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.

There are a number of recent constructions of complete Calabi-Yau metrics on $\mathbb{C}^{n}$ with $n\geq 3$, that share the unifying theme of holomorphic fibrations. The works [12][20][6] start with a holomorphic fibration given by a weighted homogeneous polynomial $F:\mathbb{C}^{n}\to\mathbb{C}$, such that $F^{-1}(0)$ carries a Sasakian-Einstein cone metric. The other fibres carry asymptotically conical Calabi-Yau metrics modelled at infinity on $F^{-1}(0)$. From this, one builds a Calabi-Yau metric on $\mathbb{C}^{n}$, whose fibrewise restrictions are approximated by these Calabi-Yau metrics on the fibres outside of a compact region, and in the horizontal direction is approximated by the pullback of the Euclidean metric on $\mathbb{C}$. Such metrics on $\mathbb{C}^{n}$ have maximal volume growth, and the tangent cone at infinity is $F^{-1}(0)\times\mathbb{C}$ with the product metric. From an algebro-geometric perspective, the singularities on $F^{-1}(0)$ are klt, which should be viewed as mild singularities. The coordinate functions on $\mathbb{C}^{n}$ all have polynomial growth with respect to the geodesic distance to the origin, even though the growth rates are typically not linear.

More recently a family of new Taub-NUT type Calabi-Yau metrics were constructed on $\mathbb{C}^{3}$ [13], using a generalized Gibbons-Hawking framework. The asymptotic geometry near infinity is generically a $T^{2}$-fibration over $\mathbb{R}^{4}$, but along three rays inside $\mathbb{R}^{4}$ the metric looks like a Taub-NUT fibration over a cylinder. These metrics are fundamentally different in that it has volume growth order $Vol(B(r))\sim O(r^{4})$ coming from the $\mathbb{R}^{4}$ direction, which is not maximal volume growth. Algebro-geometrically, these metrics are associated with the holomorphic fibration

$$F(z_{1},z_{2},z_{3})=z_{1}z_{2}z_{3}:\mathbb{C}^{3}\to\mathbb{C},$$

whose fibres are generically cylinders, contributing two dimensions to $T^{2}$ and two other dimensions to $\mathbb{R}^{4}$. The $\mathbb{C}$ factor contributes the other two dimensions to $\mathbb{R}^{4}$. Notice the singular fibres are reducible, and the nature of the singularity is much worse than klt. In terms of the growth of holomorphic functions, only $z_{1}z_{2}z_{3}$ has polynomial growth, while $z_{1},z_{2},z_{3}$ individually all have exponential type growth, meaning that $\log|z_{k}|$ has polynomial growth.

In view of these constructions, the new feature of this paper is that in the generalized Calabi ansatz, the approximately Calabi-Yau fibres do not appear as fibres of holomorphic fibrations, but rather come from the effective description of an iterated non-holomorphic fibration. The algebraic functions on $X$ have exponential type growth. The prototype of these phenomena is of course already known in the case of the Calabi ansatz, but we believe the NA MA equation points towards a much larger generality of examples, not limited to ODE reduction methods.

As we mentioned before, the generalized Calabi ansatz was discovered in an attempt to interpret the non-archimedean version of the Monge-Ampère equation in the context of polarized degenerations [14]. This relation to non-archimedean geometry remains conjectural, because at least some regularity is needed in order for the NA MA equation to admit a metric interpretation, which is unfortunately still not proven even in some cases where the Calabi-Yau metric is completely understood, such as the case studied in [19]. The reader is thus warned that the following discussions will be rather speculative; they are meant to provide a more general and higher brow perspective to section 2.4, and to motivate directions of future research.

In the polarized degeneration setting, one associates dual complexes to SNC models (or more generally dlt models) of the degeneration family. When the SNC models are related by blow ups with centres supported on the central fibre, then there exist comparison maps between the dual complexes, and the Berkovich space is the inductive limit of these dual complexes. One should think of the Berkovich space as encoding the pure birational geometry of the degeneration family. The polarization provides an extra positive line bundle structure on the Berkovich space, which one should imagine as metric information. One can make sense of the non-archimedean version of plurisubharmonic functions and semipositive metrics, and associate the non-archimedean Monge-Ampère measure. The foundational result of this non-archimedean pluripotential theory is that one can solve the non-archimedean Calabi conjecture on the Berkovich space by a variational method, as is expertly surveyed in [3].

Now in the noncompact setting, the natural analogue of SNC models is SNC pairs $(\bar{X},D)$ with $X=\bar{X}\setminus D$,¹¹ 1 In general $\bar{X}$ needs not be Fano. to which one can associate dual complexes and build up a version of the Berkovich space. To the author’s knowledge, non-archimedean pluripotential theory has not been developed in this noncompact setting, but the key point we would like to suggest is that the non-archimedean Monge-Ampère equation in this conjectural theory, should be equivalent to the differential geometric version (4), and its purpose is to prescribe the asymptotic of the Kähler potential.

The Berkovich space by itself only has the complex geometric information, and by Remark 2.8 we know that this is far from sufficient to specify the metric. Another problem with the non-compact setting, is that the Calabi-Yau volume is infinite. In the concrete setting of this paper, the problem of infinity is essentially solved by imposing the homogeneity ansatz, which reduced the NA MA equation to a boundary value problem with two ends, which is morally a compact problem. Now the geometric meaning of the homogeneity ansatz has to do with the growth order of the algebraic functions with respect to the geodesic distance. This growth information goes beyond pure complex geometry, and knows something about the metric. We would like to suggest it plays a similar role to the positive line bundle in the context of polarized degenerations. Once this is taken into account, one can at least hope for a non-archimedean Calabi conjecture type result in this noncompact setting.

The above picture fits quite well with a heuristic principle of Yau, that complete non-compact Calabi-Yau manifolds should (under mild conditions) admit a natural (quasi-)projective compactifications. When this compactification is projective, one may hope that under an additional specification of the homogeneity ansatz, the non-archimedean geometry produces a version of the NA MA solution, which one can then use to prescribe the asymptotic Kähler potential in the generic region. One then tries to find a completion of the ansatz metric, and hopes that a (highly elaborate) application of the Tian-Yau existence proof would eventually construct a Calabi-Yau metric.

This very large pool of potential examples still do not exhaust the full richness of the complete Calabi-Yau metrics. The reason is that in general Yau’s compactification is only a partial compactification into a quasi-projective variety. A typical phenomenon is that there is a holomorphic fibration to a lower dimensional variety, and the partial compactification amounts to the compactification of the fibres. For instance, the Taub-NUT metric on $\mathbb{C}^{2}$ can be compactified into the rational surface

$$\{([X_{0}:X_{1}:X_{2}],y)|X_{1}X_{2}=yX_{0}^{2}\}\subset\mathbb{CP}^{2}\times\mathbb{C}_{y},$$

where we added in two compactification divisors $D_{1}=\{X_{0}=X_{1}=0\}$ and $D_{2}=\{X_{0}=X_{2}=0\}$. These divisors encode the exponential growth of the coordinate functions in the fibre direction, and are responsible for the fact that the fibrewise metric restrictions are approximately cylindrical. A very similar phenomenon happens with the Taub-NUT type metric on $\mathbb{C}^{3}$ mentioned above. The upshot is that by mixing the holomorphic fibration with the NA MA ansatz, one can hope to generate an even larger supply of Calabi-Yau metrics.