By the concavity of φ 1,w , we have φ 1,w dν w ≤ φ 1,w (E(ν w )) for all w ∈ M K . Therefore, by (4.12) and (4.15), which proves the first inequality and completes the proof of (4.10). The inequalities in (4.11) follow directly from (4.10) and (4.9).
Proof of Theorem 4.3. Let v ∈ M K and Φ v : E → R the function defined by (3.6) with g 1,v and g 2,v as in Notation 4.2. Since the net of points (p l ) l∈I is D-small, From Lemma 4.5, we deduce that The theorem is then a direct consequence of Proposition 3.11.
To state a partial converse of Theorem 4.3, we need a further definition.
Definition 4.6. The adelic Kantorovich-Rubinstein distance W K on H K is defined, where W denotes the Kantorovich-Rubinstein distance in N R as in Definition 3.12. By the definition of H K , there is only a finite number of nonzero terms in this sum. The topology on H K induced by this distance is called the adelic KR-topology.
Theorem 4.7. With notation and hypothesis as before, let ν = (ν v ) v∈M K ∈ H K be a centered adelic measure such that Then there is a generic D-small net (p l ) l∈I of algebraic points of X 0 such that the net of measures (ν p l ) l∈I converges to ν with respect to the adelic Kantorovich-Rubinstein distance.
The proof of Theorem 4.7 is given below, after some preliminary results. The first result gives the main properties of the function η D . Proof. Let S ⊂ M K be a finite subset such that where the second inequality is given by the duality formula (3.12) and the last by the observation that Lip(ψ D,v ) = max x∈∆ D x for all v. This proves that η D is Lipschitz continuous with respect to W K . As already remarked, the functions Φ v are nonpositive. By Lemma 4.5, this implies the inequality (4.16). From the same result, it follows that the equality holds if and only if Φ v (ν v ) = 0 for all v. By Proposition 3.9, this holds if and only if supp(ν v ) ⊂ F v and E[ν v ] ∈ B v , completing the proof of the lemma.
From this lemma, we deduce as a direct consequence the next characterization of algebraic points in toric varieties realizing the essential minimum.
Let H K ⊂ v∈M K N R be the subspace defined by the equation v n v u v = 0. By sending the point (u v ) v ∈ H K to the adelic centered measure (δ uv ) v ∈ H K , we identify H K with a subspace of H K . Proof. We denote by ∆ D,max ⊂ ∆ D the set of points where ϑ D attains its maximum.
as stated.
We next show that the measures coming from algebraic points are dense in H K .
Proposition 4.11. For every ν ∈ H K there is a generic net (p l ) l∈I of algebraic points of X 0 such that the net of associated measures (ν p l ) l∈I as in (4.8), converges to ν with respect to the adelic KR-topology.
Proof. Put ν = (ν v ) v and let ε > 0 be given. Let S be a finite nonempty subset of M K such that ν v = δ 0 for all v / ∈ S. By [Vil09, Theorem 6.18], we can approach, with respect to the KR-distance, each ν v , v ∈ S, by a probability measure with finite support. Therefore, we can find d ≥ 1 big enough and, for each v ∈ S, a sequence (u v,1 , . . . , u v,d ) of points of N R , such that the probability measure Set also ν v = δ 0 for v / ∈ S. Then ν = (ν v ) v ∈ H K and W K (ν, ν ) < ε 2 . Let F/K be a finite extension of degree d such that all places in S split completely, as given by [BPS14b, Lemma 2.2]. For each v ∈ S and w ∈ M F such that w | v, we have n w = n v /d. We enumerate the places above a given place v ∈ S as w(v, j), j = 1, . . . , d.
Let H F ⊂ w∈M F N R be the subspace defined by the equation w n w u w = 0. For each v ∈ M K consider the element u ∈ H F given, for w ∈ M F , by Consider the map val F : T(F) → w∈M F N R defined by val F = (val w ) w∈M F . This is a group homomorphism and so it can be extended to a map By the product formula, the image of this map lies in the hyperplane H F and, by [BPS14b, Lemma 2.3], it is dense with respect to the L 1 -topology on H F . For α ∈ T(F) and r ∈ Q, we have for the probability measure λ v on N R × N R given by This measure has marginals ν v and ν α r ,v . Hence, the quantity in (4.17) is an upper bound for the KR-distance W (ν v , ν α r ,v ). It follows that we can choose α and r such that for every torsion point ω of T(K), the point p = ω · α r verifies for every v that . Hence W K (ν , ν p ) < ε/2 and thus W K (ν, ν p ) < ε. Since the orbit of α r under the action of the group of torsion points of T(K) is Zariski dense, we have shown that, given ε > 0 and a nonempty open subset U ⊂ X, we can choose p ∈ U (K) satisfying As in the proof of Proposition 2.5, let I be the set of hypersurfaces of X ordered by inclusion. For each Y ∈ I choose a point p Y ∈ (X \ Y )(K) such that with c(Y ) the number of components of Y . Thus, the net of algebraic points (p Y ) Y ∈I is generic and the net of probability measures (ν p Y ) Y ∈I converges to ν in the KR-topology, proving the result.
Proof of Theorem 4.7. Let ν = (ν v ) v be a centered adelic measure on N R such that each measure ν v satisfies the condition (4.5). By Lemma 4.8, it satisfies η D (ν) = µ ess D (X). Proposition 4.11 implies that there is a generic net (p l ) l∈I of points in T(K) = X 0 (K) such that (ν p l ) l∈I converges to ν with respect to the distance W K . On the other hand, by Lemma 4.8 we also have there is a generic D-small net (p l ) l∈I of algebraic points of X 0 such that the net of measures (ν p l ,v ) l∈I converges to ν v with respect to the Kantorovich-Rubinstein distance. In particular, (ν p l ,v ) l∈I it also converges to ν v in the weak- * topology with respect to C b (N R ).
Proof. Using the hypothesis that the point The statement then follows from Theorem 4.7 applied to the centered adelic measure ν = (ν v ) v .
Combining Theorems 4.3 and 4.7, we can obtain a criterion for when the direct image under the valuation map of the Galois orbits of a small net converges in the sense of measures. We show that in this case, the limit measure is concentrated in a single point. (1) for every D-small net (p l ) l∈I of algebraic points of X 0 , the net of measures (ν p l ,v ) l∈I converges in the weak- * topology with respect to C b (N R ); (2) for every generic D-small net (p l ) l∈I of algebraic points of X 0 , the net of measures (ν p l ,v ) l∈I converges in the weak- * topology with respect to C c (N R ), the space of continuous functions on N R with compact support; (3) the face F v contains only one point. When these equivalent conditions hold, the limit measures in (1) and (2) coincide with the Dirac measure at the unique point of F v .
Proof. It is clear that (1) implies (2), and Theorem 4.3 shows that (3) implies (1). Now suppose that the face F v has more than one point. Since F v is the minimal face containing B v , we can find distinct points u 0 , u 1 , u 2 ∈ F v such that The probability measures δ u0 and 1 2 δ u1 + 1 2 δ u2 satisfy the conditions (4.5). By Corollary 4.12, we can find generic D-small nets (p l ) l∈I and (q l ) l∈I such that the nets of measures (ν p l ,v ) l∈I and (ν q l ,v ) l∈I respectively converge to δ u0 and 1 2 δ u1 + 1 2 δ u2 in the KR-topology, and hence in the weak- * topology with respect to C c (N R ).
Combining these nets, we can obtain a net that does not converge in this weak- * topology. Hence the condition (2) implies the condition (3). The last statement follows from Theorem 4.3.
When any of the equivalent conditions of Corollary 4.13 holds we say that the metrized divisor D satisfies the modulus concentration property at the place v. Thus Corollary 4.13 gives us a criterion for the modulus concentration property at a place. We next give a criterion for the modulus concentration property at all places simultaneously, that can be directly read from the roof function. Before giving it, we need some preliminary results and a definition.  Recall that ∆ D,max denotes the convex set of points of ∆ D where ϑ D attains its maximum.
Proposition 4.16. The following conditions are equivalent: (1) the metrized R-divisor D is monocritical; (2) for every point x ∈ ∆ D,max , the set Proof. We prove first that (1) implies (2). Assume that D is monocritical. If the set (4.18) contains two elements u i = (u i,v ) v ∈ H K , i = 1, 2. Then the measures contradicting the hypothesis that D is monocritical, and showing that (4.18) contains a unique element.
Assume now that the set (4.18) contains a single point u = (u v ) v ∈ H K and there is a place v 0 ∈ M K such that u v0 is not a vertex of ∂ϑ D,v0 (x). Then we can find two points u v0,1 , u v0,2 ∈ ∂ϑ D,v0 (x) such that We consider the measure ν 1 = (δ uv ) v and the measure ν 2 = (ν v ) v defined by Then ν 2 is in (4.18) and, again by Lemma 4.8, we have that contradicting the hypothesis that D is monocritical, and completing the proof of (2). Assume that (2) is true and fix x ∈ ∆ D,max . Let S ⊂ M K be the finite set of places where u v = 0 or ϑ D,v is not identically zero. We have that Moreover, (2) implies that the equation has a unique solution a v = u v and this solution satisfies that a v is a vertex of ∂ϑ D,v (x). Therefore, by Lemma 3.15 we deduce that 0 is a vertex of ∂ϑ D (x). Hence (2) implies (3).
Since ∆ D,max is nonempty, (3) implies (4). Assume now that (4) is true. For each v, let g 1,v and g 2,v be the continuous concave functions on ∆ D in Notation 4.2. Since ϑ D = n v g 1,v + n v g 2,v , Lemma 3.15 and the definition of the set B v imply that this set contains one single point u v , and that this point is a vertex of both ∂g 1,v (x) and of −∂g 2,v (x). Hence B v is already a face of ∂g 1,v (x). Thus F v = B v = {u v } and so (4) implies (5).
By Lemma 4.8 it is clear that (5) implies (1) finishing the proof of the equivalence. Assume now that D is monocritical. Since by Lemma 4.8 the point u in (2) satisfies that η D (u) = min ν∈H K η D (ν), it is the critical point. Following the proof of the equivalence we deduce that F v = {u v } proving the last statement.
For a given toric metrized R-divisor, the condition of being monocritical and its critical point behave well with respect to scalar extensions. The following result follows from the compatibility of toric metrics with scalar extensions in [BPS14a, Proposition 4.3.8].
Proposition 4.17. Let X and D as before. Let F ⊂ K be a finite extension of K and write D F for the toric metrized R-divisor on X F obtained by scalar extension. If D is monocritical with critical point (u v ) v∈M K , then D F is also monocritical and its critical point (u w ) w∈M F is given by u w = u v for all v ∈ M K and w over v.
We now give the criterion for modulus concentration at every place.
Theorem 4.18. Let X and D be as before. The following conditions are equivalent: (1) for every D-small net (p l ) l∈I of algebraic points of X 0 and every place v ∈ M K , the net of measures (ν p l ,v ) l∈I converges.
(2) the metrized R-divisor D is monocritical; When these equivalent conditions hold, Proof. The theorem follows directly from Corollary 4.13 and Proposition 4.16.
When there is modulus concentration for every place, we can show that the convergence holds not only in the weak- * topology with respect to C b (N R ) but even in the stronger adelic KR-topology.
Theorem 4.19. Let X and D be as before. Assume that D is monocritical. Let u = (u v ) v be the critical point of D and set δ u = (δ uv ) v ∈ H K . Then, for every D-small net (p l ) l∈I of algebraic points of X 0 , the net of centered adelic measures (ν p l ) l∈I converges to δ u in the adelic KR-topology. In particular, for every v ∈ M K , the net of measures (ν p l ,v ) l∈I converges to δ uv in the KR-topology.
Proof. For each v ∈ M K , let f v : N R → R be the function given by This is an adelic family of bounded continuous functions on N R with f v = 0 for all but a finite number of v. Consider then the function η : H K → R given by Since the net (p l ) l∈I is D-small, By Theorem 4.18, the net of measures (ν p l ,v ) l∈I converges to δ uv . Using Corollary 4.10, we deduce that Choose a point x in the interior of ∆ D . Then there is a constant c > 0 such that, for all u ∈ N R , u ≤ −c (Ψ D − x)(u).
It follows from the definition of the Kantorovich-Rubinstein distance that, for each v ∈ M K , where the last equality follows from the product formula in Proposition 2.1(2). By (4.19), this distance converges to 0, completing the proof.

Equidistribution of Galois orbits and the Bogomolov property
We turn to the study of the limit measures of Galois orbits of D-small nets of algebraic points in toric varieties. In this section, we denote by X a proper toric variety over a global field K and D a toric metrized R-divisor on X with D big. For v ∈ M K , recall that val v : T an v → N R denotes the valuation map, defined in (4.1).
We first describe the limit measures in the monocritical case.
Definition 5.1. Given v ∈ M K and u ∈ N R , the probability measure λ Sv,u on X an v is defined as follows. ( In this case, λ Sv,u is the direct image under the translation by p of the Haar probability measure of S v . (2) When v is non-Archimedean, consider the multiplicative seminorm on the group algebra n ] that, to a Laurent polynomial m∈M α m χ m , assigns the value max m (|α m | v e − m,u ). This seminorm gives a point, denoted by θ(u), in the Berkovich space X an v . The point θ(u) is of type II or III in Berkovich's classification and lies in the preimage val −1 v (u). We then set λ Sv,u = δ θ(u) , the Dirac measure at this point. The following result corresponds to Theorem 1.2 in the introduction, and shows that modulus concentration at every place implies the equidistribution property at every place. Due to the existing equidistribution theorems in the literature, we restrict its statement to divisors (rather than R-divisors).
Theorem 5.2. Let X be a proper toric variety over K and D a semipositive toric metrized divisor on X with D big. The following conditions are equivalent: (1) for every generic D-small net (p l ) l∈I of algebraic points of X 0 and every place v ∈ M K , the net of probability measures (µ p l ,v ) l∈I on X an v converges; (2) the metrized divisor D is monocritical. When these equivalent conditions hold, the limit measure in (1) is λ Sv,uv , with u v ∈ N R the v-adic component of the critical point of D.
The proof of Theorem 5.2 is done by reduction to the quasi-canonical case. The following is the characterization of quasi-canonical toric metrized R-divisors in [BPS14b].
Proposition 5.3. Let X be a proper toric variety over K and D a semipositive toric metrized R-divisor on X with D big. The following conditions are equivalent: (1) D is quasi-canonical (Definition 2.7); (2) ϑ D is constant; Proof. The equivalence of (1) and (3) is given by [BPS14b,Corollary 4.7]. The equivalence of (1) and (2) is given in the course of the proof of [BPS14b, Proposition 4.6], recalling that vol(D) = deg D (X) and noting that, since by assumption D is semipositive, vol χ (D) = h D (X).
The following result gives the key step in the proof of Theorem 5.2.
Proposition 5.4. Let X be a proper toric variety over K and D a monocritical metrized R-divisor on X with critical point u = (u v ) v∈M K . Let D be the toric metrized R-divisor over D corresponding to the family of concave functions Then D is quasi-canonical and every D-small net of algebraic points of X 0 is also D -small.
Proof. The fact that D is quasi-canonical is given by Proposition 5.3. Let (p l ) l∈I be a D-small net of algebraic points of X 0 . By Theorem 4.19, the net of centered adelic measures (ν p l ) l∈I converges to δ u = (δ uv ) v with respect to the adelic KR-distance. By Lemma 4.8, the function η D is continuous with respect to this distance. Using (4.9), we deduce that On the other hand, ϑ D ,v = u v for each v. Since the critical point u lies in the subspace H K , we have that ϑ D = v n v u v = 0. Hence, Thus (p l ) l∈I is D -small, as stated.
Proof of Theorem 5.2. Suppose that the condition (1) holds. Given a generic Dsmall net (p l ) l∈I of algebraic points of X 0 and v ∈ M K , the net of measures (µ p l ,v ) l∈I converges weakly with respect to the space C(X an v ). Hence, the net of direct images (ν p l ,v ) l∈I converges weakly with respect to the space C c (N R ). By Corollary 4.13, for each v, the face F v contains only one point. Proposition 4.16 then implies that D is monocritical, giving the condition (2). Now suppose that the condition (2) holds. Let (Y, E) be the polarized toric variety associated to the polytope ∆ D . By the characterization of semipositive toric metrics in [BPS14a, Theorem 4.8.1], the metric in D induces a semipositive toric metric on E, and we denote by E the corresponding toric metrized divisor. We have that ψ E,v = ψ D,v for all v, and so E is also monocritical with the same critical point as D. Let be the ample divisor E on Y equipped with the quasi-canonical toric metric given by Proposition 5.4, with D replaced by E. Let (p l ) l∈I be a generic D-small net of algebraic points of X 0 = T = Y 0 . It is also a generic E-small net of algebraic points of Y 0 . By Proposition 5.4 with D replaced by E, it is also E -small. By Theorem 2.11, for each place v the net (µ p l ,v ) l∈I converges to the normalized Monge-Ampère measure Then [BPS14a, Theorem 4.8.11] implies that µ v = λ Sv,uv . Therefore, the net of measures (µ p l ,v ) l∈I on X an v converges to λ Sv,uv , giving the condition (1) and the last statement in the theorem.
Example 5.5. Let D can be a big and nef toric divisor on X equipped with the canonical metric. Following Example 4.15, this toric metrized divisor is monocritical with critical point 0 ∈ H K . Hence, it satisfies the v-adic equidistribution property with limit measure λ Sv,0 , for every v ∈ M K .
In [Bil97], Bilu gave an equidistribution theorem for Galois orbits of sequences of points of small canonical height. This result is restricted to number fields and Archimedean places. However, and in contrast with the previous example, this result holds not just for generic, but for strict sequences of points, that is, sequences that eventually avoid any given proper torsion subvariety. This stronger version of the equidistribution property was used in a crucial way in loc. cit. to prove the Bogomolov property for the canonical height.
Here we extend this version of the equidistribution property to monocritical metrized R-divisors on toric varieties (Theorem 5.7) and deduce from it the Bogomolov property (Theorem 1.5 in the introduction, or Theorem 5.12 below). Our proofs are similar to Bilu's and use Fourier analysis. Hence, for the rest of the section we restrict to the case when K is a number field and we only study the equidistribution over the Archimedean places. Following Remark 2.10, we restrict without loss of generality to sequences, instead of nets.
To formulate this extension, we have to modify slightly the notion of strict sequence. First we recall some standard terminology: a subtorus of T is an algebraic subgroup of T that is geometrically irreducible, a translate of a subtorus is a subvariety of T K that is the orbit of a point p ∈ T(K) by a subtorus, and a torsion subvariety is a translate of a subtorus by a torsion point of the group T(K) (K × ) n .
Definition 5.6. A sequence (p l ) l≥1 of algebraic points of T is strict if, for every translate of a subtorus U T K , there is l 0 ≥ 1 such that p l / ∈ U (K) for all l ≥ l 0 . Equivalently, (p l ) l≥1 is strict if, for every m ∈ M \ {0} and every point q ∈ X 0 (K), there is l 0 ≥ 1 such that χ m (p l ) = χ m (q) for all l ≥ l 0 .
Theorem 5.7. Let X be a proper toric variety over a number field K and D a monocritical metrized R-divisor on X. Then, for every strict D-small sequence (p l ) l≥1 of algebraic points of X 0 and every Archimedean place v ∈ M K , the sequence (µ p l ,v ) l≥1 converges to the probability measure λ Sv,uv , with u v ∈ N R the v-adic component of the critical point of D.
Proof. Let (p l ) l≥1 be a strict D-small sequence of algebraic points of X 0 . For each m ∈ M \ {0} consider the character Since (p l ) l≥1 is strict, the sequence (χ m (p l )) l≥1 is generic.
We embed G m,K → P 1 K as the principal open subset. Let D 0 = div(x 0 ) be the divisor at infinity on P 1 K , that we equip with the quasi-canonical toric metric corresponding to the adelic family of functions ψ D m 0 ,v : R → R given by For each v ∈ M K , there is a commutative diagram The commutativity of this diagram implies that ν χ m (p l ),v = m * ν p l ,v . By Theorem 4.19, the sequence (ν p l ) l≥1 converges in the adelic KR-topology to the centered adelic measure (δ uv ) v on N R . Hence, the sequence (ν χ m (p l ) ) l≥1 converges in the adelic KR-topology to the centered adelic measure (δ m,uv ) v on R. By (4.9) and Lemma 4.8, the sequence of points (χ m (p l )) l≥1 is D m 0 -small. Summarizing, the sequence (χ m (p l )) l≥1 of algebraic points of P 1 0 is generic and small with respect to the quasi-canonical toric metrized divisor D m 0 . Theorem 2.11 then implies that the sequence of measures (µ χ m (p l ),v ) l≥1 on the analytification P 1,an v P 1 (C) converges to λ Sv, m,uv . Assume now that v is Archimedean. Since the space of probability measures on X(C) is sequentially compact, by restricting to a subsequence we can suppose without loss of generality that (µ p l ,v ) l≥1 converges to a measure µ. Since the sequence of direct images ((val v ) * µ p l ,v ) l≥1 converges in the KR-topology to the Dirac measure on the point u v ∈ N R , we deduce that Let z be the standard affine coordinate of P 1 (C). For each m ∈ M \ {0}, let z m be a continuous function on P 1 (C) that agrees with z on a neighborhood of S 1 · χ m (e −uv ). Hence (χ m ) * (z m ) agrees with the character χ m on a neighborhood of S v · e −uv . Then where the last equality comes from Cauchy's formula. Hence χ m dµ = 0 for all m ∈ M \ {0}. By Fourier analysis, the only probability measure supported on S v · e −uv satisfying this condition is λ Sv,uv . Thus µ = λ Sv,uv , concluding the proof.
Remark 5.8. Our notion of strict sequence is stronger than the one in [Bil97]. Nevertheless, for the canonical height on a projective space, a small sequence of points is strict in our sense if and only if it eventually avoids any fixed translate of a subtorus with essential minimum equal to 0. Such a translate of a subtorus is necessarily a torsion subvariety, see for instance Example 5.16. Hence, a small sequence of points that is strict in the sense of Bilu [Bil97] is also strict in the sense of Definition 5.6. Thus Theorem 5.7 applied to the canonically metrized divisor at infinity on a projective space specializes to [Bil97, Theorem 1.1].
Remark 5.9. To the best of our knowledge, even for the canonical metric it is still not know if the equidistribution property for strict sequences holds for the non-Archimedean places of a global field.
The toric Bogomolov conjecture can be stated as follows: let X be a toric variety and D an ample divisor on X. Let V ⊂ X 0,K be a subvariety which is not torsion.
Then there exists ε > 0 such that the subset of algebraic points of V of canonical height bounded above by ε, is not dense in V . Equivalently, if V ⊂ X 0,K is a subvariety such that µ ess D can (V ) = 0, then V is a torsion subvariety. This conjecture was proved by Zhang in the number field case [Zha95]. Bilu obtained a proof of Zhang's theorem based on his equidistribution theorem. In what follows, we extend his approach to the general monocritical case over a number field.
Recall that X denotes a proper toric variety over a number field K and D a toric metrized R-divisor on X. For a subset V ⊂ X(K), we denote by µ abs D (V ) the absolute minimum of the height of its algebraic points. The fact that D is toric implies µ ess D (X) = µ abs D (X 0 ), (5.2) see [BPS14b, Lemma 3.9(1)]. Therefore, for any subvariety V ⊂ X 0,K , This motivates the following definition.
. In particular, an algebraic point p of X 0 is D-special if and only if h D (p) = µ ess D (X). We also propose the following terminology.
Definition 5.11. The toric metrized R-divisor D satisfies the Bogomolov property if every D-special subvariety of X 0,K is a translate of a subtorus.
We consider the problem of deciding if a given toric metrized R-divisor satisfies the Bogomolov property. The following result corresponds to Theorem 1.5 in the introduction, and shows that the answer is affirmative for monocritical metrics.
Theorem 5.12. Let X be a proper toric variety over a number field K and D a monocritical metrized R-divisor on X with critical point u = (u v ) v∈M K . Let V be a D-special subvariety of X 0,K . Then V is a translate of a subtorus.
Furthermore, if u v ∈ val v (T(K)) ⊗ Q for all v, then V is the translate of a subtorus by a D-special point.
Before giving the proof of this theorem, we study special points and, more generally, special translates of subtori in the monocritical case. We first give a criterion for the existence of such points.
Proposition 5.13. Let X be a proper toric variety over K and D a monocritical metrized R-divisor on X with critical point u = (u v ) v∈M K . Then there exists a D-special point if and only if If this is the case, then every D-special point is of the form q 1/ with q ∈ X 0 (K) and ≥ 1.
Proof. Suppose that there is a D-special point p ∈ X 0 (K). Choose a finite normal extension F ⊂ K of K where p is defined. Consider the norm of p relative to this extension, given by where Gal(F/K) and [F : K] i are the Galois group and the inseparable degree of the extension, respectively.
Let v ∈ M K . For every τ ∈ Gal(F/K), there is a place w ∈ M F over v such that val v (τ (p)) = val w (p). By Corollary 4.9 and Proposition 4.17, we have that val w (p) = u v for any such place. It follows that val v (τ (p)) = u v for all τ . Using Since N F K (p) ∈ T(K), we get that [F : K]u v ∈ val v (T(K)), proving the implication. Conversely, assume that the condition (5.4) holds. Let S ⊂ M K be a finite set containing the Archimedean places and those places v where u v = 0. Set In other terms, there is q ∈ T(K) S such that val v (q) = u v for all v ∈ S. By Corollary 4.9, the point p = q 1/ ∈ T(K) is D-special, proving the reverse implication.
To prove the last statement, suppose that the condition (5.4) holds and consider an arbitrary D-special point p ∈ X 0 (K). Let p be the D-special point constructed above and F ⊂ K a finite extension of K so that p, p ∈ T(F). Then val w (p p −1 ) = 0 for all w ∈ M F . By Kronecker's theorem, the point p p −1 is torsion. We conclude that some positive power of p lies in T(K), as stated.
Next we characterize the translates of subtori that are D-special. Let U = T K · p be the translate of a subtorus T ⊂ T by a point p ∈ X 0 (K). The subtorus T corresponds to a saturated sublattice Q of N ; we denote by ι : Q → N the corresponding inclusion map. Let F ⊂ K be a finite extension of K where p is defined. For each w ∈ M F , we consider the affine subspace of N R given by Indeed A U,w = val w (U an w ) and so this affine subspace depends only on U and not on a particular choice for the translating point p.
As explained in [BPS14a, §3.2], the normalization of the closure of U in X K can be given a structure of toric variety. Let Σ be the fan on N R corresponding to X and Σ Q the fan on Q R obtained by restricting Σ to this latter linear space. Then the inclusion ι : Q R → N R induces an equivariant map of toric varieties ϕ p,ι : X Σ Q ,K → X K extending the inclusion U → T K .
Proposition 5.14. Let X be a proper toric variety over a number field K and D a monocritical metrized R-divisor on X with critical point u = (u v ) v∈M K . Let U = T K · p ⊂ X 0,K be the translate of a subtorus T ⊂ T by a point p ∈ X 0 (K) defined over a finite extension F ⊂ K of K. For a place w in M F denote by v(w) the place in M K below w. Then we have the following properties.
(1) The translate U is D-special if and only if u v(w) ∈ A U,w for all w ∈ M F .
(2) If the translate U is D-special, then the metrized R-divisor ϕ * p,ι D is monocritical and its critical point is (u v(w) − val w (p)) w∈M F . Proof. By passing to a suitable large finite extension of K and applying Proposition 4.17, we can reduce to the case when U is the translate of a K-rational point, that is, U = T K · p with p ∈ X 0 (K). With this assumption, F = K and we set v := w = v(w).
Since D is a semipositive toric metrized divisor with D big, the virtual support function Ψ D is concave and its associated polytope has dimension n. Hence, there is m ∈ M R such that m, u > Ψ D (u) for all u = 0. Moreover, the metric functions ψ D,v are concave for all v ∈ M K .
Consider the toric metrized R-divisor E := ϕ * ι,p D on the toric variety X Σ Q . By [BPS14a, Proposition 4.3.19], its virtual support function and metric functions are given, for z ∈ Q R , by Therefore Ψ E is concave and satisfies ι ∨ m, z > Ψ E (z) for all z ∈ Q R \ {0}. Hence, the R-divisor E is big. Moreover, the metric functions ψ E,v are concave and so E is semipositive.
Since U is identified with a dense open subset of X Σ Q ,K , we have Since D monocritical, the minimum in the right equality is attained only at u = (u v ) v . We conclude that µ ess E (U ) = µ ess D (X) if and only if u v ∈ A U,v for all v ∈ M K , proving both statements.
Corollary 5.15. Let X be a proper toric variety over a number field K and D a monocritical metrized R-divisor on X with critical point u = (u v ) v∈M K , and suppose that u v ∈ val v (T(K)) ⊗ Q for all v ∈ M K . Then a translate of a subtorus of X 0 is D-special if and only if it is the translate of a subtorus by a D-special point.
Proof. Clearly, the translate of a subtorus by a D-special point is D-special. To prove the reverse implication, let U be a D-special translate of a subtorus and write U = T K ·p as in the statement of Proposition 5.14. By this result, the toric metrized Since p ∈ X 0 (F), Hence u v − val w (p) ∈ val w (T (F)) ⊗ Q. Extending the base field to F and restricting to X Σ Q , Proposition 5.13 implies that this toric variety contains an E-special point. Hence U contains a D-special point and it is the translate of T by this point, as stated.
Example 5.16. Let D can be a nef and big toric R-divisor on the proper toric variety X, equipped with the canonical metric. By Example 4.15, it is monocritical with critical point 0 ∈ H K . Hence, p ∈ X 0 (K) is D can -special if and only if val v (p) = 0 for every v ∈ M K . By Kronecker's theorem, this is also equivalent to the fact that p is torsion. Hence, Corollary 5.15 shows that a translate of a subtorus that is D can -special is necessarily the translate of a subtorus by a torsion point, that is, a torsion subvariety.
Proof of Theorem 5.12. Let U ⊂ X 0,K be the minimal translate of a subtorus containing the subvariety V and let Q and Σ Q be defined before Proposition 5.14. By (5.2) and (5.3), we have µ abs D (U ) = µ ess D (U ) and µ ess . Therefore, U is D-special. By Proposition 5.14(2), D pulls back to a monocritical metrized R-divisor on X Σ Q , the normalization of the closure of U in X K . Replacing X by this toric variety, we reduce to the case where U = X 0,K .
Using Proposition 2.5, we choose a sequence (p l ) l≥1 of algebraic points of V , that is generic in V and satisfies lim l h D (p l ) = µ ess D (V ). Since V is not contained in any proper translate of a subtorus, this sequence is strict and, since V is D-special, it is also D-small.
Applying Theorem 5.7 to an Archimedean place v ∈ M K , we obtain that the sequence of measures (µ p l ,v ) l≥1 converges to a measure whose support is the translate S v · e −uv of the compact subtorus, with u v the v-adic coordinate of the critical point of D. Hence S v · e −uv ⊂ V an v . Since S v · e −uv is dense in X an v with respect to the Zariski topology, it follows that V = X 0,K , proving the result. By Theorem 5.12 and Example 5.16, the canonical toric metrized R-divisor D can satisfies the Bogomolov property, and every D can -special subvariety is torsion.
Hence, Theorem 5.12 extends Zhang's theorem to the general monocritical case.
On the other hand, in §6.3 we will give examples of non-monocritical metrized divisors not satisfying the Bogomolov property.

Examples
The obtained criteria can be applied in concrete situations, to decide if a given semipositive toric metrized R-divisor satisfies properties like modulus concentration or equidistribution. In this section, we consider translates of subtori with the canonical height, and toric metrized R-divisors equipped with positive smooth metrics at the Archimedean places and canonical metrics at the non-Archimedean ones. We also give a family of counterexamples to the Bogomolov property in the non-monocritical case.
6.1. Translates of subtori with the canonical height. Let X a proper toric variety of dimension n over a global field K and D a big and nef toric R-divisor on X. Let Ψ D be its virtual support function.
We denote by D can this R-divisor equipped with the canonical metric as in By Proposition 4.16, the canonical metric is monocritical and so, by Theorem 5.2, D can satisfies the equidistribution property at every place (Example 5.5).
We next study the toric metrics on D that are obtained as inverse image by an equivariant map of a canonical metrized toric divisor on an projective space.
Let v ∈ M K . If v is Archimedean, we set λ v = 1 whereas, if v is non-Archimedean, we set λ v as the positive generator of the discrete subgroup val v (K × ) of R. A piecewise affine function is said to be λ v -rational if all its defining affine functions x, u + b satisfy x ∈ M Q and b ∈ λ v Q.
Let P r K be a standard projective space over K with homogeneous coordinates (z 0 : · · · : z r ) and H the hyperplane at infinity, defined by the equation z 0 = 0. Denote by H can this toric divisor equipped with the canonical metric. As seen in [BPS14a,Example 3.7.11], if ψ : N R → R is a concave λ v -rational piecewise affine function with |ψ − Ψ D | is bounded, then there is an integer r > 0 and a toric morphism ι : Hence, any such function ψ can be realized as the v-adic metric function of a toric metrized divisor on D. This allows us to construct many examples, both monocritical and non-monocritical, of metrized toric divisors.
In the next examples, we fix K = Q and, as before, we denote by H can the hyperplane at infinity with the canonical metric.
Example 6.1. Let ι : G m,Q → P 2 Q be the map given by ι(t) = (1 : t/2 : t). Let By Corollary 4.13, this metrized divisor does not satisfy the modulus concentration property at any place. A fortiori, it does not satisfy the equidistribution property at any place. Indeed, by (4.3) we have µ ess D (X) = log(2). Let (ω l ) l≥1 be a sequence given by a choice of a primitive l-th root of the unity, a = 2 a positive prime number and r an integer with log(a) ≤ r log(2). Choose any r-th root of a and consider the generic sequences of points p l = (1 : ω l ) and q l = (1 : 2a −1/r ω l ) for l ≥ 1.
For every v ∈ M Q , l ≥ 1, p ∈ Gal(p l ) v and q ∈ Gal(q l ) v we have (val v ) * (p) = 0 and Either by computing the local roof functions of D or the Weil height of the image of these points under the inclusion ι, we deduce that h D (p l ) = log(2) and h D (q l ) = log(2).
Therefore both sequences are D-small. For any place v, the sequence µ p l ,v converges to λ Sv,0 . By contrast, if we denote u v = (val v ) * (q) for any q ∈ Gal(q l ) v , then µ q l ,v converges to λ Sv,uv . This shows that neither the modulus concentration nor the equidistribution properties hold for the places 2, a, ∞. Varying a, we deduce that these properties do not hold at any place of Q.
The metric of D at the Archimedean place is the canonical one. The metrics at the non-Archimedean places can be interpreted in terms of integral models. Let X be the blow up of P 1 Z at the point (1 : 0) over the prime 2. The fibre of the structural map X → Spec(Z) over the point 2 has two components: the exceptional divisor of the blow up, that we denote by E, and the strict transform of the fibre of P 1 Z , that we denote Y . Consider the divisor where ∞ denotes the closure in X of the point (0 : 1) ∈ P 1 (Q). The pair (X , D) is a model of (X, D). For each non-Archimedean place v, this model induces an algebraic metric on D that agrees with the v-adic metric of D.
Let X be the normalization of the closure of ι(G m,Q ) and D = ι * (H can ). In this case, X = P 1 Q and D is the divisor at infinity plus the divisor at zero. We have ∆ D = [−1, 1]. As before, we compute the local roof functions using [BPS14a, Example 5.1.16]. Their graphs are represented in Figure 2. For x ∈ [0, 2], By Corollary 4.13, D satisfies modulus concentration for all places except the place 2. This toric metrized divisor is not monocritical, and so we cannot apply Theorem 5.2 in this case. Indeed, later we will see that D does not satisfy the equidistribution property at any other place of Q (Example 7.6). As in the previous example, the metric of D at the Archimedean place is the canonical one, and those at the non-Archimedean places can be interpreted in terms of integral models. Let X be the blow up of P 1 Z at the points (1 : 0) and (0 : 1) over the prime 2. The fibre of the structural map X → Spec(Z) over the point 2 has three components. Consider the divisor where ∞ denotes the closure in X of the point (0 : 1) ∈ P 1 (Q) and 0 the closure of the point (1 : 0). The pair (X , D) is a model of (X, D). For each non-Archimedean place v, this model induces an algebraic metric on D that agrees with the v-adic metric of D.
Let X be the normalization of the closure of ι(G m,Q ) and D = ι * (H can ). In this case, X = P 1 Q and D is three times the divisor at infinity. We have ∆ D = [0, 3] and the local roof functions are represented in Figure 3. They are given by ϑ 2 (x) = log(2) min(x, 1, 3 − x) and ϑ v (x) = 0 for v = 2. The Thus D is monocritical, by Proposition 4.16. By Corollary 4.13 and Theorem 5.2, it satisfies both the modulus concentration and the equidistribution properties for any place. By Theorem 5.12, it also satisfies the Bogomolov property.
6.2. Positive Archimedean metrics. The following result covers many of the examples considered in [BPS14a,BMPS12,BPS14b]: twisted Fubiny-Study metrics on projective spaces, metrics from polytopes, Fubini-Study metrics on toric bundles, p -metrics on toric varieties, and Fubini-Study metrics on weighted projective spaces. All of them consist of toric varieties over Q with a toric divisor equipped with a positive smooth metric at the Archimedean place and the canonical metric at the non-Archimedean ones.
Theorem 6.4. Let X a proper toric variety over a number field K and D = (D, ( · v ) v∈M K ) a semipositive toric metrized R-divisor with D big. We assume that, when v is Archimedean, · v is a positive smooth metric on the principal open subset X an 0,v whereas, when v is non-Archimedean, it is the v-adic canonical metric of D. Then D is monocritical. In particular, it satisfies the equidistribution property for every place of K.
When K = Q, the v-adic limit measure is λ Sv,0 for every v ∈ M Q .
Proof. Since the metric is smooth and positive for v Archimedean, the proof of [BPS14a,Proposition 4.4.1] implies that the metric function ψ D,v is smooth and strictly concave, in the sense that its Hessian is negative definite. Therefore ψ D,v is of Legendre type in the sense of [BPS14a, Definition 2.4.1] and, by [BPS14a, Theorem 2.4.2(2)], the local roof function ϑ D,v is of Legendre type. In particular, ϑ D,v is smooth and strictly concave on the interior of ∆ D and the sup-differential at any point of the border of the polytope is empty. For the non-Archimedean places, the metrics are canonical and so their local roof functions are zero. Hence this function is smooth and strictly concave on the interior of ∆ D , and its supdifferential at any point of the border of ∆ D is empty. This implies that there is a unique maximizing point x max ∈ ∆ D , that it lies in the interior of the polytope, and that ∂ϑ D (x max ) = {0}.
The first assertion then follows from Proposition 4.16. The rest of the statement follows from Theorem 5.2.
Example 6.5. Let X = P 1 Q and D the divisor at infinity equipped with the Fubini-Study metric at the Archimedean place and the canonical metric at the non-Archimedean ones. By Theorem 6.4, this toric metrized divisor satisfies the equidistribution property at every place. Moreover, the limit measure of the Galois orbits of any generic D-small sequence is λ Sv,0 .
Recall that the canonical metric at the non-Archimedean places corresponds to the canonical model of (P 1 Q , ∞) given by (P 1 Z , ∞), where ∞ is the closure of the point (0 : 1) ∈ P 1 (Q). If we change the integral model, different phenomena may occur. For instance, if we consider the integral model of Example 6.1, then the maximum of the global roof function is attained at the interior of the polytope. Since the global roof function is differentiable in the interior of the polytope, we deduce that the sup-differential is reduced to one point. By Proposition 4.16, this new toric metrized divisor is also monocritical.
By contrast, if we consider the divisor D = 0 + ∞ with the Fubini-Study metric at the Archimedean place and the metrics induced by the integral model of Example 6.2, then the maximum of the global roof function is attained at the point zero and the sup-differential at this point is [− log(2), log(2)]. Since zero is not a vertex of this set, by Proposition 4.16 this divisor is not monocritical. Hence it does not satisfy the equidistribution property at the Archimedean place.
6.3. Counterexamples to the Bogomolov property. In this section, we give examples of toric metrized divisors not satisfying the Bogomolov property. For simplicity, we restrict to the case K = Q. As in §6.1, we denote by H can the canonical metrized divisor at infinity on a projective space.
As in the examples in the previous section, we denote by X the normalization of the closure of the image of ι and D = ι * (H can ). In this case, X = P 2 Q and D is the divisor at infinity.
We have that ∆ D is the standard simplex of N R = R 2 and Ψ D : R 2 → R is the function given by Ψ D (u 1 , u 2 ) = min(0, u 1 , u 2 ).
By [BPS14a,Example 4.3.21], the local metric functions are given, for (u 1 , u 2 ) ∈ R 2 , by By [BPS14a, Example 5.1.16], the local roof functions are given, for (x 1 , x 2 ) ∈ ∆ D , by Hence the global roof function agrees with ϑ D,∞ . Its only maximizing point is We also have µ ess D (X) = ϑ(0, 0) = log(2). Let (z 0 : z 1 : z 2 ) be homogeneous coordinates of X and consider the curve C of equation z 0 + z 1 + z 2 = 0. In what follows, we will see that this curve is a D-special subvariety. Since C is not a translate of a subtorus, this will show that D does not satisfy the Bogomolov property.
For l ≥ 1 choose a primitive l-th root of the unity ω l . Let z 1,l be a solution of the equation z 2 + z + ω l = 0 and put z 2,l = ω l /z 1,l for the other solution. Then z 1,l + z 2,l + 1 = 0 and z 1,l z 2,l = ω l . (6.2) In particular, p l = (1 : z 1,l : z 2,l ) is an algebraic point of C.
Let v ∈ M Q and q = (1 : q 1 : q 2 ) ∈ Gal(p l ) v . If v = ∞, then the conditions (6.2) imply that If v = ∞, then these same conditions (6.2) give max(|q Moreover, by the product formula and (6.3), we have By Corollary 4.9, the conditions (6.3), (6.4) and (6.5) imply that h D (p l ) = µ ess D (X). Since the sequence (p l ) l≥1 is generic in C, we deduce µ ess D (C) = µ ess D (X) and so C is a D-special subvariety.
We generalize this example to a family of metrics on toric varieties of dimension greater than or equal to 2. Proposition 6.7. Let X be a proper toric variety over Q of dimension n ≥ 2 and D a big and nef R-divisor on X. Let u 0 ∈ N R and consider the metrized divisor D u0 over D defined by Then D u0 satisfies the Bogomolov property if and only if u 0 = 0.
Proof. When u 0 = 0 we have D u0 = D can . By Theorem 5.12 and Example 5.16, this toric metrized divisor satisfies the Bogomolov property. Suppose u 0 = 0. The local roof functions of D u0 are given, for x ∈ ∆ D , by In particular, the global roof function ϑ D coincides with ϑ D u 0 ,∞ . Fix x 0 in the relative interior of the convex subset of ∆ D on which ϑ D attains it maximum value. If we denote by ϑ 0 the constant function equal to 0 defined on ∆ D , then σ 0 = ∂ϑ 0 (x 0 ) is a cone in N R containing −u 0 in its relative interior. Moreover, It follows that 0 ∈ B v for every v, that F ∞ = u 0 + σ 0 and that F v = σ 0 for v = ∞. As in Example 6.6, to prove that D u0 does not satisfy the Bogomolov property, it is enough to exhibit a curve C in X which is D-special but not a translate of a subtorus. We identify N R R n . Since X is proper and σ 0 is a cone of the fan of X, there is a primitive vector n 0 ∈ N in σ 0 . It follows that there is ε 0 > 0 such that Choose a primitive vector a 0 ∈ N such that a 0 and n 0 generate a saturated sublattice V of N . Put b 0 = n 0 + a 0 . Then a 0 and b 0 form an integral basis of V . Fix an integer k 0 ≥ ε −1 0 and consider the linear map L : V R → R 2 defined by L(sa 0 + tb 0 ) = k 0 · (s, t).
Let S be the toric surface in X 0 associated to the saturated sublattice V . The linear map L induces a toric morphism ι : S → G 2 m,Q . Let C be the curve in G 2 m,Q of equation x + y + 1 = 0 and denote by C 0 the closure in X of the curve ι −1 (C). As in Example 6.6, for l ≥ 1 choose a primitive l-th root of unity root ω l . Let z 1,l be a solution of the equation z 2 + z + ω l = 0 and put z 2,l = ω l /z 1,l . Hence z 1,l + z 2,l + 1 = 0 and z 1,l z 2,l = ω l .
For every place v there is a commutative diagram Arguing as in Example 6.6, for every non-Archimedean place v and every point q ∈ Gal(p l ) v , we have val v (ι(q)) = 0.
Since L is injective, val v (q) = 0 and therefore ν p l ,v = δ 0 . In particular, When v = ∞, the product formula implies that On the other hand, the facts that |z 1,l | ∞ |z 1,l | ∞ = 1 and that imply that val ∞ (ι(q)) ∈ for every q ∈ Gal(p l ) ∞ . Thus This implies that supp(ν p l ,∞ ) ⊂ F ∞ . By Lemma 4.8, we have h D (p l ) = µ ess D (X). Being the sequence (p l ) l≥1 generic in C 0 , we deduce that C 0 is D-special. Since C 0 is not a translate of a subtorus, we conclude that D does not satisfy the Bogomolov property, as stated.

Potential theory on the projective line and small points
In this section, we apply potential theory on the projective line over a number field, and in particular Rumely's Fekete-Szegő theorem, to produce interesting sequences of small points in the non-monocritical case.
In the absence of modulus concentration, this allows to produce a wealth of nontoric measures that are limit measures of Galois orbits of generic sequences of points of small height. These techniques also allow to show that the absence of modulus concentration at a place can affect the equidistribution property at another place. 7.1. Limit measures in the absence of modulus concentration. We recall the basic objects of potential theory on the projective line. For most of the details and precise definitions, we refer the reader to [Tsu75] and [BR10] for the Archimedean and non-Archimedean cases, respectively.
Let K be a number field and fix a place v ∈ M K . For a subset E ⊂ C v , we denote by E its closure in A 1,an v . Moreover, for r > 0, put In particular, for a ∈ C v the set B v (a, r) is the closed ball with center a and radius r.
We denote by the function defined by δ v (z, z ) = |z − z | v for v Archimedean, and as the unique upper semicontinuous extension of the function on Given a probability measure µ on A 1,an v , the energy integral (with respect to the point at infinity) of µ is defined as (7.1) be a measurable subset. The v-adic Robin constant and capacity (with respect to the point at infinity) of K are respectively defined as In the non-Archimedean case, C v is a proper subset of A 1,an v . In general, but there is a particular case where the equality holds. If K is a closed strict affinoid domain, then K ∩ C v is an RL-domain in the sense of [Rum89]. Moreover, For instance, O v is a closed strict affinoid domain and In the Archimedean case, O v = O v and these equalities are also valid. If K is compact and cap v (K) > 0, then there exists a unique probability measure, denoted by ρ K , supported on K and realizing the infimum in (7.2), see [Tsu75, §III.2 and Theorem III.32] for the Archimedean case and [BR10, Propositions 6.6 and 7.21] for the non-Archimedean one. Hence This measure is called the equilibrium measure of K. For K = O v , it agrees with λ Sv,0 , the Haar probability measure on the unit circle when v is Archimedean, and the Dirac measure at the Gauss point of A 1,an v when v is non-Archimedean. Given an adelic set E = (E v ) v∈M K , its (global) capacity is defined as By (7.3), this product actually runs over a finite set and so the global capacity is well-defined.
The following result shows that, in the non-monocritical case, there is a wealth of measures not invariant under the action of the compact torus, that can be obtained as limit measures of Galois orbits of generic sequences of points of small height.
Theorem 7.2. Let X = P 1 K and D the divisor at infinity equipped with a semipositive toric metric. Let B v , F v be the associated subsets of N R = R as in (4.4). Let E = (E v ) v∈M K be a closed bounded adelic set with E v an RL-domain for every non-Archimedean place v, and such that cap(E) = 1. Assume that the following conditions hold: Then there is a generic D-small sequence (p l ) l≥1 of algebraic points of X 0 = G m,K such that, for every v ∈ M K , the sequence of probability measures (µ p l ,v ) l≥1 converges to ρ Ev .
The proof of this theorem will be given after two preliminary propositions. The next statement is a direct consequence of Rumely's version of the Fekete-Szegő theorem in [Rum02, Theorem 2.1].
Proposition 7.3. Let E = (E v ) v∈M K be a closed bounded adelic set such that cap(E) ≥ 1. There exists a generic sequence (p l ) l≥1 of points of K × satisfying for all l ≥ 1 and v ∈ M K . In particular, Gal(p l ) v ⊂ E v for every non-Archimedean Proof. For l ≥ 1, consider the bounded adelic neighbourhood U l = (U l,v ) v∈M K of E given by By [Rum02, Theorem 2.1], there is an infinite number of points p ∈ K × such that Gal(p) v ⊂ U l,v for all v. Inductively, for each l ≥ 1 we choose p l as one of these points which is different from p l for l ≤ l − 1.
In the notation of Proposition 7.3, when the adelic set E has capacity 1, the sequence of v-adic Galois orbits of the points p l equidistribute according to the equilibrium measure of the closure E v .
Proposition 7.4. Let E = (E v ) v∈M K be a closed bounded adelic set with E v an RLdomain for every non-Archimedean place v and such that cap(E) = 1. Let (p l ) l≥1 be a generic sequence of points of K × with Gal(p l ) v ⊂ B v (E v , 1 l ) for all l ≥ 1 and v ∈ M K . Then, for all v ∈ M K , the sequence (µ p l ,v ) l≥1 converges to the equilibrium measure of E v .
Proof. By taking a subsequence, we can suppose without loss of generality that, for each v in M K , the sequence (µ p l ,v ) l≥1 converges to a probability measure µ v .
For each l ≥ 1 and v ∈ M K , put for short G l,v = Gal(p l ) v and set Consider also the probability measure on P 1,an v × P 1,an v given by and note that (ν l,v ) l≥1 converges to µ v × µ v . The function log(δ v (·, ·)) is bounded from above on B v (E v , 1) × B v (E v , 1). Similarly as in the proof of Lemma 3.8, this property implies that ≥ log(cap v (E v )).
Together with (7.4), this implies I v (µ v ) = − log cap v (E v ). It follows that µ v is the equilibrium measure of E v , completing the proof.
Proof of Theorem 7.2. Let (p l ) l≥1 be a generic sequence of points of K × as in Proposition 7.4, that exists thanks to Proposition 7.3. Then Proposition 7.4 implies that, for every v ∈ M K , the sequence of probability measures (µ p l ,v ) l≥1 converges to ρ Ev . Here we have to show that, under the present hypotheses, this sequence of points is D-small. The measures µ p l ,v converge to ρ Ev and are supported on the closure B v (E v , 1). Also, for all v / ∈ S, we have ρ Ev = λ Sv,0 and g D,v vanishes on the support of this measure. Hence (7.5) By the condition (3) and the fact that E is an adelic set, we deduce that the collection ν = ((val v ) * ρ Ev ) v∈M K is a centered adelic measure (Definition 4.4). Moreover, g D,v = −ψ D,v • val v on A 1,an v \ {0}. By (7.5) and the fact that the equilibrium measure does not charge any individual point, we have Lemma 4.8 together with the conditions (1) and (2) implies that η D (ν) = µ ess D (X). Hence the sequence (p l ) l≥1 is D-small, as stated, finishing the proof of the theorem. 7.2. Local modulus concentration and equidistribution. Corollary 4.13 gives a criterion for a semipositive toric metrized R-divisor to satisfy the modulus concentration property at a given place. Applying it, one can immediately give examples where modulus concentration fails at that place. If this happens, then the equidistribution property also fails at that place.
Can this absence of modulus concentration affect the equidistribution property at another place? The next result on the projective line over a number field shows that this can be the case under a rationality hypothesis.
Proposition 7.5. Let X = P 1 K be the projective line over a number field K, D the divisor at infinity equipped with a semipositive toric metric, and v 0 ∈ M K . For each v ∈ M K , let B v be the set introduced in Notation 4.2. Assume that there is a point p ∈ X 0 (K) = K × such that val v (p) ∈ B v for all v ∈ M K and val v0 (p) ∈ ri(B v0 ).
If D does not satisfy the modulus concentration property at v 0 , then D does not satisfy the equidistribution property at any place of K.
Proof. Assume that D does not satisfy the modulus concentration property at v 0 . Let v ∈ M K . If v = v 0 then clearly D does not satisfy the equidistribution property at v, so we can suppose that v = v 0 . Extending scalars to a suitable large number field and translating by the point p, we can also reduce to the case when 0 ∈ ri(B v0 ) and 0 ∈ B w for all w ∈ M K .
Let F v0 and A v0 be the convex sets given in Notation 4.2. By Corollary 4.13, the set F v0 is not a single point. Since 0 ∈ ri(B v0 ) and F v0 is the minimal face of A v0 containing B v0 , there is δ > 0 such that the set The local capacities of these sets are cap v0 (E v0 ) = c, cap v (E v ) = 1/c and cap w (E w ) = 1 for w = v 0 , v, see for instance [Rum02,§3]. Hence, the global capacity of E is 1. Consider the map R : P 1 K → P 1 K given by R(z) = z + 1 z . Using the expression R(z) − 2 = (z−1) 2 z , one checks that, for w non-Archimedean, Using that z = 1 2 (R(z) ± R(z) 2 − 4), one also checks that, for w Archimedean, Note that c − √ c 2 − 1 = e −δ and c + √ c 2 − 1 = e δ . We deduce from the previous analysis that, regardless whether v 0 or w are Archimedean or not, we have v0 ([−δ, δ]) and R −1 (E w ) ⊂ val −1 w (0) for w = v 0 . We represent in Figure 4 the inverse images by R of the sets E v0 , E v and E w in the Archimedean case. The point x therein is Figure 4. Inverse images by R of the sets E v0 , E v and E w for v 0 , v and w = v, v 0 Archimedean Let (p l ) l≥1 be a generic sequence as given by Proposition 7.3 applied to the adelic set E. For each l ≥ 1, choose a point q l ∈ R −1 (p l ). After restricting to a subsequence, we can assume that the sequence (µ q l ,w ) l≥1 converges to a probability measure µ w on P 1,an v , for all w ∈ M K . By construction, for each w the supports of the direct image measures ν q l ,w = (val w ) * µ q l ,w , l ≥ 1, are contained in a fixed bounded subset of N R = R. Therefore, this sequence of measures converges in the KR-topology to the direct image (val w ) * µ w .
Let S ⊂ M K be the finite subset consisting of the Archimedean places plus v 0 and v. If w = v 0 , then Gal(q l ) w ⊂ val Thus E[(val w ) * (µ w )] = 0 ∈ B w for all w ∈ M K . By construction, it is also clear that supp((val w ) * µ w ) ⊂ F w for all w. By Lemma 4.8, the sequence (q l ) l≥1 is D-small. We have thus constructed a generic D-small sequence such that its v-adic Galois orbit converges to a measure µ v whose support is contained in the closure R −1 (E v ).
Observe that R −1 (E v ) S v = val −1 (0) because, in the Archimedean case, it does not contain the points 1 and −1, whereas in the non-Archimedean case, it does not contain the Gauss point.
On the other hand, the sequence (ω l ) l≥1 given by the choice of a primitive l-th root of unity is also D-small, but its v-adic Galois orbit converges to the measure λ S,0 . The support of this measure is strictly bigger than that of µ v . We deduce that D does not satisfy the v-adic equidistribution property, as stated.
Example 7.6. Let X = P 1 Q and D the divisor at infinity plus the divisor at zero, equipped with the semipositive toric metric from Example 6.2. As explained therein, D does not satisfy modulus concentration at the place v 0 = 2 and, by (6.1), we have 0 ∈ ri(B v ) for all v ∈ M Q . Theorem 7.2 implies that D does not satisfy the equidistribution property for any place of Q.
Remark 7.7. A rationality hypothesis like the condition that the sets B v contain the image by the valuations map of an algebraic point, is necessary for the conclusion of Proposition 7.5 to hold. Indeed, suppose that, for a given non-Archimedean . By the tree structure of the Berkovich projective line, this implies that val −1 v (u v ) consists of single point, of type III in Berkovich's classification [BR10,§1.4]. Hence, the v-adic modulus concentration at v given by Corollary 4.13, easily implies that the v-adic Galois orbits of D-small sequences of algebraic points concentrate around this point of type III, regardless of the structure of the set B v0 .