# Differentiability of Adelic Volumes and Equidistribution on Quasi-Projective
Varieties

Debam Biswas

###### Abstract

Differentiability of geometric and arithmetic volumes of Hermitian line-
bundles leads to the proof of equidistribution results on projective varieties
using the variational principle. In this article, we work in the setting of
adelic divisors on quasi-projective varieties recently introduced by Xinyi
Yuan and Shou-Wu Zhang to show that their geometric and arithmetic adelic
volume functions are differentiable on the big cone. We introduce the notions
of positive intersection products and show that the differentials are realised
as positive intersection products against integrable divisors at big points.
We show an analogue of the Fujita Approximation for restricted volumes of
adelic divisors in the geometric setting using a construction similar to those
of positive intersections and as an application of our differentiability
result we derive a slightly weaker quasi-projective analogue of the
equidistribution theorem of Berman and Boucksom which generalises the
equidistribution obtained by Yuan and Zhang for arithmetically nef adelic
divisors on quasi-projective varieties.

## Keywords:

2020 Mathematics Subject Classification : 14G40

##  1 Introduction

Let $k$ be a field and $X$ a projective variety over $k$. If $L$ is a line
bundle on $X$ then we define

$\text{vol}(L):=\limsup_{m\to\infty}\frac{d!\cdot\text{dim}_{k}(H^{0}(X,L^{\otimes
m}))}{m^{d}}$

where $d=\text{dim}(X)$. We say $L$ is big if $\text{vol}(L)>0$. Boucksom,
Favre and Jonsson showed that the function $\text{vol}(L)$ is differentiable
on the big cone in [BFJ09]. Lazarsfeld and Mustață independently showed the
differentiability result by using Okounkov bodies in [LM09]. The differential
of the volume function in a suitable direction involves the definition of
_positive intersection products_ which were first introduced in [Bou+13].
In Arakelov geometry, there is a similar notion of arithmetic volumes for
Hermitian line bundles which measures the asymptotic growth of _small global
sections_ of a given Hermitian line bundle. More precisely if
$\mathcal{X}\rightarrow\text{Spec}(\mathbb{Z})$ is a projective arithmetic
variety of dimension $d$ with generic fiber $X$ and let
$\overline{\mathcal{L}}$ be a Hermitian line bundle with generic fiber $L$,
then there is an induced Hermitian metric $|\cdot|$ on the complexified line
bundle $L_{\mathbb{C}}:=L\otimes_{\mathbb{Q}}\mathbb{C}$. This metric induces
a sup-norm $||\cdot||$ corresponding to every global section $H^{0}(X,L)$ and
we define the space of _small sections_ as

$\widehat{H}^{0}(\mathcal{X},\overline{\mathcal{L}}):=\\{s\in
H^{0}(X,L)\mid||s||\leq 1\\}\ \text{and}\
\widehat{h}^{0}(\mathcal{X},\overline{\mathcal{L}}):=\text{log}\\#\hat{H^{0}}(\mathcal{X},\overline{\mathcal{L}})$

Then we can finally define the _arithmetic volume_ of $\overline{\mathcal{L}}$
as

$\widehat{\text{vol}}(\overline{\mathcal{L}}):=\limsup_{m\to\infty}\frac{d!\cdot\widehat{h}^{0}(H^{0}(\mathcal{X},\overline{\mathcal{L}}^{\otimes
m}))}{m^{d}}$

We call $\overline{\mathcal{L}}$ to be _big_ if
$\widehat{\text{vol}}(\overline{\mathcal{L}})>0$. Analogously to the classical
volume function in algebraic geometry, Chen showed that the arithmetic volume
function is differentiable in the big cone. In [Che11], he introduces an
arithmetic analogue of the positive intersections in [Bou+13] and shows that
the differentiability at the big point is given by the arithmetic positive
intersection with a suitable direction analogously.
In their recent work [YZ21], Yuan and Zhang introduce the notion of _adelic
divisors_ on a quasi-projective (arithmetic) varietiy $U$. Their idea is to
consider all arithmetic divisors coming from some projective model of the
fixed quasi-projective variety $U$ and put a _boundary topology_ on them which
measures their growth along the boundary outside $U$. Then they define the
_adelic divisors_ as those divisors which are compactified with respect to the
boundary topology. More precisely if we fix a boundary divisor
$\overline{D_{0}}:=(D_{0},g_{0})$ _i.e_ an arithmetic divisor
$\overline{D_{0}}$ on some projective compactification $X_{0}$ of $U$ such
that $\text{Supp}(D_{0})=X_{0}\backslash U$, then an _adelic divisor_ is given
by sequence $\\{X_{i},\overline{D_{i}}\\}$ where $X_{i}$ are projective
compactifications of $U$ and $\overline{D}_{i}$ are arithmetic divisors on
$X_{i}$ satisfying a Cauchy condition _i.e_ there is a sequence of positive
rationals $\\{q_{i}\\}$ converging to 0 such that

$-q_{i}\overline{D_{0}}\leq\overline{D_{j}}-\overline{D_{i}}\leq
q_{i}\overline{D_{0}}\ \text{for all}\ j\geq i$

where the effectivity relations are understood to hold after passing to a
common projective model via birational pull-backs. The notion of adelic
divisors are closely related to the notion of _b-divisors_ introduced in
[Sho03] (see [BG22] for a nice review on b-divisors).
Yuan and Zhang also introduce the notions of volumes and arithmetic volumes
$\widehat{\text{vol}}(\overline{D})$ for an adelic divisor $\overline{D}$ on a
quasi-projective variety and show that they satisfy many properties of the
classical volumes on projective varieties like continuity, log-concavity etc
(see [YZ21, Chapter 5]. Following [BFJ09] in the geometric case and [Che11] in
the arithmetic case, we define _positive intersection products_ of adelic
divisors $\overline{D}$ against integrable directions $\overline{E}$ denoted
by $\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$ (see Definition 3.2 in
section 3). Our first main result in this article is to show that these volume
functions are differentiable. Suppose $k=(B,\Sigma)$ is a tuple where $B$ is a
number ring or a smooth projective curve with fraction field $K$ and
$\Sigma\subseteq\text{Hom}(K,\mathbb{C})$ and suppose $U$ is an essentially
quasi-projective variety over $k$. Then in 3.13 we show the following
differentiability of adelic volumes

######  Theorem (A).

Suppose $\overline{D}$ is a big adelic divisor and $\overline{E}$ is an
integrable adelic divisor on a normal essentially quasi-projective variety $U$
over $k$ such that any quasi-projective model has dimension $d$. Then the
function $t\mapsto\widehat{\emph{vol}}(\overline{D}+t\overline{E})$ is
differentiable at $t=0$ with derivative given by

$\frac{d}{dt}\widehat{\emph{vol}}(\overline{D}+t\overline{E})\mid_{t=0}=d\cdot\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$

The arguments to deduce the differentiability of both geometric and arithmetic
adelic volumes are very similar to the ones used by Boucksom, Favre and
Jonsson in [BFJ09] to deduce differentiability of geometric volumes (see the
proof of 3.13). The crucial point is using the continuity of positive
intersection products and adelic volumes to obtain a limit version of the
bounds in [BFJ09] while perturbing in small directions of the boundary
divisor. We also use Fujita approximation [YZ21, Theorem 5.2.8] and Siu’s
inequality [YZ21, Theorem 5.2.2(2)] for adelic volumes.
Given a closed sub-variety of a projective variety, there is an invariant
called _asymptotic intersection number_ defined along it for a line bundle in
[Ein+09]. The definition of this intersection number is closely related to the
definition of positive intersection products. In [Ein+09] it is showed that
these intersection numbers are related to the _restricted volumes_ of the line
bundle along the closed sub-variety provided that the closed sub-variety is
general enough with respect to the bundle. Given a closed subvariety $E$ of a
quasi-projective variety and a (geometric) adelic divisor $\overline{D}$ on
$U$, we have introduced the notions of the restricted volumes denoted by
$\widehat{\text{vol}}_{U|E}(\overline{D})$ and _augmented base locus_ of
$\overline{D}$ denoted by $B_{+}(\overline{D})$ in [Bis23]. In this article we
introduce an invariant similar to the asymptotic intersection numbers for
adelic divisors which we denote by $\langle\overline{D}^{d}\rangle\cdot E$ and
show that they are equal to the restricted volume when the closed sub-variety
is not contained in the augmented base locus. This allows us to obtain a
version of Fujita approximation for restricted volumes. Now suppose $k$ is a
field and $U$ is a quasi-projective variety over $k$. In Theorem 4.5, we show
the following

######  Theorem (B).

Suppose $\overline{D}$ is a big adelic divisor and suppose $E$ is a closed
sub-variety of $U$ such that $E\nsubseteq B_{+}(\overline{D})$. Then we have

$\widehat{\emph{vol}}_{U|E}(\overline{D})=\langle\overline{D}^{d}\rangle\cdot
E$

where $d=\emph{dim}(U)$

Next as an application of our differentiability results we consider the
variational principle useful in proving equidistribution conjectures. We work
uniformly over the base $K$ which can either be a number field or a function
field of one variable. The differentiability of both the geometric and
arithmetic adelic volumes allows us to mimick the arguments in [Che11, Section
5.2] to obtain an equdistribution for big arithmetic adelic divisors, Suppose
$U$ is now a quasi-projective variety over $K=\text{Frac}(B)$ where $B$ is
either the ring of integers of a number field or a smooth projective curve. We
fix a place $v$ on $K$ and we denote the $K_{v}$-analytic space associated to
$U$ where $K_{v}$ is the completion $K_{v}$ of $K$ at $v$. In Theorem 5.11, we
show the following

######  Theorem (C).

Suppose $U$ is a quasi-projective variety over $K$ of dimension $d§$ and
suppose $\overline{\mathcal{D}}$ is a big arithmetic adelic divisor in
$\widehat{\emph{Div}}(U,k)$. Furthermore suppose $\\{x_{m}\\}$ is a generic
sequence of geometric points in $U(\overline{K})$ which is small with respect
to $\overline{\mathcal{D}}$. Then for any place $v$ on $K$ and for any $g\in
C_{c}^{0}(U_{v})$, we have

$\lim_{m\to\infty}\int_{U_{v}}g\
d\eta_{x_{m}}=\frac{\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g)}{\widehat{\emph{vol}}(\overline{D})}$

In particular, the sequence of Radon measures $\\{\eta_{x_{m}}\\}$ converge
weakly to the Radon measure given by

$g\in
C_{c}^{0}(U_{v})\mapsto\frac{\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g)}{\widehat{\emph{vol}}(\overline{D})}$

We go on to show that this strengthens the equidistribution theorem for quasi-
projective varieties proved by Yuan and Zhang in [YZ21, Theorem 5.4.3]. The
above theorem can be thought of as an analogue of the equdistribution theorem
obtained by Berman and Boucksom in [BB10] in the quasi-projective case where
we require an extra positivity asusmption for the arithmetic adelic divisor to
be arithmetically big.
The article is organised as follows. In section 2, we start with a brief
review of the adelic divisors of Yuan and Zhang following [YZ21] which will be
necessary for our article. In section 3, we start by defining positive
intersection products for adelic divisors. We go on to obtain properties of
this product and in the end obtain the differentiability result. In section 4
we define _positive intersection products_ along a closed sub-variety and
relate them with restricted volumes under suitable hypothesis on the closed
sub-variety. In section 5 we deduce the equidistribution result as an
application of differentiability of geometric and arithmetic adelic volumes
and finally in section 6 we relate our equidistribution with two other
equidistribution results in the literature.

## Acknowledgements

The author would like to thank Sébastien Boucksom, Huyai Chen, Walter Gubler,
Roberto Gualdi and Antoine Sedillot for fruitful discussions and comments
during the preparation of the article. The author would like to think François
Ballaÿ for pointing out a relation to the equidistribution result of Berman
and Boucksom.

##  2 Review of Adelic Divisors

In this section, we recall the definitions of adelic divisors on quasi-
projective varieties over Dedekind bases. We follow the treatment along [YZ21,
Section 2.7]. We begin by considering a tuple $k:=(B,\Sigma)$ where $B$ is a
Dedekind scheme _i.e_ a Noetherian integral scheme of dimension 1 with
function field $K$ and $\Sigma\subseteq\text{Hom}(\Sigma,\mathbb{C})$. Note
that here we allow $\Sigma$ to be the empty set which is typically the case
when $B$ is a scheme over a field of positive characteristic. Note that every
$\sigma\in\Sigma$ induces an Archimedean valuation $|\cdot|_{\sigma}$ of
$\mathbb{C}$ and we have $|\cdot|_{\sigma}=|\cdot|_{\sigma^{\prime}}$ if and
only if $\sigma^{\prime}=c\circ\sigma$ where $c$ is the complex conugation on
$\mathbb{C}$. We further impose that when our scheme $B=\text{Spec}(F)$ for
some field $F$ or $B$ is a smooth projective curve, we set that $\Sigma$ is
empty, even if $F$ has characteristic 0. We do this to include the geometric
case in our uniform terminology. By a variety over $k$ we mean a flat,
integral, separated, finite-type scheme $\mathcal{X}$ over $B$. Furthermore we
say it is quasi-projective (projective) if $\mathcal{X}$ is quasi-projective
(projective) over $B$. For a quasi-projective variety $\mathcal{U}$ over $k$,
a _projective model_ of $\mathcal{U}$ means a projective variety $\mathcal{X}$
over $k$ together with an open immersion
$\mathcal{U}\xhookrightarrow{}\mathcal{X}$.
We want to include a slightly more general class of varieties that we want to
work with. In order to do so, we recall the notion of _pro-open immersions_. A
morphism between integral schemes is said to be _pro-open_ if the underlying
map of topological spaces is injective and it induces and isomorphism of local
rings at each point (see [YZ21, Section 2.3.1]). By an _essentially quasi-
projective variety_ over $k$, we mean an integral, flat, finite scheme $X$
over $B$ together with a projective variety $\mathcal{X}$ over $k$ such that
there is a pro-open immersion $X\xhookrightarrow{}\mathcal{X}$. By a
_(quasi)projective_ model of $X$ over $k$, we mean a (quasi)-projective
variety $\mathcal{X}$ over $k$ together with a pro-open immersion
$X\xhookrightarrow{}\mathcal{X}$. Suppose $\mathcal{X}$ is a projective
arithmetic variety over $k$. Then we set
$\mathcal{X}_{\Sigma}:=\coprod_{\sigma\in\Sigma}\mathcal{X}_{\sigma}$ where
$\mathcal{X}_{\sigma}$ denotes the base change of $\mathcal{X}$ to
$\mathbb{C}$ via $\sigma$. By an _arithmetic divisor_ over $\mathcal{X}$, we
mean a tuple $(\mathcal{D},g_{\mathcal{D}})$ where $\mathcal{D}$ is a divisor
and $g_{\mathcal{D}}$ is a Green function
$\mathcal{X}_{\Sigma}(\mathbb{C})\backslash|\mathcal{D}(\mathbb{C})|\rightarrow\mathbb{R}$
which is invariant under the action $F_{\infty}$ of complex conjugation. This
means that if $\overline{\sigma}=c\circ\sigma$, then we require that
$g_{\mathcal{D},\overline{\sigma}}=g_{\overline{\mathcal{D}},\sigma}$ where
$g_{\overline{\mathcal{D}},\sigma}$ denotes the induced Green’s function on
the factor $\mathcal{X}_{\sigma}$. We denote the group of arithmetic divisors
on $\mathcal{X}$ by $\widehat{\text{Div}}(\mathcal{X})$. For further notions
we refer to [YZ21, Section 2.7].
Given an open subscheme of $\mathcal{X}$, we denote the _objects of mixed
coefficients_ as

$\widehat{\text{Div}}(\mathcal{X},\mathcal{U})=\widehat{\text{Div}}(\mathcal{X})_{\mathbb{Q}}\oplus_{\text{Div}(\mathcal{U})_{\mathbb{Q}}}\text{Div}(\mathcal{U})$

Given a fixed quasi-projective arithmetic variety $\mathcal{U}$ on $k$, we can
define the group of _model adelic divisors_ as

$\widehat{\text{Div}}(\mathcal{U})_{\text{mod}}:=\lim_{\mathcal{X}}\widehat{\text{Div}}(\mathcal{X},\mathcal{U})$

where we take the filtered colimit by varying $\mathcal{X}$ across all
projective models of $\mathcal{U}$ and viewing the objects of mixed co-
efficients as filtered system via bi-rational pull-backs. We can extend the
notions of effectivity to the group of adelic divisors by passing to filtered
colimits. By a _boundary divisor_ on $\mathcal{U}$, we mean a model divisor
$\overline{\mathcal{D}}_{0}\in\widehat{\text{Div}}(\mathcal{X},\mathcal{U})$
such that $\text{Supp}(\mathcal{D}_{0})=\mathcal{X}\backslash\mathcal{U}$.
Given the choice of such a boundary divisor we can endow the group of model
divisors by a _boundary norm_ as

$||\cdot||_{\overline{\mathcal{D}}_{0}}:\widehat{\text{Div}}(\mathcal{U})_{\text{mod}}\rightarrow\mathbb{R}\cup\\{\infty\\}$
$||\mathcal{D}||_{\overline{\mathcal{D}}_{0}}:=\inf\\{q\in\mathbb{Q}\mid-q\overline{\mathcal{D}}_{0}\leq\overline{\mathcal{D}}\leq
q\overline{\mathcal{D}}_{0}\\}$

We can define the group _adelic divisors_ on $\mathcal{U}$, denoted by
$\widehat{\text{Div}}(\mathcal{U},k)$, as the Cauchy completion of
$\widehat{\text{Div}}(\mathcal{U})_{\text{mod}}$ with respect to the topology
induced by $||\cdot||_{\overline{\mathcal{D}}_{0}}$ for some boundary divisor
$\overline{\mathcal{D}}_{0}$. Finally given an essentially quasi-projective
variety $U$ over $k$, we define the group of adelic divisors on $U$, denoted
by $\widehat{\text{Div}}(U,k)$ as

$\widehat{\text{Div}}(U,k):=\lim_{\mathcal{U}}\widehat{\text{Div}}(\mathcal{U},k)$

where we take the filtered colimit by varying $\mathcal{U}$ across all quasi-
projective models of $U$ over $k$ and using birational pull-backs as
transition maps. We further have the notions of _strongly nef_ , _nef_ and
_integrable_ adelic divisors denoted by
$\widehat{\text{Div}}(U,k)_{\text{snef}}$,
$\widehat{\text{Div}}(U,k)_{\text{nef}}$ and
$\widehat{\text{Div}}(U,k)_{\text{int}}$ respectively and we refer to [YZ21,
Section 2.4] for further definitions and details.
Note that the notions of effectivity of divisors on
$\widehat{\text{Div}}(\mathcal{U})_{\text{mod}}$ induce notions of effectivity
on $\widehat{\text{Div}}(\mathcal{U},k)$ and $\widehat{\text{Div}}(U,k)$ by
passing to completions and filtered colimits. Given
$\overline{D}\in\widehat{\text{Div}}(U,k)$, Yuan and Zhang introduces the
notion of _small sections_ as

$H^{0}(U,\overline{D}):=\\{f\in
k(U)^{\times}\mid\widehat{\text{div}}(f)+\overline{D}\geq 0\\}$

where $k(U)$ denotes the function field of $U$. They are further able to show
that these spaces are finite dimensional in the geometric case and are of
finite cardinality in the arithmetic case (see [YZ21, Lemma 5.1.6])). This
allows them to define the notions of geometric and arithmetic volumes
$\widehat{\text{vol}}(\overline{D})$ which will be our main objects of
investigation in this article. We say that
$\overline{D}\in\widehat{\text{Div}}(U,k)$ is _big_ if
$\widehat{\text{vol}}(\overline{D})>0$. We refer to [YZ21, Sections 5.1-5.2]
for details and various properties of adelic volumes that are obtained.

##  3 Positive Intersection products

In this section, we define the positive intersection product of
(arithmetic)adelic divisors with an integrable adelic divisor inspired by
positive intersection products defined in [BFJ09] in the geometric case and
[Che11] in the arithmetic case. We go on to show that at big adelic divisors,
this product behaves continuously. Then using arguments like in [BFJ09] we
show that the function
$t\mapsto\widehat{\text{vol}}(\overline{D}+t\overline{E})$ is differentiable
at $t=0$ and the derivative is given by the defined positive intersection
product when $\overline{E}$ is an integrable adelic divisor.
We are going to treat the arithmetic and geometric cases parallely as in
[YZ21]. We will denote by $k:=(B,\Sigma)$ where $B$ is either the ring of
integers of a number field $K$, a smooth projective curve with function field
$K$ or any field. Furthermore if $B$ is either a smooth projective curve or a
field then we force $\Sigma$ to be empty and when $B$ is the ring of integers
then we force $\Sigma=\text{Hom}(K,\mathbb{C})$ to be the full set of
embeddings. Note that then in any of the above cases, $k$ will be a valued
Dedekind scheme as defined in the previous section and hence the formalism of
the previous section goes through. We restrict to these cases since absolute
intersection numbers are only defined in these global cases in [YZ21]. We will
denote by $U$ an essentially quasi-projective variety over $k$ and by
$\mathcal{U}$ any quasi-projective model of $U$ over $k$. Given two such
models of $U$, we say one dominates the other if there is a morphism over $U$
between them in the appropriate direction. Note that given two models
$\mathcal{U}_{1},\mathcal{U}_{2}$ of $U$ such that $\mathcal{U}_{1}$ dominates
$\mathcal{U}_{2}$, $\mathcal{U}_{1}$ is necessarily a birational modification
of $\mathcal{U}_{2}$ which is an isomorphism over $U$.
We have defined adelic divisors on $U$ in the previous section and we give a
further description here on a quasi-projective model which will be useful for
our computations later. An adelic divisor $\overline{D}$ on a quasi-projective
variety $\mathcal{U}$ over $k$ is given by the data
$\\{\mathcal{X}_{i},\overline{D_{i}},q_{i}\\}$ where $\mathcal{X}_{i}$ are
projective varieties over $k$ containing $\mathcal{U}$ as a dense open subset,
$\overline{D_{i}}\in\widehat{\text{Div}}(\mathcal{X}_{i})_{\mathbb{Q}}\oplus_{\text{Div}(\mathcal{U})_{\mathbb{Q}}}\text{Div}(\mathcal{U})=\text{Div}(\mathcal{X}_{i},\mathcal{U})$
are the objects of mixed co-efficients as defined in [YZ21, Section 2.4] such
that $\overline{D_{i}}$ and $\overline{D_{j}}$ have the same image in
$\text{Div}(\mathcal{U})_{\mathbb{Q}}$ and $\\{q_{i}\\}$ is a zero sequence of
positive rationals such that they satisfy the Cauchy condition

$-q_{j}\overline{D_{0}}\leq\overline{D_{i}}-\overline{D_{j}}\leq
q_{j}\overline{D_{0}}\ \text{in}\ \text{Div}(\mathcal{U})_{\text{mod}}\
\text{for all}\ i\geq j$

for a boundary divisor $\overline{D_{0}}$. An adelic divisor $\overline{D}$ is
called _(strongly)nef_ if $\mathcal{O}(\overline{D})$ is a (strongly) nef
adelic line bundle as defined in [YZ21, Definition 2.5.2]. Similarly we say an
adelic divisor is _integrable_ if $\mathcal{O}(\overline{E})$ is an integrable
adelic line bundle as defined in [YZ21, Definition 2.5.2]. We denote the group
of adelic divisors, nef adelic divisors and integrable adelic divisors by
$\widehat{\text{Div}}(\mathcal{U},k)$,
$\widehat{\text{Div}}(\mathcal{U},k)_{\text{nef}}$ and
$\widehat{\text{Div}}(\mathcal{U},k)_{\text{int}}$ respectively.
Finally we can define the group of adelic divisors, nef adelic divisors and
integrable adelic divisors on an essentially quasi-projective variety $U$ over
$k$ to be

$\widehat{\text{Div}}(U,k):=\varinjlim_{\mathcal{U}}\widehat{\text{Div}}(\mathcal{U},k)\
\
\widehat{\text{Div}}(U,k)_{\text{nef}}:=\varinjlim_{\mathcal{U}}\widehat{\text{Div}}(\mathcal{U},k)_{\text{nef}}\
\
\widehat{\text{Div}}(U,k)_{\text{int}}:=\varinjlim_{\mathcal{U}}\widehat{\text{Div}}(\mathcal{U},k)_{\text{int}}$

where we vary $\mathcal{U}$ over all the quasi-projective models of $U$ and we
take the filtered colimit by viewing the collections of
$\widehat{\text{Div}}(\mathcal{U},k)$ as a filtered system via birational
pull-backs.
Next we give the main definition of _positive intersection product_ of an
adelic divisor $\overline{D}$ with a nef adelic divisor $\overline{E}$. Note
that Yuan and Zhang has defined top absolute intersection numbers for
integrable adelic divisors in the case when $k$ is a field or a number field
(see [YZ21, Prop 4.1.1]. We remark here that using the same arguments we can
define absolute intersection numbers for integrable adelic divisors even in
the case when $k$ is a smooth projective curve. Our strategy will be to define
the positive intersection first for quasi-projective models and then pass to
the essentially quasi-projective case. Hence we first give the definition for
quasi-projective varieties over $k$. We first start with a definition.

######  Definition 3.1.

Suppose $\mathcal{X}$ is a projective variety over $k$ and $\overline{A}$ an
arithmetic divisor. We say that $\overline{A}$ is free if

  * •

the underlying line bundle $O(A)$ is semi-ample i.e some positive tensor power
of it is generated by global sections when $\Sigma$ is empty.

  * •

the associated curvature current of $\overline{A}$ on $\mathcal{X}_{\Sigma}$
is semi-positive and some positive tensor power of the underlying line bundle
$O(A)$ is generated by small sections when $\Sigma$ is non-empty.

######  Remark.

First we remark that it is easy to check that the definition above can be
easily extended to the case of $\mathbb{Q}$-divisors. Furthermore for any open
subset $\mathcal{U}$ of $\mathcal{X}$, $\overline{A}$ is model nef on
$\mathcal{U}$. This follows from [Che11, Prop 2.3] when $\Sigma$ is non-empty
and follows from intersection theory for the case when $\Sigma$ is empty.
Furthermore note that if $\overline{A}$ is arithmetically ample then it is
free. Moreover note that since the notion of freenes is invariant under
birational pull-backs, we can talk about free model $\mathbb{Q}$-divisors.

######  Definition 3.2.

Suppose $\mathcal{U}$ is a quasi-projective variety of dimension $d$ over $k$
and $\overline{D},\overline{E}$ adelic divisors on $\mathcal{U}$ such that
$\overline{E}$ is nef. Then we define the positive intersection product of
$\overline{D}$ with $\overline{E}$, denoted
$\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$, as

$\langle\overline{D}^{d-1}\rangle\cdot\overline{E}=\sup_{X^{\prime},\overline{A}}\overline{A}^{d-1}\cdot(\pi^{*}\overline{E})$

where $(X^{\prime},\overline{A})$ runs over all tuples such that $X^{\prime}$
is a projective model of a birational modification
$\pi\colon\mathcal{U}^{\prime}\rightarrow\mathcal{U}$ of $\mathcal{U}$ and
$\overline{A}$ is free model $\mathbb{Q}$-divisor on $X^{\prime}$ such that
$\pi^{*}\overline{D}-\overline{A}\geq 0$ as adelic divisors over
$\mathcal{U}^{\prime}$. We denote any such above tuple
$(X^{\prime},\overline{A})$ as an _admissible approximation_ of $\overline{D}$
on $\mathcal{U}$.

######  Remark.

Note that by [YZ21, Proposition 4.1.1], intersection numbers
$\overline{A}^{d-1}\cdot\pi^{*}\overline{E}$ are defined as $\overline{E}$
(and hence $\pi^{*}\overline{E}$) is nef and in particular integrable and
hence the definition makes sense. Also note that as the notions intersection
numbers and effectivity can be extended to $\mathbb{Q}$-adelic divisors, the
same can be done for positive intersection products. Note that for an arbitary
$\overline{D}$, the set of its admissible approximations might be empty and in
that case we set the positive intersection products to be $-\infty$ as a
matter of set theoretic convention. However we almost always exclusively work
with a big $\overline{D}$ in which case there always exists an admissible
approximation of $\overline{D}$. Moreover by choosing model nef divisors
$\overline{\omega}\geq\overline{D}$ it is also easy to see that the
intersection numbers $\overline{A}^{d-1}\cdot\pi^{*}\overline{E}$ are bounded
from above uniformly as we vary our admissible approximations and hence the
positive intersection is a finite number whenever $\overline{D}$ is big.

We want to extend the above definition to essentially quasi-projective
varieties over $k$. The ordinary intersection products are defined by choosing
any quasi-projective model and doing the intersection over there and it is
well-defined because of product formula (since any two quasi-projective models
of $U$ are birational). We will do something similar for positive intersection
products which is the content of our next lemma.

######  Lemma 3.3.

Suppose $\pi_{0}\colon\mathcal{U}_{1}\rightarrow\mathcal{U}_{2}$ are two
quasi-projective models with dimension $d$ of an essentially quasi-projective
variety $U$ over $k$ and with $\pi_{0}$ a birational morphism. Suppose
$\overline{D}\in\widehat{\emph{Div}}(\mathcal{U}_{2},k)$ and
$\overline{E}\in\widehat{\emph{Div}}(\mathcal{U}_{2},k)_{\emph{nef}}$. Then we
have

$\langle\pi_{0}^{*}\overline{D}^{d-1}\rangle\cdot\pi_{0}^{*}\overline{E}=\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$

###### Proof.

We first suppose that
$(\mathcal{U}_{2}^{\prime},X_{2}^{\prime},\overline{A},\pi)$ is an admissible
approximation of $\overline{D}$ on $\mathcal{U}_{2}$. Since $\mathcal{U}_{1}$
itself is a birational modification of $\mathcal{U}_{2}$, we get that the
fiber product
$\mathcal{U}_{1}^{\prime}:=\mathcal{U}_{2}^{\prime}\times_{\mathcal{U}_{2}}\mathcal{U}_{1}$
is a birational modification of $\mathcal{U}_{1}$ and a pull-back square

${\mathcal{U}_{1}^{\prime}}$${\mathcal{U}_{2}^{\prime}}$${\mathcal{U}_{1}}$${\mathcal{U}_{2}}$$\scriptstyle{\pi_{0}^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{\pi_{0}}$

Denote by
$\pi^{\prime}\colon\mathcal{U}_{1}^{\prime}\rightarrow\mathcal{U}_{1}$ be the
canonical birational morphism. Since $\pi^{*}\overline{D}\geq\overline{A}$ in
$\widehat{\text{Div}}(\mathcal{U}_{2}^{\prime},k)$, pulling back by
$\pi_{0}^{\prime}$ we get that
$\pi^{\prime*}(\pi_{0}^{*}\overline{D})=(\pi_{0}^{\prime})^{*}(\pi^{*}\overline{A})\geq\pi_{0}^{\prime*}\overline{A}$
where
$\pi_{0}^{\prime}\colon\mathcal{U}_{1}^{\prime}\rightarrow\mathcal{U}_{2}^{\prime}$
is the canonical morphism and hence
$(\mathcal{U}_{1}^{\prime},(\pi_{0}^{\prime})^{*}X_{2}^{\prime},(\pi_{0}^{\prime})^{*}\overline{A},\pi^{\prime})$
is an admissible approximation of $\pi_{0}^{*}\overline{D}$ on
$\mathcal{U}_{1}$ since $\pi_{0}^{\prime*}\overline{A}$ is free. The
projection formula clearly yields
$\pi_{0}^{\prime*}\overline{A}\cdot\pi^{\prime*}(\pi_{0}^{*}\overline{E})=\overline{A}\cdot\pi^{*}\overline{E}$.
We easily deduce from the definition of positive intersection products that

$\langle\pi_{0}^{*}\overline{D}^{d-1}\rangle\cdot\pi_{0}^{*}\overline{E}\geq\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$
(1)

On the other hand, suppose
$(\mathcal{U}_{1}^{\prime},X_{1}^{\prime},\overline{A},\pi)$ is an admissible
approximation of $\pi_{0}^{*}\overline{D}$ on $\mathcal{U}_{1}$. Then clearly
$\mathcal{U}_{1}^{\prime}$ is a birational modification of $\mathcal{U}_{2}$
as well. Moreover since $\pi^{*}(\pi_{0}^{*}\overline{D})\geq\overline{A}$, we
clearly get that
$(\mathcal{U}_{1}^{\prime},X_{1}^{\prime},\overline{A},\pi_{0}\circ\pi)$ is an
admissible approximation of $\overline{D}$ on $\mathcal{U}_{2}$. This clearly
yields

$\langle\overline{D}^{d-1}\rangle\cdot\overline{E}\geq\langle\pi_{0}^{*}\overline{D}^{d-1}\rangle\cdot\pi_{0}^{*}\overline{E}$
(2)

Equations (1) and (2) together clearly finishes the proof of the claim. ∎

With the above lemma which shows invariance of positive intersection products
with respect to birational pull-backs, we can now extend the definition of our
positive intersection products to essentially quasi-projective varieties as
follows

######  Definition 3.4.

Suppose $U$ is an essentially quasi-projective variety over $k$ such that a
quasi-projective model of $U$ has dimension $d$ and
$\overline{D}\in\widehat{\emph{Div}}(U),\overline{E}\in\widehat{\emph{Div}}(U)_{\emph{nef}}$.
Furthermore suppose $\overline{D}$ arises as an arithmetic adelic divisor
$\overline{\mathcal{D}}$ on a quasi-projective model $\mathcal{U}$ of $U$.
Then we define the positive intersection product of $\overline{D}$ against
$\overline{E}$, denoted by $\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$
as

$\langle\overline{D}^{d-1}\rangle\cdot\overline{E}:=\langle\overline{\mathcal{D}}^{d-1}\rangle\cdot\overline{E}$

where the right hand side is as defined in Definition 3.2. Note that the
definition does not depend on the choice of $\mathcal{U}$ due to Lemma 3.3.

We begin by recording some easy positivity properties of nef adelic divisors
and their intersection numbers.

######  Lemma 3.5.

Suppose $\overline{E}_{1},\ldots\overline{E}_{d}$ be nef adelic divisors and
suppose $\overline{\omega}$ is a model nef adelic divisor on an essentially
quasi-projective variety $U$ over $k$ such that
$\overline{\omega}\geq\overline{E}_{i}$ for all $i$ and suppose any quasi-
projective model of $U$ has dimension $d$. Then

$0\leq\prod_{i=1}^{d}\overline{E}_{i}\leq\overline{\omega}^{d}$

###### Proof.

We assume that all the divisors live in a common quasi-projective model
$\mathcal{U}$ of $U$ since intersection numbers are birational invariants.
Note that it is easy to check that the absolute intersection numbers defined
in [YZ21, Prop 4.1.1] is continuous in the sense that for all rational $t$ and
integrable adelic divisors $\overline{E}_{i},\overline{M}$, $\lim_{t\to
0}\prod_{i}(\overline{E}_{i}+t\overline{M})=\prod_{i}\overline{E}_{i}$ and
hence it is enough to check the above inequalities for strongly nef
$\overline{E}_{i}$’s. Note that the left hand side of the inequality is
already obtained in Propositon 4.1.1 and hence it is enough to show the right
hand side. Suppose $\overline{E}_{i}$ is given by the models
$\\{X_{j},\overline{E}_{ij}\\}$ and rationals $\\{q_{j}\to 0\\}$ such that
each $\overline{E}_{ij}$ is nef which we can always assume by passing to finer
models. Then note that we have the effectivity relation
$\overline{\omega}+q_{j}\overline{D}_{0}\geq\overline{E}_{ij}$ by the Cauchy
condition $\overline{E}_{i}\geq\overline{E}_{ij}-q_{j}\overline{D}_{0}$.
Moreover we can choose a model ample divisor
$\overline{D}_{0}^{\prime}\geq\overline{D}_{0}$ and shrinking $\mathcal{U}$ if
necessary, we can assume that $\overline{D}_{0}^{\prime}$ has support
$X_{0}\backslash\mathcal{U}$. Hence for the proof we can assume that the
boundary divisor is nef since intersection numbers do not change if we shrink
$U$ (and $\mathcal{U}$) to a dense open subset. Then clearly we have
$(\overline{\omega}+q_{j}\overline{D}_{0})^{d}\geq\prod_{i=1}^{d}\overline{E}_{ij}$
for all $j$ since each $\overline{E}_{ij}$, $\overline{\omega}$ and
$\overline{D_{0}}$ is nef. Now as indicated in the proof of Proposition 4.1.1
the intersection number $\prod_{i=1}^{d}\overline{E}_{i}$ is just given by the
limit of the products $\prod_{i=1}^{d}\overline{E}_{ij}$ as $j$ goes to
infinity and hence we conclude

$\overline{\omega}^{d}=\lim_{j\to\infty}(\overline{\omega}+q_{j}\overline{D}_{0})^{d}\geq\lim_{j\to\infty}\prod_{i=1}^{d}\overline{E}_{ij}=\prod_{i=1}^{d}\overline{E}_{i}$

where the first equality follows since $q_{j}\to 0$ by continuity of
intersection numbers and this finishes the proof of the lemma. ∎

######  Remark.

Note that the above lemma can be used easily to show that the positive
intersection product is not $+\infty$ by bounding our adelic divisor by a
large nef model divisor. In case that the set of admissible approximations is
empty, we set the positive intersection product to be $-\infty$ as a matter of
set theoretic convention. However we will later see that it is never the case
when $\overline{D}$ is big.

Next we want to show that the positive intersection product defined above is
continuous at big divisors.

######  Lemma 3.6.

Suppose $\overline{D}$ is a big adelic divisor, $\overline{E}$ is a nef adelic
divisor and $\overline{F}$ is any adelic divisor on an essentially quasi-
projective variety $U$ over $k$ such that any quasi-projective model of it has
dimension $d$. Then there is a positive integer $m$ depending only on
$\overline{D},\overline{E}$ and $\overline{F}$ such that

$(1-mt)^{d-1}\langle\overline{D}^{d-1}\rangle\cdot\overline{E}\leq\langle(\overline{D}+t\overline{F})^{d-1}\rangle\cdot\overline{E}\leq(1+mt)^{d-1}\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$

for all rational $t\geq 0$. In particular for $t$ rational

$\lim_{t\to
0}\langle(\overline{D}+t\overline{F})^{d-1}\rangle\cdot\overline{E}=\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$

###### Proof.

As usual we assume that all the divisors live in a common quasi-projective
model $\mathcal{U}$ of $U$. We begin by noting that there is a positive
integer $m$ such that $m\overline{D}\pm\overline{F}\geq 0$. Indeed if
$\overline{D},\overline{F}$ are represented by a sequence $\\{D_{j},F_{j}\\}$
on some common models $X_{j}$ and rationals $q_{j}\to 0$ as usual, then as
$\overline{D}$ is big, thanks to [YZ21, Theorem 5.2.1(2)] we can find a $j$
such that $D_{j}-q_{j}D_{0}$ is big since $\\{D_{j}-q_{j}D_{0}\\}$ is also a
sequence of models representing $\overline{D}$. Then by Kodaira’s lemma
([Laz04, Prop 2.2.6)]) for the geometric case and [Yua08, Corollary 2.4(3)]
for the arithmetic case, we can find a positive integer $m$ such that
$m(D_{j}-q_{j}D_{0})-(F_{j}+q_{j}D_{0})\geq 0$ and
$m(D_{j}-q_{j}D_{0})+(F_{j}-q_{j}D_{0})\geq 0$. Now the usual Cauchy
effectivity relations $\overline{D}\geq D_{j}-q_{j}D_{0}$ and
$F_{j}-q_{j}D_{0}\leq\overline{F}\leq F_{j}+q_{j}D_{0}$ yield
$m\overline{D}-\overline{F}\geq 0$ and $m\overline{D}+\overline{F}\geq 0$
respectively.
Next note it is easy to check from the definition of positive intersection
products and Lemma 3.5 that if $\overline{D}_{1}\leq\overline{D}_{2}$ are two
adelic divisors and $\overline{E}$ any nef adelic divisor, then

$\langle\overline{D}_{1}^{d-1}\rangle\cdot\overline{E}\leq\langle\overline{D}_{2}^{d-1}\rangle\cdot\overline{E}$

since $\overline{E}$ is nef. Clearly by the choice of $m$ we have the
effectivity relations

$(1-mt)\overline{D}\leq\overline{D}+t\overline{E}\leq(1+mt)\overline{D}$

when $0<t<\frac{1}{m}$ Then the L.H.S of the inequality claimed in the lemma
easily follows for all rational $0<t<\frac{1}{m}$ and the right hand follows
for all rational $t$ by noticing that the positive intersection product is
homogeneous with respect to positive scaling and using the above effectivity
relations. The L.H.S for $t\geq\frac{1}{m}$ follows trivially by noting that
the positive intersection product is positive.

∎

As a consequence of the continuity of positive intersection products, we
derive two corollaries which will be useful for us. The first one gives an
alternate description of the positive intersection products for big divisors
in terms of nef divisors instead of free ones.

######  Corollary 3.7.

Suppose $\overline{D}$ is a big adelic divisor and $\overline{E}$ is a nef
adelic divisor on a quasi-projective variety $\mathcal{U}$ over $k$. Then we
have

$\langle\overline{D}^{d-1}\rangle\cdot\overline{E}=\sup_{(X^{\prime},\overline{A})}\overline{A}^{d-1}\cdot\overline{E}$

where the supremum is taken as $(X^{\prime},\overline{A})$ varies over all
tuples such that $X^{\prime}$ is a projective model of a birational
modification $\pi\colon\mathcal{U}^{\prime}\rightarrow\mathcal{U}$ and
$\overline{A}$ is a nef $\mathbb{Q}$-divisor on $X^{\prime}$ such that
$\pi^{*}\overline{D}\geq\overline{A}$ in
$\widehat{\emph{Div}}(\mathcal{U},k)$.

###### Proof.

Note that since all free $\mathbb{Q}$-divisors are automatically nef by our
remark earlier, we immediately get that
$\langle\overline{D}^{d-1}\rangle\cdot\overline{E}\leq\sup_{(X^{\prime},\overline{A})}\overline{A}^{d-1}\cdot\overline{E}$.
For the converse inequality, choose any arithmetically ample (or ample in the
geometric case) divisor $\overline{\omega}$ on $X^{\prime}$. Then since
$\overline{A}$ is nef, we get that
$\overline{A}+\epsilon\cdot\overline{\omega}$ is ample and hence free for any
positive rational $\epsilon$. Note that then
$(X^{\prime},\overline{A}+\epsilon\cdot\overline{\omega})$ is an admissible
approximation of the big adelic divisor
$\overline{D}+\epsilon\cdot\overline{\omega}$. Thus we get by the definition
of positive intersection products that

$(\overline{A}+\epsilon\cdot\overline{\omega})^{d-1}\cdot\overline{E}\leq\langle(\overline{D}+\epsilon\cdot\overline{\omega})^{d-1}\rangle\cdot\overline{E}$

Then letting $\epsilon\to 0$ and noting that positive intersection products
are continuous at $\overline{D}$ since it is big (Lemma 3.6), we deduce

$\overline{A}^{d-1}\cdot\overline{E}\leq\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$

Since $(X^{\prime},\overline{A})$ was an arbitary admissible approximation,
taking supremum of the left hand side above across all such approximations
easily gives the desired converse inequality and finishes the proof. ∎

######  Remark.

Note that the above lemma shows that we can choose our approximations such
that $\overline{A}$ is just nef instead of being free. Hence for the rest of
the article, we will freely choose approximations to be either nef or free
depending on what is suitable for our argument and this in particular shows
almost by definition that for model nef divisors $\overline{D}$, the positive
intersection products are the same as usual intersection products.

As a second corollary of the continuity of positive intersections, we deduce
that for nef and big arithmetic adelic divisors, the positive intersection
products are the same as usual absolute intersection numbers.

######  Corollary 3.8.

Suppose $\overline{D}$ is a big and nef arithmetic adelic divisors on $U$ and
$\overline{E}$ be any nef arithmetic adelic divisor. Then we have

$\langle\overline{\mathcal{D}}^{d}\rangle\cdot\overline{E}=\overline{D}^{d}\cdot\overline{E}$

###### Proof.

Note that after choosing a common quasi-projective model of $U$, we can assume
that $\overline{D}$ is given by a Cauchy sequence
$\\{\overline{D}_{i},q_{i}\\}$ as usual where each $\overline{D}_{i}$ is model
nef. Then we have the effectivity relation

$\overline{D}-q_{i}\overline{D}_{0}\leq\overline{D}_{i}\leq\overline{D}+q_{i}\overline{D}_{0}\
\text{for all}\ i$

for a sequence of positive rational numbers $q_{i}$ converging to $0$. Then we
have the inequality of positive intersection products

$\langle(\overline{D}-q_{i}\overline{D}_{0})^{d}\rangle\cdot\overline{E}\leq\langle\overline{D}_{i}^{d}\rangle\cdot\overline{E}\leq\langle(\overline{D}+q_{i}\overline{D}_{0}\rangle\cdot\overline{E}$

Hence we can deduce from Lemma 3.6 that
$\lim_{i\to\infty}\langle\overline{D}_{i}^{d}\rangle\cdot\overline{E}=\langle\overline{D}^{d}\rangle\cdot\overline{E}$
since $\overline{D}$ is big. However since $\\{\overline{D}_{i}\\}$ is a
sequence of model nef divisors converging to $\overline{D}$ and $\overline{E}$
is nef we have that
$\lim_{i\to\infty}\overline{D}_{i}^{d}\cdot\overline{E}=\overline{D}^{d}\cdot\overline{E}$
by how absolute intersection numbers for integrable adelic divisors are
constructed in [YZ21, Prop 4.1.1]. Now since $\overline{D}_{i}$ are model nef,
by our remark before this Lemma. we have that
$\langle\overline{D}_{i}^{d}\rangle\cdot\overline{E}=\overline{D}_{i}^{d}\cdot\overline{E}$.
Putting everything together we conclude

$\langle\overline{D}^{d}\rangle\cdot\overline{E}=\lim_{i\to\infty}\langle\overline{D}_{i}^{d}\rangle\cdot\overline{E}=\lim_{i\to\infty}\overline{D}_{i}^{d}\cdot\overline{E}=\overline{D}^{d}\cdot\overline{E}$

which finishes the proof. ∎

We want to show that the intersection products is additive in $\overline{E}$
nef so that we extend the product to integrable $\overline{E}$. For that we
first need to show that the family of admissible approximations are filtered
under the natural relation of dominance. We essentially use an argument of
Chen in [Che11] but since we work more generally over function fields, we
restate it for clarity.

######  Lemma 3.9.

Suppose $X$ is a projective variety over a valued Dedekind scheme $k$. Suppose
$\overline{D}$ is an arithmetic divisor on $X$ and
$\overline{A}_{1},\overline{A}_{2}$ are two free $\mathbb{Q}$-arithmetic
divisors on $\mathcal{X}$ such that $\overline{D}\geq\overline{A}_{i}$ for
$i=1,2$. Then there is a birational modification $\pi\colon
X^{\prime}\rightarrow X$ and a free arithmetic $\mathbb{Q}$-divisor
$\overline{A}$ on $X^{\prime}$ such that
$\overline{A}\geq\pi^{*}\overline{A}_{i}$ for $i=1,2$ and
$\pi^{*}\overline{D}\geq\overline{A}$.

###### Proof.

Note that the arithmetic case when $\Sigma$ is non-empty is exactly the
content of [Che11, Proposition 3.1] and the geometric case follows by the same
construction by ignoring the part about Hermitian metrics. ∎

######  Lemma 3.10.

For nef adelic divisors $\overline{E}_{1}$ and $\overline{E}_{2}$ and any big
adelic divisor $\overline{D}$, we have

$\langle\overline{D}^{d-1}\rangle\cdot(\overline{E}_{1}+\overline{E}_{2})=\langle\overline{D}^{d-1}\rangle\cdot\overline{E}_{1}+\langle\overline{D}^{d-1}\rangle\cdot\overline{E}_{2}$

###### Proof.

As usual we assume that all divisors live in a common quasi-projective model
$\mathcal{U}$ of $U$. Note that since

$\overline{A}^{d-1}\cdot(\overline{E}_{1}+\overline{E}_{2})=\overline{A}^{d-1}\cdot\overline{E}_{1}+\overline{A}^{d-1}\cdot\overline{E}_{2}$

for any admissible approximation $(X^{\prime},\overline{A})$ we deduce that

$\langle\overline{D}^{d-1}\rangle\cdot(\overline{E}_{1}+\overline{E}_{2})\leq\langle\overline{D}^{d-1}\rangle\cdot\overline{E}_{1}+\langle\overline{D}^{d-1}\rangle\cdot\overline{E}_{2}$

and hence it is sufficient to prove the other side of the inequality.
Suppose $(X_{1}^{\prime},\overline{A}_{1})$ and
$(X_{2}^{\prime},\overline{A}_{2})$ are two admissible approximations of
$\overline{D}$. We can assume by passing to finer models that
$X_{1}^{\prime}=X_{2}^{\prime}=X^{\prime}$. Then note that for $i=1,2$,
$\overline{A}_{i}\leq D_{j}+q_{j}D_{0}$ for a sequence $\\{D_{j}\\}$ and
rationals $q_{j}$ representing $\overline{D}$ as usual. By Lemma 3.9 we can
find a birational $\pi_{j}\colon X_{j}^{\prime}\rightarrow X_{j}$ and a free
$\mathbb{Q}$-divisor $\overline{A}\geq\overline{A}_{i}$ for $i=1,2$ such that
$\pi_{j}^{*}\overline{D}-\overline{A}\geq 0$. Then the effectivity relations
$D_{j}+q_{j}D_{0}\leq\overline{D}+2q_{j}D_{0}$ imply that
$(X_{j}^{\prime},\overline{A})$ is an admissible approximation of
$\overline{D}+2q_{j}D_{0}$ over $U$. Since
$\overline{A},\overline{A}_{i},\overline{E}_{i}$ are all nef, from Lemma 3.5
we have

$\overline{A}^{d-1}\cdot\overline{E}_{i}\geq\overline{A}_{i}^{d-1}\cdot\overline{E}_{i}\
\text{for}\ i=1,2$

Hence we deduce

$\langle(\overline{D}+2q_{j}D_{0})^{d-1}\rangle\cdot(\overline{E}_{1}+\overline{E}_{2})\geq\overline{A}^{d-1}\cdot(\overline{E}_{1}+\overline{E}_{2})\geq\overline{A}_{1}^{d-1}\cdot\overline{E}_{1}+\overline{A}_{2}^{d-1}\cdot\overline{E}_{2}$

Since $(X_{1},\overline{A}_{1})$ and $(X_{2},\overline{A}_{2})$ were arbitary
we deduce from above that

$\langle(\overline{D}+2q_{j}D_{0})^{d-1}\rangle\cdot(\overline{E}_{1}+\overline{E}_{2})\geq\langle\overline{D}^{d-1}\rangle\cdot\overline{E}_{1}+\langle\overline{D}^{d-1}\rangle\cdot\overline{E}_{2}$

Note that since $q_{j}$ was arbitary and we know that $q_{j}\to 0$, then using
that $\overline{D}$ is big, we deduce from Lemma 3.6 that

$\langle\overline{D}^{d-1}\rangle\cdot(\overline{E}_{1}+\overline{E}_{2})\geq\langle\overline{D}^{d-1}\rangle\cdot\overline{E}_{1}+\langle\overline{D}^{d-1}\rangle\cdot\overline{E}_{2}$

which finishes the proof. ∎

######  Remark.

Note that the linearity described above allows us to extend the definition of
positive intersection product of any big adelic divisor $\overline{D}$ with an
integrable adelic divisor $\overline{E}$ which we also denote as
$\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$.

We want obtain the main inequality required for the differentiability where we
estimate terms of order at least 2. However for later purpose we want to keep
track of the constants involved in the estimates and for that we need an easy
lemma which we record next.

######  Lemma 3.11.

Suppose $\overline{E}$ is integrable. Then we can write
$\overline{E}=\overline{E}_{1}-\overline{E}_{2}$ where $\overline{E}_{i}$ is
nef and effective for $i=1,2$.

###### Proof.

As usual we choose a common quasi-projective model $\mathcal{U}$ of $U$ where
all the divisors live. By definition of integrable divisors we can choose
$\overline{E}_{i}^{\prime}$ nef such that
$\overline{E}=\overline{E}_{1}^{\prime}-\overline{E}_{2}^{\prime}$. Choose
models $E_{1}^{\prime}$ and $E_{2}^{\prime}$ on a common projective model $X$
of $U$ such that $\overline{E}_{i}^{\prime}\geq E_{i}^{\prime}$. Then we can
choose an ample divisor $\overline{A}_{0}$ on $X$ such that
$E_{i}^{\prime}+\overline{A}_{0}$ is effective for $i=1,2$ by Serre’s
finiteness theorem. Then the effectivity relation shows
$\overline{E}_{i}^{\prime}+\overline{A}_{0}\geq 0$ and nef and hence we deduce
the claim for $\overline{E}_{i}=\overline{E}_{i}^{\prime}+\overline{A}_{0}$.
∎

Next we state the main inequality required for our differentiability.

######  Lemma 3.12.

Suppose $\mathcal{U}$ is a quasi-projecive model with dimension $d$ of an
essentially quasi-projective variety $U$ over $k$ and suppose
$(X^{\prime},\overline{A})$ is an admissible approximation of $\overline{D}$
on $\mathcal{U}$ and $\overline{E}=\overline{E}_{1}-\overline{E}_{2}$ is an
integrable adelic divisors with $\overline{E}_{i}$ nef and effective for
$i=1,2$. Then

$\widehat{\emph{vol}}(\overline{A}+t\overline{E})\geq\overline{A}^{d}+(d\overline{A}^{d-1}\overline{E})\cdot
t-Ct^{2}$

for all rational $0\leq t\leq 1$ where $C=2^{d}d^{2}\overline{\omega}^{d}$ for
any model nef divisor $\overline{\omega}$ such that
$\overline{\omega}\geq\overline{D}$ and
$\overline{\omega}\geq\overline{E}_{i}$ for $i=1,2$.

######  Remark.

As volumes and intersection numbers are birational invariants, we always
identify $\pi^{*}\overline{E}$ with $\overline{E}$ harmlessly unless otherwise
stated where $\pi\colon U^{\prime}\rightarrow U$ is the modification
corresponding to the admissible approximation $(X^{\prime},\overline{A})$.

###### Proof.

Choose a model nef divisor $\overline{\omega}$ such that
$\overline{\omega}\geq\overline{D}$ and
$\overline{\omega}\geq\overline{E}_{i}$ for $i=1,2$. First note that by Siu’s
inequality ([YZ21, Theorem 5.2.2(2)]) we obtain

$\widehat{\text{vol}}(\overline{A}+t\overline{E})=\widehat{\text{vol}}(\overline{A}+t\overline{E}_{1}-t\overline{E}_{2})\geq(\overline{A}+t\overline{E}_{1})^{d}-d(\overline{A}+t\overline{E}_{1})^{d-1}\cdot(t\overline{E}_{2})\
\text{for all positive rational}\ t$ (3)

noting that $\overline{A}+t\overline{E}_{1}$ and $t\overline{E}_{2}$ are nef
when $t>0$. Next we have the binomial expansion of intersection numbers

$(\overline{A}+t\overline{E}_{1})^{d}-d(\overline{A}+t\overline{E}_{1})^{d-1}\cdot(t\overline{E}_{2})=\overline{A}^{d}+(d\overline{A}^{d-1}\cdot\overline{E})\cdot
t-\sum_{k=1}^{d-1}B_{k}(d\overline{A}^{d-1-k}\overline{E}_{1}^{k}\overline{E}_{2})t^{k+1}+\sum_{k\geq
2}C_{k}(\overline{A}^{d-k}\overline{E}_{1}^{k})t^{k}$ (4)

where $B_{k}$ and $C_{k}$ are appropriate binomial coefficients all bounded by
$2^{d}$. Since by choice we have
$\overline{\omega}\geq\overline{D}\geq\overline{A}$ and
$\overline{\omega}\geq\overline{E}_{i}$ for $i=1,2$, by Lemma 3.5 we can bound
each coefficient of $t^{k}$ in the above sum for $k\geq 2$ by
$2^{d}\overline{\omega}^{d}$. Since all the terms of the last summand above
are positive, ignoring them we have the lower bound

$(\overline{A}+t\overline{E}_{1})^{d}-d(\overline{A}+t\overline{E}_{1})^{d-1}\cdot(t\overline{E}_{2})\geq\overline{A}^{d}+(d\overline{A}^{d-1}\cdot\overline{E})\cdot
t-Ct^{2}\ \text{for all}\ 0\leq t\leq 1$ (5)

by counting the number of such higher order terms in (4).

∎

We are ready to state and prove the main theorem of differentiability in this
section

######  Theorem 3.13.

Suppose $\overline{D}$ is a big adelic divisor and $\overline{E}$ is an
integrable adelic divisor on a normal essentially quasi-projective variety $U$
over $k$ such that any quasi-projective model has dimension $d$. Then the
function $t\mapsto\widehat{\emph{vol}}(\overline{D}+t\overline{E})$ is
differentiable at $t=0$ with derivative given by

$\frac{d}{dt}\widehat{\emph{vol}}(\overline{D}+t\overline{E})\mid_{t=0}=d\cdot\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$

###### Proof.

As usual we assume that all the divisors live in a common quasi-projective
model $\mathcal{U}$ of $U$ since the quantities defined in both sides of the
equality in the theorem are defined by choosing a quasi-projective model.
Suppose $\overline{D}$ is given by the data $\\{X_{i},D_{i},q_{i}\\}$ as
usual. Then choose an integer $r$ such that $r>q_{i}$ for all $i$ and $r>1$.
Choose some model nef $\overline{\omega}\geq\overline{D}+2r\overline{D}_{0}$
and $\overline{\omega}\geq\overline{E}_{i}$ where $\overline{E}_{i}$ are nef
effective adelic divisors with
$\overline{E}=\overline{E}_{1}-\overline{E}_{2}$ as chosen in the beginning of
the proof of Lemma 3.12. Thanks to Lemma 3.12 for any admissible approximation
$(X,\overline{A})$ of $\overline{D}$ on $\mathcal{U}$ we have that

$\widehat{\text{vol}}(\overline{D}+t\overline{E})\geq\widehat{\text{vol}}(\overline{A}+t\overline{E})\geq\overline{A}^{d}+(d\overline{A}^{d-1}\overline{E}_{1})\cdot
t-(d\overline{A}^{d-1}\overline{E}_{2})\cdot t-Ct^{2}\ \text{for all}\ 0\leq
t\leq 1$ (6)

for $C=2^{d}d^{2}(\overline{\omega})^{d}>0$. Now suppose we have any other
admissible approximation $(X^{\prime},\overline{A}^{\prime})$ of
$\overline{D}$ over $\mathcal{U}$. Note that after refining the models $X$ and
$X^{\prime}$ we can assume that both $(X,\overline{A})$ and
$(X^{\prime},\overline{A}^{\prime})$ are admissible approximations of
$D_{j}+q_{j}D_{0}$ over $X_{j}$ for all $j$ due to the effectivity relation
$\overline{D}_{j}+q_{j}\overline{D}_{0}\geq\overline{D}$. Then by Lemma 3.9 we
can find an admissible approximation $(X_{0},\overline{A}_{0})$ of
$D_{j}+q_{j}D_{0}$ depending on $j$ such that
$\overline{A}_{0}\geq\overline{A}$ and
$\overline{A}_{0}\geq\overline{A}^{\prime}$ in $X_{0}$. Note then
$\overline{A}_{0}$ is then an admissible approximation of
$\overline{D}+2q_{j}\overline{D}_{0}$ due to the effectivity relation
$\overline{D}+2q_{j}\overline{D}_{0}\geq D_{j}+q_{j}D_{0}$. Then applying (6)
with $\overline{D}$ replaced by $\overline{D}+2q_{j}\overline{D}_{0}$, we
conclude

$\widehat{\text{vol}}(\overline{D}+2q_{j}\overline{D}_{0}+t\overline{E})\geq\overline{A}_{0}^{d}+(d\overline{A}_{0}^{d-1}\overline{E}_{1})\cdot
t-(d\overline{A}_{0}^{d-1}\overline{E}_{2})\cdot t-Ct^{2}$ (7)

Note here that
$\overline{\omega}\geq\overline{D}+2r\overline{D}_{0}\geq\overline{D}+2q_{j}\overline{D}_{0}$
since $q_{j}<r$ and $\overline{D}_{0}$ is effective. Moreover we have
$\overline{\omega}\geq\overline{E}_{i}$ for $i=1,2$. Hence we can use the same
constant $C$ to bound the higher order terms. Next note that by definition of
the positive intersection we have
$d(\overline{A}_{0}^{d-1}\overline{E}_{2})<d\langle(\overline{D}+2q_{j}\overline{D}_{0})^{d-1}\rangle\cdot\overline{E}_{2}$
since $\overline{D}_{0}$ is an admissible approximation of
$\overline{D}+2q_{j}\overline{D}_{0}$. Hence we can bound the R.H.S of (7) by
the R.H.S of

$\widehat{\text{vol}}(\overline{D}+2q_{j}\overline{D}_{0}+t\overline{E})\geq\overline{A}_{0}^{d}+(d\overline{A}_{0}^{d-1}\overline{E}_{1})\cdot
t-d(\langle(\overline{D}+2q_{j}\overline{D}_{0})^{d-1}\rangle\cdot\overline{E}_{2})\cdot
t-Ct^{2}$ (8)

since $t$ is non-negative. Now noting that $\overline{A}_{0}\geq\overline{A}$
and $\overline{A}_{0}\geq\overline{A}^{\prime}$ are nef together with
$\overline{E}_{1}$ we conclude from Lemma 3.5 and the above inequality that

$\widehat{\text{vol}}(\overline{D}+2q_{j}\overline{D}_{0}+t\overline{E})\geq\overline{A}^{\prime
d}+(d\overline{A}^{d-1}\overline{E}_{1})\cdot
t-d(\langle(\overline{D}+2q_{j}\overline{D}_{0})^{d-1}\rangle\cdot\overline{E}_{2})\cdot
t-Ct^{2}$ (9)

Taking the supremum of inequality (9) across all admissible approximations
$(X^{\prime},\overline{A}^{\prime})$ while keeping $(X,\overline{A})$ fixed,
we deduce

$\widehat{\text{vol}}(\overline{D}+2q_{j}\overline{D}_{0}+t\overline{E})\geq\widehat{\text{vol}}(\overline{D})+(d\overline{A}^{d-1}\overline{E}_{1})\cdot
t-d(\langle(\overline{D}+2q_{j}\overline{D}_{0})^{d-1}\rangle\cdot\overline{E}_{2})\cdot
t-Ct^{2}$ (10)

as Fujita’s approximation ([YZ21, Theorem 5.2.8]) exactly says that
$\sup_{(X^{\prime},\overline{A}^{\prime})}\overline{A}^{\prime
d}=\widehat{\text{vol}}(\overline{D})$ as the $\sup$ is taken over all
admissible approximations of $\overline{D}$ on $U$ because all model
arithmetically positive divisors are model nef and $\overline{D}$ is big. Now
taking the $\sup$ of the inequality (10) across all admissible approximations
$(X,\overline{A})$ we deduce

$\widehat{\text{vol}}(\overline{D}+2q_{j}\overline{D}_{0}+t\overline{E})\geq\widehat{\text{vol}}(\overline{D})+(d\langle\overline{D}^{d-1}\rangle\cdot\overline{E}_{1})\cdot
t-(d\langle(\overline{D}+2q_{j}\overline{D}_{0})^{d-1}\rangle\cdot\overline{E}_{2})\cdot
t-Ct^{2}$ (11)

Finally since the above inequality is true for all $j$ and as $q_{j}\to 0$,
using the continuity of volumes ([YZ21, Theorem 5.2.9]) and positive
intersection numbers as in Lemma 3.2, taking limits as $j\to\infty$ while
noting that the same model nef $\overline{\omega}$ (and hence the same $C$)
works for the bounds, we deduce

$\widehat{\text{vol}}(\overline{D}+t\overline{E})\geq\widehat{\text{vol}}(\overline{D})+(d\langle\overline{D}^{d-1}\rangle\cdot\overline{E}_{1})\cdot
t-(d\langle\overline{D}^{d-1}\rangle\cdot\overline{E}_{2})\cdot t-Ct^{2}$ (12)

Using the linearity of positive intersections as in Lemma 3.10 since
$\overline{D}$ is big, we deduce

$\widehat{\text{vol}}(\overline{D}+t\overline{E})-\widehat{\text{vol}}(\overline{D})\geq(d\langle\overline{D}^{d-1}\rangle\cdot\overline{E})\cdot
t-Ct^{2}$ (13)

Now replacing $\overline{E}$ with $-\overline{E}$, using the linearity of
positive intersections and noting that the same model nef $\overline{\omega}$
(and hence the same $C$) works for the bounds, we conclude

$\widehat{\text{vol}}(\overline{D})-\widehat{\text{vol}}(\overline{D}-t\overline{E})\leq(d\langle\overline{D}^{d-1}\rangle\cdot\overline{E})\cdot
t+Ct^{2}$ (14)

We want to apply the inequality (13) with the flips
$\overline{D}\longleftrightarrow\overline{D}+t\overline{E}$ and
$\overline{E}\longleftrightarrow-\overline{E}$ and the inequality (14) with
the flips $\overline{D}\longleftrightarrow\overline{D}-t\overline{E}$ and
$\overline{E}\longleftrightarrow-\overline{E}$ for $0<t\ll 1$ small enough
positive rational such that $\widehat{\text{vol}}(\overline{D}\pm
t\overline{E})>0$ which can be obtained by continuity of adelic volumes
([YZ21], Theorem 5.2.9). Note that
$2\overline{\omega}\geq\overline{D}+t\overline{E}_{1}\geq\overline{D}+t\overline{E}$
for $0\leq t\leq 1$ since $\overline{E}_{i}$ are effective. Replacing
$\overline{E}$ by $-\overline{E}$ we also deduce
$2\overline{\omega}\geq\overline{D}-t\overline{E}$ for $0\leq t\leq 1$. Hence
we change the constant $C$ to $2^{d}d^{2}(2\overline{\omega})^{d}=2^{d}\cdot
C$. In other words we have

$\widehat{\text{vol}}(\overline{D})-\widehat{\text{vol}}(\overline{D}-t\overline{E})\geq(d\langle(\overline{D}-t\overline{E})^{d-1}\rangle\cdot\overline{E})\cdot
t-2^{d}Ct^{2}\ \ \ \ 0\leq t\ll 1$ (15)
$\widehat{\text{vol}}(\overline{D}+t\overline{E})-\widehat{\text{vol}}(\overline{D})\leq(d\langle(\overline{D}+t\overline{E})^{d-1}\rangle\cdot\overline{E})\cdot
t+2^{d}Ct^{2}\ \ \ \ 0\leq t\ll 1$ (16)

for $C=2^{d+1}d^{2}\overline{\omega}^{d}$. Finally the claim follows from the
equations (13), (14), (15) and (16) together with continuity described in
Lemma 3.6. ∎

We end the section by deducing the _isoperimetric inequality_ for adelic
divsors which is the analogue of [Che11, Proposition 4.5]).

######  Corollary 3.14.

Suppose $\overline{D}$ and $\overline{E}$ are adelic divisors such that
$\overline{D}$ is big and $\overline{E}$ is integrable and effective. Then we
have

$\langle\overline{D}^{d-1}\rangle\cdot\overline{E}\geq\widehat{\emph{vol}}(\overline{D})^{\frac{d-1}{d}}\widehat{\emph{vol}}\cdot(\overline{E})^{\frac{1}{d}}$

where $d$ is the dimension of any quasi-projective model of $U$.

###### Proof.

By Theorem 3.13 we have

$\lim_{t\to
0}\frac{\widehat{\text{vol}}(\overline{D}+t\overline{E})-\widehat{\text{vol}}(\overline{D})}{t}=d\cdot\langle\overline{D}^{d-1}\rangle\cdot\overline{E}$
(17)

Moreover since $\overline{E}$ is effective, by [YZ21, Theorem 5.2.6] we have

$\widehat{\text{vol}}(\overline{D}+t\overline{E})\geq(\widehat{\text{vol}}(\overline{D})^{\frac{1}{d}}+t\widehat{\text{vol}}(\overline{E})^{\frac{1}{d}})^{d}\
\text{for all positive rational}\ t$

Using the binomial expansion in the above inequality we get

$\widehat{\text{vol}}(\overline{D}+t\overline{E})-\widehat{\text{vol}}(\overline{D})\geq(d\cdot\widehat{\text{vol}}(\overline{D})^{\frac{d-1}{d}}\cdot\widehat{\text{vol}}(\overline{E})^{\frac{1}{d}})\cdot
t+O(t^{2})$

which together with the equation (17) yields the claim by taking the limit as
$t\to 0$.

∎

##  4 Relation with restricted volumes in the geometric case

In this section we define a slight variant of the positive intersection
product that we defined in the last chapter inspired by the _asymptotic
intersection numbers_ defined in [Ein+09]. We show using very similar
arguments as in the previous chapter to show that these intersection numbers
are continuous as well on big divisors. As a consequence we obtain a a
generalized Fujita approximation like theorem for restricted volumes in the
case that the sub-variety is not contained in the augmented base locus as an
adelic version of [Ein+09, Theorem 2.13].
We start working in thw the geometric setting assuming $U$ is now a normal
quasi-projective variety over a field $k$. Furthermore suppose $\overline{D}$
is an adelic divisor and $E$ is a closed sub-variety of $U$ of dimension
$d+1$. Furthermore suppose $E\xhookrightarrow{}U$ be a closed sub-variety.
Then we consider a slight variant of the admissible approximations that we
considered before.

######  Definition 4.1.

Suppose $\overline{D}$ and $E$ are as described above. Then we define an _$E$
-admissible approximation_ of $\overline{D}$ with respect to $U$ by the data
$(U^{\prime},X^{\prime},\overline{A},\pi)$ where $\pi\colon
U^{\prime}\rightarrow U$ is a birational morphism such that it is an
isomorphism over the generic point of $E$, $X^{\prime}$ is a projective model
of $U^{\prime}$ and $\overline{A}$ is a nef $\mathbb{Q}$-divisor on
$X^{\prime}$ such that $\pi^{*}\overline{D}-\overline{A}\geq 0$. We generally
abbreviate the whole data by $(X^{\prime},\overline{A})$.

We have defined the _augmented base locus_ $B_{+}(\overline{D})$ and the _the
restricted volume_ $\widehat{\text{vol}}_{U|E}(\overline{D})$ along $E$ (see
[Bis23, Def 2.4] and [Bis23, Def 2.6]). Furthermore we have showed that if
$E\nsubseteq B_{+}(\overline{D})$ and $\overline{D}$ is given by models
$\\{X_{i},D_{i}\\}$ as usual, then we have
$\widehat{\text{vol}}_{U|E}(\overline{D})=\lim_{i\to\infty}\text{vol}_{X_{i}|E_{i}}(D_{i})$
where $E_{i}$ is the Zariski closure of $E$ in $X_{i}$ and the terms in the
right hand side of the equality denote the classical projective restricted
volumes as defined in [Ein+09, Definition 2.1]viewing $D_{i}$ as a
$\mathbb{Q}$-divisor in $X_{i}$ (see[Bis23, Corollary 2.17]). It is easy to
check using the Zariski density of $U$ in each $X_{i}$ that
$\widehat{\text{vol}}_{U|E}(\overline{D_{i}})=\text{vol}_{X_{i}|E_{i}}(D_{i})$
where in the left we view $D_{i}$ as model adelic divisors. We next define a
positive intersection product of a divisor $\overline{D}$ along $E$.

######  Definition 4.2.

Suppose $\overline{D}$ and $E$ are as described above. We define

$\langle\overline{D}^{d}\rangle\cdot
E=\sup_{(X^{\prime},\overline{A})}\overline{A}^{d}\cdot\tilde{E}$

where $(\pi\colon X^{\prime}\rightarrow X,\overline{A})$ varies over all
$E$-admissible approximations of $\overline{D}$ on $U$ and
$\tilde{E}=\pi^{-1}(E)$.

######  Remark.

Note that since we assume $\pi\colon U^{\prime}\rightarrow U$ to be an
isomorphism over the generic point of $E$, the strict transforms $\tilde{E}$
always have the same Krull dimension as $E$ and hence the intersection numbers
appearing in the definition actually make sense. Note further that it is easy
to check that the above product is homogenous with respect to positive scaling
in $\overline{D}$ just as for positive intersection products defined above.
Hence we can extend the notion of the above product also to
$\mathbb{Q}$-adelic divisors.

We begin by showing that this product is continuous in the sense of Lemma 3.6.
The proof will be very similar and hence we mention only the crucial points.

######  Lemma 4.3.

Suppose $\overline{D}$ is a big adelic divisor and $\overline{F}$ is any
adelic divisor on $U$. Then there is a positive integer $m$ depending only on
$\overline{D}$ and $\overline{F}$ such that

$(1-mt)^{d}\langle\overline{D}^{d}\rangle\cdot
E\leq\langle(\overline{D}+t\overline{F})^{d}\rangle\cdot
E\leq(1+mt)^{d}\langle\overline{D}^{d}\rangle\cdot\ E$

for all rational $\frac{1}{m}>t\geq 0$. In particular for $t$ rational

$\lim_{t\to 0}\langle(\overline{D}+t\overline{F})^{d}\rangle\cdot
E=\langle\overline{D}\rangle^{d}\cdot E$

###### Proof.

Note first that since $\overline{D}$ is big, using the same argument as in the
beginning of the proof of Lemma 4.3 we can find a positive integer $m$
depending on $\overline{D}$ and $\overline{F}$ such that
$m\overline{D}\pm\overline{F}\geq 0$. Next note that it is easy to check from
the definition of the positive intersection that if
$\overline{D}_{1}\leq\overline{D}_{2}$ are two adelic divisors then
$\langle\overline{D}_{1}^{d}\rangle\cdot
E\leq\langle\overline{D}_{2}^{d}\rangle\cdot E$. Now noting the sequence of
effectivity relations

$(1-mt)\overline{D}\leq\overline{D}+t\overline{F}\leq(1+mt)\overline{D}\
\text{for all positive rational}\ t$

and noting the homogneity of positive intersections with positive scaling we
obtain the claim readily for all rational $0\leq t<\frac{1}{m}$.

∎

Next suppose $D$ is a $\mathbb{Q}$-divisor on a projective model $X$ of $U$
and suppose $\overline{E}$ is the Zariski closure of $E$ in $X$. Then we can
consider the _asymptotic intersection number_ $||D^{d}\cdot\overline{E}||$ as
defined in [Ein+09, Definition 2.6] Furthermore using the alternate
description of this product when $\overline{E}\nsubseteq\overline{B_{+}}(D)$
as described in [Ein+09, Prop 2.11] and our definition in 4.2 we have
$\langle\overline{D}\rangle^{d}\cdot E=||D^{d}\cdot\overline{E}||$ provided
$\overline{E}\nsubseteq\overline{B_{+}}(D)$. Here in the left we consider $D$
as an model adelic $\mathbb{Q}$-divisor and $\overline{B_{+}}(D)$ denotes the
classical augmented base locus as defined in [Ein+09, Section 1]. With this
observation and using the continuity we show that positive intersection at a
big adelic divisor is given by limit of positive intersections of models after
perturbing them a little by the boundary divisor.

######  Lemma 4.4.

Suppose $\overline{D}$ is a big adelic divisor given by a sequence
$\\{X_{i},D_{i},q_{i}\\}$ as usual and furthermore suppose that $E\nsubseteq
B_{+}(\overline{D})$. Then we have

$\langle\overline{D}^{d}\rangle\cdot
E=\lim_{i\to\infty}\langle(\overline{D}_{i}+q_{i}\overline{D}_{0})^{d}\rangle\cdot
E=\lim_{i\to\infty}||(D_{i}+q_{i}D_{0})^{d}\cdot\overline{E}_{i}||$

where $\overline{E}_{i}$ is the Zariski closure of $E$ in $X_{i}$.

###### Proof.

Note first that since $\overline{D}\leq\overline{D}_{i}+q_{i}\overline{D}_{0}$
we have that $B_{+}(\overline{D}_{i}+q_{i}\overline{D}_{0})\subseteq
B_{+}(\overline{D})$ which in particular implies that $E\nsubseteq
B_{+}(\overline{D}_{i}+q_{i}\overline{D}_{0})$. Now using Zariski density of
$U$ in each $X_{i}$ we conclude that
$\overline{E}_{i}\nsubseteq\overline{B_{+}}(D_{i}+q_{i}D_{0})$ now looking at
the models as $\mathbb{Q}$-divisors on projective varieties. Hence we conclude
from the discussion before the lemma that
$\langle(\overline{D}_{i}+q_{i}\overline{D}_{0})^{d}\rangle\cdot
E=||(D_{i}+q_{i}D_{0})^{d}\cdot\overline{E}_{i}||$ for all $i$. Thus we just
need to prove the first equality in the above claim. Note that the effectivity
relations
$\overline{D}\leq\overline{D}_{j}+q_{j}\overline{D}_{0}\leq\overline{D}+2q_{j}\overline{D}_{0}$
yields the inequalities

$\langle\overline{D}^{d}\rangle\cdot
E\leq\langle(\overline{D}_{i}+q_{i}\overline{D}_{0})^{d}\rangle\cdot
E\leq\langle(\overline{D}+2q_{j}\overline{D}_{0})^{d}\rangle\cdot E$

by the second point listed in the proof of the previous lemma. This then
evidently yields the claim by the continuity and as $q_{j}\to 0$ by noting
that $\overline{D}$ is assumed big. ∎

Finally we can state the main theorem of this section which is a variant of
Fujita approximation for restricted volumes.

######  Theorem 4.5.

Suppose $\overline{D}$ is a big adelic divisor and suppose $E$ is a closed
sub-variety of $U$ such that $E\nsubseteq B_{+}(\overline{D})$. Then we have

$\widehat{\emph{vol}}_{U|E}(\overline{D})=\langle\overline{D}^{d}\rangle\cdot
E$

###### Proof.

We noted in the proof of the previous lemma that
$\overline{E}_{i}\nsubseteq\overline{B_{+}}(D_{i}+q_{i}D_{0})$ for all $i$.
Hence by [Ein+09, Theorem 2.13] we conclude that
$\text{vol}_{X_{i}|\overline{E}_{i}}(D_{i}+q_{i}D_{0})=||(D_{i}+q_{i}D_{0})^{d}\cdot\overline{E}_{i}||$.
Furthermore since $E\nsubseteq B_{+}(\overline{D})$ we have shown earlier that
$\widehat{\text{vol}}_{U|E}(\overline{D})=\lim_{i\to\infty}\text{vol}_{X_{i}|\overline{E}_{i}}(D_{i}+q_{i}D_{0})$
since $\\{D_{i}+q_{i}D_{0}\\}$ is a Cauchy sequence of models representing
$\overline{D}$. Thus combing the above with Lemma 4.4 we deduce the claim of
the theorem. ∎

######  Remark.

Note that when $\overline{D}$ is a nef $\mathbb{Q}$-divisor we have
$\langle\overline{D}\rangle^{d}\cdot E=\overline{D}^{d}\cdot
E=\widehat{\text{vol}}_{U|E}(\overline{D})$ and hence the above theorem states
that for general enough sub-varieties $E$ with respect to $\overline{D}$, the
restricted volume is approximated by restricted volume of nef divisors
dominated by it which is analogous to the usual Fujita approximation for
volumes. A point to note here is that just like we assume the divisor to be
big for usual Fujita approximation _i.e_ we require
$\widehat{\text{vol}}(\overline{D})>0$, we require $E\nsubseteq
B_{+}(\overline{D})$ for restricted Fujita approximation to hold above and
that readily implies $\widehat{\text{vol}}_{U|E}(\overline{D})>0$. A natural
question to ask would be whether the converse is true _i.e_ whether
$\widehat{\text{vol}}_{U|E}(\overline{D})>0\Rightarrow E\nsubseteq
B_{+}(\overline{D})$. This is always the case in the projective setting which
is obtained as the main theorem in [Ein+09, Theorem (C)]. Unfortunately the
proof heavily uses cohomological methods via separation of jets and it is not
clear how to extend these methods in the quasi-projective setting.

##  5 Applications

In this section, we simultaneously consider the arithmetic and geometric case
to obtain a differentiability result for _asymptotic slopes_ following the
arguments given by Chen for arithmetic divisors on a projective arithmetic
variety (see [Che11, Proposition 5.1]). As a principal application of it we
deduce an equidistribution result for big divisors and we show that it
generalises the equidistribution result obtained by Yuan and Zhang in [YZ21,
Theorem 5.4.3].
For the rest of the section, we work over a quasi-projective variety $U$ over
$K$ where $K$ is a number field or a function field of one variable over some
field $F$. Note that both the cases are included in the general formalism of
valued Dedekind schemes introduced in section 2. Note that then we can view
$U$ as an essentially quasi-projective variety over $k=(B,\Sigma)$ in both the
cases where $B=\text{Spec}(\mathbb{Z})$ when $K$ is a number field and
$B=\mathbb{P}_{F}^{1}$ when $K$ is a function field of one variable over $F$
and our considerations in sections 1 and 2 go through in this case. Note that
then we have the group of arithmetic adelic divisors on $U$ over $k$ denoted
by $\widehat{\text{Div}}(U,k)$ and the group of geometric adelic divisors on
$U$ over $K$ which we denote by $\widehat{\text{Div}}(U,K)$. We denote the
arithmetic adelic divisors by $\overline{\mathcal{D}}$ and the geometric
adelic divisors by $\overline{D}$ for sake of clarity. Furthermore we have the
generic fiber map
$\widehat{\text{Div}}(U,k)\rightarrow\widehat{\text{Div}}(U,K)$ as described
in [YZ21, Section 2.5.5]. Furthermore $d$ will always denote the dimension of
$U$ and since we are working assuming that $K$ is either a number field or a
function field of one variable this means that any quasi-projective model of
$U$ will always have dimension $d+1$.

######  Remark.

Suppose $\overline{\mathcal{D}}\in\widehat{\text{Div}}(U,k)$ is a big
arithmetic adelic divisor. Then from [YZ21, Theorem 5.2.1(2)] we can deduce
that there is a projective model $\mathcal{X}$ of $U$ and an arithmetic
divisor $\overline{\mathcal{D}}_{1}$ such that
$\overline{\mathcal{D}}_{1}\leq\overline{\mathcal{D}}$ in
$\widehat{\text{Div}}(U,k)$ and $\overline{\mathcal{D}}_{1}$ is big. Then from
[CM20, Prop 6.4.18] we can deduce that the generic fiber $\overline{D}_{1}$ of
$\overline{\mathcal{D}}_{1}$ is big and $\overline{D}_{1}\leq\overline{D}$ in
$\widehat{\text{Div}}(U,K)$ where $\overline{D}$ is the generic fiber of
$\overline{\mathcal{D}}$. Hence we deduce that the generic fiber
$\overline{D}$ is always big whenever $\overline{\mathcal{D}}$ is
arithmetically big.

For any big arithmetic adelic divisor
$\overline{\mathcal{D}}\in\widehat{\text{Div}}(U,k)$, we can then define the
asymptotic positive slope $\hat{\mu}_{+}^{\pi}(\overline{\mathcal{D}})$ as

$\hat{\mu}_{+}^{\pi}(\overline{\mathcal{D}})=\frac{\widehat{\text{vol}}(\overline{\mathcal{D}})}{(d+1)\widehat{\text{vol}}(\overline{D})}$

It is defined in the projective setting by Chen (see [Che11, Section 5])
working over number fields. We derive as a corollary of our differentiability
results before that this slope function is differentiable when
$\overline{\mathcal{D}}$ is big.

######  Corollary 5.1.

Suppose $\overline{\mathcal{D}}$ is a big arithmetic adelic divisor and
$\overline{\mathcal{M}}$ an integrable arithmetic adelic divisor in
$\widehat{\emph{Div}}(U,k)$. Then we have

$\lim_{t\to
0}\frac{\hat{\mu}_{+}^{\pi}(\overline{\mathcal{D}}+t\overline{\mathcal{M}})-\hat{\mu}_{+}^{\pi}(\overline{\mathcal{D}})}{t}=\frac{\langle\overline{\mathcal{D}}\rangle^{d}\cdot\overline{\mathcal{M}}}{\widehat{\emph{vol}}(\overline{D})}-\frac{d\langle\overline{D}\rangle^{d-1}\cdot\overline{M}}{\widehat{\emph{vol}}(\overline{D})}\hat{\mu}_{+}^{\pi}(\overline{\mathcal{D}})$

###### Proof.

The proof is quite straightforward by applying the following elementary
principle for Analysis 1. If $f(t),g(t)$ are two differentiable functions on
$\mathbb{R}$, then the quotient $\frac{f(t)}{g(t)}$ is also differentiable at
a point $t_{0}$ such that $g(t_{0})\neq 0$ and the derivative at $t_{0}$ is
given by
$\frac{g(t_{0})f^{\prime}(t_{0})-f(t_{0})g^{\prime}(t_{0})}{g(t_{0})^{2}}$. We
apply the above to the functions
$f(t)=\widehat{\text{vol}}(\overline{\mathcal{D}}+t\overline{\mathcal{M}})$
and $g(t)=\widehat{\text{vol}}(\overline{D}+t\overline{M})$ which are both
differentiable by Theorem 3.13 to deduce the claim since
$\mu_{+}^{\pi}(\overline{\mathcal{D}}+t\overline{\mathcal{M}})=\frac{1}{(d+1)}\frac{f(t)}{g(t)}$.
∎

######  Remark.

Note that there is a notion of vector valued heights as introduced in [YZ21,
Section 5.3.1] and in the case of a number field or function field in one
variable, by composing with the Arakelov degree on $\widehat{\text{Pic}}(K)$
we get the notion of a real valued height. When the arithmetic adelic divisor
$\overline{\mathcal{D}}$ is also nef in addition to satisfying the hypotheses
of Corollary 5.1 then it can be checked easily using [YZ21, Theorem 5.2.2(1)]
that the above asymptotic positive slope is exactly equal to
$h_{\overline{\mathcal{D}}}(U)$. However it might not be true that the
perturbations $\overline{\mathcal{D}}+t\overline{\mathcal{M}}$ are still nef
for very small $t$ and hence we cannot deduce from Corollary that the height
functions from [YZ21] are differentiable at big points.

We want to use the differentiability of the asymptotic positive slope to study
the well known variational principle. We fix a place $v$ over $K$ and we
denote by $U_{v}$ the $K_{v}$-analytic space associated to $U\times_{K}K_{v}$
where $K_{v}$ is the completion of $K$ at $v$. Note that as explained in
[Ber90, Chapter 1], they are the the usual complex analytic spaces associated
to $U\times_{K}K_{v}$ when $v$ is complex Archimedean and the quotient of the
usual complex analytic spaces of $U\times_{K}K_{v}$ by complex conjugation
when $v$ is real Archimedean and the Berkovich spaces studied in [Ber90] when
$v$ is non-Archimedean. For a geometric point $x\in U(\overline{K})$, we
consider the discrete measure

$\eta_{x}:=\frac{1}{\\#O(\tilde{x})}\sum_{y\in O(\tilde{x})}\delta_{y}$

on $U_{v}$ where $\tilde{x}$ denotes the induced closed point on
$U\times_{K}\overline{K_{v}}$ from $x$, $O(\tilde{x})$ denotes the Galois
orbit of $x$ under the action of $\text{Gal}(\overline{K_{v}}/K_{v})$ and
$\delta_{y}$ is the Dirac measure concentrated at $y$.
Now we consider a sequence of points $\\{x_{m}\\}$ of geometric points in
$U(\overline{K})$ and we recall two definitions of sequence of points crucial
in stating equidistribution conjectures. Recall that given an arithmetic
adelic divisor $\overline{\mathcal{D}}$ on $\mathcal{U}$, there is real valued
height function $h_{\overline{\mathcal{D}}}(\cdot)\colon
U(\overline{K})\rightarrow\mathbb{R}$ as explained in [YZ21, Section 5.3.1].

######  Definition 5.2.

We call a sequence of geometric points $\\{x_{m}\\}$ in $U(\overline{K})$
_generic_ if the sequence is dense in the Zariski topology of $U$ _i.e_ for
all non-empty Zariski open subset $V$ of $U$, $x_{m}\in V(\overline{K})$ for
large enough $m$. Given an arithmetic adelic divisor $\overline{\mathcal{D}}$,
we call $\\{x_{m}\\}$ is _small_ w.r.t $\overline{\mathcal{D}}$ if
$\lim_{m\to\infty}h_{\overline{\mathcal{D}}}(x_{m})=\mu_{+}^{\pi}(\overline{\mathcal{D}})$

######  Remark.

Note that the definition of a small sequence above is slightly different from
the usual definition of a small sequence in literature (or _directionally
small_ as termed by Yuan and Zhang, see [YZ21, Section 5.4.1]). However when
$\overline{\mathcal{D}}$ is nef then our definition agrees withe classical
cone since then
$\mu_{+}^{\pi}(\overline{\mathcal{D}})=h_{\overline{\mathcal{D}}}(U)$.

The equidistribution theorems and conjectures deal with the problem of
determining conditions on the sequence $\\{x_{m}\\}$ and
$\overline{\mathcal{D}}$ such that the sequence of integrals $\int_{U_{v}}g\
d\eta_{x_{m}}$ converges to some real number in $\mathbb{R}$ as $m\to\infty$
for every continuous function $g$ on $U_{v}$ with compact support.
In [YZ21, Theorem 3.6.4], Yuan and Zhang introduce the notion of compactified
Green functions as

$C^{0}_{\text{cptf}}(U_{v}):=\\{f\in C^{0}(U_{v})\mid\frac{f}{g_{0}}\to 0\
\text{along the boundary}\\}$

where $g_{0}$ is the green function of any boundary divisor. Theorem [YZ21,
Theorem 3.6.4] tells us that for any $g\in C^{0}_{\text{cptf}}(U_{v})$ there
is an adelic divisor $\mathcal{O}(g)$ in $\widehat{\text{Div}}(U,k)$.
Furthermore construction identifies $C^{0}_{\text{cptf}}(U_{v})$ with a
subgroup of $\widehat{\text{Div}}(U,k)$ and contains the subgroup of compactly
supported continuous functions $C_{c}^{0}(U_{v})$ on $U_{v}$. The following
description of the height function of $\mathcal{O}(g)$ follows easily by
construction of intersection numbers when $\mathcal{O}(g)$ is integrable and
allows us to relate the measures $\eta_{x_{m}}$ with the integrals
$\int_{U_{v}}g\ d\eta_{x_{m}}$

$h_{\mathcal{O}(g)}(x_{m})=\int_{U_{v}}g\ d\eta_{x_{m}}$ (18)

We want use the variational principle and obtain our equidistribution from
differentiability in the direction of the divisor $\mathcal{O}(g)$ but
$\mathcal{O}(g)$ is not integrable in the sense of Yuan and Zhang for any
arbitary $g$. However there are a particular class of functions for which the
associated divisor is integrable and they approximate any arbitary function
with compact support under the uniform topology which we introduce next.

######  Definition 5.3.

Suppose $g\in C^{0}(U_{v})$ is a continuous function on $U_{v}$. We say that
$g$ is a _model function_ in either of the two following cases

  1. 1.

$v$ is Archimedean and $g$ is the restriction of a smooth function on some
projective model of $U_{v}$.

  2. 2.

$v$ is non-Archimedean and there is a projective model $\mathcal{X}$ of
$U\times_{B}\emph{Spec}(O_{B,v})$ over the local ring $O_{B,v}$ of $B$ at $v$
and a vertical Cartier divisor $\mathcal{A}$ on $\mathcal{X}$ such that a
positive multiple of $g$ is the function induced by $\mathcal{A}$ on $U_{v}$
via restriction.

We denote the group of model functions on $U_{v}$ by $C_{\emph{mod}}(U_{v})$.

Note that any model function $f$ is bounded and hence in particular
$C^{0}_{\text{mod}}(U_{v})\subseteq C^{0}_{\text{cptf}}(U_{v})$. We go on to
show that the adelic divisors $\mathcal{O}(g)$ are integrable in the sense of
Yuan and Zhang when $g$ is model.

######  Lemma 5.4.

Suppose $g\in C_{c}^{0}(U_{v})$ is a smooth model function with compact
support. Then the associated arithmetic adelic divisor $\mathcal{O}(g)$ is
integrable.

###### Proof.

We first assume the case when $v$ is non-archimedean. Then since $g$ is model,
be definition there is a projective model $\mathcal{X}$ of $U$ over $k$ with
generic fiber $X$ and a vertical divisor $\mathcal{A}$ on $\mathcal{X}$ such
that $g$ is induced by the model function of $\mathcal{A}$. Here we identify
$g$ as a continuous function on $X_{v}$ and $U_{v}$ by extending it to 0
outside its support. Then using Serre’s theorem we can write
$\mathcal{A}=\mathcal{A}_{1}-\mathcal{A}_{2}$ where $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ are ample divisors on $\mathcal{X}$. Then clearly writing
$\mathcal{O}(g)=\mathcal{O}(g_{1})-\mathcal{O}(g_{2})$ where $g_{1}$ and
$g_{2}$ denotes the induced model functions from $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ respectively, we conclude that $\mathcal{O}(g)$ is
integrable.
For Archimedean $g$, we give a sketch of the argument. Like before we choose a
projective model $\mathcal{X}$ of $U$ with generic fiber $X$. Since $g$ is
smooth, $\mathcal{O}(g)$ induces a smooth Chern form on $X_{v}$. Choose an
ample divisor $A$ on $X$ with usual Fubiny-Study form whic h has positive
curvature and adding high enough powers of it to $\mathcal{O}(g)$, we get that
$\mathcal{O}(g)+m\overline{A}$ also has positive curvature. Then writing
$\mathcal{O}(g)=\mathcal{O}(g)+m\overline{A}-m\overline{A}$ we deduce the
claim. We refer to [Gub03, Prop 10.4] and [YZ21, Section 2.1.1] for more
general versions and details. ∎

######  Remark.

To highlight the importance of the model functions, we remark here that these
functions approximate arbitary continuous functions on $U_{v}$ with compact
support (looking at them as continuous functions on the analytification of
some projective model of $U$) under the uniform topology. Indeed this is the
famous Weierstrass approximation theorem in the Archimedean case and was
proved by Gubler in [Gub98, Theorem 7.12] in the non-archimedean case over
algebraically closed fields. Gubler’s arguments were adapted by Yuan [Yua08,
Section 3] for the non-algebraically closed case. In particular this shows
that the functions considered in Lemma 5.4 approximate arbitary continuous
functions uniformly in both the cases of number fields and function fields and
this fact will be important to us to reduce the case of weak convergence to
model situations. Furthermore note that the space of model functions with
compact support form a subgroup of $\widehat{\text{Div}}(U,k)$ under the
inclusion into $C^{0}_{\text{cptf}}(U_{v})$ since model functions are always
bounded and we denote this subgroup of $\widehat{\text{Div}}(U,k)$ by
$C^{0}_{\text{mod}}(U_{v})$.

Our strategy to attack the equidistribution problem is identical to the
strategy in [Che11, Section 5.1] but we still repeat the arguments for
clarity. Given a sequence of geometric points
$\\{x\\}:=\\{x_{m}\\}_{m\in\mathbb{N}}$ in $U(\overline{K})$, we begin by
defining the following function

$\phi_{\\{x\\}}\colon\widehat{\text{Div}}(U,k)\rightarrow\mathbb{R}\cup\\{\pm\infty\\}$
$\overline{\mathcal{D}}\mapsto\liminf_{m\to\infty}h_{\overline{\mathcal{D}}}(x_{m})$

The additivity of $h_{\overline{\mathcal{D}}}(\cdot)$ on geometric points
easily shows that $\phi_{\\{x\\}}(\cdot)$ is a super-additive function on
$\widehat{\text{Div}}(U,k)$. Next we record two important properties of
$\phi_{\\{x\\}}(\cdot)$ that will be relevant in our future considerations.
Recall that a sequence of Radon measures $\\{d\eta_{m}\\}$ on $U_{v}$ is said
to _weakly converge_ to another Radon measure $d\eta$ if
$\lim_{m\to\infty}\int_{U_{v}}g\ d\eta_{m}=\int_{U_{v}}g\ d\eta$ for all $g\in
C_{c}^{0}(U_{v})$. If the limit measure is not specified and the integrals
converge to some real number, then we say the sequence $\\{d\eta_{m}\\}$
converges weakly. Given a big arithmetic adelic divisor
$\overline{\mathcal{D}}$, we can define the sub-semigroup

$C(U,\overline{\mathcal{D}}):=\\{m\overline{\mathcal{D}}+\mathcal{O}(f)\mid
m\in\mathbb{N},\ m\overline{\mathcal{D}}+\mathcal{O}(f)\ \text{is big and}\
f\in C^{0}_{\text{mod}}(U_{v})\\}$

of $\widehat{\text{Div}}(\mathcal{U},k)$.
Recall that given a sub-semigroup $C$ and a subgroup $H$ of an abelian group
$G$, we say that $C$ is _open with respect to_ $H$ if for all $g\in C$ and
$h\in H$, we have that $mg+h\in C$ for all large enough positive integers $m$.
It is easy to see that $C(U,\overline{\mathcal{D}})$ is open with respect to
the sub-group $C^{0}_{\text{mod}}(U_{v})$ which will be crucial to use [Che11,
Proposition 5.4]. Indeed given $g\in C^{0}_{\text{mod}}(U_{v})$ and $f\in
C^{0}_{\text{mod}}(U_{v}),\ m\in\mathbb{N}$ such that
$m\overline{\mathcal{D}}+\mathcal{O}(f)\in C(U,\overline{\mathcal{D}})$ , for
large enough positive integer $n$ we have
$n(m\overline{\mathcal{D}}+\mathcal{O}(f))+\mathcal{O}(g)=mn\overline{\mathcal{D}}+\mathcal{O}(nf+g)\in
C(U,\overline{\mathcal{D}})$ since $nf+g\in C^{0}_{\text{mod}}(U_{v})$ for all
positive integers $n$, the adelic volume function
$\widehat{\text{vol}}(\cdot)$ is positive homogeneous and continuous and
$m\overline{\mathcal{D}}+\mathcal{O}(f)$ itself is big. Next we record two
lemmas which will be useful later.

######  Lemma 5.5.

Suppose $\overline{\mathcal{D}}\in\widehat{\emph{Div}}(U,k)$ and
$\\{x\\}:=\\{x_{m}\\}$ be a sequence of geometric points in $U(\overline{K})$
such that the sequence $\\{h_{\overline{\mathcal{D}}}(x_{m})\\}$ is
convergent. For any $f\in C^{0}_{\emph{cptf}}(U_{v})$, the following are
equivalent

  * •

The sequence $\\{\int_{U_{v}}f\ d\eta_{x_{m}})\\}$ converges as $m\to\infty$.

  * •

The function $\phi_{\\{x\\}}(\cdot)$ is differentiable at
$\overline{\mathcal{D}}$ in the direction $\mathcal{O}(f)$.

Furthermore if one of the conditions hold, then

$\lim_{m\to\infty}\int_{U_{v}}f\
d\eta_{x_{m}}=\phi_{\\{x\\},{\overline{\mathcal{D}}}}^{\prime}(f)$

where $\phi_{\\{x\\},{\overline{\mathcal{D}}}}^{\prime}(f)$ denotes the
differential of $\phi_{\\{x\\}}$ at $\overline{\mathcal{D}}$ calculated at
$\mathcal{O}(f)$.

###### Proof.

The proof of the equivalence is identical to the proof of [Che11, Theorem 5.3]
but we still sketch a proof for clarity. The series of equalities that is
crucial for the proof is

$\phi_{\\{x\\}}(n\overline{\mathcal{D}}+\mathcal{O}(f))=\liminf_{m\to\infty}(nh_{\overline{\mathcal{D}}}(x_{m})+\int_{U_{v}}f\
d\eta_{x_{m}})=n\phi_{\\{x\\}}(\overline{\mathcal{D}})+\phi_{\\{x\\}}(\mathcal{O}(f))$
(19)

for all positive integers $n$ and $f\in C^{0}_{\text{cptf}}(U_{v})$. The first
equality above easily follows from the additivity of the height function on
the geometric points. Now suppose that the first assertion holds. Then the
second equality easily follows since the second summand in the middle term of
equality (19) is convergent as $m\to\infty$. Replacing $f$ by $-f$ in the
above inequality clearly shows that the function $\phi_{\\{x\\}}$ is
differentiable at $\overline{\mathcal{D}}$ along the direction of $f$. In fact
it shows that it is linear in the direction of $f$.
Conversely if we assume that the second assertion holds, then also the second
equality of the equation (19) holds but this time since the first summand of
the middle term in equation (19) is convergent. The differentiability of
$\phi_{\\{x\\}}$ then allows us to deduce by taking limits as $n\to\infty$
that

$\phi_{\\{x\\},\overline{\mathcal{D}}}^{\prime}(\mathcal{O}(f))=\phi_{\\{x\\}}(\mathcal{O}(f))$

where the left hand-side above denotes the differential of $\phi_{\\{x\\}}$ at
$\overline{\mathcal{D}}$ in the direction of $\mathcal{O}(f)$. Since
differentials are linear maps, replacing $\mathcal{O}(f)$ by $\mathcal{O}(-f)$
we deduce from the above equality that

$\phi_{\\{x\\}}(\mathcal{O}(-f))=-\phi_{\\{x\\}}(\mathcal{O}(f))$

Recalling the definition of the function $\phi_{\\{x\\}}$ we deduce that

$\limsup_{m\to\infty}\int_{U_{v}}f\
d\eta_{x_{m}}=\liminf_{m\to\infty}\int_{U_{v}}f\
d\eta_{x_{m}}=\phi^{\prime}_{\\{x\\},\overline{\mathcal{D}}}(\mathcal{O}(f))$

which proves the first assumption. Note that equation (19) also shows the last
additional claim. ∎

######  Lemma 5.6.

If $\\{x\\}$ is a generic sequence in $U(\overline{K})$, then we have

$\phi_{\\{x\\}}(\overline{\mathcal{D}})\geq\mu_{+}^{\pi}(\overline{\mathcal{D}})$

for big arithmetic adelic divisors $\overline{\mathcal{D}}$ in
$\widehat{\emph{Div}}(U,k)$.

###### Proof.

We recall the definition of the essential minima of an arithmetic adelic
divisor, denoted by $\zeta_{\text{ess}}(\cdot)$ as

$\zeta_{\text{ess}}(\overline{\mathcal{D}}):=\sup_{V}\inf_{x\in
V(\overline{K})}h_{\overline{\mathcal{D}}}(x)$

where $V$ ranges over all proper non-empty Zariski open subsets of $U$. Then
the definition of $\phi_{\\{x\\}}$ readily shows that for any Zariski open
$V\subset U$, since $x_{m}\in V(\overline{K})$ for large enough $m$ we have
that $\phi_{\\{x\\}}\geq\zeta_{\text{ess}}$ on $\widehat{\text{Div}}(U,k)$.
Then we can conclude the claim of the lemma since
$\zeta_{\text{ess}}\geq\mu^{\pi}_{+}$ from [YZ21, Lemma 5.3.4]. ∎

We are going to use the above Lemma together with our differentiability result
to obtain an equidstribution theorem. As the above lemma suggests our strategy
will be to deduce the differentiability of $\phi_{\\{x\\}}$ at big adelic
divisors from the differentiability of $\pi_{+}^{\pi}$. We will need a result
for that on differentiability of super-additive functions as done in [Che11]
which we restate for the sake of completeness.

######  Lemma 5.7.

[Che11, Prop 5.4] Let $G$ be a group, $H$ be a sub-group, $C$ be a sub semi-
group which is open with respect to $H$ and $x\in C$. Furthermore let
$f,t\colon C\rightarrow\mathbb{R}$ be two positively homogeneous functions
satisfying

  1. 1.

For all $a,b\in C$, we have $f(a+b)\geq f(a)+f(b)$

  2. 2.

$f\geq t$, $f(x)=t(x)$

  3. 3.

$t$ is differentiable at $x$ along the directions in $H$.

Then the function $f$ is differentiable as well and we have $D_{x}f=D_{x}t$.

Next we show that the positive intersections respect effectivity relations in
terms of the direction $\overline{E}$ which helps us to control positive
intersection products against generically trivial divisors with constant
metrics.

######  Lemma 5.8.

Suppose $\overline{\mathcal{D}}$ is an arithmetic adelic divisor in
$\widehat{\emph{Div}}(U,k)$ with generic fiber $\overline{D}$ and suppose
$\overline{E}_{1},\overline{E}_{2}\in\widehat{\emph{Div}}(U,k)_{\emph{int}}$
such that $\overline{E}_{1}\leq\overline{E}_{2}$. Then we have

$\langle\overline{\mathcal{D}}^{d}\rangle\cdot\overline{E}_{1}\leq\langle\overline{\mathcal{D}}^{d}\rangle\cdot\overline{E}_{2}$

Furthermore for any $r>0\in\mathbb{R}$, we have

$\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(r)\leq
r\cdot\widehat{\emph{vol}}(\overline{D})$

###### Proof.

We begin with proof of the first statement. As usual we choose a common quasi-
projective model of $U$ and suppose $(X,\overline{\mathcal{A}})$ is an
admissible approximation and suppose $\overline{A}$ is the generic fiber. Then
since $\overline{E}_{2}-\overline{E}_{1}\geq 0$ and $\overline{\mathcal{A}}$
is nef, we have

$\overline{\mathcal{A}}^{d}\cdot(\overline{E}_{2}-\overline{E}_{1})\geq
0\Rightarrow\overline{\mathcal{A}}^{d}\cdot\overline{E}_{1}\leq\overline{\mathcal{A}}^{d}\cdot\overline{E}_{2}$

Since the positive intersection products are defined by taking the supremum of
all such products appearing as we vary $(X^{\prime},\overline{\mathcal{A}})$
across all admissible approximations in the above display we can deduce the
claim.
For the second claim, we note that $\mathcal{O}(r)$ is nef and hence in
particular integrable for all constants $r>0$ and so it makes sense to take
positive intersection products against it. As in the first paragraph we take
an admissible approximation $(X^{\prime},\overline{\mathcal{A}})$ of
$\overline{\mathcal{D}}$ on a common quasi-projective model. Then we have

$\overline{\mathcal{A}}^{d}\cdot\mathcal{O}(r)=r\cdot\overline{A}^{d}\leq
r\cdot\widehat{\text{vol}}(\overline{D})\ \text{since}\ r\geq 0$ (20)

To see the last inequality above, we note that $\overline{A}\leq\overline{D}$
in the level of generic fibers and hence
$\widehat{\text{vol}}(A)\leq\widehat{\text{vol}}(\overline{D})$. However since
$\overline{\mathcal{A}}$ was arithmetically nef (as it was an admissible
approximation of $\overline{\mathcal{D}}$), we deduce that $\overline{A}$ is
nef and hence
$\widehat{\text{vol}}(\overline{A})=\overline{A}^{d}\leq\widehat{\text{vol}}(\overline{D})$
and hence we deduce the inequality in (20). Finally to obtain our second
claim, we note that positive intersection product
$\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(r)$ is defined by
taking the supremum of the first term in (20) as we vary admissible
approximations $(X^{\prime},\overline{\mathcal{A}})$ which gives us the
desired inequality. ∎

Next we extend the definition of positive intersection products against
generically trivial adelic line bundles $\mathcal{O}(g)$ where $g\in
C_{\text{cptf}}(U_{v})$ is arbitary. Using the lemma above, we show that we
can extend positive intersection products to uniform limits of model
functions. Recall from Definition 5.3 that we say $g\in C_{c}^{0}(U_{v})$ is a
_model function_ if $g$ is smooth when $v$ is Archimedean and if $g$ is
induced by a model when $v$ is non-Archimedean.

######  Lemma 5.9.

Suppose $g_{n}$ is a sequence of uniformly convergent model functions in
$C_{\emph{mod}}(U_{v})$ converging uniformly to an arbitary continuous
function $g\in C_{\emph{cptf}}(U_{v})$ and
$\overline{\mathcal{D}}\in\widehat{\emph{Div}}(U,k)$ is big. Then we have that
the sequence

$\\{\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g_{n})\\}$

converges in $\mathbb{R}$. Furthermore the limit is independent of the
sequence of functions $g_{n}$.

###### Proof.

We only prove the first assertion as the second one can be deduced very
similarly. As we have remarked before, since the functions $g_{n}$ are model
we know that each $\mathcal{O}(g_{n})$ is integrable and hence the terms
$\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g_{n})$ make sense.
Since we have that $\\{g_{m}\\}$ is uniformly convergent, we can find a
sequence of positive rational numbers $r_{m}$ converging to 0 such that
$-r_{m}\leq g_{n}-g_{m}\leq r_{m}$ for all $n\geq m$. In terms of Hermitian
line bundles, we have then the effectivity relations

$-\mathcal{O}(r_{m})\leq\mathcal{O}(g_{n})-\mathcal{O}(g_{m})\leq\mathcal{O}(r_{m})$
(21)

Using the linearity of positive intersection products in Lemma 3.10 and the
first assertion of Lemma 5.8, we deduce

$-\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(r_{m})\leq\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g_{n})-\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g_{m})\leq\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(r_{m})$

Estimating the two extremities of the above inequality using the second
assertion of Lemma 5.8, we get

$-r_{m}\cdot\widehat{\text{vol}}(\overline{D})\leq\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g_{n})-\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g_{m})\leq
r_{m}\cdot\widehat{\text{vol}}(\overline{D})\ \text{for all}\ n\geq m$

This gives us the desired claim since $r_{m}\to 0$ as $m\to\infty$. ∎

We can then finally extend our positive intersection products against
$\mathcal{O}(g)$ where $g$ is an arbitary continuous function with compact
support.

######  Definition 5.10.

Suppose $\overline{\mathcal{D}}$ is any big arithmetic adelic divisor in
$\widehat{\emph{Div}}(U,k)$ and suppose $g\in C_{c}^{0}(U_{v})$. Then we
define

$\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g):=\lim_{m\to\infty}\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g_{m})$

where $\\{g_{m}\\}$ is any sequence of model functions on $U_{v}$ converging
uniformly to $g$.

######  Remark.

We remark that our definition above makes sense. Indeed choosing a projective
model $X$ of $U$ and viewing $g$ as a continuous function on $X_{v}$ by
extending it to 0 outside the support, we can use the Weierstrass
approximation theorem in the Archimedean case and [Gub98, Theorem 7.12] in the
algebraically closed non-Archimedean case (adapted by Yuan in [Yua08] in the
arbitary non-Archimedean case) to deduce that there is indeed such a sequence
$\\{g_{m}\\}$ approximating $g$ uniformly as above. Furthermore in the second
assertion of Lemma 5.9 we also see that the limit is independent of the
sequence $\\{g_{m}\\}$.

We can finally state our main result on equidistribution of small and generic
points with respect to a big adelic divisor.

######  Theorem 5.11.

Suppose $U$ is a quasi-projective variety over $K$ of dimension $d§$ and
suppose $\overline{\mathcal{D}}$ is a big arithmetic adelic divisor in
$\widehat{\emph{Div}}(U,k)$. Furthermore suppose $\\{x_{m}\\}$ is a generic
sequence of geometric points in $U(\overline{K})$ which is small with respect
to $\overline{\mathcal{D}}$. Then for any place $v$ on $K$ and for any $g\in
C_{c}^{0}(U_{v})$, we have

$\lim_{m\to\infty}\int_{U_{v}}g\
d\eta_{x_{m}}=\frac{\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g)}{\widehat{\emph{vol}}(\overline{D})}$

In particular, the sequence of Radon measures $\\{\eta_{x_{m}}\\}$ converge
weakly to the Radon measure given by

$g\in
C_{c}^{0}(U_{v})\mapsto\frac{\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g)}{\widehat{\emph{vol}}(\overline{D})}$

###### Proof.

We begin by noting that we can reduce to the case when $g$ is a model
function. Indeed as we have remarked before, any continuous function with
compact support on $U_{v}$ can be approximated by model functions in the
uniform topology. Note that then for such a sequence of model functions
approximating our arbitary continuous function, the L.H.S will converge to the
desired integral and the R.H.S will also converge to the desired quantity
simple by how we define the positive intersection products against
$\mathcal{O}(g)$ (see Definition 5.10 and Lemma 5.9). Hence we can safely
assume that $g$ is a model function.
Note that now since we have assumed $g$ to be model, from Lemma 5.4 we know
that each $\mathcal{O}(g)\in\widehat{\text{Div}}(U,k)_{\text{int}}$. The proof
is then a consequence of all the results that we obtained in this section so
far. We begin by considering Theorem 5.7 and suppose we set
$G=\widehat{\text{Div}}(U,k)$, $H=C^{0}_{\text{mod}}(U_{v})$,
$C=C(U,\overline{\mathcal{D}})$. We set $f=\phi_{\\{x\\}}$ and
$t=\mu_{+}^{\pi}$. We check that these functions satisfy the hypotheses of
Theorem 5.7. Indeed note that we get from positive homogeneity of heights and
volumes that both $f$ and $t$ are positively homogeneous functions. Moreover

$\phi_{\\{x\\}}(\overline{\mathcal{D}}_{1}+\overline{\mathcal{D}}_{2})=\liminf_{m\to\infty}h_{\overline{\mathcal{D}}_{1}+\overline{\mathcal{D}}_{2}}(x_{m})=\liminf_{m\to\infty}(h_{\overline{\mathcal{D}}_{1}}(x_{m})+h_{\overline{\mathcal{D}}_{2}}(x_{m}))\geq\phi_{\\{x\\}}(\overline{\mathcal{D}}_{1})+\phi_{\\{x\\}}(\overline{\mathcal{D}}_{2})$

so $f=\mu_{+}^{\pi}$ satisfies super-additivity. From Lemma 5.6 we have that
$f\geq t$ on $C(U,\overline{\mathcal{D}})$ since all its elements are big by
definition. Furthermore since the sequence $\\{x\\}$ is small with respect to
$\overline{\mathcal{D}}$, from the definitions of $\phi_{\\{x\\}}$ and
$\mu_{+}^{\pi}$ we have that $f(x)=t(x)$ and hence also the second hypothesis
of Theorem 5.7 is satisfied. Finally from Corollary 5.1 and the assumed
bigness of $\overline{\mathcal{D}}$, we deduce that $t$ is differentiable at
$x$ in the direction of $\mathcal{O}(g)$ since $\mathcal{O}(g)$ is integrable
as mentioned in the first paragraph. Furthermore we also deduce from Corollary
5.1 that the differential is given by

$D_{x}t(g)=\frac{\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g)}{\widehat{\text{vol}}(\overline{D})}$

since the underlying generic fiber of $\mathcal{O}(g)$ is trivial. Hence from
Theorem 5.7 we deduce that $f=\phi_{\\{x\\}}$ is differentiable at
$x=\overline{\mathcal{D}}$ with the differential given by the differential of
$t$ at $x$. Then from Lemma 5.5 we deduce that the sequence of Radon measures
$\\{\eta_{x_{m}}\\}$ converges weakly to the Radon measure

$g\in C^{0}_{c}(U_{v})\mapsto\lim_{m\to\infty}\int_{U_{v}}g\
d\eta_{x_{m}}=D_{x}f(g)=D_{x}t(g)=\frac{\langle\overline{\mathcal{D}}^{d}\rangle\cdot\mathcal{O}(g)}{\widehat{\text{vol}}(\overline{D})}$

which is what we wanted to prove. ∎

##  6 Relation to other Equidistribution Results

In this section, we relate our equidistribution result obtained in Theorem
5.11 to two other equidistribution results already known. The first one is due
to Yuan and Zhang obtained for nef adelic line bundles on quasi-projective
varieties as in [YZ21, Theorem 5.4.3] and we show it can be deduced by our
result. We end by remarking the relation of our result with a previously
obtained equidistribution of Berman and Boucksom
Recall that given an integrable arithmetic adelic divisor
$\overline{\mathcal{D}}\in\widehat{\text{Div}}(U,k)_{\text{int}}$, there is a
Radon measure on $U_{v}$, denoted as the _Chambert-Loir measure_ by Yuan and
Zhang, corresponding to $\overline{\mathcal{D}}$ as explained in [YZ21,
Section 3.6.6]. We denote this Radon measure by
$c_{1}(\overline{\mathcal{D}})^{d}$. They are constructed as weak limits of
the already existing measures for the projective models approximating
$\overline{\mathcal{D}}$. The property crucial for us is that when
$\overline{\mathcal{D}}$ is nef, we have for any $g\in C_{c}^{0}(U_{v})$ the
equality

$\overline{\mathcal{D}}^{d}\cdot\mathcal{O}(g)=\int_{U_{v}}g\
c_{1}(\overline{\mathcal{D}})^{d}$ (22)

Indeed note that a similar inequality holds in the projective setting by the
definition of arithmetic intersections (see [GS90]). Then the above equality
can be deduced by noting that both the left and right hand sides are
constructed by taking limits along projective models approximating
$\overline{\mathcal{D}}$ as in [YZ21, Prop 4.1.1] and [YZ21, section 3.6.6]
respectively. Then we can deduce Yuan and Zhang’s equidistribution in the next
theorem.

######  Theorem 6.1.

[YZ21, Theorem 5.4.3] Suppose
$\overline{\mathcal{D}}\in\widehat{\emph{Div}}(U,k)_{\emph{nef}}$ is a nef
arithmetic adelic divisor such that
$\widehat{\emph{vol}}(\overline{D})=\overline{D}^{d}>0$ and suppose
$\\{x_{m}\\}$ is a generic sequence in $U(\overline{K})$ which is small with
respect to $\overline{\mathcal{D}}$. Then the sequence of Radon measure
$\\{d\eta_{x_{m}}\\}$ converge weakly to the Radon measure
$\frac{c_{1}(\overline{\mathcal{D}})^{d}}{\overline{D}^{d}}$.

###### Proof.

We have to show that for all $g\in C_{c}^{0}(U_{v})$, we have

$\lim_{m\to\infty}\int_{U_{v}}g\
d\eta_{x_{m}}\mapsto\frac{1}{\overline{D}^{d}}\cdot\int_{U_{v}}g\
c_{1}(\overline{\mathcal{D}})^{d}$

The trick is to deduce it from Theorem 5.11 by twisting
$\overline{\mathcal{D}}$. Note that for any $r>0$, we have that the $r$-twist
$\overline{\mathcal{D}}(r)$ is again nef. Moreover since it is nef, we have
$\widehat{\text{vol}}(\overline{\mathcal{D}}(r))=\overline{\mathcal{D}}(r)^{d+1}=\overline{\mathcal{D}}^{d+1}+(d+1)r\overline{D}^{d}>0$
from Theorem [YZ21, Theorem 5.2.2(1)] and since $\overline{D}^{d}>0$ by
hypothesis. Hence we deduce that the $r$-twist $\overline{\mathcal{D}}(r)$ is
big for all $r>0$. From the previous equality we can also deduce
$\widehat{\text{vol}}(\overline{\mathcal{D}}(r))=\widehat{\text{vol}}(\overline{\mathcal{D}})+(d+1)r\cdot\widehat{\text{vol}}(\overline{D})\Rightarrow\hat{\mu}^{\pi}_{+}(\overline{\mathcal{D}}(r))=\hat{\mu}^{\pi}_{+}(\overline{\mathcal{D}})+r$.
Moreover we have
$h_{\overline{\mathcal{D}}(r)}(x_{m})=h_{\overline{\mathcal{D}}}(x_{m})+r$ and
hence we deduce that the sequence $\\{x_{m}\\}$ is small with respect to
$\overline{\mathcal{D}}(r)$ as well. Then we can apply Theorem 5.11 and deduce
that the sequence of measures $\\{d\eta_{x_{m}}\\}$ converges weakly to the
measure given by

$g\in
C_{c}^{0}(U_{v})\mapsto\frac{\langle\overline{\mathcal{D}}(r)^{d}\rangle\cdot\mathcal{O}(g)}{\overline{D}^{d}}$

However note that since $\overline{\mathcal{D}}(r)$ is already nef, by
Corollary 3.8 we have that in this case the positive intersection products are
the same as usual intersection products and hence

$\frac{\langle\overline{\mathcal{D}}(r)^{d}\rangle\cdot\mathcal{O}(g)}{\overline{D}^{d}}=\frac{\overline{\mathcal{D}}(r)^{d}\cdot\mathcal{O}(g)}{\overline{D}^{d}}$

However note that since $\mathcal{O}(g)$ is generically trivial and
$\mathcal{O}(r)$ is also generically trivial with constant metric, we have
that
$\overline{\mathcal{D}}(r)^{d}\cdot\mathcal{O}(g)=\overline{\mathcal{D}}^{d}\cdot\mathcal{O}(g)$
which finishes the proof together with Equation (22).

∎

######  Remark.

We end this section by relating our equidistribution with an equidistribution
result obtained by Berman and Boucksom. In [BB10, Theorem D] Berman and
Boucksom essentially show that for a smooth projective variety over
$\mathbb{C}$ and a big line bundle on it endowed with a continuous metric
denoted by $\overline{L}$, the Galois orbits of a generic sequence
$\\{x_{m}\\}$ equidistribute to the _equilibrium measure_ whenever the height
of the points with respect to $\overline{L}$ converge to the _adelic energy at
equilibrium_. They obtain it as a corollary of differentiability of _energy at
equilibrium_ as in [BB10, Theorem A]. Our equidistribution can be thought of
as a generalisation of Berman-Boucksom equidistribution in the quasi-
projective setting. However as in Chen’s equidistribution (see [Che11,
Corollary 5.5]) we need an additional assumption of the divisor
$\overline{\mathcal{D}}$ to be arithmetically big compared to the more relaxed
positivity assumptions of Berman-Boucksom.

## Contact

Address:  _Debam Biswas, Department of Mathematics, University of Regensburg
Universitatstrasse 31, 93053, Regensburg_
Email<EMAIL_ADDRESS>
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