Plurisubharmonic (psh) functions are among the objets souples (soft objects) in complex analysis according to P. Lelong [Lel85]. This is reflected in certain useful compactness properties. The global analogues of psh functions are semipositive singular metrics on holomorphic line bundles. Here “singular” means that vectors may have infinite length.

To make sense of the compactness statement we need to specify the topology on $\operatorname{PSH}({L^{\mathrm{an}}})$. In the complex case, one usually fixes a volume form $\mu$ on ${X^{\mathrm{an}}}$ and takes the topology induced by the $L^{1}$-norm: $\|\phi-\psi\|=\int_{{X^{\mathrm{an}}}}|\phi-\psi|\mu$. In the non-Archimedean case, there is typically no volume form on ${X^{\mathrm{an}}}$. Instead, we say that a net $(\phi_{j})_{j}$ in $\operatorname{PSH}({L^{\mathrm{an}}})$ converges to $\phi$ if $\lim_{j}\sup_{\Delta_{\mathcal{X}}}|\phi_{j}-\phi|=0$ for every SNC model ${\mathcal{X}}$. Implicit in this definition is that the restriction to $\Delta_{\mathcal{X}}$ of every singular metric in $\operatorname{PSH}({L^{\mathrm{an}}})$ is continuous: see Theorem 6.2 below.

In the complex case, one typically defines $\operatorname{PSH}({L^{\mathrm{an}}})$ as the set of usc singular metrics $\phi$ that are locally represented by $L^{1}$ functions and whose curvature current $dd^{c}\phi$ (computed in the sense of distributions) is a positive closed current. Thus $\phi$ is locally given as the sum of a smooth function and a psh function. Most of the statements above then follow from basic facts about plurisubharmonic functions in ${\mathbf{C}}^{n}$. The regularization result is the most difficult. On ${\mathbf{C}}^{n}$ it is easy to regularize using convolutions. With some care, one can in the global (projective) case glue together local regularizations to obtain a global one. See [Dem92] for a general result and [BK07] for a relatively simple argument applicable in our setting.

In the non-Archimedean case, we are not aware of any workable a priori definition of $\operatorname{PSH}({L^{\mathrm{an}}})$. Chambert-Loir and Ducros [CD12] have a notion of forms and currents on Berkovich spaces, but it is unclear if it gives the right objects for the purposes of the theorem above. Instead, we prove the following result:

This is proved using a rather subtle argument, involving intersection numbers on toroidal models dominating ${\mathcal{X}}$. It would be interesting to have a different proof. At any rate, Theorem 6.2 allows us to define $\operatorname{PSH}({L^{\mathrm{an}}})$ as the set of usc singular metrics $\phi$ satisfying, for every sufficiently large SNC model ${\mathcal{X}}$,

• (i)

  $(\phi-\phi_{0})\circ r_{\mathcal{X}}\geq\phi-\phi_{0}$;

• (ii)

  the restriction of $\phi$ to $\Delta_{\mathcal{X}}$ is a uniform limits of a sequence $\phi_{m}|_{\Delta_{\mathcal{X}}}$, where each $\phi_{m}$ is a semipositive model metric.

Here $\phi_{0}$ is a fixed model metric, determined by some model dominated by ${\mathcal{X}}$. The map $r_{\mathcal{X}}:{X^{\mathrm{an}}}\to\Delta_{\mathcal{X}}\subset{X^{\mathrm{an}}}$ is a natural retraction. Since $\phi$ is usc, condition (i) implies that $\phi=\phi_{0}+\lim_{\mathcal{X}}(\phi-\phi_{0})\circ r_{\mathcal{X}}$, so that $\phi$ is determined by its restrictions to all dual complexes.

With this definition, the compactness of $\operatorname{PSH}({L^{\mathrm{an}}})/{\mathbf{R}}$ follows from Theorem 6.2 and Ascoli’s theorem. Regularization, however, is quite difficult to show. We are not aware of any procedure that would replace convolution in the complex case. Instead we use algebraic geometry. Here is an outline of the proof.

Fix $\phi\in\operatorname{PSH}({L^{\mathrm{an}}})$. For any SNC model ${\mathcal{X}}$, $\phi$ naturally induces a model metric $\phi_{\mathcal{X}}$. The semipositivity of $\phi$ implies that the net $(\phi_{\mathcal{X}})_{\mathcal{X}}$, indexed by the collection of (isomorphism classes of) SNC models decreases to $\phi$. Unfortunately, except in the curve case $n=1$, $\phi_{\mathcal{X}}$ has no reason to be semipositive; this reflects the fact that the pushforward of a nef line bundle may fail to be nef. We address this by defining $\psi_{\mathcal{X}}$ as the supremum of all semipositive (singular) metrics dominated by $\phi_{\mathcal{X}}$. We then show that $\psi_{\mathcal{X}}$ is continuous and can be uniformly approximated by a sequence $(\psi_{{\mathcal{X}},m})_{m}^{\infty}$ of semipositive model metrics. From this data it is not hard to produce a decreasing net of semipositive model metrics converging to $\phi$.

Let us say a few words on the construction of the semipositive model metrics $\phi_{{\mathcal{X}},m}$ since this is a key step in the paper [BFJ12]. For simplicity assume that $L$ is base point free and that $\phi_{\mathcal{X}}$ is associated to a line bundle ${\mathcal{L}}$ (rather than an ${\mathbf{R}}$-line bundle) on ${\mathcal{X}}$. Let ${\mathfrak{a}}_{m}$ be the base ideal of $m{\mathcal{L}}$, cut out by the global sections; it is cosupported on the special fiber ${\mathcal{X}}_{0}$. The sequence $({\mathfrak{a}}_{m})_{m}$ is a graded sequence in the sense that ${\mathfrak{a}}_{l}\cdot{\mathfrak{a}}_{m}\subset{\mathfrak{a}}_{l+m}$, Each ${\mathfrak{a}}_{m}$ naturally defines a semipositive model metric $\psi_{{\mathcal{X}},m}$ on ${L^{\mathrm{an}}}$. The fact that $\psi_{{\mathcal{X}},m}$ converges uniformly to $\psi_{\mathcal{X}}$ translates into a statement that the graded sequence $({\mathfrak{a}}_{m})_{m}$ is “almost” finitely generated. This in turn is proved using multiplier ideals and ultimately reduces to the Kodaira vanishing theorem; to apply the latter it is crucial to work in residue characteristic zero.

The argument above proves that any $\phi\in\operatorname{PSH}({L^{\mathrm{an}}})$ is the limit of a decreasing net of semipositive model metrics. When $\phi$ is continuous, the convergence is uniform by Dini’s Theorem, and we can use the sup-norm to extract a decreasing sequence of model metrics converging to $\phi$. In the general case, the Monge-Ampère capacity developed in [BFJ15, §4] (and modeled on [BT82, GZ05]) can similarly be used to extract a cenvergent sequence from a net.