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 K\"ahler manifold $\oM$. 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-Amp\`ere equation when the Ricci potiential is of $O(r^{-1})$ decay on the generalized $ALG$ manifolds. Then we used the solution to this K\"ahler Ricci flat metric to prove a holomorphic splitting theorem: If $K_{\oM}=\calo(-2D)$, where $D$ can be realized as a smooth Calabi-Yau manifold, and if $\calo_{3D}(D)$ is trivial, then this K\"ahler manifold $\oM$ is biholomorphic to $\bbp^1\times D$.