Let $L$ be an ample line bundle on an $n$-dimensional complex projective variety $X$ and let $\mu$ be a smooth volume form on the associated complex manifold ${X^{{\mathrm{an}}}}$ of total mass $\deg_{L}(X)$. The Calabi conjecture claims that there is a smooth semipositive metric ${\|\ \|}$ on $L^{{\mathrm{an}}}$, unique up to positive multiples, solving the Monge–Ampère equation

(1.1)

$$c_{1}(L,{\|\ \|})^{\wedge n}=\mu.$$

This was conjectured by Calabi who proved uniqueness [Cal54, Cal57] and the existence part was solved by Yau [Yau78]. In fact, they proved a more general version in the setting of compact Kähler manifolds, but this will not be relevant for our paper.

The motivation of this paper is the study of the non-archimedean version of this conjecture. We consider a non-archimedean field $K$ with valuation ring ${K^{\circ}}$. Let $L$ be an ample line bundle on an $n$-dimensional smooth projective variety $X$ over $K$. The line bundle $L$ induces a line bundle $L^{\mathrm{an}}$ on the analytification ${X^{{\mathrm{an}}}}$ of $X$ as a Berkovich non-archimedean analytic space. In non-archimedean geometry, model metrics on $L^{{\mathrm{an}}}$ play a similar role as smooth metrics on line bundles on complex manifolds. We call a model metric on $L^{{\mathrm{an}}}$ semipositive if it is induced by an nef model. Zhang introduced continuous semipositive metrics on $L^{{\mathrm{an}}}$ as uniform limits of semipositive model metrics. For such metrics, Chambert-Loir defined a Monge–Ampère measure $c_{1}(L,{\|\ \|})^{\wedge n}$ on ${X^{{\mathrm{an}}}}$ which is a positive Radon measure of total mass $\deg_{L}(X)$. These measures play an important role in arithmetic equidistribution results (see [Yua08]). We refer to Section 2 for details about these notions.

In the non-archimedean Calabi–Yau problem, one looks for continuous semipositive metrics on $L^{{\mathrm{an}}}$ solving the Monge–Ampère equation (1.1). Yuan and Zhang [YZ17] proved that such a metric is unique up to constants. The existence of a singular semipositive solution was proven in the case of curves by Thuillier [Thu05, Cor. 3.4.13]. Liu [Liu11] proved existence of a continuous semipositive solution for totally degenerate abelian varieties $A$ if $\mu$ is a smooth volume form on the canonical skeleton of $A$.

Next, we describe the fundamental existence result of Boucksom, Favre and Jonsson [BFJ16a, BFJ15]. We assume that $K$ is a complete discretely valued field with valuation ring ${K^{\circ}}$. We recall that an SNC-model is a regular model of $X$ such that the special fiber has simple normal crossing support. Boucksom, Favre and Jonsson prove in [BFJ15, Thm. A] that the non-archimedean Calabi–Yau problem has a continuous semipositive solution ${\|\ \|}$ if the following assumptions are satisfied:

• (a)

  The characteristic of the residue field $\tilde{K}$ is zero.

• (b)

  The positive Radon measure $\mu$ is supported on the skeleton of a projective SNC-model of $X$ and satisfies $\mu({X^{{\mathrm{an}}}})=\deg_{L}(X)$.

• (c)

  The smooth projective variety $X$ is of geometric origin from a $1$-dimensional family over $\tilde{K}$.

The last condition will play an important role in this paper. More generally, we say that $X$ is of geometric origin from a $d$-dimensional family over the field $k$ if there is a codimension $1$ point $b$ in a normal variety $B$ over $k$ such that ${K^{\circ}}$ is the completion of $\mathcal{O}_{B,b}$ and such that $X$ is defined over the function field $k(B)$. In [BGJKM16, Thm. D], we have shown that (c) is not necessary for the existence of a continuous semipositive solution of the non-archimedean Calabi–Yau problem if we assume (a) and (b).

We will later look for a similar result in equicharacteristic $p>0$. To do so, we have to understand the basic ingredients in the proof of the existence result of Boucksom, Favre and Jonsson. In [BFJ16a], the authors develop a global pluripotential theory on ${X^{{\mathrm{an}}}}$ for singular semipositive metrics using the piecewise linear structure on the skeletons of SNC-models. It is here where Assumption (a) enters the first time as resolution of singularities is used to have sufficiently many SNC-models of $X$ at hand. For a continuous metric ${\|\ \|}$, we define the semipositive envelope ${P}({\|\ \|})$ by

(1.2)

$${P}({\|\ \|})\coloneqq\inf\{{\|\ \|}^{\prime}\mid\text{${\|\ \|}\leq{\|\ \|}^{\prime}$ {and ${\|\ \|}^{\prime}$ is a semipositive model metric on $L^{{\mathrm{an}}}$}}\}.$$

It is absolutely crucial for pluripotential theory to prove that ${P}({\|\ \|})$ is continuous as this is equivalent to the monotone regularization theorem (see [BFJ16a, Lemma 8.9]). The monotone regularization theorem is the basis in [BFJ15] to introduce the Monge–Ampère measure, capacity and energy for singular semipositive metrics. The proof of continuity of ${P}({\|\ \|})$ in [BFJ16a, §8] uses multiplier ideals on regular projective models. In the proof of the required properties of multiplier ideals (see [BFJ16a, Appendix B]), the authors use vanishing results which hold only in characteristic zero, and hence Assumption (a) plays an important role here as well.

A second important result is the orthogonality property for ${P}({\|\ \|})$ given in [BFJ15, Thm. 7.2]. Multiplier ideals occur again in their proof and it is here where the geometric assumption (c) is used. However, it is shown in [BGJKM16, Thm. 6.3.3] that continuity of ${P}({\|\ \|})$ is enough to prove the orthogonality property without assuming (a) or (c). Then the variational method of Boucksom, Favre and Jonsson can be applied to prove existence of a continuous semipositive solution for the non-archimedean Calabi–Yau problem.

This makes it very clear that continuity of the semipositive envelope ${P}({\|\ \|})$ plays a crucial role in the non-archimedean Calabi–Yau problem. It is the main object of study in this paper. In Section 2, we will study the basic properties of a slight generalization of ${P}({\|\ \|})$ which is called the $\theta$-psh envelope for a closed $(1,1)$-form $\theta$ on $X$. For the sake of simplicity, we will restrict our attention in the introduction to the semipositive envelope ${P}({\|\ \|})$, while all the results hold more generally for the $\theta$-psh envelope assuming that the de Rham class of $\theta$ is ample.

In Section 3, we will look at continuity of the semipositive envelope in the case of a smooth projective curve $X$ over an arbitrary complete non-archimedean field $K$. Potential theory on the curve ${X^{{\mathrm{an}}}}$ was developed in Thuillier’s thesis [Thu05]. We will use Thuillier’s results and the slope formula of Katz, Rabinoff, and Zureick-Brown [KRZB16, Thm. 2.6] to prove:

A slightly more general version will be proved in Theorem 3.1. The following is important in the proof: Let $L$ be any line bundle on the smooth projective curve $X$. We assume that $X$ has a strictly semistable model $\mathscr{X}$ such that $L$ has a model metric ${\|\ \|}_{0}$ associated to a line bundle on $\mathscr{X}$. For any metric ${\|\ \|}$ on $L^{{\mathrm{an}}}$, we consider the function $\varphi\coloneqq-\log({\|\ \|}/{\|\ \|}_{0})$. Let $p_{\mathscr{X}}\colon{X^{{\mathrm{an}}}}\to\Delta$ be the canonical retraction to the skeleton $\Delta$ associated to $\mathscr{X}$. Then

(1.3)

$${\|\ \|}_{\Delta}\coloneqq e^{-\varphi\circ p_{\mathscr{X}}}{\|\ \|}_{0}$$

is a metric on $L^{{\mathrm{an}}}$ which does not depend on the choice of ${\|\ \|}_{0}$. The following result is crucial in the proof of Theorem 1.1:

This will be proven in Propositions 3.5 and 3.8. It would make pluripotential theory and the solution of the non-archimedean Calabi–Yau problem much easier if Proposition 1.2 would also hold in higher dimensions as we could work more combinatorically on skeletons. It is still true that ${\|\ \|}_{\Delta}$ is a model metric which satisfies ${\|\ \|}_{\Delta}\leq{\|\ \|}$. Burgos and Sombra show in a two dimensional toric counterexample in the Appendix that ${\|\ \|}_{\Delta}$ does not have to be semipositive.

We show now that Proposition 1.2 is also crucial for the existence of the solution of the non-archimedean Calabi–Yau problem in the case of curves arguing as in [BFJ16b, §9]. By Thuillier [Thu05, Cor. 3.4.13], there is a semipositive metric ${\|\ \|}$ solving (1.1), but it might be singular. Here, semipositive means that the metric is an increasing pointwise limit of semipositive model metrics of the ample line bundle $L$. If we assume that the positive Radon measure $\mu$ has support in the skeleton $\Delta$ of a strictly semistable model $\mathscr{X}$ of $X$, then it follows easily from Proposition 1.2 that ${\|\ \|}_{\Delta}$ is a continuous semipositive metric solving (1.1). Burgos and Sombra show in their counterexample in the Appendix that this does not hold in higher dimensions either.

To look for solutions of the higher dimensional non-archimedean Monge–Ampère equation in positive characteristic, we will replace the use of multiplier ideals by the use of test ideals. Test ideals were introduced by Hara and Yoshida using a generalization of tight closure theory. In Section 4, we will gather the necessary facts about test ideals mainly following [Mus13] and so we work on a smooth variety $X$ over a perfect field $k$ of characteristic $p>0$. Similarly as in the case of multiplier ideals, one can define an asymptotic test ideal $\tau(\lambda\|D\|)$ of exponent $\lambda\in\mathbb{R}_{\geq 0}$ for a divisor $D$ on $X$. Crucial for us is that $\tau(\lambda\|D\|)$ satisfies a subadditivity property and the following uniform generation property:

This was proven by Mustaţă if $X$ is projective over $k$. As we will later work over discrete valuation rings, we need the more general version with $X$ only projective over $R$. This will be possible in Theorem 4.6 as we can replace the use of Fujita’s vanishing theorem in Mustaţă’s proof by Keeler’s generalization.

Now we come to the non-archimedean Calabi–Yau problem in equicharacteristic $p>0$. For the remaining part of the introduction, we now fix an $n$-dimensional smooth projective variety $X$ over a complete discretely valued field $K$ of characteristic $p$. To apply the results on test ideals, we have to require that $X$ is of geometric origin from a $d$-dimensional family over a perfect field $k$. We also fix an ample line bundle $L$ on $X$.

For the precise definition about resolution of singularities, we refer to Definition 6.1. As resolution of singularities is known in dimension $3$ over a perfect field by a result of Cossart and Piltant [CP09, Thm. p. 1839], Theorem 1.4 is unconditional if $X$ is a smooth projective surface of geometric origin from a $1$-dimensional family over $k$.

In Section 7, we will prove Theorem 1.4 in the case when ${\|\ \|}$ is a model metric associated to a model which is also defined geometrically over $k$. We will follow the proof of Boucksom, Favre, and Jonsson, replacing multiplier ideals by test ideals. As we use a rather weak notion of resolution of singularities, a rather subtle point in the argument is necessary in the proof of Lemma 7.5 which involves a result of Pépin about semi-factorial models. Theorem 1.4 will be proved in full generality in Section 8 using the $dd^{c}$-lemma and basic properties of the semipositive envelope. In fact, we will prove in Theorem 8.2 a slightly more general result.

If we use additionally that embedded resolution of singularities (see Definition 6.2) holds over $k$ in dimension $d+n$, then the family of projective SNC-models will be cofinal in the category of all models of $X$. We will see in Section 9 that this and Theorem 1.4 allow us to set up the pluripotential theory from [BFJ16a] on ${X^{{\mathrm{an}}}}$. By [BFJ15, Thm. 7.2] again, the continuity of ${P}({\|\ \|})$ yields that the orthogonality property holds for any continuous metric on $L^{{\mathrm{an}}}$. We will use this in Section 9 to show that the variational method of Boucksom, Favre and Jonnson proves the following result (see Theorem 9.3).

Cossart and Piltant [CP08, CP09] have shown resolution of singularities and embedded resolution of singularities in dimension $3$ over a perfect field, hence Theorem 1.5 holds unconditionally for a smooth projective surface $X$ of geometric origin from a $1$-dimensional family over the perfect field $k$.