ScalingStacks

For fixed $\epsilon$, after deleting terms suppressed by order $O(\frac{1}{|\log|t||})$, we have the asymptote on $E_{J,t}^{\epsilon}\subset X_{t}$ as $t\to 0$:

$$dd^{c}u\sim\sum_{0}^{p}\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||^{2}}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}+\frac{1}{\log|t|}\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i},$$

where $z_{0},\ldots z_{p}$ are the local defining functions of the divisors $E_{i}\subset\mathcal{X}$, and $E_{J}=\cap_{0}^{p}E_{i}$. Using that $t=z_{0}\ldots z_{p}$ locally around $E_{J}$, we can eliminate the $z_{0}$ variable to write

$$dd^{c}u\sim\sum_{1}^{p}\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||^{2}}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}+\frac{1}{\log|t|}\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i},$$

hence

$$\begin{split}-dd^{c}\log h_{t}^{1/2}\sim&\sum_{1}^{p}\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}\\ &-dd^{c}\log h_{\mathcal{L}}^{1/2}-\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i}.\end{split}$$

(13)

Notice that $dd^{c}\log r_{i}\sim-dd^{c}\phi_{i}$ has a smooth extension to the central fibre as $t\to 0$; the singular effect is eliminated by $dd^{c}\log|z_{i}|=0$ on $X_{t}$. The term $\sum_{1}^{p}\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}$ dominates in the directions transverse to $E_{J}$, and the term $-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}$ dominates in the directions tangential to $E_{J}$.