As before, suppose $K\simeq k(\!(t)\!)$ is a discretely valued field with valuation ring $R\simeq k[\![t]\!]$ and residue field $k$. We further assume that $K$ has residue characteristic zero, $\operatorname{char}k=0$. This implies that $R\simeq k[\![t]\!]$ and $K\simeq k(\!(t)\!)$, where $k$ is the residue field of $K$. More importantly, $X$ then admits SNC models, that is, regular models ${\mathcal{X}}$ such that the special fiber ${\mathcal{X}}_{0}$ has simple normal crossings. The dual complex $\Delta_{\mathcal{X}}$, encoding intersections between irreducible components of ${\mathcal{X}}_{0}$, then embeds as a compact subset of ${X^{\mathrm{an}}}$.

Here the condition on $X$ means that there exists a smooth projective curve $C$ over $k$, a smooth projective variety $Y$ over $C$, and a point $p\in C$ such that $X$ is isomorphic to the base change $Y\times_{k}\operatorname{Spec}K$, where $K$ is the fraction field of $\widehat{{\mathcal{O}}}_{C,p}$. This condition is presumably redundant, but is used in the proof: see §8.

To our knowledge, the first to consider the Monge-Ampère equation (or Calabi-Yau problem) in a non-Archimedean setting were Kontsevich and Tschinkel [KT00]. They outlined a strategy in the case when $\mu$ is a point mass.

The case of curves ($n=1$) was treated in detail by Thuillier in his thesis [Thu05]; see also [BR10, FJ04]. In this case, the Monge-Ampère equation is linear and one can construct fundamental solutions by exploring the topological structure of ${X^{\mathrm{an}}}$.

In higher dimensions, Yuan and Zhang [YZ13] proved the uniqueness statement (ii). Their proof, based on the method by Błocki, is valid in a more general context than stated above. The first existence result was obtained by Liu [Liu11], who treated the case when $X$ is a maximally degenerate abelian variety and $\mu$ is equivalent to Lebesgue measure on the skeleton of $X$. His approach amounts to solving a real Monge-Ampère equation on the skeleton. The existence result (i) above was proved by the authors in [BFJ15] and the companion paper [BFJ12]. We will discuss our approach below.

The geometric ramifications of the non-Archimedean Monge-Ampère equations remain to be developed.