We use additive notation for line bundles and metrics over an analytic space $X$, both in the complex and non-Archimedean setting. This amounts to the following two rules:

• (i)

  if for $i=1,2$, $\phi_{i}$ is a metric on a line bundle $L_{i}$ and $a_{i}\in{\mathbb{Z}}$, then $a_{1}\phi_{1}+a_{2}\phi_{2}$ is a metric on $a_{1}L_{1}+a_{2}L_{2}$;

• (ii)

  a metric on the trivial line bundle ${\mathcal{O}}_{X}$ is of the form $|\cdot|e^{-\phi}$ for a function $\phi$ on $X$, and we identify the metric with $\phi$.

If $s$ is a section of a line bundle $L$ on $X$, then $\log|s|$ stands for the corresponding (possibly singular) metric on $L$ in which $s$ has length 1. For any metric $\phi$ on $L$, the above rules imply that $\log|s|-\phi$ is a function on $X$, and

$$|s|_{\phi}:=|s|e^{-\phi}=\exp(\log|s|-\phi)$$

is the pointwise length of $s$ in the metric $\phi$.

A metric on a ${\mathbb{Q}}$-line bundle $L$ is a collection $(\phi_{m})_{m}$ of metrics on $mL$, for $m$ sufficiently divisible, such that $\phi_{jm}=j\phi_{m}$.

The line bundle ${\mathcal{O}}_{X}(D)$ associated to any Cartier divisor $D$ on $X$ comes with a canonical singular metric $\phi_{D}$, smooth outside $D$. This fact extends to ${\mathbb{Q}}$-divisors, by interpreting $\phi_{D}$ as a metric on a ${\mathbb{Q}}$-line bundle. In the complex case at least, the curvature current of $\phi_{D}$, correctly normalized, coincides with the integration current on $D$.