We use additive notation for line bundles and metrics over an analytic space $X$, both in the complex and non-Archimedean setting. This amounts to the following two rules:

• (i)

  if for $i=1,2$, $\phi_{i}$ is a metric on a line bundle $L_{i}$ and $a_{i}\in{\mathbb{Z}}$, then $a_{1}\phi_{1}+a_{2}\phi_{2}$ is a metric on $a_{1}L_{1}+a_{2}L_{2}$;

• (ii)

  a metric on the trivial line bundle ${\mathcal{O}}_{X}$ is of the form $|\cdot|e^{-\phi}$ for a function $\phi$ on $X$, and we identify the metric with $\phi$.

If $s$ is a section of a line bundle $L$ on $X$, then $\log|s|$ stands for the corresponding (possibly singular) metric on $L$ in which $s$ has length 1. For any metric $\phi$ on $L$, the above rules imply that $\log|s|-\phi$ is a function on $X$, and

is the pointwise length of $s$ in the metric $\phi$.

A metric on a ${\mathbb{Q}}$-line bundle $L$ is a collection $(\phi_{m})_{m}$ of metrics on $mL$, for $m$ sufficiently divisible, such that $\phi_{jm}=j\phi_{m}$.

The line bundle ${\mathcal{O}}_{X}(D)$ associated to any Cartier divisor $D$ on $X$ comes with a canonical singular metric $\phi_{D}$, smooth outside $D$. This fact extends to ${\mathbb{Q}}$-divisors, by interpreting $\phi_{D}$ as a metric on a ${\mathbb{Q}}$-line bundle. In the complex case at least, the curvature current of $\phi_{D}$, correctly normalized, coincides with the integration current on $D$.

Any finite-dimensional real vector space $V$ comes equipped with a Lebesgue (or Haar) measure $\lambda$, uniquely defined up to a multiplicative constant. Any lattice $\Lambda\subset V$ allows us to normalize $\lambda$ by $\lambda(V/\Lambda)=1$.

To any top-dimensional differential form $\omega$ on a $C^{\infty}$ manifold $X$ is associated a positive measure $|\omega|$ on $X$. For example, if $\Lambda\subset V$ is a lattice as above, $m_{1},\dots,m_{n}$ is a basis of the dual lattice, then $|dm_{1}\wedge\dots\wedge dm_{n}|$ is Lebesgue measure on $V$ normalized by $\Lambda$.

If $X$ is a complex manifold of dimension $n$, and $\Omega$ is a section of $K_{X}$, that is, a holomorphic $n$-form, we define $|\Omega|^{2}$ as the positive measure

The normalization is chosen so that the measure associated to the form $dz=dx+idy$ on ${\mathbb{C}}$ is Lebesgue measure $|dz|^{2}=|dx\wedge dy|$ on ${\mathbb{C}}\simeq{\mathbb{R}}^{2}$.

This construction induces a natural bijection between smooth metrics on the canonical bundle $K_{X}$ and (smooth, positive) volume forms on $X$, which associates to a smooth metric $\psi$ on $K_{X}$ the volume form $e^{2\psi}$ locally defined by

for any local section $\Omega$ of $K_{X}$. If $\psi^{\prime}$ is another metric on $K_{X}$, then

where $e^{2(\psi^{\prime}-\psi)}$ is the usual exponential of the smooth function $2(\psi^{\prime}-\psi)\in C^{\infty}(X)$. This can be used to make sense of $e^{2\psi}$ as a positive measure for any (possibly singular) metric $\psi$ on $K_{X}$. Similarly, $e^{2\psi/m}$ is a volume form for every metric $\psi$ on $mK_{X}$, $m\in{\mathbb{Z}}$.

Now assume $(X,B)$ is a pair in the sense of the Minimal Model Program, i.e. $X$ is a normal complex space and $B$ is a (not necessarily effective) ${\mathbb{Q}}$-Weil divisor on $X$ such that

is a ${\mathbb{Q}}$-line bundle. Denote by $\phi_{B}$ the canonical singular metric on $B|_{X_{\mathrm{reg}}}$, viewed as a ${\mathbb{Q}}$-line bundle. If $\psi$ is smooth metric on the ${\mathbb{Q}}$-line bundle $K_{(X,B)}$, then $\psi-\phi_{B}$ is a smooth metric on $K_{X_{\mathrm{reg}}\setminus B}$, and $e^{2(\psi-\phi_{B})}$ is thus a volume form on $X_{\mathrm{reg}}\setminus B$.³³ 3 Here and in what follows, we write $X\setminus D$ for the complement of the support of a (not necessarily reduced) divisor $D$ in a complex space $X$.

A pair $(X,B)$ is subklt if for some (or, equivalently, any) log resolution $\rho\colon X^{\prime}\to X$ of $(X,B)$, the unique ${\mathbb{Q}}$-divisor $B^{\prime}$ such that $\rho^{*}K_{(X,B)}=K_{(X^{\prime},B^{\prime})}$ and $\rho_{*}B^{\prime}=B$ has coefficients $<1$. The pair $(X,B)$ is klt if $B$ is further effective.

When $(X,B)$ is subklt, we may thus view $e^{2(\psi-\phi_{B})}$ as a finite positive (Radon) measure on $X$, putting no mass on Zariski closed subsets. Such measures are called adapted in [EGZ09, BBEGZ11].

The following discussion roughly follows [KKMS, p.59] and [Berk04, §1].

If $P$ is a rational polytope in ${\mathbb{R}}^{n}$, that is, the convex hull of a finite subset of ${\mathbb{Q}}^{n}$, denote by $M_{P}\subset C^{0}(P)$ the finitely generated free abelian group obtained by restricting to $P$ affine functions with coefficients in ${\mathbb{Z}}$ (constant term included). Denote by $1_{P}$ the constant function on $P$ with value $1$, and set

Denote also by $b_{P}\in{\mathbb{N}}$ the greatest integer such that $b_{P}^{-1}1_{P}\in M_{P}$.

The data of $(P,M_{P})$ modulo homeomorphism is called an (abstract) ${\mathbb{Z}}$-polytope. The functions in $M_{P}$ are called integral affine, or ${\mathbb{Z}}$-affine.

The evaluation map defines a canonical realization $P\hookrightarrow(M_{P})^{\vee}_{\mathbb{R}}$ as a codimension one rational polytope, with tangent space $T_{P}$ identified with $(\vec{M}_{P})^{\vee}_{\mathbb{R}}$. Further, the lattice $T_{P,{\mathbb{Z}}}:=\operatorname{Hom}(\vec{M}_{P},{\mathbb{Z}})\subset T_{P}$ yields a normalized Lebesgue measure $\lambda_{P}$ on $P$.

The main example for us is as follows.

A compact rational polyhedron $K$ in ${\mathbb{R}}^{n}$ is a finite union of rational polytopes $P_{i}$, which may then be arranged so that $P_{i}\cap P_{j}$ is either empty or a common face of $P_{i}$ and $P_{j}$. We then say that $(P_{i})$ is a subdivision of $K$, and call the subdivision simplicial if each $P_{i}$ is a simplex. A continuous function on $K$ is integral piecewise affine (${\mathbb{Z}}$-PA for short) if $f|_{P_{i}}\in M_{P_{i}}$ for some subdivision of $K$. These functions form a subgroup $\operatorname{PA}_{\mathbb{Z}}(K)\subset C^{0}(K)$, and the data of $(K,\operatorname{PA}_{\mathbb{Z}}(K))$ modulo homeomorphism is called a compact ${\mathbb{Z}}$-PA space.

The normalized Lebesgue measure of $K$ is defined as

for some (and hence any) subdivision into ${\mathbb{Z}}$-polytopes.

Note that a ${\mathbb{Z}}$-polytope $P$ can be regarded as a ${\mathbb{Z}}$-PA space and that $M_{P}\subset\operatorname{PA}_{\mathbb{Z}}(P)$.

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

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

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

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

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

i.e.

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

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

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

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

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

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

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

i.e.

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

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,

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.