Let $F$ be either a number field (arithmetic case), or a finite extension of the field of rational functions over a constant field (geometric case). Let $X$ be a projective variety over $F$, Let $M(F)$ be the set of normalized absolute values on $F$. Any $v\in M(F)$ gives rise to a complete valued field $F_{v}$, and to an analytic space $X_{v}$ over $F_{v}$ : if $v$ is archimedean, $X_{v}=X(\overline{F_{v}})$, while $X_{v}$ is the Berkovich analytic space attached to $X_{F_{v}}$ if $v$ is ultrametric.

If $L$ is a line bundle on $X$, an adelic metric on $L$ is a family $(\left\|{\cdot}\right\|_{v})_{v\in M(F)}$ of continuous metrics on the induced line bundles over the analytic spaces $X_{v}$. We require the following supplementary compatibility assumption : there exists a model $(\mathfrak{X},\mathscr{L},e)$ over the ring of integers of $F$ inducing the given metrics at almost all places $v$. An adelic metric is said to be semi-positive, resp. admissible if it is so at all places of $F$.

Line bundles on $X$ endowed with an adelic metric form a group $\overline{\operatorname{Pic}}(X)$ ; admissible line bundles form a subgroup $\overline{\operatorname{Pic}}_{\text{ad}}(X)$. If $f\colon Y\rightarrow X$ is any morphism, there is a natural morphism of groups $f^{*}\colon\overline{\operatorname{Pic}}(X)\rightarrow\overline{\operatorname{Pic}}(Y)$ ; it maps $\overline{\operatorname{Pic}}_{\text{ad}}(X)$ into $\overline{\operatorname{Pic}}_{\text{ad}}(Y)$.

Consider line bundles $\overline{L}_{0},\dots,\overline{L}_{n}$ with admissible adelic metrics. Let $Z$ be a subvariety of $X$ of dimension $k$ and $s_{0},\dots,s_{k}$ invertible meromorphic sections of $L_{0},\dots,L_{k}$ whose divisors hace no common intersection point on $Z$. For any $v\in M(F)$, we have recalled in Sections 1.2, 1.2 and 1.3 the definitions of the local height pairing

$$(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|Z)_{v}$$

where the index $v$ indicates the corresponding place of $F$. The global height is the sum, over all $v\in M(F)$, of these local heights :

$$(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|Z)=\sum_{v\in M(F)}(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|Z)_{v}.$$

It inherits from the local heights their multilinear symmetric character.

Let us replace $s_{k}$ by another invertible meromorphic section $fs_{k}$. Then,

	$\displaystyle(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(fs_{k})|Z)$	$\displaystyle=\sum_{v\in M(F)}(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(fs_{k})|Z)_{v}$
		$\displaystyle=\sum_{v\in M(F)}(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|Z)_{v}$
		$\displaystyle\qquad+\sum_{v\in M(F)}\int_{X_{v}}\log\left|{f}\right|^{-1}c_{1}(\overline{L}_{0})\dots c_{1}(\overline{L}_{k-1})\delta_{\mathrm{Z}_{v}}.$

In particular, if $Z$ is a point $z\in X(F)$, then $\delta_{\mathrm{Z}_{v}}=\delta_{z}$ is the Dirac mass at $z$ and

$$(\mathop{\widehat{\operatorname{div}}}(s_{0})|Z)=\sum_{v\in M(F)}\log\left\|{s_{0}}\right\|^{-1}_{v}(z).$$

Let us observe that it is independent on the choice of the chosen meromorphic section $s_{0}$, provided it is regular at $z$. Any other section has the form $fs_{0}$, for some invertible meromorphic function $f$ on $X$. Then,

	$\displaystyle(\mathop{\widehat{\operatorname{div}}}(fs_{0})|Z)$	$\displaystyle=\sum_{v\in M(F)}\log\left\|{fs_{0}}\right\|^{-1}_{v}(z)$
		$\displaystyle=\sum_{v\in M(F)}\log\left\|{s_{0}}\right\|^{-1}_{v}(z)+\sum_{v\in M(F)}\log\left|{f}\right|_{v}^{-1}(z)$
		$\displaystyle=(\mathop{\widehat{\operatorname{div}}}(fs_{0})|Z)$

since, by the product formula, the second term vanishes.

By induction on the dimension of $Z$, and using the commutativity of the local height pairings, it follows that the global height only depends on the metrized line bundles, and not on the actual chosen sections $s_{0},\dots,s_{k}$. We denote it by

$$({\widehat{c}}_{1}(\overline{L}_{0})\dots{\widehat{c}}_{1}(\overline{L}_{k})|Z).$$

Again, it is multilinear symmetric in the metrized line bundles $\overline{L}_{0},\dots,\overline{L}_{k}$. By the same argument, it only depends on their isomorphism classes in $\overline{\operatorname{Pic}}_{\text{ad}}(X)$.

It satisfies a projection formula : for any morphism $f\colon Y\rightarrow X$ and any $k$-dimensional subvariety $Z$ of $Y$,

$$({\widehat{c}}_{1}(f^{*}\overline{L}_{0})\dots{\widehat{c}}_{1}(f^{*}\overline{L}_{k})|Z)=({\widehat{c}}_{1}(\overline{L}_{0})\dots{\widehat{c}}_{1}(\overline{L}_{k})|f_{*}(Z),$$

where the cycle $f_{*}(Z)$ is defined as $\deg(Z/f(Z))f(Z)$ if $Z$ and $f(Z)$ have the same dimension, so that $f\colon Z\rightarrow f(Z)$ is generically finite, of some degree $\deg(Z/f(Z))$. If $Z$ and $f(Z)$ don’t have the same dimension, one sets $f_{*}(Z)=0$.

The height of an algebraic point is an important tool in Diophantine geometry. If $\overline{L}$ is a line bundle with an adelic metric on $X$, then for any point $P\in X(F)$, viewed as a closed subscheme of $X$, one has

$$h_{\overline{L}}(P)=({\widehat{c}}_{1}(\overline{L})|P)=\sum_{v}\log\left\|{s}\right\|^{-1}_{v}(P),$$

where $s$ is any meromorphic section on $L$ which has neither a zero nor a pole at $P$. More generally, let $P\in X(\overline{F})$ be an algebraic point and let $[P]$ be the corresponding closed point of $X$. Then,

$$h_{\overline{L}}(P)=\frac{1}{[F(P):F]}({\widehat{c}}_{1}(\overline{L})|[P])$$

is the height of $P$ with respect to the metrized line bundle $\overline{L}$. In fact, restricted to points, these definitions apply to any, not necessary admissible,

Observe also the following functorial property of the height : If $f\colon Y\rightarrow X$ is a morphism and $P\in Y(\overline{F})$, then $h_{f^{*}\overline{L}}(P)=h_{\overline{L}}(f(P))$. Finally, recall that if $F$ is a global field, then the height with respect to a metrized ample line bundle $\overline{L}$ satisfies Northcott’s finiteness property : for any integers $d$ and $B$, there are only finitely many points $P\in X(\overline{F})$ such that $[F(P):F]\leq d$ and $h_{\overline{L}}(P)\leq B$.

The essential minimum of the height $h_{\overline{L}}$ is defined as

$$e(\overline{L})=\sup_{\begin{subarray}{c}\emptyset\neq U\subset X\end{subarray}}\inf_{P\in U(\overline{F})}h_{\overline{L}}(P),$$

where the supremum runs over non-empty open subsets of $X$. If $L$ is big, then $e(\overline{L})$ is a real number. Another way to state its definition is the following : for any real number $B$, then the set

$$\{P\in X(\overline{F})\,;\,h_{\overline{L}}(P)\leq B\}$$

is Zariski dense if $B>e(\overline{L})$, and is not Zariski dense if $B<e(\overline{L})$.

Assume that $\overline{L}$ is an ample line bundle on $X$, equipped with a semi-positive adelic metric. The (geometric/arithmetic) Hilbert-Samuel theorem implies the following inequality

$$e(\overline{L})\geq\frac{({\widehat{c}}_{1}(\overline{L})^{n+1}|X)}{(n+1)(c_{1}(L)^{n}|X)}.$$

(See Zhang [59], as well as [38, 28] for more details in the geometric case). When $X$ is a curve and $F$ is a number field, Autissier [3] proved that the inequality holds for any ample line bundle with an admissible adelic metric (see [18]) ; this extends to the geometric case.