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 $f_{*}(Z)=0$.