Theorem 4.2 . [023I]

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Theorem 4.2.

[16][10, Prop. 2.9] There is a complete Ricci-flat Kähler metric

\omega_{TY}=dd^{c}\phi_{TY}=\omega_{Cal^{\prime}}+dd^{c}\phi_{TY,rel}

solving the complex Monge-Ampère equation

\omega_{TY}^{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}},

with exponential decay estimate for some constant $c>0$ depending on $D_{1},D_{2}$ :

|\nabla_{\omega_{Cal^{\prime}}}^{k}\phi_{TY,rel}|_{\omega_{Cal^{\prime}}}=O(e^{-cr_{D_{1}}^{\frac{n-1}{n}}}),\quad r_{D_{1}}\to+\infty,\quad k\geq 0.