A Holomorphic Splitting Theorem

A long-term project is to construct a complete Calabi-Yau metric on the complement of the anticanonical divisor in a compact Kähler manifold $\text{oM}$. We focus on the case where this smooth divisor has multiplicity 2 and is itself a compact Calabi-Yau manifold. Firstly we solved the Monge-Ampère equation when the Ricci potiential is of $O(r^{-1})$ decay on the generalized $ALG$ manifolds. Then we used the solution to this Kähler Ricci flat metric to prove a holomorphic splitting theorem: If $K_{\text{oM}}=\text{calo}(-2D)$, where $D$ can be realized as a smooth Calabi-Yau manifold, and if ${\text{calo}}_{3D}(D)$ is trivial, then this Kähler manifold $\text{oM}$ is biholomorphic to ${\text{bbp}}^1\times D$.