The goal of this paper is to construct continuous solutions to a non-Archimedean analogue of certain complex Monge-Ampère equations on projective manifolds, which arose in complex geometry as more degenerate versions of the by-now classical equations considered by Aubin, Calabi and Yau. More specifically, our main result can be understood as an analogue of a fundamental result by S. Kołodziej [Koł98].

Let us briefly recall the complex statement that we have in mind. Let $L$ be an ample line bundle on a smooth complex projective variety $X$ of dimension $n$. Let $\mu$ be a positive measure on $X$, of mass equal to $c_{1}(L)^{n}$. It was shown in [Koł98] that under a mild regularity assumption on $\mu$ (which is for instance satisfied as soon as $\mu$ has $L^{p}$-density with respect to Lebesgue measure for some $p>1$), there exists a continuous metric $\|\cdot\|$ on $L$, unique up to a multiplicative factor, whose curvature form $c_{1}(L,\|\cdot\|)$ is a closed positive $(1,1)$-current satisfying $c_{1}(L,\|\cdot\|)^{n}=\mu$ in the sense of pluripotential theory [BT82]. This result relied on the work of Aubin, Calabi and Yau, which culminated in the celebrated article [Yau78], where it was shown that the solution metric is smooth when $\mu$ is a smooth positive volume form on $X$.

We next turn to the non-Archimedean analogue, referring to §2 for more details. Let $K$ be a complete discrete valuation field whose residue field $k$ has characteristic zero, so that $K\simeq k(\!(t)\!)$. Let $X$ be a smooth projective variety over $K$, and write $n=\dim X$. Thanks to the non-Archimedean GAGA principle, it is reasonable to also denote by $X$ the corresponding $K$-analytic space in the sense of Berkovich, whose underlying topological space is compact Hausdorff. A model of $X$ is a normal scheme $\mathcal{X}$ that is flat and projective over $S:=\Spec k[\![t]\!]$, and whose generic fiber can be identified with $X$.

Consider a ample line bundle $L$ on $X$. A model metric on $L$ is a metric defined by a extension $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{Q}}$ of $L$ to some model $\mathcal{X}$. Such a metric is called semipositive if $\mathcal{L}$ is nef, i.e. has non-negative degree on all proper curves of the special fiber of $\mathcal{X}$. S.-W. Zhang introduced in [Zha95] the more flexible notion of semipositive continuous metric as the uniform limit of semipositive model metrics.¹¹ 1 We refer to Table 1 in §2.6 below for a comparison of our terminology with existing terminology. In this context, A. Chambert-Loir [CL06] defined the Monge-Ampère measure $c_{1}(L,\|\cdot\|)^{n}$ of a semipositive continuous metric $\|\cdot\|$ on $L$. It is a positive Radon measure on $X$, of mass $\deg L$.

V. Berkovich constructed in [Ber99] the skeleton associated to a polystable model of $X$. Since we are assuming $K$ to have residue characteristic zero, it is easier to rely on resolution of singularities and instead consider SNC models, i.e. models whose special fiber has simple normal crossing support (but is not necessarily reduced, as opposed to a semistable model). To each SNC model $\mathcal{X}$ is associated a dual complex $\Delta_{\mathcal{X}}$ that encodes the combinatorics of the intersections of the components of the special fiber, and which embeds in the Berkovich space $X$ just as skeletons do. Any finite set of divisorial points is contained in the dual complex of some SNC model; in particular $\bigcup_{\mathcal{X}}\Delta_{\mathcal{X}}$ is dense in $X$.

We can now state our main result. We say that $X$ is algebraizable if there exists a (one-variable) function field $F$ admitting $K$ as a completion and a smooth projective $F$-scheme $Y$ such that $X=Y_{K}$.

Even though the result is most likely true without this assumption, the algebraizability condition plays an essential role in our proof, as we shall explain below. Note that the line bundle $L$ is not assumed to be defined over a function field.

The uniqueness part in Theorem A follows from a result of X. Yuan and S.-W. Zhang [YZ10] asserting more generally that a continuous semipositive metric $\|\cdot\|$ is uniquely determined up to a constant by its Monge-Ampère measure. Their proof is inspired by the one given by Błocki [Bło03] in the complex setting.

Our approach does not give any information on the regularity of the metric besides continuity. It would be interesting to further investigate this issue, for instance when $\mu$ is supported on finitely many divisorial points. We refer to §9 for a discussion of this problem in the case of toric varieties, based on the recent work [BPS11].

Versions of Theorem A are already known in a few cases. For curves (and in fact over any complete non-Archimedean, non-trivially valued field), it can easily be deduced from results of A. Thuillier [Thu05], who developed a theory of singular semipositive metrics on analytic curves that is completely analogous to the complex case. Solving (1.1) for curves boils down to a system of linear equations and relies on the negativity of the intersection form of the special fiber of a suitable model, see §9. Alternatively, one can exploit the structure of the Berkovich space as a metrized graph as in [BR10, FJ04].

In higher dimensions, Y. Liu [Liu10] treated the related case when $X$ is a totally degenerate abelian variety over $\mathbf{C}_{p}$, and $\mu$ is a (smooth) measure supported on the dual complex of the canonical formal model of $X$, as constructed by Mumford. By exploiting the fact that this dual complex is a compact (real) torus, one can translate the equation $c_{1}(L,\|\cdot\|)^{n}=\mu$ into a (real) Monge-Ampère equation on this real torus, and apply Yau’s result to its complexification to obtain the metric.

A statement very close to Theorem A also appears in an unpublished set of notes by M. Kontsevich and Y. Tschinkel [KT00] dating from 2001, where the authors propose a detailed strategy of proof in the case $\mu$ is a Dirac mass at a divisorial point. Several ingredients in their approach also appear in our paper (see Remark 8.7 below).

We are now going to present an outline of our proof of Theorem A, which consists in mimicking as far as possible the variational approach to complex Monge-Ampère equations of [BBGZ09] and the $C^{0}$-estimates of [Koł98]. To that end we will rephrase Theorem A in a more analytic language. Let us thus recall the notion of quasi-plurisubharmonic function that we developed in [BFJ11] and its main properties.

As a variant of [BGS95] we first define the space of closed $(1,1)$-forms on $X$ as the direct limit

$$\mathcal{Z}^{1,1}(X):=\varinjlim_{\mathcal{X}}N^{1}(\mathcal{X}/S),$$

where $\mathcal{X}$ ranges over all models of $X$ and the space of numerical classes $N^{1}(\mathcal{X}/S)$ is defined as $\Pic(\mathcal{X})_{\mathbf{R}}$ modulo numerical equivalence on the special fiber. Each closed $(1,1)$-form $\theta\in\mathcal{Z}^{1,1}(X)$ defines a class $\{\theta\}\in N^{1}(X)$, which we refer to as its de Rham class. We say that $\theta\in\mathcal{Z}^{1,1}(X)$ is semipositive if it is determined by a nef numerical class on some model. Each model metric $\|\cdot\|$ on a line bundle $L$ over $X$ defines a closed $(1,1)$-form $c_{1}(L,\|\cdot\|)$ that we call the curvature form of the metric. The de Rham class of $c_{1}(L,\|\cdot\|)$ is just $c_{1}(L)\in N^{1}(X)$, and the model metric $\|\cdot\|$ is semipositive (in the sense of Zhang) iff its curvature is. Each model metric on the trivial line bundle is of the form $e^{-\varphi}$ for some $\varphi\in C^{0}(X)$, which is then by definition a model function. Following complex notation, we write $dd^{c}\varphi$ for the curvature form of this metric, so that $c_{1}(L,\|\cdot\|e^{-\varphi})=c_{1}(L,\|\cdot\|)+dd^{c}\varphi$.

Now let $\omega\in\mathcal{Z}^{1,1}(X)$ be a reference closed semipositive $(1,1)$-form on $X$, such that $\{\omega\}\in N^{1}(X)$ is furthermore ample. This situation arises for instance when $\omega$ is the curvature form of a semipositive model metric on an ample line bundle $L$. As was shown in [BFJ11], one may then define a class $\PSH(X,\omega)$ of $\omega$-psh functions with the following properties:

• •

  Each $\varphi\in\PSH(X,\omega)$ is an upper semicontinuous function $X\to[-\infty,+\infty[$ whose restriction to the faces of any dual complex is continuous and convex.

• •

  The set $\PSH(X,\omega)$ is convex and stable under max.

• •

  A model function $\varphi\in\mathcal{D}(X)$ is $\omega$-psh iff $\omega+dd^{c}\varphi\in\mathcal{Z}^{1,1}(X)$ is semipositive.

The two main results of [BFJ11] further state that

• •

  $\PSH(X,\omega)/\mathbf{R}$ is compact with respect to the topology of uniform convergence on dual complexes.

• •

  Every $\varphi\in\PSH(X,\omega)$ is the decreasing limit of a family of $\omega$-psh model functions.

It follows from the latter property and Dini’s lemma that every continuous $\omega$-psh function is a uniform limit over $X$ of $\omega$-psh model functions. This shows in particular that our definition of continuous semipositive metrics is compatible with Zhang’s. Chambert-Loir’s definition of the Monge-Ampère measure of a continuous semipositive metric immediately extends to our setting and enables us to associate to any $n$-tuple of continuous $\omega$-psh functions $\varphi_{1},...,\varphi_{n}\in C^{0}(X)\cap\PSH(X,\omega)$ a (mixed) Monge-Ampère measure

$$(\omega+dd^{c}\varphi_{1})\wedge...\wedge(\omega+dd^{c}\varphi_{n}),$$

a positive Radon measure on $X$ of mass $\{\omega\}^{n}$, which depends continuously on $(\varphi_{1},...,\varphi_{n})$ with respect to the topology of uniform convergence on $X$. As in the complex case, it is however not possible to define such mixed Monge-Ampère measures in a reasonable way for arbitrary $\omega$-psh functions, as soon as $n\geq 2$.

The following result is a slight generalization of Theorem A phrased in the present language.

This formulation is designed to emphasize the analogy with the complex case. However, it is important to keep in mind that the non-Archimedean Monge-Ampère operator is not a differential operator but rather defined in terms of intersection theory.

Let us now set up the variational approach we use to solve our non-Archimedean Monge-Ampère equation, following [BBGZ09]. A key feature of Monge-Ampère equations is that they may be written as Euler-Lagrange equations. This fact goes back at least to Alexandrov [Ale38] in the more classical case of real Monge-Ampère equations, while the relevant functional in the complex case has been well-known in Kähler geometry since the works of Aubin, Calabi and Yau. We introduce in our setting the energy functional

(1.3)

$$E_{\omega}(\varphi):=\frac{1}{n+1}\sum_{j=0}^{n}\int\varphi\,(\omega+dd^{c}\varphi)^{j}\wedge\omega^{n-j},$$

defined for the moment for $\varphi\in C^{0}(X)\cap\PSH(X,\omega)$. An easy computation shows that

(1.4)

$$\frac{d}{dt}\bigg|_{t=0_{+}}E_{\omega}((1-t)\varphi+t\psi)=\int(\psi-\varphi)\,(\omega+dd^{c}\varphi)^{n}$$

for any two $\varphi,\psi\in C^{0}(X)\cap\PSH(X,\omega)$, so that (1.2) is indeed the Euler-Lagrange equation of the functional

$$F_{\mu}(\varphi):=E_{\omega}(\varphi)-\int\varphi\,d\mu.$$

Observe that the compatibility condition $\mu(X)=\{\omega\}^{n}$ guarantees that $F_{\mu}$ is translation-invariant, i.e. $F_{\mu}(\varphi+c)=F_{\mu}(\varphi)$ for all $c\in\mathbf{R}$. As in the complex case, one shows that the functional $E_{\omega}$ is concave on $C^{0}(X)\cap\PSH(X,\omega)$, so that any solution $\varphi$ to (1.2) is necessarily a maximizer of $F_{\mu}$. The variational method conversely amounts to proving the existence of a maximizer of $F_{\mu}$ and showing that it satisfies (1.2). But the lack of compactness of the space $C^{0}(X)\cap\PSH(X,\omega)$ where $F_{\mu}$ is defined so far makes it hard to construct a maximizer, while it is at any rate non-obvious that such a maximizer should satisfy the Euler-Lagrange equation, since it might belong to the boundary of $C^{0}(X)\cap\PSH(X,\omega)$. In order to circumvent these difficulties we are going to argue along the following three steps.

• Step 1:

  Enlarge the space where the variational problem is being considered, in order to gain compactness and construct a maximizer $\varphi_{0}$ there.

• Step 2:

  Show that the maximizer is in a natural way a ”generalized solution” of the non-Archimedean Monge-Ampère equation (1.2).

• Step 3:

  Show the regularity (i.e. continuity) of this generalized solution using capacity estimates.

The general strategy for Steps 1 and 2 follows [BBGZ09], whereas Step 3 follows [Koł98].

The condition that $\mu$ is supported on a dual complex makes Step 1 relatively easy in our case, granted the compactness property of $\PSH(X,\omega)/\mathbf{R}$ proved in [BFJ11]. Indeed, the support condition guarantees that the linear part $\varphi\mapsto\int\varphi\,d\mu$ of $F_{\mu}$ is finite valued and continuous on the whole of $\PSH(X,\omega)$. Because of that, several complications that occurred in [BBGZ09] to handle general measures disappear, since it is enough to extend $E_{\omega}$ to a usc functional $E_{\omega}:\PSH(X,\omega)\to[-\infty,+\infty[$, which is done by setting

$$E_{\omega}(\varphi):=\inf\left\{E_{\omega}(\psi)\mid\psi\geq\varphi,\ \psi\in C^{0}(X)\cap\PSH(X,\omega)\right\}.$$

Step 2 requires much more work and constitutes the main body of the article, in particular because virtually none of the more classical results in pluripotential theory on which [BBGZ09] was able to rely were available so far in our non-Archimedean context. The only obvious information we have on the maximizer $\varphi_{0}$ of $F_{\mu}$ is that it lies in the set

$$\mathcal{E}^{1}(X,\omega):=\left\{\varphi\in\PSH(X,\omega),\,E_{\omega}(\varphi)>-\infty\right\}$$

of $\omega$-psh functions with finite energy. In the complex case, $\mathcal{E}^{1}(X,\omega)$ was introduced in [Ceg98, GZ07] as a higher dimensional and non-linear generalization of the classical Dirichlet space from potential theory. The goal of Step 2 is to show that the Monge-Ampère operator can be naturally extended to $\mathcal{E}^{1}(X,\omega)$, and that $\varphi_{0}$ satisfies

$$(\omega+dd^{c}\varphi_{0})^{n}=\mu$$

in this generalized sense.

In order to do so, we first extend the Monge-Ampère operator from continuous to bounded $\omega$-psh functions, following the fundamental work of Bedford and Taylor [BT82, BT87]. As in the complex case, this mild generalization is in fact crucial in order to develop a reasonable capacity theory, and also because the natural bounded approximants $\max\{\varphi,-m\}$, $m\in\mathbf{N}$, of a given $\omega$-psh function $\varphi$ are not continuous in general. It is however substantially more involved than the continuous case, since uniform convergence has to be replaced with monotone convergence. The fact that any (bounded) $\omega$-psh function can be written as a decreasing limit of a family of $\omega$-psh model functions, proved in [BFJ11], plays a key role at this stage.

Of crucial importance is the following locality property of the Monge-Ampère operator: if $\varphi,\psi$ are bounded $\omega$-psh functions, then the restrictions of the measures $(\omega+dd^{c}\max\{\varphi,\psi\})^{n}$ and $(\omega+dd^{c}\varphi)^{n}$ to the Borel set $\{\varphi>\psi\}$ coincide. Note that, even when $\varphi,\psi$ are model functions, this fact is not clear from the definition in terms of intersection numbers.

Next, we further extend the Monge-Ampère operator from bounded $\omega$-psh functions to functions with finite energy. The key observation, which goes back to [BT87], is the monotonicity of the sequence of measures

$$\one_{\{\varphi>-m\}}\left(\omega+dd^{c}\max\{\varphi,-m\}\right)^{n}\,(m\in\mathbf{N})$$

a direct consequence of the locality property. This allows us to define $(\omega+dd^{c}\varphi)^{n}$ as the increasing limit of this sequence of measures, which is shown to be well-behaved for $\varphi\in\mathcal{E}^{1}(X,\omega)$. More generally, mixed Monge-Ampère measures are shown to be well-defined for functions in $\mathcal{E}^{1}(X,\omega)$, and (1.3), (1.4) are still valid in this generality.

As was already pointed out, these facts are however a priori not enough to show that the maximizer $\varphi_{0}$ of $F_{\mu}$ satisfies $(\omega+dd^{c}\varphi_{0})^{n}=\mu$, because small perturbations of $\varphi_{0}$ cease to be $\omega$-psh in general. In order to handle a similar difficulty in the setting of real Monge-Ampère equations, Alexandrov devised in [Ale38] an envelope argument, an analogue of which was subsequently found in the complex case in [BBGZ09]. Following the same lead, we introduce the $\omega$-psh envelope $P_{\omega}(f)$ of a given continuous function $f$ on $X$ by setting for each $x\in X$

$$P_{\omega}(f)(x):=\sup\left\{\varphi(x)\mid\varphi\in\PSH(X,\omega),\,\varphi\leq f\right\}.$$

It follows from [BFJ11] that $P_{\omega}(f)$ is the largest $\omega$-psh function dominated by $f$ on $X$. The key point is then the following differentiability property, whose complex analogue was established in [BB10]:

(1.5)

$$\frac{d}{dt}\bigg|_{t=0}E_{\omega}\circ P_{\omega}\left(f+tg\right)=\int_{X}g\,(\omega+dd^{c}P_{\omega}(f))^{n}$$

for any two $f,g\in C^{0}(X)$, which may more vividly be written as the chain rule-like formula $(E_{\omega}\circ P_{\omega})^{\prime}=E_{\omega}^{\prime}\circ P_{\omega}$. Granted (1.5), a fairly direct argument based on the monotonicity of $E_{\omega}$ implies $(\omega+dd^{c}\varphi_{0})^{n}=\mu$ as desired.

The proof of (1.5) can be reduced by elementary arguments to the differentiability of $t\mapsto\int P_{\omega}(f+tg)\,(\omega+dd^{c}P_{\omega}(f))^{n}$, which in turn ultimately follows from the following orthogonality property:

(1.6)

$$\int_{X}(f-P_{\omega}(f))\,(\omega+dd^{c}P_{\omega}(f))^{n}=0.$$

Since $f\geq P_{\omega}(f)$, this relation means that $(\omega+dd^{c}P_{\omega}(f))^{n}$ is supported on the contact locus $\{f=P_{\omega}(f)\}$, a well-known fact in the complex case where the proof argues by balayage, using Bedford and Taylor’s solution to the Dirichlet problem for the homogeneous complex Monge-Ampère equation on the ball. Such an approach seems far beyond reach in the non-Archimedean case. We proceed instead by translating (1.6) into an intersection theoretic statement on a model of $X$, where it boils down to the orthogonality of relative asymptotic Zariski decompositions for a line bundle that is ample on the generic fiber. It is precisely at this point that we use the assumption that $X$ is algebraizable. Indeed, this allows us to choose the model where we work to be algebraic, and therefore compactifiable into a projective variety over the residue field $k$. As explained in Appendix A, we can then reduce to the absolute case of big line bundles on projective varieties treated in [BDPP04].

Finally, Step 3 is handled by adapting in a fairly direct manner the capacity estimates of Kołodziej [Koł98, Koł03] to prove that $\varphi_{0}$ is actually continuous. The proof relies on the locality property in $\mathcal{E}^{1}(X,\omega)$. This shows the existence part of Theorem A’. Uniqueness is proved following [Bło03], as in [YZ10].

Our result is not optimal, and we next discuss three important assumptions that we use in Theorems A and A’.

First, the condition that the measure $\mu$ be supported on a dual complex is probably unnecessarily strong. Relying on ideas of Cegrell [Ceg98], Guedj and Zeriahi [GZ07] have defined in the case of compact Kähler manifolds a class $\mathcal{E}(X,\omega)$ of $\omega$-psh functions where the Monge-Ampère operator is well-defined and such that the measures $(\omega+dd^{c}\varphi)^{n}$, $\varphi\in\mathcal{E}(X,\omega)$ are exactly the positive measures $\mu$ on $X$ giving zero mass to pluripolar²² 2 A subset set $A\subset X$ is pluripolar if there exists an $\omega$-psh function $\varphi$ such that $A\subset\{\varphi=-\infty\}$. sets. The function $\varphi$ is here again uniquely determined up to an additive constant by its Monge-Ampère measure, as was later shown by Dinew [Din09]. We expect the corresponding results to be true in our setting, too. The proof would probably require an even more systematic development of pluripotential theory in a non-Archimedean setting, something that is certainly of interest.

Second, as explained above, the proof of the orthogonality property (1.6) relies in a crucial way on the algebraizability assumption for $X$. It would be interesting to drop this condition, which we expect to be an unnecessary restriction.

Finally, our variational approach uses the compactness of the space $\PSH(X,\omega)/\mathbf{R}$, which was obtained in [BFJ11]. The proof of this fact relied heavily on the existence of SNC models, which are so far only available in residue characteristic zero. It seems to be a challenging task to extend our methods and results to local fields and more general complete non-Archimedean fields. See [FJ04, BFJ08] for related work in the case of a trivially valued field.

Let us end this introduction by indicating the structure of the paper.

In §2 we give the necessary background on Berkovich spaces, metrized line bundles, $\omega$-psh functions and wedge-products of closed $(1,1)$-forms. We also recall some facts from measure theory.

The next three sections, §§3-5, develop some of the basic Bedford-Taylor theory in our non-Archimedean setting. The definition of the Monge-Ampère operator on bounded functions and the continuity along decreasing families is carried out in §3. In §4 we introduce a Monge-Ampère capacity used to measures the size of subsets of $X$. We obtain the important result that any $\omega$-psh function is quasicontinuous, i.e. continuous outside a set of arbitrarily small capacity. We also strengthen the regularization theorem of [BFJ11] and prove that any $\omega$-psh function is a decreasing limit of a (countable) sequence of $\omega$-psh model functions. Finally, in §5 we prove the locality property. The results in §§3–5 and even some of the proofs parallel those in complex analysis (especially the ones on compact Kähler manifolds, see [GZ05]). However, the non-Archimedean results ultimately originate in basic properties of the intersection form on models whereas the basic results in the complex case concern differential operators.

The energy of an $\omega$-psh function is introduced in §6. Following [Ceg98, GZ07] we extend the Monge-Ampère operator to the class $\mathcal{E}^{1}(X,\omega)$ of $\omega$-psh functions with finite energy and prove that the locality property continues to hold.

In §7 we introduce $\omega$-psh envelopes and prove the related differentiability theorem. This is a key result that leads to the proof of Theorem A’ given in §8. It uses the locality property and is based on an orthogonality statement whose proof is given in Appendix A. Here the exposition is modeled on [BB10, BBGZ09].

We next explain in §8 how to get Theorem A from Theorem A’. Finally, §9 discusses the case of curves and toric varieties.