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.

We now translate the ODE asymptote at $t\to 0$ to the geometry of the region $1\ll x_{2}\ll x_{1}$. The analyticity of the ODE solution (cf. Cor. 3.5) gives a fractional power series expansion (19), which converts via (13) to

$$w(t)=(1-t)\{w_{0}+\frac{n-1}{n}w_{0}^{-\frac{3}{n-1}}(\frac{t}{1-t})^{\frac{n}{n-1}}+O((\frac{t}{1-t})^{\frac{2n}{n-1}})\}.$$

Now

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

hence we have a fractional power series

$$\begin{split}v(t)=(1-t)^{\frac{n+2}{n}}\{v_{0}+\frac{n-1}{n}a(\frac{t}{1-t})^{\frac{n}{n-1}}+\sum_{k=2}^{\infty}a_{k}(\frac{t}{1-t})^{\frac{kn}{n-1}}\}.\end{split}$$

Recall

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

We introduce the new variable

$$\tilde{x}_{1}:=x_{1}-\frac{d_{1}}{d_{2}}x_{2}=(1-t)x_{1}=\frac{d_{1}}{d_{2}}\frac{1-t}{t}x_{2}.$$

In terms of the defining sections $S_{i}\in H^{0}(\bar{X},d_{i}L_{0})$ for $D_{1},D_{2}$, we have

$$\tilde{x}_{1}=-\log|S_{1}|+\frac{d_{1}}{d_{2}}\log|S_{2}|=\frac{1}{d_{2}}\log|S_{1}^{\otimes d_{1}}/S_{1}^{\otimes d_{2}}|,$$

where

$$\xi^{\prime}:=S_{2}^{\otimes d_{1}}/S_{1}^{\otimes d_{2}}$$

is actually a holomorphic function, which means its magnitude does not involve the choice of a Hermitian metric. The geometric significance of this function is that although the normal bundle $\mathcal{O}(D_{1})=d_{1}L_{0}$ of $D_{1}\subset\bar{X}$ is nontrivial, it restricts to a line bundle on $D_{1}\setminus D_{2}$ which becomes trivial after taking finite power.

We then have an expansion for $1\ll x_{2}\ll x_{1}$,

$$u=v_{0}\tilde{x}_{1}^{\frac{n+2}{n}}+\frac{n-1}{n}a\tilde{x}_{1}^{\frac{n+2}{n}-\frac{n}{n-1}}(\frac{d_{1}x_{2}}{d_{2}})^{\frac{n}{n-1}}+\sum_{k\geq 2}a_{k}\tilde{x}_{1}^{\frac{n+2}{n}-\frac{kn}{n-1}}(\frac{d_{1}x_{2}}{d_{2}})^{\frac{kn}{n-1}}.$$

(29)

We recall some details of the Tian-Yau metric (cf. [11, section 3] for more expositions).

In our setup, $D_{1}$ is a Fano manifold in its own right with anticanonical bundle $d_{2}L_{0}|_{D_{1}}$, and $Y=D_{2}\cap D_{1}$ is an anticanonical divisor defined by some section $S\in H^{0}(D_{1},d_{2}L_{0})$, which in our context is the restriction of $S_{2}\in H^{0}(\bar{X},d_{2}L_{0})$ to $D_{1}$. Up to a multiplicative constant $S^{-1}$ can be viewed as a holomorphic volume form $\Omega_{D_{1}}$ on $D_{1}\setminus D_{2}$ with a simple pole along $Y$, which we normalize to have residue $\Omega_{Y}$ along $Y$. In local coordinates,

$$\Omega_{D_{1}}\approx d\log\xi_{2}\wedge\Omega_{Y},$$

where $\xi_{2}$ is a local defining function of $D_{2}\cap D_{1}\subset D_{1}$. Recall $h_{L_{0}}^{\otimes d_{2}}$ is the Hermitian metric on $L_{0}|_{Y}$ whose curvature form is the Calabi-Yau metric on $Y$ in the class $d_{2}c_{1}(L_{0})$. We extend $h_{L_{0}}$ to a smooth, positively curved metric on $D_{1}$, so that

$$\omega_{Cal^{\prime}}=\frac{n-1}{n}dd^{c}(-\log|S|)^{\frac{n}{n-1}}$$

defines a Kähler form on a neighbourhood of infinity in $D_{1}\setminus D_{2}$. Up to an approximately holomorphic diffeomorphism, the metric $\omega_{Cal^{\prime}}$ on $(D_{1}\setminus D_{2})$ outside a compact set, agrees with the Calabi ansatz on the total space of the line bundle $L_{0}\to Y$, up to exponentially small errors. The slightly unusual exponent $\frac{n}{n-1}$ is because $\dim Y=n-2$. Asymptotically, the distance to a fixed basepoint in $D_{1}\setminus D_{2}$ is uniformly equivalent to $r_{D_{1}}\sim(-\log|S|)^{\frac{n}{2(n-1)}}.$ We can arrange $\omega_{Cal^{\prime}}$ to extend to an exact global Kähler metric on $D_{1}\setminus D_{2}$, agreeing with the previous formula outside a compact set, still denoted $\omega_{Cal^{\prime}}$.

A technical subtlety is that $S$ is only defined up to a multiplicative constant, and correspondingly $\log|S|$ has an additive constant ambiguity, which is fixed by the integral normalization condition

$$\int_{D_{1}\setminus D_{2}}\left(\omega_{Cal^{\prime}}^{n-1}-\frac{d_{2}^{n-2}\int_{Y}c_{1}(L_{0})^{n-2}}{4\pi}\sqrt{-1}^{(n-1)^{2}}\Omega_{D_{1}}\wedge\overline{\Omega}_{D_{1}}\right)=0.$$

(30)

This makes sense because the integrand has exponential decay near infinity, and morever Stokes theorem on large compact sets shows that the integral only depends on the asymptotic information of $\omega_{cal^{\prime}}$ near infinity. The preferred normalization of $\log|S|$ resolves the additive ambiguity of $x_{2}=-\log|S_{2}|$; a completely analogous normalization on the Tian-Yau space $D_{2}\setminus D_{1}$ fixes the additive ambiguity of $x_{1}=-\log|S_{1}|$.

The main existence theorem and asymptotic information of the Tian-Yau metric is summarized as follows:

The volume growth rate of geodesic balls on the Tian-Yau space is $\text{Vol}(B(r_{D_{1}}))\sim r_{D_{1}}^{\frac{2(n-1)}{n}}$, which is less than quadratic. As such, solving the Poisson equation with potential decaying at infinity would require an integral normalization on the forcing term, which is related to (30) as the Poisson equation is the linearization of the complex Monge-Ampère equation.

To solve the Poisson equation without the integral normalization condition on the forcing term, we need to allow solutions growing at infinity. Consider the special function $(-\log|S|)^{\frac{1}{n-1}}$ defined near infinity, which we extend smoothly to a global function $u_{0}$ on the Tian-Yau space $D_{1}\setminus D_{2}$.

Next, we need to move up one dimension, and produce some ansatz metric on $X=\bar{X}\setminus D_{1}\cup D_{2}$ in the region near infinity close to $D_{1}\setminus D_{2}$, corresponding in the logarithmic coordinate to $x_{1}\gg|x_{2}|+1$, including in particular the region with $x_{2}=O(1),x_{1}\gg 1$. Pick an auxiliary smooth Hermitian metric on $\bar{X}$, such that the normal vector field to $D_{1}\subset\bar{X}$ is tangent to $D_{2}$ along $Y=D_{1}\cap D_{2}$. Then normal geodesic flow identifies a tubular neighbourhood of $D_{1}$ with the normal bundle of $D_{1}$, and the error caused by the failure of holomorphicity is exponentially suppressed in the log coordinates. We can thus regard the potential $\phi_{TY}$ of the Tian-Yau metric on $D_{1}\setminus D_{2}$ as a function on its tubular neighbourhood, via pullback.

To match with the asymptote (29) in the region $1\ll x_{2}\ll x_{1}$, we are motivated to consider the local ansatz potential near $D_{1}\setminus D_{2}$

$$\phi_{D_{1}}=v_{0}\tilde{x}_{1}^{\frac{n+2}{n}}+a(\frac{d_{1}}{d_{2}})^{\frac{n}{n-1}}\tilde{x}_{1}^{\frac{n-2}{n(n-1)}}\phi_{TY}+\sum_{k\geq 2}a_{k}\tilde{x}_{1}^{\frac{n+2}{n}-\frac{kn}{n-1}}(\frac{d_{1}x_{2}}{d_{2}})^{\frac{kn}{n-1}}.$$

(31)

Here we make a fixed choice of a positive valued smooth function on the Tian-Yau space matching with $x_{2}$ outside a compact set, still denoted as $x_{2}$, so that when $x_{2}=O(1)$ the expression $\phi_{D_{1}}$ remains smooth. The convergence of the series is valid for $x_{2}\ll x_{1}$. Of course, the ad hoc choice means that the ansatz needs to be corrected later by more refined terms.

The local geometry should be imagined as a fibration of Tian-Yau metrics with slowly changing size depending on the logarithmic variable $x_{1}$. Consider $\tilde{x}_{1}$ around a given large value $x_{1}^{\prime}\gg 1$. The dominant terms in $dd^{c}\phi_{D_{1}}$ are

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

(32)

Up to taking a finite cover (to do with taking fractional powers of $\xi^{\prime}$), this model metric describes the product of the Tian-Yau metric with length scale $O(\rho)$ corresponding to the harmonic radius scale, and a cylinder of circle length scale $O(x_{1}^{\prime\frac{2-n}{2n}})$ corresponding to the injectivity scale.

We can now set up the local Hölder norms in the nongeneric region $x_{2}\ll x_{1}$ (The case of $x_{1}\ll x_{2}$ is completely similar). When $x_{2}=O(1),x_{1}\gg 1$, we unwind the $S^{1}$ factor of the cylinder to pass to the local universal cover. On $O(\rho)$ neighbourhoods, we can use the local product metric (32) to define the local Hölder seminorm and the $C^{k,\alpha}$-norms, as in (26)(27).

The case of $1\ll x_{2}\ll x_{1}$ is already covered by section 4.1. If we had used the local product metric (32) as reference metric, it would lead to an equivalent definition of local Hölder norms up to uniform equivalence. Notice that within $O(\rho)$ neighbourhoods the values of $x_{1},x_{2}$ do not vary drastically, so expressions like $O(x_{1}^{\alpha}x_{2}^{\beta})$ are not sensitive to the choice of points in the $O(\rho)$ neighbourhood, except when $x_{2}=O(1)$, in which case we use a slightly abusive convention that $O(x_{2}^{\beta})$ stands for $O(1)$ in this region.

We start by observing the derivative bounds of basic functions, which can be read off using the leading order metric ansatz, remembering that the Tian-Yau metric is well approximated by the Calabi ansatz except when $x_{2}=O(1)$, and that the $x_{1}^{\prime}$-dependence in $\omega_{x_{1}^{\prime}}$ introduces scaling factors.

The following lemma quantifies the approximation of the local ansatz potential $\phi_{D_{1}}$ by the model product metric: