Potential theory on non-Archimedean analytic curves (over arbitrary complete valuation fields) was developed in detail by A.Thuillier in [Thu05]. We only indicate how to recover Theorem A’ when $\dim X=1$ following his approach.

Let $X$ be a smooth projective curve over $K$. Thuillier defined spaces $D^{0}(X)$ and $D^{1}(X)$ of distributions and currents on $X$ as follows. An element of $D^{0}(X)$ is an arbitrary function $X^{\mathrm{qm}}\to\mathbf{R}$ [Thu05, Proposition 3.3.3]. The $dd^{c}$-operator extends to $dd^{c}:D^{0}(X)\to D^{1}(X)$, and its image is exactly the set of currents $\rho\in D^{1}(X)$ such that $\int_{X}\rho=0$ [Thu05, Théorème 3.3.13]. By linearity, this fact easily reduces to the existence, for any two $x,y\in X^{\mathrm{div}}$, of a ’Green function’, i.e. a model function $g_{x,y}$ such that $dd^{c}g_{x,y}=\delta_{x}-\delta_{y}$. The existence of $g_{x,y}$ is in turn a consequence of the intersection form being negative definite on $\Div_{0}(\mathcal{X})_{\mathbf{R}}/\mathbf{R}\mathcal{X}_{0}$, for a model $\mathcal{X}$ such that $x$ and $y$ correspond to components of $\mathcal{X}_{0}$.

Now let $\omega$ be a $(1,1)$-form with $\int\omega>0$, and let $\mu$ be an arbitrary positive Radon measure on $X$ such that $\int\mu=\int\omega$. The previous result shows the existence of a distribution $\varphi_{\mu}$ such that

(9.1)

$$\omega+dd^{c}\varphi_{\mu}=\mu.$$

By [Thu05, Lemme 3.4.1] the positivity of the current $\omega+dd^{c}\varphi_{\mu}$ shows that $\varphi_{\mu}$ uniquely extends to a $\omega$-psh function, and we conclude that any positive Radon measure $\mu$ with $\int\mu=\int\omega$ satisfies (9.1) for some $\varphi\in\PSH(X,\omega)$, unique up to an additive constant.

Finally, assume that $\mu$ is supported on a dual complex $\Delta_{\mathcal{X}}$. In order to see that $\varphi_{\mu}\in C^{0}(X)$, we may assume that $\mathcal{X}$ is also a determination of $\omega$. In this one-dimensional setting, it is easy to check that composing with the retraction $p_{\mathcal{X}}:X\to\Delta_{\mathcal{X}}$ preserves $\omega$-psh functions, i.e. $\varphi\circ p_{\mathcal{X}}$ is $\omega$-psh for every $\omega$-psh function $\varphi$. Since $\mu$ is supported on $\Delta_{\mathcal{X}}$ we have $\left(p_{\mathcal{X}}\right)_{*}\mu=\mu$, hence $\theta+dd^{c}(\varphi_{\mu}\circ p_{\mathcal{X}})=\mu$. It follows that $\varphi_{\mu}\circ p_{\mathcal{X}}=\varphi_{\mu}$ by uniqueness up to an additive constant, since the two functions coincide on $\Delta_{\mathcal{X}}$. Now $\varphi_{\mu}|_{\Delta_{\mathcal{X}}}$ is continuous, hence the continuity of $\varphi_{\mu}$.

Let us now make the connection with the approach we followed in higher dimensions. In dimension $1$, the energy is equal to $E(\varphi)=2\int\varphi\omega+\int\varphi dd^{c}\varphi$ so that a $\omega$-psh function $\varphi$ has finite energy iff $\varphi$ is integrable with respect to the trace measure of $dd^{c}\varphi$.

Now fix a positive Radon measure $\mu$ such that the solution $\varphi_{\mu}$ to (9.1) has finite energy. Then $\varphi_{\mu}$ is the unique $\omega$-psh function realizing the infimum of the functional $E(\varphi)-\int\varphi\mu$, by [Thu05, Proposition 3.5.9].

Observe that the assumption on $\mu$ is automatically satisfied when $\mu$ is supported in some dual complex $\Delta_{\mathcal{X}}$ whence Thuillier’s result gives a stronger version than our result in dimension $1$.

We refer to [Thu05] for more on potential theory on non-Archimedean curves including the notion of harmonic functions, capacity, and the study of polar sets. See also [BR10] for the case of the projective line.

We use [Ful93, KKMS73, BPS11] as references. Let $M\simeq\mathbf{Z}^{n}$ be a free abelian group, $N$ its dual, and let $T=\Spec K[M]$ be the corresponding split $K$-torus. A projective toric $K$-variety $X$ is described by a rational fan subdivision $\Sigma$ of $N_{\mathbf{R}}$, and there is a natural embedding $j:N_{\mathbf{R}}\to X^{\mathrm{an}}$ given by monomial valuations that sends $n\in N_{\mathbf{R}}$ to the norm $\sum a_{m}m\in K[M]\mapsto\max\{|a_{m}|\exp(-\langle m,n\rangle)\}$. In particular, $j(0)=x_{G}$, the Gauss point of the open $T$-orbit.

An ample $T$-line bundle $L$ on $X$ defines a rational polytope $\Delta\subset M_{\mathbf{R}}$ with normal fan $\Sigma$, such that points of $M\cap\Delta$ identify with $T$-eigensections of $L$.

According to [BPS11] we have the following description of toric metrics on $L$. The polytope $\Delta$ is the Newton polytope of the piecewise $\mathbf{Q}$-linear convex function $g_{\Delta}=\sup_{m\in\Delta}m$ on the dual space $N_{\mathbf{R}}=M_{\mathbf{R}}^{*}$, and toric bounded (resp. model) metrics $\|\cdot\|$ on $L$ correspond to bounded (resp. piecewise $\mathbf{Q}$-affine) functions $f$ on $N_{\mathbf{R}}$ such that $f-g_{\Delta}$ is bounded. The metric $\|\cdot\|_{f}$ attached to a function $f$ is semipositive iff $f$ is convex.

The real Monge-Ampère measure of any convex function $f$ on $N_{\mathbf{R}}$ is a well-defined positive Radon measure $\MA_{\mathbf{R}}(f)$ on $N_{\mathbf{R}}$ (see e.g. [RT77]), while the growth condition $f=g_{\Delta}+O(1)$ further guarantees that

$$\int_{N_{\mathbf{R}}}\MA_{\mathbf{R}}(f)=\vol(\Delta).$$

If $f$ is a convex function on $N_{\mathbf{R}}$ with $f=g_{\Delta}+O(1)$, and if $\|\cdot\|_{f}$ is the corresponding continuous semipositive metric on $L$, then [BPS11, Theorem 5.70] relates their Monge-Ampère measures as follows:

(9.2)

$$c_{1}(L,\|\cdot\|_{f})^{n}=n!\,j_{*}\MA_{\mathbf{R}}(f).$$

Since $g_{\Delta}$ is homogeneous, $\MA_{\mathbf{R}}(g_{\Delta})$ is a Dirac mass at the origin of mass $\vol(\Delta)$, and [Ful93, p.111] implies the corresponding metric $\|\cdot\|_{g_{\Delta}}$ on $L$ to satisfy

$$c_{1}(L,\|\cdot\|_{g_{\Delta}})^{n}=c_{1}(L)^{n}\,\delta_{x_{G}}.$$

Translating in $N_{\mathbf{R}}$ we get:

In the case of atomic measures supported at toric divisorial points, we can show: