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.