2.1

A scheme is integral if it is reduced and irreducible. By a variety over a field $k$ we mean an integral scheme of finite type over $k$. If $X$ is a Noetherian integral scheme, we denote by $\operatorname {\mathrm {Div}}(X)$ or $\operatorname {\mathrm {Div}}(X)_{{\mathbb {Z}}}$ the group of Cartier divisors on $X$ and by ${ \kappa }(X)$ the field of rational functions on $X$. If $\mathbb {K}$ denotes ${\mathbb {Q}}$ or ${\mathbb {R}}$, we let $\operatorname {\mathrm {Div}}(X)_{\mathbb {K}} = \operatorname {\mathrm {Div}}(X) \otimes _{\mathbb {Z}} \mathbb {K}$ and ${ \kappa }(X)_{\mathbb {K}} = {{ \kappa }(X)^{\times }} \otimes _{\mathbb {Z}} \mathbb {K}$. The elements of $\operatorname {\mathrm {Div}}(X)_{{\mathbb {K}}}$ and ${ \kappa }(X)_{{\mathbb {K}}}$ are called ${\mathbb {K}}$-Cartier divisors and ${\mathbb {K}}$-rational functions, respectively. A ${\mathbb {K}}$-rational function $\phi \in { \kappa }(X)_{{\mathbb {K}}}$ defines a ${\mathbb {K}}$-Cartier divisor $(\phi )\in \operatorname {\mathrm {Div}}(X)_{\mathbb {K}}$. We denote by $\operatorname {\mathrm {Supp}} D$ the support of a ${\mathbb {K}}$-Cartier divisor $D$ (see [Reference MoriwakiMor16, § 1.2] for details). It is a Zariski-closed subset of $X$ [Reference MoriwakiMor16, Proposition 1.2.1].

2.2

Let $K_0$ denote either the field ${\mathbb {Q}}$ of rational numbers or the field of functions $k(T)$, where $k$ is an arbitrary field. A global field is by definition a finite extension of $K_0$. It is either a number field or the field of functions of a regular projective curve $C_K$ over $k$ equipped with a finite morphism $\varphi _K \colon C_K \rightarrow C_0 = {\mathbb {P}}^{1}_k$, unique up to $k$-isomorphism.

2.3

Let $K$ be a global field. Let $\Sigma _K$ be the set of places of $K$ and $\Sigma _{K,\infty } \subset K$ the set of archimedean places. Note that $\Sigma _{K,\infty } = \emptyset$ if $K_0 = k(T)$. For each $v \in \Sigma _K$, we denote by $K_v$ the completion of $K$ at the place $v$ and by $K_{0,v}$ the completion of $K_0$ with respect to the restriction of $v$ to $K_0$. We shall normalize absolute values associated to $v$.

Assume that $K$ is a number field. For each place $v \in \Sigma _K$, we let $|\cdot |_v$ be the unique absolute value on $K_v$ extending the usual absolute value $|\cdot |_v$ on ${\mathbb {Q}}_v$: $|p|_v = p^{-1}$ if $v$ is a finite place over a prime number $p$, and $|\cdot |_v = |\cdot |$ is the usual absolute value on ${\mathbb {R}}$ if $v$ is archimedean. For each $v \in \Sigma _K$, we let $n_v(K) = [K_v:{\mathbb {Q}}_v]/[K:{\mathbb {Q}}]$. We now turn to the function field case, so that $K_0 = k(T)$. The set of places of $K$ is in one-to-one correspondence with the set of closed points of $C_K$. For each $v \in \Sigma _K$ and each $f \in K^{\times } = {\kappa }(C_K)^{\times }$, we denote by $\operatorname {\mathrm {ord}}_v(f)$ the order of $f$ in the discrete valuation ring ${\mathcal {O}}_{C_K,v}$ and by $e_v(\varphi _K)$ the ramification index of $\varphi _K\colon C_K \rightarrow {\mathbb {P}}^{1}_k$ at $v \in C_K$. We consider the absolute value $|\cdot |_v$ on $K$ given by $|f|_v = e^{-\operatorname {\mathrm {ord}}_v(f)/e_v(\varphi _K)}$ and we let

\[ n_v(K) = \frac{e_v(\varphi_K) [\kappa(v):k]}{[K:K_0]}, \]

where $\kappa (v)$ denotes the residue field of $v$ in $C_K$.

With these conventions, we have the following product formula:

(2.1)\begin{equation} \forall x \in K^{{\times}}, \sum_{v \in \Sigma_K}n_v(K)\ln |x|_v = 0. \end{equation}

2.4

Let $X$ be a scheme on $\operatorname {\mathrm {Spec}} K$. For each $v \in \Sigma _K$, let $X_v$ be the fibre product $X \times _K \operatorname {\mathrm {Spec}} K_v$. If $v$ is non-archimedean, we denote by $X_v^{\mathrm {an}}$ the analytification of $X_v$ in the sense of Berkovich (see [Reference Burgos Gil, Philippon and SombraBPS14, § 1.2] for a short definition sufficient for our purposes). When $K$ is a number field and $v$ is archimedean, we denote by $\Sigma _v \subset K({\mathbb {C}})$ the set of embeddings $\sigma \colon K \hookrightarrow {\mathbb {C}}$ associated to $v$. Hence $\Sigma _v$ is a singleton if $v$ is real and $\Sigma _v$ consists of two conjugate embeddings if $v$ is complex. For each $\sigma \in \Sigma _v$, we let $K_\sigma = K \otimes _K^{\sigma } {\mathbb {C}}$ be the tensor product of $K$ with respect to $\sigma$ and $X_\sigma = X \times _K^{\sigma } \operatorname {\mathrm {Spec}} {\mathbb {C}}$. We define the analytification of $X_v$ by $X_v^{\mathrm {an}} = X_v({\mathbb {C}}) = \bigsqcup _{\sigma \in \Sigma _v} X_\sigma ({\mathbb {C}})$. The disjoint union

\[ X_\infty^{\mathrm{an}} := \bigsqcup_{v \in \Sigma_K, \ v | \infty} X_v^{\mathrm{an}} = \bigsqcup_{\sigma \in K({\mathbb{C}})} X_\sigma({\mathbb{C}}) =X({\mathbb{C}}) \]

is the set of morphisms of $\operatorname {\mathrm {Spec}} {\mathbb {Q}}$-schemes $x \colon \operatorname {\mathrm {Spec}} {\mathbb {C}} \rightarrow X$. We let $F_\infty \colon X_\infty ^{\mathrm {an}} \rightarrow X_\infty ^{\mathrm {an}}$ be the involution induced by the complex conjugation.

2.5

Let $x \in X$ be a closed point and ${\kappa }(x)$ its residue field. For a place $v \in \Sigma _K$, we denote by $\Sigma _v(x)$ the set of $K_v$-algebra morphisms ${\kappa }(x) \otimes _K K_v \rightarrow {\mathbb {C}}_v$, where ${\mathbb {C}}_v$ is an algebraic closure of $K_v$. For each $\sigma \in \Sigma _v(x)$, we denote by $x_\sigma$ the closed point of $X_v$ given by composition

\[ \operatorname{\mathrm{Spec}} {\mathbb{C}}_v \overset{\sigma}{\longrightarrow} \operatorname{\mathrm{Spec}} ({ \kappa}(x) \otimes_K K_v) \overset{x \times \mathrm{id}}{\longrightarrow} X_v. \]

We define the orbit $O_v(x) \subset X_v$ of $x$ by

\[ O_v(x) = \{ x_{\sigma} \mid \sigma \in \Sigma_v(x)\}. \]

Note that the cardinal of $O_v(x)$ is $[{ \kappa }(x) : K]$. Moreover, for each $z \in X_v({\mathbb {C}}_v)$, we define a point $z^{\mathrm {an}} \in X_v^{\mathrm {an}}$ as follows. If $v | \infty$, we let $z^{\mathrm {an}} = z$. If $v$ is finite, $z^{\mathrm {an}}\in X_v^{\mathrm {an}}$ is the unique valuation on $K_v(z)$ extending $v$.

2.6

Assume that $X$ is projective and geometrically integral. Let $D \in \operatorname {\mathrm {Div}}(X)_{\mathbb {R}}$, $v \in \Sigma _K$ and let $D_v \in \operatorname {\mathrm {Div}}(X_v)_{\mathbb {R}}$ be the pullback of $D$ to $X_v$. We consider an open covering $X_v = \cup _{i=1}^{\ell }U_i$ such that $D_v$ is defined by $f_i \in { { \kappa }(X_v)_{\mathbb {R}}}$ on $U_i$ for each $i \in \{1,\ldots , \ell \}$. A continuous (respectively, smooth) $D$-Green function on $X_v^{\mathrm {an}}$ is a function

\[ g_v \colon X_v^{\mathrm{an}}{\setminus} (\operatorname{\mathrm{Supp}} D_v)^{\mathrm{an}} \rightarrow {\mathbb{R}} \]

such that $g_v + \ln |f_i|^{2}_v$ extends to a continuous (respectively, smooth) function on the analytification $U_i^{\mathrm {an}}$ of $U_i$ for each $i \in \{1, \ldots , \ell \}$. We refer the reader to [Reference MoriwakiMor16, §§ 1.4 and 2.1] for more details on Green functions.

3.1 Notation

Throughout this section we fix a global field $K$ and a projective, normal and geometrically integral variety $X$ on $\operatorname {\mathrm {Spec}} K$. We let $d := \dim X$ and we fix an algebraic closure $\bar {K}$ of $K$. We define a scheme $\mathcal {S}$ as follows.

1. (i) If $K_0 = {\mathbb {Q}}$, $\mathcal {S} = \operatorname {\mathrm {Spec}} {\mathcal {O}}_K$ is the spectrum of the ring of integers ${\mathcal {O}}_K$ of $K$.

2. (ii) If $K_0 = k(T)$, $\mathcal {S} = C_K$ is a regular projective integral curve over $k$ with field of functions ${\kappa }(C_K)=K$. This implies the existence of a finite $k$-morphism $\varphi \colon C_K \rightarrow {\mathbb {P}}^{1}_k$. Moreover, the curve $C_K$ is unique up to $k$-isomorphism.

Throughout this section, $\mathbb {K}$ denotes either ${\mathbb {Z}}$, ${\mathbb {Q}}$ or ${\mathbb {R}}$. Let $D \in \operatorname {\mathrm {Div}}(X)_{\mathbb {K}}$ and let $U \subset \mathcal {S}$ be a non-empty open subset. A model $\mathcal {X}$ of $X$ over $U$ is an integral scheme together with a projective dominant morphism $\pi _{\mathcal {X}} \colon \mathcal {X} \rightarrow U$ with generic fibre $X$. We will denote a model $(\mathcal {X}, \pi _{\mathcal {X}})$ if we need to specify the associated morphism. We say that $\mathcal {X}$ is a normal model if $\mathcal {X}$ is normal. If $\mathcal {D}$ is a ${\mathbb {K}}$-Cartier divisor on $\mathcal {X}$ such that the restriction of $\mathcal {D}$ to $X$ is equal to $D$, we say that $(\mathcal {X},\mathcal {D})$ is a model of $(X,D)$ over $U$. For each non-archimedean place $v \in U$, we denote by $g_{\mathcal {D},v}$ the $D$-Green function on $X_v^{\mathrm {an}}$ induced by $\mathcal {D}$ (see [Reference MoriwakiMor16, § 0.2] for details on this construction).

3.2 Definitions

Note that an adelic ${\mathbb {R}}$-Cartier divisor $\bar {\xi }$ on $\operatorname {\mathrm {Spec}} K$ is just a collection of real numbers $\bar {\xi } = (\xi _v)_{v \in \Sigma _K}$ such that $\xi _v =0$ for all but finitely many $v \in \Sigma _K$. The normalized Arakelov degree of $\bar {\xi }$ is defined by

\[ \widehat{\deg}(\bar{\xi}) = \frac{1}{2}\sum_{v \in \Sigma_K}n_v(K) \xi_v. \]

3.3 Arithmetic volume function

Let $\bar {D} = (D, (g_v)_{v \in \Sigma _K})$ be an adelic ${\mathbb {R}}$-Cartier divisor on $X$. We consider the $K$-vector space

\[ H^{0}(X,D) := \{ \phi \in { \kappa}(X)^{{\times}} \mid D + {(\phi)} \geq 0\} \cup \{0\}. \]

For any non-zero element $\phi \in H^{0}(X,D)$ and any $v \in \Sigma _K$, the function $\|\phi \|_{v} := |\phi _v|_v\exp (-g_v/2)$ extends to a continuous function on $X_v^{\mathrm {an}}$, where $\phi _v$ is the function on $X_v^{\mathrm {an}}$ induced by $\phi$ (see [Reference MoriwakiMor16, Propositions 1.4.2 and 2.1.3]). We also define $\|\phi \|_{v,\sup } := \sup _{x \in { X_v^{\mathrm {an}}}} \|\phi \|_{v}(x)$ and

\[ \hat{H}^{0}(X,\bar{D}) := \{ \phi \in H^{0}(X,D) \mid \|\phi\|_{v,\sup}\leq 1 \ \forall v \in \Sigma_K \}. \]

We let

\[ \hat{h}^{0}(X,\bar{D}) = \left\{\begin{array}{@{}ll} \ln \# \hat{H}^{0}(X,\bar{D}) & \textrm{if}\ K_0 = {\mathbb{Q}},\\ \dim_k \hat{H}^{0}(X,\bar{D}) & \textrm{if}\ K_0 = k(T). \end{array} \right. \]

Definition 3.3 The arithmetic volume of $\bar {D}$ is the quantity

\[ \widehat{\operatorname{\mathrm{vol}}}(\bar{D}) := \limsup_{n \rightarrow +\infty} \frac{\hat{h}^{0}(X,n\bar{D})}{n^{d+1}/(d+1)!}. \]

Example 3.4 Assume that there exists a normal model $(\mathcal {X},\mathcal {D})$ over $\mathcal {S}$ such that all the $D$-Green functions of $\bar {D}$ are induced by $\mathcal {D}$. By [Reference MoriwakiMor16, Proposition 4.3.1], we have

\[ \hat{H}^{0}(X,D) = \Big\{ \phi\in H^{0}(\mathcal{X},\mathcal{D}) \mid \max_{v\in \Sigma_{K,\infty}}\|\phi\|_{v,\sup}\leq 1\Big\}. \]

In particular, if $K_0 = k(T)$ then $\hat {H}^{0}(X,D) = H^{0}(\mathcal {X},\mathcal {D})$. It follows that $\widehat {\operatorname {\mathrm {vol}}}(\bar {D})=\operatorname {\mathrm {vol}}(\mathcal {D})$ is the (geometric) volume of $\mathcal {D}$ (see [Reference LazarsfeldLaz04a, § 2.2]):

\[ \operatorname{\mathrm{vol}}(\mathcal{D})= \limsup_{n \rightarrow +\infty} \frac{\dim_k H^{0}(\mathcal{X}, n\mathcal{D})}{n^{d+1}/(d+1)!}. \]

We now recall a well-known continuity property of the volume function $\widehat {\operatorname {\mathrm {vol}}}$ due to Moriwaki.

Theorem 3.5 Let $\bar {D},\bar {D}_1, \ldots , \bar {D}_\ell$ be adelic $\mathbb {R}$-Cartier divisors on $X$. Then

\[ \lim_{\varepsilon_1 \rightarrow 0, \ldots, \varepsilon_\ell \rightarrow 0} \widehat{\operatorname{\mathrm{vol}}}(\bar{D}+\varepsilon_1\bar{D}_1 + \cdots + \varepsilon_\ell\bar{D}_\ell) = \widehat{\operatorname{\mathrm{vol}}}(\bar{D}). \]

When $K$ is a number field, this is a particular case of [Reference MoriwakiMor16, Theorem 5.2.1]. In the function field case, one can prove the result by using similar arguments and the continuity of the geometric volume function [Reference LazarsfeldLaz04a, Theorem 2.2.44]. Alternatively, Theorem 3.5 is a particular case of [Reference Chen and MoriwakiCM19, Theorem 6.4.24] due to Chen and Moriwaki, who recently established the continuity of the arithmetic volume in the much more general setting of adelic curves.

3.4 Height function

Let $\bar {D} = (D, (g_v)_{v \in \Sigma _K})$ be an adelic ${\mathbb {R}}$-Cartier divisor on $X$ and let $x \in X(\bar {K})$ be a closed point with residue field ${ \kappa }(x)$. The height of $x$ with respect to $\bar {D}$ is defined to be

\[ h_{\bar{D}}(x):={-}\frac{1}{[{ \kappa}(x) : K]}\sum_{v \in \Sigma_K}n_v(K) \sum_{z \in {O_v}(x)} \ln \|\phi\|_v({z^{\mathrm{an}}}), \]

where $\phi \in H^{0}(X,D)$ is any function with $x \notin \operatorname {\mathrm {Supp}}(D+(\phi ))$. This definition does not depend on the choice of $\phi$ (see [Reference MoriwakiMor16, § 4.2] for details).

Remark 3.6 We mention an equivalent formula for the height of $x$, which may be more usual. For each place $w \in \Sigma _{{ \kappa }(x)}$, we fix a $K$-embedding $\sigma _w \colon { \kappa }(x) \hookrightarrow {\mathbb {C}}_v$ associated to $w$, where $v$ denotes the restriction of $w$ to $K$ (note that there are exactly $[{ \kappa }(x)_w : K_v]$ such embeddings). The pair $(x,\sigma _w)$ uniquely determines a point $x_w \in X_v$, and the quantity $\|\phi \|_w(x) := \|\phi \|_v(x_w^{\mathrm {an}})$ does not depend on the choice of $\sigma _w$. With this notation, we have

\[ h_{\bar{D}}(x) ={-}\sum_{w \in \Sigma_{{ \kappa}(x)}} n_w({ \kappa}(x)) \ln \|\phi\|_w(x). \]

If we denote by $\pi \colon X \rightarrow \operatorname {\mathrm {Spec}} K$ the structural morphism, we have

(3.1)\begin{equation} h_{\bar{D} + \pi^{*}\bar{\xi}}(x) = h_{\bar{D}}(x) + \widehat{\deg}(\bar{\xi}) \end{equation}

for any $\bar {\xi } \in \widehat {\operatorname {\mathrm {Div}}}(\operatorname {\mathrm {Spec}} K)_{\mathbb {R}}$ and $x \in X(\bar {K})$. In view of Example 3.7 below, this basic fact can be interpreted as a projection formula.

Example 3.7 Assume that $K$ is a function field and that there exists a normal model $(\mathcal {X}, \mathcal {D})$ of $(X,D)$ over $\mathcal {S} = C_K$ such that for each $v \in \Sigma _K$, the $D$-Green function $g_v = g_{\mathcal {D},v}$ is induced by $\mathcal {D}$. Let $x \in X(\bar {K})$ be a closed point and let $C_x \subset \mathcal {X}$ be the Zariski closure of $x$ in $\mathcal {X}$. Note that $C_x$ is an integral projective curve, and that the restriction ${\pi _{\mathcal {X}}}_{|C_x} \colon C_x \rightarrow C_K$ is surjective. We have

\[ h_{\bar{D}}(x) = \frac{\mathcal{D} \cdot C_x}{[{ \kappa}(x) : K_0]}, \]

where ${ \kappa }(x)$ is the residue field of $X$ at $x$ and $\mathcal {D} \cdot C_x$ is the usual geometric intersection product of cycles. We will see that an analogue of this formula involving the arithmetic intersection product holds when $K$ is a number field (see § 3.7).

We end this subsection with the following well-known lemma, giving a lower bound for the height of points at which a small section does not vanish.

Proof. If $x \in X(\bar {K})$ is not contained in the support of $D + {(\phi )}$, we have

\[ h_{\bar{D}}(x) ={-} \frac{1}{[{ \kappa}(x):K]}\sum_{v \in \Sigma_K} n_v(K)\sum_{z \in O_v(x)} \ln \|\phi\|_v(z^{\mathrm{an}}) \geq 0 \]

since $\|\phi \|_v(z^{\mathrm {an}}) \leq \|\phi \|_{v,\sup } \leq 1$ for every $v \in \Sigma _K$ and $z \in O_v(x)$.

3.5 Positivity of adelic ${\mathbb {R}}$-Cartier divisors

In this section we define several notions of arithmetic positivity, following [Reference MoriwakiMor16, § 4.4] and [Reference ZhangZha95b]. For the definition of plurisubharmonic functions we refer the reader to [Reference MoriwakiMor12b, § 2.1] and [Reference MoriwakiMor16, § 1.4].

3.6 Arithmetic Cartier divisors

In this section we define arithmetic ${\mathbb {K}}$-Cartier divisors on arithmetic varieties, which are special cases of adelic ${\mathbb {K}}$-Cartier divisors for which all the Green functions at the non-archimedean places are induced by the same model. In this subsection we assume that $K$ is a number field. Let $\mathcal {X}$ be a projective arithmetic variety over $\operatorname {\mathrm {Spec}} {\mathcal {O}}_K$, that is, an integral scheme projective and flat over $\operatorname {\mathrm {Spec}} {\mathcal {O}}_K$. We let $d+1$ be the Krull dimension of $\mathcal {X}$, so that the generic fibre $X := \mathcal {X}\times _{{\mathcal {O}}_K} \operatorname {\mathrm {Spec}} K$ of $\mathcal {X}$ has dimension $d$. We say that $\mathcal {X}$ is generically smooth if $X$ is smooth.

Arithmetic ${\mathbb {K}}$-Cartier divisors on $\mathcal {X}$ form a group, denoted by $\widehat {\operatorname {\mathrm {Div}}}(\mathcal {X})_{\mathbb {K}}$. Given an arithmetic ${\mathbb {K}}$-Cartier divisor $\bar {\mathcal {D}} = (\mathcal {D},(g_v)_{v \in \Sigma _{K,\infty }})$, we denote by $\bar {\mathcal {D}}^{\mathrm {ad}}$ the adelic ${\mathbb {K}}$-Cartier divisor on the generic fibre $X$ defined by $(D,(g_v)_{v \in \Sigma _K})$, where $D = \mathcal {D}_{|X}$ is the restriction of $\mathcal {D}$ and, for every non-archimedean place $v \in \Sigma _K$, $g_v = g_{\mathcal {D},v}$ is the $D$-Green function induced by $\mathcal {D}$.

We observe that our definitions of big, pseudo-effective and nef arithmetic ${\mathbb {R}}$-divisor coincide with those of Ikoma [Reference IkomaIko15b]. We will also need the notion of ampleness used in [Reference IkomaIko15b]. We first recall the definition of the curvature current. Let $\bar {\mathcal {D}} = (\mathcal {D},(g_v)_{v}) \in \widehat {\operatorname {\mathrm {Div}}}(\mathcal {X})_{\mathbb {Z}}$. We assume that $\mathcal {X}$ is generically smooth. For $v \in \Sigma _{K, \infty }$, we denote by $D_v$ the divisor induced by $\mathcal {D}$ on $X_v$ and we consider the current $c_{1,v}(\bar{\mathcal{D}}) := ({\sqrt {-1}}/{2\pi })\partial \bar {\partial }[g_v] + \delta _{D_v}$ on $X_v^{\mathrm {an}}$. Note that $c_{1,v}(\bar {\mathcal {D}})$ is non-negative if the function $g_v$ is plurisubharmonic, and that it is a real $(1,1)$ form if $g_v$ is smooth. We let $c_1(\bar {\mathcal {D}})$ be the current on $X({\mathbb {C}})$ whose restriction to $X_v^{\mathrm {an}}$ is $c_{1,v}(\bar{\mathcal{D}})$ for each $v \in \Sigma _{K,\infty }$.

The following definition of ampleness is the one used by Ikoma in [Reference IkomaIko15b].

Definition 3.14 Let $\mathcal {X}$ be a generically smooth arithmetic variety over $\operatorname {\mathrm {Spec}} {\mathcal {O}}_K$ and let $\bar {\mathcal {A}} = (\mathcal {A}, (g_v)_{v \in \Sigma _{K,\infty }}) \in \widehat {\operatorname {\mathrm {Div}}}(\mathcal {X})_{{\mathbb {Z}}}$. We say that $\bar {\mathcal {A}}$ is ample if it satisfies the following conditions:

1. (i) $\mathcal {A}$ is ample;

2. (ii) for each $v \in \Sigma _{K,\infty }$, $g_v$ is smooth;

3. (iii) the curvature form $c_1(\mathcal {\bar {A}})$ is positive pointwise on $X({\mathbb {C}})$;

4. (iv) for $m\gg 1$, ${H}^{0}(\mathcal {X},{ m{\mathcal {A}}})$ is generated as an ${\mathcal {O}}_K$-module by sections $s$ with $\max _{v\in \Sigma _{K,\infty }}\|s\|_{v,\sup } < 1$.

An arithmetic ${\mathbb {R}}$-divisor $\bar {\mathcal {D}}$ on $\mathcal {X}$ is ample if there exist ample arithmetic ${\mathbb {Z}}$-Cartier divisors $\bar {\mathcal {A}}_1, \ldots , \bar {\mathcal {A}}_\ell$ and positive real numbers $a_1, \ldots , a_\ell$ such that

\[ \bar{\mathcal{D}} = a_1\bar{\mathcal{A}}_1+ \cdots + a_\ell\bar{\mathcal{A}}_\ell. \]

3.7 Reminder on arithmetic intersection theory

In this subsection we recall some properties of the arithmetic intersection product that we will need in the proof of Theorem 1.1. We assume that $K$ is a number field and let $\mathcal {X}$ be a normal arithmetic variety over $\operatorname {\mathrm {Spec}} {\mathcal {O}}_K$.

We say that a divisor $\bar {\mathcal {D}} \in \widehat {\operatorname {\mathrm {Div}}}(\mathcal {X})_{\mathbb {R}}$ is integrable if it belongs to the subgroup of $\widehat {\operatorname {\mathrm {Div}}}(\mathcal {X})_{\mathbb {R}}$ generated by nef arithmetic ${\mathbb {R}}$-Cartier divisors. In particular, a semi-positive arithmetic ${\mathbb {R}}$-Cartier divisor is integrable. Let $\bar {\mathcal {D}}_0,\bar {\mathcal {D}}_1, \ldots , \bar {\mathcal {D}}_d$ be integrable arithmetic ${\mathbb {R}}$-Cartier divisors on $\mathcal {X}$. In [Reference MoriwakiMor12b, § 6.4], Moriwaki defined a multilinear intersection product

(3.2)\begin{equation} \widehat{\deg}(\bar{\mathcal{D}}_0 \cdots \bar{\mathcal{D}}_d) \end{equation}

when $\mathcal {X}$ generically smooth. In [Reference IkomaIko15b], Ikoma extended the intersection product (3.2) to the cases where $\mathcal {X}$ is not generically smooth and $\bar {\mathcal {D}}_0$ is not integrable. Let $\bar {\mathcal {D}} \in \widehat {\operatorname {\mathrm {Div}}}(\mathcal {X})_{\mathbb {R}}$ be an integrable arithmetic ${\mathbb {R}}$-Cartier divisor and $\bar {D} = \mathcal {\bar {D}}^{\mathrm {ad}}$ be the induced adelic divisor on $X$. Let $x \in X(\bar {K})$ be a closed point and let $C_x \subset \mathcal {X}$ be the Zariski closure of $x$ in $\mathcal {X}$. Then we have

(3.3)\begin{equation} h_{\bar{D}}(x) = \frac{\widehat{\deg}(\bar{\mathcal{D}}_{|C_x})}{[{ \kappa}(x): K_0]} \end{equation}

(see [Reference MoriwakiMor12b, § 5.3] for details). When $\mathcal {X}$ is generically smooth and when the divisors $\bar {\mathcal {D}}_0,\bar {\mathcal {D}}_1, \ldots , \bar {\mathcal {D}}_d \in \widehat {\operatorname {\mathrm {Div}}}(\mathcal {X})_{\mathbb {Z}}$ are of $C^{\infty }$-type, the product (3.2) coincides with the usual one, used for example in [Reference ZhangZha95a, Reference ZhangZha95b]. In particular, for any non-zero function $\phi \in H^{0}(\mathcal {X}, \mathcal {D}_d)$ we have

\[ \widehat{\deg}\left(\bar{\mathcal{D}}_0 \cdots \bar{\mathcal{D}}_d\right) = \sum_{i=1}^{\ell} a_i \widehat{\deg}\left(\bar{\mathcal{D}}_{0|\mathcal{Z}_i} \cdots \bar{\mathcal{D}}_{{d-1}|\mathcal{Z}_i}\right) - \sum_{v \in \Sigma_{K,\infty}} \int_{{X_v^{\mathrm{an}}}} \ln\|\phi\|_{v} c_{1,v}(\bar{\mathcal{D}}_0) \wedge \cdots c_{1,v}(\bar{\mathcal{D}}_{d-1}), \]

where $(\phi ) +{\mathcal {D}}_d = \sum _{i=1}^{\ell } a_i \mathcal {Z}_i$ is the decomposition of the Weil divisor $(\phi )+{\mathcal {D}}_d$: for each $i \in \{1, \ldots , \ell \}$, $a_i \in {\mathbb {Z}}_{> 0}$ and $\mathcal {Z}_i \subset \mathcal {X}$ is a codimension $1$ integral subvariety.

3.8 Successive minima of an adelic ${\mathbb {R}}$-Cartier divisor

Let $K$ be global field and let $X$ be a projective, normal and geometrically integral variety on $\operatorname {\mathrm {Spec}} K$ of dimension $d \geq 1$. Let $\bar {D} = (D, (g_v)_{v \in \Sigma _K})$ be an adelic ${\mathbb {R}}$-Cartier divisor on $X$. For all $\lambda \in {\mathbb {R}}$, we denote by $X_{\bar {D}}(\lambda )$ the Zariski closure in $X$ of the set

\[ \{x \in X(\bar{K}) \mid h_{\bar{D}}(x) \leq \lambda\}. \]

For every integer $i \in \{1, \ldots , d +1\}$, we define the $i$th minimum $\zeta _i(\bar {D})$ of $\bar {D}$ by

\[ \zeta_i(\bar{D}) = \inf \{\lambda \in {\mathbb{R}} \mid \dim X_{\bar{D}}(\lambda) \geq i-1 \}. \]

These invariants were first introduced by Zhang in the number field setting [Reference ZhangZha95a]. We call $\zeta _{\mathrm {abs}}(\bar {D}) = \zeta _1(\bar {D})$ the absolute minimum and $\zeta _{\mathrm {ess}}(\bar {D}) := \zeta _{d+1}(\bar {D})$ the essential minimum of $\bar {D}$. Note that $\zeta _{\mathrm {abs}}(\bar {D}) = \inf _{x \in X(\bar {K})} h_{\bar {D}}(x)$ and

\[ \zeta_{\mathrm{ess}}(\bar{D}) = \inf \{\lambda \in {\mathbb{R}} \mid X_{\bar{D}}(\lambda) \text{ is dense} \} = \sup_{Y \subsetneq X}\inf_{x \in X(\bar{K}) {\setminus} Y} h_{\bar{D}}(x), \]

where the supremum is taken on the Zariski-closed proper subschemes $Y$ of $X$.

We gather some basic properties of the essential minimum in the following lemma.

Proof. (i) Without loss of generality, we assume that $\zeta _{\mathrm {ess}}(\bar {D}_1),\zeta _{\mathrm {ess}}( \bar {D}_2) > -\infty$. Let $\lambda _1 < \zeta _{\mathrm {ess}}(\bar {D}_1)$ and $\lambda _2 < \zeta _{\mathrm {ess}}(\bar {D}_2)$ be real numbers. By definition of the essential minimum, there exist non-empty open subsets $U_1$, $U_2$ in $X$ such that for each $i \in \{1,2\}$,

\[ \forall x \in U_i,\quad h_{\bar{D}_i}(x) \geq \lambda_i. \]

Since $X$ is integral, the open subset $U = U_1 \cap U_2$ is non-empty. It follows that

\[ \zeta_{\mathrm{ess}}(\bar{D}_1+ \bar{D}_2) \geq \inf_{x \in U(\bar{K})} (h_{\bar{D}_1}(x) +h_{\bar{D}_2}(x)) \geq \lambda_1 + \lambda_2, \]

and we conclude by letting $\lambda _1$ and $\lambda _2$ tend to $\zeta _{\mathrm {ess}}(\bar {D}_1)$ and $\zeta _{\mathrm {ess}}(\bar {D}_2)$, respectively.

(ii) By (i), we have

\[ \zeta_{\mathrm{ess}}(\bar{D}_1) =\zeta_{\mathrm{ess}}(\bar{D}_2 + \bar{D}_1 - \bar{D}_2) \geq \zeta_{\mathrm{ess}}(\bar{D}_2) + \zeta_{\mathrm{ess}}( \bar{D}_1-\bar{D}_2). \]

By Lemma 3.8, there exists a closed subset $Z_t \subsetneq X$ such that $th_{\bar {D}_1-\bar {D}_2}(x) = h_{t(\bar {D}_1-\bar {D}_2)}(x) \geq 0$ for any $x \in X(\bar {K}){\setminus} Z_t$. Therefore

\[ \zeta_{\mathrm{ess}}( \bar{D}_1-\bar{D}_2) \geq \inf_{x \in X(\bar{K}) {\setminus} Z_t} h_{\bar{D}_1-\bar{D}_2}(x) \geq 0. \]

(iii) Let $\pi \colon X \rightarrow \operatorname {\mathrm {Spec}} K$ denote the structural morphism. If we replace $\bar {D}_1$ by $\bar {D}_1 + \pi ^{*}\bar {\xi }$ for some adelic ${\mathbb {R}}$-Cartier divisor $\bar {\xi }$ on $\operatorname {\mathrm {Spec}} K$, both sides of the inequality differ by $\widehat {\deg }(\bar {\xi })$ (see the projection formula for the height (3.1)). Hence we may assume that $\bar {D}_1$ is big. Let $\varepsilon > 0$ and $t$ be real numbers. By continuity of the volume function (Theorem 3.5), the divisors

\[ (1+\varepsilon)\bar{D}_1 - (\bar{D}_1 + t\bar{D}_2) = \varepsilon\bar{D}_1 - t\bar{D}_2 \]

and

\[ (\bar{D}_1 + t\bar{D}_2)-(1-\varepsilon)\bar{D}_1 = \varepsilon\bar{D}_1 + t\bar{D}_2 \]

are both big if $t$ is sufficiently close to zero. By (ii), we have

\[ (1+\varepsilon)\zeta_{\mathrm{ess}}(\bar{D}_1) \geq \zeta_{\mathrm{ess}}(\bar{D}_1 + t\bar{D}_2) \geq (1-\varepsilon)\zeta_{\mathrm{ess}}(\bar{D}_1) \]

and the result follows.

4.1 Reduction to a fixed model

Proof. Since $\bar {D}$ is semi-positive, there exists a sequence $(\mathcal {X}_n, \mathcal {D}_n, (g_{n,v})_{v \in \Sigma _K})_{n \in {\mathbb {N}}}$ such that:

1. (i) for all $n \in {\mathbb {N}}$, $(\mathcal {X}_n, \mathcal {D}_n)$ is a normal model for $(X,D)$ with $\mathcal {D}_n$ relatively nef;

2. (ii) for all $n \in {\mathbb {N}}$, $g_{n,v}$ is a smooth plurisubharmonic $D$-Green function invariant under $F_\infty$ if $v \in \Sigma _{K,\infty }$ and $g_{n,v}=g_{\mathcal {D}_n,v}$ for every non-archimedean $v \in \Sigma _K$;

3. (iii) $\lim _{n \rightarrow +\infty } \|g_{n,v}-g_v\|_{v, \sup } = 0$ for every $v \in \Sigma _K$.

For each $n \in {\mathbb {N}}$, we denote by $\pi _n \colon \mathcal {X}_n \rightarrow \mathcal {S}$ the morphism given by the model $\mathcal {X}_n$. Let $M \in \operatorname {\mathrm {Div}}(\mathcal {S})_{\mathbb {R}}$ be an effective ${\mathbb {R}}$-Cartier divisor such that $\widehat {\deg }(\bar {\xi }_M)=1$, where $\bar {\xi }_M$ is the adelic ${\mathbb {R}}$-Cartier divisor on $\operatorname {\mathrm {Spec}} K$ induced by $M$. Let $\varepsilon > 0$ be a real number. We let $\mathcal {D}_{n,\varepsilon }:= \mathcal {D}_n+\varepsilon \pi ^{*}_nM$, $\bar {\mathcal {D}}_{n,\varepsilon } = (\mathcal {D}_{n,\varepsilon }, (g_{n,v})_{v \in \Sigma _{K,\infty }})$ and $\bar {\mathcal {D}}_{n} = (\mathcal {D}_{n}, (g_{n,v})_{v \in \Sigma _{K,\infty }})$.

By (iii) above and by the projection formula (3.1), there exists an integer $n_{\varepsilon } \geq 1$ such that

\[ h_{\bar{\mathcal{D}}^{\mathrm{ad}}_{n,\varepsilon}}(x) = h_{\bar{\mathcal{D}}^{\mathrm{ad}}_n}(x) + \varepsilon \geq h_{\bar{D}}(x) + \varepsilon/2 \]

for every $x \in X(\bar {K})$ and $n \geq n_{\varepsilon }$. It follows that $\zeta _{\mathrm {ess}}(\bar {\mathcal {D}}^{\mathrm {ad}}_{n,\varepsilon })> \zeta _{\mathrm {ess}} (\bar {D})$ for all $\varepsilon > 0$ and $n \geq n_{\varepsilon }$. Moreover, $\mathcal {D}_{n,\varepsilon }$ is relatively nef for all $n \in {\mathbb {N}}$ and $\varepsilon > 0$. All that remains is to check that we can choose $\varepsilon$ and $n \geq n_{\varepsilon }$ so that $\bar {\mathcal {D}}_{n,\varepsilon }$ is not pseudo-effective. Since $\bar {D}$ is not pseudo-effective, there exists a big adelic ${\mathbb {R}}$-divisor $\bar {B}$ on $X$ such that $\bar {D} + \bar {B}$ is not big. For all $n \in {\mathbb {N}}$, $\varepsilon > 0$ and $v \in \Sigma _K$, we let $s_{v,n,\varepsilon } = \|g_{\mathcal {D}_{n,\varepsilon },v}-g_v\|_{v,\sup }$. By construction, we can choose $n$ and $\varepsilon$ so that $\max _{v \in \Sigma _K}s_{v,n,\varepsilon }$ is arbitrarily small. By continuity of $\widehat {\operatorname {\mathrm {vol}}}$ (Theorem 3.5), it follows that there exist $\varepsilon _0>0$ and $m \geq n_{\varepsilon _0}$ such that $\bar {B}_{m, \varepsilon _0} := \bar {B} - (0, (s_{v,m,\varepsilon _0})_{v \in \Sigma _K})$ is big. If $\bar {\mathcal {D}}_{m, \varepsilon _0}$ is pseudo-effective, then

\[ \widehat{\operatorname{\mathrm{vol}}}(\bar{D}+\bar{B}) \geq \widehat{\operatorname{\mathrm{vol}}}(\bar{\mathcal{D}}^{\mathrm{ad}}_{m,\varepsilon_0} + \bar{B}_{m, \varepsilon_0})>0, \]

which is absurd. Hence $\bar {\mathcal {D}}_{m, \varepsilon _0}$ is not pseudo-effective, and we obtain the lemma with $\bar {\mathcal {D}} = \bar {\mathcal {D}}_{m, \varepsilon _0}$.

4.2 The function field case

The goal of this subsection is to prove Theorem 4.1 when $K$ is a function field, that is, when $\mathcal {S} = C_K$ is a regular integral projective curve on a field $k$. We will see that in this case, Theorem 4.1 is a consequence of the following theorem of Boucksom, Demailly, Păun and Perternell, describing the dual of the pseudo-effective cone of a projective variety.

Theorem 4.4 [Reference Boucksom, Demailly, Păun and PeternellBDPP13, Theorem 2.2]

Let $Z$ be an integral projective variety over a field and let $\ell = \dim Z -1$. A divisor $E$ on $Z$ is pseudo-effective if and only if for every birational morphism $\varphi \colon Z' \rightarrow Z$ and for all ample divisors $A_1, \ldots , A_\ell$ on $\mathcal {X'}$,

\[ E \cdot \varphi_*(A_1 \cdots A_\ell) \geq 0, \]

where $\varphi _*$ denotes the proper pushforward of cycles.

In [Reference Boucksom, Demailly, Păun and PeternellBDPP13, Theorem 2.2], this theorem is stated and proved over ${\mathbb {C}}$. Recently, Das [Reference DasDas20] observed that the proof given by Lazarsfeld in [Reference LazarsfeldLaz04b, § 11.4.C] (following [Reference Boucksom, Demailly, Păun and PeternellBDPP13]) works over an algebraically closed field of arbitrary characteristic. By the work of Cutkosky [Reference CutkoskyCut15], it turns out that the proof given in [Reference LazarsfeldLaz04b] remains valid without any assumption on the base field. More precisely, the key point in the proof of [Reference LazarsfeldLaz04b, Theorem 11.4.19] is the ‘asymptotic orthogonality of Fujita approximation’ [Reference LazarsfeldLaz04b, Theorem 11.4.21]. The latter holds for any field $k$ by [Reference CutkoskyCut15, Corollary 5.5].Footnote ¹ The rest of the proof of [Reference LazarsfeldLaz04b, Theorem 11.4.19] does not make any use of the algebraically closed or characteristic zero assumption on $k$.

We will use the following classical consequence of Theorem 4.4 (see [Reference LazarsfeldLaz04b, Lemma 11.4.18]).

We are now ready to prove Theorem 4.1 when $K$ is a function field.

Proof Proof of Theorem 4.1 (${ K_0} = k(T$))

By Lemma 3.16, it suffices to prove the implication

\[ \zeta_{\mathrm{ess}}(\bar{D}) \geq 0 \implies \bar{D} \text{ is pseudo-effective.} \]

Assume that $\bar {D}$ is not pseudo-effective. By Lemma 4.3 (see also Example 3.11), there exist a normal model $(\mathcal {X},\pi _{\mathcal {X}})$ of $X$ on $C_K$ and a ${\mathbb {R}}$-Cartier divisor $\mathcal {D}$ on $\mathcal {X}$ such that

1. (i) $\mathcal {D}$ is relatively nef,

2. (ii) $\mathcal {D}$ is not pseudo-effective,

3. (iii) $\zeta _{\mathrm {ess}}(\bar {\mathcal {D}}^{\mathrm {ad}}) > \zeta _{\mathrm {ess}} (\bar {D})$.

We only need to show that $\zeta _{\mathrm {ess}}(\bar {\mathcal {D}}^{\mathrm {ad}}) \leq 0$. Let $\mathcal {Y} \subsetneq \mathcal {X}$ be a proper closed subscheme. By Corollary 4.5, there exists an integral curve $C_{\mathcal {Y}} \in \mathcal {X}$ not contained in $\mathcal {Y}$ such that $\mathcal {D} \cdot C_{\mathcal {Y}} < 0$. Since $\mathcal {D}$ is relatively nef, we have $\mathcal {D} \cdot C \geq 0$ for any curve contained in a fibre of $\pi _{\mathcal {X}}$ above a closed point. It follows that the curve $C_{\mathcal {Y}}$ is horizontal, in the sense that the restriction ${\pi _{\mathcal {X}}}_{|C_{\mathcal {Y}}}$ is a surjective morphism. The generic point $P_{\mathcal {Y}} \in X(\bar {K})$ of $C_{\mathcal {Y}}$ satisfies

\[ h_{\bar{\mathcal{D}}^{\mathrm{ad}}}(P_{\mathcal{Y}}) = \frac{\mathcal{D}\cdot C_{\mathcal{Y}}}{[ {\kappa}(C_{\mathcal{Y}}) : K_0]} < 0. \]

We infer that

\[ \sup_{\mathcal{Y} \varsubsetneq \mathcal{X}} \inf\{h_{\bar{\mathcal{D}}^{\mathrm{ad}}}(P) \mid P \in X(\bar{K}), \ C_P \nsubseteq \mathcal{Y}\} \leq 0, \]

where, for any $P \in X(\bar {K})$, $C_P$ denotes the Zariski closure of $P$ in $\mathcal {X}$. Now the supremum on the left-hand side is the essential minimum $\zeta _{\mathrm {ess}}(\bar {\mathcal {D}}^{\mathrm {ad}})$, so the theorem is proved.

4.3 The number field case

We now assume that $K$ is a number field, that is, $\mathcal {S} = \operatorname {\mathrm {Spec}} {\mathcal {O}}_K$. The following arithmetic analogue of Theorem 4.4 is due to Ikoma.

We shall derive Theorem 4.1 from Ikoma's theorem by applying a method similar to the one we used in the function field case. In order to deduce an analogue of Corollary 4.5 from Theorem 4.6, we use an arithmetic Bertini-type theorem due to Moriwaki [Reference MoriwakiMor95, Theorem 5.3]. The latter was extended by Ikoma [Reference IkomaIko15a], and more recently by Charles [Reference CharlesCha17].

Proof of Theorem 4.1 By Lemma 3.16, it is sufficient to assume that $\bar {D}$ is not pseudo-effective and to show that $\zeta _{\mathrm {ess}}(\bar {D}) < 0$. By Lemma 4.3, there exist a normal projective arithmetic variety $\mathcal {X}$ on $\operatorname {\mathrm {Spec}} {\mathcal {O}}_K$ and an arithmetic ${\mathbb {R}}$-Cartier divisor $\bar {\mathcal {D}} = (\mathcal {D},(g_{v})_{v \in \Sigma _{K,\infty }})$ such that:

1. (i) $\bar {\mathcal {D}}$ is semi-positive of $C^{\infty }$-type;

2. (ii) $\bar {\mathcal {D}}$ is not pseudo-effective;

3. (iii) $\zeta _{\mathrm {ess}}(\bar {\mathcal {D}}^{\mathrm {ad}}) > \zeta _{\mathrm {ess}} (\bar {D})$.

It is enough to show that $\zeta _{\mathrm {ess}}(\bar {\mathcal {D}}^{\mathrm {ad}}) \leq 0$. By homogeneity and continuity of the essential minimum (Lemma 3.15(iii)), we may assume that $\bar {\mathcal {D}}^{\mathrm {ad}} \in \widehat {\operatorname {\mathrm {Div}}}(\mathcal {X})_{{\mathbb {Z}}}$. Since $\bar {\mathcal {D}}$ is not pseudo-effective, by Theorem 4.6 there exist a birational morphism $\varphi \colon \mathcal {X}' \rightarrow \mathcal {X}$, with $\mathcal {X}'$ normal and generically smooth, and an ample arithmetic ${\mathbb {Q}}$-divisor $\bar {\mathcal {H}}$ on $\mathcal {X}'$ such that

\[ \widehat{\deg}(\varphi^{*}\bar{\mathcal{D}} \cdot \bar{\mathcal{H}}^{d}) <0, \]

where $d = \dim X$. Without loss of generality, we assume that $\mathcal {H} \in \operatorname {\mathrm {Div}}(\mathcal {X}^{\prime})_{{\mathbb {Z}}}$. Let $\mathcal {Y} \subsetneq \mathcal {X}$ be a closed subscheme. Since $\bar {\mathcal {H}}$ is ample, [Reference IkomaIko15a, Theorem A] implies the existence of an integer $n\geq 1$ and a non-zero function $\phi \in H(\mathcal {X}',n\mathcal {H})$ with $\max _{v \in \Sigma _{K,\infty }}\|\phi \|_{v,\sup } <1$ such that the divisor $(\phi )+n\mathcal {H}$ is generically smooth and intersects $\mathcal {Y}':=\varphi ^{-1}(\mathcal {Y})$ properly. We write

\[ (\phi) + n\mathcal{H}= \sum_{i=1}^{\ell} a_i \mathcal{Z}_i \]

for the decomposition of $(\phi ) +n\mathcal {H}$ as a cycle: for each $i \in \{1,\ldots , \ell \}$, $a_i> 0$ is an integer and $\mathcal {Z}_i \subset \mathcal {X}'$ is a generically smooth subvariety of codimension $1$ not contained in $\mathcal {Y}'$. We have

(4.2)\begin{equation} \widehat{\deg}(\varphi^{*}\bar{\mathcal{D}} \cdot (n\bar{\mathcal{H}})^{d}) = \sum_{i=1}^{\ell} a_i\widehat{\deg}(\bar{\mathcal{D}}_{|\mathcal{Z}_i} \cdot (n\bar{\mathcal{H}}_{|\mathcal{Z}_i})^{d-1}) -\sum_{v \in \Sigma_{K,\infty}}\int_{X_v^{\mathrm{an}}} \ln \|\phi\|_v c_{1,v}(\varphi^{*}\bar{\mathcal{D}})\wedge c_{1,v}(n\bar{\mathcal{H}})^{d-1}. \end{equation}

Since $\varphi ^{*}\bar {\mathcal {D}}$ is semi-positive and $\bar {\mathcal {H}}$ is ample, the current $c_{1,v}(\varphi ^{*}\bar {\mathcal {D}})\wedge c_{1,v}(n\bar {\mathcal {H}})^{d-1}$ is non-negative for each $v \in \Sigma _{K,\infty }$. On the other hand, $\|\phi \|_{v, \sup } \leq 1$, hence we have

\[ -\int_{X_v^{\mathrm{an}}} \ln \|\phi\|_v c_{1,v}(\varphi^{*}\bar{\mathcal{D}})\wedge c_{1,v}(\overline{n\mathcal{H}})^{d-1} \geq 0 \]

for each $v \in \Sigma _{K,\infty }$. It follows from (4.2) that

\[ \sum_{i=1}^{\ell} a_i\widehat{\deg}(\bar{\mathcal{D}}_{|\mathcal{Z}_i} \cdot (n\bar{\mathcal{H}}_{|\mathcal{Z}_i})^{d-1}) \leq \widehat{\deg}(\varphi^{*}\bar{\mathcal{D}} \cdot (n\bar{\mathcal{H}})^{d}) = n^{d}\widehat{\deg}(\varphi^{*}\bar{\mathcal{D}} \cdot \bar{\mathcal{H}}^{d}) < 0. \]

Since $a_i > 0$ for all $1 \leq i \leq \ell$, there exists a codimension $1$ generically smooth subvariety $\mathcal {Z} \subset \mathcal {X}'$ not contained in $\mathcal {Y}'$ such that

\[ \widehat{\deg}(\varphi^{*}\bar{\mathcal{D}}_{|\mathcal{Z}} \cdot \bar{\mathcal{H}}_{|\mathcal{Z}}^{d-1})<0. \]

Since $\bar {\mathcal {H}}_{|\mathcal {Z}}$ is ample, we can iterate this process. Eventually we obtain an integral curve $\mathcal {C}' \subset \mathcal {X}'$ not contained in $\mathcal {Y}'$ such that

\[ \widehat{\deg}(\varphi^{*}\bar{\mathcal{D}}_{|\mathcal{C}'}) < 0. \]

We denote by $\pi _{\mathcal {X}} \colon \mathcal {X} \rightarrow \mathcal {S} = \operatorname {\mathrm {Spec}} {\mathcal {O}}_K$ the morphism defining the model $\mathcal {X}$, and we let $\pi _{\mathcal {X}'} = \pi _{\mathcal {X}}\circ \varphi$. Assume that $\pi _{\mathcal {X}'}(\mathcal {C}') \in \mathcal {S}$ is a closed point, and let $p$ be its image in $\operatorname {\mathrm {Spec}} {\mathbb {Z}}$. In this case we have

\[ \widehat{\deg}(\varphi^{*}\bar{\mathcal{D}}_{|\mathcal{C}'}) = \ln(p)\deg(\varphi^{*}\mathcal{D} \cdot \mathcal{C}') < 0, \]

which is absurd since $\varphi ^{*}\mathcal {D}$ is relatively nef. It follows that $\mathcal {C}'$ is horizontal, that is, $\pi _{\mathcal {X}'}(\mathcal {C}') = \mathcal {S}$. The image $\varphi (\eta _{\mathcal {C}'})$ of the generic point of $\mathcal {C}'$ by $\varphi$ defines a closed point $x \in X(\bar {K}){\setminus} \mathcal {Y}$ of height $h_{\bar {\mathcal {D}}^{\mathrm {ad}}}(x) < 0$. It follows that

\[ \inf \{h_{\bar{\mathcal{D}}^{\mathrm{ad}}}(x) \mid x \in X(\bar{K}),\ x \notin \mathcal{Y}\} < 0. \]

By taking the supremum on the Zariski-closed proper subschemes $\mathcal {Y} \subset \mathcal {X}$ we find $\zeta _{\mathrm {ess}}(\bar {\mathcal {D}}^{\mathrm {ad}}) \leq 0$, which concludes the proof.

5.1 Definitions and properties

Let $\bar {D} \in \widehat {\operatorname {\mathrm {Div}}}(X)_{\mathbb {R}}$. For any real number $t$ and any integer $n \geq 1$, we consider the $K$-linear subspace $V_n^{t}$ of $V_n = H^{0}(X,nD)$ generated by functions $\phi \in V_n$ such that for each place $v \in \Sigma _K$,

\[ \|\phi\|_{v,\sup} \leq \left\{\begin{array}{@{}ll} e^{{-}t} & \textrm{if}\ v | v_0,\\ 1 & \textrm{otherwise}. \end{array} \right. \]

We let

\[ \lambda_{\max,n}(\bar{D}) := \sup \{t \in {\mathbb{R}} \mid V_n^{t} \ne 0\}\quad\text{and}\quad \lambda_{\min,n}(\bar{D}) := \sup \{t \in {\mathbb{R}} \mid V_n^{t} = V_n\}. \]

Example 5.1 (1) Suppose that $K$ is a number field and that $\bar {D} = \bar {\mathcal {D}}^{\mathrm {ad}}$ for some arithmetic ${\mathbb {R}}$-Cartier divisor $\bar {\mathcal {D}}$ on a normal model $\mathcal {X}$ of $X$. Then for any $n \geq 1$ and $t \in {\mathbb {R}}$, $V_n^{t}$ identifies with the set

\[ \Big\{\phi \in H^{0}(\mathcal{X}, n \mathcal{D}) \mid \max_{v \in \Sigma_{K,\infty}}\|\phi\|_{v, \sup} \leq e^{{-}t}\Big\}. \]

(2) Suppose that $K$ is a number field and let $n \geq 1$ be an integer. Then $\exp (-\lambda _{\max ,n}(\bar {D}))$ and $\exp (-\lambda _{\min ,n}(\bar {D}))$ are respectively the first and last minima of $(V_n, ({ \|\cdot \|_{v,\sup }})_{v \in \Sigma _K})$ in the sense of Bombieri and Vaaler [Reference Bombieri and VaalerBV83, § III.1].

Definition 5.2 The asymptotic maximal slope of $\bar {D}$ is the quantity

\[ {\hat{\mu}^{\mathrm{asy}}_\mathrm{max}} (\bar{D}) = \limsup_{n \rightarrow +\infty} \frac{\lambda_{\max,n}(\bar{D})}{n}. \]

The asymptotic minimal slope of $\bar {D}$ is

\[ {\hat{\mu}^{\mathrm{asy}}_\mathrm{min}} (\bar{D}) = \liminf_{n \rightarrow +\infty} \frac{\lambda_{\min,n}(\bar{D})}{n}. \]

Remark 5.3 For any integer $n \geq 1$ the $K$-vector space $V_n$ equipped with the collections of norms $(\|\cdot \|_{v,\sup })_{v \in \Sigma _K}$ is an adelic $K$-vector space in the sense of Gaudron [Reference GaudronGau08], and we let ${\hat {\mu }_\mathrm {max}}(\bar {V}_n)$ be its maximal slope as defined in [Reference GaudronGau08, § 5.1]. By [Reference GaudronGau08, Theorem 5.20], we have ${\hat {\mu }_\mathrm {max}}(\bar {V}_n) = \lambda _{\max ,n}(\bar {D}) + o(n)$. Therefore

\[ {\hat{\mu}^{\mathrm{asy}}_\mathrm{max}} (\bar{D}) = \limsup_{n \rightarrow +\infty} \frac{{\hat{\mu}_\mathrm{max}}(\bar{V}_n)}{n}. \]

In particular, Definition 5.2 of ${\hat {\mu }^{\mathrm {asy}}_\mathrm {max}} (\bar {D})$ is consistent with that of [Reference ChenChe10b]. The same is true for the asymptotic minimal slope.

The asymptotic slopes of $\bar {D}$ are actually limits under suitable positivity conditions on $D$.

Proposition 5.4 [Reference ChenChe10b, Theorem 4.1.2]

If $D$ is big, the sequence $(\lambda _{\max ,n}(\bar {D})/n)_{n \in {\mathbb {N}}}$ converges in ${\mathbb {R}}$: ${\hat {\mu }^{\mathrm {asy}}_\mathrm {max}} (\bar {D}) = \lim _{n \rightarrow +\infty } \lambda _{\max ,n}(\bar {D})/n$. Moreover, if the graded $K$-algebra

\[ V_\bullet{=} \bigoplus_{n \in {\mathbb{N}}} V_n \]

is finitely generated, the sequence $(\lambda _{\min ,n}(\bar {D})/n)_{n \in {\mathbb {N}}}$ converges in ${\mathbb {R}}$:

\[ {\hat{\mu}^{\mathrm{asy}}_\mathrm{min}} (\bar{D}) = \lim_{n \rightarrow +\infty} \lambda_{\min,n}(\bar{D})/n. \]

The following result of Chen [Reference ChenChe10a, Proposition 3.11] characterizes the positivity of the asymptotic maximal slope in terms of the bigness of $\bar {D}$. It holds in the more general setting of adelic curves by Chen and Moriwaki [Reference Chen and MoriwakiCM19].

5.2 Proof of Theorem 1.2

The goal of this subsection is to prove the following theorem, which is a slightly more precise version of Theorem 1.2 in the introduction. We say that a divisor $D \in \operatorname {\mathrm {Div}}(X)_{\mathbb {Q}}$ is semi-ample if there exists an integer $n \geq 1$ such that $nD \in \operatorname {\mathrm {Div}}(X)_{\mathbb {Z}}$ and ${\mathcal {O}}_X(nD)$ is generated by global sections.

Note that the first assertion of this theorem implies Conjecture 1. As we will see, it is an immediate consequence of Corollary 4.2 and Proposition 5.5. We shall derive Theorem 5.6(ii) from an arithmetic Nakai–Moishezon criterion for ampleness, originally proved by Zhang when $K$ is a number field [Reference ZhangZha95a, Theorem 4.2] and extended to the much wider setting of adelic curves by Chen and Moriwaki [Reference Chen and MoriwakiCM19], under weaker positivity assumptions.

Proof. (i) Assume that $D$ is big. Let $\bar {\xi }_0 = (\xi _{0,v})_{v \in \Sigma _K}$ be the adelic divisor on $\operatorname {\mathrm {Spec}} K$ given by

\[ \xi_{0,v} = \left\{\begin{array}{@{}ll} 2 & \textrm{if}\ v | v_0,\\ 0 & \textrm{otherwise}. \end{array} \right. \]

It follows from the definitions that ${\hat {\mu }^{\mathrm {asy}}_\mathrm {max}}(\bar {D} - t\pi ^{*}\bar {\xi }_0) = {\hat {\mu }^{\mathrm {asy}}_\mathrm {max}}(\bar {D}) -t$ for any $t \in {\mathbb {R}}$. Applying Proposition 5.5 to ${\hat {\mu }^{\mathrm {asy}}_\mathrm {max}}(\bar {D} - t\pi ^{*}\bar {\xi }_0)$ when $t \in {\mathbb {R}}$ varies, we find

\begin{align*} {\hat{\mu}^{\mathrm{asy}}_\mathrm{max}}(\bar{D}) & = \sup\{t \in {\mathbb{R}} \mid \bar{D} - t\pi^{*}\bar{\xi}_0 \text{ is big}\}\\ & = \sup\{t \in {\mathbb{R}} \mid \bar{D} - t\pi^{*}\bar{\xi}_0 \text{ is pseudo-effective}\}, \end{align*}

where the second equality follows from the definition of pseudo-effectivity. We conclude with Corollary 4.2 (note that $\widehat {\deg }(\bar {\xi }_0)=1$).

(ii) For $t \in {\mathbb {R}}$, we let $\bar {D}_t := \bar {D} - t\pi ^{*}\bar {\xi }_0$. It follows from the projection formula for the height (3.1) and the definition of $\zeta _{\mathrm {abs}}(\bar {D})$ that

\[ \zeta_{\mathrm{abs}}(\bar{D}) = \sup \{t \in {\mathbb{R}} \mid \bar{D}_t \text{ is nef}\}. \]

Let $n \geq 1$ be an integer and let $(\phi _1, \ldots , \phi _{\ell _n})$ be a basis for $H^{0}(X,nD)$. Let $x \in X(\bar {K})$ be a point with residue field ${ \kappa }(x)$. Since $D$ is semi-ample, there exists $i \in \{1, \ldots , \ell _n\}$ such that $x\notin \operatorname {\mathrm {Supp}} ({ nD + (\phi _i)})$ provided that $n$ is sufficiently large and divisible. It follows that

\[ nh_{\bar{D}}(x) ={-}\sum_{w \in \Sigma_{{ \kappa}(x)}}n_w({ \kappa}(x)) \ln\|\phi_i\|_w(x) \geq{-}\sum_{w \in \Sigma_{{ \kappa}(x)}}n_w({ \kappa}(x)) \ln\|\phi_i\|_{w,\sup}. \]

We infer that $h_{\bar {D}}(x) \geq \lambda _{\min , n}(\bar {D})/n$ for every $x \in X(\bar {K})$ and $n \gg 1$ sufficiently divisible. By the projection formula (3.1), $D_t = \bar {D} -t\pi ^{*}\bar {\xi }_0$ is nef for every $t \leq {\hat {\mu }^{\mathrm {asy}}_\mathrm {min}}(\bar {D})$, so that

\[ {\hat{\mu}^{\mathrm{asy}}_\mathrm{min}}(\bar{D}) \leq \sup\{t \in {\mathbb{R}} \mid \bar{D}_t \text{ is nef}\} = \zeta_{\mathrm{abs}}(\bar{D}). \]

We now assume that $\bar {D}$ is semi-positive to prove the converse inequality. By definition, ${\hat {\mu }^{\mathrm {asy}}_\mathrm {min}}(\bar {D}_t) = {\hat {\mu }^{\mathrm {asy}}_\mathrm {min}}(\bar {D})-t$ for any $t \in {\mathbb {R}}$. By the above, there exists $t_0 \in {\mathbb {R}}$ such that $\bar {D}_t$ is nef for every $t \leq t_0$. Let $t < t_0$ be a rational number and let $Y \subset X$ be an irreducible subvariety. We have

\[ {\hat{\mu}^{\mathrm{asy}}_\mathrm{max}}(\bar{D}_{t|Y}) = \zeta_{\mathrm{ess}}({\bar{D}_{t|Y}}) \geq \zeta_{\mathrm{abs}}({\bar{D}_{t|Y}}) > \zeta_{\mathrm{abs}}(\bar{D}_{t_0}) \geq 0. \]

By definition of ${\hat {\mu }^{\mathrm {asy}}_\mathrm {max}}({\bar {D}_{t|Y}})$ there exists an integer $n_Y\geq 1$ such that $\hat {H}^{0}(Y,n{\bar {D}_{t|Y}}) \ne 0$ for every $n \geq n_Y$. By the arithmetic Nakai–Moishezon criterion [Reference Chen and MoriwakiCM19, Theorem 7.4.1], we have ${\hat {\mu }^{\mathrm {asy}}_\mathrm {min}}(\bar {D})-t = {\hat {\mu }^{\mathrm {asy}}_\mathrm {min}}(\bar {D}_t) \geq 0$, and finally

\[ {\hat{\mu}^{\mathrm{asy}}_\mathrm{min}}(\bar{D}) \geq \sup\{t \in {\mathbb{R}} \mid \bar{D}_t \text{ is nef}\} = \zeta_{\mathrm{abs}}(\bar{D}). \]

Remark 5.7 When $D$ is big, we proved that

(5.1)\begin{equation} {\hat{\mu}^{\mathrm{asy}}_\mathrm{max}}(\bar{D}) = \sup\{t \in {\mathbb{R}} \mid \bar{D} - t\pi^{*}\bar{\xi}_0 \text{ is pseudo-effective}\}, \end{equation}

where $\bar {\xi }_0 = (\xi _{0,v})_{v \in \Sigma _K} \in \widehat {\operatorname {\mathrm {Div}}}(\operatorname {\mathrm {Spec}} K)_{\mathbb {R}}$ is defined by $\xi _{0,v} = 2$ if $v |v_0$ and $\xi _{0,v} = 0$ otherwise. This equality actually holds for any $\bar {\xi } \in \widehat {\operatorname {\mathrm {Div}}}(\operatorname {\mathrm {Spec}} K)_{\mathbb {R}}$ with $\widehat {\deg }(\bar {\xi }) = 1$ (see [Reference Chen and MoriwakiCM15]). Note that (5.1) shows that Theorem 4.1 is actually equivalent to Conjecture 1.

6.1 Okounkov bodies

We briefly recall the construction of an Okounkov body attached to a graded linear series of a divisor, following [Reference Lazarsfeld and MustaţăLM09] and [Reference MoriwakiMor12a]. Let $K$ be a global field and $X \rightarrow \operatorname {\mathrm {Spec}} K$ be a projective, normal and geometrically integral variety of dimension $d := \dim X$. Let $D\in{\rm Div}(X)_{\mathbb{R}}$, and let $D_{\bar {K}}$ be the pullback of $D$ to $X_{\bar {K}} = X\times _K \operatorname {\mathrm {Spec}} \bar {K}$. The choice of a system of parameters $\mathbf {z}$ centred at a regular point $P \in X(\bar {K})$ defines a function

\[ \nu_{\mathbf{z}} \colon H^{0}(X_{\bar{K}},D_{\bar{K}}){\setminus} \{0\} \rightarrow {\mathbb{R}}^{d} \]

satisfying the following conditions (see [Reference MoriwakiMor12a, § 1] for details):

• – for every $a \in F^{\times }$ and $\phi \in H^{0}(X_{\bar {K}},D_{\bar {K}}){\setminus} \{0\}$, $\nu _{\mathbf {z}}(a\phi ) = \nu _{\mathbf {z}}(\phi )$;

• – for all $\phi _1,\phi _2 \in H^{0}(X_{\bar {K}},D_{\bar {K}}){\setminus} \{0\}$,

  \[ \nu_{\mathbf{z}}(\phi_1\phi_2) = \nu_{\mathbf{z}}(\phi_1) + \nu_{\mathbf{z}}(\phi_2) \text{ and } \nu_{\mathbf{z}}(\phi_1 +\phi_2) \geq_{\mathrm{lex}} \min\{\nu_{\mathbf{z}}(\phi_1),\nu_{\mathbf{z}}(\phi_2)\}, \]

where $\geq _{\mathrm {lex}}$ denotes the lexicographic order on ${\mathbb {R}}^{d}$. Given a graded $K$-subalgebra $W_\bullet = \bigoplus _{n \in {\mathbb {N}}}W_n$ of $V_\bullet (D):= \bigoplus _{n \in {\mathbb {N}}} H^{0}(X,nD)$, we consider the subset of ${\mathbb {R}}^{d}$ defined by

\[ \Gamma_{\mathbf{z}}(W_\bullet) = \bigcup_{n \geq 1} \frac{1}{n}\nu_{\mathbf{z}}(W_n \otimes_K \bar{K} {\setminus} \{0\}) = \biggl\{ \frac{\nu_{\mathbf{z}}(\phi)}{n} \biggm| n \geq 1, \ 0 \ne \phi \in W_n \otimes_K \bar{K} \biggr\}. \]

Definition 6.1 The Okounkov body of $W_\bullet$ with respect to the system of parameters $\mathbf {z}$ is defined to be the closure

\[ \Delta_{\mathbf{z}}(W_\bullet) = \overline{\Gamma_{\mathbf{z}}(W_\bullet)} \]

of $\Gamma _{\mathbf {z}}(W_\bullet )$ in ${\mathbb {R}}^{d}$ for the euclidean topology. We call $\Delta _{\mathbf {z}}(D) = \Delta _{\mathbf {z}}(V_\bullet (D))$ the Okounkov body of $D$ with respect to $\mathbf {z}$.

When $D$ is big, $\Delta _{\mathbf {z}}(D) \subset {\mathbb {R}}^{d}$ is a convex body (see [Reference Lazarsfeld and MustaţăLM09, Proposition 2.1]), and we have $\operatorname {\mathrm {vol}}(D) = d!\operatorname {\mathrm {vol}}(\Delta _{\mathbf {z}}(D))$.

6.2 The concave transform

Let $\bar {D} = (D,(g_v)_{v \in \Sigma _K})$ be an adelic ${\mathbb {R}}$-Cartier divisor on $X$ with $D$ big. For any $t \in {\mathbb {R}}$ and $n\in {\mathbb {N}}$, let $V_n^{t}$ be the $K$-linear subspace of $V_n = H^{0}(X,nD)$ defined in § 5. We consider the graded $K$-subalgebra $V_\bullet ^{t} \subset V_\bullet (D)$ defined by $V_\bullet ^{t} = \bigoplus _{n \in {\mathbb {N}}} V_n^{nt}$.

Definition 6.2 The concave transform of $\bar {D}$ with respect to $\mathbf {z}$ is the function $G_{\bar {D},\mathbf {z}}\colon \Delta _{\mathbf {z}}(D) \rightarrow {\mathbb {R}} \cup \{-\infty \}$ defined by

\[ G_{\bar{D},\mathbf{z}}(\alpha) = \sup \{t \mid \alpha \in \Delta_{\mathbf{z}}(V_\bullet^{t})\} \]

for every $\alpha \in \Delta _{\mathbf {z}}(D)$.

By [Reference Boucksom and ChenBC11, § 1.3], $G_{\bar {D},\mathbf {z}}$ is an upper-semicontinuous concave function and we have

\[ {\hat{\mu}^{\mathrm{asy}}_\mathrm{min}}(\bar{D}) \leq G_{\bar{D},\mathbf{z}}(\alpha) \leq {\hat{\mu}^{\mathrm{asy}}_\mathrm{max}}(\bar{D}) \]

for any $\alpha \in \Delta _{\mathbf {z}}(D)$. The maximal asymptotic slope actually coincides with the maximum of $G_{\bar {D},\mathbf {z}}$.

Lemma 6.3 Assume that $D$ is big. Then we have

\[ {\hat{\mu}^{\mathrm{asy}}_\mathrm{max}}(\bar{D}) = \max_{\alpha \in \Delta_{\mathbf{z}}(D)}G_{\bar{D},\mathbf{z}}(\alpha) . \]

Proof. We have already seen that $G_{\bar {D},\mathbf {z}}(\alpha ) \leq {\hat {\mu }^{\mathrm {asy}}_\mathrm {max}}(\bar {D})$ for any $\alpha \in \Delta _{\mathbf {z}}(D)$. For the reverse inequality, let $\bar {\xi }_0 \in \widehat {\operatorname {\mathrm {Div}}}(\operatorname {\mathrm {Spec}} K)_{\mathbb {R}}$ be the ${\mathbb {R}}$-adelic divisor defined in Remark 5.7. Let $t \in {\mathbb {R}}$ be such that $\bar {D} - t\pi ^{*}\bar {\xi }_0$ is big. For $n \geq 1$ sufficiently large, there exists a non-zero $\phi \in \hat {H}^{0}(X,n\bar {D}-nt\pi ^{*}\bar {\xi }_0) \subset V^{nt}_n$. It follows that $\alpha _\phi := ({1}/{n})\nu _{\mathbf {z}}(s) \in \Delta _{\mathbf {z}}(V_\bullet ^{t})$, hence $G_{\bar {D},\mathbf {z}}(\alpha _\phi ) \geq t$. Therefore

\begin{align*} G_{\bar{D},\mathbf{z}}(\alpha_\phi)& \geq \sup\{t \in {\mathbb{R}} \mid \bar{D} - t\pi^{*}\bar{\xi}_0 \text{ is big}\}\\ & = \sup\{t \in {\mathbb{R}} \mid \bar{D} - t\pi^{*}\bar{\xi}_0 \text{ is pseudo-effective}\}={\hat{\mu}^{\mathrm{asy}}_\mathrm{max}}(\bar{D}), \end{align*}

so that $\max _{\alpha \in \Delta _{\mathbf {z}}(D)}G_{\bar {D},\mathbf {z}}(\alpha )\geq {\hat {\mu }^{\mathrm {asy}}_\mathrm {max}}(\bar {D})$.

6.3 Concave transform and volume functions

Theorem 6.4 below gives two useful formulae due to Boucksom and Chen [Reference Boucksom and ChenBC11], expressing the volumes $\widehat {\operatorname {\mathrm {vol}}}(\bar {D})$ and $\widehat {\operatorname {\mathrm {vol}}}_{\chi }(\bar {D})$ in terms of the concave transform. We first recall the definition of $\widehat {\operatorname {\mathrm {vol}}}_{\chi }(\bar {D})$ for an adelic ${\mathbb {R}}$-Cartier divisor $\bar {D}$ on $X$. Let $\mathbb {A}_K$ be the ring of adèles of $K$. For any integer $n \geq 1$, we let $V_n= H^{0}(X,nD)$ and we consider the adelic unit ball $\mathbb {B}(V_n) \subset V_n\otimes _K \mathbb {A}_K$ defined by

\[ \mathbb{B}(V_n) = \{(\phi_v) \in V_n \otimes_K \mathbb{A}_K\mid \|\phi_v\|_{v,\text{sup}} \leq 1\ \forall v \in \Sigma_K\}. \]

We denote by $\nu$ the Haar measure on $V_n \otimes _K \mathbb {A}_K$ with $\nu ((V_n \otimes _K \mathbb {A}_K)/V_n)=1$. We define the $\chi$-volume of $\bar {D}$ by

\[ \widehat{\operatorname{\mathrm{vol}}}_{\chi}(\bar{D}) = \limsup_{n \rightarrow +\infty} \frac{\nu(\mathbb{B}(V_n))}{n^{d+1}/(d+1)!}. \]

Theorem 6.4 (Boucksom and Chen [Reference Boucksom and ChenBC11])

Let $\bar {D}$ be an adelic ${\mathbb {R}}$-Cartier divisor on $X$ such that $D$ is big. Then

\[ \widehat{\operatorname{\mathrm{vol}}}(\bar{D}) = (d+1)![K:K_0] \int_{\Delta_{\mathbf{z}}(D)} \max\{0,G_{\bar{D},\mathbf{z}}\} \,d\lambda \]

and

\[ \widehat{\operatorname{\mathrm{vol}}}_{\chi}(\bar{D}) \leq (d+1)! [K:K_0]\int_{\Delta_{\mathbf{z}}(D)} G_{\bar{D},\mathbf{z}} \,d\lambda, \]

with equality if $\inf _{\alpha \in \Delta _{\mathbf {z}}(D)} G_{\bar {D},\mathbf {z}}(\alpha ) > -\infty$.