The relevance of the generalized Calabi ansatz to the Tian-Yau problem (cf. Question 2.4) is as follows. Let $\bar{X}$ be a smooth Fano manifold, and $D=\sum D_{i}$ be an anticanonical divisor, which we assume to be simple normal crossing, with only multiplicity one components. Let $X=\bar{X}\setminus D$, then $X$ has a nowhere vanishing holomorphic volume form $\Omega$, and one can ask when this admits a complete Calabi-Yau metric.

An important intuition to keep in mind, is that most of the volume growth near the infinity of $X$ in fact concentrates near the deeper intersection strata $D_{J}=\cap_{i\in J}D_{i}$ of the component divisors $D_{i}$. Let $m$ denote the maximal $|J|$ for which $D_{J}$ is non-empty, then from the volume growth perspective, the neighbourhood of the $m$-fold intersection loci are the generic regions in the Calabi-Yau $X$. For any such subset $J$ with $|J|=m$, the neighbourhood of $D_{J}$ is essentially the total space of $\oplus_{j\in J}\mathcal{O}(D_{j})|_{D_{J}}$, which corresponds to $Z$, and an application of the adjunction formula shows $D_{J}$ is a compact Calabi-Yau, which corresponds to $Y$. Up to inessential normalization constants, the holomorphic volume form is (3) up to negligible errors. In many examples all the $\mathcal{O}(D_{i})$ are ample. The possible relevance of $L$ comes from the prescribed global Kähler class on $X$. The picture we would like to advocate, for which our present paper is a very special case, is that one can find new Tian-Yau type metrics on $X$, whose behaviour in the generic region is modelled on the generalized Calabi ansatz. Morever, further details from this paper suggests the whole problem is inductive on $m$, in the sense that what happens in non-generic regions is related to the generalized Calabi ansatz with smaller $m$.