Huayi Chen
email: huayi.chen@imj-prg.fr

Atsushi Moriwaki
email: moriwaki@math.kyoto-u.ac.jp

ARAKELOV THEORY ON ARITHMETIC SURFACES OVER A TRIVIALLY VALUED FIELD

In this article, we consider an analogue of Arakelov theory of arithmetic surfaces over a trivially valued field. In particular, we establish an arithmetic Hilbert-Samuel theorem and studies the effectivity up to R-linear equivalence of pseudoeffective metrised R-divisors.

Introduction

In Arakelov geometry, one considers an algebraic variety over the spectrum of a number field and studies various constructions and invariants on the variety such as metrised line bundles, intersection product, height functions etc. Although these notions have some similarities to those in classic algebraic geometry, their construction is often more sophisticated and get involved analytic tools.

Recently, an approach of R-filtration has been proposed to study several invariants in Arakelov geometry, which allows to get around analytic technics in the study of some arithmetic invariants, see for example [START_REF] Chen | Convergence des polygones de Harder-Narasimhan[END_REF][START_REF] Chen | Arithmetic Fujita approximation[END_REF]. Let us recall briefly this approach in the setting of Euclidean lattices for simplicity. Let E = (E, . ) be a Euclidean lattice, namely a free Z-module of finite type E equipped with a Euclidean norm . on E R = E ⊗ Z R. We construct a family of vector subspaces of E Q = E ⊗ Z Q as follows. For any t ∈ R, let F t (E) be the Q-vector subspace of E Q generated by the lattice vectors s such that s e -t . This construction is closely related to the successive minima of Minkowski. In fact, the i-th minimum of the lattice E is equal to

exp -sup{t ∈ R | rk Q (F t (E)) i} .
The family (F t (E)) t∈R is therefore called the R-filtration by minima of the Euclidean lattice E.

Classically in Diophantine geometry, one focuses on the lattice points of small length, which are analogous to global sections of a vector bundle over a smooth projective curve. However, such points are in general not stable by addition. This phenomenon brings difficulties to the study of Arakelov geometry over a number field and prevents the direct transplantation of algebraic geometry methods in the arithmetic setting. In the R-filtration approach, the arithmetic invariants are encoded in a family of vector spaces, which allows to apply directly algebraic geometry methods to study some problems in Arakelov geometry.

If we equipped Q with the trivial absolute value | . | 0 such that |a| 0 = 1 if a belongs to Q \ {0} and |0| 0 = 0, then the above R-filtration by minima can be considered as an ultrametric norm . 0 on the Q-vector space E Q such that

s 0 = exp(-sup{t ∈ R | s ∈ F t (E)}).
Interestingly, finite-dimensional ultrametrically normed vector spaces over a trivially valued field are also similar to vector bundles over a smooth projective curve. This method is especially successful in the study of the arithmetic volume function. Moreover, R-filtrations, or equivalently, ultrametric norms with respect to the trivial absolute value, are also closely related to the geometric invariant theory of the special linear group, as shown in [11, §6].

All this works suggest that there would be an Arakelov theory over a trivially valued field. From the philosophical point of view, the R-filtration approach can be considered as a correspondance from the arithmetic geometry over a number field to that over a trivially valued field, which preserves some interesting arithmetic invariants. The purpose of this article is to build up such a theory for curves over a trivially valued field (which are actually analogous to arithmetic surfaces). Considering the simplicity of the trivial absolute value, one might expect such a theory to be simple. On the contrary, the arithmetic intersection theory for adelic divisors in this setting is already highly nontrivial, which has interesting interactions with the convex analysis on infinite trees.

Let k be a field equipped with the trivial absolute value and X be a regular irreducible projective curve over Spec k. We denote by X an the Berkovich analytic space associated with X, which identifies with a tree of length 1 whose leaves correspond to closed points of X (see [3, §3.5]).

η 0 • • • x 0 • • •
For each closed point x of X, we denote by [η 0 , x 0 ] the edge connecting the root and the leaf corresponding to x. This edge is parametrised by the interval [0, +∞] and we denote by t( . ) : [η 0 , x 0 ] → [0, +∞] the parametrisation map.

Recall that an R-divisor D on X can be viewed as an element in the free real vector space over the set X (1) of all closed points of X. We denote by ord x (D) the coefficient of x ∈ X (1) in the writing of D into a linear combination of elements of X (1) . We call Green function of D any continuous map g : X an → [-∞, +∞] such that there exists a continuous function ϕ g : X an → R which satisfies the following condition ∀ x ∈ X (1) , ∀ ξ ∈ [η 0 , x 0 [, ϕ g (ξ) = g(ξ) -ord x (D)t(ξ).

The couple D = (D, g) is called a metrised R-divisor on X. Note that the set Div R (X) of metrised R-divisors on X forms actually a vector space over R.

Let D be an R-divisor on X. We denote by H 0 (D) the subset of the field Rat(X) of rational functions on X consisting of the zero rational function and all rational functions s such that D + (s) is effective as an R-divisor, where (s) denotes the principal divisor associated with s, whose coefficient of x is the order of s at x. The set H 0 (D) is actually a k-vector subspace of Rat(X). Moreover, the Green function g determines an ultrametric norm . g on the vector space H 0 (D) such that s g = exp -inf ξ∈X an (g + g (s) )(ξ) .

Let D 1 = (D 1 , g 1 ) and D 2 = (D 2 , g 2 ) be adelic R-divisors on X such that ϕ g 1 and ϕ g 2 are absolutely continuous with square integrable densities, we define a pairing of D 1 and D 2 as (see §3.3 for details)

(D 1 • D 2 ) := g 1 (η 0 ) deg(D 1 ) + g 2 (η 0 ) deg(D 1 )
-x∈X (1) [k(x) : k]

x 0 η 0 g 1 (ξ)g 2 (ξ) dt(ξ) (1.1)
Note that such pairing is similar to the local admissible pairing introduced in [27, §2] or, more closely, similar to the Arakelov intersection theory on arithmetic surfaces with L 2 1 -Green functions (see [5, §5]). This construction is also naturally related to harmonic analysis on metrised graphes introduced in [START_REF] Baker | Harmonic analysis on metrized graphs[END_REF] (see also [START_REF] Chinburg | The capacity pairing[END_REF] for the capacity pairing in this setting), although the point η 0 is linked to an infinitely many vertices. A more conceptual way to understand the above intersection pairing (under diverse extra conditions on Green functions) is to introduce a base change to a field extension k of k, which is equipped with a non-trivial absolute value extending the trivial absolute value on k. It is then possible to define a Monge-Ampère measure over X an k for the pull-back of g 1 , either by the theory of δ-forms [START_REF] Gubler | A tropical approach to nonarchimedean Arakelov geometry[END_REF][START_REF] Gubler | Positivity properties of metrics and deltaforms[END_REF], or by the non-Archimedean Bedford-Taylor theory developed in [START_REF] Chambert | Formes différentielles réelles et courants sur les espaces de berkovich[END_REF], or more directly, by the method of Chambert-Loir measure [START_REF] Chambert-Loir | Mesures et équidistribution sur les espaces de Berkovich[END_REF][START_REF] Chambert | Mesures de Mahler et équidistribution logarithmique[END_REF]. It turns out that the push-forward of this measure on X an does not depend on the choice of valued extension k /k (see [START_REF] Boucksom | Space of formes, determinant of cohomology and fekete points in non-archimedean geometry[END_REF]Lemma 7.2]). We can then interpret the intersection pairing as the height of D 2 with respect to (D 1 , g 1 ) plus the integral of g 2 with respect to the pushforward of this Monge-Ampère measure.

One contribution of the article is to describe the asymptotic behaviour of the system of ultrametrically normed vector spaces (H 0 (nD), . ng ) in terms of the intersection pairing, under the condition that the Green function g is plurisubharmonic (see Definition 6.14). More precisely, we obtain an analogue of the arithmetic Hilbert-Samuel theorem as follows (see §7 infra).

Theorem 1.1. -Let D = (D, g) be an adelic R-divisor on X. We assume that deg(D) > 0 and g is plurisubharmonic. Then one has

lim n→+∞ -ln s 1 ∧ • • • ∧ s rn ng,det n 2 /2 = (D • D),
where (s i ) rn i=1 is a basis of H 0 (nD) over k (with r n being the dimension of the k-vector space H 0 (nD)), . ng,det denotes the determinant norm associated with . ng , and

(D • D) is the self-intersection number of D.
Diverse notions of positivity, such as bigness and pseudo-effectivity, are discussed in the article. We also study the effectivity up to R-linear equivalence of pseudo-effective metrised R-divisors. The analogue of this problem in algebraic geometry is very deep. It is the core of the non-vanishing conjecture, which has applications in the existence of log minimal models [START_REF] Birkar | On existence of log minimal models II[END_REF]. It is also related to Keel's conjecture (see [21, Question 0.9] and [24, Question 0.3]) for the ampleness of divisors on a projective surface over a finite field. In the setting of an arithmetic curve associated with a number field, this problem can actually be interpreted by Dirichlet's unit theorem in algebraic number theory. In the setting of higher dimensional arithmetic varieties, the above effectivity problem is very subtle. Both examples and obstructions were studied in the literature, see for example [START_REF] Moriwaki | Toward Dirichlet's unit theorem on arithmetic varieties[END_REF][START_REF] Chen | Algebraic dynamical systems and Dirichlet's unit theorem on arithmetic varieties[END_REF] for more details.

In this article, we establish the following result.

Theorem 1.2. -Let (D, g) be a metrised R-divisor on X. For any x ∈ X (1) , we let

µ inf,x (g) := inf ξ∈]η 0 ,x 0 [ g(ξ) t(ξ) . Let µ inf (g) := x∈X (1) µ inf,x (g)[k(x) : k].
Then the following assertions hold.

(1) (D, g) is pseudo-effective if and only if µ inf (g) 0.

(2) (D, g) is R-linearly equivalent to an effective metrised R-divisor if and only if µ inf,x (g) 0 for all but finitely many x ∈ X (1) and if one of the following conditions holds:

(a) µ inf (g) > 0, (b) x∈X (1) µ inf,x (g)
x is a principal R-divisor. The article is organised as follows. In the second section, we discuss several properties of convex functions on a half line. In the third section, we study Green functions on an infinite tree. The fourth section is devoted to a presentation of graded linear series on a regular projective curve. These sections prepares various tools to develop in the fifth section an Arakelov theory of metrised R-divisors on a regular projective curve over a trivially valued field. In the sixth section, we discuss diverse notion of global and local positivity of metrised R-divisors. Finally, in the seventh section, we prove the Hilbert-Samuel theorem for arithmetic surfaces in the setting of Arakelov geometry over a trivially valued field. real number µ(g) such that the function

ϕ g : R >0 -→ R, ϕ g (t) := g(t) -µ(g)t
extends to a continuous function on [0, +∞]. The real number µ(g) satisfying this condition is unique. We call it the asymptotic slope of g. The set of asymptotically linear continuous functions forms a real vector space with respect to the addition and the multiplication by a scalar. The map µ( . ) is a linear form on this vector space.

We denote by L 2 1 (R >0 ) the vector space of continuous functions ϕ on R >0 such that the derivative (in the sense of distribution) ϕ is represented by a square-integrable function on R >0 . We say that an asymptotically linear continuous function

g on R >0 is pairable if the function ϕ g belongs to L 2 1 (R >0 ). Remark 2.1. -The functional space L 2
1 is a natural object of the potential theory on Riemann surfaces. In the classic setting of Arakelov geometry, it has been used in the intersection theory on arithmetic surfaces. We refer to [5, §3] for more details.

ϕ (t) dt.
See [2, Theorem 1.28] for a proof. In particular, the function ϕ represents the derivative of ϕ in the sens of distribution.

Proposition 2.2. -Let ϕ be a convex function on R >0 which is bounded.

(1) One has ϕ 0 on R >0 and lim t→+∞ ϕ (t) = 0. In particular, the function ϕ is decreasing and extends to a continuous function on [0, +∞].

(2) We extends ϕ continuously on [0, +∞]. The function

(t ∈ R >0 ) -→ ϕ(t) -ϕ(0) t is increasing. Moreover, one has lim t↓0 ϕ(t) -ϕ(0) t = lim t↓0 ϕ (t) ∈ [-∞, 0],
which is denoted by ϕ (0).

Proof. -(1) We assume by contradiction that ϕ (a) > 0 at certain a ∈ R >0 . By (2.1), for any x ∈ R >0 such that x > a, one has

ϕ(x) -ϕ(a) = ]a,x[ ϕ (t) dt ]a,x[ ϕ (a) dt = (x -a)ϕ (a),
so that lim x→+∞ ϕ(x) = +∞. This is a contradiction. Thus ϕ (t) 0 for all t ∈ R >0 . Therefore, one has

lim t→+∞ ϕ (t) = sup t∈R >0 ϕ (t) 0.
To show that the equality lim t→+∞ ϕ (t) = 0 holds, we assume by contradiction that there exists ε > 0 such that ϕ (t) -ε for any t ∈ R >1 . Then, by (1), for any

x ∈ R >1 , ϕ(x) -ϕ(1) = x 1 ϕ (t) dt ]1,x[ (-ε) dt = -ε(x -1).
Therefore, lim x→+∞ ϕ(x) = -∞, which leads to a contradiction.

(2) For 0 < a < b, since

ϕ(a) = ϕ((1 -a/b)0 + (a/b)b) ≤ (1 -a/b)ϕ(0) + (a/b)ϕ(b),
one has ϕ(a) -ϕ(0) a ϕ(b) -ϕ(0) b as required. Denote by ϕ (0) the limite lim s↓0 ϕ (s). Note that the equality (2.1) actually holds for (a, b) ∈ [0, +∞] 2 (by the continuity of ϕ and the monotone convergence theorem). Therefore

ϕ (0) ϕ(t) -ϕ(0) t = 1 t ]0,t[ ϕ (s) ds ϕ (t).
By passing to limit when t ↓ 0, we obtain the result.

Proposition 2.3. -Let ϕ and ψ be continuous functions on [0, +∞] which are convex on R >0 . One has

(2.2) ]0,+∞] ϕ dψ = - R >0 ψ (t)ϕ (t) dt -ϕ(0)ψ (0) ∈ [-∞, +∞[.
In particular, if ϕ(0) = ψ(0) = 0, then one has

(2.3) ]0,+∞] ϕ dψ = ]0,+∞] ψ dϕ .
Proof. -By (2.1), one has

]0,+∞] ϕ dψ = ]0,+∞] ]0,x[ ϕ (t) dt dψ (x) + ϕ(0) ]0,+∞] dψ .
By Fubini's theorem, the double integral is equal to

R >0 ϕ (t) ]t,+∞] dψ dt = - R >0 ϕ (t)ψ (t) dt.
Therefore, the equality (2.2) holds. In the case where ϕ(0) = ψ(0) = 0, one has

]0,+∞] ϕ dψ = - R >0 ψ (t)ϕ (t) dt = ]0,+∞] ψ dϕ .
Proposition 2.4. -Let ϕ be a continuous function on [0, +∞] which is convex on R >0 . One has

(2.4) R >0
x dϕ (x) = ϕ(0) -ϕ(+∞).

Proof.

-By Fubini's theorem

R >0 x dϕ (x) = R >0 ]0,x[ dt dϕ (x) = R >0 ]t,+∞[ dϕ (x) dt = - R >0 ϕ (t) dt = ϕ(0) -ϕ(+∞),
where in the third equality we have used the fact that lim t→+∞ ϕ (t) = 0 proved in Proposition 2.2.

2.3.

Transform of Legendre type. -Definition 2.5. -Let ϕ be a continuous function on [0, +∞] which is convex on R >0 . We denote by ϕ * the fonction on [0, +∞] defined as

∀ λ ∈ [0, +∞], ϕ * (λ) := inf x∈[0,+∞] (xλ + ϕ(x) -ϕ(0)).
Clearly the function ϕ * is increasing and non-positive. Moreover, one has

ϕ * (0) = inf x∈[0,+∞] ϕ(x) -ϕ(0) = ϕ(+∞) -ϕ(0).
Therefore, for any λ ∈ [0, +∞], one has

ϕ(+∞) -ϕ(0) ϕ * (λ) 0. Proposition 2.6. -Let ϕ be a continuous function on [0, +∞] which is con- vex on R >0 . For p ∈ R >1 , one has +∞ 0 (-ϕ (x)) p dx = -(p -1)p +∞ 0 λ p-2 ϕ * (λ)dλ.
In particular,

(2.5) +∞ 0 ϕ (x) 2 dx = -2 +∞ 0 ϕ * (λ) dλ.
Proof. -Since ϕ is increasing one has

ϕ * (λ) = inf x∈[0,+∞[ x 0 (λ + ϕ (t)) dt = +∞ 0 min{λ + ϕ (t), 0} dt.
Therefore, by Fubini's theorem,

+∞ 0 λ p-2 ϕ * (λ) dλ = +∞ 0 +∞ 0 λ p-2 min{λ + ϕ (t), 0} dλ dt = +∞ 0 -ϕ (t) 0 λ p-2 (λ + ϕ (t)) dλ dt = +∞ 0 λ p p + ϕ (t)λ p-1 p -1 -ϕ (t) 0 dt = -1 (p -1)p +∞ 0 (-ϕ (t)) p dt

Green functions on a tree of length 1

The purpose of this section is to establish a framework of Green functions on a tree of length 1, which serves as a fundament of the arithmetic intersection theory of adelic R-divisors on an arithmetic surface over a trivially valued field. In other words, for any x ∈ S, the restriction of t( . ) to [η 0 , x 0 ] is the inverse of the injective map ξ x . We call t( . ) the parametrisation map of T (S). (1) for any x ∈ S, ξ -1 x (U ) is an open subset of [0, +∞], (2) if η 0 ∈ U , then U contains [η 0 , x 0 ] for all but finitely many x ∈ S. By definition, all maps ξ x : [0, +∞] → T (S) are continuous. However, if S is an infinite set, then the parametrisation map t( . ) is not continuous.

Note that the topological space T (S) is compact. We can visualise it as an infinite tree of depth 1 whose root is η 0 and whose leaves are x 0 with x ∈ S.

3.2. Green functions. -Let S be a non-empty set and w : S → R >0 be a map. We call Green function on T (S) any continuous map g from T (S) to [-∞, +∞] such that, for any x ∈ S, the composition of g with ξ x | R >0 defines an asymptotically linear function on R >0 . For any x ∈ S, we denote by µ x (g) the unique real number such that the function

(u ∈ R >0 ) -→ g(ξ x (u)) -µ x (g)u.
extends to a continuous function on [0, +∞]. We denote by ϕ g : T (S) → R the continuous function on T (S) such that

ϕ g (ξ) = g(ξ) -µ x (g)t(ξ) for any ξ ∈ [η 0 , x 0 ], x ∈ S.
Remark 3.4. -Let g be a Green function on T (S). It takes finite values on T (S) \ {x 0 : x ∈ S}. Moreover, for any x ∈ S, the value of g at x 0 is finite if and only if µ x (g) = 0. As g is a continuous map, g -1 (R) contains all but finitely many x 0 with x ∈ S. In other words, for all but finitely many x ∈ S, one has µ x (g) = 0. Note that the Green function g is bounded if and only if µ x (g) = 0 for any x ∈ S. Definition 3.5. -Let g be a Green function on T (S). We denote by g can the map from T (S) to [-∞, +∞] which sends ξ ∈ [η 0 , x 0 ] to µ x (g)t(ξ). Note that the composition of g can with ξ x | R >0 is a linear function on R >0 . We call it the canonical Green function associated with g. Note that there is a unique bounded Green function ϕ g on T (S) such that g = g can + ϕ g . We call it the bounded Green function associated with g. The formula g = g can + ϕ g is called the canonical decomposition of the Green function g. If g = g can , we say that the Green function g is canonical.

Proposition 3.6. -Let g be a Green function on T (S). For all but countably many x ∈ S, the restriction of g on [η 0 , x 0 ] is a constant function.

Proof. -For any n ∈ N such that n 1, let U n be set of ξ ∈ T (S) such that

|g(ξ) -g(η 0 )| < n -1 .
This is an open subset of T (S) which contains η 0 . Hence there is a finite subset S n of S such that [η 0 , x 0 ] ⊂ U n for any x ∈ S \ S n . Let S = n∈N, n 1 S n . This is a countable subset of S. For any x ∈ S \ S and any ξ ∈ [η 0 , x 0 ], one has g(ξ) = g(η 0 ) Remark 3.7. -It is clear that, if g is a Green function on T (S), for any a ∈ R, the function ag : T (S) → [-∞, +∞] is a Green function on T (S). Moreover, the canonical decomposition of Green functions allows to define the sum of two Green functions. Let g 1 and g 2 be two Green functions on T (S). We define g 1 + g 2 as (g 1,can + g 2,can ) + (ϕ g 1 + ϕ g 2 ).

Note that the set of all Green functions, equipped with the addition and the multiplication by a scalar, forms a vector space over R.

3.3.

Pairing of Green functions. -Let S be a non-empty set and w : S → R >0 be a map, called a weight function. We say that a Green function g on T (S) is pairable with respect to w if the following conditions are satisfied:

(1) for any x ∈ S, the function

ϕ g • ξ x | R >0 belongs to L 2 1 (R >0 ) (see §2.1), (2) one has x∈S w(x) R >0 (ϕ g • ξ x | R >0 ) (u) 2 du < +∞.
For each x ∈ S we fix a representative of the function (ϕ g • ξ x | R >0 ) and we denote by

ϕ g : x∈S ]η 0 , x 0 [ -→ R the function which sends ξ ∈ ]η 0 , x 0 [ to (ϕ g • ξ x | R >0 ) (t(ξ)).
We equip x∈S [0, +∞] with the disjoint union of the weighted Lebesgue measure w(x) du, where du denotes the Lebesgue measure on [0, +∞]. We denote by ν S,w the push-forward of this measure by the projection map x∈S [0, +∞] -→ T (S).

Then the function ϕ g is square-integrable with respect to the measure ν S,w . Definition 3.8. -Note that pairable Green functions form a vector subspace of the vector space of Green functions. Let g 1 and g 2 be pairable Green functions on T (S). We define the pairing of g 1 and g 2 as

x∈S w(x) µ x (g 1 )g 2 (η 0 ) + µ x (g 2 )g 1 (η 0 ) - T (S) ϕ g 1 (ξ)ϕ g 2 (ξ) ν S,w (dξ),
denoted by g 1 , g 2 w , called the pairing of Green functions g 1 and g 2 . Note that , w is a symmetric bilinear form on the vector space of pairable Green functions.

3.4. Convex Green functions. -Let S be a non-empty set. We say that a Green function g on T (S) is convex if, for any element x of S, the function

g • ξ x on R >0 is convex. Convention 3.9.
-If g is a convex Green function on T (S), by convention we choose, for each x ∈ S, the right derivative of

ϕ g • ξ x | R >0 to represent the derivative of ϕ g • ξ x | R >0 in the sense of distribution. In other words, ϕ g • ξ x | R >0
is given by the right derivative of the function ϕ g • ξ x | R >0 . Moreover, for any x ∈ S, we denote by ϕ g (η 0 ; x) the element ϕ g•ξx (0) ∈ [-∞, 0] (see Proposition 2.2 (2)). We emphasis that ϕ g•ξx (0) could differ when x varies. Definition 3.10. -Let g be a Green function on T (S). We call convex envelop of g and we denote by ( g the continuous map from T (S) to [-∞, +∞] such that, for any x ∈ S,

( g •ξ x | R >0 is the convex envelop of g •ξ x | R >0 (see §2.4).
By Proposition 2.7, the function ( g is well defined and defines a convex Green function on T (S). Moreover, it is the largest convex Green function on T (S) which is bounded from above by g. Proposition 3.11. -Let g be a Green function on T (S). The following equalities hold:

g can = ( g can , g(η 0 ) = ( g(η 0 ), ( ϕ g = ϕ ( g .
Proof. -The first two equalities follows from Proposition 2.7. The third equality comes from the first one and the fact that

( g = g can + ( ϕ g .
3.5. Infimum slopes. -Let S be a non-empty set and g be a Green function on T (S). For any x ∈ S, we denote by µ inf,x (g) the element

inf ξ∈]η 0 ,x 0 [ g(ξ) t(ξ) ∈ R ∪ {-∞}.
Clearly one has µ inf,x (g) µ x (g). Therefore, by Remark 3.4 we obtain that µ inf,x (g) 0 for all but finitely many x ∈ S. If w : S → R 0 is a weight function, we define the global infimum slope of g with respect to w as x∈X (1)

µ inf,x (g)w(x) ∈ R ∪ {-∞}.
This element is well defined because µ inf,x (g) 0 for all but finitely many x ∈ S. If there is no ambiguity about the weight function (notably when S is the set of closed points of a regular projective curve cf. Definition 6.6), the global infimum slope of g is also denoted by µ inf (g). Proposition 3.12. -Let g be a convex Green function on T (S). For any x ∈ S one has

µ inf,x (g -g(η 0 )) = µ x (g) + ϕ g (η 0 ; x).
Proof. -This is a direct consequence of Proposition 2.2 (2).

Graded linear series

Let k be a field and X be a regular projective curve over Spec k. We denote by X (1) the set of closed points of X. By R-divisor on X, we mean an element in the free R-vector space generated by X (1) . We denote by Div R (X) the Rvector space of R-divisors on X. If D is an element of Div R (X), the coefficient of x in the expression of D into a linear combination of elements of X (1) is denoted by ord x (D). If ord x (D) belongs to Q for any x ∈ X (1) , we say that D is a Q-divisor ; if ord x (D) ∈ Z for any x ∈ X (1) , we say that D is a divisor on X. The subsets of Div R (X) consisting of Q-divisors and divisors are denoted by Div Q (X) and Div(X), respectively.

Let D be an R-divisor on X. We define the degree of D to be

(4.1) deg(D) := x∈X (1) [k(x) : k] ord x (D),
where for x ∈ X, k(x) denotes the residue field of x. Denote by Supp(D) the set of all x ∈ X (1) such that ord x (D) = 0, called the support of the R-divisor D. This is a finite subset of X (1) . Although X (1) is an infinite set, (4.1) is actually a finite sum: one has

deg(D) = x∈Supp(D) ord x (D)[k(x) : k].
Denote by Rat(X) the field of rational functions on X. If f is a non-zero element of Rat(X), we denote by (f ) the principal divisor associated with f , namely the divisor on X given by x∈X (1) ord x (f )x, where ord x (f ) ∈ Z denotes the valuation of f with respect to the discrete valuation ring O X,x . The map Rat(X) × → Div(X) is additive and hence induces an R-linear map (1) . We denote by D 0 the condition "D is effective". For any R-divisor D on X, we denote by

Rat(X) × R := Rat(X) × ⊗ Z R -→ Div R (X), which we still denote by f → (f ). Definition 4.1. -We say that an R-divisor D is effective if ord x (D) 0 for any x ∈ X
H 0 (D) the set {f ∈ Rat(X) × : (f ) + D 0} ∪ {0}.
It is a finite-dimensional k-vector subspace of Rat(X). We denote by genus(X) the genus of the curve X relatively to k. The theorem of Riemann-Roch implies that, if D is a divisor such that deg(D) > 2 genus(X) -2, then one has

(4.2) dim k (H 0 (D)) = deg(D) + 1 -genus(X).
We refer the readers to [14, Lemma 2.2] for a proof.

Let D be an R-divisor on X. We denote by

Γ(D) × R the set {f ∈ Rat(X) × R : (f ) + D 0}. This is an R-vector subspace of Rat(X) × R . Similarly, we denote by Γ(D) × Q the Q-vector subspace {f ∈ Rat(X) × Q : (f ) + D 0} of Rat(X) × Q . Note that one has (4.3) Γ(D) × Q = n∈N, n 1 {f 1 n : f ∈ H 0 (nD) \ {0}}. Definition 4.2.
-Let D be an R-divisor on X. We denote by D and D the divisors on C such that

ord x ( D ) = ord x (D) , ord x ( D ) = ord x (D) . Clearly one has deg( D ) deg(D) deg( D ). Moreover, deg( D ) > deg(D) - x∈Supp(D) [k(x) : k], (4.4) deg( D ) < deg(D) + x∈Supp(D) [k(x) : k]. (4.5) Let (D i ) i∈I be a family of R-divisors on X such that sup i∈I ord x (D i ) = 0
for all but finitely many x ∈ X (1) . We denote by sup i∈I D i the R-divisor such that ∀ x ∈ X (1) , ord x sup

i∈I D i = sup i∈I ord x (D i ). Proposition 4.3. -Let D be an R-divisor on X such that deg(D) 0. One has (4.6) lim n→+∞ dim k (H 0 (nD)) n = deg(D).
Proof. -We first assume that deg(D) > 0. By (4.4), for sufficiently positive integer n, one has deg( nD ) > 2 genus(X) -2. Therefore, (4.2) leads to

dim k (H 0 ( nD )) = deg( nD ) + 1 -genus(X). Moreover, since deg(D) > 0 one has deg( nD ) n deg(D) > 2 genus(X) -2 for sufficiently positive n ∈ N 1 . Hence (4.2) leads to dim k (H 0 ( nD )) = deg( nD ) + 1 -genus(X). Since H 0 ( nD ) ⊆ H 0 (nD) ⊆ H 0 ( nD ), we obtain deg( nD ) + 1 -genus(X) n dim k (H 0 (nD)) n deg( nD ) + 1 -genus(X) n .
Taking limit when n → +∞, by (4.4) and (4.5) we obtain (4.6).

We now consider the case where deg(D) = 0. Let E be an effective R-Cartier divisor such that deg(E) > 0. For any ε > 0 one has

lim sup n→+∞ dim k (H 0 (nD)) n lim n→+∞ dim k (H 0 (n(D + εE))) n = deg(D + εE) = ε deg(E).
Since ε is arbitrary, the equality (4.6) still holds.

Proposition 4.4. -Let D be an R-divisor on X such that deg(D) > 0. Then one has (4.7) sup s∈Γ(D) × Q (s -1 ) = D.
Proof. -For any s ∈ Γ(D) × Q one has ord x (s) + ord x (D) 0 and hence ord x (s -1 ) ord x (D).

For any x ∈ X (1) and any ε > 0, we pick an R-divisor D x,ε on X such that

D -D x,ε is effective, ord x (D x,ε ) = ord x (D) and 0 < deg(D x,ε ) < ε. Since deg(D x,ε ) > 0, the set Γ(D x,ε ) × Q is not empty (see (4.3) and Proposition 4.3). This set is also contained in Γ(D) × Q since D x,ε D. Let f be an element of Γ(D x,ε ) × Q . One has D x,ε + (f ) 0 and deg(D x,ε + (f )) = deg(D x,ε ) < ε. Therefore ord x (D + (f )) = ord x (D x,ε + (f )) ε [κ(x) : k] ,
which leads to

ord x (f -1 ) ord x (D) - ε [κ(x) : k] .
Since ε > 0 is arbitrary, we obtain

sup s∈Γ(D) × Q ord x (s -1 ) = ord x (D).
(s) = 0. Similarly, if D is an R-divisor on X such that Γ(D) × Q = ∅, then the equality sup s∈Γ(D) × Q (s -1 ) = D always holds.
Definition 4.6. -Let Rat(X) be the field of rational functions on X. By graded linear series on X, we refer to a graded sub-k-algebra

V • = n∈N V n T n of Rat(X)[T ] = n∈N Rat(X)
T n which satisfies the following conditions:

(1) V 0 = k, (2) there exists n ∈ N 1 such that V n = {0} (3) there exists an R-divisor D on X such that V n ⊆ H 0 (nD) for any n ∈ N.

If W is a k-vector subspace of Rat(X), we denote by k(W ) the extension

k({f /g : (f, g) ∈ (W \ {0}) 2 }) of k. If V • is a graded linear series on V , we set k(V • ) := k n∈N 1 {f /g : (f, g) ∈ (V n \ {0}) 2 } . If k(V • ) = Rat(X), we say that the graded linear series V • is birational. Example 4.7. -Let D be an R-divisor on X such that deg(D) > 0.
Then the total graded linear series n∈N H 0 (nD) is birational. Proposition 4.8. -Let V • be a graded linear series on X. The set

N(V • ) := {n ∈ N 1 : V n = {0}}
equipped with the additive law forms a sub-semigroup of N 1 . Moreover, for any n ∈ N(V • ) which is sufficiently positive, one has k(

V • ) = k(V n ).
Proof. -Let n and m be elements of N(V • ). If f and g are respectively nonzero elements of V n and V m , then f g is a non-zero element of

V n+m . Hence n + m belongs to N(V • ). Therefore, N(V • ) is a sub-semigroup of N 1 . In particular, if d
1 is a generator of the subgroup of Z generated by N(V • ), then there exists

N 0 ∈ N 1 such that dn ∈ N(V • ) for any n ∈ N, n N 0 . Since k ⊆ k(V • ) ⊆ Rat(X) and Rat(X) is finitely generated over k, the ex- tension k(V • )/k is finitely generated (see [7, Chapter V, §14, n • 7, Corollary 3]).
Therefore, there exist a finite family {n 1 , . . . , n r } of elements in N 1 , together with pairs

(f i , g i ) ∈ (V dn i \ {0}) 2 such that k(V • ) = k(f 1 /g 1 , . . . , f r /g r ). Let N ∈ N such that N -max{n 1 , . . . , n r } N 0 .
For any i ∈ {1, . . . , r} and

M ∈ N N , let h M,i ∈ V d(M -n i ) \ {0}. Then (h M,i f i , h M,i g i ) ∈ (V dM \ {0}) 2 , which shows that k(V • ) = k(V dM ). Definition 4.9. -If V • is a graded linear series, we define Γ(V • ) × Q as n∈N 1 {f 1 n | f ∈ V n \ {0}},
and let D(V • ) be the following R-divisor

sup s∈Γ(V•) × Q (s -1 ),
called the R-divisor generated by V • . The conditions (2) and (3) in Definition 4.6 show that the R-divisor D(V • ) is well defined and has non-negative degree.

Proposition 4.10. -Let V • be a birational graded linear series on X. One has

(4.8) lim n∈N, Vn ={0} n→+∞ dim k (V n ) n = deg(D(V • )) > 0.
Proof. -By definition, for any n ∈ N one has

V n ⊆ H 0 (nD(V • )). Therefore Proposition 4.3 leads to lim sup n→+∞ dim k (V n ) n deg(D(V • )).
Let p be a sufficiently positive integer (so that Rat(X) = k(V p )). Let

V [p] • := n∈N Im(S n V p -→ V np )T n .
Clearly one has D(V [p] • ) pD(V • ). Moreover, since Rat(X) = k(V p ), X identifies with the normalisation of Proj(V [p] • ), and the pull-back on X of the tautological line bundle on Proj(V [p] • ) identifies with

O(D(V [p] • )). This leads to 1 p deg(D(V [p] • )) = lim n→+∞ dim k (V [p] n ) pn lim inf n∈N, Vn ={0} n→+∞ dim k (V n ) n .
As the map p → 1 p D(V [p] • ) preserves the order if we consider the relation of divisibility on p, by the relation

D(V • ) = sup p 1 p D(V [p]
• ) we obtain that

deg(D(V • )) = sup p 1 p deg(D(V [p] • )) lim inf n∈N, Vn ={0} n→+∞ dim k (V n ) n .
Therefore the equality in (4.8) holds.

If p is a positive integer such that Rat(X) = k(V p ), then V p admits an element s which is transcendental over k. In particular, the graded linear series V [p] • contains a polynomial ring of one variable, which shows that

lim inf n→+∞ dim k (V n ) n > 0.

Arithmetic surface over a trivially valued field

In this section, we fix a commutative field k and we denote by | . | the trivial absolute value on k. Let X be a regular projective curve over Spec k. We denote by X an the Berkovich topological space associated with X. Recall that, as a set X an consists of couples of the form ξ = (x, | . | ξ ), where x is a scheme point of X and | . | ξ is an absolute value on the residue field κ(x) of x, which extends the trivial absolute value on k. We denote by j : X an → X the map sending any pair in X an to its first coordinate. For any ξ ∈ X an , we denote by κ(ξ) the completion of κ(j(ξ)) with respect to the absolute value | . | ξ , on which | . | ξ extends in a unique way. For any regular function f on a Zariski open subset U of X, we let |f | be the function on j -1 (U ) sending any ξ to the absolute value of f (j(ξ)) ∈ κ(j(ξ)) with respect to | . | ξ . The Berkovich topology on X an is defined as the most coarse topology making the map j and all functions of the form |f | continuous, where f is a regular function on a Zariski open subset of X.

Remark 5.1. -Let X (1) be the set of all closed points of X. The Berkovich topological space X an identifies with the tree T (X (1) ), where (a) the root point η 0 of the tree T (X (1) ) corresponds to the pair consisting of the generic point η of X and the trivial absolute value on the field of rational functions on X, (b) for any x ∈ X (1) , the leaf point x 0 ∈ T (X (1) ) corresponds to the closed point x of X together with the trivial absolute value on the residue field κ(x), (c) for any x ∈ X (1) , any ξ ∈ ]η 0 , x 0 [ corresponds to the pair consisting of the generic point η of X and the absolute value e -t(ξ) ordx( . ) , with ord x ( . ) being the discrete valuation on the field of rational functions Rat(X) corresponding to x.

5.1. Metrised divisors. -We call metrised R-divisor on X any copy (D, g), where D is an R-divisor on X and g is a Green function on T (X (1) ) such that µ x (g) = ord x (D) for any x ∈ X (1)

(see §3.2). If in addition D is a Q-divisor (resp. divisor), we say that D is a metrised Q-divisor (resp. metrised divisor ). If (D, g) is a metrised R-divisor on X and a is a real number, then (aD, ag) is also a metrised R-divisor, denoted by a(D, g). Moreover, if (D 1 , g 1 ) and (D 2 , g 2 ) are two metrised R-divisors on X, then (D 1 + D 2 , g 1 + g 2 ) is also a metrised R-divisor, denoted by (D 1 , g 1 ) + (D 2 , g 2 ).
The set Div R (X) of all metrised R-divisors on X then forms a vector space over R.

If (D, g) is a metrised R-divisor on X, we say that g is a Green function of the R-divisor D.

Remark 5.2. -(1) Let (D, g) be a metrised R-divisor on X. Note that the R-divisor part D is uniquely determined by the Green function g. Therefore the study of metrised R-divisors on X is equivalent to that of Green functions on the infinite tree T (X (1) ). The notation of pair (D, g) facilites however the presentation on the study of metrised linear series of (D, g). (2) Let D be an R-divisor on X, there is a unique canonical Green function on T (X (1) ) (see Definition 3.5), denoted by g D , such that (D, g D ) is an metrised R-divisor. Note that, for any metrised R-divisor (D, g) which admits D as its underlying R-divisor, one has g D = g can (see Definition 3.5). In particular, if (D, g) is a metrised R-divisor such that D is the zero R-divisor, then the Green function g is bounded.

Definition 5.3. -Let Rat(X) be the field of rational functions on X and Rat(X) × R be the R-vector space Rat(X) × ⊗ Z R. For any φ in Rat(X) × R , the couple ((φ), g (φ) ) is called the principal metrised R-divisor associated with φ and is denoted by (φ).

φ g := exp -inf ξ∈T (X (1) ) (g (φ) + g)(ξ) .
By convention, 0 g is defined to be zero.

5.2. Ultrametrically normed vector spaces.

-Let E be a finitedimensional vector space over k (equipped with the trivial absolute value). By ultrametric norm on E, we mean a map . : E → R 0 such that (1) for any x ∈ E, x = 0 if and only if x = 0, (2) ax = x for any x ∈ E and a ∈ k \ {0}, (3) for any (x, y) ∈ E × E, x + y max{ x , y }. Let r be the rank of E over k. We define the determinant norm associated with . the norm . det on det(E) = Λ r (E) such that

∀ η ∈ det(E), η = inf (s 1 ,...,sr)∈E r s 1 ∧...∧sr=η s 1 • • • s r .
We define the Arakelov degree of (E, . ) as

(5.2) deg(E, . ) = -ln η det ,
where η is a non-zero element of det(E). We then define the positive Arakelov degree as deg + (E, . ) := sup

F ⊂E deg(F, . F ),
where F runs over the set of all vector subspaces of E, and . F denotes the restriction of . to F .

Example 5.5. -Let (D, g) be a metrised R-divisor on X. Note that the restriction of . g to H 0 (D) defines an ultrametric norm on the k-vector space H 0 (D).

Assume that (E, . ) is a non-zero finite-dimensional ultrametrically normed vector space over k. We introduce a Borel probability measure P (E, . ) on R such that, for any t ∈ R,

P (E, . ) (]t, +∞[) = dim k ({s ∈ E : s < e -t }) dim k (E) .
Then, for any random variable Z which follows P (E, . ) as its probability law, one has

(5.3) deg(E, . ) dim k (E) = E[Z] = R t P (E, . ) (dt)

and

(5.4)

deg + (E, . ) dim k (E) = E[max(Z, 0)] = +∞ 0 t P (E, . ) (dt). 5.3. Essential infima. -Let (D, g) be a metrised R-divisor on X such that Γ(D) × R is not empty. We define λ ess (D, g) := sup φ∈Γ(D) × R inf ξ∈X an (g (φ) + g)(ξ),
called the essential infimum of the metrised R-divisor (D, g). By (5.1), we can also write λ ess (D, g) as

sup φ∈Γ(D) × R -ln φ g . Proposition 5.6. -Let (D 1 , g 1 ) and (D 2 , g 2 ) be metrised R-divisors such that Γ(D 1 ) × R and Γ(D 2 ) × R are non-empty. Then one has (5.5) λ ess (D 1 + D 2 , g 1 + g 2 ) λ ess (D 1 , g 1 ) + λ ess (D 2 , g 2 ).
Proof. -Let φ 1 and φ 2 be elements of Γ(D 1 ) × R respectively. One has

φ 1 φ 2 ∈ Γ(D 1 + D 2 ) × R . Moreover, g (φ 1 φ 2 ) = g (φ 1 ) + g (φ 2 ) . Therefore g (φ 1 φ 2 ) + (g 1 + g 2 ) = (g (φ 1 ) + g 1 ) + (g φ 2 + g 2 ),
which leads to

inf ξ∈X an g (φ 1 ) + g 1 (ξ) + inf ξ∈X an g (φ 2 ) + g 2 (ξ) inf ξ∈X an g (φ 1 φ 2 ) + (g 1 + g 2 ) (ξ) λ ess (D 1 + D 2 , g 1 + g 2 ).
Taking the supremum with respect to φ 1 ∈ Γ(D 1 ) × R and φ 2 ∈ Γ(D 2 ) × R , we obtain the inequality (5.5).

Remark 5.7. -In the literature, the essential infimum of height function is studied in the number field setting. We can consider its analogue in the setting of Arakelov geometry over a trivially valued field. For any closed point x of X, we define the height of x with respect to (D, g) as

h (D,g) (x) := ϕ g (x 0 ),
where ϕ g = g -g can is the bounded Green function associated with g (see Definition 3.5), and x 0 denotes the point of X an corresponding to the closed point

x equipped with the trivial absolute value on its residue field. In particular, for any element x ∈ X (1) outside of the support of D, one has

h (D,g) (x) = g(x 0 ).
Then the essential infimum of the height function h (D,g) is defined as

µ ess (D, g) := sup Z X inf x∈X (1) \Z h (D,g) (x),
where Z runs over the set of closed subschemes of X which are different from X (namely a finite subset of X (1) ). If Γ(D) × R is not empty, one has λ ess (D, g) sup

φ∈Γ(D) × R inf x∈X (1)
(g (φ) + g)(x 0 ).

For each φ ∈ Γ(D) × R , one has inf x∈X (1) (g (φ) + g(x 0 )) inf

x∈X (1) \(Supp(D)∪Supp((φ)))

g(x 0 ), which is clearly bounded from above by µ ess (D, g). Therefore, one has (5.6) λ ess (D, g) µ ess (D, g).

The following proposition implies that λ ess (D, g) is actually finite.

Proposition 5.8. -Let (D, g) be a metrised R-divisor on X. One has µ ess (D, g) = g(η 0 ), where η 0 denotes the point of X an corresponding to the generic point of X equipped with the trivial absolute value on its residue field.

Proof. -Let α be a real number that is > g(η 0 ). The set {ξ ∈ X an : g(ξ) < α} is an open subset of X an containing η 0 and hence there exists a finite subset S of X (1) such that g(x 0 ) < α for any x ∈ X (1) \ S. Therefore we obtain µ ess (D, g) α. Since α > g(η 0 ) is arbitrary, we get µ ess (D, g) g(η 0 ). Conversely, if β is a real number such that β < g(η 0 ), then {ξ ∈ X an : g(ξ) > β} is an open subset of X an containing η 0 . Hence there exists a finite subset S of X (1) such that g(x 0 ) > β for any x ∈ X (1) \ S . Hence µ ess (D, g) β. Since β < g(η 0 ) is arbitrary, we obtain µ ess (D, g) g(η 0 ).

Lemma 5.9. -Let r ∈ N 1 and s 1 , . . . , s r be elements of Rat(X) × Q and a 1 , . . . , a r be real numbers which are linearly independent over Q. Let s :=

s a 1 1 • • • s ar r ∈ Rat(X) × R .
Then for any i ∈ {1, . . . , r} one has Supp((s i )) ⊂ Supp((s)).

Proof. -Let x be a closed point of X which does not lie in the support of (s). One has r i=1 ord x (s i )a i = 0 and hence ord x (s 1 ) = . . . = ord x (s r ) = 0 since a 1 , . . . , a r are linearly independent over Q.

Lemma 5.10. -Let n and r be two positive integers, 1 , . . . , n be linear forms on R r of the form

j (y) = b j,1 y 1 + • • • + b j,
r y r , where (b j,1 , . . . , b j,r ) ∈ Q r and q 1 , . . . , q n be non-negative real numbers. Let a = (a 1 , . . . , a r ) be an element of R r >0 which forms a linearly independent family over Q, and such that j (a)+ q j 0 for any j ∈ {1, . . . , n}. Then, for any ε > 0, there exists a sequence

δ (m) = (δ (m) 1 , . . . , δ (m) r ), m ∈ N in R r >0
, which converges to (0, . . . , 0) and verifies the following conditions: (1) for any j ∈ {1, . . . , n}, one has j (δ (m) ) + εq j 0, (2) for any m ∈ N and any i ∈ {1, . . . , r}, one has δ

(m) i + a i ∈ Q.
Proof. -Without loss of generality, we may assume that q 1 = • • • = q d = 0 and min{q d+1 , . . . , q n } > 0. Since a 1 , . . . , a r are linearly independent over Q, for j ∈ {1, . . . , d}, one has j (a) > 0. Hence there exists an open convex cone C in R r >0 which contains a, such that j (y) > 0 for any y ∈ C and j ∈ {1, . . . , d}. Moreover, if we denote by . sup the norm on R r (where R is equipped with its usual absolute value) defined as (y 1 , . . . , y r ) sup := max{|y 1 |, . . . , |y r |}, then there exists λ > 0 such that, for any z ∈ C such that z sup < λ and any j ∈ {d + 1, . . . , n}, one has j (z) + εq j 0. Let

C λ = {y ∈ C : y sup < λ}.
It is a convex open subset of R r . For any y ∈ C λ and any j ∈ {1, . . . , n}, one has j (y) + εq j 0. Since C λ is open and convex, also is its translation by -a. Note that the set of rational points in a convex open subset of R r is dense in the convex open subset. Therefore, the set of all points δ ∈ C λ such that δ + a ∈ Q r is dense in C λ . Since (0, . . . , 0) lies on the boundary of C λ , it can be approximated by a sequence (δ (m) ) m∈N of elements in C λ such that δ (m) + a ∈ Q r for any m ∈ N.

Remark 5.11. -We keep the notation and hypotheses of Lemma 5.10. For any m ∈ N, and j ∈ {1, . . . , n} one has j (a + δ (m) ) + (1 + ε)q j 0, or equivalently,

j 1 1 + ε (a + δ (m) ) + q j 0.
Therefore, one can find a sequence (a (p) ) p∈N of elements in Q r which converges to a and such that j (a (p) ) + q j 0 hods for any j ∈ {1, . . . , n} and any p ∈ N. Proposition 5.12. -Let (D, g) be an arithmetic R-divisor on X such that

Γ(D) × Q = ∅. One has λ ess (D, g) = sup φ∈Γ(D) × Q inf ξ∈X an (g (φ) + g)(ξ) = sup φ∈Γ(D) × Q -ln φ g = sup n∈N, n 1 1 n sup s∈H 0 (nD)\{0}
-ln s ng .

(

Proof.

-By definition one has

Γ(D) × Q = n∈N, n 1 {s 1 n : s ∈ H 0 (nD) \ {0}}.
Moreover, for φ ∈ Γ(D) × Q , one has inf ξ∈X an (g (φ) + g)(ξ) = -ln φ g . Therefore the second and third equalities of (5.7) hold. To show the first equality, we denote temporarily by λ Q,ess (D, g) the second term of (5.7).

Let a be an arbitrary positive rational number. The correspondance

Γ(D) × Q → Γ(aD) × Q given by φ → φ a is a bijection. Moreover, for φ ∈ Γ(D) × Q one has φ a ag = φ a g . Hence the equality (5.8) λ Q,ess (aD, ag) = aλ Q,ess (D, g) holds.
By our assumption, we can choose φ ∈ Γ(D) × Q . For K ∈ {Q, R}, the map

α ψ : Γ(D) × K -→ Γ(D + (ψ)) × K , φ -→ φψ -1 is a bijection. Moreover, for any φ ∈ Γ(D) × K , φ g = α ψ (φ) g+g (ψ) .
Hence one has λ Q,ess (D, g) = λ Q,ess (D + (ψ), g + g (ψ) ), (5.9) λ ess (D, g) = λ ess (D + (ψ), g + g (ψ) ). (5.12) Therefore, to prove the proposition, we may assume without loss of generality that D is effective and ϕ g 0.

By definition one has λ Q,ess (D, g) λ ess (D, g). To show the converse inequality, it suffices to prove that, for any s ∈ Γ(D) × R , one has -ln s g λ Q,ess (D, g).

We choose s 1 , . . . , s r in Rat(X) × Q and a 1 , . . . , a r in R >0 such that a 1 , . . . , a r are linearly independent over Q and that s = s a 1 1 • • • s ar r . By Lemma 5.9, for any i ∈ {1, . . . , r}, the support of (s i ) is contained in that of (s). Assume that Supp((s)) = {x 1 , . . . , x n }. Since s ∈ Γ(D) × R , for j ∈ {1, . . . , n}, one has (5.13)

a 1 ord x j (s 1 ) + • • • + a r ord x j (s r ) + ord x j (D) 0.
By Lemma 5.10, for any rational number ε > 0, there exists a sequence (δ (m) 1 , . . . , δ (m) r ), m ∈ N in R r , which converges to (0, . . . , 0), and such that,

(1) for any j ∈ {1, . . . , n} and any m ∈ N, one has

δ (m) 1 ord x j (s 1 ) + • • • + δ (m)
r ord x j (s r ) + ε ord x j (D) 0.

(2) for any i ∈ {1, . . . , r} and any m ∈ N, δ

(m) i + a i ∈ Q. For any m ∈ N, let s (m) = s δ (m) 1 1 • • • s δ (m) r r ∈ Γ(εD) × R . The conditions (1) and (2) above imply that s • s (m) ∈ Γ((1 + ε)D) × Q . Hence one has inf ξ∈X an (1 + ε)g + g (s•s (m) ) (ξ) λ Q,ess ((1 + ε)D, (1 + ε)g).
Since D is effective and ϕ g 0 by s (m) ∈ Γ(εD) × R , one has εg + g (s (m) ) εϕ g 0.

Therefore we obtain

-ln s g = inf ξ∈X an (g + g (s) )(ξ) λ Q,ess ((1 + ε)D, (1 + ε)g) = (1 + ε)λ Q,ess (D, g),
where the last equality comes from (5.8). Taking the limit when ε ∈ Q >0 tends to 0, we obtain the desired inequality.

5.4. χ-volume. -Let (D, g) be a metrised R-divisor on X. We define the χ-volume of (D, g) as

vol χ (D, g) := lim sup n→+∞ deg(H 0 (nD), . ng ) n 2 /2 .
This invariant is similar to the χ-volume function in the number field setting introduced in [START_REF] Yuan | Big line bundles over arithmetic varieties[END_REF]. Note that, if deg(D) < 0, then H 0 (D) = {0}, so that

H 0 (nD) = {0} for all n ∈ Z >0 . Indeed, if f ∈ H 0 (D) \ {0}, then 0 deg(D + (f )) = deg(D) < 0, which is a contradiction. Hence vol χ (D, g) = 0.
Proposition 5.13. -Let D be an R-divisor on X, and g and g be Green functions of D. If g g , then vol χ (D, g) vol χ (D, g ).

Proof. -Note that . ng . ng on H 0 (X, nD), so that one can see that . ng,det

. ng ,det on det H 0 (X, nD). Therefore we obtain deg(H 0 (X, nD), . ng ) deg(H 0 (X, nD), . ng ) for all n 1. Thus the assertion follows. Proof. -For any n ∈ N, one has . n(g+c) = . ng+nc = e -nc . ng . Therefore, one has

deg(H 0 (nD), . n(g+c) ) = deg(H 0 (nD), . ng ) + nc dim k (H 0 (nD)).
Note that, by Proposition 4.3,

dim k (H 0 (nD)) = deg(D)n + o(n), n → +∞.
Therefore, one has

deg(H 0 (nD), . n(g+c) ) n 2 /2 = deg(H 0 (nD), . ng ) n 2 /2 +2c deg(D)+o(1), n → +∞.
Taking the superior limit when n → +∞, we obtain (5.14).

Definition 5.15. -Let (D, g) be a metrised R-divisor such that deg(D) > 0.

We denote by Γ(D, g) × R the set of s ∈ Γ(D) × R such that s g < 1. Similarly, we denote by

Γ(D, g) × Q the set of s ∈ Γ(D) × Q such that s g < 1.
For any t ∈ R such that t < λ ess (D, g), we let D g,t be the R-divisor

sup s∈Γ(D, g-t) × Q (s -1 ).
For sufficiently negative number t such that s g < e -t for any

s ∈ Γ(D) × R , one has Γ(D, g -t) × Q = Γ(D)
s ng < e -tn }T n ⊆ K[T ].
Then one has

(5.15) lim n→+∞ dim k (V t n (D, g)) n = deg(D g,t ) > 0.
Proof. -By Proposition 4.10, it suffices to show that the graded linear series V t • (D, g) is birational (see Definition 4.6). As deg(D) > 0, there exists m ∈ N 1 such that k(H 0 (mD)) = Rat(X) (see Example 4.7 and Proposition 4.8). Note that the norm . mg is a bounded function on H 0 (mD). In fact, if (s i ) rm i=1 is a basis of H 0 (kmD), as the norm . mg is ultrametric, for any (λ i ) rm i=1 ∈ k rm , one has λ 1 s 1 + • • • + λ rm s rm mg max i∈{1,...,rm} s i mg .

We choose ε > 0 such that t + ε < λ ess (D, g). By (5.7) we obtain that there exist n ∈ N 1 and s ∈ H 0 (nD) such that s ng e -n(t+ε) . Let d be a positive integer such that

d > 1 nε tm + max i∈{1,...,rm} ln s i mg .
Then, for any s ∈ H 0 (mD), one has

s d s (dn+m)g < e -(dn+m)t ,
which means that s d s ∈ V t dn+m (D, g). Therefore we obtain k(V t dn+m (D, g)) = Rat(X) since it contains k(H 0 (mD)). The graded linear series V t

• (D, g) is thus birational and (5.15) is proved.

Theorem 5.17. -Let (D, g) be a metrised R-divisor such that deg(D) > 0. Let P (D,g) be the Borel probability measure on R such that (5. [START_REF] Chinburg | The capacity pairing[END_REF])

P (D,g) (]t, +∞[) = deg(D g,t )
for t < λ ess (D, g) and P (D,g) (]t, +∞[) = 0 for t λ ess (D, g). Then one has

(5.17)

vol χ (D, g) 2 deg(D) = R t P (D,g) (dt).
Proof. -For any n ∈ N, let P n be the Borel probability measure on R such that

P n (]t, +∞[) = dim k (V t n (D, g)) dim k (H 0 (nD))
for t < λ ess (D, g) and P n (]t, +∞[) = 0 for t λ ess (D, g). By Propositions 5.16 and 4.3, one has

∀ t ∈ R, lim n→+∞ P n (]t, +∞[) = P (D,g) (]t, +∞[).
Therefore the sequence of probability measures (P n ) n∈N converges weakly to P. Moreover, if we write g as g D + f , where f is a continuous function on X an , then the supports of the probability measures

P n are contained in [inf f, g(η)].
Therefore one has

lim n→+∞ R t P n (dt) = R t P (D,g) (dt).
By (5.3), for any n ∈ N 1 such that H 0 (nD) = {0}, one has

R t P n (dt) = deg(H 0 (nD), . ng ) dim k (H 0 (nD)) .
Therefore we obtain (5.17 For any t < g(η 0 ) one has

P (D,g) (]G (D,g) (λ), +∞[) = deg(D g,G (D,g) (λ) ) deg(D) ,
namely, the probability measure P (D,g) coincides with the direct image of the uniform distribution on [0, deg(D)] by the map G (D,g) .

Proposition 5.20. -Let (D, g) be a metrised R-divisor such that deg(D) > 0. For any t ∈ R such that t < λ ess (D, g), one has

(5.18) D g,t = sup s∈Γ(D, g-t) × R (s -1 ). Proof. -Since deg(D) > 0, the set Γ(D) × Q is not empty. Let φ ∈ Γ(D) × Q and (D , g ) = (D, g) + (φ)
. By (5.9), one has λ ess (D, g) = λ ess (D , g ).

Moreover, the correspondance

s → s • φ -1 defines a bijection from Γ(D, g -t) × K to Γ(D , g -t) × K for K = Q or R.
Therefore, without loss of generality, we may assume that D is effective. Moreover, by replacing g by g -t and t by 0 we may assume that λ ess (D, g) > 0 and t = 0.

It suffices to check that D g,0 (s -1 ) for any s ∈ Γ(D, g) × R . We write s as s a 1 1 • • • s ar r , where s 1 , . . . , s r are elements of Rat(X) × Q , and a 1 , . . . , a r are positive real numbers which are linearly independent over Q. Assume that Supp((s)) = {x 1 , . . . , x n }. By Lemma 5.9, for any i ∈ {1, . . . , r}, the support of (s i ) is contained in {x 1 , . . . , x n }. For any j ∈ {1, . . . , n}, one has For any m ∈ N, let

ord x j (D) + r i=1 ord x j (s i )a i 0.
s (m) = s a (m) 1 1 • • • s a (m) r r
. (5.20) and the fact that s g < 1, for sufficiently positive m, one has s (m) g < 1 and hence D g,0 ((s (m) ) -1 ). By taking the limit when m → +∞, we obtain D g,0 (s -1 ).

By (5.19) one has s (m) ∈ Γ(D) × Q . Moreover, by
vol χ (D 1 + D 2 , g 1 + g 2 ) deg(D 1 ) + deg(D 2 ) vol χ (D 1 , g 1 ) deg(D 1 ) + vol χ (D 2 , g 2 ) deg(D 2 )
Proof. -Let t 1 and t 2 be real numbers such that t 1 < λ ess (D 1 , g 2 ) and

t 2 < λ ess (D 2 , g 2 ). For all s 1 ∈ Γ(D 1 , g 1 -t 1 ) × R and s 2 ∈ Γ(D 2 , g 2 -t 2 ) × R one has s 1 s 2 ∈ Γ(D 1 + D 2 , g 1 + g 2 -t 1 -t 2 ) × R .
Therefore, by Proposition 5.20 one has (5.21)

(D 1 + D 2 ) g 1 +g 2 ,t 1 +t 2 (D 1 ) g 1 ,t 1 + (D 2 ) g 2 ,t 2 .
As a consequence, for any

(λ 1 , λ 2 ) ∈ [0, deg(D 1 )] × [0, deg(D 2 )], one has (5.22) G (D 1 +D 2 ,g 1 +g 2 ) (λ 1 + λ 2 ) G (D 1 ,g 1 ) (λ 1 ) + G (D 2 ,g 2 ) (λ 2 ).
Let U be a random variable which follows the uniform distribution on . By Theorem 5.17 one has

vol χ (D 1 + D 2 , g 1 + g 2 ) 2(deg(D 1 ) + deg(D 2 )) = E[G (D 1 +D 2 ,g 1 +g 2 ) (U + f (U ))] since U + f (U ) follows the uniform distribution on [0, deg(D 1 ) + deg(D 2 )]
. By (5.22) we obtain

vol χ (D 1 + D 2 , g 1 + g 2 ) 2(deg(D 1 ) + deg(D 2 )) E[G (D 1 ,g 1 ) (U )] + E[G (D 2 ,g 2 ) (f (U ))] vol χ (D 1 , g 1 ) 2 deg(D 1 ) + vol χ (D 2 , g 2 ) 2 deg(D 2 ) .
The theorem is thus proved.

Finally let us consider other properties of vol χ ( . ). Note that vol χ (D, g D ) = 0, so that the assertion follows from Propositions 5.13 and 5.14.

(2) If we set c = ϕ g -ϕ g sup , then g -c g g + c, so that (2) follows from Propositions 5.13 and 5.14.

(3) is a consequence of (1). Proposition 5.24. -Let V be a finite-dimensional vector subspace of Div R (X). Then vol χ ( . ) is continuous on V .

Proof. -We denote by V + the subset of (D, g) such that deg(D) > 0. The function V + → R given by (D, g) → vol χ (D, g)/ deg(D) is concave by Corollary 5.21 and Theorem 5.22, and hence it is continuous on V + .

We fix (D, g) ∈ V . If deg(D) < 0, then there exists a neighbourhood U of (D, g) in V such that deg(D ) < 0 for any (D , g ) ∈ U . Hence vol χ ( . ) vanishes on U . If deg(D) > 0, then the above observation shows the continuity at (D, g), so that we may assume that deg(D) = 0. Then, by (3), vol χ (D, g) = 0. Therefore it is sufficient to show that lim (ε 1,n ,...,εr,n)→(0,...,0)

vol χ (ε 1,n (D 1 , g 1 ) + • • • + ε r,n (D r , g r ) + (D, g)) = 0,
where (D 1 , g 1 ), . . . , (D r , g r ) ∈ V . By using [START_REF] Baker | Harmonic analysis on metrized graphs[END_REF],

| vol χ (ε 1,n (D 1 , g 1 ) + • • • + ε r,n (D r , g r ) + (D, g))| 2 ε 1,n ϕ g 1 + • • • + ε r,n ϕ gr + ϕ g sup deg(ε 1,n D 1 + • • • + ε r,n D r + D).
On the other hand, note that note that

     lim (ε 1,n ,...,εr,n)→(0,...,0) ε 1,n ϕ g 1 + • • • + ε r,n ϕ gr + ϕ g sup = ϕ g sup , lim (ε 1,n ,...,εr,n)→(0,...,0) deg(ε 1,n D 1 + • • • + ε r,n D r + D) = deg(D) = 0.
Thus the assertion follows.

5.5. Volume function. -Let (D, g) be a metrised R-divisor on X. We define the volume of (D, g) as vol(D, g) := lim sup

n→+∞ deg + (nD, ng) n 2 /2 .
Note that this function is analogous to the arithmetic volume function introduced in [START_REF] Moriwaki | Continuity of volumes on arithmetic varieties[END_REF]. Proof. -We keep the notation introduced in the proof of Theorem 5.17. By (5.4), for any n ∈ N 1 one has

deg + (H 0 (nD), . ng ) dim k (H 0 (nD)) = R max{t, 0} P n (dt).
By passing to limit when n → +∞, we obtain the first equality. The second equality comes from the first one and (5.16) by integration by part.

Positivity

The purpose of this section is to discuss several positivity conditions of metrised R-divisors. We fix in this section a field k equipped with the trivial absolute value | . | and a regular integral projective curve X sur Spec k.

6.1. Bigness and pseudo-effectivity. -Let (D, g) be a metrised Rdivisor on X. If vol(D, g) > 0, we say that (D, g) is big; if for any big metrised R-divisor (D 0 , g 0 ) on X, the metrised R-divisor

(D + D 0 , g + g 0 ) is big, we say that (D, g) is pseudo-effective. Remark 6.1. -Let (D, g) be a metrised R-divisor. Let n ∈ N, n 1. If H 0 (nD) = {0}, then Γ(D) ×
Q is not empty. Moreover, for any non-zero element s ∈ H 0 (nD), one has -ln s g nλ ess (D, g) by (5.7), (5.6) and Proposition 5.8. In particular, one has

deg + (H 0 (nD), . ng ) n max{λ ess (D, g), 0} dim k (H 0 (nD)).
Therefore, if vol(D, g) > 0, then one has deg(D) > 0 and λ ess (D, g) > 0. Moreover, in the case where (D, g) is big, one has (6.1) vol(D, g) 2 deg(D) λ ess (D, g).

Proposition 6.2. -Let (D, g) be a metrised divisor on X. The following assertions are equivalent.

(1) (D, g) is big.

( We then prove the second assertion. Since (D 1 , g 1 ) and (D 2 , g 2 ) are big, one has deg(D

1 ) > 0, deg(D 2 ) > 0, λ ess (D 1 , g 1 ) > 0, λ ess (D 2 , g 2 ) > 0. Therefore, deg(D 1 + D 2 ) = deg(D 1 ) + deg(D 2 ) > 0.
Moreover, by (5.5) one has

λ ess (D 1 + D 2 , g 1 + g 2 ) λ ess (D 1 , g 1 ) + λ ess (D 2 , g 2 ) > 0. Therefore (D 1 + D 2 , g 1 + g 2 ) is big. Corollary 6.4. -Let (D, g) be a metrised R-divisor on X such that deg(D) > 0. Then (D, g) is pseudo-effective if and only if λ ess (D, g) 0.
Proof. -Suppose that (D, g) is pseudo-effective. Since deg(D) > 0, by (5.12) there exists c > 0 such that λ ess (D, g + c) > 0 (and thus (D, g + c) is big by Proposition 6.2). Hence for any ε ∈ ]0, 1[,

(1 -ε)(D, g) + ε(D, g + c) = (1 -ε) (D, g) + ε 1 -ε (D, g + c)
is big. Therefore,

λ ess (1 -ε)(D, g) + ε(D, g + c) = λ ess (D, g + εc) = λ ess (D, g) + εc > 0.
Since ε ∈ ]0, 1[ is arbitrary, we obtain λ ess (D, g) 0.

In the following, we assume that λ ess (D, g) 0 and we prove that (D, g) is pseudo-effective. For any big metrised R-divisor (D 1 , g 1 ) one has

deg(D + D 1 ) = deg(D) + deg(D 1 ) > 0
and, by (5.5),

λ ess (D + D 1 , g + g 1 ) λ ess (D, g) + λ ess (D 1 , g 1 ) > 0.
Therefore (D + D 1 , g + g 1 ) is big. Proposition 6.5. -Let (D, g) be a metrised R-divisor on X which is pseudoeffective. Then one has deg(D) 0 and g(η 0 ) 0.

Proof. -Let (D 1 , g 1 ) be a big metrised R-divisor. For any ε > 0, the metrised R-divisor (D + εD 1 , g + εg 1 ) is big. Therefore, by Proposition 6.2, one has

deg(D + εD 1 ) = deg(D) + ε deg(D 1 ) > 0.
Moreover, by Proposition 6.2, the inequality (5.6) and Proposition 5.8, one has g(η 0 ) + εg 1 (η 0 ) λ ess (D + εD 1 , g + εg 1 ) > 0.

Since ε > 0 is arbitrary, we obtain deg(D) 0 and g(η 0 ) 0.

6.2. Criteria of effectivity up to R-linear equivalence. -Let (D, g) be a metrised R-divisor on X. We say that (D, g) is effective if D is effective and g is a non-negative function. We say that two metrised R-divisor are R-linear equivalent if there exists an element ϕ ∈ Rat(X

) × R such that (D 2 , g 2 ) = (D 1 , g 1 ) + (ϕ).
By Proposition 6.2, if (D, g) is big, then it is R-linearly equivalent to an effective metrised R-divisor. Definition 6.6. -Let (D, g) be a metrised R-divisor on X. We denote by µ inf (g) the value x∈X (1)

µ inf,x (g)[k(x) : k] ∈ R ∪ {-∞},
where by definition (see §3.5)

µ inf,x (g) = inf ξ∈]η 0 ,x 0 [ g(ξ) t(ξ)
.

Note that

µ inf,x (g) lim ξ→x 0 g(ξ) t(ξ) = ord x (D).
Therefore, (6.2) µ inf (g)

x∈X (1) ord

x (D)[k(x) : k] = deg(D).
Moreover, if D 1 is an R-divisor and g D 1 is the canonical Green function associated with D 1 , then one has

(6.3) ∀ x ∈ X (1) , µ inf,x (g + g D 1 ) = µ inf,x (g) + ord x (D 1 )
and hence (6.4)

µ inf (g + g D 1 ) = µ inf (g) + deg(D 1 ).
The invariant µ inf ( . ) is closely related to the effectivity of a metrised Rdivisor. Proposition 6.7. -Let (D, g) be a metrised R-divisor. Assume that there exists an element φ ∈ Γ(D) × R such that g + g (φ) 0. Then for all but a finite number of x ∈ X (1) one has µ inf,x (g) = 0. Moreover, µ inf (g) 0.

Proof. -By (6.3) , for any x ∈ X (1) one has µ inf,x (g + g (φ) ) = µ inf,x (g) + ord x (φ).

Therefore, for all but a finite number of x ∈ X (1) , one has

µ inf,x (g) = µ inf,x (g + g (φ) ) 0.
Note that µ inf,x (g) ord x (D) for any x ∈ X (1) , and hence µ inf,x (g) 0 for x ∈ X (1) \ Supp(D). We then deduce that µ inf,x (g) vanishes for all but finitely many x ∈ X (1) . Moreover, by (6.4) one has

µ inf (g) = µ inf (g + g (φ) ) 0. Proposition 6.8. -Let (D, g) be a metrised R-divisor on X.
(1) (D, g) is R-linearly equivalent to an effective metrised R-divisor if and only if there exists s ∈ Γ(D) × R with s g 1.

(2) If (D, g) is R-linearly equivalent to an effective metrised R-divisor, then (D, g) is pseudo-effective.

(3) Assume that µ inf,x (g) 0 for all but finitely many x ∈ X (1) and µ inf (g) > 0, then (D, g) is R-linearly equivalent to an effective metrised R-divisor. (4) Assume that µ inf,x (g) 0 for all but finitely many x ∈ X (1) , and µ inf (g) = 0, then (D, g) is R-linearly equivalent to an effective metrised R-divisor if and only if the R-divisor x∈X (1) µ inf,x (g)x is principal.

Proof. -(1) Let s be an element of Γ(D) × R , one has (D, g) + (s) = (D + (s), g (s) + g).

By definition, D + (s) is effective. Moreover, -ln s g = inf(g (s) + g).

Therefore, s g 1 if and only if g (s) + g 0.

(2) Since there exists s ∈ Γ(D) × R such that s g 1, one has λ ess (D, g) 0 and deg(D) 0. Let (D 1 , g 1 ) be a big metrised R-divisor. By Proposition 6.

λ ess (D + D 1 , g + g 1 ) λ ess (D, g) + λ ess (D 1 , g 1 ) > 0.
Still by Proposition 6.2, we obtain that (D + D 1 , g + g 1 ) is big.

(3) Let S be a finite subset of X (1) which contains Supp(D) and all x ∈ X (1) such that µ inf,x (g) < 0, and which satisfies the inequality

x∈S µ inf,x (g)[k(x) : k] > 0.
Since the R-divisor x∈S µ inf,x (g)x has a positive degree, there exists an element ϕ of Rat(X) × R such that (6.5)

ord x (ϕ) -µ inf,x (g), if x ∈ S, 0, if x ∈ X (1) \ S.
Note that µ inf,x (g) ord x (D) for any x ∈ X (1) . Hence ϕ ∈ Γ(D) × R . Moreover, by (6.5) one has g + g (ϕ) 0.

Hence (D, g) + (ϕ) is effective.

(4) Note that µ inf,x (g) ord x (D) = 0 for any x ∈ X (1) \ Supp(D), we obtain that µ inf,x (g) = 0 for all but finitely many x ∈ X (1) . Therefore x∈X (1) µ inf,x (g)x is well-defined as an R-divisor on X.

Taking the limit when m → +∞, we obtain ln |s|(ξ) -ln s g g 0 (ξ).

The proposition is thus proved. Proposition 6.13. -Let D be an R-divisor on X such that Γ(D) × R is not empty. For any Green function g of D, the function g extends on X an to a convex Green function of D which is bounded from above by g.

Proof.

-We first show that g is bounded from above by g. For any s ∈ Γ(D) × R one has ∀ ξ ∈ X an , -ln s g = inf(g (s) + g) g(ξ) -ln |s|(ξ), so that ∀ ξ ∈ X an , ln |s|(ξ) -ln s g g(ξ). It remains to check that g extends by continuity to a convex Green function of D.

We first treat the case where deg(D) = 0. By Remark 4.5 we obtain that Γ(D) × R contains a unique element s and one has D = -(s). Therefore g = ln |s| -ln s g = g D -ln s g , which clearly extends to a convex Green function of D.

In the following, we assume that deg(D) > 0. Let x be an element of X (1) . The function g • ξ x | R >0 (see §3.1) can be written as

(t ∈ R >0 ) -→ sup s∈Γ(D) × R -t ord x (s) -ln s g ,
which is the supremum of a family of affine functions on t > 0. Therefore g • ξ x | R >0 is a convex function on R >0 . This expression also shows that, for any s ∈ Γ(D) × R , one has lim inf ξ→x 0 g(ξ) t(ξ) ord x (s -1 ).

By Proposition 4.4 (see also Remark 4.5), one has

lim inf ξ→x 0 g(ξ) t(ξ) sup s∈Γ(D) × R ord x (s -1 ) = ord x (D).
Moreover, since g g and since g is a Green function of D, one has

lim sup ξ→x 0 g(ξ) t(ξ) lim ξ→x 0 g(ξ) t(ξ) = ord x (D).
Therefore one has

lim ξ→x 0 g(ξ) t(ξ) = ord x (D).
Step 2: In the following, we assume that deg(D) > 0. By replacing g by g -g(η 0 ) it suffices to check that, in the case where g(η 0 ) = 0, the equality g(η 0 ) = 0 holds if and only if µ inf (g) 0. By definition one has

g(η 0 ) = sup s∈Γ(D) × R (-ln s g ).
Step 2.1: We first assume that g(η 0 ) = 0 and show that µ inf (g) 0. Let s be an element of Γ(D) × R . By definition one has -ln s g = inf ξ∈X an (g + g (s) )(ξ).

Let (D 1 , g 1 ) be a big metrised R-divisor. We fix s 1 ∈ Γ(D 1 ) × R such that s 1 g 1 < 1 (see Proposition 6.2 for the existence of s 1 ). Since g(η 0 ) = 0, there exists s ∈ Γ(D) × R such that

ss 1 g+g 1 s g • s 1 g 1 < 1.
Therefore λ ess (D + D 1 , g + g 1 ) > 0 and hence (D + D 1 , g + g 1 ) is big (see Proposition 6.2). We then obtain that (D, g) is pseudo-effective and hence µ inf (g) 0 (see Proposition 6.5).

Step 2.2: We now show that µ inf (g) > 0 implies g(η 0 ) = 0. For ε > 0, let

U ε := {ξ ∈ X an : g(ξ) > -ε}.
This is an open subset of X an which contains η 0 . Hence there exists a finite set X

ε of closed points of X, which contains the support of D and such that, for any closed point x of X lying outside of X [START_REF] Baker | Harmonic analysis on metrized graphs[END_REF] ε , one has g| [η 0 ,x 0 ] > -ε. Moreover, for any x ∈ X (1) \ Supp(D) one has µ inf,x (g) 0 since g is bounded on [η 0 , x 0 ]. Therefore, the condition µ inf (g) > 0 implies that (6.8)

x∈X (1) ε µ inf,x (g)[k(x) : k] > 0.
We let s ε be an element of Rat(X) × R such that ord x (s ε ) -µ inf,x (g) for any x ∈ X (1) ε and that ord x (s ε ) 0 for any x ∈ X (1) \ X [START_REF] Baker | Harmonic analysis on metrized graphs[END_REF] ε . This is possible by the inequality (6.8). In fact, the R-divisor

E = x∈X (1) ε µ inf,x (g) • x
has a positive degree, and hence Γ(E) × R is not empty. Note that µ inf,x (g) ord x (D) for any x ∈ X (1) . Therefore D + (s ε ) is effective. Moreover, for any x ∈ X (1) \ X

ε . This leads to g(η 0 ) = 0 since ε is arbitrary.

Step 2.3: We assume that µ inf (g) > 0 and show that ( g = g. By definition, for any x ∈ X (1) , the function g • ξ x | R >0 can be written as the supremum of a family of affine functions, hence it is a convex fonction on R >0 bounded from above by g. In the following, we fix a closed point x of X.

Without loss of generality, we may assume that x belongs to X

(1) ε

for any ε > 0. Note that for any ξ ∈ [η 0 , x 0 ] one has

g(ξ) ln |s ε |(ξ) -ln s ε g µ inf,x (g)t(ξ) -ε.
Since ε > 0 is arbitrary, one has g(ξ) µ inf,x (g)t(ξ).

Let a and b be real numbers such that at(ξ) + b g(ξ) for any ξ ∈ [η 0 , x 0 [. Then one has b 0 since g(η 0 ) = 0. Moreover, one has

a = lim ξ→x 0 at(ξ) + b t(ξ) lim ξ→x 0 g(ξ) t(ξ) = ord x (D).
We will show that at(ξ) + b g(ξ) for any ξ ∈ [η 0 , x 0 [. This inequality is trivial when a µ x (g) since g(ξ) µ inf,x (g)t(ξ) and b 0. In the following, we assume that a > µ x (g).

For any ε > 0, we let s a,b ε an element of Rat(X) × R such that

ord y (s a,b ε )      -a if y = x, -µ inf,y (g) if y ∈ X (1) ε , y = x, 0 if y ∈ X (1) \ X (1)
ε . This is possible since µ inf (g) > 0 and a > µ inf,x (g). Note that s a,b ε belongs to

Γ(D) × R . Moreover, for ξ ∈ [η 0 , x 0 [, one has g(ξ) -ln |s a,b ε |(ξ) g(ξ) -at(ξ) b;
for any y ∈ X

ε \ {x}, one has Step 3: In this step, we assume that deg(D) > 0 and µ inf (g -g(η 0 )) = 0. We show that and ( g = g. Without loss of generality, we assume that g(η 0 ) = 0.

g(ξ) -ln |s a,b ε |(ξ) = g(ξ) -µ inf,y (

Since deg(D) =

x∈X (1) µ x (g)[k(x) : k] > 0, there exists y ∈ X (1) such that µ inf,y (g) < ord x (D) = µ y (g).

We let g 0 be the bounded Green function on T (X (1) ) such that g 0 (ξ) = 0 for ξ ∈

x∈X (1) , x =y [ξ 0 , x 0 ], and g 0 (ξ) = min{t(ξ), 1}, for ξ ∈ [η 0 , y 0 ].

One has g 0 0, and sup ξ∈X (1) g 0 (ξ) 1.

For any ε > 0, we denote by g ε the Green function g + εg 0 . One has Note that for any s ∈ Γ(D) × R one has e ε s gε s g s gε .

µ inf,x (g ε ) > µ inf,x (
Hence we obtain g ε -ε g g ε .

Since g ε (η 0 ) = 0 for any ε > 0, we obtain g(η 0 ) = 0. Finally, the inequalities

g ε -ε g g ε leads to ( g ε -ε ( g ( g ε .
By what we have shown in Step 2.3, one has g ε = ( g ε for any ε > 0. Therefore the equality g = ( g holds. Corollary 6.17. -Let (D, g) be a metrised R-divisor on X such that Γ(D) × R = ∅. Then g is plurisubharmonic if and only if it is convex and µ inf (g -g(η 0 )) 0. 6.5. Global positivity conditions under metric positivity. -Let X be a regular projective curve over Spec k and (D, g) be a metrised R-divisor. In this section, we consider global positivity conditions under the hypothesis that g is plurisubharmonic.

Proposition 6.18. -Let (D, g) be a metrised R-divisor such that Γ(D) × R is not empty and that the Green function g is plurisubharmonic.

(1) (D, g) is pseudo-effective if and only if g(η 0 ) 0.

(2) One has λ ess (D, g) = g(η 0 ).

(3) The metrised R-divisor (D, g) is big if and only if deg(D) > 0 and g(η 0 ) > 0.

Proof. -(1) We have already seen in Proposition 6.5 that, if (D, g) is pseudoeffective, then g(η 0 ) 0. It suffices to prove that g(η 0 ) 0 implies that (D, g) is pseudo-effective. Since g is plurisubharmonic, by Theorem 6.16 one has µ inf (g) µ inf (g -g(η 0 )) 0.

By Proposition 6.11, one obtains that (D, g) is pseudo-effective (2) By (5.6) and Proposition 5.8, it suffices to prove that g(η 0 ) λ ess (D, g). In the case where deg(D) = 0, the hypotheses that Γ(D) × R is not empty and g is plurisubharmonic imply that D is a principal R-divisor, Γ(D) × R contains a unique element s with D = -(s), and g -g(η 0 ) is the canonical Green function of D (see the first step of the proof of Theorem 6.16). Therefore one has λ ess (D, g) = -ln s g = g(η 0 ).

In the following we treat the case where deg(D) > 0. Since g is plurisubharmonic, by Theorem 6.16 one has µ inf (g -g(η 0 )) 0, so that (D, g -g(η 0 )) is pseudo-effective (see Proposition 6.11). As deg(D) > 0, by Corollary 6.4 and (5.12), one has λ ess (g -g(η 0 )) = λ ess (g) -g(η 0 ) 0.

(3) follows from (2) and Proposition 6.2.

Hilbert-Samuel formula on curves

Let k be a field equipped with the trivial valuation. Let X be a regular and irreducible projective curve of genus R over k. The purpose of this section is to prove a Hilbert-Samuel formula for metrised R-divisors on X. Definition 7.1. -We identify X an with the infinite tree T (X (1) ) and consider the weight function w : X

(1) → ]0, +∞[ defined as w(x) = [k(x) : k]. If D 1 = (D 1 , g 1 )
and D 2 = (D 2 , g 2 ) are metrised R-divisors on X such that g 1 and g 2 are both pairable (see Definition 3.8) we define (D 1 • D 2 ) as the pairing g 1 , g 2 w , namely

(D 1 • D 2 ) = g 2 (η 0 ) deg(D 1 ) + g 1 (η 0 ) deg(D 2 ) - x∈X (1) [k(x) : k] +∞ 0 ϕ g 1 •ξx (t)ϕ g 2 •ξx (t) dt. (7.1) Remark 7.2. -Assume that s is an element of Rat(X) × R such that D 2 = (s) = ((s), g (s) ).
One has (see Definition 3.8)

(D 1 , D 2 ) = g 1 , g (s) w = g 1 (η 0 ) deg((s)) = 0. Theorem 7.3. -Let D = (D, g) be a metrised R-divisor on X such that Γ(D) × R = ∅ and g is plurisubharmonic. Then vol χ (D) = (D • D). Remark 7.4.
-Let g D be the canonical admissible Green function of D and ϕ g := g -g D (considered as a continuous function on X an ). Note that a plurisubharmonic Green function is convex (see Proposition 6.13). Therefore, by Proposition 3.12, one has µ inf,x (g -g(η 0 )) = g (η 0 ; x) = ord x (D) + ϕ g (η 0 ; x). Theorem 6.16 shows that (7.2) µ inf (g -g(η 0 )) = deg(D) +

x∈X (1) ϕ g (η 0 ; x)[k(x) : k] 0.

In the case where deg(D) = 0, one has g = g(η 0 ) + g D (see Step 1 in the proof of Theorem 6.16). Therefore one has

(D • D) = 2g(η 0 ) deg(D) = 0 = vol χ (D),
where the last equality comes from (3) of Proposition 5.23. Therefore, to prove Theorem 7.3, it suffices to treat the case where deg(D) > 0.

Assumption 7.5. -Let Σ be the set consisting of closed points x of X such that ϕ g is not a constant function on [η 0 , x 0 ]. Note that Σ is countable by Proposition 3.6. Here we consider additional assumptions (i) -(iv).

(i) D is a divisor. (ii) Σ is finite. (iii) ϕ g (η 0 ) = 0. (iv) µ inf (g -g(η 0 )) 0.
These assumptions actually describes a special case of the setting of the above theorem, but it is an essential case because the theorem in general is a a consequence of its assertion under these assumptions by using the continuity of vol χ ( . ). Before starting the proof of Theorem 7.3 under the above assumptions, we need to prepare several facts. For a moment, we proceed with arguments under Assumption 7.5. Let L = O X (D) and h be the continuous metric of L given by exp(-ϕ g ). For x ∈ Σ, let a x := ϕ g (η 0 ; x) and ϕ x := ϕ g•ξx .

For x ∈ Σ and n ∈ N 1 , we set a x,n = -na x . One has

a x,n -na x < a x,n + 1 and lim n→∞ a x,n n = -a x .
Moreover, as

x∈Σ a x [k(x) : k] + deg(L) > 0 by our assumptions, there exists n 0 ∈ N 1 such that 2(genus(X) -1) + x∈Σ a x,n [k(x) : k] + x∈Σ [k(x) : x] n 2(genus(X) -1) + x∈Σ [k(x) : x] n - x∈Σ a x [k(x) : k] < deg(L)
holds for any integer n n 0 , that is,

(7.3) ∀ n ∈ N n 0 , 2(genus(X) -1) + x∈Σ (a x,n + 1)[k(x) : k] < n deg(L).
We set

D n = x∈Σ (a x,n + 1)x and D x,n = D n -(a x,n + 1)x. Note that        H 0 (X, nL ⊗ O X (-D n )) = {s ∈ H 0 (X, nL) : ord x (s) a x,n + 1 (∀x ∈ Σ)}, H 0 (X, nL ⊗ O X (-D x,n -ix)) = s ∈ H 0 (X, nL) : ord y (s) a y,n + 1 (∀y ∈ Σ \ {x}) and ord x (s) i Lemma 7.6.
-For any integer n such that n 0, the following assertions hold.

(

) x∈Σ H 0 (X, nL ⊗ O X (-D x,n )) = H 0 (X, nL). 1
(2) One has

H 0 (X, nL)/H 0 (X, nL ⊗ O X (-D n )) = x∈Σ H 0 (X, nL ⊗ O X (-D x,n ))/H 0 (X, nL ⊗ O X (-D n )) Proof. -(1) Let us consider a natural homomorphism x∈Σ nL ⊗ O X (-D x,n ) → nL.
Note that the above homomorphism is surjective and the kernel is isomorphic to (nL ⊗ O X (-D n )) ⊕ card(Σ)-1 . Moreover, by Serre's duality,

H 1 (X, nL ⊗ O X (-D n )) H 0 (X, ω X ⊗ -nL ⊗ O X (D n )) ∨ and deg(ω X ⊗ -nL ⊗ O X (D n )) = 2(genus(X) -1) -n deg(L) + x∈Σ (a x,n + 1)[k(x) : k] < 0, so that H 1 (X, nL ⊗ O X (-D n )) = 0. Therefore one has (1).
(2) By (1), it is sufficient to see that if

x∈Σ s x ∈ H 0 (X, nL ⊗ O X (-D n )) and ∀ x ∈ Σ, s x ∈ H 0 (X, nL ⊗ O X (-D x,n )), then s x ∈ H 0 (X, nL ⊗ O X (-D n ))
for all x ∈ Σ. Indeed, as

∀ y ∈ Σ \ {x}, s y ∈ H 0 (X, O X (-(a x,n + 1)x)) and y∈Σ s y ∈ H 0 (X, O X (-(a x,n + 1)x)), we obtain s x ∈ H 0 (X, O X (-(a x,n + 1)x)), so that s x ∈ H 0 (X, O X (-D n )), as required. Lemma 7.7. -For x ∈ Σ and i ∈ {0, . . . , a x,n }, dim k H 0 (X, nL⊗O X (-D x,n -ix))/H 0 (X, nL⊗O X (-D x,n -(i+1)x)) = [k(x) : k].
Proof. -Let us consider an exact sequence x,i ∈ H 0 (X, nL ⊗ O X (-D x,n -ix)) \ H 0 (X, nL ⊗ O X (-D x,n -(i + 1)x)) whose classes form a basis of H 0 (X, nL ⊗ O X (-D x,n -ix))/H 0 (X, nL ⊗ O X (-D x,n -(i + 1)x)) for i = 0, . . . , a x,n . Moreover we choose a basis {t 1 , . . . , t en } of H 0 (X, nL ⊗ O X (-D n )). Then, by Lemma 7.6, ∆ n := {t 1 , . . . , t en } ∪ x∈Σ s ( )

0 → nL ⊗ O X (-D x,n -(i + 1)x) → nL ⊗ O X (-D x,n -ix) → k(x) → 0, so that, since deg(ω X ⊗ -nL ⊗ O X (D x,n + (i + 1)x)) = 2(
x,0 , . . . , s ( ) x,ax,n : ∈ {1, . . . , [k(x) : k]} forms a basis of H 0 (X, nL). Proof. -First of all, note that, for s ∈ H 0 (X, nL) \ {0} and ξ ∈ X an , -ln |s| nh (ξ) = t(ξ) ord x (s) ≥ 0 if ξ ∈ [η 0 , x 0 ] and x ∈ Σ, n t(ξ)(ord x (s)/n) + ϕ x (t(ξ)) if ξ ∈ [η 0 , x 0 ] and x ∈ Σ, so that Proof of Theorem 7.3 without additional assumptions. -First of all, note that Σ is a countable set (cf. Proposition 3.6).

Step 1: (the case where D is Cartier divisor, Σ is finite and ϕ g (η 0 ) + deg(D) > 0). By the previous observation, vol χ (D, g -g(η 0 )) = (D, g -g(η 0 )) • (D, g -g(η 0 )) . Thus the assertion follows.

Step 2: (the case where D is Cartier divisor and Σ is finite). For 0 < ε < 1, we set g ε = g can D + εϕ g . If ϕ g (η 0 ) = 0, then Σ = ∅, so that the assertion is obvious. Thus we may assume that ϕ g (η 0 ) < 0. As ϕ g (η 0 ) + deg(D) 0, we have εϕ g (η 0 ) + deg(D) > 0. Therefore, by Step 1, vol χ (D, g ε ) = (D, g ε ) • (D, g ε ) .

Thus the assertion follows by Proposition 5.24.

Step 3: (the case where D is Cartier divisor and Σ is infinite). We write Σ in the form of a sequence {x 1 , . . . , x n , . . . , }. For any n ∈ Z 1 , let g n be the Green function defined as follows:

∀ ξ ∈ X an , g n (ξ) = g D (ξ) + ϕ g (ξ) if ξ ∈ n i=1 [η 0 , x i,0 ], g(η 0 ) otherwise. Step 5: (general case). By our assumption, there are adelic Q-Cartier divisors (D 1 , g 1 ), . . . , (D r , g r ) and a 1 , . . . , a r ∈ R >0 such that D 1 , . . . , D r are semiample, g 1 , . . . , g r are plurisubharmonic, and (D, g) = a 1 (D 1 , g 1 ) + • • • + a r (D r , g r ). We choose sequences {a 1,n } ∞ n=1 , . . . , {a r,n } ∞ n=1 of positive rational numbers such that lim n→∞ a i,n = a i for i = 1, . . . , r. We set (D n , g n ) = a 1,n (D 1 , g 1 ) + . . . + a r,n (D r , g r ). Then we may assume that deg(D n ) > 0. By Step 4, then vol χ (D n , g n ) = (D n , g n ) 2 . On the other hand, by Proposition 5.24, vol χ (D, g) = lim n→∞ vol χ (D n , g n ). Moreover,

Note that lim

(D, g) • (D, g) = lim n→∞ (D n , g n ) • (D n , g n ) .
Thus the assertion follows. This inequality actually holds without plurisubharmonic condition (namely it suffices that g 1 and g 2 are pairable). In fact, by (7.1) one has In the case where (D 1 • D 2 ) and (D 2 • D 2 ) sont non-negative, by the inequality of arithmetic and geometric means, we obtain that

(D i • D i ) deg(D i ) = 2g i (η 0 ) - x∈X (1)
(D 1 • D 2 ) (D 1 • D 1 )(D 2 • D 2 ),
where the equality holds if and only if D 1 and D 2 are proportional up to Rlinear equivalence. This could be considered as an analogue of the arithmetic Hodge index inequality of Faltings [START_REF] Faltings | Calculus on arithmetic surfaces[END_REF]Theorem 4] and Hriljac [20,Theorem 3.4], see also [START_REF] Zhang | Positive line bundles on arithmetic surfaces[END_REF]Theorem 7.1] and [5, §5.5].

2. 2 .

 2 Convex function on [0, +∞]. -Let ϕ be a convex function on R >0 . Then ϕ is continuous on R >0 . Moreover, for any t ∈ R >0 , the right derivative of ϕ at t, given by lim ε↓0 ϕ(t + ε) -ϕ(t) ε , exists in R. By abuse of notation, we denote by ϕ the right derivative function of ϕ on R >0 . It is a right continuous increasing function. We refer to [2, Theorem 1.26] for more details. Moreover, for any (a, b) ∈ R 2 >0 , one has (2.1) ϕ(b) -ϕ(a) = ]a,b[

(

  g(0) = g(0).

3. 1 .

 1 Tree of length 1 associated with a set. -Let S be a non-empty set. We denote by T (S) the quotient set of the disjoint union x∈S [0, +∞] obtained by gluing the points 0 in the copies of [0, +∞]. The quotient map from x∈S [0, +∞] to T (S) is denoted by π. For each x ∈ S, we denote by ξ x : [0, +∞] → T (S) the restriction of the quotient map π to the copy of [0, +∞] indexed by x. The set T (S) is the union of ξ x ([0, +∞]), x ∈ S. Notation 3.1. -Note that the images of 0 in T (S) by all maps ξ x are the same, which we denote by η 0 . The image of +∞ by the map ξ x is denoted by x 0 . If a and b are elements of [0, +∞] such that a < b, the images of the intervals [a, b], [a, b[, ]a, b], ]a, b[ by ξ x are denoted by [ξ x (a), ξ x (b)], [ξ x (a), ξ x (b)[, ]ξ x (a), ξ x (b)], ]ξ x (a), ξ x (b)[ respectively. Definition 3.2. -We denote by t : T (S) → [0, +∞] the map which sends an element ξ ∈ ξ x ([0, +∞]) to the unique number a ∈ [0, +∞] such that ξ x (a) = ξ.

Definition 3 . 3 .

 33 -We equip T (S) with the following topology. A subset U of T (S) is open if and only if the conditions below are simultaneously satisfied:

Remark 4 . 5 .

 45 -Let D be an R-divisor on X. Note that one has sup s∈Γ(D) × R (s -1 ) D. Therefore, the above proposition implies that, if deg(D) > 0, then sup s∈Γ(D) × R (s -1 ) = D. This equality also holds when deg(D) = 0 and Γ(D) × R = ∅. In fact, if s is an element of Γ(D) × R , then one has D + (s) 0. Moreover, since deg(D) = 0, one has deg(D + (s)) = deg(D) + deg((s)) = 0 and hence D +

Definition 5 . 4 .

 54 -If (D, g) is a metrised R-divisor, for any φ ∈ Γ(D) × R , we define(5.1)

  any c ∈ R, one has λ Q,ess (D, g + c) = λ Q,ess (D, g) + c, (5.11) λ ess (D, g + c) = λ ess (D, g) + c.

Proposition 5 . 14 .

 514 -Let (D, g) be a metrised R-divisor such that deg(D) 0. For any c ∈ R, one has (5.14) vol χ (D, g + c) = 2c deg(D) + vol χ (D, g).

). Remark 5 . 18 ./ 2 .

 5182 -Theorem 5.17 and Proposition 4.3 show that the sequence defining the χ-volume function has a limit. More precisely, if (D, g) is a metrised R-divisor such that deg(D) > 0, then one has vol χ (D, g) = lim n→+∞ deg(H 0 (nD), . ng ) n 2 Definition 5.19. -Let (D, g) be a metrised R-Cartier divisor on X such that deg(D) > 0. We denote by G (D,g) : [0, deg(D)] → R the function sending u ∈ [0, deg(D)] to sup{t ∈ R <g(η) : deg(D g,t ) > u}.

By Lemma 5 .

 5 10 and Remark 5.11, there exists a sequence of vectors a (m) = (a (m) 1 , . . . , a (m) r ), m ∈ N in Q r such that (5.19) ord x j (D) + r i=1 ord x j (s i )a lim m→+∞ a (m) = (a 1 , . . . , a r ).

Corollary 5 .

 5 21. -Let (D, g) be a metrised R-Cartier divisor such that deg(D) > 0. For any a > 0 one hasdeg χ (aD, ag) = a 2 deg χ (D, g).Proof. -By Proposition 5.20 one has (aD) ag,at = aD g,t .By (5.17) one has vol χ (aD, ag) = 2 λess(D,g) -M deg((aD) ag,at ) dat + 2aM deg(D) = 2a 2 λess(D,g) -M deg(D g,t ) dt + 2a 2 M deg(D) = a 2 deg χ (D, g). Theorem 5.22. -Let (D 1 , g 1 ) and (D 2 , g 2 ) be metrised R-Cartier divisors such that deg(D 1 ) > 0 and deg(D 2 ) > 0. One has

  [0, deg(D 1 )]. Let f : [0, deg(D 1 )] → [0, deg(D 2 )] be the linear map sending u to u deg(D 2 )/ deg(D 1 )

Proposition 5 . 23 . 3 )

 5233 -Let D be an R-divisor on X such that deg(D) 0, and g and g be Green functions of D. Then one has the following: (1) 2 deg(D) min ξ∈X an {ϕ g (ξ)} vol χ (D, g) 2 deg(D) max ξ∈X an {ϕ g (ξ)}. (2) | vol χ (D, g) -vol χ (D, g )| 2 ϕ g -ϕ g sup deg(D). (If deg(D) = 0, then vol χ (D, g) = 0. Proof. -(1) If we set m = min ξ∈X an {ϕ g (ξ)} and M = max ξ∈X an {ϕ g (ξ)}, then g D + m g g D + M.

Proposition 5 . 25 .

 525 -Let (D, g) be a metrised R-divisor such that deg(D) > 0. Let P (D,g) be the Borel probability measure on R defined in Theorem 5.17. Then one has vol(D, g) 2 deg(D) = R max{t, 0} P (D,g) (dt), (5.23) vol(D, g) = +∞ 0 deg(D g,t ) dt. (5.24)

  ) deg(D) > 0 and λ ess (D, g) > 0 (3) deg(D) > 0 and there exists s ∈ Γ(D) × R such that s g < 1. (4) deg(D) > 0 and there exists s ∈ Γ(D) × Q such that s g < 1. Proof. -"(1) ⇔ (2)" We have seen in the above Remark that, if (D, g) is big, then deg(D) > 0 and λ ess (D, g) > 0. The converse comes from the equality vol(D, g) = +∞ 0 deg(D g,t ) dt. proved in Proposition 5.25. Note that the function t → deg(D g,t ) is decreasing. Moreover, by Proposition 5.16, one has deg(D g,t ) > 0 once t < λ ess (D, g). Therefore, if λ ess (D, g) > 0, then vol(D, g) > 0. "(2)⇔(3)" comes from the definition of λ ess (D, g). "(2)⇔(4)" comes from Proposition 5.12. Corollary 6.3. -(1) If (D, g) is a big metrised R-divisor on X, then, for any positive real number ε, the metrised R-divisor ε(D, g) = (εD, εg) is big. (2) If (D 1 , g 1 ) and (D 2 , g 2 ) are two metrised R-divisor on X which are big, then (D 1 + D 2 , g 1 + g 2 ) is also big. Proof. -The first assertion follows from Proposition 6.2 and the equalities deg(εD) = ε deg(D) and λ ess (ε(D, g)) = ελ ess (D, g).

  2, one has deg(D) > 0 and λ ess (D, g) > 0. Therefore, deg(D + D 1 ) = deg(D) + deg(D 1 ) > 0, and, by Proposition 5.6,

( 1 )

 1 ε and ξ ∈ [η 0 , x 0 [ one has (g -ln |s ε |)(ξ) g(ξ) -ε.Therefore we obtain s ε e ε since g -ln |s ε | 0 on [η 0 , x 0 [ for any x ∈ X

  g)t(ξ) 0;for any y ∈ X(1) \ X ε , one has g(ξ) -ln |s a,b ε |(ξ) g(ξ) -ε. Therefore we obtain -ln s a,b ε min{-ε, b}.As a consequence, for any ξ ∈ [η 0 , x 0 [, one has g(ξ) ln |s a,b ε |(ξ) -ln s g = at(ξ) + min{-ε, b}. Since b 0 and since ε > 0 is arbitrary, we obtain g(ξ) at(ξ) + b.

  g) 0. Moreover, by definition g ε (η 0 ) = 0. Therefore, by what we have shown in Step 2.2, one has g ε (η 0 ) = sup s∈Γ(D) × R -ln s gε = 0.

  genus(X) -1) -n deg(L)+ (i + 1) -(a x,n + 1) [k(x) : k] + y∈Σ (a y,n + 1)[k(y) : k] 2(genus(X) -1) -n deg(L) + y∈Σ (a y,n + 1)[k(y) : k] < 0,one has the assertion as before. By Lemma 7.7, for each x ∈ Σ, there ares ( ) x,0 , . . . , s ( ) x,ax,n ∈ H 0 (X, nL ⊗ O X (-D x,n )), ∈ {1, . . . , [k(x) : k]}such that the classes of s ( )x,0 , . . . , s ( )x,ax,n form a basis ofH 0 (X, nL ⊗ O X (-D x,n ))/H 0 (X, nL ⊗ O X (-D n ))

Lemma 7 . 8 . 2 )

 782 -(1) The equality s ( )x,i nh = exp(-nϕ * x (i/n))holds for x ∈ Σ, ∈ {1, . . . , [k(x) : k]} and i ∈ {0, . . . , a x,n }. Moreover t j nh = 1 for all j ∈ {1, . . . , e n }. (The basis ∆ n of H 0 (X, nL) is orthogonal with respect to . nh .

(7. 4 )( 1 ) 2 0 (ϕ x ) 2

 41202 s nh = max 1, max x∈Σ {exp(-nϕ * x (ord x (s)/n))} . The assertion follows from (7.4) because ϕ * x(λ) = 0 if λ -a x . (2) Fix s ∈ H 0 (X, nL) \ {0}. We set s = b 1 t 1 + • • • + b en t en + If s ∈ H 0 (X, nL ⊗ O X (-D n )), then c ( )x,i = 0 for all x, i and . Thus 1 = max j∈{1,...,en}{|b j | • t j nj } = s nh . Next we assume that s ∈ H 0 (X, nL ⊗ O X (-D n )). If we set T = {x ∈ Σ : ord x (s) a x,n }, then T = ∅ and, for x ∈ Σ and ∈ {1, . . . , [k(x) : k]}, . . . , 0) if x ∈ T . ..,en {|b j | • t j nh }, max x∈Σ, i=0,...,ax,n {|c x,i | • s x,i nh } (s) nh } = max x∈Σ {exp(-nϕ * x (ord x (s)/n))} =s nh , as required. Let us begin the proof of Theorem 7.3 under Assumption 7.5. By Lemma 7.8 together with Definition 3.8 and Proposition 2.6, lim n→∞ deg H 0 (X, nL), . nh n dt = (D • D), as required.

  On the other hand, by Proposition 5.14, one hasvol χ (D, g) = vol χ (D, g -g(η 0 )) + 2 deg(D)g(η 0 ).Moreover, by the bilinearity of the arithmetic intersection pairing, one hasD • D = (D, g -g(η 0 )) • (D, g -g(η 0 )) + 2 deg(D)g(η 0 ).

  gn (ξ) -ϕ g (ξ)| = 0. Indeed, as ϕ g is continuous at η 0 , for any ε > 0, there is an open set U of X an such that η 0 ∈ U and |ϕ g (ξ) -ϕ g (η 0 )| ε for any ξ ∈ U . Since η 0 ∈ U , one can find N such that [η 0 , x n,0 ] ⊆ U for all n N . Then, for n N ,|ϕ g (ξ) -ϕ gn (ξ)| ε if ξ ∈ [η 0 , x i,0] for some i > n, = 0 othertwise, as required. Thus, by (2) in Proposition 5.23, the assertion is a consequence of Step 2.

Step 4 :

 4 (the case where D is Q-Cartier divisor). Choose a positive integer a such that aD is Cartier divisor. Then, by Step 3, vol χ (aD) = (aD • aD) = a 2 (D • D). By Corollary 5.21, one has vol χ (aD) = a 2 vol χ (D). Hence the equality vol χ (D) = (D • D) holds.

Remark 7 . 9 .

 79 -Let D 1 = (D 1 , g 1 ) and D 2 = (D 2 , g 2 ) be adelic R-divisors such that deg(D 1 ) > 0 and deg(D 2 ) > 0. Let D = (D 1 + D 2 , g 1 + g 2 ). If g 1 and g 2 are plurisubharmoic, then Theorems 5.22 and 7.3 lead to the following inequality (7.5) (D • D) deg(D) (D 1 • D 1 ) deg(D 1 ) + (D 2 • D 2 ) deg(D 2 ) .

x∈X ( 1 )(ϕ g 1 ϕ g 1 ϕ g 2 2 +∞ 0 ϕ g 1

 1112201 [k(x) : k] deg(D i ) +∞ 0 ϕ g i •ξx (t) 2 dt for i ∈ {1, 2},and(D • D) deg(D) = 2(g 1 (η 0 ) + g 2 (η 0 )) -[k(x) : k] deg(D 1 ) + deg(D 2 ) +∞ 0 •ξx (t) + ϕ g 2 •ξx (t)) 2 dt, which leads to (deg(D 1 ) + deg(D 2 )) (D • D) •ξx (t) 2 dt + deg(D 1 ) deg(D 2 ) +∞ 0 •ξx (t) 2 dt -•ξx (t)ϕ g 2 •ξx (t) dt 0,by using Cauchy-Schwarz inequality and the inequality of arithmetic and geometric means.The inequality (7.5) leads to2(D 1 • D 2 ) deg(D 2 ) deg(D 1 ) (D 1 • D 1 ) + deg(D 1 ) deg(D 2 ) (D 2 • D 2 ).

  ,

	extends continuously on [0, +∞]. It remains to show that g(0) =	( g(0). Let
	ε > 0. The function t → (g(t) -g(0) + ε)/t is continuous on ]0, +∞] and one
	has
				lim t↓0	g(t) -g(0) + ε t	= +∞.
	Therefore this function is bounded from below by a real number α. Hence the
	function g is bounded from below on R >0 by the affine function
					t -→ αt + g(0) -ε,
	which implies that	( g(0)	g(0) -ε. Since g	( g and since ε is arbitrary, we
	obtain
	as required.
	2.4. Convex envelop of asymptotically linear functions. -Let g :
	R >0 → R be an asymptotically linear continuous function (see §2.1). We
	define the convex envelop of g as the largest convex function	( g on R >0 which
	is bounded from above by g. Note that	( g identifies with the supremum of all
	affine functions bounded from above by g.
	Proposition 2.7. -Let g : R >0 → R be an asymptotically linear contin-
	uous function. Then	( g is also an asymptotically linear continuous function.
	Moreover, one has µ(g) = µ( g) and g(0) = (	( g(0).
	Proof. -Let ϕ g : [0, +∞] → R be the continuous function such that ϕ g (t) =
	g(t) -µ(g)t on R >0 . Let M be a real number such that |ϕ g (t)| M for any
	t ∈ [0, +∞]. One has
			µ(g)t -M g(t) µ(g)t + M.
	Therefore,
			µ(g)t -M	( g(t) µ(g)t + M.
	By Proposition 2.2, the function
		ϕ	( g : R >0 → R, ϕ	( g (t) :=	( g(t) -µ(g)t

×

  Qand hence, by Proposition 4.4, D g,t = D. If t λ ess (D, g), by convention we let D g,t be the zero R-divisor.

Proposition 5.16. -Let (D, g) be a metrised R-divisor such that deg(D) > 0, and t ∈ R such that t < λ ess (D, g). Let V t • (D, g) := n∈N {s ∈ H 0 (nD) :

Acknowledgement: We are grateful to Walter Gubler and Klaus Künnemann for discussions.

Asymptotically linear functions

2.1. Asymptotic linear functions on R >0 . -We say that a continuous function g : R >0 → R is asymptotically linear (at the infinity) if there exists a

Assume that the R-divisor x∈S µ inf,x (g)x is principal, namely of the form (ϕ) for some ϕ ∈ Rat(X) × R . Then the metrised R-divisor (D, g) -(ϕ) is effective. Conversely, if φ is an element of Rat(X) × R which is different from -x∈X (1) µ inf,x (g)x, then there exists x ∈ X (1) such that ord x (φ) < -µ inf,x (g) since x∈X (1) ord x (φ)[k(x) : k] = -x∈X (1) µ inf,x (g)[k(x) : k] = 0.

Therefore the function g + g (φ) can not be non-negative.

Combining Propositions 6.7 and 6.8, we obtain the following criterion of effectivity up to R-linear equivalence for metrised R-divisors. Theorem 6.9. -Let (D, g) be a metrised R-divisor on X. Then (D, g) is Rlinearly equivalent to an effective metrised R-divisor if and only if µ inf,x (g) = 0 for all but finitely many x ∈ X (1) and if one of the following conditions holds:

6.3. Criterion of pseudo-effectivity. -By using the criteria of effectivity up to R-linear equivalence in the previous subsection, we prove a numerical criterion of pseudo-effectivity in terms of the invariant µ inf ( . ).

Lemma 6.10. -Let (D, g) be a metrised R-divisor. Assume that (D, g + ε) is pseudo-effective for any ε > 0. Then (D, g) is also pseudo-effective.

Proof. -Let (D 1 , g 1 ) be a big metrised R-divisor. By Proposition 6.2, one has deg(D 1 ) > 0 and λ ess (D 1 , g 1 ) > 0. Let ε be a positive number such that ε < λ ess (D 1 , g 1 ). By (5.12) one has

Hence (D 1 , g 1 -ε) is big (by Proposition 6.2). Therefore,

Proof. -"⇐=": For any ε > 0, one has µ inf (g + ε) > 0. By Theorem 6.9, (D, g + ε) is R-linearly equivalent to an effective metrised R-divisor, and hence is pseudo-effective (see Proposition 6.8 (2)). By Lemma 6.10, we obtain that (D, g) is pseudo-effective.

"=⇒": We begin with the case where deg(D) > 0. If (D, g) is pseudoeffective, then by Corollary 6.4, one has λ ess (D, g) 0. Hence (D, g + ε) is big for any ε > 0 (by (5.12) and Proposition 6.2). In particular, one has µ inf (g + ε)

0 for any ε > 0. For each x ∈ X (1) , the function (ε > 0) → µ inf,x (g + ε) is decreasing and bounded from below by µ inf,x (g). Moreover, for any

Therefore we obtain inf

By the monotone convergence theorem we deduce that

We now treat the general case. Let y be a closed point of X. We consider y as an R-divisor on X and denote it by D y . Let g y be the canonical Green function associated with D y . As D y is effective and g y 0, we obtain that (D y , g y ) is effective and hence pseudo-effective. Therefore, for any δ > 0,

Since δ > 0 is arbitrary, one obtains µ inf (g) 0.

6.4. Positivity of Green functions. -Let D be an R-divisor on X such that Γ(D) × R is not empty. For any Green function g of D, we define a map g :

Proposition 6.12. -Let D be an R-divisor on X such that Γ(D) × Q is not empty. For any ξ ∈ X an \ {x 0 : x ∈ X (1) } one has

Proof.

-Without loss of generality, we may assume that D is effective. For clarifying the presentation, we denote temporarily by

Let s be an element of Γ(D) × R , which is written in the form s a 1 1 • • • s ar r , where s 1 , . . . , s r are elements of Rat(X) × Q and a 1 , . . . , a r are positive real numbers, which are linearly independent over Q. Let {x 1 , . . . , x n } be the support of (s). By Lemma 5.9, for any i ∈ {1, . . . , r}, the support of (s i ) is contained in {x 1 , . . . , x n }. Since s belongs to Γ(D) × Q , for j ∈ {1, . . . , n}, one has a 1 ord x j (s 1 ) + • • • + a r ord x j (s r ) + ord x j (D) 0.

By Lemma 5.10 and Remark 5.11, there exist a sequence (ε (m) ) m∈N in Q >0 and a sequence δ (m) = (δ

>0 which satisfy the following conditions (1) the sequence (ε (m) ) m∈N converges to 0, (2) the sequence (δ (m) ) m∈N converges to (0, . . . , 0), (3) if we denote by u (m) the element

Q , and hence it belongs to Γ(D) × Q . Note that one has

The proposition is thus proved. Definition 6.14. -Let (D, g) be a metrised R-divisor on X such that Γ(D) × R is not empty. We call g the plurisubharmonic envelope of the Green function g. In the case where the equality g = g holds, we say that the Green function g is plurisubharmonic. Note that g is bounded from above by the convex envelope ( g of g.

Remark 6.15. -If we set ϕ = g -g, then ϕ is a non-negative continuous function on X an , so that, in some sense, the decomposition (D, g) = (D, g) + (0, ϕ) gives rise to a Zariski decomposition of (D, g) on X. Theorem 6.16. -Let (D, g) be an adelic R-Cartier divisor on X such that Γ(D) × R is not empty. Then g(η 0 ) = g(η 0 ) if and only if µ inf (g -g(η 0 )) 0. Moreover, in the case where these equivalent conditions are satisfied, g identifies with the convex envelop Note that g(η 0 ) = ϕ g (η 0 ). Therefore, the equality g(η 0 ) = g(η 0 ) holds if and only if ϕ g attains its minimal value at η 0 , or equivalently ∀ x ∈ X (1) , µ inf,x (g -g(η 0 )) = ord x (g).

In particular, if g(η 0 ) = g(η 0 ), then µ inf (g -g(η 0 )) =

x∈X (1) ord x (g)[k(x) : k] = 0.

Conversely, if µ inf (g -g(η 0 )) 0, then by (6.2) one obtains that µ inf (g -g(η 0 )) = 0 and the equality µ inf,x (g -g(η 0 )) = ord x (g) holds for any x ∈ X (1) . Hence g(η 0 ) = g(η 0 ). If ϕ is a bounded Green function on X an , which is bounded from above by ϕ g , by Proposition 2.2 one has ϕ(ξ) ϕ(η 0 ) ϕ g (η 0 ) = g(η 0 ) for any ξ ∈ X an . In the case where the inequality g(η 0 ) = g(η 0 ) holds, the function g = g D + g(η 0 ) is the largest convex Green function of D which is bounded from above by g, namely the equality g = ( g holds.