ScalingStacks

We then modify $h_{t}$ into $\tilde{h}_{t}=h_{t}e^{-2\phi_{x}}$, where $x=\text{Log}_{\mathcal{X}}(z)$ on $X_{t}$. The metric asymptote (13) then implies that on $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))\subset X_{t}$ for $|t|\ll 1$, up to $O(\frac{1}{|\log|t||})$ relative error,

$$\begin{split}-dd^{c}\log\tilde{h}_{t}^{1/2}\sim&\sum_{1}^{p}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}\\ &-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}+dd^{c}\phi_{x}.\end{split}$$

(20)

The normalisation ambiguity on $\phi_{x}$ is suppressed by $O(\frac{1}{|\log|t||})$ relative to the term $\sum_{1}^{p}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}$. In particular, we see $-dd^{c}\log\tilde{h}_{t}^{1/2}$ is positive definite, so defines a Kähler metric.