We will ignore all exponentially small errors coming from identifying tubular neighbourhoods with normal bundles. We compute $dd^{c}\phi_{D_{1}}$ by the Leibniz rule:

$$dd^{c}\tilde{x}_{1}^{\frac{n+2}{n}}=\frac{2(n+2)}{n^{2}}\tilde{x}_{1}^{\frac{n+2}{n}-2}\frac{\sqrt{-1}}{4\pi d_{2}^{2}}d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}},$$

$$\begin{split}&dd^{c}(\tilde{x}_{1}^{\frac{n-2}{n(n-1)}}\phi_{TY})=\frac{n-2}{n(n-1)}(\frac{n-2}{n(n-1)}-1)\tilde{x}_{1}^{\frac{n-2}{n(n-1)}-2}\phi_{TY}\frac{\sqrt{-1}}{4\pi d_{2}^{2}}d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}\\ &+\frac{n-2}{n(n-1)}\tilde{x}_{1}^{\frac{n-2}{n(n-1)}-1}(d\tilde{x}_{1}\wedge d^{c}\phi_{TY}+d\phi_{TY}\wedge d^{c}\tilde{x}_{1})+\tilde{x}_{1}^{\frac{n-2}{n(n-1)}}dd^{c}\phi_{TY}.\end{split}$$

and similarly with $dd^{c}(\tilde{x}_{1}^{\frac{n+2}{n}-\frac{kn}{n-1}}x_{2}^{\frac{kn}{n-1}})$.

Comparing $v_{0}dd^{c}\tilde{x}_{1}^{\frac{n+2}{n}}$ and the term $\frac{2(n+2)v_{0}}{n^{2}d_{2}^{2}}x_{1}^{\prime-\frac{n-2}{n}}\frac{\sqrt{-1}}{4\pi}d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}$, the deviation is of order $O(\frac{|x_{2}|+|x_{1}-x_{1}^{\prime}|}{x_{1}^{\prime}})$. Notice for $n\geq 3$, we have $\frac{n}{2(n-1)}<1$ and so $\frac{|x_{2}|}{x_{1}^{\prime}}\leq\left(\frac{x_{2}}{x_{1}}\right)^{\frac{n}{2(n-1)}}$. For $|x_{1}-x_{1}^{\prime}|\leq x_{1}^{\prime\frac{n-2}{2(n-1)}}$, the error $O(\frac{|x_{1}-x_{1}^{\prime}|}{x_{1}^{\prime}})=O(x_{1}^{-\frac{n}{2(n-1)}})$, which is absorbed into $O((\frac{x_{2}}{x_{1}})^{\frac{n}{2(n-1)}})$.

Using the lemmas, we can estimate the terms appearing in $dd^{c}(\tilde{x}_{1}^{\frac{n-2}{n(n-1)}}\phi_{TY})$ in the local $C^{k,\alpha}$-norms:

$$\left\lVert\phi_{TY}x_{1}^{\frac{n-2}{n(n-1)}-2}d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}\right\rVert_{k,\alpha,loc}=O(x_{2}^{\frac{n}{(n-1)}}x_{1}^{\frac{n-2}{n(n-1)}-2}x_{1}^{\frac{n-2}{n}})=O((\frac{x_{2}}{x_{1}})^{\frac{n}{n-1}}).$$

The cross terms have order

$$\left\lVert x_{1}^{\frac{n-2}{n(n-1)}-1}d\log\xi^{\prime}\wedge d^{c}\phi_{TY}\right\rVert_{k,\alpha,loc}=O(x_{1}^{\frac{n-2}{n(n-1)}-1}x_{1}^{\frac{n-2}{2n}}x_{1}^{-\frac{n-2}{2n(n-1)}}x_{2}^{\frac{n}{2(n-1)}})=O((\frac{x_{2}}{x_{1}})^{\frac{n}{2(n-1)}}).$$

So does its complex conjugate. The last error comes from the deviation between $\tilde{x}_{1}^{\frac{n-2}{n(n-1)}}dd^{c}\phi_{TY}$ from $x_{1}^{\prime\frac{n-2}{n(n-1)}}dd^{c}\phi_{TY}$. This error is again of order $O(\frac{|x_{2}|+|x_{1}-x_{1}^{\prime}|}{x_{1}^{\prime}})$.

The remainder terms have leading order contribution $dd^{c}(\tilde{x}_{1}^{\frac{n+2}{n}-\frac{2n}{n-1}}x_{2}^{\frac{2n}{n-1}}).$ Its main contribution is $\tilde{x}_{1}^{\frac{n+2}{n}-\frac{2n}{n-1}}dd^{c}x_{2}^{\frac{2n}{n-1}}$, which has magnitude $O((\frac{x_{2}}{x_{1}})^{\frac{n}{n-1}})$. Combining all the errors, the deviation between $dd^{c}\phi_{D_{1}}$ and the local product metric (32) has magnitude bounded by $O((\frac{x_{2}}{x_{1}})^{\frac{n}{2(n-1)}}).$ ∎

The volume form error will be smaller than the metric deviation due to extra cancellation effects. Recall the computation of $dd^{c}\phi_{D_{1}}$ from Lemma 4.9 above. Notice $(dd^{c}\phi_{D_{1}})^{n}$ is a volume form, so must take one $d\log\xi^{\prime}$ and $d\overline{\log\xi^{\prime}}$ from either a pair of cross derivative terms, or from a factor $dd^{c}\tilde{x}_{1}^{\frac{n+2}{n}-\frac{kn}{n-1}}$ for $k\geq 0$. The differentiation of $\phi_{TY}$ and the powers of $x_{2}$ would only produce factors on $D_{1}\setminus D_{2}$ without $\tilde{x}_{1}$ dependence. By thinking about all the possible ways to take wedge products contributing to $(dd^{c}\phi_{D_{1}})^{n}$, we get an absolutely convergent series within $x_{2}\ll x_{1}$:

$$(dd^{c}\phi_{D_{1}})^{n}\approx\sum_{k\geq 0}\tilde{x}_{1}^{-\frac{kn}{n-1}}\sqrt{-1}d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}\wedge\mathcal{I}_{k},$$

(33)

where the coefficients $\mathcal{I}_{k}$ are top degree forms on the $D_{1}\setminus D_{2}$ factor, without $\tilde{x}_{1}$ dependence. Here we write $\approx$ as a reminder that we have ignored the exponentially small errors from identifying the tubular neighbourhood with the normal bundles of $D_{1}\setminus D_{2}$, which is holomorphically trivial up to finite cover.

We now identify the leading term $d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}\wedge\mathcal{I}_{0}$. By thinking about form types, this comes from the binomial expansion term

$$nv_{0}dd^{c}\tilde{x}_{1}^{\frac{n+2}{n}}\wedge\left(a(\frac{d_{1}}{d_{2}})^{\frac{n}{n-1}}\tilde{x}_{1}^{\frac{n-2}{n(n-1)}}dd^{c}\phi_{TY}\right)^{n-1},$$

which is

$$\frac{2(n+2)v_{0}}{nd_{2}^{2}}\frac{\sqrt{-1}}{4\pi}d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}\wedge a^{n-1}(\frac{d_{1}}{d_{2}})^{n}(dd^{c}\phi_{TY})^{n-1}.$$

From the explicit formula (24) for $v_{0},a$,

$$v_{0}a^{n-1}=(\frac{n}{n+2})^{3},$$

and recalling the complex Monge-Ampère equation for the Tian-Yau metric in Theorem 4.2, the above simplifies to

$$\frac{2n^{2}\int_{Y}c_{1}(L_{0})^{n-2}}{(n+2)^{2}d_{2}^{2}}(\frac{d_{1}}{d_{2}})^{n}d_{2}^{n-2}\frac{\sqrt{-1}}{4\pi}d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}\wedge\sqrt{-1}^{(n-1)^{2}}\Omega_{D_{1}}\wedge\overline{\Omega}_{D_{1}}.$$

Up to exponentially small errors from complex structure identifications, in terms of the local defining functions $\xi_{1},\xi_{2}$ for $D_{1},D_{2}$,

$$\Omega_{D_{1}}\approx d\log\xi_{2}\wedge\Omega_{Y},\quad\log\xi^{\prime}\approx d_{2}\log\xi_{1}-d_{1}\log\xi_{2},\quad\Omega\approx d\log\xi_{1}\wedge d\log\xi_{2}\wedge\Omega_{Y},$$

and the above reduce to $K_{0}\sqrt{-1}^{n^{2}}\Omega\wedge\overline{\Omega},$ for the constant $K_{0}$ in (28). In short, the leading term cancels with $K_{0}\sqrt{-1}^{n^{2}}\Omega\wedge\overline{\Omega}$.

The subleading terms are suppressed by the factor $\tilde{x}_{1}^{-\frac{n}{n-1}}$, and the fast convergence of the power series means we only need to consider $\mathcal{I}_{1}$. In the Tian-Yau core region $x_{2}=O(1)$, the crude information is that $d\log\xi^{\prime}\wedge d\overline{\log\xi^{\prime}}\wedge\mathcal{I}_{1}$ has local $C^{k,\alpha}$ norm $O(1)$. This explains the $O(x_{1}^{-\frac{n}{n-1}})$ decay in the $x_{2}=O(1)$ subregion.

For $1\ll x_{2}\ll x_{1}$, the deviation between the Tian-Yau potential $\phi_{TY}$ and the Calabi ansatz potential is $O(e^{-cx_{2}^{1/2}})$, namely $O(e^{-cr_{D_{1}}^{\frac{n-1}{n}}})$ (for some changing constant $c>0$). After replacing $\phi_{TY}$ by $\frac{n-1}{n}x_{2}^{\frac{n}{n-1}}$, we recover the potential $u$ in the generic region. Ignoring exponentially small complex structure errors as usual, then $u$ is by construction a solution to the complex Monge-Ampère equation. This explains the exponential decay $\mathcal{I}_{k}=O(e^{-cx_{2}^{1/2}})$ for $k\geq 1$. ∎