Let $X$ be a topological space together with a sheaf of local rings $\mathscr{O}_{X}$ (“analytic functions”) ; let also $\mathscr{C}_{X}$ be the sheaf of continuous functions on $X$. In analytic geometry, local functions have an absolute value which is a real valued continuous function, satisfying the triangle inequality. Let us thus assume that we have a morphism of sheaves $\mathscr{O}_{X}\rightarrow\mathscr{C}_{X}$, written $f\mapsto\left|{f}\right|$, such that $\left|{fg}\right|=\left|{f}\right|\left|{g}\right|$, $\left|{1}\right|=1$, and $\left|{f+g}\right|\leq\left|{f}\right|+\left|{g}\right|$.

A line bundle on $(X,\mathscr{O}_{X})$ is a sheaf $L$ of $\mathscr{O}_{X}$-modules which is locally isomorphic to $\mathscr{O}_{X}$. In other words, $X$ is covered by open sets $U$ such that $\mathscr{O}_{U}\simeq L|U$ ; such an isomorphism is equivalent to a non-vanishing section $\varepsilon_{U}\in\Gamma(U,L)$, also called a local frame of $L$.

If $s$ is a section of a line bundle $L$ on an open set $U$, the value of $s$ at a point $x\in U$ is only well-defined as an element of the stalk $L(x)$, which is a $\kappa(x)$-vector space of dimension $1$. (Here, $\kappa(x)$ is the residue field of $\mathscr{O}_{X}$ at $x$.) Prescribing a metric on $L$ amounts to assign, in a coherent way, the norms of these values. Formally, a metric on $L$ is the datum, for any open set $U\subset X$ and any section $s\in\Gamma(U,L)$, of a continuous function $\left\|{s}\right\|_{U}\colon U\rightarrow{\mathbf{R}}_{+}$, satisfying the following properties :

1. (1)

   for any open set $V\subset U$, $\left\|{s}\right\|_{V}$ is the restriction to $V$ of the function $\left\|{s}\right\|_{U}$ ;

2. (2)

   for any function $f\in\mathscr{O}_{X}(U)$, $\left\|{fs}\right\|=\left|{f}\right|\left\|{s}\right\|$ ;

3. (3)

   if $s$ is a local frame on $U$, then $\left\|{s}\right\|$ doesn’t vanish on $U$.

One usually writes $\overline{L}$ for the pair $(L,\left\|{\cdot}\right\|)$ of a line bundle $L$ and a metric on it.

Observe that the trivial line bundle $\mathscr{O}_{X}$ has a natural “trivial” metric, for which $\left\|{1}\right\|=1$. In fact, a metric on the trivial line bundle $\mathscr{O}_{X}$ is equivalent to the datum of a continuous function $h$ on $X$, such that $\left\|{1}\right\|=e^{-h}$.

Isomorphism of metrized line bundles are isomorphisms of line bundles which respect the metrics ; they are called isometries. Constructions from tensor algebra extend naturally to the framework of metrized line bundles, compatibly with isometries. The tensor product of two metrized line bundles $\overline{L}$ and $\overline{M}$ has a natural metrization such that $\left\|{s\otimes t}\right\|=\left\|{s}\right\|\left\|{t}\right\|$, if $s$ and $t$ are local sections of $L$ and $M$ respectively. Similarly, the dual of a metrized line bundle has a metrization, and the obvious isomorphism $L\otimes L^{\vee}\simeq\mathscr{O}_{X}$ is an isometry. Consequently, isomorphism classes of metrized line bundles on $X$ form an Abelian group $\overline{\operatorname{Pic}}(X)$. This group fits in an exact sequence

$$0\rightarrow\mathscr{C}(X)\rightarrow\overline{\operatorname{Pic}}(X)\rightarrow\operatorname{Pic}(X)\rightarrow 0,$$

where the first map associates to a real continuous function $h$ on $X$ the trivial line bundle endowed with the metric such that $\left\|{1}\right\|=e^{-h}$, and the second associates to a metrized line bundle the underlying line bundle. It is surjective when any line bundle has a metric (this certainly holds if $X$ has partitions of unity).

Similarly, we can consider pull-backs of metrized line bundle. Let $\varphi\colon Y\rightarrow X$ be a morphism of locally ringed spaces such that $\left|{\varphi^{*}f}\right|=\left|{f}\right|\circ\varphi$ for any $f\in\mathscr{O}_{X}$. Let $\overline{L}$ be a metrized line bundle on $X$. Then, there is a canonical metric on $\varphi^{*}L$ such that $\left\|{\varphi^{*}s}\right\|=\left\|{s}\right\|\circ\varphi$ for any section $s\in\Gamma(U,L)$. This induces a morphism of Abelian groups $\varphi^{*}\colon\overline{\operatorname{Pic}}(X)\rightarrow\overline{\operatorname{Pic}}(Y)$.