We now glue the potential $u$ in the generic region $1\ll\min\{x_{1},x_{2}\}$, and the potential $\phi_{D_{1}}^{(2)}$ in the region $x_{1}\gg|x_{2}|+1$ (and a completely similar potential $\phi_{D_{2}}^{(2)}$ in the region $x_{2}\gg|x_{1}|+1$). We now take a smooth cutoff function $\eta:\mathbb{R}\to[0,1]$, with

$$\eta(x)=\begin{cases}1,\quad x\leq 1,\\ 0,\quad x\geq 2.\end{cases}$$

The glued potential is defined for $|x_{1}|+|x_{2}|\gg 1$,

$$\phi_{glue}=\eta(\tilde{C}\frac{x_{2}}{x_{1}})(\phi_{D_{1}}^{(2)}-u)+\eta(\tilde{C}\frac{x_{1}}{x_{2}})(\phi_{D_{2}}^{(2)}-u)+u,$$

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where $\tilde{C}\gg 1$ is some large fixed constant, specifying the gluing region $x_{1}\sim\tilde{C}x_{2}$ (resp. $x_{2}\sim\tilde{C}x_{1}$). For large $x_{1}\geq\tilde{C}\tilde{x}_{2}$, then $\phi_{glue}$ agrees with $\phi_{D_{1}}^{(2)}$, while for $2\tilde{C}\tilde{x}_{2}\leq x_{1}\leq(2\tilde{C})^{-1}x_{2}$, then $\phi_{glue}$ agrees with $u$.

We also need to extend the gluing ansatz to a Kähler metric over the compact region with $x_{1},x_{2}=O(1)$. We can first extend $\phi_{glue}$ to a smooth potential over $X$, which may not be Kähler inside a fixed compact set. Taking a very ample linear system $H^{0}(\bar{X},m(d_{1}+d_{2})L_{0})$ for $m\gg 1$, we can construct a Fubini-Study metric on $\bar{X}$. Using the defining section $S_{1}S_{2}\in H^{0}(\bar{X},(d_{1}+d_{2})L_{0})$ of the divisor $D_{1}+D_{2}$, we can regard the Fubini-Study metric as a function on $X=\bar{X}\setminus D_{1}\cup D_{2}$, of the form

$$\phi_{FS}=\log\sum_{i}|\frac{s_{i}}{(S_{1}S_{2})^{m}}|^{2}.$$

Clearly $\phi_{FS}$ tends to infinity near $D_{1}\cup D_{2}$ at a speed comparable to $x_{1}+x_{2}$. We take some large constant $A\gg 1$, and $R\gg 1$ depending on $A$, and add to $\phi_{glue}$ the term $A\phi_{FS}\eta(\frac{\phi_{FS}}{R}).$ Intuitively, the cutoff function $\eta$ turns off the Fubini-Study potential outside a large compact subset. By making $A$ large enough, we can improve $\phi_{glue}$ to be Kähler in a fixed compact set. For $R\gg 1$, the cutoff error in the region $\phi_{FS}\sim R$ is suppressed by $dd^{c}\phi_{glue}$, and the metric remains positive. By a slight abuse, we shall continue to use $\phi_{glue}$ to refer to the global Kähler metric on $X$.

Define the volume error function $Err_{2}$ by

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

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