Following Yau’s solution of the Calabi conjecture [22], the analytic approach to the construction of complete Calabi-Yau metrics on noncompact manifolds was initiated by the seminal works of Tian-Yau [16, 17] who proved the following fundamental theorem

The asymptotic geometry, from the holomorphic viewpoint, is modelled on the normal bundle $\mathcal{O}(D)|_{D}=-K_{X}|_{D}\to D$. This total space carries a model metric known as the Calabi ansatz, which specifies the asymptotic behaviour of the desired Calabi-Yau metric. The main gist of Tian and Yau’s work, which is improved later by Hein [8] among others, is that once we have a good asymptotic ansatz, then running a non-compact version of Yau’s proof of the Calabi conjecture would produce an actual Calabi-Yau metric on $X$.

In the current paper we are motivated by the following well-known question, which dates back to the work of Tian-Yau;

This setup has plenty of examples, for instance $\bar{X}$ can be $\mathbb{CP}^{n}$, and $D$ is a multiplicity one simple normal crossing divisor in $\mathcal{O}(n+1).$ Morally speaking, we are asking ‘what would happen to the Tian-Yau construction once $D$ breaks up into several components’. Our main result is a solution of the Tian-Yau problem in the case of two proportional divisors.

More precisely, the holomorphic geometry in the generic region near infinity is modeled on the total space of the bundle $d_{1}L_{0}\oplus d_{2}L_{0}\to D_{1}\cap D_{2}$. We will explain that there is a generalized Calabi ansatz on (an open subset of) this total space, which is associated with a PDE we call the non-archimedean Monge-Ampère equation. This story has its origin in the context of polarized degenerations, non-archimedean geometry and the SYZ conjecture [14]. In our specialized setting, this equation can be dimensionally reduced to an ODE, whose solutions can be rather explicitly given in terms of hypergeometric functions.

The infinity of $X$ also contains non-generic regions, which complex geometrically correspond to the neighbourhood of $D_{1}\setminus D_{2}$ (resp. $D_{2}\setminus D_{1}$) inside $\bar{X}$. Notice that $D_{1}\cap D_{2}$ is an anticanonical divisor inside the Fano manifold $D_{1}$ (resp. $D_{2}$), and the metric geometry in the non-generic region involves a fibration by the Tian-Yau metrics on $D_{1}\setminus D_{2}$ (resp. $D_{2}\setminus D_{1}$) with some power law scalings. Matching the generalized Calabi ansatz with the metric behaviour in the non-generic region involves a nontrivial ODE matching problem.

The main strategy to construct the Calabi-Yau metric, is to first produce an approximate metric ansatz, and after suitably improving the decay rate of the volume form error, we appeal to Tian-Yau-Hein’s existence package. This strategy has been used in a number of recent works, notably [10][12][13][6][20].

Our construction suggests that the Tian-Yau problem has an inductive structure in terms of the depth of intersections of the divisors $D_{i}$ on $\bar{X}$. For instance, in the case of three divisors $D_{1},D_{2},D_{3}$ with nonempty intersections, we expect the Tian-Yau problem on $D_{1}\setminus(D_{2}\cup D_{3})\cap D_{1}$ (and the cyclic permutations) to appear in the non-generic region at infinity. The generic region would involve a generalized Calabi ansatz metric, associated with a more complicated non-archimedean Monge-Ampère equation, whose boundary conditions are prescribed by the need to match with the nongeneric regions. We think the principal remaining difficulty is then to solve the non-archimedean Monge-Ampère equation, which can in general no longer be reduced to an ODE as we increase the number of divisors. Another interesting direction is to further develop the link with non-archimedean geometry (cf. section 2.8.3).

The organization of this paper is as follows. Section 2 contains most of the geometric aspects. It introduces the generalized Calabi ansatz, the ODE reduction of the non-archimedean Monge-Ampère equation, and the boundary conditions. We also discuss further relations to the literature and future directions. Section 3 explicitly solves the ODE and implements the matching problem. Section 4 is concerned with producing an approximate metric ansatz, and the rather technical issue of error estimates. Section 5 reviews the Tian-Yau-Hein existence package, and hammers a few final nails.