Let $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$ be an snc degeneration, i.e. a proper, surjective holomorphic map from a connected complex manifold to the unit disc in ${\mathbb{C}}$, whose restriction to $X:=\pi^{-1}({\mathbb{D}}^{*})$ is a submersion and such that ${\mathcal{X}}_{0}:=\pi^{-1}(0)=\sum_{i\in I}b_{i}E_{i}$ has snc support. Note that $X_{t}:=\pi^{-1}(t)$ is non-singular for $t\in{\mathbb{D}}^{*}$. The dual complex $\Delta({\mathcal{X}})$ is defined as that of ${\mathcal{X}}_{0}$; it is equipped with its natural ${\mathbb{Z}}$-PA structure. The logarithmic canonical bundle of ${\mathcal{X}}$ is

$$K^{\mathrm{log}}_{\mathcal{X}}:=K_{\mathcal{X}}+{\mathcal{X}}_{0,\mathrm{red}}.$$

Setting $K^{\mathrm{log}}_{{\mathbb{D}}}:=K_{\mathbb{D}}+[0]$, we define the relative logarithmic canonical bundle as

$$K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}:=K^{\mathrm{log}}_{\mathcal{X}}-\pi^{*}K^{\mathrm{log}}_{{\mathbb{D}}}=K_{{\mathcal{X}}/{\mathbb{D}}}+{\mathcal{X}}_{0,\mathrm{red}}-{\mathcal{X}}_{0}.$$

Now suppose we are given a ${\mathbb{Q}}$-line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ extending $K_{X/{\mathbb{D}}^{*}}$. We then have a unique decomposition

$$K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}={\mathcal{L}}+\sum_{i\in I}a_{i}E_{i}$$

with $a_{i}\in{\mathbb{Q}}$. Set $\kappa_{i}:=a_{i}/b_{i}$ and $\kappa_{\min}:=\min_{i}\kappa_{i}$.

In general, $\Delta({\mathcal{L}})$ is neither connected nor pure dimensional. We say that a face of $\Delta({\mathcal{L}})$ is maximal if it is not contained in a larger face of $\Delta({\mathcal{L}})$.

If $\psi$ is a continuous metric on ${\mathcal{L}}$, $\psi|_{Y}$ may thus be viewed as a metric on $K_{(Y,B^{\mathcal{L}}_{Y})}$. When $\sigma$ is a maximal face of $\Delta({\mathcal{L}})$, the pair $(Y,B^{\mathcal{L}}_{Y})$ is subklt, and Lemma 1.1 applies. This leads to the following notion.

This measure can be more explicitly described as follows. At each point $\xi\in Y$, pick local coordinates $(z_{0},\dots,z_{n})$ such that $z_{0},\dots,z_{p}$ are local equations for the components $E_{0},\dots,E_{p}$ of ${\mathcal{X}}_{0}$ that pass through $\xi$, indexed so that $J=\{0,\dots,d\}$, where $0\leq d\leq p$, and such that $t=\prod_{j=0}^{p}z_{j}^{b_{j}}$ The logarithmic form

$$\Omega:=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}$$

is a local trivialization of $K^{\mathrm{log}}_{{\mathcal{X}}}$, and hence induces a local trivialization $\Omega^{\mathrm{rel}}=\Omega\otimes(dt/t)^{-1}$ of $K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}$. We may then view $\tau:=\prod_{i=0}^{p}z_{i}^{a_{i}}\Omega^{\mathrm{rel}}$ as a local ${\mathbb{Q}}$-generator of ${\mathcal{L}}$. Under the identification ${\mathcal{L}}|_{Y}=K_{(Y,B^{\mathcal{L}}_{Y})}$, we have

$$\tau|_{Y}=\prod_{i=d+1}^{p}z_{i}^{a_{i}-\kappa_{\min}b_{i}}\operatorname{Res}_{Y}(\Omega)$$

with

$$\operatorname{Res}_{Y}(\Omega)=\frac{dz_{d+1}}{z_{d+1}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}\bigg|_{Y}.$$

We infer

$$\operatorname{Res}_{Y}(\psi)=|\tau|^{-2}_{\psi}\prod_{i=d+1}^{p}|z_{i}|^{2(a_{i}-\kappa_{\min}b_{i}-1)}\bigg|\bigwedge_{i=d+1}^{n}dz_{i}\bigg|^{2}.$$

(3.1)

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.

As in §3.1, we introduce the logarithmic form

$$\Omega:=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n},$$

and the corresponding local trivialization $\Omega^{\mathrm{rel}}=\Omega\otimes(dt/t)^{-1}$ of $K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}$. The restriction $\Omega_{t}$ of $\Omega^{\mathrm{rel}}$ to the fiber $U_{t}:={\mathcal{X}}_{t}\cap{\mathcal{U}}$ is a trivializing section of $K_{U_{t}}$, explicitly given by

$$\Omega_{t}=\frac{1}{p+1}\sum_{j=0}^{p}\frac{(-1)^{j}}{b_{j}}\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\widehat{\frac{dz_{j}}{z_{j}}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}\bigg|_{U_{t}}.$$

For $t\in{\mathbb{D}}^{*}$ close to 0, consider the map $\operatorname{Log}_{t}\colon U_{t}\to\sigma\times(Y\cap{\mathcal{U}})$ defined by

$$\operatorname{Log}_{t}=(\operatorname{Log}_{{\mathcal{U}}},y)=\left(\frac{\log|z_{0}|}{\log|t|},\dots,\frac{\log|z_{p}|}{\log|t|},z_{p+1},\dots,z_{n}\right).$$

Note the similarity to the situation considered in §1.4. More precisely, view $U:={\mathcal{U}}\cap X$ as embedded in $T\times{\mathbb{C}}^{n-p}$, where $T=({\mathbb{C}}^{*})^{p+1}$, and consider the character $\chi=\prod_{i=0}^{p}z_{i}^{b_{i}}$ on $T$. If $L\colon T\to{\mathbb{R}}^{p+1}$ is the tropicalization map, then

$$\operatorname{Log}_{t}=(\lambda(t)^{-1}L(z^{\prime},z^{\prime\prime}),y).$$

Each fiber $\operatorname{Log}_{t}^{-1}(w,y)$ is a torsor for the (possibly disconnected) compact Lie group

$$K=\left\{\theta\in({\mathbb{R}}/{\mathbb{Z}})^{p+1}\mid\sum_{i}b_{i}\theta_{i}=0\right\};$$

hence carries a unique $K$-invariant probability measure $\rho_{t,w,y}$.

The analysis in §1.4 now gives the following expression for the volume form $|\Omega_{t}|^{2}$ on $U_{t}$ in logarithmic polar coordinates:

As before, view $\tau:=\prod_{i=0}^{p}z_{i}^{a_{i}}\Omega^{\mathrm{rel}}$ as a local ${\mathbb{Q}}$-generator of ${\mathcal{L}}$, and set

$$g:=-\log|\tau|_{\psi}\in C^{0}({\mathcal{U}}).$$

By definition, we have $\mu_{t}=(2\pi)^{-d}\lambda(t)^{d}|\Omega_{t}|^{2}/|\Omega_{t}|^{2}_{\psi_{t}}$, and hence

$$(2\pi)^{d-p}\lambda(t)^{p-d}|t|^{-2\kappa_{0}}\int\limits_{U_{t}}h\mu_{t}\\ =\int\limits_{\sigma\times(Y\cap{\mathcal{U}})}|t|^{2\sum_{i=q+1}^{p}b_{i}w_{i}(\kappa_{i}-\kappa_{0})}b_{\sigma}^{-1}\lambda_{\sigma}(dw)\otimes|dy|^{2}\int he^{2g}\,\rho_{t,w,y}\\ =\int\limits_{\sigma\times(Y\cap{\mathcal{U}})}e^{-2\lambda(t)^{-1}\sum_{i=q+1}^{p}b_{i}w_{i}(\kappa_{i}-\kappa_{0})}b_{\sigma}^{-1}\lambda_{\sigma}\otimes|dy|^{2}\int he^{2g}\,\rho_{t,w,y}$$

(3.3)

for every $h\in C^{0}_{c}({\mathcal{U}})$, thanks to Lemma 3.7.

We use the following change of variables. For $t\in{\mathbb{D}}^{*}$, consider the polytope

$$\sigma_{t}:=\{(w^{\prime},x^{\prime\prime})\in{\mathbb{R}}_{+}^{q+1}\times{\mathbb{R}}_{+}^{p-q}\mid b^{\prime}\cdot w^{\prime}=1,\ b^{\prime\prime}\cdot x^{\prime\prime}\leq\lambda(t)^{-1}\}\subset\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-q}\subset{\mathbb{R}}_{+}^{p+1},$$

where $b^{\prime}=(b_{0},\dots,b_{q})$ and $b^{\prime\prime}=(b_{q+1},\dots,b_{p})$.

Using the map $Q_{t}$ and the fact that $\kappa_{i}-\kappa_{0}>0$ for $i>q$, it is easy to see that

$$\int_{\sigma}e^{-2\lambda(t)^{-1}\sum_{i=q+1}^{p}b_{i}w_{i}(\kappa_{i}-\kappa_{0})}\lambda_{\sigma}(dw)=O(\lambda(t)^{p-q}).$$

By (3.3), it follows that

$$\mu_{t}(U_{t})=O(\lambda(t)^{d-q}|t|^{2\kappa_{0}}),$$

(3.4)

and hence $\mu_{t}(U_{t})\to 0$ unless $\kappa_{0}=0$ and $q=d$, which we henceforth assume. Given $\varphi\in C^{0}(\sigma)$, our goal is now to show

$$\int_{U_{t}}(\varphi\circ\operatorname{Log}_{{\mathcal{U}}})\chi\,\mu_{t}\to\left(\int_{\sigma^{\prime}}\varphi b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}\right).\left(\int_{Y^{\prime}}\chi\operatorname{Res}_{Y^{\prime}}(\psi)\right).$$

(3.5)

Let us first express both sides of (3.5) in logarithmic polar coordinates. We start by the left-hand side. Set $f:=\chi e^{2g}\in C^{0}({\mathcal{U}})$. By (3.3) and Lemma 3.8 we have

$$(2\pi)^{d-p}\int_{U_{t}}(\varphi\circ\operatorname{Log}_{{\mathcal{U}}})\chi\,\mu_{t}\\ =\lambda(t)^{d-p}\int\limits_{\sigma\times(Y\cap{\mathcal{U}})}\varphi(w)e^{-2\lambda(t)^{-1}a^{\prime\prime}\cdot w^{\prime\prime}}b_{\sigma}^{-1}\lambda_{\sigma}(dw)\otimes|dy|^{2}\int f\,\rho_{t,w,y}\\ =\int\limits_{\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-d}\times(Y\cap{\mathcal{U}})}H_{t}(w^{\prime},x^{\prime\prime})b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}(dw^{\prime})\otimes|dx^{\prime\prime}|\otimes|dy|^{2}\int f\,\rho_{t,w^{\prime},x^{\prime\prime},y},$$

(3.6)

where

$$H_{t}(w^{\prime},x^{\prime\prime})=\mathbf{1}_{\sigma_{t}}\varphi(Q_{t}(w^{\prime},x^{\prime\prime}))e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}(1-\lambda(t)b^{\prime\prime}\cdot x^{\prime\prime})^{d},$$

and $\rho_{t,w^{\prime},x^{\prime\prime},y}$ is the same measure as $\rho_{t,w,y}$ via the identification $Q_{t}(w^{\prime},x^{\prime\prime})=w$.

Note that $\lim_{t\to 0}Q_{t}(w^{\prime},x^{\prime\prime})=(w^{\prime},0)$, so

$$\lim_{t\to 0}H_{t}(w^{\prime},x^{\prime\prime})=\mathbf{1}_{\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-d}}\varphi(w^{\prime},0)e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}.$$

Consider the tropicalization map

$$S\colon Y^{\prime}\cap{\mathcal{U}}\to{\mathbb{R}}_{+}^{p-d}\times(Y\cap{\mathcal{U}})$$

given by $S=(-\log|z_{d+1}|,\dots,-\log|z_{p}|,y)$. Each fiber $S^{-1}(x^{\prime\prime},y)$ is a torsor for the compact torus $({\mathbb{R}}/{\mathbb{Z}})^{p-d}$ and hence carries a unique invariant probability measure $\rho_{x^{\prime\prime},y}$. As $t\to 0$, the probability measure $\rho_{t,w^{\prime},x^{\prime\prime},y}$ converges weakly to $\rho_{x^{\prime\prime},y}$ for any $w^{\prime}\in\sigma^{\prime}$.

By dominated convergence it follows that

$$\lim_{t\to 0}(2\pi)^{d-p}\int_{U_{t}}(\varphi\circ\operatorname{Log}_{{\mathcal{U}}})\chi\,d\mu_{t}\\ =\int_{\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-d}\times(Y\cap{\mathcal{U}})}\varphi(w^{\prime},0)e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}(dw^{\prime})\otimes|dx^{\prime\prime}|\otimes|dy|^{2}\int f\,\rho_{x^{\prime\prime},y}\\ =\left(\int_{\sigma^{\prime}}\varphi b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}\right)\left(\int_{{\mathbb{R}}_{+}^{p-d}}e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}|dx^{\prime\prime}|\int_{Y\cap{\mathcal{U}}}|dy|^{2}\int f\,\rho_{x^{\prime\prime},y}\right).$$

(3.7)

It only remains to compare the second factor of (3.7) to the second factor in (3.5). To this end, we again use logarithmic polar coordinates. We have

$$\chi\operatorname{Res}_{Y^{\prime}}(\psi)=f\prod_{i=d+1}^{p}|z_{i}|^{2a_{i}-2}|dz^{\prime\prime}|^{2}\otimes|dy|^{2}.$$

(3.8)

For $d<j\leq p$, set $z_{j}=e^{-x_{j}+2\pi i\theta_{j}}$ with $x^{\prime\prime}\in{\mathbb{R}}_{+}^{p-d}$ and $\theta^{\prime\prime}\in({\mathbb{R}}/{\mathbb{Z}})^{p-d}$. Then

$$\int_{Y^{\prime}\cap{\mathcal{U}}}\chi\operatorname{Res}_{Y^{\prime}}(\psi)=(2\pi)^{p-d}\int_{{\mathbb{R}}_{+}^{p-d}}e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}|dx^{\prime\prime}|\int_{Y\cap{\mathcal{U}}}|dy|^{2}\int f\,\rho_{x^{\prime\prime},y},$$

(3.9)

which completes the proof of (3.5), and hence of Theorem 3.4.