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 $\overline{\operatorname{Pic}}_{\text{ad}}(Y)$.