The non-archimedean analogue of the Calabi conjecture is still an open problem in non-archimedean geometry. In the complex case it states that for a complex compact $n$-dimensional manifold $M$ with a Kähler form $\omega$ and $f\in C^{\infty}(M)$, $f>0$ such that $\int_{M}f\omega^{n}=\int_{M}\omega^{n}$ there exists a unique up to constant $\varphi\in C^{\infty}(M)$ such that $\omega+dd^{c}\varphi>0$ and $(\omega+dd^{c}\varphi)^{n}=f\omega^{n}$. This was solved by Calabi (uniqueness, [Cal57]) and Yau (existence, [Yau78]). In the non-archimedean setting, we fix a non-archimedean, non-trivially valued field $K$ and a smooth projective variety $X$ over $K$ of dimension $n$ with a line bundle $L$ on $X$ and consider the corresponding $K$-analytic space $X^{\textup{an}}$ with the line bundle $L^{\textup{an}}$ in the sense of Berkovich. To any continuous semipositive metric $\|\cdot\|$ on $L^{\textup{an}}$ one can associate a positive Radon measure $c_{1}(L,\|\cdot\|)^{n}$ on $X^{\textup{an}}$, called the Monge-Ampère measure, which was introduced by Chambert-Loir in [Cha06]. In a non-archimedean analogue of the Calabi conjecture one asks for a solution of $c_{1}(L,\|\cdot\|)^{n}=\mu$ for a positive Radon measure $\mu$ on $X^{\textup{an}}$ of mass $L^{n}$ when $L$ is ample. The uniqueness up to addition of a constant of such a solution was proved by Yuan and Zhang in [YZ16]. The existence was proved by Liu in [Liu11] for the case of a totally degenerate abelian variety $X$ under some regularity assumptions on the measure by reducing to the complex case. The best known existence result is due to Boucksom, Favre and Jonsson [BFJ15, Theorem A]. They prove existence of a solution to the non-archimedean Monge-Ampère equation if $K$ is discretely valued of residue characteristic zero and $\mu$ is supported on the dual complex of some SNC model of $X$. Note that they assumed also an algebraicity condition which was later removed by Burgos Gil, Gubler, Jell, Künnemann and Martin [BGJ+, Theorem D]. As such a dual complex consists of faces which look like simplices in $\mathbb{R}^{n}$ it would be tempting to observe a connection of the non-archimedean Monge-Ampère operator with the real one. This is the aim of the paper at hand. In particular we will prove the following result (a precise definition of the occurring measures is given in section 4.):

The paper is organized as follows: In Section 2 we give an overview over basic concepts in formal geometry. We recall the definition of a strongly nondegenerate strictly polystable formal scheme and its associated skeleton introduced in [Ber99] and explain the stratum face correspondence developed in [Gub10]. At the end of the section we construct a Cartier divisor from a piecewise affine linear function on the skeleton and prove an important lemma dealing with the degree with respect to this divisor in the case of an affine linear function.

In Section 3 we collect basic definitions and facts on metrized line bundles. Following [GM19] we introduce piecewise linear, algebraic and formal metrics and the notion of semipositivity for them. We also recall some useful properties and the situations in which the definitions coincide.

In Section 4 we recall the definitions of the real and non-archimedean Monge-Ampère measure but we define the latter locally on open subsets of the analytification of a separated scheme of finite type over the field $K$. In order to do so, we prove a local convergence result. This will allow us to formulate Theorem 1.1 in a more general setting where everything is defined locally, see Corollary 5.7.

Section 5 is subject to the proof of Theorem 1.1. In fact in Corollary 5.7, we prove a local generalization of this result. It will follow from Lemma 4.8 and Corollary B.4 that Corollary 5.7 implies Theorem 1.1. We will also generalize the local result in Corollary 5.10 to strongly nondegenerate polystable formal models of $X^{\textup{an}}$ i.e. we will prove:

The proof is inspired by the proof of [Gub10, Theorem 5.18]. In order to reduce to the toric situation, a key ingredient will be Lemma 2.13, showing that affine linear functions on a closed face of a skeleton induce numerically trivial vertical Cartier divisors on a suitable part of the corresponding formal model.

Finally, in Section 6, we apply Theorem 1.1 to obtain two regularity results for solutions to the non-archimedean Calabi-Yau problem. For example we will prove in Proposition 6.4:

Here $C^{k}(\tau)$ is the space of $k$ times continuously differentiable functions on $\tau$. Theorem 1.3 follows from Theorem 1.1 and regularity of the real Monge-Ampère equation.
Terminology. In the following, $K$ denotes a complete, non-archimedean, non-trivially valued field and $K^{\circ}$ its corresponding valuation ring with maximal ideal $K^{\circ\circ}$. All schemes are assumed to be locally of finite type.
Acknowledgements. I thank Walter Gubler for his constant advice and many helpful discussions. I am also grateful to Sébastien Boucksom for helpful discussions and to Antoine Ducros for suggesting a generalization of [CD, Lemme 6.5.1]. Furthermore I would like to thank Klaus Künnemann and Antoine Chambert-Loir for helpful comments, Thomas Fenzl for answering my questions about skeletons and Florent Martin and Walter Gubler for the permission to use their unpublished notes on convexity of psh-functions.