There is a diffeomorphism $\Phi:\mathcal{C}\setminus K^{\prime}\rightarrow X\setminus K$, where $K\subset X$ is compact and $K^{\prime}=\{|\xi|_{h}\geq\frac{1}{2}\}$, such that the following hold uniformly for all large enough values of $z$.

1. (a)

   We have the complex structure asymptotics

   (3.12)

   $$|\nabla_{g_{\mathcal{C}}}^{k}(\Phi^{*}J_{X}-J_{\mathcal{C}})|_{g_{\mathcal{C}}}=O(e^{-(\frac{1}{2}-\epsilon)z^{n}})\ \text{for all}\ k\geq 0,\epsilon>0.$$

2. (b)

   We have the holomorphic $n$-form asymptotics

   (3.13)

   $$|\nabla_{g_{\mathcal{C}}}^{k}(\Phi^{*}\Omega_{X}-\Omega_{\mathcal{C}})|_{g_{\mathcal{C}}}=O(e^{-(\frac{1}{2}-\epsilon)z^{n}})\ \text{for all}\ k\geq 0,\epsilon>0.$$

3. (c)

   There is some positive constant

   (3.14)

   $$\underline{\delta}>0$$

   such that for all $k\geq 0$ the Ricci-flat Kähler metric $\omega_{TY}$ satisfies the asymptotics

   (3.15)

   $$|\nabla_{g_{\mathcal{C}}}^{k}(\Phi^{*}\omega_{TY}-\omega_{\mathcal{C}})|_{g_{\mathcal{C}}}=O(e^{-\underline{\delta}z^{n/2}}).$$