It will be convenient to introduce the quantity

$$\lambda(t):=(\log|t|^{-1})^{-1},$$

for $t\in{\mathbb{D}}^{*}$. Note that $\lambda(t)\to 0$ as $t\to 0$.

Let ${\mathcal{X}}^{\mathrm{hyb}}:=X\coprod\Delta({\mathcal{X}})$ be the locally compact hybrid space constructed in §2. It comes with a proper map $\pi\colon{\mathcal{X}}^{\mathrm{hyb}}\to\Delta$ extending $\pi\colon X\to{\mathbb{D}}^{*}$ and such that $\Delta({\mathcal{X}})=\pi^{-1}(0)$. The next result implies Theorem A in the introduction.

We start by making a few reductions. First, we may—and will—assume in what follows that $\kappa_{\min}=0$. Indeed, $t$ defines a nonvanishing section of ${\mathcal{O}}_{\mathcal{X}}({\mathcal{X}}_{0})$, and hence a smooth metric $\log|t|$, so we may replace ${\mathcal{L}}$ and $\psi$ with ${\mathcal{L}}-\kappa_{\min}{\mathcal{X}}_{0}$ and $\psi-\kappa_{\min}\log|t|$, respectively, and end up with $\kappa_{\min}=0$.

Since $\min_{i}a_{i}/b_{i}=\kappa_{\min}=0$, we then have $a_{i}\geq 0$, with equality if and only if $E_{i}$ corresponds to a vertex of $\Delta({\mathcal{L}})$.

Next we reduce the assertion of Theorem 3.4 to a local problem. Let $Y\subset{\mathcal{X}}_{0}$ be the stratum of an arbitrary face $\sigma$ of $\Delta({\mathcal{X}})$, and denote by $E_{0},\dots,E_{p}$ the components of ${\mathcal{X}}_{0}$ cutting out $Y$, ordered so that

$$\kappa_{0}=\dots=\kappa_{q}<\kappa_{q+1}\leq\dots\leq\kappa_{p}.$$

We can then make the identification

$$\sigma=\left\{w\in{\mathbb{R}}_{+}^{p+1}\mid b\cdot w=1\right\}$$

with $b=(b_{0},\dots,b_{p})\in{\mathbb{Z}}_{>0}^{p+1}$. Set $b^{\prime}=(b_{0},\dots,b_{q})\in{\mathbb{Z}}_{>0}^{q+1}$ and

$$\sigma^{\prime}:=\left\{w^{\prime}\in{\mathbb{R}}_{+}^{q+1}\mid b^{\prime}\cdot w^{\prime}=1\right\}.$$

Then $\sigma^{\prime}$ is a face of $\sigma$ under the embedding ${\mathbb{R}}_{+}^{q+1}\hookrightarrow{\mathbb{R}}_{+}^{p+1}$ given by $w^{\prime}\to(w^{\prime},0)$. Let $Y^{\prime}\supset Y$ be the corresponding stratum of ${\mathcal{X}}_{0}$.

Note that $\sigma$ contains a face of $\Delta({\mathcal{L}})$ if and only if $\kappa_{0}=0$; in that case, the face is unique, equal to $\sigma^{\prime}$ (which then implies $q\leq d$).

Pick $x\in\mathring{Y}$, and choose local coordinates $z=(z_{0},\dots,z_{n})$ at $x$ such that $z_{i}$ is a local equation of $E_{i}$ for $0\leq i\leq p$ and

$$t=\prod_{i=0}^{p}z_{i}^{b_{i}}$$

We may assume that $z$ is defined on a polydisc ${\mathcal{U}}\simeq{\mathbb{D}}(r)^{p+1}\times{\mathbb{D}}^{n-p}$ with $0<r\ll 1$. Decompose

$$z=(z_{0},\dots,z_{n})\in{\mathcal{U}}\simeq{\mathbb{D}}(r)^{p+1}\times{\mathbb{D}}^{n-p}$$

as

$$z=(z^{\prime},z^{\prime\prime},y)\in{\mathbb{D}}(r)^{q+1}\times{\mathbb{D}}(r)^{p-q}\times{\mathbb{D}}^{n-p},$$

where we view $y$ as a point of ${\mathcal{U}}\cap Y\simeq{\mathbb{D}}^{n-p}$, and $(z^{\prime\prime},y)$ as a point of ${\mathcal{U}}\cap Y^{\prime}\simeq{\mathbb{D}}(r)^{p-q}\times{\mathbb{D}}^{n-p}$.

The coordinate chart $({\mathcal{U}},z)$ is adapted to ${\mathcal{X}}_{0}$ in the sense of §2.2, with

$$\operatorname{Log}_{{\mathcal{U}}}\colon{\mathcal{U}}\setminus{\mathcal{X}}_{0}\to\sigma$$

given by

$$\operatorname{Log}_{{\mathcal{U}}}=\left(\frac{\log|z_{i}|}{\log|t|}\right)_{0\leq i\leq p}.$$

We aim to establish the following result.

Granted this result, let us show how to prove Theorem 3.4. For $0<r\ll 1$, ${\mathcal{V}}:=\pi^{-1}(\overline{{\mathbb{D}}}_{r})\subset{\mathcal{X}}$ is an compact neighborhood of ${\mathcal{X}}_{0}$ with a map $\operatorname{Log}_{\mathcal{V}}\colon{\mathcal{V}}^{\mathrm{hyb}}\to\Delta({\mathcal{X}})$ as in Proposition 2.1. We will use

By Lemma 3.6 we must show that

$$(\operatorname{Log}_{\mathcal{V}})_{*}\mu_{t}\to\mu_{0}=\sum_{\sigma^{\prime}}\left(\int_{Y^{\prime}}\operatorname{Res}_{Y^{\prime}}(\psi)\right)b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}},$$

where $\sigma^{\prime}$ ranges over $d$-dimensional simplices in $\Delta({\mathcal{L}})$. But this is easily seen to follow from Lemma 3.5, using a partition of unity argument as in the proof of Proposition 2.1.