The material in this section is surely well known, but we include the details for lack of a suitable reference. The calculations here are used in the proof of Theorem 3.4 (which implies Theorem A).

Let $N\simeq{\mathbb{Z}}^{p+1}$ be a lattice, $M=\operatorname{Hom}(N,{\mathbb{Z}})$ the dual lattice, ${\mathbb{C}}[M]$ the semigroup ring and $T=\operatorname{Spec}{\mathbb{C}}[M]=N\otimes{\mathbb{C}}^{*}$ the algebraic torus. A basis for $N$ induces a dual basis $(m_{0},\dots,m_{p})$ for $M$ and elements $z_{i}\in{\mathbb{C}}[M]$, $0\leq i\leq p$, such that ${\mathbb{C}}[M]={\mathbb{C}}[z_{0}^{\pm 1},\dots,z_{p}^{\pm 1}]$ and $T\simeq({\mathbb{C}}^{*})^{p+1}$.

Let $\Omega\in H^{0}(T,K_{T})$ be the $T$-invariant global section given in coordinates by

$$\Omega=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}.$$

Note that $\Omega$ is independent of the choice of coordinates, up to a sign. Its associated measure

$$\rho:=|\Omega|^{2}$$

is $T$-invariant, and hence a Haar measure on $T$.

We can write this measure in (logarithmic) polar coordinates via the canonical tropicalization map $L\colon T\to N_{\mathbb{R}}$, given in the basis above by

$$L=(-\log|z_{0}|,\dots,-\log|z_{p}|).$$

Note that $L$ sits in the exact sequence $1\to K\to T\to N_{\mathbb{R}}$ obtained by tensoring with $N$ the exact sequence $1\to S^{1}\to{\mathbb{C}}^{*}\to{\mathbb{R}}\to 0$ induced by $z\mapsto-\log|z|$. In particular, $K=N\otimes S^{1}\simeq(S^{1})^{p+1}$ is a compact torus, and $L\colon T\to N_{\mathbb{R}}$ is a principal $K$-bundle.

On the one hand, let $\omega$ be the translation invariant real $(p+1)$-form on the tropical torus $N_{\mathbb{R}}\simeq{\mathbb{R}}^{p+1}$ given by

$$\omega=dm_{0}\wedge\dots\wedge dm_{p}.$$

This form is again independent of the choice of basis, up to a sign, and its associated measure $\lambda:=|\omega|$ is the Lebesgue (or Haar) measure on $N_{\mathbb{R}}$ normalized by $N$.

On the other hand, since $L:T\to N_{\mathbb{R}}$ is a principal $K$-bundle, each fiber $K_{w}=L^{-1}(w)$ has a unique $K$-invariant probability measure $\rho_{w}$. Then $\rho$ has a fiber decomposition

$$\rho=(2\pi)^{p+1}\lambda(dw)\otimes\rho_{w},$$

i.e.

$$\int_{T}f\,d\rho=(2\pi)^{p+1}\int_{N_{\mathbb{R}}}\left(\int_{K_{w}}f\,d\rho_{w}\right)\lambda(dw),$$

(1.1)

for any $f\in C^{0}_{c}(T)$. Concretely, we can use logarithmic polar coordinates on $T$:

$$z_{j}=\exp(-w_{j}+2\pi i\theta_{j})$$

for $0\leq j\leq p$; then $\rho_{w}=|d\theta_{0}\wedge\dots\wedge d\theta_{p}|$, and

$$\rho=\left|\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\right|^{2}=(2\pi)^{p+1}|dw_{1}\wedge\dots\wedge dw_{n}|\otimes\rho_{w}.$$

We will need the same analysis on certain subgroups of $T$. Fix an element $m\in M$ and let $\chi=\chi^{m}\colon T\to{\mathbb{C}}^{*}$ be the corresponding character. Let $b\in{\mathbb{Z}}_{>0}$ be the largest integer such that $b^{-1}m\in M$. In the bases above, we can write $m=\sum_{i=0}^{p}b_{i}m_{i}$ and $\chi=\prod_{i}z_{i}^{b_{i}}$, where $b_{i}\in{\mathbb{Z}}$; then $b=\gcd_{i}b_{i}$. On the other hand, we can pick a basis such that $m=bm_{0}$ and $\chi=z_{0}^{b}$. This is useful for computations.

For $t\in{\mathbb{C}}^{*}$, $T_{t}:=\chi^{-1}(t)$ is a complex manifold with $b$ connected components. Note that $T^{\prime}:=X_{1}$ is an algebraic subgroup of $T$ and that $T_{t}$ is a torsor for $T^{\prime}$ for any $t\in{\mathbb{C}}^{*}$. The $T$-invariant $(p+1)$-form $\Omega$ induces in a canonical way a $T^{\prime}$-invariant $p$-form $\Omega_{t}$ on $T_{t}$, obtained as the restriction to $T_{t}$ of any choice of holomorphic $p$-form $\Omega^{\prime}$ on $T$ such that $\frac{d\chi}{\chi}\wedge\Omega^{\prime}=\Omega$. In general coordinates as above, we can pick

$$\Omega^{\prime}=\frac{1}{\#J}\sum_{j\in J}\frac{(-1)^{j}}{b_{j}}\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\widehat{\frac{dz_{j}}{z_{j}}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}},$$

where $J=\{j\mid b_{j}\neq 0\}$. In special coordinates, so that $m=bm_{0}$ and $\chi=z_{0}^{b}$, we then have $\Omega^{\prime}=\frac{1}{b}\frac{dz_{1}}{z_{1}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}$, and hence

$$T_{t}=\bigcup_{u^{b}=t}\{z_{0}=u\}{\quad\text{and}\quad}\Omega_{t}=\frac{1}{b}\frac{dz_{1}}{z_{1}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\bigg|_{T_{t}}.$$

Note that $\rho_{1}:=|\Omega_{1}|^{2}$ is Haar measure on $T^{\prime}$, whereas $\rho_{t}:=|\Omega_{t}|^{2}$ is a $T^{\prime}$-invariant measure on $T_{t}$. In the special case $p=0$, $T_{t}$ consists of $b$ points, and $\rho_{t}$ gives mass $\frac{1}{b^{2}}$ to each of them.

Next we study the analogous situation in the tropical torus $N_{\mathbb{R}}$. Viewing $m$ as a linear form on $N_{\mathbb{R}}$, set $H_{s}:=m^{-1}(s)$ for $s\in{\mathbb{R}}$. The lattice $N^{\prime}=\operatorname{Ker}m\subset N$ defines an integral affine structure on $H_{s}$, and hence a normalized Lebesgue measure $\lambda_{s}$. Note that

$$|\omega^{\prime}|_{H_{s}}|=\frac{1}{b}\lambda_{s}$$

for any choice of $p$-form $\omega^{\prime}$ on $N_{\mathbb{R}}$ such that $dm\wedge\omega^{\prime}=\omega$. In general coordinates, we pick

$$\omega^{\prime}=\frac{1}{\#J}\sum_{j\in J}\frac{(-1)^{j}}{b_{j}}dm_{0}\wedge\dots\wedge\widehat{dm_{j}}\wedge\dots\wedge dm_{p},$$

where $J=\{j\mid b_{j}\neq 0\}$. In special coordinates, $\omega^{\prime}=\frac{1}{b}dm_{1}\wedge\dots\wedge dm_{p}$.

Finally we describe $\rho_{t}$ in polar coordinates. The tropicalization map $L\colon T\to N_{\mathbb{R}}$ induces a principal $T^{\prime}\cap K$-bundle $T_{t}\to H_{s}$ with $s=-\log|t|$, and hence an invariant probability measure on $\rho_{t,w}$ on each fiber $K_{t,w}:=T_{t}\cap K_{w}$. We claim that

$$\rho_{t}=\frac{(2\pi)^{p}}{b}\lambda_{s}(dw)\otimes\rho_{t,w},$$

i.e.

$$\int_{T_{t}}f\,d\rho_{t}=\frac{(2\pi)^{p}}{b}\int_{H_{s}}\left(\int_{K_{t,w}}f\,\rho_{t,w}\right)\lambda_{s}(dw),$$

(1.2)

for any $f\in C^{0}_{c}(T_{t})$, where $s=\log|t|^{-1}$.

The proof is essentially the same as that of (1.1). We work in special coordinates, so that $\chi=z_{0}^{b}$ and $m=bm_{0}$. Then $T_{t}=\{z_{0}^{b}=t\}$ has $b$ connected components $T_{t}^{(l)}$, $1\leq l\leq b$, and

$$\rho_{t}=|\Omega_{t}|^{2}=\frac{1}{b^{2}}\left|\frac{dz_{1}}{z_{1}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\right|^{2}.$$

The restriction of the tropicalization map to $T_{t}^{(l)}$ amounts to the change of coordinates $z_{j}=u_{j}^{(l)}\exp(-w_{j}+2\pi i\theta_{j})$ for $1\leq j\leq p$, where the $u_{j}^{(l)}$ are constants with $|u_{j}^{(l)}|=1$. In these coordinates,

$$\rho_{t}|_{T_{t}^{(l)}}=\frac{(2\pi)^{p}}{b^{2}}|dm_{1}\wedge\dots\wedge dm_{n}|\otimes|d\theta_{1}\wedge\dots\wedge d\theta_{p}|.$$

Here $\frac{1}{b}|d\theta_{1}\wedge\dots\wedge d\theta_{p}|$ induces the measure $\rho_{t,w}$ on $K_{t,w}$, whereas $|dm_{1}\wedge\dots\wedge dm_{s}|$ is Lebesgue measure $\lambda_{s}$ on $H_{s}$. Hence (1.2) follows.