(3.2.1) In the following subsections, we will make use of the weight function

$$\mathrm{wt}_{\omega}:X_{K}^{\mathrm{an}}\to\mathbb{R}\cup\{+\infty\}$$

associated to a non-zero $m$-pluricanonical form on $X_{K}$, for any $m>0$. Its construction and main properties are described in [MN13, 4.4.5]. For us, its most important features are the following: if $\mathscr{X}$ is an $snc$-model of $X$ over $\mathscr{C}$ and $x$ is a point of $\mathrm{Sk}(\mathscr{X})$, then

$$\mathrm{wt}_{\omega}(x)=v_{x}(\mathrm{div}_{\mathscr{X}}(\omega)+m(\mathscr{X}_{s})_{\mathrm{red}})$$

(here we use the notation recalled in (2.1)). Moreover, for every point $y$ of $\widehat{\mathscr{X}}_{\eta}$, we have

$$\mathrm{wt}_{\omega}(y)\geq\mathrm{wt}_{\omega}(\rho_{\mathscr{X}}(y))$$

with equality if and only if $y$ lies on $\mathrm{Sk}(\mathscr{X})$. In [MN13, 4.4.5] there is no properness assumption on $X_{K}$; this allows us to deal with rational pluricanonical forms by removing the locus of poles from $X_{K}$.