L’approximation arithmétique de Fujita sur courbes adéliques

# Arithmetic Fujita approximation over adelic curves

Chunhui Liu  Institute for Advanced Study in Mathematics
Harbin Institute of Technology
150001 Harbin
P. R. China<EMAIL_ADDRESS>
###### Abstract

In this paper, we will prove an analogue of Fujita’s approximation theorem
under the framework of Arakelov theory over adelic curves, which proves a
conjecture of Huayi Chen and Atsushi Moriwaki.

Dans cet article, on démontrera une analogie du théorème de l’approximation de
Fujita sous le cadre de la théorie d’Arakelov sur courbes adéliques, qui
démontre une conjecture de Huayi Chen et Atsushi Moriwaki.

###### Contents

  1. 1 Introduction
    1. 1.1 Arithmetic analogue over number fields
      1. 1.1.1
      2. 1.1.2
    2. 1.2 Arakelov geometry over adelic curves
      1. 1.2.1
      2. 1.2.2
    3. 1.3 Main result
      1. 1.3.1
      2. 1.3.2
    4. 1.4 Organization of article
  2. 2 Arakelov theory over adelic curves
    1. 2.1 Foundations of Arakelov geometry over adelic curves
      1. 2.1.1 Adelic curve
      2. 2.1.2 Adelic vector bundle
      3. 2.1.3 Some Arakelov invariants
    2. 2.2 Harder-Narasimhan filtration from $\mathbb{R}$-filtration
      1. 2.2.1 $\mathbb{R}$-filtrations on vector spaces
      2. 2.2.2 A measure induced by $\mathbb{R}$-filtration
      3. 2.2.3 Harder-Narasimhan filtration
    3. 2.3 Filtrations on graded algebras
      1. 2.3.1 Approximable graded algebras
      2. 2.3.2 Convergence of measures on an approximable algebra
      3. 2.3.3 Limits of a measure from filtrated algebras
  3. 3 Graded linear series of global section spaces
    1. 3.1 Adelic line bundles on an arithmetic variety
      1. 3.1.1 Adelic line bundle
      2. 3.1.2 Semi-positive metric
      3. 3.1.3 Ampleness
      4. 3.1.4 Nefness
      5. 3.1.5 Effectivity
    2. 3.2 Algebraic construction of graded linear series
      1. 3.2.1 Graded linear series containing an ample divisor
      2. 3.2.2 Big line bundles
    3. 3.3 Arithmetic invariants of graded linear series
      1. 3.3.1
      2. 3.3.2
      3. 3.3.3
    4. 3.4 Graded linear series over trivially valued fields
      1. 3.4.1
      2. 3.4.2
      3. 3.4.3
      4. 3.4.4
      5. 3.4.5
  4. 4 Fujita approximation of graded linear series
    1. 4.1 Approximatible graded linear series
      1. 4.1.1
      2. 4.1.2
    2. 4.2 Fujita approximation of graded linear series
  5. 5 Fujita approximation of adelic line bundles
    1. 5.1 Fujita approximation of big adelic line bundles
    2. 5.2 Case of $\mathbb{K}$-line bundles
      1. 5.2.1
      2. 5.2.2

##  1 Introduction

Fujita approximation is an approximative version of Zariski decomposition of
pseudo-effective divisors [25]. More precisely, let $X$ be a projective
variety over a field $k$ and $L$ be a line bundle on $X$. The volume of $L$ is
defined as

$\operatorname{vol}(L)=\limsup_{D\rightarrow\infty}\frac{\dim_{k}H^{0}(X,L^{\otimes
D})}{D^{\dim(X)}/(\dim(X))!}.$

We say that $L$ is a big line bundle if $\operatorname{vol}(L)>0$. Fujita’s
approximation theorem states that, for all $\epsilon>0$, there exist a
$p\in\mathbb{N}$ and a birational morphism $v:X^{\prime}\rightarrow X$, such
that $v^{*}(L^{\otimes p})$ can be decomposed the tensor product of an ample
line bundle $A$ and an effective line bundle $M$, which satisfy

$p^{-\dim(X)}\operatorname{vol}(A)\geqslant\operatorname{vol}(L)-\epsilon.$

In fact, this result asserts that a big line bundle can be arbitrarily closely
approximated by ample line bundles.

This approximated decomposition theorem is proved by T. Fujita [10] for the
case of characteristic $0$, and by S. Takagi [21] for an arbitrary
characteristic. We refer the readers to [15, §11.4] for a survey on this
topic.

###  1.1 Arithmetic analogue over number fields

It is quite nature to find an analogue of Fujita’s approximation theorem under
the framework of Arakelov geometry, since we can define arithmetic
positivities and arithmetic volume of a normed line bundle over arithmetic
varieties.

#### 1.1.1

The arithmetic analogue of volume function and arithmetic bigness under the
framework of Arakelov geometry are introduced in [17, 18] by A. Moriwaki. Soon
after these, voluminous works appeared on this topic, such as [19, 4, 24, 12,
20, 13].

Let $K$ be a number field, $\mathscr{X}$ be an arithmetic variety over
$\operatorname{Spec}\mathcal{O}_{K}$ of dimension $d$, and
$\overline{\mathscr{L}}$ be a Hermitian line bundle on $\mathscr{X}$. We
define

$\widehat{h}^{0}(\mathscr{X},\overline{\mathscr{L}}^{\otimes
D})=\log\\#\\{s\in H^{0}(\mathscr{X},\mathscr{L}^{\otimes
D})\mid\|s\|_{\sup,\sigma}\leqslant 1,\;\forall\sigma\in M_{K,\infty}\\},$

and

$\widehat{\operatorname{vol}}(\overline{\mathscr{L}})=\limsup_{n\to\infty}\frac{\widehat{h}^{0}(\mathscr{X},\overline{\mathscr{L}}^{\otimes
n})}{n^{\dim(\mathscr{X})}/\dim(\mathscr{X})!}.$

By the works mentioned above, the arithmetic volume function
$\widehat{\operatorname{vol}}(\raisebox{1.72218pt}{.})$ has many similar
behaviors to those of the geometric setting
$\operatorname{vol}(\raisebox{1.72218pt}{.})$. So it is expected that an
arithmetically big line bundle $\overline{\mathscr{L}}$, which satisfies
$\widehat{\operatorname{vol}}(\overline{\mathscr{L}})>0$, also has the Fujita
approximation. In fact, this used to be a conjecture of A. Moriwaki.

#### 1.1.2

In [2, 23], H. Chen and X. Yuan proved the arithmetic Fujita approximation
over number field at almost the same time. Although they both applied [16,
§3], but their methods are essential different. X. Yuan [23] takes pluri-
subharmonic envelope to deal with the infinite part, which is well-known in
complex analysis.

H. Chen applied his $\mathbb{R}$-filtration method developed in [3]. More
precisely, by considering the successive minima on
$E_{n}=H^{0}(\mathscr{X},\mathscr{L}^{\otimes n})$, we can make that
$\bigoplus\limits_{n\geqslant 0}E_{n}$ is a filtrated graded linear series.
Then by combining some properties of approximable graded linear series in
[16], we apply some convergent properties of $\mathbb{R}$-filtrations to
construct an ample line bundle which approximates $\overline{\mathscr{L}}$
well.

###  1.2 Arakelov geometry over adelic curves

In [7], H. Chen and A. Moriwaki developed a kind of Arakelov theory over so-
called adelic curves. Based on this, they built the arithmetic intersection
theory in [8], and studied the arithmetic positivities in [9]. Under this
setting, we can study the arithmetic geometry over number fields, function
fields and other bases with adelic structures by a united method.

#### 1.2.1

Briefly speaking, an adelic curve $S=(K,(\mathcal{A},\Omega,\nu),\phi)$ is an
arbitrary field $K$ which is equipped with a parameter set $\Omega$ of all its
absolute values. We require the set $\Omega$ parameterized by $\phi$ has a
$\sigma$-algebra structure $\mathcal{A}$ and a measure $\nu$.

Under this setting, we can generalize many classic results and topics on
Arakelov geometry, including the arithmetic volume function, arithmetic
positivity, and Hilbert-Samuel formula. Since the adelic structure becomes
more complicated, the obstructions original from measurability, integrability,
and non-discreteness make the developing process more technique. In fact, some
classic properties are not valid any longer in this framework.

#### 1.2.2

Inspired by [2, 23], it is quite natural to ask whether the Fujita
approximation is still valid over adelic curves. Over a general adelic curve,
we are lack of the Minkowskian property, so we cannot apply the filtrated
method by successive minina. At the same time, some original constructions do
not work any longer without the help of integral models.

In [9, §8.5], H. Chen and A. Moriwaki proved a similar result to the
arithmetic Fujita approximation over adelic curves, which is called the
relative Fujita approximation. More precisely, let an adelic line bundle
$\overline{L}$ where $L$ is big on $X$. Then for any
$t<\widehat{\mu}_{\max}^{\operatorname{asy}}(\overline{L})$, there exists a
$p\in\mathbb{N}$ sufficiently large and a birational morphism
$\varphi:X^{\prime}\rightarrow X$, such that $(\varphi^{*}L)^{\otimes
p}=\overline{M}\otimes\overline{N}$, where $\overline{N}$ is relatively ample
and $\overline{M}$ is effective, and
$\widehat{\mu}_{\min}^{\operatorname{asy}}(\overline{L})>pt$.

Compared with this relative version, the classic arithmetic Fujita
approximation seems to take an "average" at every minima of minimal slope, and
approximates all of them simultaneously. In fact, this is a conjecture
proposed by Chen-Moriwaki in [9, Remark 9.2.7].

###  1.3 Main result

In this article, we prove the following result on the arithmetic Fujita
approximation over adelic curves under the framework [7, 9]. More general than
the conjecture proposed in [9, Remark 9.2.7], we prove the case of adelic
$\mathbb{R}$-line bundle.

######  (Corollary 5.1, 5.2.1, Theorem 5.2.2).

Let $S=(K,(\Omega,\mathcal{A},\nu),\phi)$ be a proper adelic curve, which
satisfies either $\mathcal{A}$ is discrete or has a countable sub-field which
is dense in $K_{w}$ for all $w\in\Omega$. Let $X$ be a projective variety over
$\operatorname{Spec}K$, and $\overline{L}$ be an arithmetically big adelic
$\mathbb{R}$-line bundle on $X$ over $S$. Then for any $\epsilon>0$, there
exist a $p\in\mathbb{N}$ and a birational morphism $v:X^{\prime}\rightarrow
X$, such that $v^{*}(\overline{L}^{\otimes
p})=\overline{A}\otimes\overline{M}$, where $\overline{A}$ is arithmetically
ample and $\overline{M}$ is arithmetically effective. Moreover, the inequality

$p^{-(\dim(X)+1)}\widehat{\operatorname{vol}}(\overline{A})\geqslant\widehat{\operatorname{vol}}(\overline{L})-\epsilon$

is verified.

#### 1.3.1

In [9], they develop a method which construct a graded linear series over the
adelic curve of trivially valued field from the general setting by its Harder-
Narasimhan filtration. In addition, we can modify the original norms by its
spectral norm, which makes the graded linear series to be filtrated. Under
this construction, the new graded linear series has some asymptotic invariants
same as the original one. Then some ideas in [2, 16] can be absorbed in this
work.

Once we prove the case of integral and $\mathbb{Q}$-line bundles, we can
approximate an $\mathbb{R}$-line bundle by a $\mathbb{Q}$-line bundle in an
effective direction, and then we can prove the case of $\mathbb{R}$-line
bundle.

#### 1.3.2

In the application of $\mathbb{R}$-filtration method, we need to take some
limits of some graded linear series and measures, and most operations only
work when the objects are measurable and integrable. If either $\mathcal{A}$
is discrete or has a countable sub-field which is dense in $K_{w}$ for all
$w\in\Omega$, there will be no obstructions on the measurability or
integrability, and see [7, §4] for more details of the necessity. Of course,
in [2, 23], there is no such problem since the above condition is satisfied
automatically.

###  1.4 Organization of article

In §2, we introduce the notions of Arakelov theory over adelic curves and
$\mathbb{R}$-filtration. In §3, we consider graded linear series over a
general adelic curve and construct those over the adelic curve of trivially
valued field filtrated by the Harder-Narasimhan filtration. In §4, we prove
the property of Fujita approximation of certain graded linear series. In §5,
we prove the arithmetic Fujita approximation of adelic line bundles.

### Acknowledgements

I would like to thank Prof. Atsushi Moriwaki for introducing this subject to
me. In addition, I would like to thank Prof. Huayi Chen for many useful
discussions and suggestions, and Dr. Wenbin Luo for the advices on some key
techniques. Parts of this work was accomplished during my visit to Institute
for Theoretical Sciences, Westlake University, and I would like to thank their
generous reception. Chunhui Liu was supported by National Science Foundation
of China No. 12201153.

##  2 Arakelov theory over adelic curves

In this section, we will introduce the foundations of Arakelov over adelic
curves and $\mathbb{R}$-filtration. The main references are [7, 9].

###  2.1 Foundations of Arakelov geometry over adelic curves

In this part, we recall some useful knowledge on the Arakelov geometry over
adelic curves, see [7] for a systematical introduction to this subject.

####  2.1.1 Adelic curve

Let $K$ be a field. We denote by $M_{K}$ the set of all absolute values on
$K$.

######  (Adelic curve).

We say $(\Omega,\mathcal{A},\nu)$ an adelic structure on $K$ equipped with a
mapping

$\begin{array}[]{rrcl}\phi:&\Omega&\longrightarrow&M_{K}\\\
&\omega&\mapsto&|\raisebox{1.72218pt}{.}|_{\omega}\end{array}$

which satisfies the properties below:

  1. (1)

$\mathcal{A}$ is a $\sigma$-algebra on $\Omega$, and $\nu$ is a measure on
$(\Omega,\mathcal{A})$;

  2. (2)

for all $a\in K^{\times}$, the function

$\begin{array}[]{rcl}\Omega&\longrightarrow&\mathbb{R}\\\
w&\mapsto&\log|a|_{w}\end{array}$

is $\mathcal{A}$-measurable, integrable with respect to $\nu$.

In addition, the data $S=(K,(\Omega,\mathcal{A},\nu),\phi)$ is called an
adelic curve.

We denote by $\Omega_{f}$ the subset of $\Omega$ whose images in $M_{K}$ with
respect to $\phi$ is non-archimedean, and by $\Omega_{\infty}$ the sbuset of
$\Omega$ whose image in $M_{K}$ is archimedean. We have
$\Omega=\Omega_{f}\sqcup\Omega_{\infty}$ by definition directly.

######  (Proper adelic curve).

We say that an adelic curve $S=(K,(\Omega,\mathcal{A},\nu),\phi)$ is proper if
for all $a\in K^{\times}$, we always have

$\int_{S}\log|a|_{w}\nu(\mathrm{d}w)=0,$

and the above equality is called the product formula of $S$.

We refer the readers to [7, §3] for more details on this notion.

######  .

Let $K$ be a number field or a function field. Then the usual adelic ring
$\mathbb{A}_{K}$ gives an adelic structure, and constructs examples of proper
adelic curves.

Let $K=\mathbb{Q}(\varphi_{1},\ldots,\varphi_{d})$ be a finitely
transcendental extension over $\mathbb{Q}$. In [17], A. Moriwaki constructed a
special adelic structure over $K$, and defined a height function with respect
to it. This height function has been applied to study many arithmetic geometry
subjects.

######  (Trivially valued field).

Let $K$ be an arbitrary field, and $|\raisebox{1.72218pt}{.}|_{0}$ be the
trivial absolute value over $K$, which means

$|a|_{0}=\begin{cases}1,\text{ if }a\neq 0,\\\ 0,\text{ if }a=0.\end{cases}$

We note $S_{0}=(K,\\{0\\},|\raisebox{1.72218pt}{.}|_{0})$ the adelic curve
which counts the absolute value $|\raisebox{1.72218pt}{.}|_{0}$ only once.
This example will be applied later.

We refer the readers to [7, §3.2] for more examples of adelic curve.

####  2.1.2 Adelic vector bundle

Let $S=(K,(\Omega,\mathcal{A},\nu),\phi)$ be an adelic curve, and $E$ be a
vector space over $K$. For a family of norms
$(\|\raisebox{1.72218pt}{.}\|_{w})_{w\in\Omega}$ on $E_{w}=E\otimes_{K}K_{w}$
over $K_{w}$, if the norm family
$(\|\raisebox{1.72218pt}{.}\|_{w})_{w\in\Omega}$ on $E$ and its dual norm
family $(\|\raisebox{1.72218pt}{.}\|^{\vee}_{w})_{w\in\Omega}$ on $E^{\vee}$
are both $\mathcal{A}$-measurable on $\Omega$, then we say
$\overline{E}=(E,(\|\raisebox{1.72218pt}{.}\|_{w})_{w\in\Omega})$ is an adelic
vector bundle over $S$. If $\dim(E)=1$, we say that $\overline{E}$ is an
adelic line bundle over $S$.

We say that an adelic vector bundle
$\overline{F}=(F,(\|\raisebox{1.72218pt}{.}\|^{\prime}_{w})_{w\in\Omega})$ is
an adelic sub-bundle of
$\overline{E}=(E,(\|\raisebox{1.72218pt}{.}\|_{w})_{w\in\Omega})$ over $S$, if
$F$ is a $K$-vector subspace of $E$, and
$\|\raisebox{1.72218pt}{.}\|^{\prime}_{w}$ is the restriction of
$\|\raisebox{1.72218pt}{.}\|_{w}$ on $F$ for each $w\in\Omega$.

An adelic bundle
$\overline{G}=(G,(\|\raisebox{1.72218pt}{.}\|^{\prime\prime}_{w})_{w\in\Omega})$
is said to be an adelic quotient bundle of
$\overline{E}=(E,(\|\raisebox{1.72218pt}{.}\|_{w})_{w\in\Omega})$ over $S$, if
$G$ is a quotient vector space of $E$, and
$\|\raisebox{1.72218pt}{.}\|^{\prime\prime}_{w}$ is the quotient norm
$\|\raisebox{1.72218pt}{.}\|_{w}$ with respect to the canonical homomorphism
$E\rightarrow G$ for each $w\in\Omega$. If $G$ is the quotient space of $E$
modulo $F$, and equip with the quotient metrics from $\overline{F}$, then we
denote $\overline{G}=\overline{E}/\overline{F}=\overline{E/F}$.

For an adelic vector bundle
$\overline{E}=(E,(\|\raisebox{1.72218pt}{.}\|_{w})_{w\in\Omega})$ over $S$, we
define
$\det(\overline{E})=(\det(E),(\|\raisebox{1.72218pt}{.}\|_{\det,w})_{w\in\Omega})$
as follows: the vector $K$-space $\det(E)=\bigwedge^{\dim(E)}(E)$ is the
determinant space of $E$, and $\|\raisebox{1.72218pt}{.}\|_{\det,w}$ is the
determinant norm on $\det(E_{w})$ of $\|\raisebox{1.72218pt}{.}\|_{w}$ for
each $w\in\Omega$. We refer the readers to [7, §1.2.3] for the details of the
construction of $\|\raisebox{1.72218pt}{.}\|_{\det,w}$.

######  .

In general, for an adelic vector bundle $\overline{E}$ over
$S=(K,(\Omega,\mathcal{A},\nu),\phi)$, we cannot make sure that all its
subspace equipped with the subspace norms is still an adelic vector bundle,
and its quotient spaces admit the same situation. However, if $\mathcal{A}$ is
a discrete $\sigma$-algebra or $K$ has a countable sub-field which is dense in
$K_{w}$ for all $w\in\Omega$, we will get adelic vector bundles from all the
above constructions. See [7, Proposition 4.1.24] for more details.

####  2.1.3 Some Arakelov invariants

Let $S=(K,(\Omega,\mathcal{A},\nu),\phi)$ be a proper adelic curve, and
$\overline{E}=(E,(\|\raisebox{1.72218pt}{.}\|_{w})_{w\in\Omega})$ be an adelic
vector bundle over $S$. For the determinant bundle
$\det(\overline{E})=(\det(E),(\|\raisebox{1.72218pt}{.}\|_{\det,w})_{w\in\Omega})$
of $\overline{E}$, we select an $s\in\det(E)\smallsetminus\\{0\\}$, and define

$\widehat{\deg}(\overline{E})=-\int_{\Omega}\log\|s\|_{\det,w}\nu(\mathrm{d}w),$

which is well-defined due to [7, §4.2]. This is independent of the choice of
$s$ by the product formula of $S$.

We define

$\widehat{\deg}_{+}(\overline{E})=\sup_{F\subseteq
E}\widehat{\deg}(\overline{F}),$

where $\overline{F}$ runs over all the adelic sub-bundles of $\overline{E}$.

For a non-zero adelic bundle $\overline{E}$, we define the slope of
$\overline{E}$ as

$\widehat{\mu}(\overline{E})=\frac{\widehat{\deg}(\overline{E})}{\dim(E)}.$

In addition, we define the maximal slope of $\overline{E}$ as

$\widehat{\mu}_{\max}(\overline{E})=\sup_{F\subseteq
E}\widehat{\mu}(\overline{F}),$

where $\overline{F}$ runs over all adelic sub-bundles of $\overline{E}$.
Similarly, we define the minimal slope of $\overline{E}$ as

$\widehat{\mu}_{\min}(\overline{E})=\inf_{E\twoheadrightarrow
G}\widehat{\mu}(\overline{G}),$

where $\overline{G}$ runs over all adelic quotient bundles of $\overline{E}$.

We refer the readers to [7, §4.3] for more details and properties of these
invariants.

###  2.2 Harder-Narasimhan filtration from $\mathbb{R}$-filtration

In this part, we introduce a method to filtrate an adelic vector bundle by its
minimal slopes, which is called the Harder-Narasimhan filtration of an adelic
vector bundle. See [3] for its origin and [7] for the version over adelic
curve.

####  2.2.1 $\mathbb{R}$-filtrations on vector spaces

Let $K$ be a field, and $W$ be a $K$-vector space of finite dimension. For an
$\mathbb{R}$-filtration on $W$, we mean a sequence
$\mathcal{F}=\left(\mathcal{F}^{t}W\right)_{t\in\mathbb{R}}$ of vector
subspaces of $W$, which satisfies the following conditions:

  1. (1)

if $t\leqslant s$, then $\mathcal{F}^{s}W\subseteq\mathcal{F}^{t}W$;

  2. (2)

$\mathcal{F}^{t}W=0$ for sufficiently large $t\in\mathbb{R}$, and
$\mathcal{F}^{t}W=W$ for sufficiently small $t$;

  3. (3)

the function

$\begin{array}[]{rcl}\mathbb{R}&\longrightarrow&\mathbb{R}\\\
t&\mapsto&\dim_{K}(\mathcal{F}^{t}W)\end{array}$

is left continuous with respect to Euclidean topology.

####  2.2.2 A measure induced by $\mathbb{R}$-filtration

Let $K$ be a field. For a $K$-vector space $W$ of finite dimension equipped
with an $\mathbb{R}$-filtration $\mathcal{F}$, we denote by
$\nu_{(W,\mathcal{F})}$ (or by $\nu_{W}$ if there is no ambiguity) the Borel
probability measure on $\mathbb{R}$, which is induced from the derivative of
the function

$\begin{array}[]{rcl}\mathbb{R}&\longrightarrow&\mathbb{R}\\\
t&\mapsto&-\frac{\dim(\mathcal{F}^{t}W)}{\dim(W)}.\end{array}$

In fact, $\nu_{W}$ is a linear combination of some Dirac measure. More
precisely, if the $\mathbb{R}$-filtration $\mathcal{F}$ consists of a flag

$W=W_{0}\supsetneq W_{1}\supsetneq\cdots\supsetneq W_{n}=0$

together with the jumps at the real numbers $\lambda_{1}<\cdots<\lambda_{n}$,
then we have

$\nu_{(W,\mathcal{F})}=\sum_{i=1}^{n}\frac{\dim\left(W_{i-1}/W_{i}\right)}{\dim(W)}\delta_{\lambda_{i}}.$

In addition, we define the function

$\begin{array}[]{rrcl}\lambda:&W&\longrightarrow&\mathbb{R}\cup\\{+\infty\\}\\\
&x&\mapsto&\sup\\{t\in\mathbb{R}\mid x\in\mathcal{F}^{t}W\\}.\end{array}$ (1)

Then we have $\lambda(W)\subset\operatorname{supp}(\nu_{W})\cup\\{+\infty\\}$,
and $\lambda(x)$ is finite if $x\neq 0$.

In addition, we define

$\lambda_{+}(W)=\int_{0}^{+\infty}x\nu_{W}(\mathrm{d}x)\text{, and
}\lambda_{\max}(W)=\max\limits_{x\in W\smallsetminus\\{0\\}}\lambda(x).$ (2)

####  2.2.3 Harder-Narasimhan filtration

By the notion of $\mathbb{R}$-filtration, we introduce a filtration by minimal
slopes of adelic bundles. This is original from an analogical study of vector
bundles on a smooth projective curve [11].

######  (Harder-Narasimhan filtration).

Let $\overline{E}$ be a non-zero adelic vector bundle over a proper adelic
curve $S=(K,(\Omega,\mathcal{A},\nu),\phi)$, where either $\mathcal{A}$ is a
discrete $\sigma$-algebra or $K$ has a countable sub-field which is dense in
$K_{w}$ for all $w\in\Omega$. For any $t\in\mathbb{R}$, let

$\mathcal{F}_{\operatorname{HN}}^{t}\overline{E}=\bigcap_{\epsilon>0}\sum_{\begin{subarray}{c}\\{0\\}\neq
F\in\Theta(E)\\\ \widehat{\mu}_{\min}(\overline{F})\geqslant
t-\epsilon\end{subarray}}F=\sum_{\begin{subarray}{c}\\{0\\}\neq
F\in\Theta(E)\\\ \widehat{\mu}_{\min}(\overline{F})\geqslant
t\end{subarray}}F,$

where $\Theta(E)$ denotes the set of all vector sub-spaces of $E$, and
$\overline{F}$ is equipped with the sub-space norms. This
$\mathbb{R}$-filtration $\mathcal{F}_{\operatorname{HN}}$ is called the
Harder-Narasimhan filtration of $\overline{E}$.

The last equality in Definition 2.2.3 if from [7, Proposition 4.3.36]. In
fact, the Harder-Narasimhan $\mathbb{R}$-filtration of $\overline{E}$
corresponds to an increasing flag of vector subspaces

$\\{0\\}=E_{1}\subsetneq E_{2}\subsetneq\cdots\subsetneq E_{n}=E$ (3)

and a decreasing sequence of real numbers $\mu_{1}>\cdots>\mu_{n}$ represents
the jumps of the $\mathbb{R}$-filtration
$(\mathcal{F}_{\operatorname{HN}}^{t}\overline{E})_{t\in\mathbb{R}}$, where

$\mu_{i}=\sup\left\\{t\in\mathbb{R}\mid\dim_{K}(\mathcal{F}^{t}_{\operatorname{HN}}\overline{E})\geqslant
i\right\\}.$

In particular, still by [7, Proposition 4.3.36], we have
$\mu_{n}=\widehat{\mu}_{\min}(\overline{E})$. The uniqueness of (3) is from
[7, Theorem 4.3.58].

###  2.3 Filtrations on graded algebras

In this part, we study some asymptotic properties of graded linear algebras by
$\mathbb{R}$-filtraion.

####  2.3.1 Approximable graded algebras

The notation of approximable graded algebra is from [2], whose idea is
original from [16].

######  (Approximable graded algebra).

Let $K$ be a field, and $E_{\bullet}=\bigoplus\limits_{D\geqslant 0}E_{D}$ be
a graded $K$-algebra. We define $E_{\bullet}$ is approximable if:

  1. (1)

for each $D\in\mathbb{N}$, $E_{D}$ is of finite dimension, and $E_{D}\neq 0$
for sufficiently large $D\in\mathbb{N}$;

  2. (2)

for every $\epsilon\in(0,1)$, there exists a $p_{0}\in\mathbb{N}^{+}$, such
that for any $p\geqslant p_{0}$, we have

$\liminf\limits_{n\to\infty}\frac{\dim\left(\operatorname{Im}\left(S^{n}E_{p}\rightarrow
E_{np}\right)\right)}{\dim(E_{np})}>1-\epsilon,$

where $S^{n}B_{p}\rightarrow E_{np}$ is the canonical homomorphism defined by
the algebra structure on $E_{\bullet}$.

####  2.3.2 Convergence of measures on an approximable algebra

For a graded algebra, we equip an $\mathbb{R}$-filtration with all its
homogeneous part. If it is an approximable graded algebras, we will have some
properties of convergence.

Let $E_{\bullet}=\bigoplus\limits_{D\geqslant 0}E_{D}$ be an approximable
graded algebra. Suppose for every $D\geqslant 0$, the vector space $E_{D}$ is
equipped with an $\mathbb{R}$-filtration $\mathcal{F}$. We ignore the index
$D\in\mathbb{N}$ since there will be no ambiguity.

######  .

With all the above notations and conditions. If for every homogeneous elements
$x_{1},x_{2}\in E_{\bullet}$, we have

$\lambda(x_{1}x_{2})\geqslant\lambda(x_{1})+\lambda(x_{2}),$

where the function $\lambda(\raisebox{1.72218pt}{.})$ is defined at (1) and
the homogeneous degrees are ignored, then we say $E_{\bullet}$ is filtrated
with respect to $\mathcal{F}$.

For every $\epsilon>0$, let $T_{\epsilon}$ be the operator on the space of
Borel probability measures on $\mathbb{R}$ by mapping $x\mapsto\epsilon x$.

We have the following result of filtrated graded algebras.

######  (Theorem 2.11, [2]).

With all the above notations. Suppose that
$E_{\bullet}=\bigoplus\limits_{D\geqslant 0}E_{D}$ is a filtrated graded
algebra with respect to $\lambda(\raisebox{1.72218pt}{.})$ defined at (1), and

$\sup_{n\geqslant 1}\frac{\lambda_{\max}(E_{n})}{n}<+\infty,$

where $\lambda_{\max}(\raisebox{1.72218pt}{.})$ is defined at (2). Then the
sequence $\left(\frac{\lambda_{\max}(E_{n})}{n}\right)_{n\in\mathbb{N}^{+}}$
converges in $\mathbb{R}$, and the sequence of measures
$\left(T_{\frac{1}{n}}(\nu_{E_{n}})\right)_{n\geqslant 1}$ converges weakly to
a Borel probability measure on $\mathbb{R}$.

####  2.3.3 Limits of a measure from filtrated algebras

We consider the filtrated approximable graded $K$-algebra
$E_{\bullet}=\bigoplus\limits_{D\geqslant 0}E_{D}$. Let $\nu_{E_{\bullet}}$ be
the limit of the sequence of measures
$\left(T_{\frac{1}{n}}(\nu_{E_{n}})\right)_{n\geqslant 1}$, and
$\lambda^{\operatorname{asy}}_{\max}(E_{\bullet})=\lim\limits_{n\to\infty}\frac{\lambda_{\max}(E_{n})}{n}$,
whose existences are assured by Proposition 2.3.2. In addition, we have

$\lambda_{+}^{\operatorname{asy}}(E_{\bullet}):=\int_{0}^{+\infty}x\nu_{E_{\bullet}}(\mathrm{d}x)=\lim\limits_{n\to\infty}\frac{\lambda_{+}(E_{n})}{n},$
(4)

where $\lambda_{+}(\raisebox{1.72218pt}{.})$ is defined at (2). See [2, §2.4]
for more details.

We have the following result.

######  (Theorem 2.13, [2]).

Let $E_{\bullet}$ be a filtrated approximable graded $K$-algebra. Then
$\lambda_{+}^{\operatorname{asy}}(E_{\bullet})>0$ if and only if
$\lambda_{\max}^{\operatorname{asy}}(E_{\bullet})>0$.

##  3 Graded linear series of global section spaces

Let $S$ be an adelic curve, $X$ be a projective scheme over
$\operatorname{Spec}K$, and $\overline{L}$ be an adelic line bundle on $X$
over $S$. For our target, we will construct some graded linear series
consisting of a particular kind of adelic vector bundles over $S$ from the
above data.

###  3.1 Adelic line bundles on an arithmetic variety

In this part, we recall some notions about arithmetic positivity of the adelic
line bundles on an arithmetic variety over adelic curve. We refer the readers
to [9, §6 – §9] for a self-contained introduction to these notions.

####  3.1.1 Adelic line bundle

Let $S=(K,(\mathcal{A},\Omega,\nu),\phi)$ be an adelic curve, and $X$ be a
projective scheme over $\operatorname{Spec}K$. We say that
$\overline{L}=(L,\varphi)$ is an adelic line bundle on $X$, if $L$ is a line
bundle on $X$, and $\varphi=(\|\raisebox{1.72218pt}{.}\|_{w})_{w\in\Omega}$ is
a family of metrics on $L_{w}$, which is dominated and measurable.

In addition, we suppose that either $\mathcal{A}$ is a discrete
$\sigma$-algebra or $K$ contains a sub-field which is dense in $K_{w}$ for all
$w\in\Omega$. Then for every $w\in\Omega$ and $x\in
X_{w}^{\operatorname{an}}$, we define

$\begin{array}[]{rrcl}|\raisebox{1.72218pt}{.}|_{\varphi_{w}}(x):&H^{0}(X_{w},L_{w})&\longrightarrow&\mathbb{R}\\\
&s&\mapsto&|s|_{\varphi_{w}}(x)\end{array}$

as the value of $s$ at the point $x$ with respect to the metric, and

$\begin{array}[]{rrcl}\|\raisebox{1.72218pt}{.}\|_{\varphi_{w}}:&H^{0}(X_{w},L_{w})&\longrightarrow&\mathbb{R}\\\
&s&\mapsto&\sup\limits_{x\in
X^{\operatorname{an}}_{w}}\\{|s|_{\varphi_{w}}(x)\\}.\end{array}$ (5)

We denote $\varphi=(\|\raisebox{1.72218pt}{.}\|_{\varphi_{w}})_{w\in\Omega}$
the norm family on $H^{0}(X,L)$ over $S$ if there is no ambiguity. By [7,
Theorem 6.1.13, Theorem 6.1.32], $(H^{0}(X,L),\varphi)$ is an adelic vector
bundle over $S$.

Let $\overline{L}=(L,\varphi)$ and $\overline{M}=(M,\psi)$ be two adelic line
bundles on $X$. We denote by $\overline{L}\otimes\overline{M}=(L\otimes
M,\varphi+\psi)$ the adelic line bundle, where $\varphi+\psi$ is the tensor
product of metrics. In particular, for $n\in\mathbb{N}$, we denote by
$\overline{L}^{\otimes n}=(L^{\otimes n},n\varphi)$ the adelic line bundle of
$n$-times tensor product of $\overline{L}$, and by $(H^{0}(X,L^{\otimes
n}),n\varphi)$ the induced adelic vector bundle over $S$. By [7, Proposition
6.1.28, Proposition 6.1.31], all the above constructions are well-defined.

We refer the readers to [7, §6.1] for a precise explanation of §3.1.1, see
also [9, §2.6 – 2.8].

####  3.1.2 Semi-positive metric

Let $S=(K,(\mathcal{A},\Omega,\nu),\phi)$ be an adelic curve, $X$ be a
projective scheme over $\operatorname{Spec}K$, $L$ be a line bundle on $X$,
$\varphi=(\varphi_{w})_{w\in\Omega}$ and $\psi=(\psi_{w})_{w\in\Omega}$ be two
families of metrics on $L$ such that $(L,\varphi)$ and $(L,\psi)$ are both
adelic line bundles. Then we define the distance between $\varphi$ and $\psi$
as

$d(\varphi,\psi)=\int_{\Omega}\sup_{x\in
X^{\operatorname{an}}_{w}}\left|\log\frac{|\raisebox{1.72218pt}{.}|_{\varphi_{w}}(x)}{|\raisebox{1.72218pt}{.}|_{\psi_{w}}(x)}\right|\nu(\mathrm{d}w).$

If $L$ is semiample, and if there exist a positive integer $m$ and a sequence
$(\varphi_{n})_{n\in\mathbb{N}}$ of quotient metric families (where
$\varphi_{n}$ a metric family of $L^{\otimes mn}$), such that

$\lim_{n\to\infty}\frac{1}{mn}d(nm\varphi,\varphi_{n})=0,$

we say that the metric family $\varphi$ is semi-positive.

####  3.1.3 Ampleness

Let $\overline{L}=(L,\varphi)$ be an adelic line bundle on $X$ over $S$. We
say $\overline{L}$ is relatively ample if $L$ is ample and $\varphi$ is semi-
positive.

In addition, an adelic line bundle $\overline{L}$ on $X$ is ample if it is
relatively ample and if there exists an $\epsilon>0$ such that the inequality

$(\overline{L}|_{Y}^{\dim(Y)+1})\geqslant\epsilon\deg_{L}(Y)(\dim(Y)+1)$

holds for any integral closed subscheme $Y$ of $X$, where
$(\overline{L}|_{Y}^{\dim(Y)+1})$ is the arithmetic intersection number
studied in [8]. We also call it arithmetically ample to distinguish it from
the geometric case.

######  .

Compared with the arithmetically ample definition of S. Zhang in [26], an
arithmetically ample Hermitian line bundle on an arithmetic variety requires
the metrics on the line bundle to be smooth, while the definition in [9,
Definition 9.1.1] over an adelic curve only require continuous metrics. In
this article, we follow the definition given in [9, Definition 9.1.1].

####  3.1.4 Nefness

Let $\overline{L}=(L,\varphi)$ be an adelic line bundle on $X$ over $S$. We
say that $\overline{L}$ is relatively nef if there exists a relatively ample
adelic line bundle $\overline{A}$ on $X$ and a positive integer $N$ such that,
for any $n\geqslant N$, the tensor product $\overline{L}^{\otimes
n}\otimes\overline{A}$ is relatively ample.

An adelic line bundle $\overline{L}$ is said to be nef or arithmetically nef
if there exist an ample adelic line bundle $\overline{A}$ and a positive
integer $M$ such that $\overline{L}^{\otimes m}\otimes\overline{A}$ is ample
for any $m\geqslant M$ in the arithmetic sense.

####  3.1.5 Effectivity

Let $\overline{L}=(L,\varphi)$ be an adelic line bundle on $X$ over $S$. If
$s\in H^{0}(X,L)$ is a non-zero global section such that
$\|s\|_{\varphi_{w}}\leqslant 1$ for any $w\in\Omega$, we say that the global
section $s$ is effective. We say that $\overline{L}$ is effective or
arithmetically effective if it admits at least one effective global section by
the above definition.

###  3.2 Algebraic construction of graded linear series

In this part we only consider the geometric setting, which means everything
works over an arbitrary field $K$. For every $D\in\mathbb{N}^{+}$, we consider
the global section space $H^{0}(X,L^{\otimes D})$, and the graded $K$-algebra
$\bigoplus\limits_{n\geqslant 0}H^{0}(X,L^{\otimes n})$.

######  .

Let $X$ be a projective variety over $K$, and $L$ be a line bundle on $X$. For
a graded linear series of $L$, we mean a graded sub-$K$-algebra of
$\bigoplus\limits_{n\geqslant 0}H^{0}(X,L^{\otimes n})$.

####  3.2.1 Graded linear series containing an ample divisor

The following definition is original from [16, Definition 2.9], where it is
called "Condition (C)". This notion plays a key role in the arithmetic Fujita
approximation of [2].

######  .

Let $X$ be a projective variety, and $L$ be a line bundle on $X$. Suppose that
there exists a $p\in\mathbb{N}^{+}$, an ample line bundle $A$, an effective
line bundle $M$ on $X$, and a non-zero section $s\in H^{0}(X,M)$, such that
$L^{\otimes p}=A\otimes M$. We say a graded linear series
$W_{\bullet}=\bigoplus\limits_{n\geqslant 0}W_{n}$ of $L$ contains an ample
divisor if the homomorphism of graded algebras

$\bigoplus_{n\geqslant 0}H^{0}(X,A^{\otimes
n})\rightarrow\bigoplus_{n\geqslant 0}H^{0}(X,L^{\otimes np})$

induced by $s$ can factor through $\bigoplus\limits_{n\geqslant 0}W_{np}$.

In §2.3, we have seen that an approximable graded linear series has many
convergent properties. The following property is original from [16], which is
applied in [2] to judge whether a graded linear series is approximable.

######  (Theorem 3.4, [2]).

If $W_{\bullet}=\bigoplus\limits_{n\geqslant 0}W_{n}$ contains an ample
divisor and $W_{n}\neq 0$ for sufficiently large $n$, then $W_{\bullet}$ is
approximable.

####  3.2.2 Big line bundles

Let $W_{\bullet}=\bigoplus\limits_{n\geqslant 0}W_{n}$ be a graded linear
series of $L$. We denote

$\operatorname{vol}(W_{\bullet})=\limsup_{n\to\infty}\frac{\dim_{K}(W_{n})}{n^{\dim(X)}/\dim(X)!}.$

If $W_{n}=H^{0}(X,L^{\otimes n})$, we denote
$\operatorname{vol}(L)=\operatorname{vol}(W)$.

If $\operatorname{vol}(L)>0$, we say that $L$ is a big line bundle on $X$.

###  3.3 Arithmetic invariants of graded linear series

In this part, we consider arithmetic constructions.

#### 3.3.1

Let $\overline{L}=(L,\varphi)$ be an adelic line bundle on $X$ over a proper
adelic curve $S=(K,(\Omega,\mathcal{A},\nu),\phi)$, where either $\mathcal{A}$
is a discrete $\sigma$-algebra or $K$ contains a countable sub-field which is
dense in $K_{w}$ for all $w\in\Omega$. In §3.1.1, we defined a norm family on
$H^{0}(X,L^{\otimes D})$ by (5) indexed by $\Omega$, which is denoted by
$D\varphi$ with no ambiguity. By [7, §6.4.1],
$\|\raisebox{1.72218pt}{.}\|_{D\varphi_{w}}$ is ultrametric for every
$w\in\Omega_{f}$. Then $(H^{0}(X,L^{\otimes D}),D\varphi)$ is an adelic vector
bundle over $S$.

Let $\overline{E}_{D}$ be an adelic vector sub-bundle $H^{0}(X,L^{\otimes D})$
over $S$. In this case, we say that
$\overline{E}_{\bullet}=\bigoplus\limits_{D\in\mathbb{N}}\overline{E}_{D}$ is
a normed graded linear series over $S$ if $E_{D}\cdot E_{D^{\prime}}\subseteq
E_{D+D^{\prime}}$ for $D,D^{\prime}\in\mathbb{N}$.

#### 3.3.2

Let $\overline{E}_{\bullet}=\bigoplus\limits_{D\in\mathbb{N}}\overline{E}_{D}$
be a normed graded linear series over $S$. We define the arithmetic volume of
$\overline{E}_{\bullet}$ as

$\widehat{\operatorname{vol}}(\overline{E}_{\bullet})=\limsup_{n\to\infty}\frac{\widehat{\deg}_{+}\left(\overline{E}_{n}\right)}{n^{\dim(X)+1}/(\dim(X)+1)!}.$
(6)

In particular, if $\overline{E}_{D}=\left(H^{0}(X,L^{\otimes
D}),D\varphi\right)$ as above, we denote
$\widehat{\operatorname{vol}}(\overline{L})=\widehat{\operatorname{vol}}(\overline{E}_{\bullet})$
for simplicify.

If $\widehat{\operatorname{vol}}(\overline{L})>0$, we say that $\overline{L}$
is big on $X$, or we call it arithmetically big to distinguish it from the
geometric case.

#### 3.3.3

For every $D\in\mathbb{N}$, Let
$\left(\mathcal{F}^{t}_{\operatorname{HN}}(\overline{E}_{D})\right)_{t\in\mathbb{R}}$
be the $\mathbb{R}$-filtration induced by the Harder-Narasimhan filtration of
$\overline{E}_{D}$ following Definition 2.2.3, where we ignore the homogeneous
degree $D\in\mathbb{N}$.

We denote by $\nu_{\overline{E}_{D}}$ the Borel probability measure induced by
the Harder-Narasimhan filtration of $\overline{E}_{D}$ constructed in the
sense of §2.2.2. Then we have the result below from Proposition 4.3.50,
Proposition 4.3.51 and Corollary 4.3.52 of [7], see also [9, §2.5.4].

######  .

With all the above notations and conditions. We have

$0\leqslant\widehat{\deg}_{+}(\overline{E}_{D})-\int_{0}^{+\infty}x\nu_{\overline{E}_{D}}(\mathrm{d}x)\leqslant\frac{1}{2}\nu(\Omega_{\infty})\dim_{K}(E_{D})\log(\dim_{K}(E_{D})).$

###  3.4 Graded linear series over trivially valued fields

Let $K$ be a field, $S=(K,(\Omega,\mathcal{A},\nu),\phi)$ be a proper adelic
curve where either $\mathcal{A}$ is a discrete $\sigma$-algebra or $K$ has a
sub-field which is dense in $K_{w}$ for all $w\in\Omega$, and
$S_{0}=(K,\\{0\\},|\raisebox{1.72218pt}{.}|_{0})$ be the adelic curve of
trivially valued field introduced in Example 2.1.1.

In fact, for any normed vector space over $(K,|\raisebox{1.72218pt}{.}|_{0})$
of finite dimension, its norm takes only finitely many values (cf. [9, §3.1]).

In this part, we will study the adelic vector bundles and graded linear series
over $S_{0}$ following that in [9, §3]. Some constructions are from those over
$S$.

#### 3.4.1

Let $\overline{V}$ be an adelic vector bundle over an adelic curve
$S=(K,(\Omega,\mathcal{A},\nu),\phi)$, and
$\left(\mathcal{F}^{t}_{\operatorname{HN}}(\overline{V})\right)_{t\in\mathbb{R}}$
be the Harder-Narasimhan filtration of $\overline{V}$ defined in Definition
2.2.3. We define the norm $\|\raisebox{1.72218pt}{.}\|_{\operatorname{HN}}$ on
$V$ over $(K,|\raisebox{1.72218pt}{.}|_{0})$ as

$\begin{array}[]{rrcl}\|\raisebox{1.72218pt}{.}\|_{\operatorname{HN}}:&V&\longrightarrow&\mathbb{R}^{+}\\\
&s&\mapsto&\exp\left(-\sup\\{t\in\mathbb{R}\mid
s\in\mathcal{F}^{t}_{\operatorname{HN}}(\overline{V})\\}\right)\end{array}.$
(7)

Then
$\overline{V}_{\operatorname{HN}}=\left(V,\|\raisebox{1.72218pt}{.}\|_{\operatorname{HN}}\right)$
is an adelic vector bundle over $S_{0}$.

#### 3.4.2

Let $\overline{E}_{\bullet}=\bigoplus\limits_{D\in\mathbb{N}}\overline{E}_{D}$
be a normed graded linear series over $S=(K,(\Omega,\mathcal{A},\nu),\phi)$.
For every $\overline{E}_{D}$, we denote by
$\|\raisebox{1.72218pt}{.}\|_{D}^{\operatorname{HN}}$ the norm (7) on $E_{D}$.
Then $(E_{D},\|\raisebox{1.72218pt}{.}\|_{D}^{\operatorname{HN}})$ an Adélie
vector bundle over $S_{0}$.

By [9, Proposition 4.1.3], for every $s\in E_{m}$ and $t\in E_{n}$, we have

$\|s\cdot
t\|^{\operatorname{HN}}_{m+n}\leqslant\exp(f(m)+f(n))\|s\|^{\operatorname{HN}}_{m}\|t\|^{\operatorname{HN}}_{n},$

where $f(n)=\frac{3}{2}\nu(\Omega_{\infty})\log n$, and $\nu(\Omega_{\infty})$
is the measure of achimedean absolute values of
$S=(K,(\Omega,\mathcal{A},\nu),\phi)$.

In order to obtain a new family of norms which are sub-multiplicative, we
introduce the spectral norm $\|\raisebox{1.72218pt}{.}\|_{\mathrm{sp},D}$ of
$\|\raisebox{1.72218pt}{.}\|^{\operatorname{HN}}_{D}$ as

$\begin{array}[]{rrcl}\|\raisebox{1.72218pt}{.}\|_{\mathrm{sp},D}:&H^{0}(X,L^{\otimes
D})&\longrightarrow&\mathbb{R}_{\geqslant 0}\\\
&s&\mapsto&\lim\limits_{n\rightarrow\infty}\left(\|s^{n}\|_{D}^{\operatorname{HN}}\right)^{1/n}.\end{array}$
(8)

Then $\left(E_{D},\|\raisebox{1.72218pt}{.}\|_{D,\mathrm{sp}}\right)$ is an
adelic vector bundle over $S_{0}$. By [9, Proposition 3.2.2 (3)], for every
$s\in E_{m}$ and $t\in E_{n}$, we have

$\|s\cdot
t\|_{\mathrm{sp},m+n}\leqslant\|s\|_{\mathrm{sp},m}\|t\|_{\mathrm{sp},n}.$

Let
$\overline{E}_{\operatorname{HN},D}=\left(E_{D},\|\raisebox{1.72218pt}{.}\|_{\mathrm{sp},D}\right)$,
then

$\overline{E}_{\operatorname{HN},\bullet}=\bigoplus\limits_{D\in\mathbb{N}}\overline{E}_{\operatorname{HN},D}$
(9)

is a filtrated graded linear series (see Definition 2.3.2 for the definition)
over $S_{0}$, where we take
$\lambda(\raisebox{1.72218pt}{.})=-\log\|\raisebox{1.72218pt}{.}\|_{\mathrm{sp},D}$.

We denote by $\mathcal{F}_{\mathrm{sp}}$ the $\mathbb{R}$-filtration on
$\overline{E}_{\operatorname{HN},D}$ induced by the filtrated structure
$\lambda(\raisebox{1.72218pt}{.})=-\log\|\raisebox{1.72218pt}{.}\|_{\mathrm{sp},D}$,
where we ignore the index $D\in\mathbb{N}$.

#### 3.4.3

Some asymptotic behaviors of $\overline{E}_{\operatorname{HN},\bullet}$ are
useful for the study of $\overline{E}_{\bullet}$. Since
$\overline{E}_{\operatorname{HN},\bullet}$ is filtrated, Proposition 2.3.2 and
Proposition 2.3.3 will be applied in this process.

######  .

With all the notations in §3.4.2, we have

$\widehat{\operatorname{vol}}(\overline{E}_{\bullet})=\widehat{\operatorname{vol}}(\overline{E}_{\operatorname{HN},\bullet}).$

###### Proof.

It is from [9, Corollary 3.4.4, (4.1.2), Corollary-Definition 4.1.4] directly.
∎

By Proposition 2.3.2, the sequence
$\\{\frac{1}{n}\widehat{\mu}_{\max}(\overline{E}_{n})\\}_{n\geqslant 0}$
converges to a real number, and we denote it by
$\widehat{\mu}_{\max}^{\operatorname{asy}}(\overline{E}_{\bullet})$. If
$E_{n}=H^{0}(X,L^{\otimes n})$ equipped with the superior norms, it is denoted
by $\widehat{\mu}_{\max}^{\operatorname{asy}}(\overline{L})$ for simplicity.

We have the following result from [7, Proposition 6.4.18], which is an
equivalent of arithmetically big adelic line bundle.

######  .

Let $\overline{L}$ be an adelic line bundle on $X$ over an adelic curve $S$.
If $L$ is a big line bundle on $X$, then
$\widehat{\operatorname{vol}}(\overline{L})>0$ if and only if
$\widehat{\mu}_{\max}^{\operatorname{asy}}(\overline{L})>0$.

From Proposition 3.3.3, we have the following result by an elementary
calculation.

######  .

Let $\overline{E}^{\bullet}=\bigoplus\limits_{D\geqslant 0}\overline{E}_{D}$
be a normed graded linear series over $S$. If
$\operatorname{vol}(E^{\bullet})>0$, then we have

$\lambda_{+}^{\operatorname{asy}}(\overline{E}_{\operatorname{HN}}^{\bullet})=\frac{\widehat{\operatorname{vol}}(\overline{E}^{\bullet})}{(d+1)\operatorname{vol}(E^{\bullet})},$

where $\lambda_{+}^{\operatorname{asy}}(\raisebox{1.72218pt}{.})$ is defined
at (4) with respect to the canonical filtrated structure as (9) over $S_{0}$,
and $d=\dim(X)$.

#### 3.4.4

Let $\overline{L}=(L,\varphi)$ be an adelic line bundle on $X$ over an
arbitrary curve $S$. We consider $\widehat{\mu}_{\min}(H^{0}(X,L^{\otimes
n}),n\varphi)$, and we define

$\widehat{\mu}_{\min}^{\operatorname{asy}}(\overline{L})=\liminf_{n\to\infty}\frac{\widehat{\mu}_{\min}(H^{0}(X,L^{\otimes
n}),n\varphi)}{n}.$

We recall some results on the criteria of arithmetic positivities via
$\widehat{\mu}_{\max}^{\operatorname{asy}}(\raisebox{1.72218pt}{.})$ and
$\widehat{\mu}_{\min}^{\operatorname{asy}}(\raisebox{1.72218pt}{.})$ in [9]
below.

######  (Proposition 9.1.2, [9]).

Let $\overline{L}$ be a relatively ample adelic line bundle. Then the
following properties are equivalent:

  1. (1)

$\overline{L}$ is ample;

  2. (2)

$\widehat{\mu}_{\min}^{\operatorname{asy}}(\overline{L})>0$;

  3. (3)

there exists an $\epsilon>0$, such that for any closed sub-scheme $Y$ of $X$,
we have
$\widehat{\mu}_{\max}^{\operatorname{asy}}(\overline{L}|_{Y})>\epsilon$.

See §3.1.3 for the definitions of relatively ample and ample adelic line
bundles.

######  (Proposition 9.1.7, [9]).

Let $\overline{L}$ be an adelic line bundle. Then the following properties are
equivalent:

  1. (1)

$\overline{L}$ is nef;

  2. (2)

$\overline{L}$ is relatively nef and
$\widehat{\mu}_{\min}^{\operatorname{asy}}(\overline{L})\geqslant 0$.

See §3.1.4 for the definitions of relatively nef and nef adelic line bundles.

#### 3.4.5

The following result shows that a power of an arithmetically big line bundle
over $S_{0}$ can be decomposed as the sum of an ample and an effective line
bundles, which is an analogue of [22, Corollary 2.4].

######  .

Let $X$ be a projective variety over $\operatorname{Spec}K$, and
$\overline{L}$ be arithmetically big on $X$ over
$S_{0}=(K,\\{0\\},|\raisebox{1.72218pt}{.}|_{0})$. Then there exists a
$p\in\mathbb{N}$, such that $\overline{L}^{\otimes
p}=\overline{M}\otimes\overline{N}$, where $\overline{M}$ is arithmetically
ample and $\overline{N}$ is arithmetically effective.

###### Proof.

By [14, Corollary 2.2.7], there exists a $p_{1}\in\mathbb{N}$, such that for
all $t>p_{1}$, we are able to find an ample line bundle $M$ and an effective
line bundle $M$ satisfying $L^{\otimes t}\cong A\otimes M$.

We equip $L$ with the metric $\varphi$ given in the statement such that
$\overline{L}$ is arithmetically big, and equip $A$ with an arbitrary metric
$\psi$ such that $\overline{M}$ is arithmetically ample. Then by [7, Corollary
6.4.10], there exists $p_{2}\in\mathbb{N}$, such that for all $t>p_{2}$, we
have

$\widehat{\deg}_{+}\left(H^{0}(X,L^{\otimes t}\otimes
M^{\vee}),t\varphi-\psi\right)>0.$

Since $S_{0}$ only has one place, then from the above inequality we deduce
that $\overline{L}^{\otimes t}\otimes\overline{M}^{\vee}$ contains an
effective section over $S_{0}$ in the sense of §3.1.5, so it is arithmetically
effective over $S_{0}$ by the adelic structure of $S_{0}$.

We choose $p\geqslant\max\\{p_{1},p_{2}\\}$ which satisfying the requirement.
∎

######  .

The geometric version of Proposition 3.4.5 is firstly proved by K. Kodaira,
see [14, Proposition 2.2.6, Corollary 2.2.7] for a brief introduction. An
arithmetic version over number fields was proved by X. Yuan [22, Theorem 2.3,
Corollary 2.4] via Siu’s inequality. It is an interesting question whether
Proposition 3.4.5 is verified over a general adelic curve, or add some
technical conditions. In fact, over a general adelic curve, we cannot deduce
that an adelic line bundle $(L,\varphi)$ on $X$ contains an effective section
from the inequality $\widehat{\deg}_{+}\left((H^{0}(X,L),\varphi)\right)>0$
directly.

##  4 Fujita approximation of graded linear series

In this section, we denote $E_{D}=H^{0}(X,L^{\otimes D})$ for simplicity, and
all the constructions will be based on this notation.

###  4.1 Approximatible graded linear series

Let $S=(K,(\Omega,\mathcal{A},\nu),\phi)$ be a proper adelic curve, where
either $\mathcal{A}$ is a discrete $\sigma$-algebra or $K$ has a countable
which is dense in $K_{w}$ for all $w\in\Omega$. Let
$S_{0}=(K,\\{0\\},|\raisebox{1.72218pt}{.}|_{0})$ be the adelic curve of
trivially valued field. In this part, we will consider some approximated
properties of normed graded linear series over $S$ and $S_{0}$.

Let $\lambda\in\mathbb{R}_{\geqslant 0}$,
$\overline{E}_{\operatorname{HN},0}^{\lambda}=K$, and

$\overline{E}_{\operatorname{HN},D}^{\lambda}=\operatorname{Vect}_{K}\left\\{s\in
H^{0}(X,L^{\otimes
D})\mid\|s\|_{\mathrm{sp},D}\leqslant\exp(-D\lambda)\right\\}$

for $D\geqslant 1$. So we have

$\overline{E}_{\operatorname{HN},D}^{\lambda}=\mathcal{F}_{\mathrm{sp}}^{D\lambda}\overline{E}_{\operatorname{HN},D},$

where the $\mathbb{R}$-filtration $\mathcal{F}_{\mathrm{sp}}$ is defined in
§2.3.2, and $\overline{E}_{\operatorname{HN},D}$ follows the definition in (9)
by taking $E_{D}=H^{0}(X,L^{\otimes D})$. In addition, we define

$\overline{E}^{\lambda}_{\operatorname{HN},\bullet}=\bigoplus_{D\geqslant
0}\overline{E}_{\operatorname{HN},D}^{\lambda},$ (10)

which is a graded linear series over $S_{0}$.

We will prove that (10) is approximable from the fact that it contains an
ample divisor, where the strategy is similar to [2, Proposition 3.12].

#### 4.1.1

We have the following proposition from the definition directly.

######  .

(10) is a filtrated graded linear series over $S_{0}$.

#### 4.1.2

For $\lambda\in\mathbb{R}$, we define the adelic line bundle

$\overline{\mathcal{O}}_{\lambda}=(K,\|\raisebox{1.72218pt}{.}\|_{\lambda})$
(11)

over $S_{0}=(K,\\{0\\},|\raisebox{1.72218pt}{.}|_{0})$, where the norm
satisfies $\|\mathbbm{1}\|_{\lambda}=e^{-\lambda}$.

######  .

Let $\pi:X\rightarrow\operatorname{Spec}K$ be a projective morphism, and
$\overline{L}$ be an adelic line bundle over
$S_{0}=(K,\\{0\\},|\raisebox{1.72218pt}{.}|_{0})$. With all the notations and
assumption same as in (10). If $\overline{L}$ is arithmetically big, then for
all
$\lambda\in\left[0,\widehat{\mu}^{\operatorname{asy}}_{\max}(\overline{L})\right)$,
$\overline{E}^{\lambda,\bullet}_{\operatorname{HN}}$ contains an ample
divisor.

###### Proof.

By the choice of $\lambda$, we have
$\widehat{\mu}^{\operatorname{asy}}_{\max}(\overline{L}\otimes\pi^{*}\overline{\mathcal{O}}_{-\lambda})>0$
by definition directly, so
$\overline{L}\otimes\pi^{*}\overline{\mathcal{O}}_{-\lambda}$ is an
arithmetically big line bundle on $X$ by Proposition 3.4.3. Then for a
sufficiently large $D\in\mathbb{N}$, $L^{\otimes D}$ has a global section
$s_{D}$ satisfying $\|s_{D}\|_{\mathrm{sp},D}\leqslant\exp(-D\lambda)$, which
proves $\overline{E}_{\operatorname{HN},D}^{\lambda}\neq 0$ for $D\gg 0$.

Since $\overline{L}\otimes\pi^{*}\overline{\mathcal{O}}_{-\lambda}$ is
arithmetically big on $X$, then by Propopsition 3.4.5, there exists
$p\in\mathbb{N}$, such that
$\left(\overline{L}\otimes\pi^{*}\overline{\mathcal{O}}_{-\lambda}\right)^{\otimes
p}=\overline{M}\otimes\overline{N}$, where $\overline{M}$ is arithmetically
ample and $\overline{N}$ is arithmetically effective, in other words
$\overline{M}$ contains an effective section $s$.

The remainder part of this proof is quite similar to that of [2, Proposition
3.12]. We are able to take the above $p\in\mathbb{N}$ divisible enough such
that the graded $K$-algebra
$\bigoplus\limits_{n\in\mathbb{N}}H^{0}(X,M^{\otimes n})$ is generated by
effective sections. These sections are considered as those of
$\overline{M}\otimes\pi^{*}(\overline{\mathcal{O}}_{\lambda}^{\otimes p})$,
whose norms are smaller than $\exp(-\lambda p)$. Then the homomorphism

$\bigoplus_{D\geqslant 0}H^{0}(X,M^{\otimes
D})\longrightarrow\bigoplus_{D\geqslant 0}H^{0}(X,L^{\otimes pD})$

induced by $s$ factors through $\bigoplus\limits_{n\geqslant
0}\overline{E}_{\operatorname{HN},np}^{\lambda}$. ∎

Same as the argument in [2, Corollary 3.13], we have the following result.

######  .

The graded linear series (10) over
$S_{0}=(K,\\{0\\},|\raisebox{1.72218pt}{.}|_{0})$ is approximable for any
$\lambda\in\left[0,\widehat{\mu}^{\operatorname{asy}}_{\max}(\overline{L})\right)$.

###### Proof.

By Proposition 3.2.1, we have the assertion from Proposition 4.1.2 directly. ∎

###  4.2 Fujita approximation of graded linear series

Let $p\in\mathbb{N}$ such that $E_{\operatorname{HN},p}^{0}\neq 0$, and
$\overline{E}_{\bullet}^{(p)}=\bigoplus\limits_{n\geqslant
0}\overline{E}^{(p)}_{\operatorname{HN},np}$ be the graded $K$-algebra
generated by $E_{\operatorname{HN},p}^{0}$, which is considered as a normed
graded linear series over $S_{0}$. At the same time, it is also able to be
considered as a normed graded linear series over a general proper adelic curve
$S=(K,(\Omega,\mathcal{A},\nu),\phi)$ equipped with the subspace norms of
$\bigoplus\limits_{D\geqslant 0}\overline{E}_{D}$. In the both senses, they
have the same arithmetic volume by Proposition 3.4.3 if $\mathcal{A}$ is a
discrete $\sigma$-algebra or $K$ has a countable sub-field which is dense in
$K_{w}$ for all $w\in\Omega$.

The following result is an analogue of [2, Theorem 4.1] over $S_{0}$, whose
proof is quite similar to the ancient one.

######  .

Let $X$ be a projective variety over $\operatorname{Spec}K$, and
$\overline{L}$ be an arithmetically big line bundle on $X$ over a proper
adelic curve $S=(K,(\Omega,\mathcal{A},\nu),\phi)$, where either $\mathcal{A}$
is a discrete $\sigma$-algebra or $K$ has a countable sub-field which is dense
in $K_{w}$ for all $w\in\Omega$. With all the above notations and
constructions, we have

$\widehat{\operatorname{vol}}(\overline{L})=\sup_{p\in\mathbb{N}}\widehat{\operatorname{vol}}(\overline{E}_{\bullet}^{(p)}),$

where $\overline{E}_{\bullet}^{(p)}$ is defined as above, which is a normed
graded linear series over $S_{0}$.

###### Proof.

For any $n\in\mathbb{N}^{+}$, let
$\nu_{\mathrm{sp},n}=T_{\frac{1}{n}}\nu_{\left(E_{\operatorname{HN},n},\mathcal{F}_{\mathrm{sp}}\right)}$,
where the notation
$T_{\frac{1}{n}}\nu_{\left(E_{\operatorname{HN},n},\mathcal{F}_{\mathrm{sp}}\right)}$
is introduced in §2.3.2, and the $\mathbb{R}$-filtration
$\mathcal{F}_{\mathrm{sp}}$ is defined in §3.4.2 with respect to the spectral
norms. Then by Proposition 2.3.2, the sequence
$(\nu_{\mathrm{sp},n})_{n\geqslant 1}$ converges weakly to a Borel probability
measure, which is denoted by $\nu_{\mathrm{sp}}$.

Similarly, let
$\nu_{\mathrm{sp},n}^{(p)}=T_{\frac{1}{np}}\nu_{\left(E^{(p)}_{\operatorname{HN},np},\mathcal{F}_{\mathrm{sp}}\right)}$,
which also converges weakly to a Borel probability measure noted by
$\nu_{\mathrm{sp}}^{(p)}$ by Proposition 4.1.1.

In order to prove the assertion, we will prove that the restriction of
$\nu_{\mathrm{sp}}$ on $[0,+\infty)$ is able to be approximated well by
$\nu_{\mathrm{sp}}^{(p)}$.

Step I. – Simultaneous approximations.

Let

$D:\;0=t_{0}<t_{1}<\cdots<t_{m}<\widehat{\mu}_{\max}^{\operatorname{asy}}(\overline{L})$

be a division of the interval
$[0,\widehat{\mu}_{\max}^{\operatorname{asy}}(\overline{L}))$ such that

$\nu_{\mathrm{sp}}(\\{t_{1},\ldots,t_{m}\\})=0.$ (12)

We define the function

$\begin{array}[]{rrcl}h_{D}:&\mathbb{R}&\longrightarrow&\mathbb{R}\\\
&x&\mapsto&\sum\limits_{i=0}^{m-1}t_{i}\mathbbm{1}_{[t_{i},t_{i+1})}(x)+t_{m}\mathbbm{1}_{[t_{m},+\infty)}(x).\end{array}$

By Corollary 4.1.2 and Definition 2.3.1, for every $\epsilon>0$, there exists
a $p\in\mathbb{N}$ sufficiently large, such that $E_{\bullet}^{(p)}$
approximates all $E^{t_{i}}_{\operatorname{HN},\bullet}$ defined at (10)
simultaneously, where $i=0,\ldots,m$. In other words, there exists an
$N_{0}\in\mathbb{N}$, such that for every $n\geqslant N_{0}$, we have

$\inf_{0\leqslant i\leqslant
m}\frac{\dim_{K}\left(\operatorname{Im}\left(S^{n}E_{\operatorname{HN},p}^{t_{i}}\rightarrow
E_{\operatorname{HN},np}^{t_{i}}\right)\right)}{\dim_{K}(E_{\operatorname{HN},np}^{t_{i}})}\geqslant
1-\epsilon$

for all $i=0,\ldots,m$. Then by the above inequality, we obtain

$\displaystyle\dim_{K}(\mathcal{F}_{\mathrm{sp}}^{npt_{i}}(E_{\operatorname{HN},np}^{(p)}))$
$\displaystyle\geqslant$
$\displaystyle\dim_{K}\left(\operatorname{Im}\left(S^{n}E_{\operatorname{HN},p}^{t_{i}}\rightarrow
E_{\operatorname{HN},np}^{t_{i}}\right)\right)$ $\displaystyle\geqslant$
$\displaystyle(1-\epsilon)\dim_{K}(E_{\operatorname{HN},np}^{t_{i}})$

for all $i=0,\ldots,m$.

Step II. – An upper bound of integrations.

We note that

$\displaystyle
np\dim_{K}(E_{\operatorname{HN},np}^{(p)})\int_{0}^{+\infty}t\nu_{\mathrm{sp},np}^{(p)}(\mathrm{d}t)=-\int_{0}^{+\infty}t\mathrm{d}\dim_{K}(\mathcal{F}_{\mathrm{sp}}^{t}E_{\operatorname{HN},np}^{(p)})$
$\displaystyle=$
$\displaystyle\int_{0}^{+\infty}\dim_{K}(\mathcal{F}_{\mathrm{sp}}^{t}E_{\operatorname{HN},np}^{(p)})\mathrm{d}t\geqslant\sum_{i=1}^{m}np(t_{i}-t_{i-1})\dim_{K}(\mathcal{F}_{\mathrm{sp}}^{npt_{i}}E_{\operatorname{HN},np}^{(p)})$
$\displaystyle\geqslant$ $\displaystyle
np(1-\epsilon)\sum_{i=1}^{m}(t_{i}-t_{i-1})\dim_{K}(E_{\operatorname{HN},np}^{t_{i}}),$

where the first inequality is from the decreasing of
$\mathcal{F}_{\mathrm{sp}}$, and the last inequality is from (4.2) combined
with the fact
$E_{\operatorname{HN},np}^{t_{i}}=\mathcal{F}_{\mathrm{sp}}^{npt_{i}}E_{\operatorname{HN},np}$.

In addition, by applying Abel’s summation formula to the last term of the
above inequality, we have

$\dim_{K}(E_{\operatorname{HN},np}^{(p)})\int_{0}^{+\infty}t\nu_{\mathrm{sp},np}^{(p)}(\mathrm{d}t)\geqslant(1-\epsilon)\dim_{K}(E_{\operatorname{HN},np})\int_{0}^{+\infty}h_{D}(t)\nu_{\mathrm{sp},np}(\mathrm{d}t).$

Step III. – Estimates of arithmetic volumes.

By Proposition 3.4.3, we have

$\displaystyle\lim_{n\to\infty}\frac{d+1}{(np)^{d+1}/(d+1)!}\dim_{K}(E_{\operatorname{HN},np}^{(p)})\int_{0}^{+\infty}t\nu_{\mathrm{sp},np}^{(p)}(\mathrm{d}t)$
$\displaystyle=$
$\displaystyle(d+1)\operatorname{vol}(E_{\bullet}^{(p)})\int_{0}^{+\infty}t\nu_{\mathrm{sp}}^{(p)}(\mathrm{d}t)=\widehat{\operatorname{vol}}(E_{\bullet}^{(p)}).$

Then we have

$\displaystyle\widehat{\operatorname{vol}}(E_{\bullet}^{(p)})$
$\displaystyle\geqslant$
$\displaystyle\lim_{n\to\infty}\frac{d+1}{(np)^{d+1}/(d+1)!}(1-\epsilon)\dim_{K}(E_{\operatorname{HN},np})\int_{0}^{+\infty}h_{D}\mathrm{d}\nu_{\mathrm{sp},np}$
$\displaystyle=$
$\displaystyle(d+1)(1-\epsilon)\operatorname{vol}(L)\int_{\mathbb{R}}h_{D}\mathrm{d}\nu_{\mathrm{sp}},$

where the last equality is from [1, Chap. IV §5 $n^{\circ}$ 12 Proposition
22].

We choose a sequence of subdivisions $(D_{j})_{j\in\mathbb{N}}$ satisfying
(12), such that the sequence of functions $(h_{D_{j}}(t))_{j\in\mathbb{N}}$
converges uniformly to

$\max\\{t,0\\}-\max\\{t-\widehat{\mu}_{\max}^{\operatorname{asy}}(\overline{L}),0\\}$

when $j\to\infty$. By definition, the support of $\nu_{\mathrm{sp}}$ is
bounded from above by
$\widehat{\mu}_{\max}^{\operatorname{asy}}(\overline{L})$, we obtain

$\sup_{p\in\mathbb{N}}\widehat{\operatorname{vol}}(E_{\bullet}^{(p)})\geqslant(d+1)(1-\epsilon)\operatorname{vol}(L)\int_{0}^{+\infty}t\nu_{\mathrm{sp}}(\mathrm{d}t)=(1-\epsilon)\widehat{\operatorname{vol}}(\overline{L}),$

where the last equality is from Proposition 3.4.3 on the equality of
arithmetic volumes over $S$ and $S_{0}$. ∎

######  .

For a general graded linear series, if it satisfies the property of
Proposition 4.2, we say that this graded linear series satisfies Fujita
approximation. We would like to emphasize that this notation is different from
those in [5, Definition 3.7] and [6, §6].

##  5 Fujita approximation of adelic line bundles

From this Fujita approximation of graded linear series proved in Proposition
4.2, we will prove the arithmetic Fujita approximation of an arithmetically
big line bundle.

###  5.1 Fujita approximation of big adelic line bundles

In this part, we will prove the arithmetic Fujita approximation of usual
adelic line bundles. First, we prove a nef version.

######  .

Let $S=(K,(\Omega,\mathcal{A},\nu),\phi)$ be a proper adelic curve, where
either $\mathcal{A}$ is a discrete $\sigma$-algebra or $K$ has a countable
sub-field which is dense in $K_{w}$ for all $w\in\Omega$. Let $X$ be a
projective variety with $\dim(X)=d$, and $\overline{L}$ be an arithmetically
big line bundle on $X$ over $S$. Then for every $\epsilon>0$, there exists a
birational morphism $v:X^{\prime}\rightarrow X$ together with a positive
integer $p$, such that the decomposition $v^{*}(\overline{L}^{\otimes
p})=\overline{M}\otimes\overline{G}$ satisfies:

  1. (1)

$\overline{M}$ is effective and $\overline{G}$ is arithmetically nef;

  2. (2)

$p^{-(d+1)}\widehat{\operatorname{vol}}(\overline{G})\geqslant\widehat{\operatorname{vol}}(\overline{L})-\epsilon$.

###### Proof.

We devide the proof into three parts.

Step I. – Determining the geometric decomposition.

Let $\pi:X\rightarrow\operatorname{Spec}K$ be the structural morphism. For any
$p\geqslant 1$ such that $E_{\operatorname{HN},p}^{0}\neq 0$ (see Proposition
4.1.2 for the existence of such $p\in\mathbb{N}$), let

$\phi_{p}:X_{p}\rightarrow X$

be the blow-up along the base locus of $E_{\operatorname{HN},p}^{0}$, which
means

$X_{p}=\operatorname{Proj}\left(\operatorname{Im}\left(\bigoplus_{n\geqslant
0}\pi^{*}(E_{\operatorname{HN},np}^{p})\longrightarrow\bigoplus_{n\geqslant
0}L^{\otimes np}\right)\right).$

Let $G_{p}=\mathcal{O}_{X_{p}}(1)$, and $M_{p}$ be the line bundle on $X_{p}$
with respect to the exceptional divisor. Let $s$ be a global section of
$M_{p}$ which trivializes $M_{p}$ outside the exceptional divisor. Then by
definition, we have

$\phi_{p}^{*}(L^{\otimes p})=G_{p}\otimes M_{p}.$

The canonical morphism

$\phi_{p}^{*}\pi^{*}(E_{\operatorname{HN},p}^{0})\rightarrow G_{p}$

induces a projective embedding
$i_{p}:X_{p}\rightarrow\mathbb{P}\left(E_{\operatorname{HN},p}^{0}\right)$
with $i^{*}\left(\mathcal{O}_{E_{\operatorname{HN},p}^{0}}(1)\right)=G_{p}$.
Then the restriction of global sections of
$\mathcal{O}_{E_{\operatorname{HN},p}^{0}}(n)$ on $X_{p}$ gives an injective
homomorphism

$\operatorname{Im}\left(S^{n}E_{\operatorname{HN},p}^{0}\rightarrow
E_{\operatorname{HN},np}\right)=E_{\operatorname{HN},np}^{(p)}\rightarrow
H^{0}(X_{p},G_{p}^{\otimes n}),$ (14)

where $H^{0}(X_{p},G_{p}^{\otimes n})$ is considered as a sub-$K$-vector space
of $H^{0}(X_{p},\phi_{p}^{*}(L^{\otimes p}))$ via the section $s$.

Step II. – Constructing the metrics of line bundles.

For any $n\geqslant 1$ and $w\in\Omega$, let
$\|\raisebox{1.72218pt}{.}\|_{n,w}$ be the quotient norm on
$G_{p,w}=G_{p}\times_{\operatorname{Spec}K}\operatorname{Spec}K_{w}$ induced
by the surjective homomorphism

$\phi_{p}^{*}\pi^{*}(E_{\operatorname{HN},np}^{(p)})\rightarrow G_{p}^{\otimes
n},$

where $E_{\operatorname{HN},np}^{(p)}$ is equipped with the superior norm
$\|\raisebox{1.72218pt}{.}\|_{\sup,w}$ with respect to $w\in\Omega$.

In fact, for any $x\in X_{p,w}(\overline{K}_{w})$ outside the exceptional
divisor, it corresponds to a $1$-dimensional quotient of
$E_{\operatorname{HN},p}^{0}$. For any $n\geqslant 1$, this quotient induces a
$1$-dimensional quotient $\ell_{n,x}$ of $E_{\operatorname{HN},np}^{(p)}$. By
Hahn-Banach theorem, every linear form on
$\ker(E_{\operatorname{HN},np}^{(p)}\rightarrow\ell_{n,x})$ can be extended to
a linear form on $E_{\operatorname{HN},np}^{(p)}$. In other words, there
exists a section $w\in E_{\operatorname{HN},np}^{(p)}$ whose image in
$G_{p,w}^{\otimes n}(x)$ has norm $\|w\|_{\sup,w}$. As a section of
$L_{w}^{\otimes np}$ on $X_{v}(\overline{K}_{w})$, we have
$\|w(x)\|_{w}\leqslant\|w\|_{\sup,w}$, where $\|\raisebox{1.72218pt}{.}\|_{w}$
is the metric on $\overline{L}$ indexed by $v\in\Omega$. Then for any section
$u\in G_{p,w}$ over a neighborhood of $x$, we may assume that $u(x)^{\otimes
n}$ equals to the image of $w(x)$ in $G_{p,w}^{\otimes n}(x)$ by dilation,
then we have

$\|u(x)\|_{n,w}=\|w\|_{\sup,w}^{1/n}\geqslant\|w(x)\|_{w}^{1/n}=\|u(x)\otimes
s(x)\|_{w}.$

Let $\alpha_{n}=(\|\raisebox{1.72218pt}{.}\|_{n,w})_{W\in\Omega}$ be an adelic
metric on $G_{p}$, and we define
$(M_{p},\beta_{n})=\phi_{p}^{*}(\overline{L})\otimes(G_{p},\alpha_{n})^{\vee}$.
In this sense, $s$ is an effective section in $(M_{p},\beta_{n})$, then it is
an effective line bundle.

Step III. – The relation among arithmetic volumes.

By the construction in Step II, for any $n\geqslant 1$, we have

$p^{-(d+1)}\widehat{\operatorname{vol}}(G_{p},\alpha_{n})\geqslant\widehat{\operatorname{vol}}(E_{\bullet}^{(p)},\alpha_{n}),$

where

$\widehat{\operatorname{vol}}(E_{\bullet}^{(p)},\alpha_{n})=\lim_{m\rightarrow\infty}\frac{\widehat{\deg}_{+}(E_{\operatorname{HN},mp}^{(p)},(\|\raisebox{1.72218pt}{.}\|_{n,w,\sup})_{w\in\Omega})}{m^{d+1}/(d+1)!},$

and $\|\raisebox{1.72218pt}{.}\|_{n,w,\sup}$ is the superior norm induced by
$\alpha_{n}=(\|\raisebox{1.72218pt}{.}\|_{n,w})_{w\in\Omega}$ on $G_{p}$. Then
by [2, Corollary A.2], we have

$\limsup_{n\to\infty}\widehat{\operatorname{vol}}(E_{\bullet}^{(p)},\alpha_{n})\geqslant\widehat{\operatorname{vol}}(\overline{E}_{\bullet}^{(p)}).$

By Proposition 4.2, for any $\epsilon>0$, there exists a $p\in\mathbb{N}$,
such that

$\limsup_{n\to\infty}p^{-(d+1)}\widehat{\operatorname{vol}}(G_{p},\alpha_{n})\geqslant\widehat{\operatorname{vol}}(\overline{L})-\frac{\epsilon}{2}.$

Then there exists an $N_{0}\in\mathbb{N}$, such that

$p^{-(d+1)}\widehat{\operatorname{vol}}(G_{p},\alpha_{n})-\frac{\epsilon}{2}\geqslant\widehat{\operatorname{vol}}(\overline{L})-\frac{\epsilon}{2}$

for infinitely many $n\geqslant N_{0}$. In addition, by Proposition 3.4.4 we
obtain that $(G_{p},\alpha_{n})$ is arithmetically nef, since
$\widehat{\mu}_{\min}(\overline{E}_{\operatorname{HN},np}^{(p)})\geqslant 0$
for all $n\geqslant 1$, then we deduce the positive asymptotic minimal slope.

To sum up, we obtain the assertion. ∎

Next, we obtain the following ample version from Theorem 5.1 directly.

######  .

We keep all the notations and conditions in Theorem 5.1. Then we are able to
require $\overline{G}$ to be arithmetically ample in the term (1) of Theorem
5.1.

###### Proof.

Let $\overline{B}$ be an arithmetically big line bundle on $X$. Then by [7,
Corollary 6.4.10], for $m\in\mathbb{N}$ sufficiently large, we have
$\overline{L}^{\otimes m}\otimes\overline{B}^{\vee}$ is arithmetically big. By
Theorem 5.1, for any $\epsilon>0$, there exists a birational morphism
$\phi:X^{\prime}\rightarrow X$, a $p\in\mathbb{N}$, and a decomposition

$\phi^{*}(\overline{L}^{\otimes
mp}\otimes\overline{B}^{\otimes(-p)})=\overline{G}\otimes\overline{M}$

with $\overline{G}$ arithmetically nef and $\overline{M}$ effective, such that

$p^{-(d+1)}\widehat{\operatorname{vol}}(\overline{G})\geqslant\widehat{\operatorname{vol}}(\overline{L}^{\otimes
m}\otimes\overline{B}^{\vee})-\epsilon.$

Let $\overline{B}^{\prime}$ be an arithmetically ample line bundle on
$X^{\prime}$. Since $\phi^{*}(\overline{B})$ is arithmetically big, there
exists a $q\in\mathbb{N}$, such that $\phi^{*}(\overline{B}^{\otimes
q})\otimes\overline{B}^{\prime\vee}$ is effective. We consider

$\phi^{*}(\overline{L}^{\otimes mpq})=(\overline{G}^{\otimes
q}\otimes\overline{B}^{\prime})\otimes(\overline{M}^{\otimes
q}\otimes\phi^{*}(\overline{B}^{\otimes
pq})\otimes\overline{B}^{\prime\vee}).$

In the above decomposition, $\overline{G}^{\prime}:=\overline{G}^{\otimes
q}\otimes\overline{B}^{\prime}$ is arithmetically ample,
$\overline{M}^{\otimes q}\otimes\phi^{*}(\overline{B}^{\otimes
pq})\otimes\overline{B}^{\prime\vee}$ is effective, and

$(mpq)^{-(d+1)}\widehat{\operatorname{vol}}(\overline{G}^{\prime})\geqslant(mpq)^{-(d+1)}\widehat{\operatorname{vol}}(\overline{G}^{\otimes
q})\geqslant m^{-(d+1)}(\widehat{\operatorname{vol}}(\overline{L}^{\otimes
m}\otimes\overline{B}^{\vee})-\epsilon).$

By the continuity of the arithmetic volume function (see [7, Corollary
6.4.10]), we have the result. ∎

###  5.2 Case of $\mathbb{K}$-line bundles

In this part we will prove the arithmetic Fujita approximation is verified for
adelic $\mathbb{Q}$-line bundles and $\mathbb{R}$-line bundles over adelic
curve.

#### 5.2.1

First, we prove the case of adelic $\mathbb{Q}$-line bundle, which is a quite
straightforward from the usual case.

######  .

Let $S=(K,(\Omega,\mathcal{A},\nu),\phi)$ be a proper adelic curve, where
either $\mathcal{A}$ is a discrete $\sigma$-algebra or $K$ has a countable
sub-field which is dense in $K_{w}$ for all $w\in\Omega$. Let $X$ be a
projective variety over $\operatorname{Spec}K$ with $\dim(X)=d$, and
$\overline{L}$ be an arithmetically big adelic $\mathbb{Q}$-line bundle on $X$
over $S$. For every $\epsilon>0$, there exists a birational morphism
$\phi:X^{\prime}\rightarrow X$ together with a $p\in\mathbb{N}^{+}$, such that
the decomposition $\phi^{*}(\overline{L}^{\otimes
p})=\overline{M}\otimes\overline{G}$ satisfies:

  1. (1)

$\overline{M}$ and $\overline{G}$ are adelic $\mathbb{Q}$-line bundles, where
$\overline{M}$ is effective and $\overline{G}$ is arithmetically ample;

  2. (2)

$p^{-(d+1)}\widehat{\operatorname{vol}}(\overline{G})\geqslant\widehat{\operatorname{vol}}(\overline{L})-\epsilon$.

###### Proof.

By definition, there exists an $m\in\mathbb{N}$, such that
$\overline{L}^{\otimes m}$ is a usual arithmetically big adelic line bundle.
Then we obtain the assertion by applying Corollary 5.1 to
$\overline{L}^{\otimes m}$. ∎

#### 5.2.2

Next, we prove the case of adelic $\mathbb{R}$-line bundle, and see [7,
§6.4.3] for an introduction to this notion. For this target, we prove an
auxiliary lemma first, which allows us to approximate an arithmetically big
$\mathbb{R}$-line bundle by an arithmetically big $\mathbb{Q}$-line bundle
along an effective direction.

######  .

Let $S=(K,(\Omega,\mathcal{A},\nu),\phi)$ be a proper adelic curve, and $X$ be
a projective variety over $\operatorname{Spec}K$. Let $\overline{L}$ be an
arithmetically big $\mathbb{R}$-line bundle on $X$. Then for all $\epsilon>0$,
there exists an arithmetically big $\mathbb{Q}$-line bundle
$\overline{L}^{\prime}$, such that $\overline{L}-\overline{L}^{\prime}$ is
effective and

$0<\widehat{\operatorname{vol}}(\overline{L})-\widehat{\operatorname{vol}}(\overline{L}^{\prime})<\epsilon.$

###### Proof.

The adelic $\mathbb{R}$-line bundle $\overline{L}$ is arithmetically big, so
$L$ is a big $\mathbb{R}$-line bundle. By an equivalent definition [14,
Definition 2.2.21], we have $L=\sum\limits_{i=1}^{r}a_{i}L_{i}$, where
$a_{1},\ldots,a_{r}>0$ and $L_{1},\ldots,L_{r}$ are usual big line bundles. By
Kodaira’s decomposition [14, Corollary 2.2.7], there exists an
$m\in\mathbb{N}$, such that for all $i=1,\ldots,r$, we have
$mL_{i}=A_{i}+M_{i}$, where $A_{i}$ is ample and $M_{i}$ is effective.

We equip metrics on $A_{1},\ldots,A_{r},M_{1},\ldots,M_{r}$ over $S$, such
that $\overline{A}_{1},\ldots,\overline{A}_{r}$ are arithmetically ample, and
$\overline{M}_{1},\ldots,\overline{M}_{r}$ are arithmetically effective. Then
for any $b_{1},\ldots,b_{r},c_{1},\ldots,c_{r}\in\mathbb{R}^{+}$, we have
$b_{1}\overline{A}_{1}+\cdots+b_{r}\overline{A}_{r}+c_{1}\overline{M}_{1}+\cdots+c_{r}\overline{M}_{r}$
is effective.

By the continuity of $\widehat{\operatorname{vol}}(\raisebox{1.72218pt}{.})$
from [7, Corollary 6.4.24], for all $\epsilon>0$, there exist
$\delta_{1},\ldots,\delta_{r},\eta_{1},\ldots,\eta_{r}>0$, such that for all
$t_{i}\in(0,\delta_{i})$, $s_{i}\in(0,\eta_{i})$ for $i=1,\ldots,r$, we always
have

$0<\widehat{\operatorname{vol}}(m\overline{L})-\widehat{\operatorname{vol}}\left(m\overline{L}-\sum_{i=1}^{r}t_{i}\overline{A}_{i}-\sum_{i=1}^{r}s_{i}\overline{M}_{i}\right)<\epsilon.$

If there does not exist an arithmetically big $\mathbb{Q}$-line bundle
satisfying the requirements of the lemma, then
$m\overline{L}-\sum\limits_{i=1}^{r}t_{i}\overline{A}_{i}-\sum\limits_{i=1}^{r}s_{i}\overline{M}_{i}$
can never be a $\mathbb{Q}$-line bundle for all $t_{i}\in(0,\delta_{i})$ and
$s_{i}\in(0,\eta_{i})$ $i=1,\ldots,r$. It means there is no $\mathbb{Q}$-point
in the region

$(ma_{1}-\delta_{1},ma_{1})\times\cdots\times(ma_{r}-\delta_{r},ma_{r})\times(ma_{1}-\eta_{1},ma_{1})\times\cdots\times(ma_{r}-\eta_{r},ma_{r})\subset\mathbb{R}^{2r},$

which has a strict positive Lebesgue measure. Then it contradicts to the fact
that $\mathbb{Q}^{2r}$ is dense in $\mathbb{R}^{2r}$ with respect to the
Euclidean topology.

By choosing the constant $\epsilon$ sufficiently small, we pick a family
$t_{1},\ldots,t_{r},s_{1},\ldots,s_{r}\in\mathbb{R}^{+}$ such that
$m\overline{L}-\sum\limits_{i=1}^{r}t_{i}\overline{A}_{i}-\sum\limits_{i=1}^{r}s_{i}\overline{M}_{i}$
is an arithmetically big $\mathbb{Q}$-line bundle. By the above construction,
$\sum\limits_{i=1}^{r}t_{i}\overline{A}_{i}+\sum\limits_{i=1}^{r}s_{i}\overline{M}_{i}$
is effective. Let

$\overline{L}^{\prime}=\overline{L}-\frac{1}{m}\sum\limits_{i=1}^{r}t_{i}\overline{A}_{i}-\frac{1}{m}\sum\limits_{i=1}^{r}s_{i}\overline{M}_{i},$

which satisfies the requirement. ∎

We prove the Fujita approximation of $\mathbb{R}$-line bundles via Lemma
5.2.2.

######  .

Let $S=(K,(\Omega,\mathcal{A},\nu),\phi)$ be a proper adelic curve, where
either $\mathcal{A}$ is a discrete $\sigma$-algebra or $K$ has a countable
sub-field which is dense in $K_{w}$ for all $w\in\Omega$. Let $X$ be a
projective variety over $\operatorname{Spec}K$ with $\dim(X)=d$, and
$\overline{L}$ be an arithmetically big adelic $\mathbb{R}$-line bundle on $X$
over $S$. For every $\epsilon>0$, there exists a birational morphism
$\phi:X^{\prime}\rightarrow X$ together with a $p\in\mathbb{N}^{+}$, such that
the decomposition $\phi^{*}(\overline{L}^{\otimes
p})=\overline{M}\otimes\overline{G}$ satisfies:

  1. (1)

$\overline{M}$ and $\overline{G}$ are adelic $\mathbb{R}$-line bundles, where
$\overline{M}$ is effective and $\overline{G}$ is arithmetically ample;

  2. (2)

$p^{-(d+1)}\widehat{\operatorname{vol}}(\overline{G})\geqslant\widehat{\operatorname{vol}}(\overline{L})-\epsilon$.

###### Proof.

By Lemma 5.2.2, for all $\epsilon>0$, there exists an arithmetically big
$\mathbb{Q}$-line bundle $\overline{L}^{\prime}$, such that
$\overline{L}-\overline{L}^{\prime}$ is effective and

$0<\widehat{\operatorname{vol}}(\overline{L})-\widehat{\operatorname{vol}}(\overline{L}^{\prime})<\epsilon.$

Then we apply Corollary 5.2.1 to $\overline{L}^{\prime}$, and we obtain the
assertion. ∎

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