Let $K$ be a field endowed with a complete discrete absolute value. Boucksom, Favre, and Jonsson have shown in [BFJ16a, BFJ15] that the Monge–Ampère equation for a Radon measure supported on the skeleton of a smooth projective variety over $K$ has a solution if the variety is of geometric origin from a one-dimensional family over a field of characteristic zero. In this section, we will explain that the same is true in characteristic $p>0$ if we assume resolution of singularities (see Section 6 for precise definitions).

In the following, we work under the following assumptions:

• (A1)

  The $n$-dimensional smooth projective variety $X$ over $K$ is of geometric origin from a $d$-dimensional family over a perfect field $k$ of characteristic $p>0$.

• (A2)

  Resolution of singularities holds over $k$ in dimension $d+n$.

• (A3)

  Embedded resolution of singularities holds over $k$ in dimension $d+n$.

Note that assumptions (A2) and (A3) are unconditional for $n=2$ and $d=1$ by Theorem 6.3 of Cossart and Piltant. For the following, it is crucial to have in mind that models of $X$ can be defined geometrically which follows from Proposition 5.2.

To transfer the results from [BFJ16a, BFJ15], it is essential to note that every projective ${K^{\circ}}$-model of $X$ is dominated by a projective SNC-model of $X$. To see this, we note first that we may assume that the given ${K^{\circ}}$-model is of geometric origin over the perfect field $k$ by Proposition 5.2. Using resolution of singularities in dimension $d+n$ similarly as in the third step of the proof of Lemma 7.5, we deduce that there is a regular projective scheme ${{\mathscr{X}}}_{R}$ over $R$ as in Definition 8.1 dominating the given model. Applying in the same way embedded resolution of singularities in dimension $d+n$ to the non-smooth fibers of ${{\mathscr{X}}}_{R}$ over $R$, we may assume that the singular fibers of ${{\mathscr{X}}}_{R}$ have the same support as a strict normal crossing divisor. Then base change to ${K^{\circ}}$ yields the claim as base change of the discrete valuation ring $\mathcal{O}_{B,b}$ to its completion ${K^{\circ}}$ preserves regularity [Sta17, Tag 0BG4] and strict normal crossing support.

Having sufficiently many projective SNC-models of $X$ at hand, the density results of skeletons in ${X^{{\mathrm{an}}}}$ given in [BFJ16a, Sect. 3] also hold in our case of equicharacteristic $p$. Given a closed $(1,1)$-form $\theta$ with ample de Rham class $\{\theta\}$, the notion of $\theta$-psh functions on ${X^{{\mathrm{an}}}}$ introduced in [BFJ16a, Sect. 7] keeps the same properties in our case. In fact, the results of [BFJ16a, Sect. 1–7] and their proofs carry over to our setting.

Note that we have already proven the continuity of the $\theta$-psh envelope in Theorem 8.2, which is the analogue of [BFJ16a, Thm. 8.3], by using test ideals instead of multiplier ideals. For $u\in C^{0}({X^{{\mathrm{an}}}})$, we recall from 2.8 that we have used a different definition of the $\theta$-psh envelope ${P}_{\theta}(u)$ than in [BFJ16a, Def. 8.1]. Both definitions agree in the equicharacteristic zero situation by [BFJ16a, Thm. 8.3 and Lemma 8.9]. If the characteristic of $K$ is positive and the Assumptions (A1)–(A3) hold then we have explained above how to define $\theta$-psh functions. We claim now that in this case both definitions of the envelope agree as well. Indeed, it follows from [BFJ16a, Lemma 8.4] that the definitions agree on quasi-monomial points of ${X^{{\mathrm{an}}}}$. For any $x\in{X^{{\mathrm{an}}}}$, we consider the net $p_{\mathscr{X}}(x)$ with $\mathscr{X}$ ranging over all SNC models of $X$. By [BFJ16a, Cor. 3.9], this net of quasi-monomial points converges to $x$. It follows from continuity that the net ${P}_{\theta}(u)(p_{\mathscr{X}}(x))$ converges to ${P}_{\theta}(u)(x)$ for our definition of the envelope. By [BFJ16a, Thm. 7.11, Prop. 8.2(i)], the same convergence holds for their envelope and hence both definitions agree.

This yields now in the same way as in [BFJ16a, Thm. 8.7] that the following monotone regularization holds:

In [BFJ15, Sect. 3], the monotone regularization is the basic ingredient to generalize the Monge–Ampère operator from $\theta$-psh model functions to bounded $\theta$-psh functions and hence it applies also to our setting leading to the same results as in [BFJ15, Sect. 3]. For a bounded $\theta$-psh function $\varphi$, we denote by ${\rm MA}_{\theta}(\varphi)$ the associated Monge-Ampère measure on $X^{\mathrm{an}}$. The definitions, results and arguments from [BFJ15, Sect. 4–6] carry over without change. In particular, we may choose a decreasing sequence of $\theta$-psh model functions on ${X^{{\mathrm{an}}}}$ in the monotone regularization from Corollary 9.1 similarly as in [BFJ15, Prop. 4.7].

A crucial step is now to prove the following orthogonality property:

As a consequence of the orthogonality property, we get differentiability of $E\circ{P}_{\theta}$ as in [BFJ15, Thm. 7.2] where $E$ is the energy from [BFJ15, Sect. 6]. We have now all ingredients available to solve the following Monge–Ampère equation.