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)