We start from the solution $w(t)$ of the ODE (12) with the matching condition $w^{\prime}(1)=\frac{1}{2}w(1)$, which guarantees the existence of the solution for $0<t<+\infty$. As discussed in section 3, this specifies the parameter choices in (11)

$$w_{0}=\left(\frac{1}{2}\right)^{\frac{n+1}{n+2}}\left(\frac{n}{(n-1)}\right)^{\frac{1}{n+2}}\left(\frac{1}{\sqrt{\pi}}\frac{\Gamma(\frac{1}{2}-\frac{1}{n})}{\Gamma(1-\frac{1}{n})}\right)^{\frac{n}{n+2}},\quad b_{1}=w_{0}^{-\frac{3}{n-1}}.$$

The boundary behaviour near $t\to 0$ is prescribed by (11). The $t\to+\infty$ boundary is determined from $w(t)=tw(1/t)$. Since the geometry of the $t\to+\infty$ limit is essentially identical to $t\to 0$, we will later only focus on $t\to 0$.

Reversing the previous ODE reductions, let

$$v(t)=(\frac{nw}{n+2})^{\frac{n+2}{n}},\quad u(x_{1},x_{2})=x_{1}^{\frac{n+2}{n}}v(t),\quad t=\frac{d_{1}x_{2}}{d_{2}x_{1}}.$$

Then after restoring a few unpleasant constants

$$(vv^{\prime\prime}-\frac{2}{n+2}v^{\prime 2})(\frac{n+2}{n}v+(1-t)v^{\prime})^{n-2}=\frac{1}{n-1}(\frac{n}{n+2})^{3},$$

$$\det(D^{2}u)(d_{1}\frac{\partial u}{\partial x_{1}}+d_{2}\frac{\partial u}{\partial x_{2}})^{n-2}=\frac{2n}{(n+2)^{2}(n-1)}(\frac{d_{1}}{d_{2}})^{2}d_{1}^{n-2}.$$

In the asymptotic formula (9) for $v$ at $t\to 0$, we can recover the constants

$$v_{0}=(\frac{nw_{0}}{n+2})^{\frac{n+2}{n}},\quad a=b_{1}v_{0}^{\frac{2}{n+2}}=(\frac{n}{n+2})^{2/n}w_{0}^{-\frac{n+2}{n(n-1)}}.$$

(24)

Let $S_{1}\in H^{0}(\bar{X},d_{1}L_{0})$ (resp. $S_{2}\in H^{0}(\bar{X},d_{2}L_{0})$) be the defining section of $D_{1}\subset\bar{X}$ (resp. $D_{2}$). Recall $h_{L_{0}}$ is the Hermitian metric on the line bundle $L_{0}$ over $Y$, whose curvature form is the Calabi-Yau metric on $Y$ in the class $c_{1}(L_{0})$. We extend $h_{L_{0}}$ smoothly over $\bar{X}$. This induces Hermitian metrics on $d_{1}L_{0}$ and $d_{2}L_{0}$, and in particular we can make sense of $x_{i}=-\log|S_{i}|.$ A technical subtlety is that $S_{1},S_{2}$ have the ambiguity of a multiplicative constant, which corresponds to the ambiguity of additive constants on $x_{1},x_{2}$ of order $O(1)$, to be fixed in section 4.3. Very large $x_{1},x_{2}$ corresponds to the generic region near infinity on the noncompact manifold $X$. The function $u(x_{1},x_{2})$ can be regarded as a Kähler potential on the tubular neighbourhood of $Y$ (with $Y$ deleted). Up to exponentially small error in the $x_{1},x_{2}$ variables, the Kähler metric is modelled on the generalized Calabi ansatz.

The normal bundle of $Y=D_{1}\cap D_{2}\subset\bar{X}$ is $\mathcal{O}(D_{1})\oplus\mathcal{O}(D_{2})|_{Y}=d_{1}L_{0}\oplus d_{2}L_{0}|_{Y}$. Using an auxiliary smooth Hermitian metric on $\bar{X}$, we can identify this normal bundle with a tubular neighbourhood of $Y\subset\bar{X}$ (eg. via normal geodesic flow). The choices of the identifications only produce errors which are exponentially small in the logarithmic variables, which will be negligible since the distance scales of the ansatz metric have power law dependence on the log variables (cf. section 2.6).

Setting up the Hölder norms require a little care, since the injectivity radius of the $T^{2}$-fibres tends to zero in the generic region near infinity. However, for $1\ll x_{2}\leq x_{1}$ (the case of $1\ll x_{1}\leq x_{2}$ being entirely similar), the harmonic radius grows like $O(x_{1}^{1/n}t^{\frac{1}{2(n-1)}})=O(x_{1}^{\frac{1}{n}-\frac{1}{2(n-1)}}x_{2}^{\frac{1}{2(n-1)}})$. This motivates the weighting function

$$\rho=\begin{cases}(|x_{1}|+1)^{\frac{n-2}{2n(n-1)}}(|x_{2}|+1)^{\frac{1}{2(n-1)}},\quad x_{2}\leq x_{1},\\ (|x_{2}|+1)^{\frac{n-2}{2n(n-1)}}(|x_{1}|+1)^{\frac{1}{2(n-1)}},\quad x_{1}\leq x_{2}.\end{cases}$$

(25)

We shall only use $\rho$ up to a uniform equivalence constant; no derivative control on $\rho$ is required. Each $Y$-fibre is covered by $O(1)$ number of charts each of length scale $O(\rho)$, such that the $T^{2}$-bundle is trivialized over the charts. Given a tensor field $T$ on an $O(\rho)$ neighbourhood inside $X$, we can pass to the local universal cover of the charts by unwrapping the $T^{2}$ factors. The local Hölder seminorm is

$$[T]_{\alpha}:=\sup_{dist(P,Q)\lesssim\rho}\rho^{\alpha}\frac{|T(P)-T(Q)|}{|P-Q|^{\alpha}},$$

(26)

using geodesic parallel transport with respect to the generalized Calabi ansatz metric on the local universal cover of the charts. The local $C^{k,\alpha}$ norm is

$$\left\lVert T\right\rVert_{k,\alpha,loc}:=\sum_{j=0}^{k}\sup_{dist(P,Q)\lesssim\rho}\rho^{j}|\nabla^{j}T|+\rho^{k}[\nabla^{k}T]_{\alpha}.$$

(27)

There is up to constant multiple a natural holomorphic volume form $\Omega$ on $X=\bar{X}\setminus D_{1}\cup D_{2}$. We take the normalization such that near $Y\subset\bar{X}$ (cf. (3) for the model case)

$$\Omega=(1+f_{\Omega})\prod_{1}^{2}d\log\xi_{i}\wedge\Omega_{Y},$$

where $\xi_{1},\xi_{2}$ are local defining functions of the smooth divisors $D_{1},D_{2}$, and $f_{\Omega}$ is a local holomorphic function near $Y$, of order $O(|\xi_{1}|+|\xi_{2}|)$, which is exponentially small in the $x_{1},x_{2}$ variables. Higher order derivatives of $f_{\Omega}$ are also exponentially small by holomorphicity.

We write

$$(dd^{c}u)^{n}=K_{0}(1+Err_{1})\sqrt{-1}^{n^{2}}\Omega\wedge\overline{\Omega},$$

(28)

where $Err_{1}$ is some error function, and $K_{0}$ is the (unfortunately complicated) proportionality constant in the model case of generalized Calabi ansatz

$$K_{0}=\frac{\int_{Y}c_{1}(L_{0})^{n-2}}{(4\pi)^{2}}\frac{2n^{2}}{(n+2)^{2}}(\frac{d_{1}}{d_{2}})^{2}d_{1}^{n-2}.$$

Recall that Lemma 2.1 says the model case is exactly Calabi-Yau.