ScalingStacks

Proof. We need to obtain upper bound on $\phi_{CY,J,t}$. Consider $z\in U_{J,t,\tau}$ and $x=\text{Log}_{\mathcal{X}}(z)$. Let $r\ll\tau$ be a parameter to be fixed, so $B(x,3r)\subset\text{Int}(\Delta_{J})$. The function $\phi_{0}$ has an a priori Lipschitz estimate on $\Delta_{\mathcal{X}}$, so the oscillation of $\phi_{0}$ on $B(x,3r)$ is less than $Cr\ll\kappa$ by choosing $r$ small enough.

Now we apply the mean value inequality to the psh function $\phi_{CY,J,t}$ on a ball in the local covering space of $U_{J,t,\tau}\subset(\mathbb{C}^{*})^{n}$, which projects to $B(x,r)$ via $\text{Log}_{\mathcal{X}}$. We have

$$\phi_{CY,J,t}(z)\leq\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{ball}\phi_{CY,J,t},$$

hence

$$\begin{split}(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})(z)&\leq\text{osc}_{B(r)}\phi_{0}+\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{ball}(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})\\ &\leq\frac{\kappa}{10}+\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{ball}(\phi_{CY,J,t}-\text{Log}_{\mathcal{X}}).\end{split}$$

But on the ball $\phi_{CY,J,t}-\text{Log}_{\mathcal{X}}<\kappa/4$, except on a subset of the ball with $d\mu_{t}$-percentage $\leq C\lambda r^{-n}$, on which we use the coarser bound $\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\leq C$. Combining these,

$$(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})(z)<\kappa/2+C\lambda r^{-n}<\kappa,$$

by choosing $\lambda$ sufficiently small depending on $\kappa,\tau$. ∎