Local Calabi–Yau threefold from a canonical line bundle

On every holomorphic trivializing open set $U\subset S$, the total space is $U\times \mathbb{C}$. The differential of the projection has kernel equal to the vertical tangent line, canonically identified with the pullback ${\pi }^{\ast }{K}_S$. The base tangent map is surjective in these product charts, so the local exact sequences glue to $0\to {\pi }^{\ast }{K}_S\to T_X\to {\pi }^{\ast }{T}_S\to 0$. The same charts show that $X$ is a smooth complex manifold of dimension $2+1=3$.

Taking determinants in this exact sequence gives $\det T_X\cong {\pi }^{\ast }(K_S\otimes \det T_S)$. Since $\det T_S=K_S^{-1}$ by the definition of $K_S$, $\det T_X\cong {\mathcal{O}}_X$; dualizing yields $K_X=\det {\mathrm{\Omega }}_X^1\cong {\mathcal{O}}_X$.

For $S={\mathbb{P}}^1\times {\mathbb{P}}^1$, every fiber of $\pi$ is isomorphic to $\mathbb{C}$ and is closed in $X$. If $X$ were compact, that closed fiber would be compact, contradicting noncompactness of $\mathbb{C}$. Thus this local threefold is noncompact.