The disadvantage of the variational approach to the Monge-Ampère equation is that it gives very little control on the solution beyond continuity. Here we shall make the solution more concrete in the case of curves; the next section deals with toric varieties.

Thus assume $X$ is a smooth projective curve over $K$. In this case, the Monge-Ampère operator (which one would normally refer to as the Laplacian) is linear: if $\phi_{i}$ is a metric on $L_{i}$, $i=1,2$, then $\operatorname{MA}(\phi_{1}+\phi_{2})=\operatorname{MA}(\phi_{1})+\operatorname{MA}(\phi_{2})$. Furthermore, as we shall see, the Monge-Ampère operator is naturally defined on any singular semipositive metric on ${L^{\mathrm{an}}}$, for an ample line bundle $L$, and we can solve $\operatorname{MA}(\phi)=\mu$ for any positive measure $\mu$ of mass $\deg L$.

Let us first explain this in the complex case; ${X^{\mathrm{an}}}$ is then a compact Riemann surface. Fix a smooth metric $\phi_{0}$ on ${L^{\mathrm{an}}}$. The curvature form $\omega_{0}:=dd^{c}\phi_{0}$ is a volume form of mass $\deg L$. A singular metric $\phi$ on ${L^{\mathrm{an}}}$ is then semipositive iff $\varphi:=\phi-\phi_{0}$ is an $\omega_{0}$-psh function, that is, a locally integrable function $\varphi$ that is locally the sum of a smooth function and a psh function, and such that $\omega_{0}+dd^{c}\varphi$ is a positive measure. We then set $\operatorname{MA}(\phi):=\omega_{0}+dd^{c}\phi$; this definition does not depend on the choice of $\phi_{0}$.

Now we explain how to solve the equation $\operatorname{MA}(\phi)=\mu$ for any positive measure $\mu$ of mass $d_{L}:=\deg L$. Writing $\phi=\phi_{0}+\varphi$ as above, we must solve $dd^{c}\varphi=\mu-\omega_{0}$, where $\omega_{0}:=dd^{c}\phi_{0}$. It sufficies to do this when $\mu=d_{L}\delta_{x}$ for some $x\in{X^{\mathrm{an}}}$: indeed, if we normalize the solution $\varphi_{x}$ to $dd^{c}\varphi_{x}=\mu-d_{L}\delta_{x}$ by $\int_{{X^{\mathrm{an}}}}\varphi_{x}\,\omega_{0}=0$, then the function $\varphi_{\mu}$ defined by $\varphi_{\mu}(y):=d_{L}^{-1}\int_{{X^{\mathrm{an}}}}\varphi_{x}(y)\,d\mu(x)$ satisfies $dd^{c}\varphi_{\mu}=\omega_{0}-\mu$ and is normalized by $\int_{{X^{\mathrm{an}}}}\varphi_{\mu}\,\omega_{0}=0$.

The function $\varphi_{x}$ can be “physically” interpreted as the voltage (suitably normalized) when putting a charge of $+d_{L}$ at the point $x$ and a total charge of $-d_{L}$ spread out according to the measure $\omega_{0}$. Mathematically, Perron’s method describes it as the supremum of all $\omega_{0}$-subharmonic functions $\varphi$ on ${X^{\mathrm{an}}}$ satisfying $\int_{{X^{\mathrm{an}}}}\varphi\,\omega_{0}=0$ and $\varphi\leq d_{L}\log|z|+O(1)$, where $z$ is a local coordinate at $x$.

Now we consider the non-Archimedean case. As before, let us assume that $K$ is a discretely valued field of residue characteristic zero, even though this is not really necessary in the one-dimensional case.³³ 3 Indeed, Thuillier [Thu05] systematically develops a potential theory on Berkovich curves in a very general setting.

The main point is that any Berkovich curve has the structure of a generalized⁴⁴ 4 This means that some distances may be infinite. metric graph. We will not describe this in detail, but here is the idea. The dual graph $\Delta_{\mathcal{X}}$ of any SNC model ${\mathcal{X}}$ is a connected, one-dimensional simplicial complex. As before, we view it as a subset of ${X^{\mathrm{an}}}$. It carries a natural integral affine structure, inducing a metric. If ${\mathcal{X}}^{\prime}$ is an SNC model dominating ${\mathcal{X}}$ (in the sense that the canonical birational map ${\mathcal{X}}\dashrightarrow{\mathcal{X}}^{\prime}$ is a morphism) then $\Delta_{\mathcal{X}}$ is a subset of $\Delta_{{\mathcal{X}}^{\prime}}$ and the inclusion $\Delta_{\mathcal{X}}\to\Delta_{{\mathcal{X}}^{\prime}}$ is an isometry. There is a also a (deformation) retraction $r_{\mathcal{X}}:{X^{\mathrm{an}}}\to\Delta_{\mathcal{X}}$, and ${X^{\mathrm{an}}}\simeq\varprojlim_{\mathcal{X}}\Delta_{\mathcal{X}}$. In this way, the metrics on the dual complexes induce a generalized metric on ${X^{\mathrm{an}}}$.

The structure of each $\Delta_{\mathcal{X}}$ and of ${X^{\mathrm{an}}}$ as metric graphs allows us to define a Laplacian on these spaces, by combining the real Laplacian on segments and the combinatorial Laplacian at branch points (and endpoints). This Laplacian allows us to understand both semipositive singular metrics and the Monge-Ampère operator.

Namely, fix a model metric $\phi_{0}$ on ${L^{\mathrm{an}}}$. It is represented by a ${\mathbf{Q}}$-line bundle on some SNC model ${\mathcal{X}}^{(0)}$. The measure $\omega_{0}:=dd^{c}\phi_{0}$ is supported on the vertices of $\Delta_{{\mathcal{X}}^{(0)}}$. Now, a singular metric $\phi$ is semipositive iff for every SNC model ${\mathcal{X}}$ dominating ${\mathcal{X}}^{(0)}$, the restriction of the function $\phi-\phi_{0}$ to $\Delta_{\mathcal{X}}$ is a $\omega_{0}$-subharmonic function in the sense that $\Delta((\phi-\phi_{0})|_{\Delta_{\mathcal{X}}})=\mu_{\mathcal{X}}-\omega_{0}$, where $\mu_{\mathcal{X}}$ is a positive measure on $\Delta_{\mathcal{X}}$ of mass $d_{L}$. In this case, there further exists a unique measure $\mu$ on ${X^{\mathrm{an}}}$ of mass $d_{L}$ such that $(r_{\mathcal{X}})_{*}\mu=\mu_{\mathcal{X}}$ for all ${\mathcal{X}}$, and we have $\operatorname{MA}(\phi)=\mu$.

To solve the equation $\operatorname{MA}(\phi)=\mu$ for a positive measure $\mu$ of mass $d_{L}$, it suffices by linearity to treat the case $\mu=d_{L}\delta_{x}$ for a point $x\in{X^{\mathrm{an}}}$. In this case, the function $\varphi:=\phi-\phi_{0}$ will be locally constant outside the convex hull of $\{x\}\cup\Delta_{{\mathcal{X}}^{(0)}}$. The latter is essentially a finite metric graph on which we need to find a function whose Laplacian is equal to $d_{L}\delta_{x}-\omega_{0}$. This can be done in a quite elementary way.

An interesting example of semipositive metrics, both in the complex and non-Archimedean case, comes from dynamics [Zha95]. Suppose $f:(X,L)\circlearrowleft$ is an polarized endomorphism of degree $\lambda>1$. In other words, $f:X\to X$ is an endomorphism and $f^{*}L$ is linearly equivalent to $\lambda L$. Then there exists a unique canonical metric $\phi_{\operatorname{can}}$ on ${L^{\mathrm{an}}}$, satisfying $f^{*}\phi=\lambda\phi$. This metric is continuous and semipositive but usually not a model metric.

As a special case, suppose $X$ is an elliptic curve and that $f$ is the map given by multiplication by $\lambda$. In the complex case, ${X^{\mathrm{an}}}\simeq{\mathbf{C}}/\Lambda$ is a torus and $\mu_{\operatorname{can}}:=\operatorname{MA}(\phi_{\operatorname{can}})$ is given by a multiple of Haar measure on ${X^{\mathrm{an}}}$. In the non-Archimedean case, there are two possibilities. If $X$ has good reduction over $\operatorname{Spec}R$, then $\mu_{\operatorname{can}}$ is a point mass. Otherwise, $\mu_{\operatorname{can}}$ is proportional to Lebesgue measure on the skeleton $\mathrm{Sk}({X^{\mathrm{an}}})$, a subset homeomorphic to a circle. A similar description of the measure $\mu_{\operatorname{can}}$ in the case of higher-dimensional abelian varieties is given in [Gub10].