For general facts about toric varieties, see [Ful93, KKMS, BPS14]. In this section we briefly describe how the complex and non-Archimedean points of view elegantly come together in the toric setting and translate into statements about convex functions and the real Monge-Ampère operator. As before, we only consider the non-Archimedean field $K=k(\!(t)\!)$ with $\operatorname{char}k=0$; however, most of what we say here should be true in a more general context: see [Gub13a].

Let $M\simeq{\mathbf{Z}}^{n}$ be a free abelian group, $N$ its dual, and let $T=\operatorname{Spec}K[M]$ be the corresponding split $K$-torus. A polarized toric variety $(X,L)$ is then determined by a rational polytope $\Delta\subset M_{\mathbf{R}}$. The variety $X$ is described by the normal fan to $\Delta$ in $N_{\mathbf{R}}$ and the points of $M\cap\Delta$ are in 1-1 correspondence with equivariant sections of $L$; we write $\chi^{u}$ for the section of $L$ associated to $u\in M$. This description is completely general and holds over any field as well as over ${\mathbf{Z}}$.

There is also a “tropical” space ${X^{\mathrm{trop}}}$ associated to $X$. As a topological space, it is compact and contains $N_{\mathbf{R}}$ as an open dense subset.⁵⁵ 5 In our setting, ${X^{\mathrm{trop}}}$ can be identified with the (moment) polytope $\Delta$ in such a way that $N_{\mathbf{R}}$ corresponds to the interior of $\Delta$, but this identification does not preserve the affine structure on $N_{\mathbf{R}}$. For any valued field $K$, there is a tropicalization map $\operatorname{trop}:{X^{\mathrm{an}}}\to{X^{\mathrm{trop}}}$, where ${X^{\mathrm{an}}}$ refers to the analytification with respect to the norm on $K$. The inverse image of $N_{\mathbf{R}}$ is the torus ${T^{\mathrm{an}}}$.

There is a natural correspondence between equivariant metrics on ${L^{\mathrm{an}}}$ and functions on $N_{\mathbf{R}}$. Let $\phi$ is an equivariant metric on ${L^{\mathrm{an}}}$. For every $u\in M$, $\chi^{u}$, is a nonvanishing section of $L$ on $T$ so $\phi-\log|\chi^{u}|$ defines a function on ${T^{\mathrm{an}}}$ that is constant on the fibers of the tropicalization map. In particular, picking $u=0$, we can write

$$\phi-\log|\chi^{0}|=g\circ\operatorname{trop}$$

(10.1)

for some function $g$ on $N_{\mathbf{R}}$. Conversely, given a function $g$ on $N_{\mathbf{R}}$, (10.1) defines an equivariant metric on the restriction of ${L^{\mathrm{an}}}$ to ${T^{\mathrm{an}}}$.

We now go from the torus $T$ to the polarized variety $(X,L)$. After replacing $L$ by a multiple, we may assume that all the vertices of $\Delta$ belong to $M$. Set

$$\phi_{\Delta}:=\max_{u\in\Delta}\log|\chi^{u}|.$$

This is a semipositive, equivariant model metric on ${L^{\mathrm{an}}}$. Its restriction to ${T^{\mathrm{an}}}$ corresponds to the homogeneous, nonnegative, convex function

$$g_{\Delta}:=\max_{u\in\Delta}u$$

on $N_{\mathbf{R}}$. In general, an equivariant singular metric $\phi$ on ${L^{\mathrm{an}}}$ corresponds to a convex function $g$ on $N_{\mathbf{R}}$ such that $g\leq g_{\Delta}+O(1)$. It is bounded iff $g-g_{\Delta}$ is bounded on $N_{\mathbf{R}}$.

The real Monge-Ampère measure of any convex function $g$ on $N_{\mathbf{R}}$ is a well-defined positive measure $\operatorname{MA}_{\mathbf{R}}(g)$ on $N_{\mathbf{R}}$ (see e.g. [RT77]). When $g=g_{\Delta}+O(1)$, its total mass is given by

$$\int_{N_{\mathbf{R}}}\operatorname{MA}_{\mathbf{R}}(g)=\operatorname{Vol}(\Delta)=\frac{(L^{n})}{n!},$$

where the last equality follows from [Ful93, p.111].

We now wish to relate the real Monge-Ampère measure of $g$ and the Monge-Ampère measure of the corresponding semipositive metric $\phi$ on ${L^{\mathrm{an}}}$.

First consider the non-Archimedean case, in which there is a natural embedding $j:N_{\mathbf{R}}\to{T^{\mathrm{an}}}\subset{X^{\mathrm{an}}}$ given by monomial valuations that sends $v\in N_{\mathbf{R}}$ to the norm

$$\sum_{u\in M}a_{u}u\in K[M]\mapsto\max_{u\in M}\{|a_{u}|\exp(-\langle u,v\rangle)\}.$$

In particular, $j(0)=x_{G}$, the Gauss point of the open $T$-orbit.

If $g$ is a convex function on $N_{\mathbf{R}}$ with $g=g_{\Delta}+O(1)$, and if $\phi$ is the corresponding continuous semipositive metric on $L$, then [BPS14, Theorem 4.7.4] asserts that

$$\operatorname{MA}(\phi)=n!\,j_{*}\operatorname{MA}_{\mathbf{R}}(g).$$

For a compactly supported positive measure $\nu$ on $N_{\mathbf{R}}$ of mass $(L^{n})$, solving the Monge-Ampère equation $\operatorname{MA}(\phi)=j_{*}(\nu)$ therefore amounts to solving the real Monge-Ampère equation $\operatorname{MA}_{\mathbf{R}}(g)=\nu/n!$. This can be done explicitly when $\nu$ is a point mass, say supported at $v_{0}\in N_{\mathbf{R}}$. Indeed, the function $g_{v_{0}}:N\to{\mathbf{R}}$ defined by $g=g_{\Delta}(\cdot-v_{0})$ is convex and satisfies $g=g_{\Delta}+O(1)$. Further, for every point $v\neq v_{0}$ there exists a line segment in $N_{\mathbf{R}}$ containing $v$ in its interior and on which $g$ is affine. This implies that $\operatorname{MA}_{\mathbf{R}}(g)$ is supported at $v_{0}$. As a a consequence, the corresponding continuous metric $\phi$ on ${L^{\mathrm{an}}}$ satisfies $\operatorname{MA}_{\mathbf{R}}(\phi)=(L^{n})\delta_{j(u_{0})}$.

This solution can be shown to tie in well with the construction at the end of §8, but is of course much more explicit. For example, when $u_{0}\in N_{\mathbf{Q}}$, so that $j(u_{0})\in{X^{\mathrm{an}}}$ is divisorial, the function $g_{u_{0}}$ is ${\mathbf{Q}}$-piecewise linear so that the corresponding metric $\phi$ is a model metric.

Finally we consider the complex case. In this case we cannot embed $N_{\mathbf{R}}$ in ${T^{\mathrm{an}}}$. However, the preimage of any point $v\in N_{\mathbf{R}}$ under the tropicalization is a real torus of dimension $n$ in ${T^{\mathrm{an}}}$ on which the multiplicative group $(S^{1})^{n}$ acts transitively. To any compactly supported positive measure $\nu$ on $N_{\mathbf{R}}$ of mass $(L^{n})/n!$ we can therefore associate a unique measure $\mu$ on ${T^{\mathrm{an}}}$, still denoted $\mu:=j_{*}\nu$, that is invariant under the action of $(S^{1})^{n}$ and satisfies $\operatorname{trop}_{*}\mu=\nu$.

If $\phi$ is an equivariant semipositive metric on ${L^{\mathrm{an}}}$, corresponding to a convex function $g$ on $N_{\mathbf{R}}$, we then have

$$\operatorname{MA}(\phi)=n!\,j_{*}\operatorname{MA}_{\mathbf{R}}(g).$$

For $(S^{1})^{n}$-invariant measures $\mu$ on ${L^{\mathrm{an}}}$ of mass $(L^{n})$, solving the complex Monge-Ampère equation $\operatorname{MA}(\phi)=\mu$ thus reduces to solving the real Monge-Ampère equation $\operatorname{MA}_{\mathbf{R}}(g)=\frac{1}{n!}\operatorname{trop}_{*}\mu$.