In this section we prove Theorem 6.1. In §7.3 we give a slight refinement of
this result in the case where $E=0$.

###  7.1. A consequence of Yuan’s inequality

The following proposition is a crucial ingredient in our proof. It is a
variant of Lemma 4.5, with a precise estimate of the error term involving an
inradius.

######  Proposition 7.1.

Let $\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$.
Assume that there exists a nef and big adelic $\mathbb{R}$-divisor
$\overline{A}$ such that $\overline{A}+\overline{E}$ is pseudo-effective and
$\overline{A}-\overline{E}$ is nef. Let $\overline{P}$ be a nef adelic
$\mathbb{R}$-divisor with $P$ big. For every $\lambda\geq 0$, we have

$\widehat{\operatorname{\mathrm{vol}}}_{\chi}(\overline{P}+\lambda\overline{E})\geq(\overline{P}^{d+1})+(d+1)\lambda(\overline{P}^{d}\cdot\overline{E})-C_{d}\lambda\times(\overline{P}^{d}\cdot\overline{A})\max_{i\in\\{1,d\\}}\left(\frac{\lambda}{r(P;A)}\right)^{i},$

where $C_{d}$ is a constant depending only on $d$.

The proof combines Yuan’s Theorem 3.11 with the following technical lemma,
which is a consequence of the arithmetic Hodge index theorem of Yuan and Zhang
[YZ17]. The first point is due to Ikoma [Iko15, Theorem 2.7(2)], and the
second one relies on similar arguments.

######  Lemma 7.2.

Let $\overline{P}$ and $\overline{A}$ be nef adelic $\mathbb{R}$-divisors with
big geometric $\mathbb{R}$-divisors $P$ and $A$. Let $i\in\\{1,\ldots,d\\}$.

  1. (1)

Let $\overline{B}$ be a DSP adelic $\mathbb{R}$-divisor on $X$. If

$(\overline{B}\cdot\overline{P}^{d+1-i}\cdot\overline{A}^{i-1})=0,$

then $(\overline{B}^{2}\cdot\overline{P}^{d-i}\cdot\overline{A}^{i-1})\leq 0$.

  2. (2)

We have

$(P^{d+1-i}\cdot A^{i-1})\times(\overline{P}^{d-i}\cdot\overline{A}^{i+1})\leq
2(\overline{P}^{d+1-i}\cdot\overline{A}^{i})\times(P^{d-i}\cdot A^{i}).$

  3. (3)

We have

$(\overline{P}^{d-i}\cdot\overline{A}^{i+1})\leq(\overline{P}^{d}\cdot\overline{A})\times\left(\frac{2}{r(P;A)}\right)^{i}$

###### Proof.

(1) This is a special case of [Iko15, Theorem 2.7 (2)]. We recall the argument
for the convenience of the reader. Since $P$ and $A$ are nef and big, we have
$(P^{d})=\operatorname{\mathrm{vol}}(P)>0$ and
$(A^{d})=\operatorname{\mathrm{vol}}(A)>0$. This implies that $(P^{d+1-i}\cdot
A^{i-1})>0$ by [Laz04, Theorem 1.6.1]. Let

$\alpha=\frac{(B\cdot P^{d-i}\cdot A^{i-1})}{(P^{d+1-i}\cdot A^{i-1})}.$

Then $((B-\alpha P)\cdot P^{d-i}\cdot A^{i-1})=0$. By the arithmetic Hodge
index theorem [YZ17, Theorem 2.2] (which remains valid for adelic
$\mathbb{R}$-divisors, see [Iko15, Theorem 2.7 (1)]), we have

$(\overline{B}^{2}\cdot\overline{P}^{d-i}\cdot\overline{A}^{i-1})+\alpha^{2}(\overline{P}^{d+2-i}\cdot\overline{A}^{i-1})=((\overline{B}-\alpha\overline{P})^{2}\cdot\overline{P}^{d-i}\cdot\overline{A}^{i-1})\leq
0,$

and therefore
$(\overline{B}^{2}\cdot\overline{P}^{d-i}\cdot\overline{A}^{i-1})\leq 0$.

(2) Let

$\alpha=\frac{(\overline{P}^{d+1-i}\cdot\overline{A}^{i})}{(P^{d+1-i}\cdot
A^{i-1})}$

and $\overline{A}(\alpha)=\overline{A}-\alpha[\infty]$. By (3.3) we have

$(\overline{A}(\alpha)\cdot\overline{P}^{d+1-i}\cdot\overline{A}^{i-1})=(\overline{P}^{d+1-i}\cdot\overline{A}^{i})-\alpha(P^{d+1-i}\cdot
A^{i-1})=0.$

Applying (1) with $\overline{B}=\overline{A}(\alpha)$, we obtain

$(\overline{A}(\alpha)^{2}\cdot\overline{P}^{d-i}\cdot\overline{A}^{i-1})\leq
0.$

To conclude, we observe that

$(\overline{A}(\alpha)^{2}\cdot\overline{P}^{d-i}\cdot\overline{A}^{i-1})=(\overline{P}^{d-i}\cdot\overline{A}^{i+1})-2\alpha\,(P^{d-i}\cdot
A^{i}).$

(3) Note that the result holds trivially for $i=0$. We deduce the general case
by induction on $i$, using (2) and the fact that

$(P^{d-i}\cdot A^{i})\leq\frac{1}{r(P;A)}(P^{d+1-i}\cdot A^{i-1})$

for every $i\in\\{1,\ldots,d\\}$. This inequality follows from (2.1) since
$P-r(P,A)A$ is pseudo-effective. ∎

###### Proof of Proposition 7.1.

Let $\overline{B}=\overline{A}-\overline{E}$. Then $\overline{B}$ is nef. By
Yuan’s Theorem 3.11, we have

$\widehat{\operatorname{\mathrm{vol}}}_{\chi}(\overline{P}+\lambda\overline{E})=\widehat{\operatorname{\mathrm{vol}}}_{\chi}(\overline{P}+\lambda\overline{A}-\lambda\overline{B})\geq((\overline{P}+\lambda\overline{A})^{d+1})-\lambda(d+1)((\overline{P}+\lambda\overline{A})^{d}\cdot\overline{B}).$

After expanding the right-hand side, we find that it equals

$(\overline{P}^{d+1})+\lambda(d+1)(\overline{P}^{d}\cdot\overline{E})+\sum_{i=2}^{d+1}\binom{d+1}{i}\lambda^{i}(\overline{P}^{d+1-i}\cdot\overline{A}^{i})\\\
-(d+1)\sum_{i=1}^{d}\binom{d}{i}\lambda^{i+1}(\overline{P}^{d-i}\cdot\overline{A}^{i}\cdot\overline{B}).$

Since $\overline{P}$ and $\overline{A}$ are nef, the first sum is non-negative
and therefore

$\widehat{\operatorname{\mathrm{vol}}}_{\chi}(\overline{P}+\lambda\overline{E})\geq(\overline{P}^{d+1})+\lambda(d+1)(\overline{P}^{d}\cdot\overline{E})-(d+1)\sum_{i=1}^{d}\binom{d}{i}\lambda^{i+1}(\overline{P}^{d-i}\cdot\overline{A}^{i}\cdot\overline{B}).$

Since $2\overline{A}-\overline{B}=\overline{A}+\overline{E}$ is pseudo-
effective, we have
$(\overline{P}^{d-i}\cdot\overline{A}^{i}\cdot\overline{B})\leq
2(\overline{P}^{d-i}\cdot\overline{A}^{i+1})$ for every $i\in\\{1,\ldots,d\\}$
by Lemma 3.8, and

$(\overline{P}^{d-i}\cdot\overline{A}^{i+1})\leq(\overline{P}^{d}\cdot\overline{A})\times\left(\frac{2}{r(P;A)}\right)^{i}$

by Lemma 7.2(3). Therefore,
$\widehat{\operatorname{\mathrm{vol}}}_{\chi}(\overline{P}+\lambda\overline{E})$
is bounded from below by

$(\overline{P}^{d+1})+\lambda(d+1)(\overline{P}^{d}\cdot\overline{E})\\\
-\lambda(\overline{P}^{d}\cdot\overline{A})(d+1)\sum_{i=1}^{d}\binom{d}{i}2^{i+1}\left(\frac{\lambda}{r(P;A)}\right)^{i}.$

∎

The following consequence of Proposition 7.1 plays a central role in our
proof.

######  Corollary 7.3.

Let $\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$.
Assume that there exists a nef adelic $\mathbb{R}$-divisor $\overline{A}$ such
that $A$ is big, $\overline{A}+\overline{E}$ is pseudo-effective, and
$\overline{A}-\overline{E}$ is nef. Let $\overline{P}$ be a nef adelic
$\mathbb{R}$-divisor with $P$ big. There exists a constant $C_{d}$ depending
only on $d$ such that

$\mu^{\mathrm{ess}}(\overline{P}+\lambda\overline{E})\geq\frac{(\overline{P}^{d+1})}{(d+1)\operatorname{\mathrm{vol}}(P+\lambda
E)}+\lambda\frac{(\overline{P}^{d}\cdot\overline{E})}{(P^{d})}-C_{d}\times\frac{(\overline{P}^{d}\cdot\overline{A})}{(P^{d})}\times\frac{\lambda^{2}}{r(P;A)}$

for every $\lambda\in[0,r(P;A)/2)$. In particular, if $E=0$ then

(7.1)
$\mu^{\mathrm{ess}}(\overline{P}+\lambda\overline{E})\geq\frac{(\overline{P}^{d+1})}{(d+1)(P^{d})}+\lambda\frac{(\overline{P}^{d}\cdot\overline{E})}{(P^{d})}-C_{d}\times\frac{(\overline{P}^{d}\cdot\overline{A})}{(P^{d})}\times\frac{\lambda^{2}}{r(P;A)}.$

###### Proof.

Let $\lambda$ be a real number with $0\leq\lambda<r(P;A)/2$. By Lemma 2.4, we
have

(7.2)
$\left(1-\frac{\lambda}{r(P;A)}\right)^{d}(P^{d})\leq\operatorname{\mathrm{vol}}(P+\lambda
E)\leq\left(1+\frac{\lambda}{r(P;A)}\right)^{d}(P^{d}).$

In particular, $P+\lambda E$ is big. By Zhang’s inequality (Theorem 3.17) we
have

$\mu^{\mathrm{ess}}(\overline{P}+\lambda\overline{E})\geq\frac{\widehat{\operatorname{\mathrm{vol}}}_{\chi}(\overline{P}+\lambda\overline{E})}{(d+1)\operatorname{\mathrm{vol}}(P+\lambda
E)}.$

Therefore Proposition 7.1 implies that there exists a constant $C_{d}>0$
depending only on $d$ such that

(7.3)
$\mu^{\mathrm{ess}}(\overline{P}+\lambda\overline{E})\geq\frac{(\overline{P}^{d+1})}{(d+1)\operatorname{\mathrm{vol}}(P+\lambda
E)}+\lambda\frac{(\overline{P}^{d}\cdot\overline{E})}{\operatorname{\mathrm{vol}}(P+\lambda
E)}-C_{d}\frac{\lambda^{2}}{r(P,A)}\times\frac{(\overline{P}^{d}\cdot\overline{A})}{\operatorname{\mathrm{vol}}(P+\lambda
E)}.$

By (7.2) we have

$\frac{(\overline{P}^{d}\cdot\overline{A})}{\operatorname{\mathrm{vol}}(P+\lambda
E)}\leq\frac{(\overline{P}^{d}\cdot\overline{A})}{(P^{d})}\times\left(1-\frac{\lambda}{r(P;A)}\right)^{-d}\leq
2^{d}\frac{(\overline{P}^{d}\cdot\overline{A})}{(P^{d})}.$

On the other hand, by Lemma 3.8 we have
$|(\overline{P}^{d}\cdot\overline{E})|\leq(\overline{P}^{d}\cdot\overline{A})$
since $\overline{P}$ is nef and both $\overline{A}+\overline{E}$ and
$\overline{A}-\overline{E}$ are pseudo-effective. Let $a=1$ if
$(\overline{P}^{d}\cdot\overline{E})\geq 0$, and $a=-1$ otherwise. By (7.2),
we have

$\displaystyle\frac{(\overline{P}^{d}\cdot\overline{E})}{\operatorname{\mathrm{vol}}(P+\lambda
E)}\geq\frac{(\overline{P}^{d}\cdot\overline{E})}{(P^{d})}\times\left(1+a\frac{\lambda}{r(P;A)}\right)^{-d}$
$\displaystyle\geq\frac{(\overline{P}^{d}\cdot\overline{E})}{(P^{d})}-C^{\prime}_{d}\frac{\lambda}{r(P;A)}\frac{|(\overline{P}^{d}\cdot\overline{E})|}{(P^{d})}$
$\displaystyle\geq\frac{(\overline{P}^{d}\cdot\overline{E})}{(P^{d})}-C^{\prime}_{d}\frac{\lambda}{r(P;A)}\frac{(\overline{P}^{d}\cdot\overline{A})}{(P^{d})}$

for some constant $C^{\prime}_{d}$ depending only on $d$. The result follows
by combining these inequalities with (7.3). ∎

###  7.2. Proof of Theorem 6.1

Let $\overline{D}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$
with $D$ big. To prove Theorem 6.1, we fix a sequence
$(\phi_{n},\overline{Q}_{n})_{n}$ of semipositive approximations of
$\overline{D}$ satisfying (6.1).

######  Lemma 7.4.

For every big $\mathbb{R}$-divisor $A$ on $X$, we have

$\lim_{n\rightarrow\infty}\frac{\mu^{\mathrm{ess}}(\overline{D})-\mu^{\mathrm{abs}}(\overline{Q}_{n})}{r(Q_{n};\phi_{n}^{*}A)}=0.$

###### Proof.

This a straightforward consequence of (6.1) thanks to Lemma 2.1. ∎

For every $n\in\mathbb{N}$, we let
$\widetilde{Q}_{n}=\overline{Q}_{n}-\mu^{\mathrm{abs}}(\overline{Q}_{n})[\infty]$.
Note that
$\mu^{\mathrm{abs}}(\widetilde{Q}_{n})=\mu^{\mathrm{abs}}(\overline{Q}_{n})-\mu^{\mathrm{abs}}(\overline{Q}_{n})=0$
and therefore $\widetilde{Q}_{n}$ is nef.

######  Lemma 7.5.

For every
$\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$ and
$n\in\mathbb{N}$, we have

$\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})}=\frac{(\overline{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})-d\,\mu^{\mathrm{abs}}(\overline{Q}_{n})\times(Q_{n}^{d-1}\cdot\phi_{n}^{*}E)}{(Q_{n}^{d})}$

and moreover

$\lim_{n\to\infty}\left(\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})}-\frac{(\overline{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})-d\,\mu^{\mathrm{ess}}(\overline{D})\times(Q_{n}^{d-1}\cdot\phi_{n}^{*}E)}{(Q_{n}^{d})}\right)=0.$

###### Proof.

The first equality follows from the multilinearity of arithmetic intersection
products and (3.3). For $n\in\mathbb{N}$, let

$\beta(n)=(\mu^{\mathrm{ess}}(\overline{D})-\mu^{\mathrm{abs}}(\overline{Q}_{n}))\times\frac{(Q_{n}^{d-1}\cdot\phi_{n}^{*}E)}{(Q_{n}^{d})}$

The second statement just means that $\lim_{n\to\infty}\beta(n)=0$. To see
this, let $A$ be an ample divisor such that $A\pm E$ are both big. For every
$n\in\mathbb{N}$ we have
$|(Q_{n}^{d}\cdot\phi_{n}^{*}E)|\leq(Q_{n}^{d}\cdot\phi_{n}^{*}A)$ by (2.1).
By Lemma 2.2 we have

$|\beta(n)|\leq(\mu^{\mathrm{ess}}(\overline{D})-\mu^{\mathrm{abs}}(\overline{Q}_{n}))\times\frac{(Q_{n}^{d-1}\cdot\phi_{n}^{*}A)}{(Q_{n}^{d})}\leq\frac{\mu^{\mathrm{ess}}(\overline{D})-\mu^{\mathrm{abs}}(\overline{Q}_{n})}{r(Q_{n},\phi_{n}^{*}A)},$

and we conclude with Lemma 7.4. ∎

######  Lemma 7.6.

For every nef and big
$\overline{A}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$, we
have

$\sup_{n\in\mathbb{N}}\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{A})}{Q_{n}^{d}}<\infty.$

###### Proof.

Replacing $\overline{D}$ by $\overline{D}+t[\infty]$ for $t$ sufficiently
large, we assume that $\overline{D}$ is big without loss of generality.
Moreover, we assume that $\overline{D}-\overline{A}$ is pseudo-effective,
after replacing $\overline{A}$ by a sufficiently small multiple if necessary.
For every $n\in\mathbb{N}$ and $\lambda>0$ we have

$\lambda(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{A})\leq\frac{1}{d+1}((\widetilde{Q}_{n}+\lambda\phi_{n}^{*}\overline{A})^{d+1}).$

By Zhang’s inequality (Theorem 3.17),

$((\widetilde{Q}_{n}+\lambda\phi_{n}^{*}\overline{A})^{d+1})\leq(d+1)\mu^{\mathrm{ess}}(\widetilde{Q}_{n}+\lambda\phi_{n}^{*}\overline{A})\times((Q_{n}+\lambda\phi_{n}^{*}A)^{d}).$

Note that
$\mu^{\mathrm{ess}}(\widetilde{Q}_{n}+\lambda\phi_{n}^{*}\overline{A})=\mu^{\mathrm{ess}}(\overline{Q}_{n}+\lambda\phi_{n}^{*}\overline{A})-\mu^{\mathrm{abs}}(\overline{Q}_{n})$.
Since $\phi_{n}^{*}\overline{D}-\overline{Q}_{n}$ and
$\overline{D}-\overline{A}$ are pseudo-effective, we have

$\mu^{\mathrm{ess}}(\widetilde{Q}_{n}+\lambda\phi_{n}^{*}\overline{A})=\mu^{\mathrm{ess}}(\overline{Q}_{n}+\lambda\phi_{n}^{*}\overline{A})-\mu^{\mathrm{abs}}(\overline{Q}_{n})\leq(1+\lambda)\mu^{\mathrm{ess}}(\overline{D})-\mu^{\mathrm{abs}}(\overline{Q}_{n}).$

Therefore

(7.4)
$\lambda(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{A})\leq(\mu^{\mathrm{ess}}(\overline{D})-\mu^{\mathrm{abs}}((\overline{Q}_{n}))(Q_{n}+\lambda\phi_{n}^{*}A)^{d})+\lambda\mu^{\mathrm{ess}}(\overline{D})((Q_{n}+\lambda\phi_{n}^{*}A)^{d}).$

If $\lambda\leq r(Q_{n};\phi_{n}^{*}A)$, then $Q_{n}-\lambda\phi_{n}^{*}A$ is
pseudo-effective and therefore $((Q_{n}+\lambda\phi_{n}^{*}A)^{d})\leq
2^{d}(Q_{n}^{d})$. Applying (7.4) to $\lambda=r(Q_{n};\phi_{n}^{*}A)$ gives

$\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{A})}{(Q_{n}^{d})}\leq
2^{d}\left(\frac{\mu^{\mathrm{ess}}(\overline{D})-\mu^{\mathrm{abs}}(\overline{Q}_{n})}{r(Q_{n};\phi_{n}^{*}A)}\right)+2^{d}\mu^{\mathrm{ess}}(\overline{D}).$

This concludes the proof since the right-hand side is upper-bounded by some
constant independent of $n$ by Lemma 7.4. ∎

We are now ready to prove the theorem.

###### Proof of Theorem 6.1.

Recall from §5 that for every
$\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$, the
one-sided derivative
$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})$ exists and
$-\partial_{-\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})\geq\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})$.
Therefore we only need to show that

(7.5)
$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})\geq\limsup_{n\rightarrow\infty}\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})}$

for any
$\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$.
Indeed, if this holds then we can apply (7.5) to $-\overline{E}$ and
$\overline{E}$ to obtain

$\liminf_{n\rightarrow\infty}\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})}\geq-\partial_{-\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})\geq\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})\geq\limsup_{n\rightarrow\infty}\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})},$

and Theorem 6.1 follows by Lemmas 4.2 and 7.5.

We first show (7.5) when $\overline{E}$ is DSP. In that case there exists a
nef and big
$\overline{A}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$ such
that $\overline{A}\pm\overline{E}$ are nef by Remark 3.10. By Lemma 7.6,

$C_{0}\coloneqq\sup_{n\in\mathbb{N}}\frac{((\widetilde{Q}_{n})^{d}\cdot(\phi_{n}^{*}\overline{A}))}{(Q_{n}^{d})}$

is a real number. Let $n\in\mathbb{N}$. Since $\widetilde{Q}_{n}$ is nef, we
have $(\widetilde{Q}_{n})^{d+1}\geq 0$. Moreover, for any $\lambda\geq 0$ such
that $D+\lambda E$ is big we have

$\mu^{\mathrm{ess}}(\overline{D}+\lambda\overline{E})-\mu^{\mathrm{abs}}(\overline{Q}_{n})\geq\mu^{\mathrm{ess}}(\overline{Q}_{n}+\lambda\phi_{n}^{*}\overline{E})-\mu^{\mathrm{abs}}(\overline{Q}_{n})=\mu^{\mathrm{ess}}(\widetilde{Q}_{n}+\lambda\phi_{n}^{*}\overline{E})$

by Lemma 3.15(4). Therefore Corollary 7.3 applied with
$\overline{P}=\widetilde{Q}_{n}$ gives

(7.6)
$\mu^{\mathrm{ess}}(\overline{D}+\lambda\overline{E})-\mu^{\mathrm{abs}}(\overline{Q}_{n})\geq\lambda\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})}-C_{d}C_{0}\times\frac{\lambda^{2}}{r(Q_{n};\phi_{n}^{*}A)},$

for every $\lambda\in(0,r(Q_{n};\phi_{n}^{*}A)/2)$, where $C_{d}$ is a
constant depending only on $d$. Let

$\gamma(n)=\frac{\mu^{\mathrm{ess}}(\overline{D})-\mu^{\mathrm{abs}}(\overline{Q}_{n})}{r(Q_{n},\phi_{n}^{*}A)}$

if $\mu^{\mathrm{ess}}(\overline{D})\neq\mu^{\mathrm{abs}}(\overline{Q}_{n})$
and $\gamma(n)=1/n$ otherwise, and set
$\lambda_{n}=\sqrt{\gamma(n)}r(Q_{n},\phi_{n}^{*}A)$. By Lemma 7.4 we have
$\lim_{n\to\infty}\gamma(n)=0$, hence $\lim_{n\to\infty}\lambda_{n}=0$.
Applying (7.6) with $\lambda=\lambda_{n}$ gives

$\frac{\mu^{\mathrm{ess}}(\overline{D}+\lambda_{n}\overline{E})-\mu^{\mathrm{ess}}(\overline{D})}{\lambda_{n}}+\sqrt{\gamma(n)}\geq\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})}-C_{d}C_{0}\sqrt{\gamma(n)},$

and we obtain (7.5) by letting $n$ tend to infinity.

We now consider the general case. Let $\varepsilon>0$. By Lemma 3.7, there
exists a DSP
$\overline{E}_{\varepsilon}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$
such that $\overline{E}-\overline{E}_{\varepsilon}$ and
$\overline{E}_{\varepsilon}-\overline{E}+\varepsilon[\infty]$ are both pseudo-
effective. The first condition implies that
$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})\geq\partial_{\overline{E}_{\varepsilon}}\,\mu^{\mathrm{ess}}(\overline{D})$,
and the second one together with Lemma 3.8 gives

$(\widetilde{Q}_{n}^{d}\cdot\overline{E}_{\varepsilon})\geq(\widetilde{Q}_{n}^{d}\cdot(\overline{E}-\varepsilon[\infty]))=(\widetilde{Q}_{n}^{d}\cdot\overline{E})-\varepsilon(Q_{n}^{d})$

for every $n\in\mathbb{N}$. Therefore

$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})\geq\partial_{\overline{E}_{\varepsilon}}\,\mu^{\mathrm{ess}}(\overline{D})\geq\limsup_{n\to\infty}\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E}_{\varepsilon})}{(Q_{n}^{d})}\geq\limsup_{n\to\infty}\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})}-\varepsilon,$

and we conclude by letting $\varepsilon$ tend to zero. ∎

###  7.3. A variant of Theorem 6.1

In the course of the proof, we neglected a term of the form

$\frac{(\widetilde{Q}_{n}^{d+1})}{\operatorname{\mathrm{vol}}(Q_{n}+\lambda\phi_{n}^{*}E)}$

when applying Corollary 7.3. It turns out that taking this term into account
does not improve our approach in general, but it does in the case where $E=0$
for which we can use the second part of Corollary 7.3. This leads to the
following slight refinement of Theorem 6.1 in that special case.

######  Theorem 7.7.

Assume that there exists a sequence $(\phi_{n},\overline{Q}_{n})_{n}$ of
semipositive approximations of $\overline{D}$ such that

$\lim_{n\rightarrow\infty}\frac{1}{r(Q_{n};\phi_{n}^{*}D)}\Big{(}\mu^{\mathrm{ess}}(\overline{D})-\frac{(\overline{Q}_{n}^{d+1})}{(d+1)(Q_{n}^{d})}\Big{)}=0\sup_{n\in\mathbb{N}}\frac{\mu^{\mathrm{ess}}(\overline{D})-\mu^{\mathrm{abs}}(\overline{Q}_{n})}{r(Q_{n};\phi_{n}^{*}D)}<\infty.$

Then $\overline{D}$ satisfies the equidistribution property at every place
$v\in\mathfrak{M}_{K}$. The $v$-adic equidistribution measure is the weak
limit $\nu_{\overline{D},v}=\lim_{n\to\infty}\nu_{n,v}$, where $\nu_{n,v}$
denotes the pushforward to $X_{v}^{\mathrm{an}}$ of the normalized Monge-
Ampère measure ${c_{1}(\overline{Q}_{n,v})^{\wedge d}}/{(Q_{n}^{d})}$.

We only outline the proof of this theorem, as it is almost the same as the one
of Theorem 6.1. Let $(\phi_{n},\overline{Q}_{n})$ be a sequence satisfying the
conditions of Theorem 7.7. As above, we let
$\widetilde{Q}_{n}=\overline{Q}_{n}-\mu^{\mathrm{abs}}(\overline{Q}_{n})[\infty]$
for every $n$. With this notation, the proof of Lemma 7.6 remains valid: we
have

$C_{0}\coloneqq\sup_{n\in\mathbb{N}}\frac{(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{A})}{(Q_{n}^{d})}<\infty.$

By Proposition 5.6, it suffices to show that

(7.7)
$-\partial_{-\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})=\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})=\lim_{n\to\infty}\frac{(\overline{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})}$

for any $\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$
over $0$. We only treat the case where $\overline{E}$ is DSP, as the general
one follows as in the proof of Theorem 6.1. Let
$\overline{A}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$ be big
and nef with $\overline{A}\pm\overline{E}$ nef. By Corollary 7.3, there exists
a constant $C_{d}$ such that

(7.8)
$\mu^{\mathrm{ess}}(\overline{D}+\lambda\overline{E})-\mu^{\mathrm{abs}}(\overline{Q}_{n})\geq\frac{(\widetilde{Q}_{n}^{d})}{(d+1)(Q_{n}^{d})}+\frac{\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})}-C_{0}C_{d}\times\frac{\lambda^{2}}{r(Q_{n},\phi_{n}^{*}A)}$

for every $n$ and $\lambda\in(0,r(Q_{n},\phi_{n}^{*}A)/2)$. On the other hand,
since $E=0$ we have

$(\widetilde{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})=(\overline{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})(\widetilde{Q}_{n}^{d})=(\overline{Q}_{n}^{d})-(d+1)(Q_{n}^{d})\times\mu^{\mathrm{abs}}(\overline{Q}_{n})$

by (3.3). Combining this with (7.8) and dividing by $\lambda$ gives

$\frac{\mu^{\mathrm{ess}}(\overline{D}+\lambda\overline{E})-\mu^{\mathrm{ess}}(\overline{D})}{\lambda}\geq\frac{1}{\lambda}\Big{(}\frac{(\overline{Q}_{n}^{d})}{(d+1)(Q_{n}^{d})}-\mu^{\mathrm{ess}}(\overline{D})\Big{)}+\frac{(\overline{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})}\\\
-C_{0}C_{d}\times\frac{\lambda}{r(Q_{n},\phi_{n}^{*}A)}.$

As before, a suitable choice of $\lambda=\lambda_{n}$ permits to conclude that

$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})\geq\limsup_{n\to\infty}\frac{(\overline{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})}{(Q_{n}^{d})},$

and we obtain (7.7) by applying this to $-\overline{E}$.

######  Remark 7.8.

Compared to Theorem 6.1, Theorem 7.7 gives more flexibility to construct the
sequence $(\phi_{n},\overline{Q}_{n})_{n}$. For example, one can deduce Yuan’s
equidistribution theorem from Theorem 7.7 without using Theorem 3.18. However,
it turns out that starting with a sequence $(\phi_{n},\overline{Q}_{n})_{n}$
satisfying the conditions of Theorem 7.7, one can modify it to construct a
sequence with (6.1). This can be done with arguments similar to the ones we
used in the proof of Corollary 4.7. Since we do not need this in the remainder
of the text, we leave the details to the interested reader.

##  8\. Logarithmic equidistribution

Let $\overline{D}$ be an adelic $\mathbb{R}$-divisor on $X$ with big geometric
$\mathbb{R}$-divisor $D$. In this section we assume that the condition of
Theorem 6.1 is satisfied, namely that there exists a sequence $(\phi_{n}\colon
X_{n}\to X,\overline{Q}_{n})_{n}$ of semipositive approximations of
$\overline{D}$ satisfying (6.1). By Theorem 6.1 and Lemma 7.5, the essential
minimum function is differentiable at $\overline{D}$ and

(8.1)
$\begin{split}\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})&=\lim_{n\rightarrow\infty}\,\frac{(\overline{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})-d\,\mu^{\mathrm{ess}}(\overline{D})\times(Q_{n}^{d-1}\cdot\phi_{n}^{*}E)}{(Q_{n}^{d})}\\\
&=\lim_{n\rightarrow\infty}\frac{(\overline{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})-d\,\mu^{\mathrm{abs}}(\overline{Q}_{n})\times(Q_{n}^{d-1}\cdot\phi_{n}^{*}E)}{(Q_{n}^{d})}\end{split}$

for every
$\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$. In
particular, $\overline{D}$ satisfies the equidistribution property at every
$v\in\mathfrak{M}_{K}$ with equidistribution measure
$\nu_{\overline{D},v}=\lim_{n\to\infty}\nu_{n,v}$, where $\nu_{n,v}$ be the
pushforward to $X_{v}^{\mathrm{an}}$ of the normalized $v$-adic Monge-Ampère
measure of $\overline{Q}_{n}$. Following [CLT09], we show that this
equidistribution phenomenon extends to functions with logarithmic
singularities along certain divisors. We closely follow [CLT09, Proof of
Theorem 1.2], adapting the arguments to our setting.

######  Definition 8.1.

Let $E\in\operatorname{\mathrm{Div}}(X)$ be an effective (Cartier) divisor on
$X$ and let $v\in\mathfrak{M}_{K}$. We say that a function $\varphi\colon
X_{v}^{\mathrm{an}}\to\mathbb{R}\cup\\{\pm\infty\\}$ has _at most logarithmic
singularities_ along $E$ if it is continuous on
$X_{v}^{\mathrm{an}}\setminus(\mathrm{Supp}\,E)_{v}^{\mathrm{an}}$ and if
every $x\in X_{v}^{\mathrm{an}}$ has a neighborhood $U\subset
X_{v}^{\mathrm{an}}$ with the following property: there exists an equation
$f_{U}$ of $E_{v}^{\mathrm{an}}$ on $U$ and a real number $c_{U}$ such that
$|\varphi|_{v}\leq c_{U}\log|f_{U}|_{v}^{-1}$ on $U$.

We start with two preliminary lemmas.

######  Lemma 8.2.

For every
$\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$ with
$E$ effective, we have

$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})\geq\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{\overline{D},v}.$

###### Proof.

Let $n\in\mathbb{N}$. By Lemma 3.14 we have

$h_{\overline{Q}_{n}}(\phi_{n}^{*}E)\geq
d\,\mu^{\mathrm{abs}}(\overline{Q}_{n})\times(Q_{n}^{d-1}\cdot\phi_{n}^{*}E),$

where the left-hand side denotes the height of the $\mathbb{R}$-Weil divisor
associated to $\phi_{n}^{*}E$. By the arithmetic Bézout formula (3.2) we
obtain

$\displaystyle\frac{(\overline{Q}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})-d\,\mu^{\mathrm{abs}}(\overline{Q}_{n})\times(Q_{n}^{d-1}\cdot\phi_{n}^{*}E)}{(Q_{n}^{d})}$
$\displaystyle\geq\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{n,v}$
$\displaystyle\geq\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}\min\\{a,g_{\overline{E},v}\\}\,d\nu_{n,v}$

for any $a\in\mathbb{R}$. Since $E$ is effective, $g_{\overline{E},v}$ is
bounded from below and therefore $\min\\{a,g_{\overline{E},v}\\}\in
C(X_{v}^{\mathrm{an}})$ for every $v\in\mathbb{R}$. Letting $n$ tend to
infinity, (8.1) then gives

$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})\\\
\geq\sum_{v\in\mathfrak{M}_{K}}n_{v}\liminf_{n\to\infty}\int_{X_{v}^{\mathrm{an}}}\min\\{a,g_{\overline{E},v}\\}\,d\nu_{n,v}=\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}\min\\{a,g_{\overline{E},v}\\}\,d\nu_{\overline{D},v}.$

Letting $a$ tend to $\infty$, the result follows by monotone convergence. ∎

######  Lemma 8.3.

For every
$\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$ and
$v\in\mathfrak{M}_{K}$, $g_{\overline{E},v}$ is integrable with respect to
$\nu_{\overline{D},v}$.

###### Proof.

We can write $\overline{E}=\overline{E}_{1}-\overline{E}_{2}$ with
$\overline{E}_{1},\overline{E}_{2}$ effective. By linearity, we may therefore
assume that $\overline{E}$ is effective without loss of generality. Since
$g_{\overline{E},w}\geq 0$ for every $w\in\mathfrak{M}_{K}$, we have

$\infty>\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})\geq
n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{\overline{D},v}=n_{v}\int_{X_{v}^{\mathrm{an}}}|g_{\overline{E},v}|\,d\nu_{\overline{D},v}$

by Lemma 8.2. ∎

The main result in this paragraph is the following.

######  Theorem 8.4.

Let
$\overline{E}=(E,(g_{\overline{E},v})_{v\in\mathfrak{M}_{K}})\in\widehat{\operatorname{\mathrm{Div}}}(X)$.
Assume that $E$ is effective and that

(8.2)
$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})=\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{\overline{D},v}.$

Then the same equality holds for every adelic divisor $\overline{E}^{\prime}$
over $E$. Moreover, for every $\overline{D}$-small generic sequence
$(x_{\ell})_{\ell}$ and $v\in\mathfrak{M}_{K}$ we have

$\lim_{\ell\to\infty}\int_{X_{v}^{\mathrm{an}}}\varphi\,d\delta_{O(x_{\ell})_{v}}=\int_{X_{v}^{\mathrm{an}}}\varphi\,d\nu_{\overline{D},v}$

for any function $\varphi\colon
X_{v}^{\mathrm{an}}\rightarrow\mathbb{R}\cup\\{\pm\infty\\}$ with at most
logarithmic singularities along $E$.

###### Proof.

Let $\overline{E}^{\prime}$ be another adelic divisor over $E$. Then there
exist a finite set $S\subset\mathfrak{M}_{K}$ and a collection
$(\varphi_{v})_{v\in S}$ such that $\varphi_{v}\in
C(X_{v}^{\mathrm{an}})^{G_{v}}$ for every $v\in S$ and
$\overline{E}^{\prime}-\overline{E}=\sum_{v\in S}\overline{0}^{\varphi_{v}}$.
Since the essential minimum function is differentiable at $\overline{D}$, we
have

$\partial_{\overline{E}^{\prime}}\,\mu^{\mathrm{ess}}(\overline{D})=\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})+\sum_{v\in
S}\partial_{\overline{0}^{\varphi_{v}}}\,\mu^{\mathrm{ess}}(\overline{D}).$

We let $\varphi_{v}=0$ for $v\in\mathfrak{M}_{K}\setminus S$. Proposition 5.6
together with (8.2) then gives

$\partial_{\overline{E}^{\prime}}\,\mu^{\mathrm{ess}}(\overline{D})=\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}(g_{\overline{E},v}+\varphi_{v})\,d\nu_{\overline{D},v}=\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E}^{\prime},v}\,d\nu_{\overline{D},v}.$

To prove the second statement, it suffices to consider the case where
$\varphi=g_{\overline{E},v}$ by [CLT09, Lemma 6.3]. By (8.2), Proposition 5.3
and Theorem 6.1 we have

$\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{\overline{D},v}=\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})=\lim_{\ell\to\infty}h_{\overline{E}}(x_{\ell})=\lim_{\ell\to\infty}\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\delta_{O(x_{\ell})_{v}}.$

Therefore

$\displaystyle 0$
$\displaystyle=\lim_{\ell\to\infty}\sum_{v\in\mathfrak{M}_{K}}n_{v}\Big{(}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\delta_{O(x_{\ell})_{v}}-\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{\overline{D},v}\Big{)}$
$\displaystyle\geq\sum_{v\in\mathfrak{M}_{K}}n_{v}\Big{(}\liminf_{\ell\to\infty}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\delta_{O(x_{\ell})_{v}}-\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{\overline{D},v}\Big{)}.$

On the other hand, since $\min\\{a,g_{\overline{E},v}\\}\in
C(X_{v})^{\mathrm{an}}$ for every $v$ and $a\in\mathbb{R}$ we have

$\displaystyle\liminf_{\ell\to\infty}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\delta_{O(x_{\ell})_{v}}\geq\liminf_{\ell\to\infty}\int_{X_{v}^{\mathrm{an}}}\min\\{a,g_{\overline{E},v}\\}\,d\delta_{O(x_{\ell})_{v}}=\int_{X_{v}^{\mathrm{an}}}\min\\{a,g_{\overline{E},v}\\}\,d\nu_{\overline{D},v}$

by the equidistribution property. Letting $a$ tend to infinity, it follows
that

$\liminf_{\ell\to\infty}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\delta_{O(x_{\ell})_{v}}-\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{\overline{D},v}\geq
0.$

Since the sum of these terms over all the places $v$ is non-positive, we
obtain

$\liminf_{\ell\to\infty}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\delta_{O(x_{\ell})_{v}}=\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{\overline{D},v}$

for every $v\in\mathfrak{M}_{K}$. We conclude by observing that this equality
remains true when $(x_{\ell})_{\ell}$ is replaced by an arbitrary infinite
subsequence (since the latter remains generic and $\overline{D}$-small).
Therefore we have

$\lim_{\ell\to\infty}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\delta_{O(x_{\ell})_{v}}=\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{\overline{D},v}$

as desired. ∎

######  Remark 8.5.

Assume that for every $v\in\mathfrak{M}_{K}$, the sequence of probability
measures $(\nu_{n,v})_{n}$ is eventually constant, that is
$\nu_{n,v}=\nu_{\overline{D},v}$ for every sufficiently large $n$. As we shall
see in §10, this condition is satisfied in the case of dynamical systems and
semi-abelian varieties. Then for every
$\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)$, the condition (8.2)
is equivalent to

$\lim_{n\to\infty}\frac{h_{\overline{Q}_{n}}(\phi_{n}^{*}E)-d\,\mu^{\mathrm{ess}}(\overline{D})\times(Q_{n}^{d-1}\cdot\phi_{n}^{*}E)}{(Q_{n}^{d})}=0.$

Indeed, by (8.1) and the arithmetic Bézout formula (3.2) we have

$\displaystyle\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})$
$\displaystyle=\lim_{n\to\infty}\Big{(}\frac{h_{\overline{Q}_{n}}(\phi_{n}^{*}E)-d\,\mu^{\mathrm{ess}}(\overline{D})\times(Q_{n}^{d-1}\cdot\phi_{n}^{*}E)}{(Q_{n}^{d})}+\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{n,v}\Big{)}$
$\displaystyle=\lim_{n\to\infty}\frac{h_{\overline{Q}_{n}}(\phi_{n}^{*}E)-d\,\mu^{\mathrm{ess}}(\overline{D})\times(Q_{n}^{d-1}\cdot\phi_{n}^{*}E)}{(Q_{n}^{d})}+\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\nu_{\overline{D},v}.$

Theorem 8.4 implies [CLT09, Theorem 1.2] as follows. Assume that
$\overline{D}$ is semipositive and that

$\mu^{\mathrm{ess}}(\overline{D})=\frac{(\overline{D}^{d+1})}{(d+1)(D^{d})}.$

By Theorem 3.18 we have
$\mu^{\mathrm{ess}}(\overline{D})=\mu^{\mathrm{abs}}(\overline{D})$, and
therefore the condition of Theorem 6.1 is trivially satisfied for the constant
sequence $(\phi_{n},\overline{Q}_{n})_{n}$ equal to
$(\operatorname{Id}_{X},\overline{D})$. By Remark 8.5, the conclusion of
Theorem 8.4 is satisfied for any effective divisor
$E\in\operatorname{\mathrm{Div}}(X)_{\mathbb{R}}$ such that

$\frac{h_{\overline{D}}(E)}{d(D^{d-1}\cdot
E)}=\mu^{\mathrm{ess}}(\overline{D})=\frac{(\overline{D}^{d+1})}{(d+1)(D^{d})}.$

##  9\. Toric varieties

In this section we study the differentiability of the essential minimum
function for toric adelic $\mathbb{R}$-divisors on toric varieties, and
explore its consequences for the equidistribution of the Galois orbits of
small generic sequences of points.

In the first and second subsections we explain the basic constructions and
results from the algebraic and the Arakelov geometry of toric varieties,
referring to [BPS14, BPS15, BMPS16, BPRS19] for the proofs and more
information, and to Appendix A for the necessary definitions from convex
analysis. As a small addition to the theory we give a formula for the positive
arithmetic intersection numbers of toric adelic $\mathbb{R}$-divisors
(Proposition 9.6). In the third subsection we apply these elements to prove
our differentiability and equidistribution results in the toric setting
(Theorems 9.8 and 9.15).

###  9.1. Geometric aspects

For $d\geq 1$ let $\mathbb{T}\simeq\mathbb{G}_{\mathrm{m}}^{d}$ be a split
$d$-dimensional torus over $K$. A _toric variety_ is a normal variety $X$ over
$K$ that contains $\mathbb{T}$ as an open subset and is equipped with an
action of this torus extending its action onto itself by translations. An
$\mathbb{R}$-divisor $D$ on $X$ is _toric_ if it is invariant under this
action. A modification $\phi\colon X^{\prime}\to X$ is _toric_ if $X^{\prime}$
is also a toric variety with the same torus $\mathbb{T}$ and the restriction
of $\phi$ to this torus coincides with the identity on $\mathbb{T}$.

Toric varieties and $\mathbb{R}$-divisors can be constructed and classified
with polyhedral objects like fans and $\mathbb{R}$-virtual support functions.
To this end, first set

$M=\operatorname{\mathrm{Hom}}(\mathbb{T},\mathbb{G}_{\mathrm{m}})N=\operatorname{\mathrm{Hom}}(\mathbb{G}_{\mathrm{m}},\mathbb{T})$

for the lattices of characters and of co-characters of $\mathbb{T}$,
respectively. They are both isomorphic to $\mathbb{Z}^{d}$ and are dual of
each other, that is $M=N^{\vee}$ and $N=M^{\vee}$. Set
$N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}$ and
$M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$. These spaces are also dual
of each other, and for $u\in N_{\mathbb{R}}$ and $x\in M_{\mathbb{R}}$ we
denote their pairing by either $\langle x,u\rangle$ or $\langle u,x\rangle$.
Let $K[M]$ be the group algebra of $M$, and for each $m\in M$ set $\chi^{m}\in
K[M]$ for the corresponding monomial.

A _fan_ $\Sigma$ on $N_{\mathbb{R}}$ is a polyhedral complex of strongly
convex cones defined over $N$. To each cone $\sigma\in\Sigma$ we associate the
semigroup in $M$ defined by

$M_{\sigma}=\\{m\in M\mid\langle u,m\rangle\geq 0\text{ for all
}u\in\sigma\\}$

and we set $K[M_{\sigma}]$ for its semigroup algebra. Then
$X_{\sigma}=\operatorname{\mathrm{Spec}}(K[M_{\sigma}])$ is an affine toric
variety. Moreover, for each pair $\sigma,\sigma^{\prime}\in\Sigma$ we have
that $X_{\sigma\cap\sigma^{\prime}}$ can be naturally seen an open subset of
both $X_{\sigma}$ and $X_{\sigma^{\prime}}$. The toric variety $X_{\Sigma}$ is
then defined by gluing these affine toric varieties:

$X_{\Sigma}=\bigcup_{\sigma\in\Sigma}X_{\sigma}.$

The affine toric variety $X_{0}$ corresponding to the zero cone is canonically
isomorphic to the torus $\mathbb{T}$, and is called the _principal open
subset_ of $X_{\Sigma}$.

An _$\mathbb{R}$ -virtual support function_ $\Psi$ on $\Sigma$ is a real-
valued function on the subset of $N_{\mathbb{R}}$ covered by $\Sigma$ that is
linear on each of the cones of this fan. For each $\sigma\in\Sigma$ choose
$m_{\sigma}\in M_{\mathbb{R}}$ such that $\Psi(u)=\langle m_{\sigma},u\rangle$
for all $u\in\sigma$. For each pair $\sigma,\sigma^{\prime}\in\Sigma$ the
$\mathbb{R}$-rational function
$\chi^{-m_{\sigma}}/\chi^{-m_{\sigma^{\prime}}}$ restricted to the overlap
$X_{\sigma}\cap X_{\sigma^{\prime}}$ is an $\mathbb{R}$-linear combination of
regular functions, and so

(9.1) $D_{\Psi}=\\{(X_{\sigma},\chi^{-m_{\sigma}})\\}_{\sigma\in\Sigma}$

defines an $\mathbb{R}$-divisor on $X_{\Sigma}$ that is toric and independent
of the choice of these vectors.

The assignment $\Sigma\mapsto X_{\Sigma}$ is a bijection between the set of
fans on $N_{\mathbb{R}}$ and that of toric varieties with torus $\mathbb{T}$.
Similarly, for a fixed fan $\Sigma$ the assignment $\Psi\mapsto D_{\Psi}$ is a
bijection between the set of $\mathbb{R}$-virtual support functions on
$\Sigma$ and that of toric $\mathbb{R}$-divisors on $X_{\Sigma}$. In the
sequel we revert these notations, and for a toric variety $X$ with torus
$\mathbb{T}$ and a toric $\mathbb{R}$-divisor $D$ on it we respectively denote
its fan and $\mathbb{R}$-virtual support function by $\Sigma_{X}$ and
$\Psi_{D}$.

The toric variety $X$ is projective if and only if its fan is complete and
regular, namely it covers the whole of $N_{\mathbb{R}}$ and can be defined as
the linearity locus of a piecewise linear concave function on this vector
space. We suppose this is the case from now on. We then associate to $D$ the
subset of $M_{\mathbb{R}}$ defined as

$\Delta_{D}=\\{x\in M_{\mathbb{R}}\mid\langle u,x\rangle\geq\Psi_{D}(u)\text{
for all }u\in N_{\mathbb{R}}\\}.$

It is a quasi-rational polytope, that is a polytope with rational slopes.

The positivity invariants and properties of a toric $\mathbb{R}$-divisor can
be directly read from its $\mathbb{R}$-virtual support function and polytope.
For instance,

$\operatorname{\mathrm{vol}}(D)=d!\operatorname{\mathrm{vol}}_{M}(\Delta_{D})$

where $\operatorname{\mathrm{vol}}_{M}$ denotes the Haar measure on
$M_{\mathbb{R}}$ normalized so that $M$ has covolume $1$. In particular, if
$D$ is nef then $(D^{d})=d!\operatorname{\mathrm{vol}}_{M}(\Delta_{D})$. More
generally, for a family $D_{i}$, $i=1,\dots,d$, of toric nef
$\mathbb{R}$-divisors on $X$ we have

(9.2) $(D_{1}\cdots
D_{d})=\operatorname{\mathrm{MV}}_{M}(\Delta_{D_{1}},\dots,\Delta_{D_{d}}),$

where $\operatorname{\mathrm{MV}}_{M}$ denotes the mixed volume function with
respect to $\operatorname{\mathrm{vol}}_{M}$.

The $\mathbb{R}$-divisor $D$ is pseudo-effective if and only if
$\Delta_{D}\neq\emptyset$, and it is big if and only if $\dim(\Delta_{D})=d$.
In addition, $D$ is nef if and only if $\Psi_{D}$ is concave. For another
toric $\mathbb{R}$-divisor $E$ that is nef we have that $D-E$ is pseudo-
effective if and only if there is $x\in M_{\mathbb{R}}$ such that
$x+\Delta_{E}\subset\Delta_{D}$. These properties are either contained in
[BMPS16, Propositions 4.6, 4.7 and 4.9] or follow easily from these results.

This characterization of pseudo-effectivity readily implies the next
proposition, showing that the inradius of a toric $\mathbb{R}$-divisor with
respect to another can be computed in terms of the inradius of the associated
polytopes, in the sense of convex geometry (Definition A.1). This connection
was first pointed out by Teissier for toric line bundles [Tei82, §2.1 and
§2.2].

######  Proposition 9.1.

Let $D$ and $A$ be toric $\mathbb{R}$-divisors on $X$ such that $D$ is big and
$A$ is big and nef. Then $r(D;A)=r(\Delta_{D};\Delta_{A})$.

The toric modifications of $X$ can be classified by the complete and regular
refinements of the fan. Let $X^{\prime}$ be another projective toric variety
with torus $\mathbb{T}$ such that $\Sigma_{X^{\prime}}$ refines $\Sigma_{X}$.
Then for each cone $\sigma^{\prime}\in\Sigma_{X^{\prime}}$ there is another
cone $\sigma\in\Sigma$ such that $\sigma^{\prime}\subset\sigma$. The
associated semigroups verify that $M_{\sigma^{\prime}}\supset M_{\sigma}$, and
this inclusion induces a morphism of affine toric varieties

$X^{\prime}_{\sigma^{\prime}}=\operatorname{\mathrm{Spec}}(K[M_{\sigma^{\prime}}])\longrightarrow
X_{\sigma}=\operatorname{\mathrm{Spec}}(K[M_{\sigma}]).$

These morphisms glue together into a toric modification, that is a morphism
$X^{\prime}\to X$ whose restriction to the respective principal open subsets
is the identity on $\mathbb{T}$. Indeed, every such modification of $X$ arises
in this manner [BPS14, Theorem 3.2.4].

###  9.2. Arithmetic aspects

Throughout this subsection we denote by
$\mathbb{T}\simeq\mathbb{G}_{\mathrm{m}}^{d}$ a split torus and $X$ a
projective toric variety with torus $\mathbb{T}$. For each place
$v\in\mathfrak{M}_{K}$ we denote by $\mathbb{S}_{v}$ the compact torus of the
$v$-adic analytic torus $\mathbb{T}_{v}^{\mathrm{an}}$. In the Archimedean
case $\mathbb{S}_{v}$ is isomorphic to the real torus $(S^{1})^{d}$, whereas
in the non-Archimedean case it is an analytic subgroup of
$\mathbb{T}_{v}^{\mathrm{an}}$ in the sense of Berkovich. An adelic
$\mathbb{R}$-divisor $\overline{D}$ on $X$ is _toric_ if its geometric
$\mathbb{R}$-divisor is toric and all its $v$-adic Green functions are
invariant under the action of $\mathbb{S}_{v}$.

Toric adelic $\mathbb{R}$-divisors over $D$ can be constructed and classified
with adelic families of functions on $N_{\mathbb{R}}$ whose behavior at
infinity is governed by the $\mathbb{R}$-virtual support function $\Psi_{D}$.
To this end, let $v\in\mathfrak{M}_{K}$ and consider the _valuation map_

$\operatorname{\mathrm{val}}_{v}\colon\mathbb{T}_{v}^{\mathrm{an}}=X_{0,v}^{\mathrm{an}}\longrightarrow
N_{\mathbb{R}}.$

Using a splitting of the torus, we can identify the dense subset
$\mathbb{T}_{v}^{\mathrm{an}}(\mathbb{C}_{v})$ with
$(\mathbb{C}_{v}^{\times})^{d}$ and the vector space $N_{\mathbb{R}}$ with
$\mathbb{R}^{d}$. In these coordinates, the map writes down for a point of
$\mathbb{T}_{v}^{\mathrm{an}}(\mathbb{C}_{v})$ as

$\operatorname{\mathrm{val}}_{v}(x_{1},\dots,x_{n})=(-\log|x_{1}|_{v},\dots,-\log|x_{n}|_{v}).$

This valuation map extends to a proper continuous, $\mathbb{S}_{v}$-invariant
and similarly noted map of topological spaces

$\operatorname{\mathrm{val}}_{v}\colon X_{v}^{\mathrm{an}}\longrightarrow
N_{\Sigma_{X}},$

where $N_{\Sigma_{X}}$ denotes the Mumford compactification of
$N_{\mathbb{R}}$ induced by the fan, see [BPS14, Section 4.1] for details.

Now let $\psi_{v}\colon N_{\mathbb{R}}\to\mathbb{R}$,
${v\in\mathfrak{M}_{K}}$, be a family of functions such that
$\psi_{v}-\Psi_{D}$ extends to a continuous function on $N_{\Sigma_{X}}$ for
all $v$ and such that $\psi_{v}=\Psi_{D}$ for all but a finite number of these
places. Then for each $v$ we can consider the function $g_{\psi_{v}}\colon
X_{0,v}^{\mathrm{an}}\to\mathbb{R}$ defined as

$g_{\psi_{v}}(x)=-\psi_{v}(\operatorname{\mathrm{val}}_{v}(x)).$

The fact that $\psi_{v}-\Psi_{D}$ admits a continuous extension to
$N_{\Sigma_{X}}$ implies that for every cone $\sigma\in\Sigma_{X}$ and
defining vector $m_{\sigma}$ as in (9.1) we have that
$g_{\psi_{v}}+\log|\chi^{-m_{\sigma}}|_{v}$ extends to a continuous function
on the affine chart $X_{\sigma,v}^{\mathrm{an}}$. Hence $g_{\psi_{v}}$ is an
$\mathbb{S}_{v}$-invariant $v$-adic Green function for $D$. Furthermore, the
fact that $\psi_{v}=\Psi_{D}$ for all but a finite number of places implies
that the pair $(D,(g_{\psi_{v}})_{v\in\mathfrak{M}_{K}})$ is a toric adelic
$\mathbb{R}$-divisor [BPS14, Example 4.5.4].

This construction gives a bijection between the set of such adelic families of
functions on $N_{\mathbb{R}}$ and that of toric adelic $\mathbb{R}$-divisors
over $D$. Reverting it, for a toric adelic $\mathbb{R}$-divisor $\overline{D}$
we denote by

$\psi_{\overline{D},v}\colon
N_{\mathbb{R}}\longrightarrow\mathbb{R},\quad{v\in\mathfrak{M}_{K}},$

the associated family of _metric functions_. We also associate to
$\overline{D}$ its family of local roof functions

${\vartheta_{\overline{D},v}\colon\Delta_{D}\longrightarrow\mathbb{R}},\quad{v\in\mathfrak{M}_{K}}.$

For each $v\in\mathfrak{M}_{K}$, the $v$-adic roof function is the continuous
concave function on the polytope defined as

(9.3) $\vartheta_{\overline{D},v}(x)=\inf_{u\in N_{\mathbb{R}}}\langle
u,x\rangle-\psi_{\overline{D},v}(u)\quad\text{ for }x\in\Delta_{D}.$

We have $\vartheta_{\overline{D},v}=0|_{\Delta_{D}}$, the zero function on the
polytope, for all but a finite number of places. We consider then the _global
roof function_ $\vartheta_{\overline{D}}\colon\Delta_{D}\to\mathbb{R}$,
defined as the weighted sum

(9.4)
$\vartheta_{\overline{D}}=\sum_{v\in\mathfrak{M}_{K}}n_{v}\vartheta_{\overline{D},v}.$

We also consider the compact convex set where this concave function is
nonnegative:

(9.5)
$\Gamma_{\overline{D}}=\\{x\in\Delta_{D}\mid\vartheta_{\overline{D}}(x)\geq
0\\}.$

In analogy with the geometric case, the positivity invariants and properties
of a toric adelic $\mathbb{R}$-divisor can be read from its metric and roof
functions. For instance, the essential minimum of $X$ with respect to
$\overline{D}$ is the maximum of global roof function [BPS15, Theorem 1.1]:

$\mu^{\mathrm{ess}}(\overline{D})=\max_{x\in\Delta_{D}}\vartheta_{\overline{D}}(x).$

Moreover, if $\overline{D}$ is semipositive then
$\mu^{\mathrm{abs}}(\overline{D})=\min_{x\in\Delta_{D}}\vartheta_{\overline{D}}(x)$
[BPS15, Remark 3.15].

The arithmetic volume and the $\chi$-arithmetic volume of $\overline{D}$ can
be respectively computed as [BMPS16, Theorem 5.6]

$\widehat{\operatorname{\mathrm{vol}}}(\overline{D})=(d+1)!\int_{\Gamma_{D}}\vartheta_{\overline{D}}\,d\operatorname{\mathrm{vol}}_{M}\widehat{\operatorname{\mathrm{vol}}}_{\chi}(\overline{D})=(d+1)!\int_{\Delta_{D}}\vartheta_{\overline{D}}\,d\operatorname{\mathrm{vol}}_{M}.$

In particular, if $\overline{D}$ is semipositive then
$(\overline{D}^{d+1})=(d+1)!\int_{\Delta_{D}}\vartheta_{\overline{D}}\,d\operatorname{\mathrm{vol}}_{M}$.
More generally, the arithmetic intersection number of a family of semipositive
toric adelic $\mathbb{R}$-divisors $\overline{D}_{i}$, $i=0,\dots,d$, can be
computed as

(9.6)
$(\overline{D}_{0}\cdots\overline{D}_{d})=\sum_{v\in\mathfrak{M}_{K}}n_{v}\operatorname{\mathrm{MI}}_{M}(\vartheta_{\overline{D}_{0},v},\dots,\vartheta_{\overline{D}_{d},v}),$

where $\operatorname{\mathrm{MI}}_{M}$ denotes the mixed integral function
with respect to the Haar measure $\operatorname{\mathrm{vol}}_{M}$ on
$M_{\mathbb{R}}$ [BPS14, Theorem 5.2.5].

We have that $\overline{D}$ is pseudo-effective if and only if there is
$x\in\Delta_{D}$ such that ${\vartheta_{\overline{D}}(x)\geq 0}$, and it is
big if and only if $\dim(\Delta_{D})=d$ and there is $x\in\Delta_{D}$ such
that $\vartheta_{\overline{D}}(x)>0$. In addition, $\overline{D}$ is
semipositive if and only if $\psi_{\overline{D},v}$ is concave for all
$v\in\mathfrak{M}_{K}$. If this is the case, then $\overline{D}$ is nef if and
only of $\vartheta_{\overline{D}}(x)\geq 0$ for all $x\in\Delta_{D}$. For
another adelic $\overline{E}$ semipositive that is semipositive we have that
$\overline{D}-\overline{E}$ is pseudo-effective if and only if
$\Delta_{E}\subset\Delta_{D}$ and
$\vartheta_{\overline{E},v}(x)\leq\vartheta_{\overline{D},v}(x)$ for all
$v\in\mathfrak{M}_{K}$ and $x\in\Delta_{E}$. The listed properties can be
found in [BMPS16, Sections 5 and 6].

######  Example 9.2.

Let $D$ be a big and nef toric $\mathbb{R}$-divisor on $X$, and choose a
family of vectors $u_{v}\in N_{\mathbb{R}}$, $v\in\mathfrak{M}_{K}$, with
$u_{v}=0$ for all but a finite number of $v$ and such that
$\sum_{v}n_{v}u_{v}=0$. Choose also a family of constants
$c_{v}\in\mathbb{R}$, $v\in\mathfrak{M}_{K}$, with $c_{v}=0$ for all but a
finite number of $v$. We consider then the semipositive toric adelic
$\mathbb{R}$-divisor $\overline{D}$ on $X$ defined by the concave metric
functions

$\psi_{v}(u)=\Psi_{D}(u-u_{v})-c_{v}\quad\text{ for all }v\text{ and }u\in
N_{\mathbb{R}}.$

Its local roof functions are the affine functions given by
$\vartheta_{v}(x)=\langle u_{v},x\rangle+c_{v}$ for all $v$ and
$x\in\Delta_{D}$, as it follows from their definition in (9.3) and the basic
properties of the Legendre-Fenchel duality of concave functions, see for
instance [BPS14, Proposition 2.3.3 and Example 2.2.1]. Thus its global roof
function is the constant function

$\vartheta_{\overline{D}}=c|_{\Delta_{D}}\quad\text{ for
}c=\sum_{v\in\mathfrak{M}_{K}}n_{v}c_{v}\in\mathbb{R}.$

In particular
$\mu^{\mathrm{ess}}(\overline{D})=\mu^{\mathrm{abs}}(\overline{D})=c,$ and so
Zhang’s lower bound is attained for $\overline{D}$.

######  Remark 9.3.

Every semipositive toric adelic $\mathbb{R}$-divisor on $X$ for which Zhang’s
lower bound for the essential minimum is attained is of this type for a
suitable choice of vectors and constants [BPRS19, Proposition 5.3].

For any semipositive toric adelic $\mathbb{R}$-divisor on $X$ it is always
possible to find an upper bound as that required in Proposition 6.13. To this
end we consider the notion of balanced family of sup-gradients for the
decomposition of $\vartheta_{\overline{D}}$ into local roof functions. This is
a family of vectors

$u_{v}\in N_{\mathbb{R}},\quad v\in\mathfrak{M}_{K},$

with $u_{v}=0$ for all but a finite number of $v$ and verifying
$\sum_{v}n_{v}u_{v}=0$, and such that there is a point $x_{0}\in\Delta_{D}$
maximizing $\vartheta_{\overline{D}}$ with $u_{v}$ lying in
$\partial\vartheta_{\overline{D},v}(x_{0})$ the sup-differential of the
$v$-adic roof function at this point (Definition A.7).

######  Proposition 9.4.

Let $\overline{D}$ be a semipositive toric adelic $\mathbb{R}$-divisor on $X$
with $D$ big. Then there is toric semipositive adelic $\mathbb{R}$-divisor
$\widetilde{D}$ such that $\widetilde{D}-\overline{D}$ is pseudo-effective and
$\mu^{\mathrm{ess}}(\widetilde{D})=\mu^{\mathrm{abs}}(\widetilde{D})=\mu^{\mathrm{ess}}(\overline{D})$.

###### Proof.

Let $(u_{v})_{v}$ be a balanced family of sup-gradients for
$\vartheta_{\overline{D}}$, which does exist thanks to Proposition A.8. Take a
point $x_{0}\in\Delta_{D}$ maximizing $\vartheta_{\overline{D}}$. Then for
each $v\in\mathfrak{M}_{K}$ we have

(9.7) $\vartheta_{\overline{D},v}(x)\leq\langle
u_{v},x\rangle+c_{v}\quad\text{ for all }x\in\Delta_{D}$

with $c_{v}=\vartheta_{\overline{D},v}(x_{0})-\langle u_{v},x_{0}\rangle$. We
have

(9.8)
$\sum_{v\in\mathfrak{M}_{K}}n_{v}c_{v}=\sum_{v\in\mathfrak{M}_{K}}n_{v}\,(\vartheta_{\overline{D},v}(x_{0})-\langle
u_{v},x_{0}\rangle)=\vartheta_{\overline{D}}(x_{0})=\mu^{\mathrm{ess}}(\overline{D}).$

Set $\widetilde{D}$ for the semipositive toric adelic $\mathbb{R}$-divisor on
$X$ from Example 9.2 corresponding to the data $(u_{v})_{v}$ and
$(c_{v})_{v}$. The inequality in (9.7) implies that
$\widetilde{D}-\overline{D}$ is pseudo-effective, and from (9.8) we get
$\mu^{\mathrm{ess}}(\widetilde{D})=\mu^{\mathrm{abs}}(\widetilde{D})=\mu^{\mathrm{ess}}(\overline{D}).$
∎

Now suppose that $\overline{D}$ is big. By the toric arithmetic Fujita
approximation theorem [BMPS16, Theorem 7.2] there is a sequence of nef
approximations of $\overline{D}$

(9.9) $(\phi_{n}\colon X_{n}\to X,\overline{P}_{n})_{n}$

such that both $\phi_{n}$ and $\overline{P}_{n}$ are toric and
$\lim_{n\to\infty}\widehat{\operatorname{\mathrm{vol}}}(\overline{P}_{n})=\widehat{\operatorname{\mathrm{vol}}}(\overline{D})$.
It can be constructed by taking any sequence $\Delta_{n}$, $n\geq 1$, of
quasi-rational polytopes contained in $\Gamma_{\overline{D}}$ and arbitrarily
approaching this convex body. Each $\Delta_{n}$ induces a toric modification
$\phi_{n}\colon X_{n}\to X$ by considering a complete and regular refinement
of $\Sigma_{X}$ that is compatible with the normal fan of $\Delta_{n}$, and it
also induces a nef toric adelic $\mathbb{R}$-divisor $\overline{P}_{n}$ on
$X_{n}$ by considering the restrictions to $\Delta_{n}$ of the local roof
functions of $\overline{D}$.

The next result gives a lower bound for the inradius of $\overline{D}$ in
terms of the convex-geometric inradius of its associated convex body.

######  Proposition 9.5.

Let $\overline{D}$ be a toric adelic $\mathbb{R}$-divisor on $X$ that is big,
and $A$ a toric $\mathbb{R}$-divisor on $X$ that is big and nef. Then
$\rho(\overline{D};A)\geq r(\Gamma_{\overline{D}};\Delta_{A})$.

###### Proof.

For the sequence of toric nef approximations in (9.9) we have that

$\rho(\overline{D};A)\geq
r(P_{n};\phi_{n}^{*}A)=r(\Delta_{n};\Delta_{A})\quad\text{ for }n\geq 1,$

by Proposition 9.1 and the fact that the polytope of a toric
$\mathbb{R}$-divisor is invariant with respect to toric modifications. Since
the sequence $(\Delta_{n})_{n}$ approaches $\Gamma_{\overline{D}}$ from the
inside, we get
$\rho(\overline{D};A)\geq\sup_{n}r(\Delta_{n};\Delta_{A})=r(\Gamma_{\overline{D}};\Delta_{A})$
as stated. ∎

Finally, we give a convex-analytic formula for the positive arithmetic
intersection numbers in the toric setting.

######  Proposition 9.6.

Let $\overline{D}$ and $\overline{E}$ be toric adelic $\mathbb{R}$-divisors on
$X$ with $\overline{D}$ big and $\overline{E}$ semipositive. Then

$\operatorname{\mathrm{vol}}(R^{0}(\overline{D}))=d!\operatorname{\mathrm{vol}}(\Gamma_{\overline{D}}),\quad(\langle\overline{D}^{d}\rangle\cdot\overline{E})=\sum_{v\in\mathfrak{M}_{K}}n_{v}\operatorname{\mathrm{MI}}_{M}(\vartheta_{\overline{D},v}|_{\Gamma_{\overline{D}}},\dots,\vartheta_{\overline{D},v}|_{\Gamma_{\overline{D}}},\vartheta_{\overline{E},v}).$

###### Proof.

The first formula follows from Corollary 4.6 using the sequence of toric nef
approximations in (9.9) and the formula for toric intersection numbers in
(9.2).

For the second we use the same sequence of toric nef approximations. Applying
the definition of positive arithmetic intersection numbers in (4.3) and the
formula for the toric arithmetic intersection numbers in (9.6) we get

$(\overline{P}_{n}^{d}\cdot\phi_{n}^{*}\overline{E})=\sum_{v\in\mathfrak{M}_{K}}n_{v}\operatorname{\mathrm{MI}}_{M}(\vartheta_{\overline{D},v}|_{\Delta_{n}},\dots,\vartheta_{{\overline{D},v}|_{\Delta_{n}}},\vartheta_{\overline{E},v}).$

We conclude by taking the limit $n\to\infty$ and applying Proposition A.9. ∎

###  9.3. Equidistribution on toric varieties

Here we apply the constructions and results explained in §9.1, §9.2 and
Appendix A to state and prove our toric differentiability and equidistribution
results. In spite these preparations, we still need to assume a certain
working knowledge in the Arakelov geometry of toric varieties.

Throughout this section we denote by $X$ a projective toric variety with torus
$\mathbb{T}$ and $\overline{D}$ a toric adelic $\mathbb{R}$-divisor on $X$
with $D$ big. For each $t\leq\mu^{\mathrm{ess}}(\overline{D})$ we consider the
_sup-level set_ of its global roof function, that is

$S_{t}(\vartheta_{\overline{D}})=\\{x\in\Delta_{D}\mid\vartheta_{\overline{D}}(x)\geq
t\\}.$

It is a nonempty compact subset of the polytope $\Delta_{D}$ that is
$d$-dimensional whenever $t<\mu^{\mathrm{ess}}(\overline{D})$. Set also
$\Delta_{D,\max}=S_{\mu^{\mathrm{ess}}(\overline{D})}(\vartheta_{\overline{D}})$.

The global roof function is said to be _wide_ if after fixing an arbitrary
norm on $M_{\mathbb{R}}$, the width of these sup-level sets remains relatively
large as the level approaches its maximum (Definition A.6). By Proposition
A.3, this can be alternatively expressed as the condition that the inradius of
these sup-level sets with respect to any fixed convex body remains relatively
large as the level approaches the maximum. By the same result, it is also
equivalent to the fact that for any maximizing point $x_{0}\in\Delta_{D,\max}$
the vector $0\in N_{\mathbb{R}}$ is a vertex of the sup-differential
$\partial\vartheta_{\overline{D}}(x_{0})\subset N_{\mathbb{R}}$ (Definition
A.2). When this is the case, by Proposition A.8 we can associate to
$\vartheta_{\overline{D}}$ a unique balanced family of sup-gradients

$u_{v}\in N_{\mathbb{R}},\quad v\in\mathfrak{M}_{K},$

with respect to its decomposition into local roof functions in (9.4).

We next describe the probability measures that appear as equidistribution
measures in the toric setting. To this end, when $v$ is non-Archimedean we
consider the section of the $v$-adic valuation map

(9.10) $\zeta_{v}\colon N_{\Sigma}\longrightarrow X_{v}^{\mathrm{an}}$

defined as the composition $\zeta_{v}=\theta_{\Sigma}\circ\bm{e}$ for the maps
$\bm{e}$ and $\theta_{\Sigma}$ respectively defined in [BPS14, §4.1 and §4.2].
By Proposition-Definition 4.2.12 in loc. cit., the section $\zeta_{v}$ is
continuous and proper. For $u\in N_{\mathbb{R}}$ we have that $\zeta_{v}(u)\in
X_{0,v}^{\mathrm{an}}$, and this point can be explicitly written down as the
multiplicative seminorm on $K[M]$ defined as

$|f|_{\zeta_{v}(u)}=\max_{m}|\alpha_{m}|_{v}\,e^{-\langle
m,u\rangle}\quad\text{ for }f=\sum_{m}\alpha_{m}\chi^{m}\in K[M].$

In particular, $\zeta_{v}(0)$ is the Gauss point of
$X_{0,v}^{\mathrm{an}}=\mathbb{T}_{v}^{\mathrm{an}}$.

######  Definition 9.7 ([BPRS19, Definition 5.1]).

Given $v\in\mathfrak{M}_{K}$ and $u\in N_{\mathbb{R}}$, the probability
measure $\eta_{v,u}$ on $X_{{v}}^{\mathrm{an}}$ is defined as follows.

  1. (1)

When $v$ is Archimedean, we have that
$\operatorname{\mathrm{val}}_{v}^{-1}(u)=\mathbb{S}_{v}\cdot x$ for any
$x\in\operatorname{\mathrm{val}}_{v}^{-1}(u)$. Then $\eta_{v,u}$ is the direct
image under the translation by $x$ of the Haar probability measure of the real
torus $\mathbb{S}_{v}\simeq(S^{1})^{d}$.

  2. (2)

When $v$ is non-Archimedean, $\eta_{v,u}$ is defined as the Dirac measure at
the point $\zeta_{v}(u)\in X_{v}^{\mathrm{an}}$.

The following theorem is the central result of this section. It corresponds to
Theorem 1.3 together with the height convergence property (1.5) in the
introduction.

######  Theorem 9.8.

If $\vartheta_{\overline{D}}$ is wide then the essential minimum function is
differentiable at $\overline{D}$, and for every
$\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$ with
$E$ toric we have

$\partial_{\overline{E}}\mu^{\mathrm{ess}}(\overline{D})=\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\eta_{v,u_{v}}$

where $u_{v}\in N_{\mathbb{R}}$, $v\in\mathfrak{M}_{K}$, is the balanced
family of sup-gradients for $\vartheta_{\overline{D}}$.

In particular $\overline{D}$ satisfies the equidistribution property, and for
each $v\in\mathfrak{M}_{K}$ we have $\nu_{\overline{D},v}=\eta_{v,u_{v}}$.

This is an application of Theorem 6.7, or rather of its reformulation in terms
of positive arithmetic intersection numbers in Theorem 6.7. It also depends on
a key lemma about the decay of the sup-level sets of concave functions on
convex bodies (Proposition A.3) and on some further auxiliary constructions
and results that we next explain.

Following [BPS14, Definition 4.3.3], we describe an averaging process allowing
to pass from an arbitrary adelic $\mathbb{R}$-divisor on $X$ to a toric one.

######  Definition 9.9.

Let $\overline{E}$ be an adelic $\mathbb{R}$-divisor on $X$ with $E$ toric.
For $v\in\mathfrak{M}_{K}$ we consider the function $\widehat{g}_{v}\colon
X_{v}^{\mathrm{an}}\setminus E_{v}^{\mathrm{an}}\to\mathbb{R}$ defined as

$\widehat{g}_{v}(x)=\begin{cases}\displaystyle{\int_{\mathbb{S}_{v}}g_{\overline{E},v}(t\cdot
x)\,d\operatorname{Haar}_{v}(t)}&\text{ if }v\text{ is Archimedean},\\\
g_{\overline{E},v}(\zeta_{v}(\operatorname{\mathrm{val}}_{v}(x)))&\text{ if
}v\text{ is non-Archimedean},\end{cases}$

with $\operatorname{Haar}_{{v}}$ the Haar probability measure of the real
torus $\mathbb{S}_{v}$ when $v$ is Archimedean, and $\zeta_{v}$ the section of
the $v$-adic valuation map in (9.10) when $v$ is non-Archimedean. We then set
${\overline{E}}\vphantom{L}^{\mathrm{tor}}=(E,(\widehat{g}_{v})_{v\in\mathfrak{M}_{K}}),$
which is a toric adelic $\mathbb{R}$-divisor on $X$.

Using the probability measures from Definition 9.7, for every
$v\in\mathfrak{M}_{K}$ the corresponding Green function of
${\overline{E}}\vphantom{L}^{\mathrm{tor}}$ can be alternatively expressed as

(9.11)
$\widehat{g}_{v}(x)=\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\eta_{v,u}\quad\text{
for }u=\operatorname{\mathrm{val}}_{v}(x).$

The relevant arithmetic intersection numbers (both standard and positive) are
invariant with respect to this averaging process.

######  Proposition 9.10.

For any toric adelic $\mathbb{R}$-divisor $\overline{P}$ on $X$ we have
$(\overline{P}^{d}\cdot\overline{E})=(\overline{P}^{d}\cdot{\overline{E}}\vphantom{L}^{\mathrm{tor}}).$
In particular, if $\overline{D}$ is big then
$(\langle\overline{D}^{d}\rangle\cdot\overline{E})=(\langle\overline{D}^{d}\rangle\cdot{\overline{E}}\vphantom{L}^{\mathrm{tor}}).$

###### Proof.

With notation as in Definition 9.9, by the arithmetic Bézout formula we have

$(\overline{P}^{d}\cdot{\overline{E}}\vphantom{L}^{\mathrm{tor}})=h_{\overline{P}}(E)+\sum_{v}n_{v}\int_{X_{v}^{\mathrm{an}}}\widehat{g}_{v}\,c_{1}(\overline{P}_{v})^{\wedge
d}.$

Let $v\in\mathfrak{M}_{K}$. When $v$ is Archimedean we have

$\int_{X_{v}^{\mathrm{an}}}\widehat{g}_{v}\,c_{1}(\overline{P}_{v})^{\wedge
d}=\int_{X_{v}^{\mathrm{an}}}\bigg{(}\int_{\mathbb{S}_{v}}g_{\overline{E},v}(t\cdot
x)\,d\operatorname{Haar}_{v}(t)\bigg{)}c_{1}(\overline{P}_{v})^{\wedge
d}(x)\\\
=\int_{\mathbb{S}_{v}}\bigg{(}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}(x)\,t_{*}c_{1}(\overline{P}_{v})^{\wedge
d}(x)\bigg{)}d\operatorname{Haar}_{v}(t)=\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}(x)\,c_{1}(\overline{P}_{v})^{\wedge
d},$

using Fubini’s theorem, the change of variable formula and the invariance of
the $v$-adic Monge-Ampère measure of $\overline{P}$ under the action of
$\mathbb{S}_{v}$. On the other hand, when $v$ is non-Archimedean we have

$\int_{X_{v}^{\mathrm{an}}}\widehat{g}_{v}\,c_{1}(\overline{P}_{v})^{\wedge
d}=\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}(\zeta_{v}(\operatorname{\mathrm{val}}_{v}(x)))\,c_{1}(\overline{P}_{v})^{\wedge
d}(x)=\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}(x)\,c_{1}(\overline{P}_{v})^{\wedge
d}(x),$

by the characterization of the Monge-Ampère measure of semipositive toric
adelic divisors in [BPS14, Theorem 4.8.11].

For the second, suppose that $\overline{D}$ is big and consider a sequence of
arithmetic Fujita approximations $(\phi_{n}\colon X_{n}\to
X,\overline{P}_{n})$, $n\in\mathbb{N}$, as in (9.9). The averaging process
commutes with toric modifications as it can be read from its definition on the
principal open subset, which remains unchanged under these modifications.
Hence applying the first statement, for each $n\in\mathbb{N}$ we get

$(\overline{P}_{n}^{d}\cdot\phi_{n}^{*}{\overline{E}}\vphantom{L}^{\mathrm{tor}})=(\overline{P}_{n}^{d}\cdot(\phi_{n}^{*}\overline{E})\vphantom{L}^{\mathrm{tor}})=(\overline{P}_{n}^{d}\cdot\phi_{n}^{*}\overline{E}).$

Then this second statement follows from the definition of these positive
arithmetic intersection numbers in (4.3) by taking the limit $n\to\infty$. ∎

Recall that $[\infty]$ denotes the adelic $\mathbb{R}$-divisor over the zero
divisor of $X$ with $v$-adic Green function equal to $1$ if $v$ is Archimedean
and to $0$ is $v$ is non-Archimedean. As in §6.3, for each
$t<\mu^{\mathrm{ess}}(\overline{D})$ we consider the twist

$\overline{D}(t)=\overline{D}-t[\infty].$

We have $\vartheta_{\overline{D}(t)}=\vartheta_{\overline{D}}-t$, as it can be
verified from the definitions of this twist and of the global roof function.
In particular we have
$\Gamma_{\overline{D}(t)}=S_{t}(\vartheta_{\overline{D}})$ for the convex body
in (9.5).

###### Proof of Theorem 9.8.

Set $\mu=\mu^{\mathrm{ess}}(\overline{D})$ for short and let $A$ be a big and
nef $\mathbb{R}$-divisor on $X$. Since $\vartheta_{\overline{D}}$ is wide we
have

(9.12)
$\lim_{t\to\mu}\frac{\mu-t}{r(S_{t}(\vartheta_{\overline{D}});\Delta_{A})}=0.$

For each $t<\mu$ we have
$S_{t}(\vartheta_{\overline{D}})=\Gamma_{\overline{D}(t)}$ and so
$r(S_{t}(\vartheta_{\overline{D}});\Delta_{A})\leq\rho(\overline{D}(t);A)$ by
Proposition 9.5. Hence the limit in (9.12) gives
$\lim_{t\to\mu}(\mu-t)/\rho(\overline{D}(t);A)=0.$ By Theorem 6.7 this implies
that the essential minimum function is differentiable at $\overline{D}$ and
that for each
$\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$ we have

$\partial_{\overline{E}}\mu^{\mathrm{ess}}(\overline{D})=\lim_{t\to\mu}\frac{\langle\overline{D}(t)^{d}\cdot\overline{E}\rangle}{\operatorname{\mathrm{vol}}(R^{t}(\overline{D}))}.$

The formula for this derivative will follow from this limit together with the
convex analysis interpretation of positive intersection numbers in the toric
setting and the averaging process previously described.

First suppose that $\overline{E}$ is both semipositive and toric. Let
$\mathfrak{S}\subset\mathfrak{M}_{K}$ be a finite set of places such that
$\psi_{\overline{D},v}=\Psi_{D}$ and $u_{v}=0$ for all $v\notin\mathfrak{S}$.
By possibly enlarging $\mathfrak{S}$ we can also assume that
$\vartheta_{\overline{E},v}=0|_{\Delta_{E}}$ for all $v\notin\mathfrak{S}$.

Let $t<\mu$ and set for short $S_{t}=S_{t}(\vartheta_{\overline{D},v})$. Set
$\varepsilon_{v}=1$ if $v$ is Archimedean and $\varepsilon_{v}=0$ if $v$ is
non-Archimedean, and set also
$\theta_{v,t}=(\vartheta_{\overline{D},v}-\varepsilon_{v}t)|_{S_{t}}$ for each
$v$. By Proposition 9.6 we have

(9.13)
$\operatorname{\mathrm{vol}}(R^{t}(\overline{D}))=d!\operatorname{\mathrm{vol}}(S_{t})\langle\overline{D}(t)^{d}\cdot\overline{E}\rangle=\sum_{v\in\mathfrak{S}}n_{v}\operatorname{\mathrm{MI}}_{M}(\theta_{v,t},\dots,\theta_{v,t},\vartheta_{\overline{E},v}).$

For $v\in\mathfrak{S}$ we have $\vartheta_{\overline{D},v}(x)\leq\langle
u_{v},x\rangle+c_{v}$ for $x\in\Delta_{D}$ with
$c_{v}=\vartheta_{\overline{D},v}(x_{0})-\langle u_{v},x_{0}\rangle$ for any
$x_{0}\in\Delta_{D,\max}$. Setting also $\kappa_{v}=\max_{x\in S_{t}}(\langle
u_{v},x\rangle+c_{v}-\vartheta_{\overline{D},v}(x))\geq 0$ we have the
estimates

(9.14) $\langle
u_{v},x\rangle+c_{v}-\kappa_{v}-\varepsilon_{v}t\leq\theta_{v,t}(x)\leq\langle
u_{v},x\rangle+c_{v}-\varepsilon_{v}t\quad\text{ for }x\in S_{t}.$

We have
$\sum_{v}n_{v}c_{v}=\sum_{v}n_{v}\,(\vartheta_{\overline{D},v}(x_{0})-\langle
u_{v},x_{0}\rangle)=\vartheta_{\overline{D}}(x_{0})=\mu$. On the other hand,
for each $v_{0}\in\mathfrak{S}$ and any $x\in S_{t}$ we have

$\mu-t\geq\mu-\vartheta_{\overline{D},v}(x)=\sum_{v\in\mathfrak{S}}n_{v}(c_{v}+\langle
u_{v},x\rangle-\vartheta_{\overline{D},v}(x))\geq n_{v_{0}}(c_{v_{0}}+\langle
u_{v_{0}},x\rangle-\vartheta_{\overline{D},v_{0}}(x))$

using the fact that $(u_{v})_{v}$ is balanced together with the previous upper
bound for the $v$-adic roof functions for $v\neq v_{0}$. Since this holds for
every $x\in S_{t}$ we deduce that $0\leq n_{v_{0}}\kappa_{v_{0}}\leq\mu-t$,
and in particular that
$0\leq\sum_{v}n_{v}\kappa_{v}\leq\\#\mathfrak{S}(\mu-t)$ where
$\\#\mathfrak{S}$ denote the cardinality of this finite set of places.

By the monotonicity of the mixed integral we have

$\operatorname{\mathrm{MI}}_{M}(\theta_{v,t},\dots,\theta_{v,t},\vartheta_{\overline{E},v})\leq\operatorname{\mathrm{MI}}_{M}((\langle
u_{v},x\rangle+c_{v}-\varepsilon_{v}t)|_{S_{t}},\dots,(\langle
u_{v},x\rangle+c_{v}-\varepsilon_{v}t)|_{S_{t}},\vartheta_{\overline{E},v}).$

By Lemma A.10, the mixed integral in the right-hand side of this inequality
can be computed as

$-d!\operatorname{\mathrm{vol}}(S_{t})\,\psi_{\overline{E},v}(u_{v})+d(c_{v}-\varepsilon_{v}t)\operatorname{\mathrm{MV}}_{M}(S_{t},\dots,S_{t},\Delta_{E})+\langle
x_{1},u_{v}\rangle$

for a point $x_{1}\in M_{\mathbb{R}}$ which does not depend on $v$, as
explained in Remark A.11. Summing over all these places, it follows from the
formulae in (9.13) that

$\frac{\langle\overline{D}(t)^{d}\cdot\overline{E}\rangle}{\operatorname{\mathrm{vol}}(R^{t}(\overline{D}))}\leq-\sum_{v}n_{v}\psi_{\overline{E},v}(u_{v})+\frac{d(\mu-t)\operatorname{\mathrm{MV}}_{M}(S_{t},\dots,S_{t},\Delta_{E})}{d!\operatorname{\mathrm{vol}}(S_{t})},$

and proceeding similarly with the lower bound in (9.14) we also obtain

$\frac{\langle\overline{D}(t)^{d}\cdot\overline{E}\rangle}{\operatorname{\mathrm{vol}}(R^{t}(\overline{D}))}\geq-\sum_{v}n_{v}\psi_{\overline{E},v}(u_{v})-\frac{d(\\#\mathfrak{S}-1)(\mu-t)\operatorname{\mathrm{MV}}_{M}(S_{t},\dots,S_{t},\Delta_{E})}{d!\operatorname{\mathrm{vol}}(S_{t})},$

Using the comparison between the inradius and the quotient of intersection
numbers in Lemma 2.1 together with the interpretation of mixed volumes as
intersection numbers of toric $\mathbb{R}$-divisors in (9.2) and the limit in
(9.12) we get

(9.15)
$\partial_{\overline{E}}\mu^{\mathrm{ess}}(\overline{D})=-\sum_{v}n_{v}\psi_{\overline{E},v}(u_{v}).$

By the additivity of both the derivative and the metric functions, this
formula readily extends to the case when $\overline{E}$ is just DSP, and by
density it extends to any toric adelic $\mathbb{R}$-divisor on $X$.

For an arbitrary adelic $\mathbb{R}$-divisor $\overline{E}$ with $E$ toric we
apply the averaging process in Definition 9.9. By the invariance of the
positive arithmetic intersection numbers with respect to this process
(Proposition 9.10) we deduce from the identities in (9.15) and (9.11) that

$\partial_{\overline{E}}\mu^{\mathrm{ess}}(\overline{D})=\partial_{{\overline{E}}\vphantom{L}^{\mathrm{tor}}}\mu^{\mathrm{ess}}(\overline{D})=\sum_{v\in\mathfrak{M}_{K}}n_{v}\,\widehat{g}_{v}(x_{v})=\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\eta_{v,u_{v}}$

where each $x_{v}\in X_{v}^{\mathrm{an}}$ is a point in the fiber
$\operatorname{\mathrm{val}}_{v}^{-1}(u_{v})$.

Finally, by Proposition 5.6 this readily implies that $\overline{D}$ satisfies
the equidistribution property with $\nu_{\overline{D},v}=\eta_{v,u_{v}}$ for
all $v$. ∎

We can extend the range of Theorem 9.8 to compute the derivative of the
essential minimum function along any adelic $\mathbb{R}$-divisor on $X$ by
reducing to the situation considered therein.

######  Remark 9.11.

Let $\overline{E}$ be an adelic $\mathbb{R}$-divisor on $X$. The $K$-algebra
$\mathcal{O}(\mathbb{T})=K[M]$ is factorial and so we can choose
$f\in\operatorname{\mathrm{Rat}}(X)^{\times}_{\mathbb{R}}$ defining the
restriction of $E$ to the torus. Set then

$\overline{F}=\overline{E}-\widehat{\operatorname{div}}(f)$

Its geometric divisor $F$ is toric because its support is contained in the
boundary $X\setminus X_{0}$ and its $v$-adic Green function is related to that
of $\overline{E}$ by $g_{\overline{F},v}=g_{\overline{E},v}+\log|f|_{v}.$

Since the essential minimum function is invariant under linear equivalence,
its derivative in the direction of $\overline{E}$ coincides with that in the
direction of $\overline{F}$. Hence we can apply Theorem 9.8 to compute it as

$\partial_{\overline{E}}\mu^{\mathrm{ess}}(\overline{D})=\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\eta_{v,u_{v}}+\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}\log|f|_{v}\,d\eta_{v,u_{v}}.$

In the semipositive case, we can prove the following converse of Theorem 9.8.
It is a direct consequence of Theorem 9.8 together with Propositions 9.4 and
6.13.

######  Corollary 9.12.

With notation as in Theorem 9.8, the following conditions are equivalent:

  1. (1)

$\vartheta_{\overline{D}}$ is wide,

  2. (2)

$\overline{D}$ satisfies the equidistribution property.

When any of them is satisfied, for each $v\in\mathfrak{M}_{K}$ we have
$\nu_{\overline{D},v}=\eta_{v,u_{v}}$.

######  Remark 9.13.

A key notion in [BPRS19] is that of monocritical semipositive toric adelic
$\mathbb{R}$-divisor. Assuming that $\overline{D}$ is semipositive, this
condition is expressed as the fact that certain associated function on a space
of adelic measures has a unique minimizing adelic point. Then [BPRS19, Theorem
1.1] states that $\overline{D}$ is monocritical if and only if it verifies the
equidistribution property at every place, with equidistribution measures
determined by this adelic point when it exists.

By Proposition 4.15 in loc.cit., the condition that $\overline{D}$ is
monocritical can be reformulated in more elementary terms as the fact that $0$
is not a vertex of $\partial\vartheta_{\overline{D}}(x_{0})$ for any
$x_{0}\in\Delta_{D,\max}$, with the minimizing adelic point given by the
balanced family of sup-gradients for $\vartheta_{\overline{D}}$. By
Proposition A.3 this condition is equivalent to the fact that the global roof
function is wide, and so Corollary 9.12 recovers this toric equidistribution
theorem. On the other hand, Theorem 9.8 extends the sufficient condition in
this criterion to the case when $\overline{D}$ is not necessarily semipositive
and strengthens its conclusion to include the differentiability of the
essential minimum function. Furthermore, it adds the even more elementary
interpretation of this condition in terms of the decay of the width sup-level
sets of the global roof function.

For the rest of this subsection we assume that $\vartheta_{\overline{D}}$ is
wide and we denote by $(u_{v})_{v}$ the associated balanced family of sup-
gradients. Then for each $f\in K[M]\setminus\\{0\\}$ we consider the quantity

$m_{\overline{D}}(f)=\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}\log|f|_{v}\,d\eta_{v,u_{v}}\in\mathbb{R}.$

Taking into account the definition of the probability measures
$\eta_{v,u_{v}}$ and writing $f=\sum_{m\in M}\alpha_{m}\chi^{m}$, this can be
alternatively expressed as

(9.16)
$m_{\overline{D}}(f)=\sum_{v\in\mathfrak{M}_{K}^{\infty}}n_{v}\int_{\mathbb{S}_{v}}\log|f(x\cdot
t_{v})|_{v}\,d\operatorname{Haar}_{{v}}(x)\\\
+\sum_{v\in\mathfrak{M}_{K}\setminus\mathfrak{M}_{K}^{\infty}}n_{v}\log\max_{m}(e^{-\langle
u_{v},m\rangle}|\alpha_{m}|_{v})$

where as before $\operatorname{Haar}_{v}$ denotes the Haar probability measure
of the real torus $\mathbb{S}_{v}$, and
$t_{v}\in\mathbb{T}_{v}^{\mathrm{an}}(\mathbb{C}_{v})$ is any point in the
fiber $\operatorname{\mathrm{val}}_{v}^{-1}(u_{v})$. Hence the defined
quantity is an extension of the classical logarithmic Gauss-Mahler measure of
a Laurent polynomial, which corresponds to the case when $u_{v}=0$ for all
$v$.

######  Lemma 9.14.

The following properties hold:

  1. (1)

for $f\in K[M]\setminus\\{0\\}$ we have $m_{\overline{D}}(f)\geq 0$,

  2. (2)

for $m\in M$ and $\alpha\in K^{\times}$ we have
$m_{\overline{D}}(\alpha\chi^{m})=0$.

  3. (3)

for $m\in M\setminus\\{0\\}$ and $\gamma\in K^{\times}$ we have

$m_{\overline{D}}(\chi^{m}-\gamma)=\sum_{v\in\mathfrak{M}_{K}}n_{v}\log\max(1,e^{-\langle
u_{v},m\rangle}|\gamma|_{v}).$

###### Proof.

The statement (1) might be deduced from Lemma 8.2 and the expression for the
derivatives of the essential minimum in Remark 9.11. It can be alternatively
deduced from standard properties of the Mahler measure, as we next explain.

Let $m_{0}$ be a vertex of the Newton polytope of $f$. Then for each
$v\in\mathfrak{M}_{K}$ we have

$\int_{X_{v}^{\mathrm{an}}}\log|f|_{v}\,d\eta_{v,u_{v}}\geq\log|\alpha_{m_{0}}|_{v}-\langle
u_{v},m_{0}\rangle.$

In case $v$ is Archimedean this follows from the fact that the Mahler measure
of a Laurent polynomial is bounded below by the absolute value of any of its
vertex coefficients. On the other hand, in case $v$ is non-Archimedean this
inequality is given by the explicit expression of this local term in (9.16).
Hence

$m_{\overline{D}}(f)\geq\sum_{v\in\mathfrak{M}_{K}}n_{v}\,(\log|\alpha_{m_{0}}|_{v}-\langle
u_{v},m_{0}\rangle)=0$

by the product formula of $K$ and the fact that $(u_{v})_{v}$ is balanced.

For (2), for each $v$ we have
$\int_{X_{v}^{\mathrm{an}}}\log|\alpha\chi^{m}|_{v}\,d\eta_{v,u_{v}}=\log|\alpha|_{v}-\langle
u_{v},m\rangle,$ and so the statement follows again from the product formula
and the fact that $(u_{v})_{v}$ is balanced.

For (3), for each $v$ we have

$\int_{X_{v}^{\mathrm{an}}}\log|\chi^{m}-\gamma|_{v}\,d\eta_{v,u_{v}}=\log\max(1,|\chi^{m}(t_{v})\gamma|_{v})=\log\max(1,e^{-\langle
u_{v},m\rangle}|\gamma|_{v})$

by Jensen’s formula for the Mahler measure in the Archimedean case, and the
explicit expression (9.16) in the non-Archimedean case. The statement follows
by considering the weighted sum of these terms. ∎

Combining Theorem 9.8 with the results in §8 we can strengthen the toric
equidistribution property to include test functions with logarithmic
singularities along effective divisors satisfying a numerical condition.

######  Theorem 9.15.

Let $\overline{D}$ be a toric adelic $\mathbb{R}$-divisor on $X$ with $D$ big
and such that $\vartheta_{\overline{D}}$ is wide. Let $E$ be an effective
divisor on $X$ whose restriction to $X_{0}$ is defined by a Laurent polynomial
$f\in K[M]\setminus\\{0\\}$ with $m_{\overline{D}}(f)=0$. Then for every
$\overline{D}$-small generic sequence $(x_{\ell})_{\ell}$ in $X(\overline{K})$
and $v\in\mathfrak{M}_{K}$ we have

$\lim_{\ell\to\infty}\int_{X_{v}^{\mathrm{an}}}\varphi\,d\delta_{O(x_{\ell})_{v}}=\int_{X_{v}^{\mathrm{an}}}\varphi\,d\eta_{v,u_{v}}$

for any function $\varphi\colon
X_{v}^{\mathrm{an}}\rightarrow\mathbb{R}\cup\\{\pm\infty\\}$ with at most
logarithmic singularities along $E$.

In particular, this holds if each irreducible component of $E$ is either
contained in the boundary $X\setminus X_{0}$, or is the closure of the zero
set of an irreducible binomial $\chi^{m}-\gamma$ with $m\in M\setminus\\{0\\}$
and $\gamma\in K^{\times}$ such that $\log|\gamma|_{v}=\langle u_{v},m\rangle$
for all $v$.

###### Proof.

By Remark 9.11 we have

$\partial_{\overline{E}}\mu^{\mathrm{ess}}(\overline{D})=\sum_{v\in\mathfrak{M}_{K}}n_{v}\int_{X_{v}^{\mathrm{an}}}g_{\overline{E},v}\,d\eta_{v,u_{v}}+m_{\overline{D}}(f).$

The first statement is then a direct application of Theorem 8.4.

For the second statement, let $[E]$ denote the associated Weil divisor and
consider its decomposition into irreducible components $[E]=\sum_{i\in
I}k_{i}W_{i}.$ For each $i\in I$ we choose a Laurent polynomial $f_{i}$
defining the restriction $W_{i}\cap X_{0}$. Then $f=\prod_{i\in
I}f_{i}^{k_{i}}$ defines the restriction of $E$ to the torus. Using Lemma 9.14
we deduce that $m_{\overline{D}}(f_{i})=0$ for each $i$ and that

$m_{\overline{D}}(f)=\sum_{i\in I}k_{i}\,m_{\overline{D}}(f_{i})=0,$

and so this statement follows from the first. ∎

It is natural to try to interpret in terms of heights the numerical condition
imposed on the effective divisor by the previous theorem. To this end, first
note that for every point $x\in X_{0}(\overline{K})$ we have
$h_{\overline{D}}(x)\geq\mu^{\mathrm{ess}}(\overline{D})$ [BPS15, Lemma
3.8(1)], and so for every subvariety $V\subset X_{0,\overline{K}}$ we have

$\mu^{\mathrm{ess}}(\overline{D}|_{\overline{V}})\geq\mu^{\mathrm{ess}}(\overline{D})$

where $\overline{V}$ denotes its closure. Following [BPRS19, Definition 5.10],
we say that $V$ is $\overline{D}$-special if this lower bound is an equality.

Using the characterization of the Bogomolov property for monocritical
semipositive toric adelic $\mathbb{R}$-divisors in [BPRS19, §5] we derive the
following logarithmic equidistribution theorem for the semipositive case.

######  Corollary 9.16.

With notation as in Theorem 9.15, assume furthermore that $\overline{D}$ is
semipositive. Let $E$ be an effective divisor on $X$ such that each of its
geometric irreducible components is either contained in $X\setminus X_{0}$ or
is the closure of a $\overline{D}$-special hypersurface of
$X_{0,\overline{K}}$. Then for every $\overline{D}$-small generic sequence
$(x_{\ell})_{\ell}$ in $X(\overline{K})$ and $v\in\mathfrak{M}_{K}$ we have

$\lim_{\ell\to\infty}\int_{X_{v}^{\mathrm{an}}}\varphi\,d\delta_{O(x_{\ell})_{v}}=\int_{X_{v}^{\mathrm{an}}}\varphi\,d\eta_{v,u_{v}}.$

for any function $\varphi\colon
X_{v}^{\mathrm{an}}\rightarrow\mathbb{R}\cup\\{\pm\infty\\}$ with at most
logarithmic singularities along $E$.

###### Proof.

Since $\overline{D}$ is semipositive and $\vartheta_{\overline{D}}$ is wide,
$\overline{D}$ is monocritical in the sense of [BPRS19], Remark 9.13. Let
$W\subset X_{\overline{K}}$ be a geometric irreducible component of $E$ such
that $W_{0}\coloneqq W\cap X_{0,\overline{K}}$ is $\overline{D}$-special. By
the Bogomolov property for monocritical adelic $\mathbb{R}$-divisors [BPRS19,
Theorem 5.12], $W_{0}$ is the translate of a subtorus. Since $W_{0}$ is a
hypersurface, this implies that there exist $m\in M$ and $x_{0}\in
X_{0,\overline{K}}(\overline{K})$ such that

$W_{0}=Z(\chi^{m}-1)\cdot x_{0}.$

After extending the base field if necessary, we assume without loss of
generality that $x_{0}\in X_{0}(K)$. Note that $W_{0}=Z(\chi^{m}-\gamma)$ for
$\gamma=\chi^{m}(x_{0})$. By [BPRS19, Proposition 5.14(1)], the fact that
$W_{0}$ is $\overline{D}$-special implies that

$u_{v}\in
m^{\bot}_{\mathbb{R}}+\operatorname{\mathrm{val}}_{v}(x_{0})\quad\text{ for
all }v,$

which is equivalent to the fact that $\langle
u_{v},m\rangle=\langle\operatorname{\mathrm{val}}_{v}(x_{0}),m\rangle=\log|\gamma|_{v}$
for all $v$. We conclude with Theorem 9.15. ∎

##  10\. Dynamical systems and semiabelian varieties

Here we study the adelic $\mathbb{R}$-divisors that arise as sums of several
canonical adelic $\mathbb{R}$-divisors with different regimes with respect to
an algebraic dynamical system. In this setting, Zhang’s lower bound for the
essential minimum might be strict, in which case Yuan’s theorem cannot be
applied. In spite of this situation, the essential minimum function is
differentiable at such adelic $\mathbb{R}$-divisors, and for every place of
$K$ the Galois orbits of the points in a small generic sequence converge
towards the normalized Monge-Ampère measure (Theorem 10.8). We also show that
this equidistribution still holds for test functions with logarithmic
singularities along hypersurfaces containing a dense subset of preperiodic
points (Theorem 10.10).

These results apply in the context of semiabelian varieties, giving the
differentiability of the essential minimum function (Theorem 10.11) and
allowing to recover Kühne’s semiabelian equidistribution theorem (Remark
10.12). Furthermore, they imply that this equidistribution also holds for test
functions with logarithmic singularities along torsion hypersurfaces (Theorem
10.15).

###  10.1. Canonical adelic $\mathbb{R}$-divisors

Zhang’s canonical metrics for dynamical systems in [Zha95b, §2] extend to the
setting of adelic $\mathbb{R}$-divisors, as shown by Chen and Moriwaki [CM15].
Here we recall this notion and study some of its basic positivity properties.

Let $X$ be a normal projective variety over $K$ of dimension $d\geq 1$ and
$\phi\colon X\rightarrow X$ a surjective endomorphism. Let
$D\in{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$ such that $\phi^{*}D\equiv
qD$ for a real number $q>1$. To define the associated canonical adelic
$\mathbb{R}$-divisor, choose
$f\in\operatorname{\mathrm{Rat}}(X)_{\mathbb{R}}^{\times}$ such that

$\phi^{*}D=qD+\operatorname{div}(f).$

By [CM15, §4], for every $v\in\mathfrak{M}_{K}$ there is a unique $v$-adic
Green function $g_{v}$ on $D$ such that
${\phi_{v}^{\mathrm{an},*}g_{v}=q\,g_{v}-\log|f|_{v}},$ and moreover
$\overline{D}\coloneqq(D,(g_{v})_{v\in\mathfrak{M}_{K}})$ is an adelic
$\mathbb{R}$-divisor on $X$ which verifies that

(10.1) $\phi^{*}\overline{D}=q\overline{D}+\widehat{\operatorname{div}}(f).$

This adelic $\mathbb{R}$-divisor is independent of the choice of $f$ up to a
summand of the form $\widehat{\operatorname{div}}(\lambda)$ with $\lambda\in
K^{\times}_{\mathbb{R}}$. Indeed, if
$f^{\prime}\in\operatorname{\mathrm{Rat}}(X)_{\mathbb{R}}$ also verifies that
$\phi^{*}D=q\,D+\operatorname{div}(f^{\prime})$ then necessarily
$f^{\prime}=\gamma f$ with $\gamma\in K^{\times}_{\mathbb{R}}$. Hence for each
$v\in\mathfrak{M}_{K}$ the $v$-adic Green function induced by $f^{\prime}$
coincides with $g_{v}+\log|\lambda|_{v}$ for $\lambda=\gamma^{1/(q-1)}$ and
the adelic $\mathbb{R}$-divisor induced by $f^{\prime}$ is
$\overline{D}+\widehat{\operatorname{div}}(\lambda)$.

######  Definition 10.1.

The _canonical_ adelic $\mathbb{R}$-divisor of $D$, denoted by
${\overline{D}}\vphantom{L}^{\operatorname{can}}$, is any adelic
$\mathbb{R}$-divisor on $X$ such that

(10.2) $\phi^{*}{\overline{D}}\vphantom{L}^{\operatorname{can}}\equiv
q{\overline{D}}\vphantom{L}^{\operatorname{can}}\quad\text{ on
}\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}.$

It is unique up to summand of the form $\widehat{\operatorname{div}}(\lambda)$
with $\lambda\in K^{\times}_{\mathbb{R}}$.

The associated height function is not affected by this indeterminacy, thanks
to the product formula. By (10.2), it verifies that

(10.3)
$h_{{\overline{D}}\vphantom{L}^{\operatorname{can}}}(\phi(x))=q\,h_{{\overline{D}}\vphantom{L}^{\operatorname{can}}}(x)\quad\text{
for }x\in X(\overline{K}).$

A point $x\in X(\overline{K})$ is _preperiodic_ if its orbit with respect to
$\phi$ is finite or equivalently, if there are positive integers $j<k$ such
that $\phi^{\circ j}(x)=\phi^{\circ k}(x)$. The functoriality in (10.3)
implies that $h_{{\overline{D}}\vphantom{L}^{\operatorname{can}}}(x)=0$
whenever $x$ is preperiodic.

It is well-known that if $D$ is ample then
${\overline{D}}\vphantom{L}^{\operatorname{can}}$ is nef and both the absolute
and the essential minima vanish. In a similar vein, we give a simple condition
ensuring the pseudo-effectivity of the canonical adelic $\mathbb{R}$-divisor
and the vanishing of its essential minimum.

######  Proposition 10.2.

If $R(D)\neq\\{0\\}$, then ${\overline{D}}\vphantom{L}^{\operatorname{can}}$
is pseudo-effective and
$\mu^{\mathrm{ess}}({\overline{D}}\vphantom{L}^{\operatorname{can}})=0$.

###### Proof.

We prove that the essential minimum vanishes. First note that it is finite
because $R(D)\neq\\{0\\}$. We have
$\mu^{\mathrm{ess}}({\overline{D}}\vphantom{L}^{\operatorname{can}})=\mu^{\mathrm{ess}}(\phi^{*}{\overline{D}}\vphantom{L}^{\operatorname{can}})$
because the essential minimum is invariant with respect to dominant and
generically finite morphisms [BPS15, Proposition 3.5(1)]. Furthermore

$\mu^{\mathrm{ess}}(\phi^{*}{\overline{D}}\vphantom{L}^{\operatorname{can}})=\mu^{\mathrm{ess}}(q{\overline{D}}\vphantom{L}^{\operatorname{can}})=q\,\mu^{\mathrm{ess}}({\overline{D}}\vphantom{L}^{\operatorname{can}}),$

by the functoriality of the canonical adelic $\mathbb{R}$-divisor and the
invariance of the essential minimum under linear equivalence. Hence
$\mu^{\mathrm{ess}}({\overline{D}}\vphantom{L}^{\operatorname{can}})=q\,\mu^{\mathrm{ess}}({\overline{D}}\vphantom{L}^{\operatorname{can}})$,
which gives
$\mu^{\mathrm{ess}}({\overline{D}}\vphantom{L}^{\operatorname{can}})=0$ as
stated.

The pseudo-effectivity of ${\overline{D}}\vphantom{L}^{\operatorname{can}}$
then follows from Theorem 3.16. For the convenience of the reader, we give an
elementary proof of this fact. Let $s=(f,eD)$ be a nonzero global section of
$eD$ for some $e\geq 1$, which exists because $R(D)\neq\\{0\\}$. Up to
multiplying $s$ by a nonzero scalar, we can suppose that $\|s\|_{v,\sup}\leq
1$ for every non-Archimedean place $v$. Then given $\varepsilon>0$ take $k\geq
1$ such that $\log\|s\|_{v,\sup}\leq\varepsilon\,e\,q^{k}$ for every
Archimedean place $v$.

The pullback $\phi^{\circ k,*}s=(\phi^{\circ k,*}f,e\,\phi^{\circ k,*}D)$ is a
nonzero global section of $e\,\phi^{\circ k,*}D$ and since $\phi$ is
surjective, it has the same $v$-adic sup-norms as $s$. Since
$\phi^{*}{\overline{D}}\vphantom{L}^{\operatorname{can}}\equiv
q\,{\overline{D}}\vphantom{L}^{\operatorname{can}}$, there is a nonzero global
section $s_{k}$ of $eq^{k}D$ with the same $v$-adic sup-norms. Hence

$\log\|s_{k}\|_{eq^{k}{\overline{D}}\vphantom{L}^{\operatorname{can}},v,\sup}=\log\|s\|_{{\overline{D}}\vphantom{L}^{\operatorname{can}},v,\sup}\leq\begin{cases}\varepsilon\,e\,q^{k}&\text{
if }v\text{ is Archimedean,}\\\ 0&\text{ otherwise,}\end{cases}$

and so $s_{k}\in
R^{-\varepsilon}({\overline{D}}\vphantom{L}^{\operatorname{can}})$. Therefore
$R^{-\varepsilon}({\overline{D}}\vphantom{L}^{\operatorname{can}})\neq\\{0\\}$
for every $\varepsilon>0$ and
${\overline{D}}\vphantom{L}^{\operatorname{can}}$ is pseudo-effective. ∎

We also have the next condition for its effectivity.

######  Proposition 10.3.

If $D$ is effective and $\phi^{*}D=qD$, then
${\overline{D}}\vphantom{L}^{\operatorname{can}}$ is effective.

###### Proof.

Since $D$ is effective, its canonical Green functions are bounded below.
Furthermore, by the construction in (10.1) the equality $\phi^{*}D=qD$ lifts
to an equality of adelic $\mathbb{R}$-divisors
$\phi^{*}{\overline{D}}\vphantom{L}^{\operatorname{can}}=q{\overline{D}}\vphantom{L}^{\operatorname{can}}$.
Hence for each $v\in\mathfrak{M}_{K}$ we have
$\phi^{*}g_{{\overline{D}}\vphantom{L}^{\operatorname{can}},v}=q\,g_{{\overline{D}}\vphantom{L}^{\operatorname{can}},v}$
and so

$\inf_{x\in
X_{v}^{\mathrm{an}}}g_{{\overline{D}}\vphantom{L}^{\operatorname{can}}}(x)=q\inf_{x\in
X_{v}^{\mathrm{an}}}g_{{\overline{D}}\vphantom{L}^{\operatorname{can}}}(x)$

because $\phi$ is surjective. Hence this infimum is zero and so
${\overline{D}}\vphantom{L}^{\operatorname{can}}$ is effective. ∎

###  10.2. Equidistribution for sums of canonical adelic $\mathbb{R}$-divisors

Let $\phi\colon X\to X$ be a surjective endormorphism of a normal variety over
$K$ as in §10.1. This endomorphism is finite [Fak03, Lemma 5.6] and we denote
by $\deg(\phi)$ its degree.

For $i=1,\dots,s$ let $D_{i}\in\operatorname{\mathrm{Div}}(X)_{\mathbb{R}}$
with $\phi^{*}D_{i}\equiv q_{i}D_{i}$ for a real number $q_{i}>1$ such that
$R(D_{i})\neq\\{0\\}$ and
${\overline{D}}\vphantom{L}^{\operatorname{can}}_{i}$ is semipositive. Set

(10.4)
$\overline{D}=\sum_{i=1}^{s}{\overline{D}}\vphantom{L}^{\operatorname{can}}_{i},$

and suppose furthermore that $D$ is ample. Up to reordering, we can also
suppose that $1<q_{1}\leq q_{2}\leq\cdots\leq q_{s}$.

Since ${\overline{D}}\vphantom{L}^{\operatorname{can}}_{i}$ is semipositive,
we have that $D_{i}$ is nef for each $i$ and so

$\phi^{*}{D}-{D}=\sum_{i=1}^{s}(q_{i}-1)D_{i}=(q_{1}-1)D+\sum_{i=1}^{s}(q_{i}-q_{1})D_{i}$

is ample, because it is the sum of an ample $\mathbb{R}$-divisor with a nef
one. By Fakhruddin’s theorem [Fak03, Theorem 5.1], this implies that the set
of periodic points of $\phi$ is dense.

Next we study the basic properties of this adelic $\mathbb{R}$-divisor,
starting with its its essential minimum, intersection numbers and Monge-Ampère
measures.

######  Proposition 10.4.

We have $\mu^{\mathrm{ess}}(\overline{D})=0$.

###### Proof.

For every preperiodic point $x$ we have
$h_{\overline{D}}(x)=\sum_{i=1}^{s}h_{{\overline{D}}\vphantom{L}^{\operatorname{can}}_{i}}(x)=0.$
Since the set of such points is dense, we get
$\mu^{\mathrm{ess}}(\overline{D})\leq 0.$ On the other hand, by Proposition
10.2 we have
$\mu^{\mathrm{ess}}({\overline{D}}\vphantom{L}^{\operatorname{can}}_{i})=0$
for each $i$, and so Lemma 3.15(1) gives the reverse inequality
$\mu^{\mathrm{ess}}(\overline{D})\geq\sum_{i=1}^{s}\mu^{\mathrm{ess}}({\overline{D}}\vphantom{L}^{\operatorname{can}}_{i})=0$.
∎

Denote by $\mathbb{N}^{s}_{d}\subset\mathbb{N}^{s}$ the set of $s$-tuples of
nonnegative integers whose components sum up to $d$. For each
$a\in\mathbb{N}^{s}_{d}$ set $q^{a}=\prod_{i=1}^{s}q_{i}^{a_{i}}$ and consider
the subset

(10.5) $I=\\{a\in\mathbb{N}^{s}_{d}\mid q^{a}=\deg(\phi)\\}.$

######  Proposition 10.5.

We have

$(D^{d})={\sum_{a\in I}{d\choose
a}\Big{(}\prod_{i=1}^{s}{D}_{i}^{a_{i}}\Big{)}}>0c_{1}(\overline{D}_{v})^{\wedge
d}=\sum_{a\in I}{d\choose
a}\bigwedge_{i=1}^{s}c_{1}({\overline{D}}\vphantom{L}^{\operatorname{can}}_{i,v})^{\wedge
a_{i}}.$

Hence if $(\prod_{i=1}^{s}{D}_{i}^{a_{i}})>0$ for some
$a\in\mathbb{N}^{s}_{d}$ then $a\in I$. In particular $I\neq\emptyset$.

###### Proof.

We have $\deg(\phi)^{k}(D^{d})=((\phi^{\circ
k,*}D)^{d})=((\sum_{i=1}^{s}q_{i}^{k}D_{i})^{d})$ by the projection formula
[Ful98, Proposition 2.3(c)]. Therefore

$(D^{d})=\sum_{a\in\mathbb{N}_{d}^{s}}{d\choose
a}\frac{q^{ka}}{\deg(\phi)^{k}}\Big{(}\prod_{i=1}^{s}{D}_{i}^{a_{i}}\Big{)}.$

Since the right-hand side is constant with respect to $k$, this has to be the
case for the left-hand side. Hence the terms corresponding to exponents
outside $I$ vanish, which gives the first formula. Its positivity comes from
the fact that $D$ is ample.

On the other hand, the multilinearity of the Monge-Ampère operator gives

$c_{1}(\overline{D}_{v})^{\wedge d}=\sum_{a\in\mathbb{N}_{d}^{s}}{d\choose
a}\bigwedge_{i=1}^{s}c_{1}({\overline{D}}\vphantom{L}^{\operatorname{can}}_{i,v})^{\wedge
a_{i}}.$

Since $\overline{D}_{i}$ is semipositive, for each $a\in\mathbb{N}^{s}_{d}$ we
have that
$\bigwedge_{i=1}^{s}c_{1}({\overline{D}}\vphantom{L}^{\operatorname{can}}_{i,v})^{\wedge
a_{i}}$ is a measure of total mass $(\prod_{i=1}^{s}{D}_{i}^{a_{i}})$. Hence
this measure is zero unless $a\in I$, which gives the second formula.

The last statement comes readily from the first formula and its positivity. ∎

The next proposition summarizes the properties of the pullback with respect to
$\phi$ of this adelic $\mathbb{R}$-divisor.

######  Proposition 10.6.

We have

  1. (1)

$r(\phi^{*}D;D)\geq q_{1}$,

  2. (2)

$\mu^{\mathrm{abs}}(\phi^{*}\overline{D})={\mu^{\mathrm{abs}}(\overline{D})}$,

  3. (3)

$(\phi^{*}D^{d})=\deg(\phi)\times(D^{d})$,

  4. (4)

$c_{1}(\phi^{*}\overline{D}_{v})^{\wedge
d}=\deg(\phi)\,c_{1}(\overline{D}_{v})^{\wedge d}$.

###### Proof.

For (1) we have $R(\phi^{*}D-q_{1}D)\neq\\{0\\}$ because
$\phi^{*}{D}-q_{1}D_{1}=\sum_{i=1}^{s}(q_{i}-q_{1}){D}_{i}$ and
$R(D_{i})\neq\\{0\\}$ for all $i$. This gives $r(\phi^{*}D;D)\geq q_{1}$, as
stated.

The other statements hold in greater generality: (2) is the invariance of the
absolute minimum with respect to pullbacks by surjective morphisms, whereas
(3) and (4) follow from the functoriality of the intersection numbers and of
the Monge-Ampère operator, see for instance [BPS14, Proposition 1.4.8]. ∎

######  Remark 10.7.

Let $v\in\mathfrak{M}_{K}$. As usual we can consider the pushforward of a
measure on $X_{v}^{\mathrm{an}}$ with respect to the analytification
$\phi_{v}^{\mathrm{an}}$, and since $\phi$ is finite we can also consider its
pullback, as explained in [Cha06, §2.8]. In our present situation, the
$v$-adic Monge-Ampère measure of $\overline{D}$ is invariant with respect to
the dynamical system, in the sense that

$\phi_{v,*}^{\mathrm{an}}c_{1}(\overline{D}_{v})^{\wedge
d}=c_{1}(\overline{D}_{v})^{\wedge
d}\phi_{v}^{\mathrm{an},*}c_{1}(\overline{D}_{v})^{\wedge
d}=\deg(\phi)\,c_{1}(\overline{D}_{v})^{\wedge d}.$

This follows readily from Proposition 10.6(4) and the behavior of Monge-Ampère
measures with respect to these operations, see for instance [Cha06, §2.8] for
the non-Archimedean case.

We now state the central theorem of this section, which gives the
differentiability of the essential minimum at the adelic $\mathbb{R}$-divisor
in (10.4). Recall that $I$ denote the index subset in (10.5).

######  Theorem 10.8.

The essential minimum function is differentiable at $\overline{D}$, and for
each $\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$ we
have

$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})=\frac{1}{(D^{d})}\sum_{a\in
I}{d\choose
a}\Big{(}\overline{E}\cdot\prod_{i=1}^{s}({\overline{D}}\vphantom{L}^{\operatorname{can}}_{i})^{a_{i}}\Big{)}.$

In particular, $\overline{D}$ satisfies the $v$-adic equidistribution property
at every $v\in\mathfrak{M}_{K}$ with
$\nu_{\overline{D},v}={c_{1}(\overline{D}_{v})^{\wedge d}}/{(D^{d})}$.

The next lemma gives the specific sequence of semipositive approximations that
we will use and its properties.

######  Lemma 10.9.

For $n\in\mathbb{N}$ set
$\overline{Q}_{n}=\sum_{i=1}^{s}(q_{i}/q_{s})^{n}\,{\overline{D}}\vphantom{L}^{\operatorname{can}}_{i}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$.
Then $Q_{n}$ is ample, $\overline{Q}_{n}$ is semipositive and
$R(D-Q_{n})\neq\\{0\\}$. In particular, $\overline{Q}_{n}$ is a semipositive
approximation of $\overline{D}$. Furthermore

  1. (1)

$r(Q_{n};D)\geq{q_{s}^{-n}}{q_{1}^{n}}$,

  2. (2)

$\mu^{\mathrm{abs}}(\overline{Q}_{n})=q_{s}^{-n}{\mu^{\mathrm{abs}}(\overline{D})}$,

  3. (3)

$(Q_{n}^{d})=q_{s}^{-dn}\deg(\phi)^{n}\times(D^{d})$,

  4. (4)

$c_{1}(\overline{Q}_{n,v})^{\wedge
d}=q_{s}^{-dn}\deg(\phi)^{n}\,c_{1}({\overline{D}}\vphantom{L}^{\operatorname{can}}_{v})^{\wedge
d}$ for every $v\in\mathfrak{M}_{K}$.

In particular, the normalized $v$-adic Monge-Ampère measures of
$\overline{Q}_{n}$ and ${\overline{D}}\vphantom{L}^{\operatorname{can}}$
coincide.

###### Proof.

We have that $Q_{n}-({q_{1}}/{q_{s}})^{n}D$ is a nonnegative linear
combination of the $D_{i}$’s. Since $D$ is ample and the $D_{i}$’s are nef,
this implies that $Q_{n}$ is ample. Clearly $\overline{Q}_{n}$ is semipositive
because it is a nonnegative linear combination of semipositive adelic
$\mathbb{R}$-divisors. Moreover $D-Q_{n}$ is a nonnegative linear combination
of the $D_{i}$’s, thus $R(D-Q_{n})\neq\\{0\\}$ since $R(D_{i})\neq\\{0\\}$ for
all $i$.

The other statements follow by iteratively applying Proposition 10.6, noting
that $\overline{Q}_{n}={q_{s}^{-n}}\,\phi^{\circ n,*}\overline{D}.$ ∎

###### Proof of Theorem 10.8.

For each $n$ let $\overline{Q}_{n}$ be the semipositive approximation of
$\overline{D}$ given by Lemma 10.9. By this result and Proposition 10.4 we get

$0\leq\frac{\mu^{\mathrm{ess}}(\overline{D})-\mu^{\mathrm{abs}}(\overline{Q}_{n})}{r(Q_{n};D)}\leq\frac{-q_{s}^{-n}\mu^{\mathrm{abs}}(\overline{D})}{q_{s}^{-n}q_{1}^{n}}=\frac{-\mu^{\mathrm{abs}}(\overline{D})}{q_{1}^{n}}.$

We have $\mu^{\mathrm{abs}}(\overline{D})>-\infty$ because $D$ is ample, and
so the above ratio converges to $0$ as $n\to\infty$. Then Theorem 6.1 gives
that the essential minimum function is differentiable at $\overline{D}$, and
that for
$\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$ the
corresponding derivative is

(10.6)
$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})=\lim_{n\rightarrow\infty}\frac{(\overline{Q}_{n}^{d}\cdot\overline{E})}{(Q_{n}^{d})}.$

By Lemma 10.9(3) we have $(Q_{n}^{d})=q_{s}^{-dn}\deg(\phi)^{n}\times(D^{d})$
and so

$\frac{(\overline{Q}_{n}^{d}\cdot\overline{E})}{(Q_{n}^{d})}=\frac{1}{(D^{d})}\sum_{a\in\mathbb{N}_{d}^{s}}{d\choose
a}\Big{(}\frac{q^{a}}{\deg(\phi)}\Big{)}^{n}\times\Big{(}\overline{E}\cdot\prod_{i=1}^{s}({\overline{D}}\vphantom{L}^{\operatorname{can}}_{i})^{a_{i}}\Big{)}.$

We obtain the formula for the derivative letting $n\to\infty$, thanks to the
existence of the limit in (10.6). The formula for the equidistribution
measures follows then from their expression in terms of derivatives of the
essential minimum function (Proposition 5.6) together with the formula in
(3.5) and Proposition 10.5. ∎

We also have the following logarithmic equidistribution result for
$\overline{D}$. Recall that a subvariety $Y\subset X$ is _preperiodic_ if
there are two positive integers $j<k$ such that $\phi^{\circ j}(Y)=\phi^{\circ
k}(Y)$.

######  Theorem 10.10.

Let $(x_{\ell})_{\ell}$ be a $\overline{D}$-small generic sequence in
$X(\overline{K})$ and $Y\subset X$ a hypersurface such that $Y(\overline{K})$
contains a dense subset of preperiodic points. Then for every
$v\in\mathfrak{M}_{K}$ and any function ${\varphi\colon
X_{v}^{\mathrm{an}}\rightarrow\mathbb{R}\cup\\{\pm\infty\\}}$ with at most
logarithmic singularities along $Y$ we have

$\lim_{\ell\to\infty}\int_{X_{v}^{\mathrm{an}}}\varphi\,d\delta_{O(x_{\ell})_{v}}=\int_{X_{v}^{\mathrm{an}}}\varphi\,\frac{c_{1}(\overline{D}_{v})^{\wedge
d}}{(D^{d})}.$

In particular, this holds when $Y$ is preperiodic.

###### Proof.

Consider again the sequence $(\overline{Q}_{n})_{n}$ of semipositive
approximations of $\overline{D}$ given by Lemma 10.9. By this result,

(10.7)
$\lim_{n\to\infty}\frac{\mu^{\mathrm{abs}}(\overline{Q}_{n})}{r(Q_{n};D)}=0\frac{c_{1}(\overline{Q}_{n,v})^{\wedge
d}}{(Q_{n}^{d})}=\frac{c_{1}(\overline{D}_{v})^{\wedge d}}{(D^{d})}.$

Up to switching to linearly equivalent divisors, we can suppose that $Y$ is
not contained in the support of any of the $D_{i}$’s, and so we can consider
the restriction of $\overline{Q}_{n}$’s to $Y$. Since the set of preperiodic
points of $Y(\overline{K})$ is dense we have
$\mu^{\mathrm{ess}}(\overline{Q}_{n}|_{Y})=0$, and by Zhang’s inequality
(Theorem 3.17) we get

$h_{\overline{Q}_{n}}(Y)\leq
d\,\mu^{\mathrm{ess}}(\overline{Q}_{n}|_{Y})\times(Q_{n}^{d-1}\cdot Y)=0.$

On the other hand, let $A$ be an ample divisor on $X$ such that $A-Y$ is
pseudo-effective. Since $Q_{n}$ is nef we have $(Q_{n}^{d-1}\cdot
Y)\leq(Q_{n}^{d-1}\cdot A)$, by the inequality in (2.1). Combining this with
Lemmas 3.14 and 2.2 we deduce that

${h_{\overline{Q}_{n}}(Y)}\geq
d\,\mu^{\mathrm{abs}}(\overline{Q}_{n})\times(Q_{n}^{d-1}\cdot A)\geq
d\,\frac{\mu^{\mathrm{abs}}(\overline{Q}_{n})}{r(Q_{n};A)}\times{(Q_{n}^{d})}.$

These inequalities with the limit in (10.7) imply that
$\lim_{n\to\infty}h_{\overline{Q}_{n}}(Y)/(Q_{n}^{d})=0$. The result then
follows from Theorem 8.4 and Remark 8.5.

When $Y$ is preperiodic, there is $j>0$ such that
$Y^{\prime}\coloneqq\phi^{\circ j}(Y)$ is periodic with period $k_{0}>0$.
Hence the iteration $\phi^{\circ k_{0}}$ induces a dynamical system on
$Y^{\prime}$. Up to linear equivalence we can restrict of $D$ to $Y^{\prime}$,
and ${\phi^{\circ k_{0},*}{D}|_{Y^{\prime}}-{D}|_{Y^{\prime}}}$ is ample.
Hence the set of periodic point of $Y^{\prime}(\overline{K})$ is dense [Fak03,
Theorem 5.1] and so is that of preperiodic points of $Y(\overline{K})$. ∎

###  10.3. Equidistribution on semiabelian varieties

In this subsection we specialize Theorems 10.8 and 10.10 in the semiabelian
setting. We first recall the basic constructions and properties that are
needed to this end, referring to [Cha00, Küh22] for the proofs and more
details.

Let $G$ be a semiabelian variety over $K$ that is the extension of an abelian
variety $A$ of dimension $g$ by a split torus $\mathbb{G}_{\mathrm{m}}^{r}$,
so that there is an exact sequence of commutative algebraic groups over $K$

$0\longrightarrow\mathbb{G}_{\mathrm{m}}^{r}\longrightarrow G\longrightarrow
A\longrightarrow 0.$

We consider the compactification $\overline{G}$ of $G$ induced by toric
compactification $(\mathbb{P}^{1})^{r}$ of $\mathbb{G}_{\mathrm{m}}^{r}$. To
construct it, one endows the product variety $G\times(\mathbb{P}^{1})^{r}$
with the action of this split torus defined at the level of points by

$t\cdot(x,y)=(t\cdot_{G}x,t^{-1}\cdot_{(\mathbb{P}^{1})^{r}}y)$

and defines $\overline{G}$ as the categorical quotient
$G\times(\mathbb{P}^{1})^{r}/\mathbb{G}_{\mathrm{m}}^{r}$. It is a smooth
variety over $K$ containing $G$ as a dense open subset, and the projection
$G\to A$ extends to a morphism $\pi\colon\overline{G}\rightarrow A$ allowing
to consider this compactification as a $(\mathbb{P}^{1})^{r}$-bundle over $A$.

For a given integer $\ell>1$ the multiplication-by-$\ell$ on $G$ extends to a
morphism $[\ell]_{\overline{G}}\colon\overline{G}\rightarrow\overline{G}$ of
degree $\ell^{r+2g}$. If we denote by $[\ell]_{A}$ the multiplication-
by-$\ell$ on $A$, then there is a commutative diagram

(10.8)
$\textstyle{\overline{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\ell]_{\overline{G}}}$$\scriptstyle{\pi}$$\textstyle{\overline{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\ell]_{A}}$$\textstyle{A}$

The boundary $\overline{G}\setminus G$ is an effective Weil divisor, and we
denote by $M$ its associated (Cartier) divisor on $\overline{G}$. It is
relatively ample with respect to $\pi$ and verifies that

(10.9) $[\ell]_{\overline{G}}^{*}M=\ell M\quad\text{ on
}{\operatorname{\mathrm{Div}}}(\overline{G}).$

Let $N$ be an ample symmetric divisor on $A$, which therefore verifies that
$[\ell]_{A}^{*}N\equiv\ell^{2}N$ on $\operatorname{\mathrm{Div}}(A)$. Then its
pullback $\pi^{*}N$ is semiample, and by (10.8) it verifies

$[\ell]_{\overline{G}}^{*}\pi^{*}N\equiv\ell^{2}\pi^{*}N\quad\text{ on
}{\operatorname{\mathrm{Div}}}(\overline{G}).$

Furthermore, the sum $D=M+\pi^{*}N$ is an ample divisor on $\overline{G}$.

Let
${\overline{M}}\vphantom{L}^{\operatorname{can}}\in\widehat{\operatorname{\mathrm{Div}}}(\overline{G})$
and
${\overline{N}}\vphantom{L}^{\operatorname{can}}\in\widehat{\operatorname{\mathrm{Div}}}(A)$
be the canonical adelic divisors of $M$ and $N$ for the dynamical systems
$[\ell]_{\overline{G}}$ and $[\ell]_{A}$, respectively. By [Cha00, Proposition
3.4] the adelic divisor ${\overline{M}}\vphantom{L}^{\operatorname{can}}$ does
not depend of the choice of $\ell$, and the same holds for
${\overline{N}}\vphantom{L}^{\operatorname{can}}$. By the commutativity in
(10.8) we have that

(10.10)
$[\ell]_{\overline{G}}^{*}{\overline{M}}\vphantom{L}^{\operatorname{can}}=\ell{\overline{M}}\vphantom{L}^{\operatorname{can}}[\ell]_{\overline{G}}^{*}\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}}\equiv\ell^{2}\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}}\quad\text{
on }\widehat{\operatorname{\mathrm{Div}}}(\overline{G}).$

We have that ${\overline{M}}\vphantom{L}^{\operatorname{can}}$ is
semipositive, as shown by Chambert-Loir in [Cha00, Proposition 3.6] relying on
some specific regular models of abelian varieties constructed by Künnemann.
Furthermore, Proposition 10.3 together with the equality in (10.9) implies
that this adelic divisor is effective. On the other hand
${\overline{N}}\vphantom{L}^{\operatorname{can}}$ is nef because $N$ is ample
and so this is also the case for
$\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}}$.

Finally set
$\overline{D}={\overline{M}}\vphantom{L}^{\operatorname{can}}+\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}}\in\widehat{\operatorname{\mathrm{Div}}}(\overline{G})$.
By (10.10), its height function verifies that

$h_{\overline{D}}([\ell]_{\overline{G}}x)=\ell\,h_{{\overline{M}}\vphantom{L}^{\operatorname{can}}}(x)+\ell^{2}\,h_{{\overline{N}}\vphantom{L}^{\operatorname{can}}}(\pi(x))\quad\text{
for }x\in\overline{G}(\overline{K}).$

It is nonnegative on $G(\overline{K})$ and vanishes on the torsion points, and
so ${\mu^{\mathrm{ess}}(\overline{D})=0}$. On the other hand, this height
function might take negative values at the points in the boundary
$\overline{G}\setminus G$ [Cha00, Corollaire 4.6]. In these cases we have
$\mu^{\mathrm{abs}}(\overline{D})<0$ and so $\overline{D}$ is outside of the
scope of Yuan’s theorem.

The next result is a direct application of Theorem 10.8.

######  Theorem 10.11.

The essential minimum function is differentiable at $\overline{D}$ with

$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})=\frac{(({\overline{M}}\vphantom{L}^{\operatorname{can}})^{r}\cdot(\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}})^{g}\cdot\overline{E})}{(M^{r}\cdot\pi^{*}N^{g})}\quad\text{
for
}\overline{E}\in\widehat{\operatorname{\mathrm{Div}}}(\overline{G})_{\mathbb{R}}.$

###### Proof.

We check that the conditions in Theorem 10.8 are met in our current situation.
First $R(M)\neq\\{0\\}$ because $M$ is effective, and
${R(\pi^{*}N)\neq\\{0\\}}$ because $\pi^{*}N$ is semiample. As explained, both
${\overline{M}}\vphantom{L}^{\operatorname{can}}$ and
$\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}}$ are semipositive and
$D$ is ample. Then this theorem gives the stated differentiability for the
essential minimum function.

To apply the formula therein for the derivative
$\partial_{\overline{E}}\,\mu^{\mathrm{ess}}(\overline{D})$ we need to
determine the elements $a\in\mathbb{N}^{2}_{r+g}$ which verify that
$\ell^{a_{1}+2a_{2}}=\deg([\ell]_{\overline{G}})=\ell^{r+2g}.$ The only one is
$a=(r,g)$, and so this formula boils down to

$\frac{{r+g\choose
r}(({\overline{M}}\vphantom{L}^{\operatorname{can}})^{r}\cdot(\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}})^{g}\cdot\overline{E})}{{r+g\choose
r}(M^{r}\cdot\pi^{*}N^{g})}=\frac{(({\overline{M}}\vphantom{L}^{\operatorname{can}})^{r}\cdot(\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}})^{g}\cdot\overline{E})}{(M^{r}\cdot\pi^{*}N^{g})}.$

∎

######  Remark 10.12.

This result gives Theorem 1.6 in the introduction, as it implies that
$\overline{D}$ satisfies the $v$-adic equidistribution property at every
$v\in\mathfrak{M}_{K}$ with

(10.11)
$\nu_{\overline{D},v}=\frac{c_{1}({\overline{M}}\vphantom{L}^{\operatorname{can}}_{v})^{\wedge
r}\wedge
c_{1}(\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}}_{v})^{\wedge
g}}{({M}^{r}\cdot\pi^{*}{N}^{g})}=\frac{c_{1}(\overline{D}_{v})^{\wedge
r+g}}{(D^{r+g})}.$

Using Theorem 10.11, the first formula for this equidistribution measure
follows from its expression in terms of derivatives of the essential minimum
function (Proposition 5.6), whereas the second formula follows from the first
together with Proposition 10.5. Theorem 10.11 together with Proposition 5.3
also implies the height convergence property in (1.6).

######  Remark 10.13.

When $v$ is Archimedean, the equidistribution measure in (10.11) coincides
with the Haar probability measure on the maximal compact subgroup
$\mathbb{S}_{v}\simeq(S^{1})^{r+2g}$ of $G_{v}^{\mathrm{an}}$, see for
instance [Küh22, Lemma 5.2].

When $v$ is non-Archimedean, the description of this measure seems more
complicated. For abelian varieties, they were described by Gubler in terms of
convex geometry [Gub10] but the extension to the semiabelian case is still
pending.

For completeness, we extend this equidistribution result to the closure of a
subvariety of $G$ with vanishing essential minimum. By the semiabelian
Bogomolov conjecture, proved by David and Philippon [DP00], these subvarieties
are translates of semiabelian subvarieties by torsion points, and so they do
not provide examples of equidistribution phenomena beyond those already
obtained. But as explained in the introduction, this extension is the
centerpiece of Kühne’s approach to this conjecture [Küh22, Proposition 4.1]
and it is worth showing that it can also be derived from our results.

Let $Y\subset\overline{G}$ be the closure of a subvariety of $G$, and set
$e=\dim(Y)$ and $e^{\prime}=\dim(\pi(Y))$. Then $Y$ is not contained in the
support of $M$, and after possibly replacing the divisor
$N\in\operatorname{\mathrm{Div}}(A)$ by a linearly equivalent one, we assume
without loss of generality that $Y$ is neither contained in the support of
$\pi^{*}N$. Hence we can consider the restriction $\overline{D}|_{Y}$, defined
as the pullback of $\overline{D}$ with respect to the inclusion morphism
$\iota\colon Y\hookrightarrow\overline{G}$.

######  Proposition 10.14.

With notation as above, assume that $\mu^{\mathrm{ess}}(\overline{D}|_{Y})=0$.
Then $\overline{D}|_{Y}$ satisfies the $v$-adic equidistribution property for
every $v\in\mathfrak{M}_{K}$ with

$\nu_{\overline{D}|_{Y},v}=\frac{c_{1}({\overline{M}}\vphantom{L}^{\operatorname{can}}_{v})^{\wedge
e-e^{\prime}}\wedge
c_{1}(\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}}_{v})^{\wedge
e^{\prime}}\wedge\delta_{Y_{v}^{\mathrm{an}}}}{(M^{e-e^{\prime}}\cdot\pi^{*}{N}^{e^{\prime}}\cdot
Y)}.$

###### Proof.

For each ${n\in\mathbb{N}}$ consider the semipositive approximation
$\overline{Q}_{n}=\ell^{-n}{\overline{M}}\vphantom{L}^{\operatorname{can}}+\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}}$
of $\overline{D}$ given by Lemma 10.9 in the semiabelian setting. In our
current situation we have that $Q_{n}$ is ample, $\overline{Q}_{n}$ is
semipositive and $\overline{D}-\overline{Q}_{n}$ is effective. Let
$\widetilde{Y}$ be the normalization of the subvariety $Y$, and denote by
$\overline{D}|_{\widetilde{Y}}$ and $\overline{Q}_{n}|_{\widetilde{Y}}$ the
adelic $\mathbb{R}$-divisors on $\widetilde{Y}$ obtained by pullback with
respect to the normalization morphism. Since this morphism is birational and
$Y$ is not contained in the support of $M$ and $\pi^{*}N$, we deduce that
$\overline{Q}_{n}|_{\widetilde{Y}}$ is a semipositive approximation of
$\overline{D}|_{\widetilde{Y}}$.

Its absolute minimum can be estimated as

$0=\mu^{\mathrm{ess}}(\overline{D}|_{Y})=\mu^{\mathrm{ess}}(\overline{D}|_{\widetilde{Y}})\geq\mu^{\mathrm{abs}}(\overline{Q}_{n}|_{\widetilde{Y}})\geq\mu^{\mathrm{abs}}(\overline{Q}_{n})=\ell^{-2n}\mu^{\mathrm{abs}}(\overline{D}),$

where the last equality comes from Lemma 10.9(2). On the other hand we have
that
$Q_{n}|_{\widetilde{Y}}-\ell^{-n}D|_{\widetilde{Y}}=(1-\ell^{-n})\pi^{*}N|_{\widetilde{Y}}$
is effective and so

$r(Q_{n}|_{\widetilde{Y}};D|_{\widetilde{Y}})\geq\ell^{-n}.$

Hence
$\lim_{n\to\infty}{\mu^{\mathrm{abs}}(\overline{Q}_{n}|_{\widetilde{Y}})}/{r(Q_{n}|_{\widetilde{Y}};D|_{\widetilde{Y}})}=0$,
and so Theorem 6.1 together with Remark 6.2 implies that $\overline{D}|_{{Y}}$
satisfies the equidistribution property for every $v\in\mathfrak{M}_{K}$ with

$\nu_{\overline{D}|_{Y},v}=\lim_{v\to\infty}\frac{c_{1}((\overline{Q}_{n}|_{Y})_{v})^{\wedge
e}}{((Q_{n}|_{Y})^{e})}=\lim_{v\to\infty}\frac{c_{1}(\overline{Q}_{n,v})^{\wedge
e}\wedge\delta_{Y_{v}^{\mathrm{an}}}}{(Q_{n}^{e}\cdot Y)}.$

To compute this limit, first note that

$(Q_{n}^{e}\cdot{Y})=\sum_{j=0}^{e}\ell^{-n(e-j)}{e\choose
j}(M^{e-j}\cdot\pi^{*}N^{j}\cdot{Y}).$

For each $j$ consider the intersection product $[M^{e-j}\cdot{Y}]$ in the Chow
group of $j$-dimensional cycles of ${Y}$. By the projection formula [Ful98,
Proposition 2.3(c)] we have

$(M^{e-j}\cdot\pi^{*}N^{j}\cdot{Y})=(\pi_{*}[M^{e-j}\cdot{Y}]\cdot N^{j}),$

where the left intersection number is computed over ${Y}$ and the right over
$\pi({Y})$. Hence this quantity vanishes when $j>e^{\prime}=\dim(\pi(Y))$,
whereas for $j=e^{\prime}$ it equals $(M^{e-e^{\prime}}\cdot
F)\times(N^{e^{\prime}})$ for a general fiber $F$ of the projection
$Y\rightarrow\pi(Y)$, and therefore it is positive because $M$ is relatively
ample and $N$ is ample. Hence

(10.12) $(Q_{n}^{e}\cdot Y)=\ell^{-n(e-e^{\prime})}{e\choose
e^{\prime}}(M^{e-e^{\prime}}\cdot\pi^{*}N^{e^{\prime}}\cdot
Y)+O(\ell^{-n(e-e^{\prime}+1)}),$

and the dominant term in this asymptotics is positive.

On the other hand, the measure $c_{1}(\overline{Q}_{n,v})^{\wedge
e}\wedge\delta_{Y_{v}^{\mathrm{an}}}$ is zero whenever $j>e^{\prime}$ because
its total mass vanishes, and therefore

$c_{1}(\overline{Q}_{n,v})^{\wedge
e}\wedge\delta_{Y_{v}^{\mathrm{an}}}=\sum_{j=0}^{e^{\prime}}\ell^{-n(e-j)}{e\choose
j}c_{1}({\overline{M}}\vphantom{L}^{\operatorname{can}}_{v})^{\wedge
e-j}\wedge
c_{1}(\pi^{*}{\overline{N}}\vphantom{L}^{\operatorname{can}}_{v})^{\wedge
j}\wedge\delta_{Y_{v}^{\mathrm{an}}}.$

The statement then follows by taking the limit for $n\to\infty$ of the ratio
between this asymptotics and that in (10.12). ∎

Finally we strengthen the equidistribution property in (10.11) to include test
functions with logarithmic singularities along the closure of a torsion
hypersurface or an irreducible component of the boundary. Recall that a
hypersurface of ${G}$ is _torsion_ if it is the translate of a semiabelian
hypersurface of $G$ by a torsion point.

######  Theorem 10.15.

Let $(x_{\ell})_{\ell}$ be a $\overline{D}$-small generic sequence in
$\overline{G}(\overline{K})$ and $Y\subset\overline{G}$ a hypersurface that is
either the closure of a torsion hypersurface of ${G}$ or an irreducible
component of $\overline{G}\setminus G$. Then for every $v\in\mathfrak{M}_{K}$
and any function
$\varphi\colon\overline{G}_{v}^{\mathrm{an}}\rightarrow\mathbb{R}\cup\\{\pm\infty\\}$
with at most logarithmic singularities along $Y$ we have

$\lim_{\ell\to\infty}\int_{\overline{G}_{v}^{\mathrm{an}}}\varphi\,d\delta_{O(x_{\ell})_{v}}=\int_{\overline{G}_{v}^{\mathrm{an}}}\varphi\,\frac{c_{1}(\overline{D}_{v})^{\wedge
r+g}}{(D^{r+g})}.$

###### Proof.

This follows from Theorem 10.10 noting that $Y$ is a preperiodic hypersurface
for the endomorphism $[\ell]_{\overline{G}}$ for any $\ell>1$. ∎

##  11\. Generalization to quasi-projective varieties

In this section we extend Theorem 1.2 to the setting of adelic divisors on
quasi-projective varieties in the sense of Yuan and Zhang [YZ23] (Theorem
11.8). We first recall the construction and important facts concerning adelic
divisors on quasi-projective varieties from [YZ23]. We closely follow the
presentation in [BK24].

###  11.1. Adelic divisors on quasi-projective varieties

####  11.1.1. Geometric case

Let $k$ be a field, $X$ a normal projective variety over $k$ of dimension
$d\geq 1$, $B\in\operatorname{\mathrm{Div}}(X)$ an effective divisor on $X$,
and $U=X\setminus\mathrm{supp}(B)$. We denote by $R(X,U)$ the category of
normal modifications of $X$ which are isomorphisms over $U$. Given such a
modification $\pi\colon X_{\pi}\rightarrow X$, we write $(X_{\pi},\pi)$ or
simply $\pi$ for the corresponding object in $R(X,U)$. The space of _model
$\mathbb{R}$-divisors_ on $U$ is the direct limit

$\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\mathrm{mod}}=\varinjlim_{\pi\in
R(X,U)}\operatorname{\mathrm{Div}}(X_{\pi})_{\mathbb{R}}.$

Given two model $\mathbb{R}$-divisors $D,D^{\prime}$ on $U$, we write $D\geq
D^{\prime}$ or $D^{\prime}\leq D$ if there exists $(X_{\pi},\pi)\in R(X,U)$
such that $D,D^{\prime}\in\operatorname{\mathrm{Div}}(X_{\pi})_{\mathbb{R}}$
and $D-D^{\prime}$ is effective. The $B$-adic norm $\|.\|$ on
$\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\mathrm{mod}}$ is defined by
setting

$\|D\|=\inf\\{\varepsilon\in\mathbb{R}_{>0}\ |\ -\varepsilon B\leq
D\leq\varepsilon B\\}$

for $D\in\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\mathrm{mod}}$.

The space $\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\lim}$ is the
completion of $\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\mathrm{mod}}$ for
the $B$-adic topology. In [YZ23, BK24], elements of
$\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\lim}$ are called adelic
divisors on $U$. We prefer not to use this terminology here in order to avoid
confusion with the arithmetic setting. The group
$\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\lim}$ only depends on the open
set $U$ by [YZ23, Lemma 2.4.1], [BK24, Remark 3.3].

Let $D\in\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\lim}$. By definition,
$D$ can be represented by a sequence of model $\mathbb{R}$-divisors
$(D_{i})_{i}$ on $U$ that is Cauchy for the $B$-adic norm. It follows from
[YZ23, Theorems 5.2.1 and 5.2.9] that the limit
$\lim_{i\to\infty}\operatorname{\mathrm{vol}}(D_{i})$ exists in $\mathbb{R}$
and does not depend on the choice of $(D_{i})_{i}$. We call this limit the
_volume_ of $D$ and we denote it by $\operatorname{\mathrm{vol}}(D)$. We say
that $D$ is _big_ if $\operatorname{\mathrm{vol}}(D)>0$.

We say that $D$ is _nef_ if the sequence $(D_{i})_{i}$ can be chosen such that
$D_{i}$ is nef for every $i\in\mathbb{N}$. We say that $D$ is _integrable_ if
it is of the form $D=A_{1}-A_{2}$ with
$A_{1},A_{2}\in\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\lim}$ nef.

The subspace of integrable adelic divisors is denoted
$\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\mathrm{int}}$. The intersection
product defined in [YZ23, §4.1.1] extends by multilinearity to a product

$(\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\mathrm{int}})^{d}\longrightarrow\mathbb{R},\
(D_{1},\ldots,D_{d})\longmapsto(D_{1}\cdot\ldots\cdot D_{d})$

with the following property [BK24, Theorem 3.7]. Let
$(D_{1},\ldots,D_{d})\in(\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\mathrm{int}})^{d}$
and let $(X_{i},\pi_{i})_{i}$ be a sequence in $R(X,U)$ such that for each
$j\in\\{1,\ldots,j\\}$, $D_{j}$ is represented by a Cauchy sequence
$(D_{j,i})_{i}$ with
$D_{j,i}\in\operatorname{\mathrm{Div}}(X_{i})_{\mathbb{R}}$ for every
$i\in\mathbb{N}$. Then we have

$(D_{1}\cdot\ldots\cdot D_{d})=\lim_{i\to\infty}(D_{1,i}\cdot\ldots\cdot
D_{d,i}).$

Moreover, $\operatorname{\mathrm{vol}}(D)=(D^{d})$ for every nef
$D\in\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\lim}$.

Let $P,A\in\operatorname{\mathrm{Div}}(U)^{\lim}_{\mathbb{R}}$ be big. The
_inradius_ of $P$ with respect to $A$ is the quantity

$r(P;A)=\sup\\{\lambda\in\mathbb{R}\ |\ P-\lambda A\text{ is big}\\}.$

It is easy to check that

(11.1) $r(P;A)=\lim_{i\to\infty}r(P_{i};A_{i})$

for any sequence $(X_{i},\pi_{i})_{i}$ in $R(X,U)$ such that $P$ and $A$ are
represented by Cauchy sequences $(P_{i})_{i}$ and $(A_{i})_{i}$ with
$P_{i},A_{i}\in\operatorname{\mathrm{Div}}(X_{i})_{\mathbb{R}}$ for every
$i\in\mathbb{N}$.

####  11.1.2. Arithmetic case

Let $K$ be a number field with ring of integers $\mathcal{O}_{K}$. By a
_projective arithmetic variety_ (respectively a _quasi-projective arithmetic
variety_) we mean an integral scheme $\mathcal{X}$ which is flat and
projective (respectively quasi-projective) over
$\operatorname{\mathrm{Spec}}\mathcal{O}_{K}$. Let $\mathcal{X}$ be a normal
projective arithmetic variety. Then its generic fiber
$X=\mathcal{X}\times_{\mathcal{O}_{K}}\operatorname{\mathrm{Spec}}K$ is also
normal. We let $d+1$ be the Krull dimension of $\mathcal{X}$, so that $X$ has
dimension $d$.

######  Definition 11.1.

An _arithmetic $\mathbb{R}$-divisor_ on $\mathcal{X}$ is a pair
$\overline{\mathcal{D}}=(\mathcal{D},(g_{v})_{v\in{\mathfrak{M}_{K}^{\infty}}})$,
where
$\mathcal{D}\in\operatorname{\mathrm{Div}}(\mathcal{X})_{\mathbb{R}}=\operatorname{\mathrm{Div}}(\mathcal{X})\otimes_{\mathbb{Z}}\mathbb{R}$
is an $\mathbb{R}$-Cartier divisor on $\mathcal{X}$ and for each
$v\in{\mathfrak{M}_{K}^{\infty}}$, $g_{v}$ is a continuous $v$-adic Green
function on $\mathcal{D}_{|X}$. The space of arithmetic $\mathbb{R}$-divisors
on $\mathcal{X}$ is denoted
$\widehat{\operatorname{\mathrm{Div}}}(\mathcal{X})_{\mathbb{R}}$.

We let
$[\infty]\in\widehat{\operatorname{\mathrm{Div}}}(\mathcal{X})_{\mathbb{R}}$
be the arithmetic divisor on $\mathcal{X}$ given by $\mathcal{D}=0$ and
$g_{v}=1$ for every $v\in\mathfrak{M}_{K}^{\infty}$.

To any
$\overline{\mathcal{D}}=(\mathcal{D},(g_{v})_{v\in{\mathfrak{M}_{K}^{\infty}}})\in\widehat{\operatorname{\mathrm{Div}}}(\mathcal{X})_{\mathbb{R}}$
we can associate an adelic $\mathbb{R}$-divisor
$\overline{\mathcal{D}}^{\mathrm{ad}}=(D,(g_{v})_{v\in\mathfrak{M}_{K}})\in\widehat{\operatorname{\mathrm{Div}}}(X)_{\mathbb{R}}$
by setting $D=\mathcal{D}_{|X}$ and $g_{v}=g_{\mathcal{D},v}$ for
$v\in\mathfrak{M}_{K}\setminus\mathfrak{M}_{K}^{\infty}$ (see §3.1). We denote
by
$h_{\overline{\mathcal{D}}}=\mathrm{h}_{\overline{\mathcal{D}}^{\mathrm{ad}}}\colon
X(\overline{K})\rightarrow\mathbb{R}$ the associated height function.

We say that $\overline{\mathcal{D}}$ is _effective_ (respectively _strictly
effective_) if $\mathcal{D}$ is effective and if $g_{v}\geq 0$ (respectively
$g_{v}>0$) on $X_{v}^{\mathrm{an}}\setminus\mathrm{supp}(D)_{v}^{\mathrm{an}}$
for every $v\in\mathfrak{M}_{K}^{\infty}$. Let
$\overline{\mathcal{B}}=(\mathcal{B},(g_{\mathcal{B},v})_{v})$ be a strictly
effective arithmetic $\mathbb{R}$-divisor on $\mathcal{X}$ and
$\mathcal{U}=\mathcal{X}\setminus\mathrm{supp}(\mathcal{B})$. We also let
$B=\mathcal{B}_{|X}\in\operatorname{\mathrm{Div}}(X)_{\mathbb{R}}$ and
$U=X\setminus\mathrm{supp}(B)$. We denote by $R(\mathcal{X},\mathcal{U})$ the
category of normal modifications of $\mathcal{X}$ over
$\operatorname{\mathrm{Spec}}\mathcal{O}_{K}$ which are isomorphisms over
$\mathcal{U}$. By definition, the objects of $R(\mathcal{X},\mathcal{U})$ are
pairs $(\pi,\mathcal{X}_{\pi})$ (also denoted $\pi$ for short) where
$\mathcal{X}_{\pi}$ is a normal projective arithmetic variety and
$\pi\colon\mathcal{X}_{\pi}\rightarrow\mathcal{X}$ is a morphism that induces
an isomorphism over $\mathcal{U}$. The space of _model arithmetic
$\mathbb{R}$-divisors_ on $\mathcal{U}$ is defined as the direct limit

$\widehat{\operatorname{\mathrm{Div}}}(\mathcal{U})_{\mathbb{R}}^{\mathrm{mod}}=\varinjlim_{\pi\in
R(\mathcal{X},\mathcal{U})}\widehat{\operatorname{\mathrm{Div}}}(\mathcal{X}_{\pi})_{\mathbb{R}}.$

For every
$\overline{\mathcal{D}}_{1},\overline{\mathcal{D}}_{2}\in\widehat{\operatorname{\mathrm{Div}}}(\mathcal{U})_{\mathbb{R}}^{\mathrm{mod}}$,
we write $\overline{\mathcal{D}}_{1}\geq\overline{\mathcal{D}}_{2}$ or
$\overline{\mathcal{D}}_{1}\leq\overline{\mathcal{D}}_{2}$ if
$\overline{\mathcal{D}}_{1}-\overline{\mathcal{D}}_{2}$ is effective on a
model where $\overline{\mathcal{D}}_{1}$ and $\overline{\mathcal{D}}_{2}$ are
defined. The $\overline{\mathcal{B}}$-adic norm is defined by letting

$\|\overline{\mathcal{D}}\|=\inf\\{\varepsilon\in\mathbb{R}_{>0}\ |\
-\varepsilon\overline{\mathcal{B}}\leq\overline{\mathcal{D}}\leq\varepsilon\overline{\mathcal{B}}\\}$

for
$\overline{\mathcal{D}}\in\widehat{\operatorname{\mathrm{Div}}}(\mathcal{U})_{\mathbb{R}}^{\mathrm{mod}}$.

######  Definition 11.2.

The space
$\widehat{\operatorname{\mathrm{Div}}}(\mathcal{U})_{\mathbb{R}}^{\mathrm{adel}}$
of _adelic $\mathbb{R}$-divisors_ on $\mathcal{U}$ is the completion of
$\widehat{\operatorname{\mathrm{Div}}}(\mathcal{U})_{\mathbb{R}}^{\mathrm{mod}}$
with respect to the $\overline{\mathcal{B}}$-adic topology.

As in the geometric case,
$\widehat{\operatorname{\mathrm{Div}}}(\mathcal{U})_{\mathbb{R}}^{\mathrm{adel}}$
depends only on $\mathcal{U}$ [BK24, Proposition 3.36]. Let
$\overline{D}\in\widehat{\operatorname{\mathrm{Div}}}(\mathcal{U})_{\mathbb{R}}^{\mathrm{adel}}$.
Then $\overline{D}$ can be represented by a sequence
$(\overline{\mathcal{D}}_{i})_{i}$ in
$\widehat{\operatorname{\mathrm{Div}}}(\mathcal{U})_{\mathbb{R}}^{\mathrm{mod}}$
which is Cauchy for the $\mathcal{\overline{B}}$-adic norm. For every
$i\in\mathbb{N}$, let $(\pi_{i},\mathcal{X}_{i})\in
R(\mathcal{X},\mathcal{U})$ such that
$\overline{\mathcal{D}}_{i}\in\widehat{\operatorname{\mathrm{Div}}}(\mathcal{X}_{i})_{\mathbb{R}}$
and let
$X_{i}=\mathcal{X}_{i}\times_{\mathcal{O}_{K}}\operatorname{\mathrm{Spec}}K$,
$\overline{D}_{i}=\overline{\mathcal{D}}_{i}^{\mathrm{ad}}\in\widehat{\operatorname{\mathrm{Div}}}(X_{i})_{\mathbb{R}}$,
and $D_{i}={\mathcal{D}_{i}}_{|X_{i}}$. Note that $(D_{i})_{i}$ is a sequence
in $\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\mathrm{mod}}$ which is
Cauchy for the $B$-adic topology in the sense of §11.1.1, hence defines an
object $D\in\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\lim}$. In the
sequel, given an adelic $\mathbb{R}$-divisor
$\overline{A}\in\widehat{\operatorname{\mathrm{Div}}}(\mathcal{U})_{\mathbb{R}}^{\mathrm{adel}}$
we always use the same letter $A$ to denote the corresponding object in
$\operatorname{\mathrm{Div}}(U)_{\mathbb{R}}^{\lim}$.

We say that $\overline{D}$ is _pseudo-effective_ (respectively _semipositive_
, respectively _nef_) if the sequence $(\overline{\mathcal{D}}_{i})_{i}$ can
be chosen such that