ScalingStacks

Proof. Combine the uniform $C^{0}$-estimate Theorem 2.2 and the uniform Skoda estimate Theorem 2.1, we know the CY potential with respect to any fixed choice of Fubini-Study background metric is uniformly bounded for small $t$. Here the implicit choice of $\epsilon$ in $\omega_{FS,t}$ does not matter because of Lemma 4.1. ∎

Proof. We wish to compare $\omega_{CY,t}$ with $\omega_{\psi,t}$ from Lemma 4.2 by an $L^{1}$-stability estimate. Pick a parameter $c$ such that

Since the potential $\phi_{CY,t}$ has a uniform bound, Theorem 2.1 implies another uniform Skoda estimate with modified constants

We also have the $L^{1}$-stability property for $\omega_{\psi,t}$ in Lemma 4.2:

We then apply the uniform $L^{1}$-stability estimate Theorem 2.6, with $Y=X_{t}$, $\omega=\omega_{CY,t}$ and $\phi=\psi_{t}-\phi_{CY,t}-c$. In this construction $\delta,\epsilon,t$ are sufficiently small, chosen successively depending on $\lambda$ and $\kappa$. We conclude

Now $|\psi_{t}|\leq 3\epsilon\ll\kappa$, so $\inf_{X_{t}}\phi_{CY,t}>-c-\kappa/10.$

Taking the contrapositive, if we choose $c=-\inf_{X_{t}}\phi_{CY,t}-\kappa/10$, then

whence for $\epsilon\ll\kappa$, using again $|\psi_{t}|\leq 3\epsilon$,

∎

Proof. By construction in Lemma 4.1, the local potential $\phi_{J,t}$ of $\omega_{FS,t}$ is $\epsilon$-close to $\phi_{0}\circ\text{Log}_{\mathcal{X}}$, and since $\epsilon\ll\kappa$ these two are practically the same. Up to an overall normalisation constant, which is fixed by $\inf=0$, we have $\phi_{CY,J,t}=\phi_{CY,t}+\phi_{J,t},$ so the measure bound follows from the previous result.

The uniform $L^{\infty}$ bound follows from Prop. 4.3 and Lemma 4.1 without any reference to $\lambda,\kappa$. ∎

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

hence

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,

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