ScalingStacks

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

$$d\mu_{t}(\{\psi_{t}-\phi_{CY,t}-c\leq 0\})\geq\lambda.$$

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

$$\int_{X_{t}}e^{-\alpha u}d\mu_{t}\leq A,\quad\forall u\in PSH(X_{t},\omega_{CY,t})\text{ with }\sup_{X_{t}}u=0.$$

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

$$\int_{X_{t}}|d\mu_{t}-\frac{\omega_{\psi,t}^{n}}{\text{Vol}(X_{t},\omega_{\psi,t})}|<\delta.$$

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

$$\sup_{X_{t}}(\psi_{t}-\phi_{CY,t}-c)\leq C(\lambda)\delta^{1/(2n+3)}\ll\kappa.$$

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

$$d\mu_{t}(\{\psi_{t}-\phi_{CY,t}-c\leq 0\})<\lambda,$$

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

$$d\mu_{t}(\{\phi_{CY,t}-\inf\phi_{CY,t}\geq\kappa/5\})<\lambda.$$

∎