ScalingStacks

Proof. By the calculations in the proof of Lemma 4.2,

$$(dd^{c}\phi_{0}\circ\text{Log}_{\mathcal{X}})^{n}=n!\det(D^{2}\phi_{0})\prod_{i}\frac{1}{4\pi}\sqrt{-1}d\zeta_{i}\wedge d\bar{\zeta}_{i},$$

while the CY condition gives (cf. section 3.1)

$$\begin{split}&(dd^{c}\phi_{CY,J,t})^{n}=\omega_{CY,t}^{n}=\frac{(L^{n})}{|\log|t||^{n}}d\mu_{t}=\frac{(L^{n})}{|\log|t||^{n}\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}\\ =&\frac{(L^{n})|\log|t||^{n}}{\int_{X_{t}}\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}}|u_{J}|^{2}\prod_{i}\sqrt{-1}d\zeta_{i}\wedge d\bar{\zeta}_{i},\end{split}$$

where $u_{J}$ is a holomorphic function of the defining functions $z_{0},\ldots z_{n}$ of the divisors $E_{i}$, with limiting value $u_{J}(E_{J})\neq 0$. The two expressions are matched by the condition that $\text{Log}_{\mathcal{X}*}d\mu_{t}$ converge to $d\mu_{0}$ as $t\to 0$, which boils down to

$$(dd^{c}\phi_{CY,J,t})^{n}=n!\det(D^{2}\phi_{0})\frac{|u_{J}|^{2}}{|u_{J}(E_{J})|^{2}}\prod_{i}\frac{1}{4\pi}\sqrt{-1}d\zeta_{i}\wedge d\bar{\zeta}_{i}.$$

Since $u_{J}$ has a Taylor expansion in $z_{0},\ldots z_{n}$, we see that $\frac{|u_{J}|^{2}}{|u_{J}(E_{J})|^{2}}=1+f$ for some smooth function $f$ in $\zeta_{1},\ldots,\zeta_{n}$ with exponentially small $C^{k}$-norm bound

$$\left\lVert f\right\rVert_{C^{k}(\text{Log}_{\mathcal{X}}^{-1}(W_{\delta}))}\lesssim_{k}\exp(-c(W_{\delta})|\log|t||)$$

for some exponent $c(W_{\delta})>0$ depending on $W_{\delta}$.

We focus on balls in the local universal cover of $\text{Log}_{\mathcal{X}}^{-1}(W_{\delta})$ with definite size in the $\zeta_{i}$ coordinates. For sufficiently small $t$, then the volume relative error $f$ has arbitrarily small $C^{k}$-norm bound, and Theorem 4.7 says the $C^{0}$-norm of $\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}$ on the ball is also arbitrarily small. Thus we can apply Savin’s theorem 2.8, to deduce that $\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{k}}$ is arbitrarily small on shrinked balls. Since $W_{\delta}$ for varying $\delta$ give an exhaustion of the regular locus of $\phi_{0}$, this shrinking can be compensated by starting with a larger $W_{\delta}$, and we deduce the $C^{k}$-convergence estimate as required. ∎