# On the structure of the complement of skeleton

Morgan Brown  Department of Mathematics, University of Miami, Ungar Bldg, 1365
Memorial Dr #515, Coral Gables, FL 33146, USA<EMAIL_ADDRESS>,
Jiachang Xu  Institute of Mathematics and Informatics, Bulgarian Academy of
Sciences, Bulgaria, Sofia 1113, Acad. G. Bonchev Str., Bl. 8
<EMAIL_ADDRESS>and  Muyuan Zhang  Department of Mathematics,
University of Miami, Ungar Bldg, 1365 Memorial Dr #515, Coral Gables, FL
33146, USA<EMAIL_ADDRESS>
###### Abstract.

We study the higher dimensional geometry of Berkovich spaces using open fiber
disks, which are given by open disks in a relative dimension $1$ fibration.
Inspired by birational geometry, we conjecture that the Berkovich skeleton is
the complement of the union of all open fiber disks, and prove this conjecture
for $\mathcal{X}$ admitting a strictly semistable model with semiample
canonical class.

###### 2000 Mathematics Subject Classification:

14G22,14E30

##  1\. Introduction

Throughout the paper, we fix a non-Archimedean field $K$ with valuation ring
$K^{\circ}$, its residue field $\widetilde{K}$, and a pseudo-uniformizer
$\varpi_{K}$. The purpose of this article is to study the structure of the
complement of skeleton of $K$-analytic spaces in the sense of Berkovich
[Ber90]. The development of the related study started from [BL85] by Bosch and
Lütkebohmert for semi-stable reduction of curves in the rigid geometry
setting, then by [Duc08] and [BPR13], the authors described the structure of
the complement of the Berkovich skeleton as a union of unit disks for
Berkovich curves which admit semi-stable models. The present work aims at
accomplishing two goals: First, we wish to elucidate the structure of higher
dimensional Berkovich spaces. Our second goal is to extend the analogy between
Berkovich geometry and birational geometry developed in the study of the
essential skeleton, which guides our approach to the first goal.

######  1.1.

The Essential Skeleton

The essential skeleton $\textsf{Sk}^{\textsf{ess}}(\mathcal{X}^{\textsf{an}})$
was first defined by Kontsevich and Soibelman [KS06] to study Mirror Symmetry
for K3 surfaces. Their definition was extended by Mustaţă and Nicaise [MN15]
for any variety admitting global nonvanishing pluricanonical forms. The
essential skeleton is a birational invariant of $\mathcal{X}$ and doesn’t
depend on the choice of a model over the valuation ring. However should
$\mathcal{X}$ admit a good dlt minimal model over the valuation ring then the
essential skeleton is identified with the Berkovich skeleton of that model
[NX16].

######  1.2.

Open Fiber Disks

Inspired by Mori’s conjecture, which states that a characteristic $0$
projective variety $\mathcal{X}$ is uniruled (has a dense algebraic family of
rational curves) if and only if the Kodaira dimension of $\mathcal{X}$ is
$-$$\infty$. This is to say that the absence of pluricanonical forms on $X$
should imply the presence of a large family rational curves. We introduce open
fiber disks 2.10 to be the Berkovich analogue of these families. The idea is
that an open fiber disk in $\mathcal{X}^{\textsf{an}}$ comes from expressing
$\mathcal{X}^{\textsf{an}}$ locally as a relative dimension $1$ fibration, by
finding a open disk over a Berkovich point in the base, which will in general
correspond to a valued field extension with a large transcendence degree over
$K$.

######  1.3.

Main Conjecture

We conjecture in general that the complement of the Berkovich skeleton is a
union of open fiber disks in $\mathcal{X}^{\textsf{an}}$:

######  Conjecture 1.4.

(Conjecture 5.1) Let $\mathcal{X}$ be a smooth proper integral variety with
dimension $n$ over a complete discrete valued field $K$ with equal
characteristic $0$, then for any point $x\in\mathcal{X}^{\textsf{an}}$, $x$ is
contained in an open fiber disk if and only if $x$ is not contained in the
essential skeleton.

We are able to prove this conjecture under the condition that $\mathcal{X}$
admits a strictly semi-stable model with semiample canonical class. While this
is a restrictive condition, this theorem provides evidence for the soundness
of our approach.

######  Theorem 1.5.

(Main Theorem 5.2) Let ${{\mathcal{X}}}$ be a proper and strictly semi-stable
scheme over complete discrete valuation ring $K^{\circ}$ such that the
canonical divisor $K_{\mathcal{X}}$ is semiample. Then for any point
$x\in\mathcal{X}_{K}^{\textsf{an}}$, $x$ is contained in an open fiber disk if
and only if $x$ is not contained in the essential skeleton.

######  1.6.

We prove Theorem 5.2 in two parts. We use the local picture of
$\mathbf{G}^{n,\textsf{an}}_{m,K}$ fibered over
$\mathbf{G}^{n-1,\textsf{an}}_{m,K}$ to establish the existence of open fiber
disks outside the Berkovich skeleton in the semi-stable case.

######  Theorem 1.7.

(Theorem 3.17). Let ${{\mathcal{X}}}$ be a strictly semi-stable scheme over
$K^{\circ}$, the complement of Berkovich skeleton
$\textsf{Sk}({\widehat{\mathcal{X}}}^{\textsf{rig}})^{\textsf{c}}$ is a union
of open fiber disks.

######  1.8.

On the other hand, we must exclude the existence of open fiber disks from the
essential skeleton. This procedure is analogous to the exclusion of families
of rational curves from varieties admitting a global pluricanonical form. A
key technical ingredient is Temkin’s work on metrization of the pluricanonical
forms [Tem16]

######  Theorem 1.9.

(Main Theorem 4.13) Let $\mathcal{X}$ be a smooth proper integral variety with
dimension $n$ over $K$, then for any point $x$ of the essential skeleton
$\textsf{Sk}^{\textsf{ess}}(\mathcal{X}^{\textsf{an}})$ there exists no open
fiber disk in $\mathcal{X}$ containing $x$.

######  1.10.

When $\mathcal{X}$ itself is uniruled, we consider the essential skeleton to
be empty, so we conjecture that $\mathcal{X}^{\textsf{an}}$ is covered by open
fiber disks after a finite field extension.

######  Conjecture 1.11.

Let $\mathcal{X}$ be a uniruled variety over $K$, then
$\mathcal{X}^{\textsf{an}}$ can be covered by open fiber disks after a finite
field extension $K^{\prime}|K$.

By using the theorem 3.17, we have the following result:

######  Theorem 1.12.

$\mathbf{P}_{K}^{n,\textsf{an}}$ can be covered by open fiber disks after a
finite field extension $K^{\prime}|K$.

######  1.13.

An alternative description for the complement of Berkovich skeleton of semi-
stable variety is motivated by a simple observation of V. Berkovich [Ber07],
that is, the maximal analytic continuation of Iwasawa log function
$\mathsf{Log}^{\lambda}$ coincides with the complement of the essential
skeleton of the one-dimensional torus. By using the local model of semi-stable
reduction, we generalized the observation to the following proposition:

######  Proposition 1.14.

(Proposition 6.9) Let ${{\mathcal{X}}}$ be a strictly semi-stable scheme over
$K^{\circ}$, the complement of Berkovich skeleton
$\textsf{Sk}({\widehat{\mathcal{X}}}^{\textsf{rig}})^{\textsf{c}}$ is a union
of analyticity sets of the naive analytic function $\mathsf{Log}^{\lambda}$.

### Acknowledgements.

Morgan Brown was supported by the Simons Foundation Collaboration Grant
524003. Morgan Brown is very grateful to the Institute for Mathematics and
Informatics at the Bulgarian Academy of Sciences (IMI-BAS) for hosting his
visit during the completion of this project. During this visit Morgan Brown
was supported by the Bulgarian Ministry of Education and Science, Scientific
Programme "Enhancing the Research Capacity in Mathematical Sciences (PIKOM)",
No. DO1-67/05.05.2022. Jiachang Xu is supported by the National Science Fund
of Bulgaria, National Scientific Program "VIHREN", Project no. KP-06-DV-7.
Jiachang Xu is very grateful for the support of Morgan Brown and Ludmil
Katzarkov for many years and the inspiring discussion with Morgan Brown and
Ludmil Katzarkov during the completion of most of this project. Jiachang Xu
also thanks Andrei Bengus-lasnier for helpful discussions. Muyuan Zhang is
grateful to Morgan Brown as his advisor and is honored to work on this project
as part of his PhD dissertation. We also thank T. Y. Yu for his helpful
suggestion on this project.

##  2\. Preliminaries

######  2.1.

For our purposes a non-Archimedean field means a complete topological field
$K$ with a non-trivial non-Archimedean valuation $|-|_{K}$. There exists a
nonzero element $\varpi$ pseudo-uniformizer such that $|\varpi|_{K}<1$ and
$\varpi$-adic topology coincides with the valuation topology. We denote
$\widehat{K^{\textsf{alg}}}$ as the completion of algebraic closure of $K$.

######  2.2.

Let $K$ be a non-Archimedean field. All $K$-analytic spaces are assumed to be
Hausdorff and strictly $K$-analytic.

######  2.3.

Let $\mathcal{X}$ be a locally finite type scheme over $K$, we denote by
$\mathcal{X}^{\textsf{an}}$ the associated $K$-analytic space defined in
[Ber90, 3.4.1. Theorem]. Let $\mathfrak{X}$ be a formal scheme locally
finitely presented over the ring of integers $K^{\circ}$, we denote by
$\mathfrak{X}^{\textsf{rig}}$ the associated $K$-analytic space which is
called the Berthelot generic fiber in the sense of [Ber94, § 1]. There is a
canonical open immersion
${\mathfrak{X}}^{\textsf{rig}}\hookrightarrow\mathcal{X}^{\textsf{an}}$ and
${\mathfrak{X}}^{\textsf{rig}}=\mathcal{X}^{\textsf{an}}$ when $\mathcal{X}$
is proper.

######  2.4.

Let $L$ be a non-Archimedean $K$-field. For the algebraic extension part, we
denote $e_{L|K}$111Ramification index if $L|K$ is finite. and $f_{L|K}$ be the
cardinality of the quotient of value groups $|L^{\times}|/|K^{\times}|$ and
$[\widetilde{L}:\widetilde{K}]$ respectively. For the transcendental extension
part, we denote
$s_{L|K}:=\textsf{rank}_{\mathbf{Q}}(|L^{\times}|/|K^{\times}|\otimes_{\mathbf{Z}}\mathbf{Q})$222$\sqrt{|K^{\times}|}=|K^{\textsf{sep},\times}|=\mathbf{Q}\otimes_{\mathbf{Z}}|K^{\times}|$,
$t_{L|K}:=\textsf{tr.deg.}(\widetilde{L}|\widetilde{K})$ and
$d_{L|K}:=s_{L|K}+t_{L|K}$. For a point $x$ belonging to a $K$-analytic space
$X$, we denote $d_{K}(x)$, $s_{K}(x)$ and $t_{K}(x)$ as
$d_{\mathscr{H}(x)|K}$, $s_{\mathscr{H}(x)|K}$ and $t_{\mathscr{H}(x)|K}$
respectively. The set $X_{\textsf{st}}:=\\{x\in X\,|\,d_{K}(x)=0\\}$ has the
following simple property under étale morphisms:

######  Proposition 2.5.

Let $f:X\longrightarrow Y$ be an étale morphism of $K$-analytic spaces, then
we have $f^{-1}(Y_{\textsf{st}})=X_{\textsf{st}}$.

###### Proof.

Note that morphisms of $K$-analytic spaces do not increase the value of
$s_{K}(x)$ and $t_{K}(x)$, so it suffices to show the $d_{K}(f(x))=d_{K}(x)$
which is equivalent to $d_{\mathscr{H}(f(x))}(\mathscr{H}(x))=0$. This holds
since $\mathscr{H}(x)|\mathscr{H}(f(x))$ is finite. ∎

######  Definition 2.6.

Let $X$ be an $K$-analytic space, a point $x\in X$ is called monomial if
$d_{K}(x)=\mathsf{dim}_{x}(X)$. A monmial point $x\in X$ is called divisorial
point if $t_{K}(x)=\mathsf{dim}_{x}(X)$.

######  2.7.

For $X=\mathcal{X}^{\textsf{an}}$ when $\mathcal{X}$ is integral, we call the
preimage of the generic point of canonical morphism
$\tau:\mathcal{X}^{\textsf{an}}\longrightarrow\mathcal{X}$ is birational
points of $\mathcal{X}$ and denote by $\mathcal{X}^{\textsf{bir}}$, which can
be also interpreted as the set of real valuations
$\textsf{K}(X)\longrightarrow\mathbf{R}$ that extend the valuation $|-|_{K}$
on the ground field $K$. When $x\in\mathcal{X}^{\textsf{an}}$ is divisorial if
and only if there exists a formal model $\mathfrak{X}$ such that $x$ is the
preimage of a generic point of $\mathfrak{X}_{\textsf{s}}$ under the reduction
map. For a given semi-stable formal model $\mathfrak{X}$ of
$\widehat{\mathcal{X}}^{\textsf{rig}}$, we have associated Berkovich skeleton
denoted as
$\textsf{Sk}(\mathfrak{X})\subseteq\mathcal{X}^{\textsf{mon}}\subseteq\mathcal{X}^{\textsf{bir}}$,
for more details of constructions associated to strictly semi-stable
(polystable) formal scheme, please see [GRW16, 4.6.] or [Ber99]. Throughout
the paper, we will use $\textsf{Sk}^{\textsf{ess}}(\mathcal{X}^{\textsf{an}})$
to represent the essential skeleton defined in [MN15, Definition 4.10.] for a
smooth projective integral scheme over a complete discrete valued field $K$
with equal characteristic $0$, and
($\overline{\textsf{Sk}}^{\textsf{k\"{a}h}}(X)$)
$\textsf{Sk}^{\textsf{k\"{a}h}}(X)$ to represent the (geometric) Kähler
skeleton for a quasi-smooth $K$-analytic space $X$, see Definition 4.6. The
construction of skeleton $\textsf{Sk}(\mathbf{G}^{n,\textsf{an}}_{m,K})$ comes
from [Tem16, Remark 7.2.2.] coincides with its Kähler skeleton.

######  Proposition 2.8.

[MN15, Proposition 2.4.8.] Let $x$ be a monomial point of
$\mathcal{X}^{\textsf{an}}$. Then the following are equivalent:

  1. (1)

The point $x$ is divisorial.

  2. (2)

The valuation $|-|_{x}$ is discrete.

  3. (3)

The analytic space $\mathcal{X}^{\textsf{an}}$ has rational rank one at $x$.

When $x$ is a divisorial point, we have $(\mathscr{H}(x),|-|_{x})$ is a
complete discrete valuation field, extending $|-|_{k}$.

We review the notations of discs and annuli throughout this paper:

######  2.9.

Discs and annuli. Let $X$ be a smooth algebraic $K$-analytic curve. For $a\in
X_{0}$, $r_{1}<r_{2}\in\mathbf{R}_{\geq 0}$ and $(f_{a})=\mathfrak{m}_{a}$ be
the maximal ideal of local ring $\mathscr{O}_{X,a}$, we define closed disc
$\mathbf{E}(a;r)_{X,K}:=\\{x\in X\,|\,|f_{a}|_{x}\leq
r_{1}^{[\kappa(a):K]}\\}$, open disc $\mathbf{D}(a,r)_{X,K}:=\\{x\in
X\,|\,|f_{a}|_{x}<r_{1}^{[\kappa(a):K]}\\}$ and closed annulus
$\mathbf{B}(a;r_{1},r_{2})_{X,K}:=\\{x\in
X\,|\,r_{1}^{[\kappa(a):K]}\leq|f_{a}|_{x}\leq r_{2}^{[\kappa(a):K]}\\}$, an
open disk after proper finite field extension, it can be represented as a
disjoint union of as standard form as
$\mathbf{D}(a,|r|)_{K^{\prime}}=\\{x\in\mathbf{A}_{K^{\prime}}^{1,\textsf{an}}\,|\,|T-a|_{x}<|r|\\}$.

######  2.10.

Open fiber disk. Throughout this paper, we refer to the open unit disks in 2.9
defined in the following two ways as open fiber disks : Analytic locally,
consider a smooth morphism of $K$-analytic spaces $f:X\longrightarrow Y$ of
relative dimension $1$. Then we can get a family of smooth
$\mathscr{H}(y)$-analytic curves $X_{y}$ for $y\in Y$. For the open sets
$\mathbf{D}_{z}(X_{y})$ where $z$ is a $\mathscr{H}(y)$-rational point, there
exists
$T_{z}:\mathbf{D}_{z}(X_{y})\xrightarrow{\raisebox{-4.30554pt}[0.0pt][0.0pt]{$\sim$}}\mathbf{D}(0,1)_{\mathscr{H}(y)}$
such that $T_{z}(z)=0$. On the other hand, consider an open unit disk
$\mathbf{D}(0,1)_{L}$ as a subset in a smooth $K$-analyitc space $X$ with
dimension $n$ such that $L|K$ with $\textsf{tr.deg}(L|K)=n-1$, then analytic
locally, there exist a smooth morphism of $K$-analytic spaces
$f^{\prime}:X\longrightarrow Y$ and $y\in Y$ such that
$\mathbf{D}(0,1)_{L}\subseteq X_{y}$ as an open subset(See 4.12 for details).

##  3\. The complement of skeleton for higher dimensional $K$-analytic space

###  3.1. Semi-stable reduction

######  Definition 3.1.

Let $\mathcal{X}$ be a scheme locally of finite presentation over $K^{\circ}$.
For an open subscheme $\mathcal{U}_{\mathcal{X}}$ of $\mathcal{X}$, a pair
$(\mathcal{X},\mathcal{U}_{\mathcal{X}})$ is called strictly semi-stable over
$K^{\circ}$ if there exists an (Zariski) étale covering
$\mathscr{V}=\\{\mathcal{V}\\}$ if we have (open) étale maps:

(3.2)
${(\mathcal{V},\mathcal{U}_{\mathcal{V}})}$${(\mathcal{X},\mathcal{U}_{\mathcal{X}})}$${\big{(}\mathcal{M}:=\textsf{Spec}\,K^{\circ}{[T_{0},\dots,T_{n}]}/(T_{0}\cdots
T_{r}-\varpi),\mathcal{U}\big{)}}$$\scriptstyle{f\,\textsf{open}}$$\scriptstyle{g\,\textsf{\'{e}t}}$

such that
$f^{-1}(\mathcal{U}_{\mathcal{X}})=\mathcal{U}_{\mathcal{V}}=g^{-1}(\mathcal{U})$
and

$\mathcal{U}\cong\textsf{Spec}\,K^{\circ}[T_{0},\dots,T_{n},T^{-1}_{0},\dots,T^{-1}_{m}]/(T_{0}\cdots
T_{r}-\varpi)$

for $0\leqslant r\leqslant m\leqslant n$ and $\varpi$ is a fixed uniformizer
of the complete discrete valuation ring $K^{\circ}$.

From the description above, we have the morphisms above are strict log étale
and

$\displaystyle\mathcal{U}$
$\displaystyle\cong\textsf{Spec}\,K^{\circ}[T_{0},\dots,T_{n},T^{-1}_{0},\dots,T^{-1}_{m},\varpi^{-1}]/(T_{0}\cdots
T_{r}-\varpi)$
$\displaystyle\cong\textsf{Spec}\,K[T_{0},\dots,T_{n},T^{-1}_{0},\dots,T^{-1}_{m}]/(T_{0}\cdots
T_{r}-\varpi)$

thus
$D_{\mathcal{X}}:=\mathcal{X}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\mathcal{U}_{\mathcal{X}}\subset\mathcal{X}_{K}$
and it is a normal crossing divisor by
$D:=\mathcal{M}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\mathcal{U}=V(T_{0}\cdots
T_{m})$. It’s easy to see when $m=r$, we have $D=\mathcal{X}_{\widetilde{K}}$.
Now we take the completion along special fibers for 3.1, étale locally, we
have an étale morphism of $K^{\circ}$-formal schemes
$\widehat{g}:(\widehat{\mathcal{X}},\widehat{\mathcal{U}_{\mathcal{X}}})\longrightarrow(\widehat{\mathcal{M}},\widehat{\mathcal{U}})$,
then take the Berthelot functor on $\widehat{g}$, we have:

$\widehat{g}^{\textsf{rig}}:(\widehat{\mathcal{X}}^{\textsf{rig}},\widehat{\mathcal{U}_{\mathcal{X}}}^{\textsf{rig}})\longrightarrow(\mathcal{M}\Big{(}K\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}T_{0},\dots,T_{n}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}/(T_{0}\cdots
T_{r}-\varpi)\Big{)},\widehat{\mathcal{U}}^{\textsf{rig}})$

Let $\widehat{\mathcal{M}}^{\textsf{rig}}:=\mathcal{G}(n,r,m)$, then we have
$\mathcal{G}(n,r,m)\cong\mathcal{G}(r,r,r)\times_{K}\widehat{\mathbf{A}}^{n-r,\textsf{rig}}_{K}$,
for $\mathcal{G}(r,r,r)$

(3.3)
$\varphi_{\mathcal{M}}:\mathcal{G}(r,r,r)\hookrightarrow\mathbf{G}^{r,\textsf{an}}_{{m},K}$

such that $\mathcal{G}(r,r,r)$ be an affinoid domain of
$\mathbf{G}^{r,\textsf{an}}_{{m},K}$ and for $\widehat{\mathbf{A}}^{n-r}_{K}$
has an open covering consists of formal tori. From the description above, we
can summarize the strictly semi-stable as Zariski locally, there is a smooth
morphism

$\mathcal{X}\longrightarrow\textsf{Spec}\,K^{\circ}[T_{0},\dots,T_{n}]/(T_{0}\cdots
T_{n}-\varpi)$

.

######  Remark 3.4.

When $K^{\circ}$ is discrete-valued, then any strictly semi-stable (formal)
scheme is a nondegenerate polystable (formal) scheme [Ber07, 3.1].

###  3.2. The Complement of Skeleton for Curves

In this subsection, we briefly review the structure of the skeleton of curves
studied in [Duc08] and [BPR13]. For a smooth and connected analytic $K$-curve
$X$, a triangulation of $X$ is a discrete and closed subset $\mathbf{V}$
consists of rigid or type-2 or type-3 points such that
$X\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\mathbf{V}$ gives connected
component decomposition as a disjoint union of virtual open disks and a finite
disjoint union of virtual open annuli. When the base field $K$ is
algebraically closed, we have the connected component decomposition as:

$X\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\mathbf{V}\cong\coprod\mathbf{D}(0,1)_{K}\smallcoprod\coprod^{n}_{i=1}\mathbf{B}(0,a_{i},1)_{K}$

The skeleton of $X$ associated to $\mathbf{V}$ is the set:

$\textsf{Sk}(X,\mathbf{V}):=\mathbf{V}\smallcoprod\coprod^{n}_{i=1}\textsf{Sk}(\mathbf{B}(0,a_{i},1)_{K})$

######  3.5.

Let $\mathfrak{C}$ be a semi-stable formal curve over $K^{\circ}$, consider
the reduction map
$\textsf{red}_{\mathfrak{C}}:\mathfrak{C}^{\textsf{rig}}\longrightarrow|\mathfrak{C}_{\widetilde{K}}|=|\mathfrak{C}|$,
we denote $\mathfrak{C}^{0}_{\widetilde{K}}$ be the set of generic points of
the irreducible components of the special fiber
$\mathfrak{C}_{\widetilde{K}}$, then the preimage
$\textsf{red}^{-1}(\mathfrak{C}^{0}_{\widetilde{K}})$ gives a triangulation
$\mathbf{V}_{\mathfrak{C}}$ of $\mathfrak{C}^{\textsf{rig}}$. In this case, we
have:

######  Proposition 3.6.

[BPR13, Proposition 4.10.] The Berkovich skeleton
$\textsf{Sk}({\mathfrak{C}}^{\textsf{rig}})$ is naturally identified with the
skeleton
$\textsf{Sk}({\mathfrak{C}}^{\textsf{rig}},\mathbf{V}_{{\mathfrak{C}}})$.

######  Lemma 3.7.

[BPR13, Lemma 3.4.] The connected components of
$\mathfrak{C}^{\textsf{rig}}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\textsf{Sk}({\mathfrak{C}}^{\textsf{rig}}):=\textsf{Sk}({\mathfrak{C}}^{\textsf{rig}})^{\textsf{c}}$
are open unit disks.

######  Remark 3.8.

Essentially, the structure of the complement of the skeleton of a curve is
induced by the complement of the skeleton of an open annulus [BPR13, Lemma
2.12.]

For a non-algebraically closed field $K$, consider a $K$-analytic curve $C$
admitting a semi-stable formal model $\mathfrak{C}$, then for the formal curve
$\mathfrak{C}_{\widehat{K^{\textsf{alg}}}^{\circ}}$, and the morphism
$\pi:\mathfrak{C}_{\widehat{K^{\textsf{alg}}}^{\circ}}\longrightarrow\mathfrak{C}$,
we have the following lemma:

######  Lemma 3.9.

For generic fiber
$\pi^{\textsf{rig}}:C_{\widehat{K^{\textsf{alg}}}}\longrightarrow C$, then
$\pi^{\textsf{rig},-1}(\textsf{Sk}(\mathfrak{C}^{\textsf{rig}}))=\textsf{Sk}(\mathfrak{C}^{\textsf{rig}}_{\widehat{K^{\textsf{alg}}}^{\circ}})$.

######  Remark 3.10.

Note that $\pi^{\textsf{rig}}$ is surjective, we have
$\pi^{\textsf{rig}}(\textsf{Sk}(\mathfrak{C}^{\textsf{rig}}_{\widehat{K^{\textsf{alg}}}^{\circ}})^{\textsf{c}})=\textsf{Sk}(\mathfrak{C}^{\textsf{rig}})^{\textsf{c}}$,
thus
$\textsf{Sk}(\mathfrak{C}^{\textsf{rig}})^{\textsf{c}}=\bigcup\pi^{\textsf{rig}}(\mathbf{D}(0,1)_{\widehat{K^{\textsf{alg}}}})=\bigcup\mathbf{D}(0,1)_{K}$,
but this not a connected component decomposition.

###  3.3. The complement of skeleton for higher dimensional $K$-analytic space

In this subsection, we will show that for a $K$-analytic space of dimension
greater than 1 which admits a strictly semi-stable formal model, for every
point of the complement of its Berkovich skeleton we may find a relative
dimension 1 fibration such that this point is not contained in the skeleton of
its fiber. The key observation is the following result:

######  Lemma 3.11.

With the notation in 3.1, for the projections
$\textsf{fp}_{i}:\mathbf{G}_{m,K}^{n,\textsf{an}}\longrightarrow\mathbf{G}_{m,K}^{n-1,\textsf{an}}$,
where $1\leq i\leq n$, by forgetting the $i$-th component, consider the
following compositions: $\varphi_{i}:=\textsf{fp}_{i}\circ\varphi_{M}$ and
$g_{i}:=\varphi_{i}\circ\widehat{g}^{\textsf{rig}}$, then the induced
morphisms on the fibers have :

  1. (1)

$\varphi_{M,x,i}:\varphi_{i}^{-1}(x)\longrightarrow\mathbf{G}_{m,\mathscr{H}(x)}^{1,\textsf{an}}$
is an analytic locally monomial chart such that
$\varphi^{-1}_{M,x,i}(\textsf{Sk}(\mathbf{G}_{m,\mathscr{H}(x)}^{1,\textsf{an}}))$
is a skeleton of $\varphi_{i}^{-1}(x)$ for any
$x\in\mathbf{G}_{m,K}^{n-1,\textsf{an}}$.

  2. (2)

Analytic locally,
$\widehat{g}^{\textsf{rig}}:g^{-1}_{i}(x)\longrightarrow\varphi^{-1}_{i}(x)$
is a generic fiber of étale morphism of strictly semi-stable formal schemes.

###### Proof.

  1. (1)

Without loss of generality, let’s assume $i=n$, note that

$M=\\{x\in\mathbf{G}^{n,\textsf{an}}_{m,K}\,|\,|T_{1}\cdots
T_{n}|_{x}\geq|\varpi|;|T_{i}|_{x}\leq 1\,\,\text{for}\,\,1\leq i\leq n\\}$

we have

$\displaystyle\textsf{fp}_{n}(M)$
$\displaystyle=\bigcup_{t}\\{x\in\mathbf{G}^{n-1,\textsf{an}}_{m,K}\,|\,|T_{1}\cdots
T_{n-1}|_{x}\geq\frac{|\varpi|}{t};|T_{i}|_{x}\leq 1\,\,\text{for}\,\,1\leq
i\leq n-1\\}$

where $t=|T_{n}|_{x}$ for $x\in M$. Note that $\frac{|\varpi|}{t}$ does not
necessarily belong to the divisible value group $\sqrt{|K^{\times}|}$. Still,
we can always take the union of strictly affinoid subdomains in the form of
$\mathcal{U}:=\bigcup_{\alpha}\mathcal{U}_{\alpha}=\bigcup_{\alpha}\\{x\in\mathbf{G}^{n-1,\textsf{an}}_{m,K}\,|\,|T_{1}\cdots
T_{n-1}|_{x}\geq|\varpi_{\alpha}|;|T_{i}|_{x}\leq 1\,\,\text{for}\,\,1\leq
i\leq n-1\\}$ to cover $\textsf{fp}_{n}(M)$ such that $\varpi_{\alpha}\in
K^{\times}$ and $|\varpi_{\alpha}|\leq 1$. Meanwhile we have
$M=\bigcup_{\alpha}\textsf{fp}^{-1}_{n}(\mathcal{U}_{\alpha})\cap
M=\bigcup_{\alpha}\\{x\in\mathbf{G}^{n,\textsf{an}}_{m,K}\,|\,|T_{1}\cdots
T_{n-1}|_{x}\geq|\varpi_{\alpha}|;|T_{i}|_{x}\leq 1\,\,\text{for}\,\,1\leq
i\leq n;|T_{n}|_{x}\geq|\varpi\varpi_{\alpha}^{-1}|\\}$. Thus

$\displaystyle\varphi_{n}^{-1}(x)$
$\displaystyle\cong\mathcal{M}\Big{(}\frac{K\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}X_{0},\dots,X_{n}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}}{(X_{0}\cdots
X_{n}-\varpi)}\Big{)}\times_{\mathbf{G}_{m,K}^{n-1,\textsf{an}}}\mathcal{M}(\mathscr{H}(x))$
$\displaystyle\cong\mathcal{M}\Big{(}\frac{K\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}X_{0},\dots,X_{n}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}}{(X_{0}\cdots
X_{n}-\varpi)}\Big{)}\times_{\mathcal{U}}\mathcal{M}(\mathscr{H}(x))$
$\displaystyle\cong\bigcup_{\alpha}(\textsf{fp}^{-1}_{n}(\mathcal{U}_{\alpha})\cap
M)\times_{\mathcal{U}_{\alpha}}\mathcal{M}\big{(}\mathscr{H}(x)\big{)}$
$\displaystyle\cong\bigcup_{\alpha}\mathcal{M}\Big{(}\frac{K\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}X_{0},\dots,X_{n-1},X_{n},\varpi\varpi^{-1}_{\alpha}X^{-1}_{n}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}}{(X_{0}\cdots
X_{n-1}-\varpi_{\alpha})}\Big{)}\times_{\mathcal{U}_{\alpha}}\mathcal{M}\big{(}\mathscr{H}(x)\big{)}$
$\displaystyle\cong\bigcup_{\alpha}\mathcal{M}\big{(}\mathscr{H}(x)\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}T,\varpi\varpi^{-1}_{\alpha}T^{-1}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}\big{)}$
$\displaystyle\cong\bigcup_{\alpha}\mathbf{B}(0;|\varpi\varpi^{-1}_{\alpha}|,1)_{\mathscr{H}(x)}$

the morphism $\varphi_{M,x,n}$ is flat. Moreover, by [BPR13, 2.3]

$\displaystyle\varphi^{-1}_{M,x,n}(\textsf{Sk}(\mathbf{G}_{m,\mathscr{H}(x)}^{1,\textsf{an}}))$
$\displaystyle=\bigcup_{\alpha}\textsf{Sk}\Big{(}\textsf{Spf}^{\textsf{rig}}\,\frac{\mathscr{H}(X)^{\circ}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}X_{0},X_{1}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}}{(X_{0}X_{1}-\varpi\varpi_{\alpha}^{-1})}\Big{)}$
$\displaystyle=\bigcup_{\alpha}\textsf{Sk}\big{(}\mathbf{B}(0;|\varpi\varpi^{-1}_{\alpha}|,1)_{\mathscr{H}(x)}\big{)}$
$\displaystyle=\textsf{Sk}\big{(}\varphi^{-1}_{n}(x)\big{)}$

  2. (2)

Since
$g_{n}^{-1}(x)\cong\mathfrak{Y}^{\textsf{rig}}\times_{\mathbf{G}^{n-1,\textsf{an}}_{m,K}}\mathcal{M}\big{(}\mathscr{H}(x)\big{)}\cong\bigcup_{\alpha}\big{(}\mathfrak{Y}^{\textsf{rig}}\times_{M}({\textsf{fp}^{-1}_{n}(\mathcal{U}_{\alpha}})\cap
M)\big{)}\times_{\mathcal{U}_{\alpha}}\mathcal{M}\big{(}\mathscr{H}(x)\big{)}$,
now we take:

(3.12)
${\mathfrak{g}_{n,\alpha,x}:\mathfrak{G}_{n,\alpha,x}:=\big{(}\mathfrak{Y}\times_{\mathcal{Z}_{M}}\textsf{Spf}\frac{K^{\circ}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}X_{0},\dots,X_{n-1},X_{n},\varpi\varpi^{-1}_{\alpha}X^{-1}_{n}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}}{(X_{0}\cdots
X_{n-1}-\varpi_{\alpha})}\big{)}\times_{\textsf{Spf}\,\frac{\mathscr{H}(X)^{\circ}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}X_{0},\dots,X_{n-1}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}}{(X_{0}\cdots
X_{n-1}-\varpi_{\alpha})}}\textsf{Spf}\,\mathscr{H}(x)^{\circ}}$${\textsf{Spf}\,\frac{\mathscr{H}(X)^{\circ}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}X_{0},X_{1}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}}{(X_{0}X_{1}-\varpi\varpi_{\alpha}^{-1})}}$ét

Then $\mathfrak{g}_{n,\alpha,x}$ is étale and
$(\widehat{g})_{x}^{\textsf{rig}}|_{g^{-1}_{n}(\mathcal{U}_{\alpha})}=\mathfrak{g}_{n,\alpha,x}^{\textsf{rig}}$,
thus $\mathfrak{G}_{n,\alpha,x}$ is strictly semi-stable. In this case, we
have
$\textsf{Sk}(\mathfrak{G}^{\textsf{rig}}_{n,\alpha,x})=\mathfrak{g}_{n,\alpha,x}^{\textsf{rig},-1}(\textsf{Sk}\big{(}\mathbf{B}(0;|\varpi\varpi^{-1}_{\alpha}|,1)_{\mathscr{H}(x)}\big{)})$
and
$\textsf{Sk}(g^{-1}_{n}(x))=\bigcup_{\alpha}\textsf{Sk}(\mathfrak{G}_{n,\alpha,x}^{\textsf{rig}})$
by [Ber99, Theorem 5.2.]

∎

###  3.4. The Complement of Skeleton

######  3.13.

We start with the torus model
$\mathbf{G}^{n,\textsf{an}}_{{m},K}=\textsf{Spec}^{\textsf{an}}\,K[t_{1}^{\pm},t_{2}^{\pm},\dots,t_{n}^{\pm}]$,
we have
$\textsf{Sk}(\mathbf{G}^{n,\textsf{an}}_{{m},K})=\bigtimes^{n}_{i=1}\textsf{Sk}(\mathbf{G}^{1,\textsf{an}}_{{m},K})$.
Then we have
$\textsf{Sk}(\mathbf{G}^{n,\textsf{an}}_{{m},K})^{\textsf{c}}=\mathbf{G}^{n,\textsf{an}}_{{m},K}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\Big{(}\bigtimes^{n}_{i=1}\textsf{Sk}(\mathbf{G}^{1,\textsf{an}}_{{m},K})\Big{)}$.
Now we consider the projection maps
$\textsf{fp}_{i}:\mathbf{G}^{n,\textsf{an}}_{{m},K}\longrightarrow\mathbf{G}^{n-1,\textsf{an}}_{{m},K}$
for $1\leq i\leq n$, and we have
$\textsf{fp}_{i}:\textsf{Sk}(\mathbf{G}^{n,\textsf{an}}_{{m},K})\longrightarrow\textsf{Sk}(\mathbf{G}^{n-1,\textsf{an}}_{{m},K})$.
Now we show the following lemma:

######  Lemma 3.14.

The complement of skeleton
$\textsf{Sk}(\mathbf{G}^{n,\textsf{an}}_{{m},K})^{\textsf{c}}$ is a union of
open fiber disks.

###### Proof.

Let’s begin by writing
$\mathbf{G}_{m,K}^{n,\textsf{an}}\cong\mathbf{G}_{m,K}^{n-1,\textsf{an}}\times_{K}\mathbf{G}_{m,K}^{1,\textsf{an}}$.
We denote $\textsf{pr}_{1}:=\textsf{fp}_{n}$ and $\textsf{pr}_{2}$ as the
projections from $\mathbf{G}_{m,K}^{n,\textsf{an}}$ to
$\mathbf{G}_{m,K}^{n-1,\textsf{an}}$ and $\mathbf{G}_{m,K}^{1,\textsf{an}}$
respectively. When $n=1$, this is a result of [BPR13, Lemma 2.12.]. We proceed
by induction on $n$. Assuming the complement of skeleton
$\textsf{Sk}(\mathbf{G}^{n-1,\textsf{an}}_{{m},K})^{\textsf{c}}$ is a union of
open fiber disks, let
$x\in\textsf{Sk}(\mathbf{G}^{n,\textsf{an}}_{{m},K})^{\textsf{c}}$, then we
will show our claim by testing whether $\\{\textsf{pr}_{i}(x)\\}_{i}$ belongs
to the skeletons of the images of $\\{\textsf{pr}_{i}\\}_{i}$ or not. Without
loss of generality, we only discuss the following case: If
$\textsf{pr}_{1}(x)\notin\textsf{Sk}(\mathbf{G}^{n-1,\textsf{an}}_{{m},K})$
and $\textsf{pr}_{2}(x)\in\textsf{Sk}(\mathbf{G}^{1,\textsf{an}}_{{m},K})$,
then the fiber
$\textsf{pr}^{-1}_{2}(\textsf{pr}_{2}(x))\cong\mathbf{G}^{n-1,\textsf{an}}_{{m},\mathscr{H}(\textsf{pr}_{2}(x))}$
we claim that $x\notin\textsf{Sk}(\textsf{pr}^{-1}_{2}(\textsf{pr}_{2}(x)))$.
This can be seen by noting that
$\textsf{Sk}(\textsf{pr}^{-1}_{2}(\textsf{pr}_{2}(x)))=\textsf{Sk}(\mathbf{G}^{n-1,\textsf{an}}_{{m},K})\times\mathcal{M}(\mathscr{H}(\textsf{pr}_{2}(x)))$.
Then there is an open unit disk $\mathbf{D}(0,1)_{L}$ such that
$x\in\mathbf{D}(0,1)_{L}\subseteq\textsf{Sk}(\mathbf{G}^{n-1,\textsf{an}}_{{m},\mathscr{H}(\textsf{pr}_{2}(x))})^{\textsf{c}}$
with field extension $L|\mathscr{H}(\textsf{pr}_{2}(x))$. Meanwhile, assume
there exists
$x_{0}\in\mathbf{D}(0,1)_{L}\cap\textsf{Sk}(\mathbf{G}^{n,\textsf{an}}_{{m},K})$,
then
$x_{0}\in\textsf{Sk}(\mathbf{G}^{n-1,\textsf{an}}_{{m},\mathscr{H}(\textsf{pr}_{2}(x))})$,
which leads to a contradiction. Thus we have
$\textsf{Sk}(\mathbf{G}^{n,\textsf{an}}_{{m},K})^{\textsf{c}}=\bigcup\mathbf{D}(0,1)_{L}$
with complete non-Archimedean field extension $L|K$. In fact, from the
induction procedure, we can see that each disk $\mathbf{D}(0,1)_{L}$ comes
from a certain fiber $\textsf{fp}^{-1}_{i}(\textsf{fp}_{i}(x))$.

∎

######  Lemma 3.15.

For a strictly semi-stable scheme $\mathcal{X}$ such that $r=n$ in 3.1, the
complement of Berkovich skeleton
$\textsf{Sk}({\widehat{\mathcal{X}}}^{\textsf{rig}})^{\textsf{c}}$ is a union
of open fiber disks.

###### Proof.

It suffices to study
$\textsf{Sk}({\widehat{\mathcal{V}}}^{\textsf{rig}})^{\textsf{c}}$, then for
any $x\in\textsf{Sk}({\widehat{\mathcal{V}}}^{\textsf{rig}})^{\textsf{c}}$,
consider
$x^{\prime}:=\varphi_{M}\circ(\widehat{g})^{\textsf{rig}}(x)\in\textsf{Sk}(\mathbf{G}^{n,\textsf{an}}_{m,K})^{\textsf{c}}$,
by Lemma 3.14, there exists $i$ for $1\leq i\leq n$ such that
$x^{\prime}\in\textsf{Sk}(\textsf{fp}_{i}^{-1}(\textsf{fp}_{i}(x^{\prime}))^{\textsf{c}}$,
then by Lemma 3.11,
$x\in\textsf{Sk}\big{(}g^{-1}_{i}(\textsf{fp}_{i}(x^{\prime}))\big{)}^{\textsf{c}}$.
∎

######  Remark 3.16.

In fact, Lemma 3.15 still holds for
$\mathcal{M}:=\textsf{Spec}\,K^{\circ}{[T_{0},\dots,T_{n}]}/(T_{0}\cdots
T_{n}-a)$ with $|a|=1$. In this case,
$\widehat{\mathcal{M}}\cong\textsf{Spf}(K^{\circ}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}T^{\pm
1}_{1},\dots T_{n}^{\pm
1}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}})$
and
$\widehat{\mathcal{M}}^{\textsf{rig}}\cong\big{(}\mathbf{B}(0;1,1)_{K}\big{)}^{n}$
which is a product of generalized annulus and contractible. Consider a
fibration similar to we did in Lemma 3.11, then we get a family of étale
morphisms
$\\{\widehat{g}_{L}^{\textsf{rig}}:\widehat{\mathcal{V}}_{L}^{\textsf{rig}}\longrightarrow\mathbf{B}(0;1,1)_{L}\\}_{L}$
for field extension $L|K$, and we have
$\widehat{g}_{L}^{\textsf{rig},-1}(\textsf{Sk}(\mathbf{B}(0;1,1))_{L})=\textsf{Sk}(\widehat{\mathcal{V}}_{L}^{\textsf{rig}})$
by [Thu05, Lemme 2.2.14.]. The rest part are the same, and we are done.

######  Theorem 3.17.

Let ${{\mathcal{X}}}$ be a strictly semi-stable scheme over complete discrete
valuation ring $K^{\circ}$ defined in 3.1, the complement of Berkovich
skeleton $\textsf{Sk}({\widehat{\mathcal{X}}}^{\textsf{rig}})^{\textsf{c}}$ is
a union of open fiber disks.

###### Proof.

Zariski locally, note that $g$ could be factorized as

${(\mathcal{V},U_{\mathcal{V}})}$${(\mathbf{A}_{\mathcal{M}}^{s},U_{\mathbf{A}_{\mathcal{M}}^{n}})}$${(\mathcal{M},U_{\mathcal{M}})}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{\textsf{pr}_{\mathcal{M}}}$

where $h$ is étale and $s$ is the relative dimension of $g$. Now let’s study
the structure of
$\textsf{Sk}({\widehat{\mathcal{V}}}^{\textsf{rig}})^{\textsf{c}}$. We first
note that the $K^{\circ}$-scheme
${\mathbf{A}}^{s}\times_{K^{\circ}}{\mathcal{M}}$ is a strictly nondegenerate
polystable scheme. More precisely, we have a locally finite covering of open
formal subscheme $\\{\widehat{\mathcal{W}}\\}$ for
$\widehat{\mathbf{A}}^{s}\times_{K^{\circ}}\widehat{\mathcal{M}}$ satisfying
the following diagram:

(3.18)
${{\mathfrak{W}^{\prime}:=\widehat{\mathcal{V}}\times_{\widehat{\mathbf{A}}^{s}\times_{K^{\circ}}\widehat{\mathcal{M}}}\widehat{\mathcal{W}}}}$${{\widehat{\mathcal{V}}}}$${{\widehat{\mathcal{W}}}}$${{\widehat{\mathbf{A}}^{s}\times_{K^{\circ}}\widehat{\mathcal{M}}}}$${{\textsf{Spf}(A_{0})\times\cdots\times\textsf{Spf}(A_{p})}}$openététopenét

where each $\textsf{Spf}(A_{i})$ is of form
$\textsf{Spf}\big{(}K^{\circ}\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}T_{0},\dots,T_{n_{i}}\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}}/(T_{0}\cdots
T_{n_{i}}-a)\big{)}$ for $0\neq a\in K^{\circ}$. Then through the construction
of [GRW16, 4.6.], it suffices to study the structure of
$\textsf{Sk}({\mathfrak{W}^{\prime}}^{\textsf{rig}})^{\textsf{c}}$, but this
is a immediate result follows from Lemma 3.15 and Remark 3.16.

∎

######  3.19.

It’s natural to study the structure of the complement of the Berkovich
skeleton for a log regular formal scheme as a generalization. Let
$\mathcal{X}$ be a proper log regular scheme over $K$ and $\mathfrak{X}$ be a
log regular formal model of $\mathcal{X}$ over $K^{\circ}$. In [BM19], the
authors show one can embed the Kato fans $\textsf{F}(\mathfrak{X})$ defined in
[Kat94] to $\mathcal{X}^{\textsf{an}}$ as a generalization of Berkovich
skeleton of a strictly semi-stable model defined in [Ber99]. The elementary
example is a generalized semistable family as follows:

Consider
$M:=\mathsf{Spec}\,\,K^{\circ}[T_{0},\dots,T_{n}]/(T^{e_{0}}_{0}\cdots
T^{e_{n}}_{n}-\varpi)$ such that $(e_{i},p)=1$ for $0\leq i\leq n$, where
$p=\textsf{char}(\widetilde{K})$. Then for the following morphism:

$\displaystyle
f:\widetilde{M}:=\mathsf{Spec}\,\,\frac{K^{\circ}[X_{0},\dots,X_{n}]}{(X^{e}_{0}\cdots
X^{e}_{n}-\varpi)}\longrightarrow\mathsf{Spec}\,\,\frac{K^{\circ}[T_{0},\dots,T_{n}]}{(T^{e_{0}}_{0}\cdots
T^{e_{n}}_{n}-\varpi)}$

which is induced by $T_{i}\longmapsto X^{\frac{e}{e_{i}}}_{i}$ we have:

######  Proposition 3.20.

[Dat12, cf.Theorem 2.2.] $f$ is an fppf covering space such that its generic
fiber $f_{K}$ is a Galois étale covering space. Moreover, the preimage of
$\textsf{Sk}(\widehat{M}^{\textsf{rig}})$ of $\widehat{f}^{\textsf{rig}}$ is
$\textsf{Sk}((\widehat{\widetilde{M}})^{{\textsf{rig}}})$.

###### Proof.

  1. (1)

The surjectivity of $f$ is easy to verify on the closed points of restriction
of generic fiber and special fiber.

  2. (2)

The étaleness of $f_{K}$ is standard and
$\textsf{Aut}(\widetilde{M}_{K}|M_{K})\cong\prod_{i}\bm{\mu}_{\frac{e}{e_{i}}}$.

  3. (3)

By the construction of the Berkovich skeleton, we have for any
$x\in\textsf{Sk}(\widehat{M}^{\textsf{rig}})$, we have
$|\sum_{I}a_{I}\underline{T}_{I}^{m_{I}}|_{x}=\textsf{max}_{I}|a_{I}|\textsf{exp}(-\sum^{n}_{i=0}m_{i}\cdot
v_{i})$ for a fixed $v_{I}=(v_{i})_{0\leq i\leq n}\in\mathbf{R}_{>0}^{n+1}$
satisfies $\sum_{i}e_{i}\cdot v_{i}=1$ and for
$x^{\prime}\in\textsf{Sk}((\widehat{\widetilde{M}})^{{\textsf{rig}}})$, we
have
$|\sum_{I}a^{\prime}_{I}\underline{X}_{I}^{r_{I}}|_{x^{\prime}}=\textsf{max}_{I}|a^{\prime}_{I}|\textsf{exp}(-\sum^{n}_{i=0}r_{i}\cdot
v^{\prime}_{i})$ for a fixed $v^{\prime}_{I}=(v^{\prime}_{i})_{0\leq i\leq
n}\in\mathbf{R}_{>0}^{n+1}$ satisfies $\sum_{i}e\cdot v^{\prime}_{i}=1$. Then
the result is obvious by direct calculation.

∎

######  3.21.

Let $G:=\prod_{i}\bm{\mu}_{\frac{e}{e_{i}}}$ as an étale abelian sheaf, then
we study the $G$-action on
$(\widehat{\widetilde{\mathcal{M}}})^{\textsf{rig}}$ The complement of the
skeleton
$\textsf{Sk}((\widehat{\widetilde{M}})^{{\textsf{rig}}})^{\textsf{c}}$ is
$G$-stable. To see this, firstly by 3.20, we have the generic fiber $f_{K}$ of
$f$ is a Galois-étale covering space and
$\mathcal{M}_{K}\cong\widetilde{\mathcal{M}}_{K}/G$. After taking the
analytification and Berthelot generic fiber for $f$ and $f_{K}$, we have the
following diagram:

(3.22)
${(\widehat{\widetilde{\mathcal{M}}})^{\textsf{rig}}}$${\widehat{\mathcal{M}}^{\textsf{rig}}}$${\widetilde{\mathcal{M}}^{\textsf{an}}_{K}}$${\mathcal{M}^{\textsf{an}}_{K}}$$\scriptstyle{\widehat{f}^{\textsf{rig}}}$$\scriptstyle{f^{\textsf{an}}_{K}}$

Now, by [Ayo15, Corollaire 1.1.50.] and [Ber96, (0.3.5) Proposition], we have
$\widehat{f}^{\textsf{rig}}$ is is finite, étale and surjective. Thus we have
$\widehat{\mathcal{M}}^{\textsf{rig}}\cong\widehat{\widetilde{\mathcal{M}}}^{\textsf{rig}}/G$
as the orbit space by [Zav24, Theorem 1.3.6.] Furthermore, consider the
restriction of $\widehat{f}^{\textsf{rig}}$ on
$\textsf{Sk}(\widehat{\widetilde{\mathcal{M}}}^{\textsf{rig}})^{\textsf{c}}$,
we have
$\textsf{Sk}(\widehat{\widetilde{\mathcal{M}}}^{\textsf{rig}})^{\textsf{c}}/G\cong\textsf{Sk}(\widehat{\mathcal{M}}^{\textsf{rig}})^{\textsf{c}}$,
thus, the complement of skeleton
$\textsf{Sk}(\widehat{\mathcal{M}}^{\textsf{rig}})^{\textsf{c}}$ is the
$G$-orbit space of the complement of skeleton
$\textsf{Sk}(\widehat{\widetilde{\mathcal{M}}}^{\textsf{rig}})^{\textsf{c}}$.
Unfortunately, we still don’t know how the disks decomposition behaves under
the group action $G$, even for the curve case.

##  4\. Characterization of essential skeleton

In this section, we assume any $K$-analytic spaces to be smooth of pure
dimension $n$. Then we give a criterion for points belonging to the essential
skeleton of a proper smooth variety in terms of open fiber disks when the
ground field $K$ is a fixed non-Archimedean field with equal characteristic
$0$.

###  4.1. Geometric Kähler Skeleton

In this subsection, we first review the basics of Kähler seminorm defined in
[Tem16] and prepare some necessary lemmas we need later.

######  4.1.

Kähler Seminorm.

Consider $X\longrightarrow S$ be a quasi-smooth of $K$-analytic spaces with
relative dimension $n$, and the relative canonical sheaf
$\mathscr{K}_{{X}/{S}}:=\bigwedge^{n}\Omega_{{X}/{S}}$ and we have
corresponding seminorm $\lvert\lvert-\lvert\lvert_{\Omega}$ and
$\lvert\lvert-\lvert\lvert_{\mathscr{K}}$. Now for a given non-zero global
pluricanonical form $\omega\in\Gamma(X,\mathscr{K}_{X/S})$, let
$\textsf{M}_{\omega}$ and $\overline{\textsf{M}}_{\omega}$ be the maximality
of locus and geometric maximality locus respectively.

######  Lemma 4.2.

Let $f:X\longrightarrow Y$ be an étale morphism of quasi-smooth $K$-analytic
spaces, then $f^{-1}(\textsf{M}_{\omega})\subseteq\textsf{M}_{f^{*}\omega}$
for any $\omega\in\Gamma(Y,\mathscr{K}_{Y})$.

###### Proof.

Note that $f$ is étale, we have $f^{*}(\mathscr{K}_{Y})\cong\mathscr{K}_{X}$,
thus $\widehat{\mathscr{K}}_{Y,\mathscr{H}(f(x))|K}$ is isometric to
$\widehat{\mathscr{K}}_{X,\mathscr{H}(x)|K}$ for any $x\in X$, then we have
$\lvert\lvert\omega\lvert\lvert_{f(x_{0})}=\lvert\lvert\omega\lvert\lvert_{\widehat{\mathscr{K}},\mathscr{H}(f(x_{0}))|K}=\lvert\lvert
f^{*}\omega\lvert\lvert_{\widehat{\mathscr{K}},\mathscr{H}(x_{0})|K}\geq\lvert\lvert
f^{*}\omega\lvert\lvert_{\widehat{\mathscr{K}},\mathscr{H}(x)|K}$ for
$f(x_{0})\in\textsf{M}_{\omega}$. ∎

######  4.3.

Geometric Kähler Seminorm.

Consider a quasi-smooth $K$-analytic space $X$ with dimension $n$, then for
the canonical morphism
$g:\overline{X}:=X_{\widehat{K^{\textsf{alg}}}}\longrightarrow X$, the
geometric Kähler seminorm $\lvert\lvert-\lvert\lvert_{\overline{\Omega},X}$ is
defined as the restriction of
$g_{*}\lvert\lvert-\lvert\lvert_{\Omega,\overline{X}}$ on $\Omega_{X}$.

######  Example 4.4.

Let $X$ be the unit open disk
$\mathcal{M}(K\mathopen{\hbox{\set@color${\langle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\langle}$}}T\mathclose{\hbox{\set@color${\rangle}$}\kern-1.94444pt\leavevmode\hbox{\set@color${\rangle}$}})$
with equal characteristic $0$, then for $\lvert\lvert
dT\lvert\lvert_{\overline{\Omega},x}=\lvert\lvert
dT\lvert\lvert_{\Omega,x^{\prime}}$ such that $g(x^{\prime})=x$ for $x\in X$
and $x^{\prime}\in\overline{X}$. Note that the maximal locus of $g_{*}dT$ is
the maximal point of $\overline{X}$, thus the maximal locus of $dT$ is the
maximal point of $X$. Furthermore, for any non-zero global $1$-form
$\omega=f(T)dT$, the maximal locus of $\omega$ remains the maximal point.

In general, these two seminorms may be different for some arithmetic non-
Archimedean field $K$.

######  Corollary 4.5.

Let $f:X\longrightarrow Y$ be an étale morphism of quasi-smooth $K$-analytic
spaces, then
$f^{-1}(\overline{\textsf{M}}_{\omega})\subseteq\overline{\textsf{M}}_{f^{*}\omega}$
for any $\omega\in\Gamma(Y,\mathscr{K}_{Y})$ .

###### Proof.

The proof is the same as Lemma 4.2 since $\overline{f}$ is étale. ∎

######  Definition 4.6.

Let $f:X\longrightarrow S$ be a quasi-smooth morphism of $K$-analytic spaces.
The Kähler skeleton of $f$ is defined as
$\textsf{Sk}^{\text{k\"{a}h}}(X/S):=\bigcup_{\omega\neq 0}\textsf{M}_{\omega}$
and geometric Kähler skeleton of $f$ is defined as
$\overline{\textsf{Sk}}^{\textsf{k\"{a}h}}(X/S):=\bigcup_{\omega\neq
0}\overline{\textsf{M}}_{\omega}$. For $S=\mathcal{M}(K)$, we denote
$\textsf{Sk}^{\textsf{k\"{a}h}}(X)$ and
$\overline{\textsf{Sk}}^{\textsf{k\"{a}h}}(X)$ as the Kähler skeleton and
geometric Kähler skeleton respectively.

######  Example 4.7.

$\overline{\textsf{Sk}}^{\textsf{k\"{a}h}}(\mathbf{D}(0,1)_{K})=\varnothing$.
To see this, we need to note that we can get two closed disks
$\mathbf{E}(0,r)_{K}\subsetneq\mathbf{E}(0,r^{\prime})_{K}\subsetneq\mathbf{D}(0,1)_{K}$
for $0<r<r^{\prime}<1$, assume
$x_{0}\in\overline{\textsf{Sk}}^{\textsf{k\"{a}h}}(\mathbf{D}(0,1)_{K})\cap\mathbf{E}(0,r)_{K}$,
then by Corollary 4.5, $x_{0}$ is a maximal point of both
$\mathbf{E}(0,r)_{K}$ and $\mathbf{E}(0,r^{\prime})_{K}$ which is impossible.

######  Lemma 4.8.

Let $X$ be an irreducible quasi-smooth analytic curve over $K$ with equal
characteristic $0$, then $\overline{\textsf{Sk}}^{\textsf{k\"{a}h}}(X)$
consists of points cannot admit an open neighborhood which is isomorphic to
$\mathbf{D}(0,1)_{K}$.

###### Proof.

Let $x_{0}\in\overline{\textsf{M}}_{\omega_{0}}$ for $\omega_{0}\neq 0$, then
by Corollary 4.5 $x_{0}\in\overline{\textsf{M}}_{\tau^{*}\omega_{0}}$ where
$\tau:\mathbf{D}(0,1)_{K}\hookrightarrow X$, note that in this case
$\tau^{*}\omega_{0}\neq 0$ and
$\overline{\textsf{Sk}}^{\textsf{k\"{a}h}}(\mathbf{D}(0,1)_{K})=\varnothing$
by Example 4.7. ∎

###  4.2. Relation with Essential Skeleton

In this subsection, we assume $K$ is a complete discrete valued field with
equal characteristic $0$.

The relation between the Kähler norm and the weight norm of Mustaţă–Nicaise is
established by M. Temkin by the following theorem in [Tem16].

######  Theorem 4.9.

[Tem16, Theorem 8.3.3.] If $K$ is discretely valued, $X$ is quasi-smooth and
$x\in X$ is a divisorial point, i.e. a monomial point with discretely valued
$\mathscr{H}(x)$, then the Kähler and the weight norms on $m$-canonical forms
are related by:

$\|-\|_{\textsf{wt}^{\otimes
m},x}=|\varpi_{K}|^{m}(\delta^{\textsf{log}}_{\mathscr{H}(x)|K})^{m}\|-\|_{\mathscr{K}_{X|K}^{\otimes
m},x}$

######  Remark 4.10.

When is a fixed non-Archimedean field with equal characteristic $0$,
$\delta^{\textsf{log}}_{\mathscr{H}(x)|K}=1$, in this case, weight norm of
Mustaţă–Nicaise coincides with Kähler norm.

######  Theorem 4.11 (M.Temkin).

Let $\mathcal{X}$ be a smooth proper integral variety over a complete discrete
valued field $K$ with equal characteristic $0$, then the associated essential
skeleton $\textsf{Sk}^{\textsf{ess}}(\mathcal{X}^{\textsf{an}})$ coincides
with Kähler skeleton
$\textsf{Sk}^{\textsf{k\"{a}h}}(\mathcal{X}^{\textsf{an}})$ as well as
geometric Kähler skeleton
$\overline{\textsf{Sk}}^{k\"{a}h}(\mathcal{X}^{\textsf{an}})$.

###### Proof.

This is a direct consequence following [Tem16, Theorem 8.2.9.], [Tem16,
Theorem 8.3.3.] and Remark 4.10. ∎

Now we demonstrate that no point of the essential skeleton may be contained in
an open fiber disk:

######  4.12.

Disk Interpretation of Essential Skeleton.

Let $X$ be a $K$-analytic space of dimension $n$, $L|K$ be a non-Archimedean
field extension of $K$ such that $d_{L|K}\leq\textsf{tr.deg}(L|K)=n-1$ and an
unit disk $\mathbf{D}(0,1)_{L}\subseteq X$, then there exist a family of
elements $\\{f_{i}\\}_{1\leq i\leq n-1}$ such that for any
$x\in\mathbf{D}(0,1)_{L}$

  * •

$\\{f_{i}\\}_{1\leq i\leq n-1}\subseteq
L\subseteq\kappa(x)\subseteq\mathscr{H}(x)$ are transcendental over $K$.

  * •

The images of $\\{f_{i}\\}_{1\leq i\leq s_{L|K}}$ in
$\widetilde{L}\subseteq\widetilde{\mathscr{H}(x)}$ are transcendental over
$\widetilde{K}$.

Then there exists an open neighborhood $U_{x}$ of $x$ and a morphism of
relative dimension $1$ by

${\varphi_{x}:U_{x}}$${\mathbf{G}^{n-1,\textsf{an}}_{m,K}}$$\scriptstyle{(f_{1}\text{,}\dots\text{,}f_{n-1})}$

Now it’s easy to see that we can glue $\varphi_{x}$ on $U_{L}:=\bigcup U_{x}$
as a smooth morphism
$\varphi_{L}:U_{L}\longrightarrow\mathbf{G}^{n-1,\textsf{an}}_{m,K}$ such that
$\mathbf{D}(0,1)_{L}\subseteq\varphi_{L}^{-1}(L,|-|_{L}):=C_{L}$ as an open
subset.

######  Theorem 4.13.

Let $\mathcal{X}$ be a smooth proper integral variety with dimension $n$ over
$K$, then for any point $x$ of the essential skeleton
$\textsf{Sk}^{\textsf{ess}}(\mathcal{X}^{\textsf{an}})$ there exists no open
fiber disk in $\mathcal{X}$ containing $x$.

###### Proof.

Let
$x_{0}\in\textsf{M}_{\phi_{0}}\subseteq\textsf{Sk}^{\textsf{ess}}(\mathcal{X}^{\textsf{an}})$
such that $x_{0}\in\mathbf{D}(0,1)_{L}$, and $x_{0}\in U_{L}$, now by Lemma
4.2, we have
$x_{0}\in\textsf{M}_{\phi_{0}|_{U_{L}}}\subseteq\textsf{Sk}^{\textsf{k\"{a}h}}(U_{L}/K)$.
We claim that
$x_{0}\in\textsf{Sk}^{\textsf{k\"{a}h}}(U_{L}/\mathbf{G}_{m,K}^{n-1,\textsf{an}})$,
to see this, let’s denote the non-zero section
$\omega_{0}:=\phi_{0}|_{U_{L}}\in\Gamma(U_{L},\mathscr{K}_{U_{L}/K})$ and then
consider the exact sequence of locally free $\mathscr{O}_{U_{L}}$-sheaves of
finite rank on $U_{L}$ [Ber93, 3.5.4. Corollary and 3.5.6. Corollary] as the
following:

(4.14)
${0}$${\varphi_{L}^{*}\Omega_{\mathbf{G}_{m,K}^{n-1,\textsf{an}}/K}}$${\Omega_{U_{L}/K}}$${\Omega_{U_{L}/\mathbf{G}_{m,K}^{n-1,\textsf{an}}}}$${0}$

We have
$\mathscr{K}_{U_{L}/K}\cong\varphi^{*}_{L}\mathscr{K}_{\mathbf{G}_{m,K}^{n-1,\textsf{an}}}\otimes_{\mathscr{O}_{U_{L}}}\mathscr{K}_{U_{L}/\mathbf{G}^{n-1,\textsf{an}}_{m,K}}$.
Now take $\\{U^{i}_{L}\\}_{i}$ to be a covering of open affinoid varieties
such that the restrictions of
$\Omega_{U_{L}/\mathbf{G}^{n-1,\textsf{an}}_{m,K}}$ on $\\{U_{L}^{i}\\}_{i}$
are free, then

$\displaystyle\mathscr{K}_{U_{L}/K}(U_{L}^{i})$
$\displaystyle\cong\varphi^{*}_{L}\mathscr{K}_{\mathbf{G}^{n-1,\textsf{an}}_{m,K}}(U_{L}^{i})\otimes_{\Gamma(U^{i}_{L})}\mathscr{K}_{U_{L}/\mathbf{G}^{n-1,\textsf{an}}_{m,K}}(U_{L}^{i})$
$\displaystyle\cong\varinjlim_{\varphi_{L}(U_{L}^{i})\subseteq
V_{i}}\mathscr{K}_{\mathbf{G}^{n-1,\textsf{an}}_{m,K}}(V_{i})\otimes_{\Gamma(U^{i}_{L})}\mathscr{K}_{U_{L}/\mathbf{G}^{n-1,\textsf{an}}_{m,K}}(U_{L}^{i})$

We can now write
$\omega_{0}|_{U_{L}^{i}}=[\frac{dt_{1}}{t_{1}}\wedge\cdots\wedge\frac{dt_{n-1}}{t_{n-1}}]\otimes\phi_{i}$.
By local calculation, we have $\phi_{i}|_{U_{L}^{i}\cap
U^{j}_{L}}=\phi_{j}|_{U_{L}^{i}\cap U^{j}_{L}}$. Let $\omega_{1}$ be the non-
zero section of
$\Gamma(U_{L},\mathscr{K}_{U_{L}/\mathbf{G}^{n-1,\textsf{an}}_{m,K}})$ by
gluing $\\{\phi_{i}\\}_{i}$, we claim that $x_{0}\in\textsf{M}_{\omega_{1}}$.
To see this, note that we have
$\lvert\lvert\omega_{0}\lvert\lvert_{\mathscr{K}_{U_{L}/K},x_{0}}=\lvert\lvert\phi_{i}\lvert\lvert_{\mathscr{K}_{U_{L}/\mathbf{G}^{n-1,\textsf{an}}_{m,K}},x_{0}}$
by assuming $x_{0}\in U^{i}_{L}$, thus by [Tem16, Corollary 6.3.12.],
$x_{0}\in\textsf{M}_{\omega_{1}|_{C_{L}}}\subseteq\textsf{Sk}^{\textsf{k\"{a}h}}(C_{L})$,
but this contradicts Lemma 4.8. ∎

##  5\. Applications

We have established that points outside the Berkovich skeleton of a strictly
semistable model can be covered by open fiber disks, whereas no point of the
essential skeleton can be contained in an open fiber disk. In this section, we
assume $K$ is a complete discrete valued field with equal characteristic $0$.

######  Conjecture 5.1.

Let $\mathcal{X}$ be a smooth proper integral variety with dimension $n$ over
$K$, then for any point $x\in\mathcal{X}^{\textsf{an}}$, $x$ is contained in
an open fiber disk if and only if $x$ is not contained in the essential
skeleton.

We can prove this conjecture when we have a strcitly semistable model whose
Berkovich skeleton is identified with the essential skeleton.

######  Theorem 5.2.

Let ${{\mathcal{X}}}$ be a proper and strictly semi-stable scheme over
complete discrete valuation ring $K^{\circ}$ such that the canonical divisor
$K_{\mathcal{X}}$ is semiample. Then for any point
$x\in\mathcal{X}_{K}^{\textsf{an}}$, $x$ is contained in an open fiber disk if
and only if $x$ is not contained in the essential skeleton.

###### Proof.

The condition that the canonical divisor is semiample on our strictly semi-
stable model implies that the Berkovich skeleton of the model and the
essential skeleton are identical, by [NX16, Theorem 3.3.3]. The result then
follows from combining Theorem 4.13 and Theorem 3.17. ∎

In the case where $\mathcal{X}$ admits no global pluricanonical forms we
expect by Mori’s conjecture that $\mathcal{X}$ is uniruled. In this case it is
natural to say the essential skeleton is empty. We wish to formulate the non-
Archimedean analog interpretation of uniruled variety over a complete discrete
valued field $K$ with equal characteristic $0$.

Let’s first recall the classical setting. A proper variety $\mathcal{X}$ is
called uniruled if there is a dominant rational map
$\varphi:\mathcal{Y}\times\mathbf{P}^{1}\dashrightarrow\mathcal{X}$, where
$\mathcal{Y}$ is a variety of dimension $\textsf{dim}(\mathcal{X})-1$.
$\mathcal{X}$ being uniruled obviously implies that rational curves cover
$\mathcal{X}$. On the other hand, if $\mathcal{X}$ is smooth and uniruled,
then on $\mathcal{X}$, there cannot exist any pluricanonical form, thus by
definition, the essential skeleton
$\textsf{Sk}^{\textsf{ess}}(\mathcal{X}^{\textsf{an}})=\emptyset$. Motivated
by Theorem 4.13, we propose the following definition and conjecture:

######  Definition 5.3.

Let $X$ be a proper geometrically integral smooth $K$-analytic space, $X$ is
called non-Archimedean uniruled if $X$ can be covered by open fiber disks
after finite field extension $K^{\prime}|K$.

######  Conjecture 5.4.

Let $\mathcal{X}$ be a uniruled variety over $K$, then
$\mathcal{X}^{\textsf{an}}$ is non-Archimedean uniruled.

Conjecture 5.4 is the special case of Conjecture 5.1 when $\mathcal{X}$ is
uniruled and therefore has empty essential skeleton.

######  Example 5.5.

The projective line $\mathbf{P}^{1}$ is non-Archimedean uniruled. To see this,
just note that
$\mathbf{P}^{1,\textsf{an}}=\textsf{trop}^{-1}[0,r_{1})\cup\textsf{trop}^{-1}(r_{2},\infty]$
with $r_{1}>r_{2}$.

######  Example 5.6.

Any ruled variety $\mathcal{X}$ is non-Archimedean uniruled, since
$\mathcal{X}=\mathcal{Y}\times\mathbf{P}^{1}$, then
$\mathcal{X}^{\textsf{an}}\cong\mathcal{Y}^{\textsf{an}}\times\mathbf{P}^{1,\textsf{an}}\cong\bigcup_{y\in\mathcal{Y}^{\textsf{an}}}\mathbf{P}_{\mathscr{H}(y)}^{1,\textsf{an}}$

######  5.7.

$\mathbf{P}^{n}$ is non-Archimedean uniruled.

Let us consider a non-Archimedean analytification of a blowing up
$\pi:\widetilde{\mathcal{X}}:=\textsf{Bl}_{\mathcal{Z}}\mathcal{X}\longrightarrow\mathcal{X}$
where $\mathcal{Z}$ is a smooth closed subscheme of $\mathcal{X}$, then we
have the following result:

######  Lemma 5.8.

Let $\mathcal{X}$ be a smooth geometrically integral proper variety over $K$,
then $\mathcal{X}^{\textsf{an}}$ is non-Archimedean uniruled if and only if
$\widetilde{\mathcal{X}}^{\textsf{an}}$ is non-Archimedean uniruled.

###### Proof.

To prove sufficiency, let us assume $\mathcal{X}$ is non-Archimedean uniruled
and $K^{\prime\prime}|K$ be a finite field extension such that
$\mathcal{X}_{K^{\prime\prime}}^{\textsf{an}}$ can be covered by open fiber
disks and $\widetilde{\mathcal{X}}_{K^{\prime\prime}}^{\textsf{}}$ admits a
proper strictly semi-stable model. Note that analytification commutate with
fiber product and by [GW20, Proposition 13.91.], we have
$\mathcal{X}_{K^{\prime\prime}}^{\textsf{an}}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\mathcal{Z}_{K^{\prime\prime}}^{\textsf{an}}=\bigcup\mathbf{D}(0,1)_{L}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\mathcal{Z}_{K^{\prime\prime}}^{\textsf{an}}\cong\widetilde{\mathcal{X}}_{K^{\prime\prime}}^{\textsf{an}}\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}\pi_{K^{\prime\prime}}^{\textsf{an},-1}(\mathcal{{Z}_{K^{\prime\prime}}^{\textsf{an}}})$,
meanwhile, we have the fiber
$\widetilde{\mathcal{X}}^{\textsf{an}}_{K^{\prime\prime},x}\cong\mathbf{P}^{r-1,\textsf{an}}_{\mathscr{H}(x)}$
for
$r=\mathsf{dim}_{x}(\widetilde{\mathcal{X}}_{K^{\prime\prime}}^{\textsf{an}})-\mathsf{dim}_{x}(\mathcal{X}_{K^{\prime\prime}}^{\textsf{an}})$
and $x\in\mathcal{Z}_{K^{\prime\prime}}^{\textsf{an}}$, thus
$x\in\widetilde{\mathcal{X}}^{\textsf{an}}_{K^{\prime\prime},x}$. Note that
$\pi_{K^{\prime\prime}}^{\textsf{an},-1}(\mathcal{Z}_{K^{\prime\prime}}^{\textsf{an}})$
has no intersection with any Berkovich skeleton
$\textsf{Sk}(\widetilde{\mathfrak{X}})$ for a fixed strictly semi-stable model
$\widetilde{\mathfrak{X}}$ of $\widetilde{\mathcal{X}}_{K^{\prime\prime}}$,
thus by Theorem 3.17, we have
$\widetilde{\mathcal{X}}_{K^{\prime\prime}}^{\textsf{an}}=\bigcup\mathbf{D}(0,1)_{L}$.
For the necessity, let’s assume there exists a point
$x_{0}\in\mathcal{X}^{\textsf{an}}$ such that any unit disks cannot cover it
after any finite field extension of $K$, for a sufficient large finite field
extension $K^{\prime}|K$, by Theorem 3.17 again, we can assume $x_{0}$ is a
birational point, then by assumption, we have
$x_{0}\in\mathbf{D}(0,1)_{L}\subseteq\widetilde{\mathcal{X}}_{K^{\prime}}^{\textsf{an}}$
and
$x_{0}\notin\pi_{K^{\prime}}^{\textsf{an},-1}(\mathcal{Z}_{K^{\prime}}^{\textsf{an}})$
for $L|K^{\prime}$ be a non-Archimedean field extension with transcendental
degree $n-1$, thus we can choose
$\mathbf{D}(0,1)_{L}\cap\pi_{K^{\prime}}^{\textsf{an},-1}(\mathcal{Z}_{K^{\prime}}^{\textsf{an}})=\emptyset$
which leads
$x_{0}\in\mathbf{D}(0,1)_{L}\subseteq\mathcal{X}_{K^{\prime}}^{\textsf{an}}$,
contradiction. ∎

######  Theorem 5.9.

$\mathbf{P}_{K}^{n}$ is non-Archimedean uniruled.

###### Proof.

Recall that the moduli space of stable genus zero curves with $n$-marked
points $\overline{\textsf{M}}_{0,n}$ can be constructed as a smooth iterated
blowing ups of the Cartesian power $(\mathbf{P}_{K}^{1})^{n-3}$ [Kee92,
Theorems 1 and 2] and a smooth iterated blow-up of $\mathbf{P}_{K}^{n-3}$
[Kap93]. On the other hand
$(\mathbf{P}_{K}^{1})^{n-3,\textsf{an}}\cong(\mathbf{P}^{1,\textsf{an}}_{K})^{n-3}$
is non-Archimedean uniruled, thus by Lemma 5.8, we are done. ∎

##  6\. Complement of the Skeleton via Analytic Functions

In the appendix, we first review the naive analytic functions sheaf developed
by V. Berkovich [Ber07]. We give an alternative description of the complement
of Berkovich skeleton in terms of analyticity sets of Iwasawa log functions.

######  6.1.

Let $X$ be a $K$-analytic space. For
$\mathscr{F}\in\textsf{Sh}(X_{\textsf{\'{e}t}})$ an étale sheaf, we define an
associated presheaf $\widetilde{\mathscr{F}}$ as
$\widetilde{\mathscr{F}}(Y):=\varinjlim\mathscr{F}(V)$, where V are open
neighborhoods of $Y_{\textsf{st}}$ in $Y$.

######  6.2.

Let $L$ be a filtered $K$-algebra, i.e., a commutative $K$-algebra provided by
an increasing sequence of $K$-vector subspaces $L^{0}\subseteq
L^{1}\subseteq\cdots$ such that $L^{i}\cdot L^{j}\subseteq L^{i+j}$ and
$L=\bigcup^{\infty}_{i=0}L^{i}$. For a $K$-analytic space $X$, we denote
$\mathscr{O}_{X}^{L,i}:=\mathscr{O}_{X}{\otimes}_{K}L^{i}$. The sheaf of
$\mathfrak{N}^{L,i}$-analytic functions $\mathfrak{N}_{X}^{L,i}$ is the
associated sheaf $\widetilde{\mathscr{O}_{X}^{L,i}}$ and we define the sheaf
of $\mathfrak{N}^{L}$-analytic function $\mathfrak{N}_{X}^{L}$ as
$\mathfrak{N}_{X}^{L}:=\varinjlim\mathfrak{N}_{X}^{L,i}$ as a sheaf of
filtered $\mathscr{O}_{X}$-algebra. For $f\in\mathfrak{N}_{X}^{L}(X)$, the
analyticity set of $f$ is the associated unique maximal open set $U_{f}$ such
that $f$ comes from $\mathscr{O}_{X}^{L}(X)$ and $X_{\textsf{st}}\subseteq
U_{f}\subseteq X$. When we take $L=K$, we denote $\mathfrak{N}_{X}^{K}$ by
$\mathfrak{n}_{X}$.

######  6.3.

Pull back of naive analytic functions sheaf $\mathfrak{N}^{L}$

Consider a pair $(f,g)$, where $f$ is a morphism of $K$-analytic spaces
$X\longrightarrow Y$ and $g$ is a morphism of filtered algebra
$L\longrightarrow S$ over a field extension $K\subseteq
K^{\prime}\subseteq\widehat{K^{\textsf{alg}}}$, then we can define the
canonical morphism
$f^{\sharp}:f^{*}\mathfrak{N}_{Y}^{S}\longrightarrow\mathfrak{N}^{L}_{X}$. To
see this, it suffices to show there exist well-defined morphisms
$f_{i}^{\flat}:\mathfrak{N}^{S_{i}}_{Y}\longrightarrow
f_{*}\mathfrak{N}^{L_{i}}_{X}$ for any $i\geq 0$, now for
$V\in\textsf{\'{E}t}(Y)$ and $\varphi\in\mathfrak{N}^{S_{i}}_{Y}(V)$, assume
$\varphi\in\mathscr{O}^{S_{i}}_{Y}(U_{\varphi})$ where
$V_{\textsf{st}}\subseteq U_{\varphi}\subseteq V$, then we have
$f^{\flat}(\varphi)\in\mathscr{O}^{L_{i}}_{X}(U_{\varphi}\times_{Y}X)$,
meanwhile we have $f(X_{\textsf{st}})\subseteq Y_{\textsf{st}}$ and
$\big{(}V\times_{Y}X\big{)}_{\textsf{st}}=\textsf{fp}_{V}^{-1}(V_{\textsf{st}})$
by Proposition 2.5, thus $\big{(}V\times_{Y}X\big{)}_{\textsf{st}}\subseteq
U_{\varphi}\times_{Y}X$ and $f_{i}^{\flat}(\varphi)\in
f_{*}\mathfrak{N}^{L_{i}}_{X}(V)$, we will denote $f_{i}^{\flat}(\varphi)$ as
$f^{*}(\varphi)$ in the following texts. We should note that
$U_{\varphi}\times_{Y}X\subsetneq U_{f^{*}(\varphi)}$ in general.

Now consider the canonical morphism of étale abelian sheaves
$\textsf{Log}^{\lambda}:\mathbf{G}^{1,\textsf{an}}_{m,X}\longrightarrow\mathfrak{N}^{L,1}_{X}$
induced by a fixed logaritmic character $\lambda$. For
$U\in\textsf{\'{E}t}(X)$,
$f\in\mathbf{G}^{1,\textsf{an}}_{m,X}(U)=\mathscr{O}(U)^{\times}$, then
$\textsf{Log}^{\lambda}(f)\in\mathfrak{N}_{X}^{L,1}(U)$ as the construction
above.

######  6.4.

Recall that A branch of the logarithm over $L$ is an
$\mathfrak{N}^{L,1}$-analytic function $f$ on
$\mathbf{G}^{1,\textsf{an}}_{m,K}$ such that $df=\frac{dT}{T}$ and
$m^{*}(f)=\textsf{fp}_{1}^{*}(f)+\textsf{fp}^{*}_{2}(f)$. Now note that
$\mathbf{G}^{1,\textsf{an}}_{m,K}(K)=\mathbf{G}^{1}_{m,K}(K)=K^{\times}$ is
contained inside the analytic set $U_{f}$ of $f$, then $f(a\cdot
b)=m^{*}(f)(a,b)=\textsf{fp}^{*}_{1}(f)(a,b)+\textsf{fp}^{*}_{2}(f)(a,b)=f(a)+f(b)$
and $f\in\Gamma(\mathscr{O}^{L,1}_{\mathbf{G}^{1,\textsf{an}}_{m,K}},U_{f})$,
such an $f$ gives a $\textsf{Gal}(K^{\textsf{alg}}|K)$-homomorphism
$\lambda_{f}:(K^{\textsf{alg}})^{\times}\longrightarrow{(K^{\textsf{alg}})^{\times}}\otimes_{K}L^{1}$
of abelian groups which is called logarithmic character with values in $L$. By
[Ber07, Lemma 1.4.1.], there is a one-to-one correspondence between
$\lambda_{f}$ and $f$ which we denote by $[\textsf{Log}^{\lambda_{f}}]$, and
for any $\lambda$,
$\textsf{Log}^{\lambda}\in\mathscr{O}(U_{\textsf{log}^{\lambda}})$ where
$U_{\textsf{log}^{\lambda}}=\bigcup_{x\in(\mathbf{G}^{1,\textsf{an}}_{m,K})_{0}}\mathbf{D}(x,|T|_{x})$,
it’s easy to see that $\textsf{Log}^{\lambda}$ is well defined, for any
$\mathbf{D}(b_{1},|b_{1}|)\cap\mathbf{D}(b_{2},|b_{2}|)\neq\varnothing$, we
have $|b_{1}|=|b_{2}|$ ,then

$\displaystyle\,\,\,\,\,\,\,\lambda(b_{1})+\textsf{Log}(\frac{T}{b_{1}})-\lambda(b_{2})-\textsf{Log}(\frac{T}{b_{2}})$
$\displaystyle=\lambda(\frac{b_{1}}{b_{2}})+\textsf{Log}(\frac{b_{2}}{b_{1}})=0$

since $|\frac{b_{1}}{b_{2}}|=1$. So $U_{\textsf{Log}^{\lambda}}$ is the
maximal analytic continuation of Log.

######  Proposition 6.5.

[Ber07, Lemma 1.4.1.] The analyticity set of any brunch of Iwasawa log
function $\mathsf{Log}^{\lambda}$ on $\mathbf{G}^{1,\textsf{an}}_{m,K}$ is
equal to the complement of the skeleton
$\textsf{Sk}(\mathbf{G}^{1,\textsf{an}}_{m,K})$.

######  Definition 6.6.

Let $X$ be a quasi-smooth $K$-analytic space, then for any morphism
$f:X\longrightarrow\mathbf{G}^{1,\textsf{an}}_{m,K}$, we can get a naive
analytic function $\mathsf{Log}^{\lambda}(f):=f^{*}\mathsf{Log}^{\lambda}$ on
$X$ which is called Iwasawa log function on $X$.

Consider
$\textsf{fp}_{i}:\mathbf{G}^{n,\textsf{an}}_{m,K}\longrightarrow\mathbf{G}^{1,\textsf{an}}_{m,K}$,
take
$\textsf{Log}^{\lambda}\in\mathfrak{N}^{K,{1}}_{\mathbf{G}^{1,\textsf{an}}_{m,K}}(\mathbf{G}^{1,\textsf{an}}_{m,K})$,
we have
$\textsf{fp}^{*}_{i}(\textsf{Log}^{\lambda})=\textsf{Log}^{\lambda}(T_{i})\in\mathfrak{N}^{K,{1}}_{\mathbf{G}^{n,\textsf{an}}_{m,K}}(\mathbf{G}^{n,\textsf{an}}_{m,K})$

######  Proposition 6.7.

The union of analyticity set of the naive analytic function
$\textsf{Log}^{\lambda}(T_{i})$ is the complement of the skeleton of
$\mathbf{G}^{n,\textsf{an}}_{m,K}$

###### Proof.

Note that the analyticity set $U_{\textsf{Log}^{\lambda}(T_{i})}$ of
$\textsf{Log}^{\lambda}(T_{i})$ is

$\mathbf{G}^{1,\textsf{an}}_{m,K}{\times}_{K}\mathbf{G}^{1,\textsf{an}}_{m,K}\cdots\mathbf{G}^{1,\textsf{an}}_{m,K}{\times}_{K}U_{\textsf{Log}^{\lambda}}{\times}_{K}\mathbf{G}^{1,\textsf{an}}_{m,K}\cdots{\times}_{K}\mathbf{G}^{1,\textsf{an}}_{m,K}$

, then by Lemma 3.14, we have
$\textsf{Sk}(\mathbf{G}^{n,\textsf{an}}_{m,K})^{\textsf{c}}=\bigcup_{\alpha}U_{\textsf{Log}^{\lambda}(T_{i})}$.
∎

######  Proposition 6.8.

For the monomial chart
$\varphi_{M}:M\hookrightarrow\mathbf{G}^{n,\textsf{an}}_{m,K}$ in 3.3, then
$\textsf{Sk}(\widehat{\mathcal{M}}^{\textsf{rig}})^{\textsf{c}}$ the
complement of skeleton $\textsf{Sk}(\widehat{\mathcal{M}}^{\textsf{rig}})$ is
$\bigcup U_{\varphi^{*}_{M}(\mathsf{Log}^{\lambda}(T_{i}))}$.

###### Proof.

By 6.7, it suffices to show
$\varphi^{-1}_{M}(U_{\mathsf{Log}^{\lambda}(T_{i})})=U_{\varphi_{M}^{*}(\mathsf{Log}^{\lambda}(T_{i}))}$,
without loss of generality, we show the case for $i=1,n=2$, then
$U_{\mathsf{Log}^{\lambda}(T_{1})}=U_{\mathsf{Log}^{\lambda}}\times_{K}\mathbf{G}^{1,\textsf{an}}_{m,K}=\bigcup
U_{\mathsf{Log}^{\lambda},\mathscr{H}(x)}$ which could reduce the case to show
$\varphi^{-1}_{M,x}(U_{\mathsf{Log}^{\lambda},\mathscr{H}(x)})=U_{\varphi^{*}_{M,x}(\mathsf{Log}^{\lambda}),\mathscr{H}(x)}$,
meanwhile by Lemma 3.3, analytic locally, we have
$M_{x}\cong\mathbf{B}(0;r,1)_{\mathscr{H}(x)}$ for
$r\in|\mathscr{H}(x)^{\times}|$ which is an analytic domain
$\\{x\in\mathbf{G}^{1,\textsf{an}}_{m,K}\,|\,r\leq|T|_{x}\leq 1\\}$ ∎

######  Proposition 6.9.

Let $X$ be a scheme that admits semi-stable reduction, the complement of its
Berkovich skeleton is a union of analyticity sets of the naive analytic
function $\mathsf{Log}^{\lambda}$.

###### Proof.

By Proposition 6.8, it suffices to show that for étale morphism
$g:\mathfrak{V}^{\textsf{rig}}\longrightarrow\mathbf{B}(0;r_{1},r_{2})_{K}\subseteq\mathbf{G}^{1,\textsf{an}}_{m,K}$,
we have $g^{-1}(U_{\mathsf{Log}^{\lambda}})=U_{\mathsf{Log}^{\lambda}(g)}$.
Denote $D^{\prime}:=\\{x\in\mathfrak{V}^{\textsf{rig}}\,|\,|g(x)-1|<1\\}$,
then we have $\mathsf{Log}^{\lambda}(g)|_{D^{\prime}}=\mathsf{Log}(g)$. Assume
$U_{\mathsf{Log}^{\lambda}(g)}\neq g^{-1}(U_{\mathsf{Log}^{\lambda}})$, then
there is a connected open subset $U^{\prime}$ of $\mathfrak{V}^{\textsf{rig}}$
such that strictly contains $D^{\prime}$ and
$\mathsf{Log}^{\lambda}(g)|_{U^{\prime}}\in\mathscr{O}(U^{\prime})$, which is
impossible by the same argument of [Ber07, Lemma 1.4.1]. ∎

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## Index