Sun and Zhang [19] studied the degenerating Calabi-Yau metric on the family of hypersurfaces

$$X_{t}=\{F_{1}F_{2}+tF=0\}\subset\mathbb{CP}^{n},\quad 0<|t|\ll 1,$$

where $F_{1},F_{2}$ define two transverse degree $d_{1},d_{2}$ smooth irreducible hypersurfaces $D_{1},D_{2}$, with $d_{1}+d_{2}=n+1$, and $F$ defines a generic hypersurface of degree $n+1$, so that the $F=0$ locus in $D_{1}\cap D_{2}$ is smooth and irreducible. A concrete special case, studied previously by [11], is when $X_{t}$ is a family of quartic K3 surfaces degenerating into the union of two quadrics.

The Calabi-Yau metric on $X_{t}$ is fibred over an interval. The ends of the interval correspond to the two regions $D_{1}\setminus D_{2}$ and $D_{2}\setminus D_{1}$, and the metrics therein are modelled on the Tian-Yau metrics, whose asymptotes match up with the Calabi ansatz. The transition between the two Calabi ansatzs on the two ends is modelled on an Ooguri-Vafa type metric, obtained by a generalized Gibbons-Hawking type ansatz.

Now algebro-geometrically, our Tian-Yau type space $\mathbb{CP}^{n}\setminus D_{1}\cup D_{2}$ can be imagined as the limit of $\mathbb{CP}^{n}\setminus X_{t}$ as $t\to 0$. It is then natural (but somewhat naïve) to imagine taking the usual Tian-Yau metric construction on $\mathbb{CP}^{n}\setminus X_{t}$, and try to extract limits. It is then not surprising that the Tian-Yau metric on $D_{1}\setminus D_{2}$ and $D_{1}\setminus D_{2}$ should appear in the asymptotic description of the metric on $\mathbb{CP}^{n}\setminus D_{1}\cup D_{2}$, even though the precise scaling factors of these Tian-Yau regions do not seem to be predicted by this naïve limit. However, the Ooguri-Vafa type region in [19] has no direct relation to the generalized Calabi ansatz in our construction.

There is a further way our construction is related to a natural generalization of [19]:

$$X_{t}=\{F_{1}F_{2}F_{3}+tF=0\}\subset\mathbb{CP}^{n},\quad 0<|t|\ll 1.$$

The algebro-geometric limit as $t\to 0$ is the union of transversely intersecting hypersurfaces $D_{1},D_{2},D_{3}$. One can similarly ask for the description of the Calabi-Yau metric on $X_{t}$ for small $t$. It is quite conceivable that the metric model in the region $D_{1}\setminus D_{2}\cup D_{3}$ (and the cyclic permutations) is provided by our construction, although how the transition happens between these three ends is an interesting open problem, which likely involves a further generalization of the Ooguri-Vafa type metric in [19].

As a more general remark, we think the higher $m$ version of the NA MA equation in this paper is the noncompact analogue of the NA MA equation appearing in the collapsing case of polarized degeneration of Calabi-Yau metrics explained in [14], and we expect the generalized Calabi ansatz to be relevant for local metric models in the polarized degenerations.