Let us first treat the case when $(X,B)$ is log smooth. In this case we need not assume that $X\to{\mathbb{D}}^{*}$ is projective. It follows from the normal crossings condition that $(X_{t},B_{t})$ is subklt for $0<|t|\ll 1$. After reparametrizing we may assume this is true for all $t\in{\mathbb{D}}^{*}$, that is, $Z=\emptyset$. Set

This is a positive measure on $X_{t}$, smooth outside the support of $B_{t}$. Pick an snc model $({\mathcal{X}},{\mathcal{B}})$ of $(X,B)$, where ${\mathcal{B}}$ is the closure of $B$ in ${\mathcal{X}}$, such that $\psi$ extends to a continuous metric on a ${\mathbb{Q}}$-line bundle ${\mathcal{L}}$ on ${\mathcal{X}}$ extending $K_{(X,B)/{\mathbb{D}}^{*}}$.

We can then prove a version of Theorem A inside the hybrid space ${\mathcal{X}}^{\mathrm{hyb}}$. By letting ${\mathcal{X}}$ vary, we obtain Theorem 8.4 as a consequence, just as Corollary B follows from Theorem A.

The proof is very similar to the proof of Theorem A, so we will only indicate the modifications needed. Let us write

with $a_{i}\in{\mathbb{Q}}$. Set $\kappa_{i}:=a_{i}/b_{i}$ and $\kappa_{\min}:=\min_{i}\kappa_{i}$. Here ${\mathcal{X}}_{0}=\sum_{i}b_{i}E_{i}$ as before.

Define $\Delta({\mathcal{L}})$ as the subcomplex of $\Delta({\mathcal{X}})$ spanned by the vertices such that $\kappa_{i}=\kappa_{\min}$. This will be the support of the measure $\mu_{0}$. For every stratum $Y$ corresponding to a maximal face of $\Delta({\mathcal{L}})$, define a subklt pair $(Y,B_{Y}^{\mathcal{L}})$ using

The residual measure $\operatorname{Res}_{Y}(\psi)$ is given by

Finally set

where $\sigma$ ranges over the $d$-dimensional faces of $\Delta({\mathcal{L}})$, with $d=\dim\Delta({\mathcal{L}})$.

We then prove a version of Theorem 3.4. Namely, if

then we show that $\mu_{t}$ converges to $\mu_{0}$ in ${\mathcal{X}}^{\mathrm{hyb}}$ as $t\to 0$. This is done via a local convergence result as in Lemma 3.5. Namely, given a point $\xi\in{\mathcal{X}}_{0}$, we choose local coordinates $(z_{0},\dots,z_{n})$ at $\xi$ as in §3.2, but further require that these coordinates also cut out the irreducible components of ${\mathcal{B}}$ containing $\xi$. More precisely, there exist $m$ with $p\leq m\leq n$ such that these irreducible components are given by $B_{i}=\{z_{i}=0\}$ for $p<i\leq m$. Also set $c_{i}:=\operatorname{ord}_{B_{i}}({\mathcal{B}})<1$.

A local ${\mathbb{Q}}$-generator for ${\mathcal{L}}$ at $\xi$ is then given by

with $\Omega^{\mathrm{rel}}$ as before. For a stratum $Y$ corresponding to a $d$-dimensional simplex in $\Delta({\mathcal{L}})$, the residual measure is given by

The measure $\mu_{t}$ can be written near $\xi$ as

The proof now proceeds exactly as in §3.3 except that we need to insert a factor $\prod_{i=p+1}^{m}|z_{i}|^{-2c_{i}}$ in the last two lines of (3.3) and (3.6), the second line and the second factor of the last line of (3.7), and the right-hand sides of (3.8) and (3.9). This completes the proof in the log smooth case.

Now we consider the general case, assuming $X\to{\mathbb{D}}^{*}$ is projective. Pick a log resolution $q\colon(X^{\prime},B^{\prime})\to(X,B)$. Since $(X^{\prime}_{t},B^{\prime}_{t})$ is subklt for $0<|t|\ll 1$, the same is true for $(X_{t},B_{t})$. We have an induced continuous map $q^{\mathrm{hyb}}\colon(X^{\prime})^{\mathrm{hyb}}\to X^{\mathrm{hyb}}$. By what precedes, there exist $\kappa\in{\mathbb{Q}}$ and $d\in{\mathbb{N}}$ such that the measure $\mu^{\prime}_{t}:=\frac{e^{2(\psi^{\prime}_{t}-\phi_{B^{\prime}_{t}})}}{|t|^{2\kappa_{\min}}(2\pi\log|t|^{-1})^{d}}$ on $X^{\prime}_{t}=X^{\mathrm{hyb}}$ converges to a nonzero positive measure $\mu^{\prime}_{0}$ on $(X^{\prime})^{\mathrm{hyb}}$. By continuity, it follows that $\mu_{t}=q^{\mathrm{hyb}}_{*}\mu^{\prime}_{t}$ converges to the nonzero positive measure $\mu_{0}=q^{\mathrm{hyb}}_{*}\mu^{\prime}_{0}$ on $X^{\mathrm{hyb}}$. This completes the proof. ∎

As recalled above, $K_{X_{t}}$ is torsion for each fixed $t$. Equivalently, $h^{0}(X_{t},rK_{X_{t}})=1$ for some positive integer $r$. Since $t\mapsto h^{0}(X_{t},rK_{X_{t}})$ is upper semicontinuous in the Zariski topology, it follows that $rK_{X_{t}}$ is trivial for a fixed $r$ independent of $t$. Given any snc model $\pi\colon{\mathcal{X}}\to{\mathbb{D}}$, $\pi_{*}{\mathcal{O}}(rK_{{\mathcal{X}}/{\mathbb{D}}})$ is torsion free of rank one, and hence a line bundle. The choice of a trivializing section yields a holomorphic section $\eta$ of $K_{{\mathcal{X}}/{\mathbb{D}}}$, inducing a holomorphic family $\eta_{t}$ of trivializing sections of $rK_{X_{t}}$ for $t\neq 0$. As a consequence, the family of volume forms $\nu_{t}:=|\eta_{t}|^{2/r}$ has analytic singularities at $t=0$, and the result is thus a consequence of Theorem A, since $\mu_{t}=\nu_{t}/\nu_{t}(X_{t})$. ∎