In this final section we indicate some possible extensions of our work and make a few general remarks.

First of all, it would be nice to have a local theory for semipositive singular metrics. Indeed, while the global approach in [BFJ12, BFJ15] serves works well for the Calabi-Yau problem, it has some unsatisfactory features. For example, it is not completely trivial to prove that the Monge-Ampère operator is local in the sense that if $\phi_{1},\phi_{2}$ are two (say) continuous semipositive metrics that agree on an open subset $U\subset{X^{\mathrm{an}}}$, then $\operatorname{MA}(\phi_{1})=\operatorname{MA}(\phi_{2})$ on $U$. We prove this in [BFJ15] using the Monge-Ampère capacity. Still, it would be desirable to say that the restriction of a semipositive metric to (say) an open subset of ${X^{\mathrm{an}}}$ remains semipositive!

In contrast, in the complex case, the classical approach is local in nature. Namely, one first defines and studies psh functions on open subsets of ${\mathbf{C}}^{n}$ and then defines singular semipositive metrics as global analogues. By construction, the Monge-Ampère operator is a local (differential) operator.⁶⁶ 6 However, one also needs to verify that the Monge-Ampère operator is local for the plurifine topology. This is nontrivial in both the complex and non-Archimedean case.

In a general non-Archimedean setting, Chambert-Loir and Ducros [CD12] (see also [Gub13b, GK14]) define psh functions as continuous functions $\varphi$ such that $d^{\prime}d^{\prime\prime}\varphi$ is a positive closed current (in their sense), for suitable operators $d^{\prime}$, $d^{\prime\prime}$ analogous to their complex counterparts and modeled on notions due to Lagerberg [Lag12]. While this leads to a very nice theory, that moreover works for general Berkovich spaces, the crucial compactness and regularization results are so far missing. At any rate, the tropical charts used in [CD12] may be a good substitute for dual complexes of SNC models.

Going back to the projective setting, there are several open questions and possible extensions, even in the case of a discretely valued ground field of residue characteristic zero.

First, when solving the Monge-Ampère equation, we needed to assume that the variety $X$ was obtained by base change from a variety over a $k$-curve. This assumption was made in order to use the orthogonality result in [BDPP13], but is presumably redundant.

Second, one should be able to solve the Monge-Ampère equation $\operatorname{MA}(\phi)=\mu$ for more general measures $\mu$. In the complex setting, this is done in [GZ07, Din09] for non-pluripolar measures $\mu$. The analogous result should be valid in the non-Archimedean setting, too, although some countability issues seem to require careful attention. Having such a general result would allow for a nice Legendre duality, as explored in [BBGZ13, Berm13] in the complex case.

Third, one could try to get more specific information about the solution. We already mentioned at the end of §8 that we don’t know whether the solution to the equation $\operatorname{MA}(\phi)=d_{L}\,\mu_{x}$ is a model function for $x$ a divisorial point (and $d_{L}=(L^{n})$). In a different direction, one could consider the case when $X$ is a Calabi-Yau variety, in the sense that $K_{X}\simeq{\mathcal{O}}_{X}$. Then there exists a canonical subset $\mathrm{Sk(X)}\subset{X^{\mathrm{an}}}$, the Kontsevich-Soibelman skeleton, see [KS06, MN12, NX13]. It is a subcomplex of the dual complex of any SNC model and comes equipped with an integral affine structure, inducing a volume form on each face. One can solve $\operatorname{MA}(\phi)=\mu$, for linear combinations of these volume forms, viewed as measures on ${X^{\mathrm{an}}}$. Can we say anything concrete about the solution $\phi$, as in the case of maximally degenerate abelian varieties considered in [Liu11]?

It would obviously be interesting to work over other types of non-Archimedean fields, such as ${\mathbf{Q}}_{p}$. Here there are several challenges. First, we systematically use SNC models, which are only known to exist in residue characteristic zero (except in low dimensions). It is possible that the tool of SNC models can, with some additional effort, be replaced by alterations, tropical charts or other methods. However, we also crucially use the assumption of residue characteristic zero when applying the vanishing theorems that underly the regularization theorem for singular semipositive metrics. Here some new ideas are needed.

A simpler situation to handle is that of a trivially valued field. This is explored in [BJ15] and can be briefly explained as follows. Let $k$ be any field of characteristic zero, equipped with the trivial norm. Let $(X,L)$ be a polarized variety over $k$. In this setting, the notion of model metrics and model functions seemingly does not take us very far, as the only model of $X$ is $X$ itself! Instead, the idea is to use a non-Archimedean field extension. Set $K=k(\!(t)\!)$, $X_{K}:=X\otimes_{k}K$ etc. The multiplicative group $G:={\mathbf{G}}_{m,k}$ acts on $X_{K}^{\mathrm{an}}$ and ${X^{\mathrm{an}}}$ can be identified with the set of $G$-equivariant points in $X_{K}^{\mathrm{an}}$. Similarly, singular semipositive metrics on ${L^{\mathrm{an}}}$ are defined as $G$-invariant singular semipositive metrics on $L_{K}^{\mathrm{an}}$. In this way, the main results about $\operatorname{PSH}({L^{\mathrm{an}}})$ follow from the corresponding results about $\operatorname{PSH}(L_{K}^{\mathrm{an}})$ and the same is true for the solution of the Monge-Ampère equation.

A primary motivation for studying the trivially valued case, at least in the case $k={\mathbf{C}}$, is that the space of singular semipositive metrics on ${L^{\mathrm{an}}}$ naturally sits “at the boundary” of the space of positive (Kähler) metrics on the holomorphic line bundle $L$. As such, it can be used to study questions on $K$-stability and may be useful for the study of the existence of constant scalar curvature metrics, see [BHJ15a, BHJ15b]. A different scenario where a complex situation degenerates to a non-Archimedean one occurs in [Jon14].

In yet another direction, one could try to consider line bundles that are not necessarily ample, but rather big and nef, or simply big. In the complex case this was done in [EGZ09, BEGZ10]. One motivation for such a generalization is that it is invariant under birational maps and would hence allow us to study singular varieties.

Finally, it would be interesting to have transcendental analogues. Indeed, it the complex case, one often starts with a Kähler manifold $X$ together with a Kähler class $\omega$, rather than a polarized pair $(X,\omega)$. A notion of Kähler metric is proposed in [KT00, Yu14], but it is not clear whether or not this plays the role of a (possibly) transcendental Kähler metric.