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,

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

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

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 $D_{1}\setminus D_{2}$.