# Stability of pullback of orbifold bundles

Soumyadip Das  School of Mathematics, Tata Institute of Fundamental Research,
Mumbai, Maharastra 400005<EMAIL_ADDRESS>and  Snehajit Misra
Department of Mathematics, Indian Institute of Technology, Bombay, Powai,
Mumbai, Maharashtra 400076<EMAIL_ADDRESS>
###### Abstract.

In this article, we prove a categorical equivalence between proper formal
orbifold curves and proper orbifold curves in the sense of Deligne-Mumford
stacks. Using this identification, we define the notion of $P$-(semi)stability
of vector bundles on proper formal orbifold curves $(X,P)$. Finally, we show
that the pullback of a $P$-stable vector bundle under a morphism
$f\colon(Y,Q)\longrightarrow(X,P)$ between two formal orbifold curves is
$Q$-stable if and only if the underlying finite cover $f\colon
Y\longrightarrow X$ is genuinely ramified.

###### Key words and phrases:

Formal orbifold curve, orbifold bundle, orbifold slope, stability

###### 2020 Mathematics Subject Classification:

14D20, 14D23, 14H60 (Primary) ; 14D21, 14D22 (Secondary)

##  1\. Introduction

We work over a fixed algebraically closed field $k$ of arbitrary
characteristic. Let $f\colon Y\rightarrow X$ be a finite, generically
separable map of smooth projective connected $k$-curves (henceforth referred
to as a finite cover). For any semistable vector bundle $E$ on $X$, the
pullback vector bundle $f^{*}E$ on $Y$ is also semistable. But this property
need not be true for stable bundles. By [2, Proposition 5.1], if the map $f$
above is a finite étale map, there is a stable vector bundle $E$ on $X$ whose
pullback $f^{*}E$ is not stable on $Y$. [2, Theorem 5.3] classifies all finite
covers $f$ as above, called the genuinely ramified maps, such that $f^{*}E$ is
stable for all stable vector bundle $E$ on $X$. For other classifications of a
genuinely ramified map, see Section § 3.4.

Formal orbifold curves were defined in [8], [2]. Roughly, a formal orbifold
curve is a pair $(X,P)$ consisting of a smooth $k$-curve $X$ together with a
data given by certain local Galois extensions attached to finitely many points
of the curve. Vector bundles on a formal orbifold curve were introduced and
studied in [6]. Our first objective is to introduce and study the stability
condition for such bundles. To do this, we show a categorical equivalence
between proper formal orbifold curves and proper orbifold curves in the sense
of Deligne-Mumford stacks (see Section § 4). This identification allows us to
borrow the definition and theories developed for stacks in the formal orbifold
curve set up. We are now able to define the ‘degree’ ($\text{\rm deg}_{P}$) of
a vector bundle on $(X,P)$, its ‘slope’ ($\mu_{P}$), and hence a definition of
(semi)stability ($P$-(semi)stability; Definition 5.4). We show that this
definition of $P$-(semi)stability satisfies some analogue properties as the
usual notion of (semi)stability for curves.

This definition of $P$-(semi)stability is not not an artificial one. When $P$
is the trivial branch data $\mathcal{O}_{(X,P)}$, that is $(X,P)$ is just the
smooth connected projective $k$-curve, our notion is the same as the usual
(semi)stability. When $(X,P)$ is obtained from a connected $G$-Galois
(possibly ramified) cover $Z\longrightarrow X$ of smooth projective connected
curves, we have a natural identification of vector bundles on $(X,P)$ with the
$G$-equivariant bundles on $Z$. Under this identification, $P$-(semi)stability
is the same as the $G$-(semi)stability (i.e. (semi)stability on $Z$ which
preserve the $G$-actions; Corollary 5.7). Moreover, when $k$ is a field of
characteristic $0$, a proper connected formal orbifold curve $(X,P)$ is
uniquely determined by $X$ and finitely many points $x_{1},\ldots,x_{n}\in X$
together with natural numbers $m_{1},\ldots,m_{n}$. It was observed in [6,
Proposition 5.15] that the category of vector bundles on $(X,P)$ is equivalent
to the category of parabolic vector bundles on $X$ with respect to the divisor
$\sum_{\begin{subarray}{c}1\leq i\leq n\end{subarray}}x_{i}$ and with rational
weights above each point $x_{i}$ that are of the form $a/m_{i}$ for $0\leq
a<m_{i}$. We also have the notion of parabolic slopes and parabolic
(semi)stability on the later category. In this situation, the parabolic slopes
$\mu_{\text{\rm par}}$ are the same as the $P$-slopes $\mu_{P}$, and the
$P$-(semi)stable bundles are the same as the parabolic (semi)stable bundles
(Remark 5.10).

We also study how this new stability criterion behaves with respect to
pullback under morphisms of formal orbifold curves. We show that the pullback
of a $P$-semistable vector bundle is $Q$-semistable under a morphism
$f\colon(Y,Q)\longrightarrow(X,P)$ of formal orbifold curves, whereas pullback
of a $P$-stable bundle need not be $Q$-stable (Theorem 6.5). It turns out that
the exact criterion is the same as that of [2, Theorem 5.3] for the underlying
morphism of curves $Y\longrightarrow X$.

######  Theorem 1.1 (Theorem 6.4, Theorem 6.5).

Let $f\colon(Y,Q)\longrightarrow(X,P)$ be a morphism of connected projective
orbifold curves. Then the induced finite cover $f\colon Y\longrightarrow X$ of
smooth projective connected $k$-curves is genuinely ramified if and only if
for every $P$-stable vector bundle $E_{*}$ on $(X,P)$, the pullback
$f^{*}E_{*}$ on $(Y,Q)$ is $Q$-stable.

The ‘if’ part of the theorem is Theorem 6.5. For the other direction, we
closely follow the strategy from [2]. We first show that the above result can
be reduced to the geometric set up: $P$ is a geometric branch data on $X$ such
that $P\geq B_{f}$ (Definition 3.6, Definition 3.1) and $Q=f^{*}P$, the
pullback of the branch data under the map $f\colon Y\longrightarrow X$
(Example 3.3). This amounts to the following commutative diagram of formal
orbifold curves where $Z$ is a smooth connected projective $k$-curve, all the
morphisms are étale and the morphisms $g$ and $h$ are also Galois étale.

${Z}$${(Y,f^{*}P)}$${(X,P)}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{f}$

We prove the following technical results analogous to [2, Proposition 3.5]

######  Lemma 1.2 (Lemma 6.1, Lemma 6.3).

Under the above notation, suppose that the underlying finite cover $f\colon
Y\longrightarrow X$ of curves is a Galois genuinely ramified map. Then in
$\text{\rm Vect}(Y,f^{*}P)$ we have

$f^{*}\left(\left(f_{*}\mathcal{O}_{(Y,f^{*}P)}\right)/\mathcal{O}_{(X,P)}\right)\subset\bigoplus_{\begin{subarray}{c}1\leq
i\leq\text{\rm deg}(f)-1\end{subarray}}M_{i}$

where each $M_{i}\in\text{\rm Vect}(Y,f^{*}P)$ is a line bundle of negative
$f^{*}p$-degree.

Moreover, if $E_{*}$, $F_{*}$ are $P$-semistable vector bundles on $(X,P)$
with $\mu_{P}(E_{*})=\mu_{P}(F_{*})$, we have

$\text{\rm Hom}_{\text{\rm
Vect}(Y,f^{*}P)}\left(f^{*}E_{*},f^{*}F_{*}\right)=\text{\rm Hom}_{\text{\rm
Vect}(X,P)}\left(E_{*},F_{*}\right).$

This allows us to construct certain bundles on formal orbifold curves, and
finally a descent argument proves the theorem.

The structure of the paper is as follows. Section § 3.1 and § 3.3 recall the
definition and properties of formal orbifold curves and vector bundles on
them. We also introduce a general definition (Definition 3.15) of pullback of
vector bundles which is necessary for our case. We recall the definition and
characterization of genuinely ramified maps in Section § 3.4. In Section § 4,
we show the equivalence of categories of formal orbifold curves and Deligne-
Mumford orbifold curves. Further, the equivalence between vector bundles on
such objects are shown. After recalling the definition and results concerning
divisors, line bundles on a Deligne-Mumford orbifold curve (Section § 5.1), in
Section § 5.2, we introduce our main objects to study—(semi)stability
criterion for vector bundles on formal orbifold curves. We prove several
results (Lemma 5.5–Proposition 5.11). The main results are discussed in
Section § 6, where we also prove the above mentioned technical lemma.

In a later paper, we will study the construction of the moduli space of
$P$-(semi)stable bundles on an orbifold curve $(X,P)$.

##  2\. Notation and Convention

Throughout this article, we work over an algebraically closed field $k$ of
arbitrary characteristics. In this paper, the curves we consider are smooth
$k$-curves. For any $k$-scheme $W$ and a closed point $w\in W$, we denote the
completion of the local ring $\mathcal{O}_{W,w}$ of $W$ at $w$ by
$\widehat{\mathcal{O}}_{W,w}$. When the local ring is a domain, we set
$K_{W,w}$ as the quotient field $QF(\widehat{\mathcal{O}}_{W,w})$. For a
smooth $k$-curves $X$, any finite extension $L/K_{X,x}$ is understood as an
extension in a fixed separable closure $K^{\text{Sep}}_{X,x}$ of $K_{X,x}$.

A cover or a finite cover $f\colon Y\longrightarrow X$ of smooth $k$-curves
refers to a finite morphism $f$ that is generically separable. For a finite
group $G$, a $G$-Galois cover $Y\longleftarrow X$ is a finite cover together
with a $G$-action on $Y$ such that $G$ acts simply transitively on each
generic geometric fiber. Since the curves are assumed to be smooth, any finite
cover (in particular, a Galois cover) is étale away from finitely many points
on the base curve, which may even be empty.

For a $G$-Galois cover $f\colon Y\longrightarrow X$, the group $G$ acts
transitively on the fiber $f^{-1}(x)\subset Y$ for each point $x\in X$; the
stabilizer groups at points in $f^{-1}(x)$ are conjugate to each other in $G$.
So (up to conjugacy) we define an inertia group above a point $x\in X$ to be a
stabilizer group $\text{\rm Stab}_{G}(y)$ for some $y\in f^{-1}(x)$. In
particular, the cover $f$ is étale above $x\in X$ if and only if the inertia
group above $x$ is the trivial group. When the order of the inertia group at
$x$ is invertible in $k$, we say that the cover $f$ is tamely ramified over
the point $x$.

##  3\. Preliminaries

###  3.1. Formal Orbifold Curves

Let $k$ be an algebraically closed field. The notion of a formal orbifold
curve was introduced in [8] when $k=\mathbb{C}$, and was later generalized
over fields of arbitrary characteristic in [5]. In this section, we recall the
definitions of the formal orbifold curves, the morphisms among them and some
properties.

######  Definition 3.1.

Let $k$ be an algebraically closed field and $X$ be a smooth $k$-curve.

  1. (a)

A quasi-branch data on $X$ is a function $P$ that to every closed point $x\in
X$ associates a finite Galois extension $P(x)$ of $K_{X,x}$ (in some fixed
separable algebraic closure of $K_{X,x}$).

For two quasi-branch data $P^{\prime}$ and $P$ on $X$, we define
$P^{\prime}\geq P$ if $P^{\prime}(x)\supset P(x)$ as finite Galois extensions
of $K_{X,x}$ for all closed points $x\in X$.

The support $\text{\rm Supp}(P)$ of a quasi-branch data $P$ on $X$ is defined
to be

$\text{\rm Supp}(P)\coloneqq\left\\{x\in X\hskip 5.69046pt|\hskip
5.69046ptP(x)\text{ is a nontrivial extension of }K_{X,x}\right\\}.$

A quasi-branch data $P$ on $X$ is said to be a branch data if $\text{\rm
Supp}(P)\subset X$ is a finite set.

  2. (b)

A formal orbifold curve is an ordered pair $(X,P)$ where $X$ is a smooth
$k$-curve $X$ and $P$ is a branch data $P$ on $X$.

A formal orbifold curve $(X,P)$ is said to be connected (respectively,
projective) if the $k$-curve $X$ is connected (respectively, projective).

Thus a formal orbifold curve is a smooth curve $X$ together with the data of a
finite set $B$ (which may be empty) of closed points in $X$, and for each
$x\in B$, a finite Galois extension $P(x)/K_{X,x}$. When $\text{\rm
char}(k)=0$ or $|\text{\rm Gal}\left(P(x)/K_{X,x}\right)|$ is coprime to
$p=\text{\rm char}(k)$ for each closed point $x\in X$, the Galois extensions
$P(x)/K_{X,x}$ are uniquely determined by their order. In these situations,
$(X,P)$ is determined by $X$ together with finitely many points and a positive
integer (that is invertible in $k$) attached to each of these points. We call
$P$ a tame branch data.

The branch data with empty support is denoted by $O$ and is called the trivial
branch data. We identify the formal orbifold curve $(X,0)$ with the $k$-curve
$X$.

######  Example 3.2.

For any finite surjective morphism $f\colon Y\longrightarrow X$ of smooth
$k$-curves, we can associate a branch data $B_{f}$ on $X$ as follows. For any
closed point $x\in X$, define $B_{f}(x)$ to be the compositum of the Galois
closures of $K_{Y,y}/K_{X,x}$ for all $y\in f^{-1}(x)\subset Y$ (where the
compositum is taken in some fixed algebraic separable closure of $K_{X,x}$).
Since $f$ is a finite cover of smooth $k$-curves, it is branched only at
finitely many points (possibly empty) of $X$. So $B_{f}$ is a branch data on
$X$ and $\text{\rm Supp}(B_{f})$ is the branch locus $\text{BL}(f)$ of $f$.

Let $\bar{f}\colon\bar{Y}\longrightarrow X$ be the Galois closure of $f$. Then
it follows that $B_{f}=B_{\bar{f}}$.

######  Example 3.3.

For any finite morphism $f\colon Y\longrightarrow X$ of smooth $k$-curves and
a branch data $P$ on $X$, we can define the pullback of the branch data $P$
under $f$ on $Y$, denoted by $f^{*}P$. For any closed point $y\in Y$, set

$f^{*}P(y)=P(f(y))\cdot K_{Y,y}$

as the compositum of extensions over $K_{X,f(y)}$. Then $f^{*}P(y)$ is a
Galois extension of $K_{Y,y}$, and the Galois group $\text{\rm
Gal}\left(f^{*}P(y)/K_{Y,y}\right)$ is a subgroup of $\text{\rm
Gal}\left(P(f(y))/K_{X,f(y)}\right)$. So $\text{\rm Supp}(f^{*}P)\subseteq
f^{-1}(\text{\rm Supp}P)$ is a finite set, and $f^{*}P$ is a branch data on
$Y$.

Morphisms of formal orbifold curves are defined as follows.

######  Definition 3.4.

Let $(Y,Q)$ and $(X,P)$ be two formal orbifold curves. A morphism

$f\colon(Y,Q)\longrightarrow(X,P)$

is defined to be a finite morphism $f\colon Y\longrightarrow X$ such that for
all closed point $y\in Y$, we have $P(f(y))\subset Q(y)$ as extensions of
$K_{X,f(y)}$.

The morphism $f\colon(Y,Q)\longrightarrow(X,P)$ is called étale at $y$ if
$Q(y)=P(f(y))$. It is called étale if it is étale for all closed point $y\in
Y$.

Let $(X,P)$ be a formal orbifold curve. A finite morphism $f\colon
Y\longrightarrow X$ of smooth $k$-curves is said to be an essentially étale
cover of $(X,P)$ if for any closed point $y\in Y$,

$K_{Y,y}\subset P(f(y))\hskip 14.22636pt\text{as extensions of }K_{X,f(y)}.$

Note that if a finite cover $f\colon Y\longrightarrow X$ of a smooth
projective $k$-curves is an essentially étale cover of the orbifold $(X,P)$,
then its Galois closure $\bar{f}\colon\bar{Y}\longrightarrow Y$ is also an
essentially étale cover of $(X,P)$.

######  Example 3.5.

Let $P$ and $P^{\prime}$ be two branch data on $X$. The identity map
$\text{\rm Id}_{X}$ defines a morphism of formal orbifolds
$\iota\colon(X,P^{\prime})\longrightarrow(X,P)$ if and only if $P^{\prime}\geq
P$. The morphism $\iota$ is étale if and only if $P^{\prime}=P$.

In Example 3.3 we saw that to any finite morphism $f\colon Y\longrightarrow X$
and a branch data $P$ on $X$, we obtain a branch data $f^{*}P$ on $Y$. For any
closed point $y\in Y$, we have $f^{*}P(y)=P(f(y))\cdot K_{Y,y}\supset P(f(y))$
as field extensions of $K_{X,f(y)}$. So $f$ induces a morphism
$f\colon(Y,f^{*}P)\longrightarrow(X,P)$ of formal orbifold curves. By [5,
Lemma 2.12], the following hold.

  1. (1)

There is a morphism $f\colon(Y,Q)\longrightarrow(X,P)$ of formal orbifold
curves if and only if $Q\geq f^{*}P$.

  2. (2)

For any finite cover $f\colon Y\longrightarrow X$ of smooth $k$-curves, the
induced morphism $f\colon(Y,f^{*}P)\longrightarrow(X,P)$ of formal orbifold
curves is étale if and only if $f$ is an essentially étale cover of $(X,P)$.

When $f\colon Y\longrightarrow X$ is a finite Galois cover of smooth connected
$k$-curves, the induced morphism $f\colon(Y,O)\longrightarrow(X,B_{f})$ is
étale ([5, Corollary 2.13]). In view of this, we have the following
definition.

######  Definition 3.6.

A connected formal orbifold curve $(X,P)$ is said to be geometric if there
exists a connected étale cover $(Y,O)\longrightarrow(X,P)$ of formal orbifold
curves where $O$ is the trivial branch data on $Y$. In this case, $P$ is
called a geometric branch data on $X$.

By [5, Proposition 2.30], given any branch data $P$ on $X$, we can always find
a branch data $Q$ such that $Q\geq P$ and $Q$ is geometric.

###  3.2. Equivariant bundles and stability

In this short section, we review the properties of an equivariant vector
bundle. Let $Z$ be a smooth $k$-curve with an action of a finite group $G$ on
it.

######  Definition 3.7.

Suppose that $\delta\colon G\times Z\longrightarrow Z$ is an action of a
finite group $G$ on a smooth $k$-curve $Z$. A $G$-equivariant vector bundle
(or a $G$-bundle) on $Z$ is a vector bundle $V$ on $Z$ such that there is an
isomorphism

$\Delta\colon\text{pr}_{Z}^{*}V\overset{\sim}{\longrightarrow}\delta^{*}V$

of sheaves on $G\times Z$ satisfying the following co-cycle condition. For
each $g\in G$, via restriction we have $\Delta_{\\{g\\}\times Z}\colon
V\overset{\sim}{\longrightarrow}\delta_{g}^{*}V$ where $\delta_{g}\colon
Z\longrightarrow Z$ is the isomorphism induced by $\delta$. By identifying
$\\{g\\}\times Z$ with $Z$, we consider this as an isomorphism of sheaves on
$Z$ and denote it by $\Delta(g)$. The co-cycle condition is
$\Delta(e)=\text{\rm Id}_{V}$ and for any $g,\,h\in G$,

$\Delta(hg)=\delta_{g}^{*}(\Delta(h))\circ\Delta(g)\colon
V\longrightarrow\delta_{g}^{*}V\longrightarrow\delta_{h}^{*}\delta_{g}^{*}V=\delta_{hg}^{*}V.$

We denote the category of $G$-bundles on $Z$ by $\text{\rm Vect}_{G}(Z)$.

The rank and degree of a $G$-bundle $V\in\text{\rm Vect}_{G}(Z)$ is the usual
rank and degree of the underlying bundle. We have the following definition of
stability in $\text{\rm Vect}_{G}(Z)$.

######  Definition 3.8.

A $G$-equivariant vector bundle $V$ on $Z$ is said to be $G$-(semi)stable if
for each $G$-sub-bundle $W\subset V$ (i.e. a $G$-equivariant injection such
that $V/W$ is a vector bundle) with $0<\text{\rm rank}(W)<\text{\rm rank}(V)$,
we have

$\mu(W)\hskip 5.69046pt(\leq)\hskip 5.69046pt\mu(V).\,$

A polystable vector bundle $V\in\text{\rm Vect}_{G}(Z)$ is a $G$-semistable
vector bundle that is a finite direct sum of $G$-stable vector bundles.

In the above, if $G$ is the trivial group, we say that $V$ is (semi)stable.

It is well known (for example, see [1, Lemma 2.7]) that the notion of
$G$-semistability (respectively, $G$-polystability) coincides with the usual
notion of semistability (respectively, polystability) for the underlying
vector bundle, by forgetting the $G$-action. Whereas, the $G$-stability is not
the usual stability. We sketch a proof below.

######  Proposition 3.9.

Let $G$ be a finite group acting on a smooth $k$-curve $Z$. Let $V\in\text{\rm
Vect}_{G}(Z)$. Then $V$ is $G$-semistable (respectively, $G$-polystable) if
and only if $V$ is a semistable (respectively, polystable) vector bundle.

Suppose that $H\leq G$ is a subgroup. If $V$ is $H$-stable with the induced
$H$-action on $V$, then $V$ is also $G$-stable.

###### Proof.

The Harder-Narasimhan filtration of a vector bundle and the socle of a
semistable bundle are unique, and so they are preserved under any automorphism
of $Z$. In particular, they are also invariant under the $G$-action. So $V$ is
$G$-semistable if and only if $V$ is equal to its maximal destabilizing sub-
bundle. This is equivalent to $V$ being semistable in the usual sense.
Similarly, $V$ is $G$-polystable if and only if $V$ is semistable and is equal
to its socle $\text{\rm gr}(V)$. This is equivalent to $V$ being polystable.

Suppose that $V$ is $H$-stable with the induced $H$-action for some subgroup
$H\leq G$. Let $W\subset V$ be a $G$-sub-bundle, $0<\text{\rm
rank}(W)<\text{\rm rank}(V)$. By definition, $W$ is a sub-bundle of $V$ and
the injection $W\hookrightarrow V$ of sheaves is $G$-equivariant. Then with
the induced $H$-action, $W$ is an $H$-sub-bundle of $V$. Thus $\mu(H)\hskip
5.69046pt<\hskip 5.69046pt\mu(G)$, and the result follows. ∎

###  3.3. Vector Bundles on Formal Orbifold Curves

In this section, we recall the definition and properties of vector bundles and
their morphisms on an orbifold curve following [6].

######  Definition 3.10 ([6, Definition 4.4, Definition 4.5]).

Let $(X,P)$ be a formal orbifold curve, $B=\text{\rm Supp}(P)$. For $x\in B$,
let $G_{x}=\text{\rm Gal}\left(P(x)/K_{x,x}\right)$, and let $R_{x}$ be the
integral closure of $\widehat{\mathcal{O}}_{X,x}$ in $P(x)$; let the
$G_{x}$-action on $R_{x}$ be defined by the group homomorphism $\phi_{x}\colon
G_{x}\longrightarrow\text{\rm Aut}_{\text{\rm Ring}}(R_{x})$.

A vector bundle on $(X,P)$ is a triple $E_{*}=(E,\\{\Phi_{x}\\}_{x\in
B},\\{\eta_{x}\\}_{x\in B})$ where $E$ is a vector bundle on $X$ together with
the following data for each $x\in B$.

  1. (i)

A group homomorphism $\Phi_{x}\colon G_{x}\longrightarrow\text{\rm
Aut}_{\text{\rm Ab}}\left(E_{x}\otimes_{\mathcal{O}_{X,x}}R_{x}\right)$
satisfying the following condition.

$\Phi_{x}(g)(r\cdot m)=\phi_{x}(g)(r)\cdot\Phi_{x}(g)(m)\,\text{ for any }g\in
G_{x},r\in R_{x},m\in E_{x}\otimes_{\mathcal{O}_{X,x}}R_{x}.$

  2. (ii)

For the induced generic actions

$\displaystyle\Phi^{0}_{x}\colon
G_{x}\overset{\Phi_{x}}{\longrightarrow}\text{\rm Aut}_{\text{\rm
Ab}}\left(E_{x}\otimes_{\mathcal{O}_{X,x}}R_{x}\right)\longrightarrow\text{\rm
Aut}_{\text{\rm Ab}}\left(E_{x}\otimes_{\mathcal{O}_{X,x}}P(x)\right)$
$\displaystyle\text{and }\phi^{0}_{x}\colon
G_{x}\overset{\phi_{x}}{\longrightarrow}\text{\rm Aut}_{\text{\rm
Ring}}\left(R_{x}\right)\longrightarrow\text{\rm Aut}_{\text{\rm
Ring}}\left(P(x)\right),$

$\eta_{x}$ is a $G_{x}$-equivariant isomorphism

$\eta_{x}\colon
E_{x}\otimes_{\mathcal{O}_{X,x}}P(x)\overset{\sim}{\longrightarrow}E_{x}\otimes_{\mathcal{O}_{X,x}}P(x),$

where $G_{x}$ acts on the left via $\text{\rm Id}_{E_{x}}\otimes\phi^{0}_{x}$,
and on the right via $\Phi^{0}_{x}$.

For vector bundles $E_{*}=\left(E,\\{\Phi_{x}\\}_{x\in B},\\{\eta_{x}\\}_{x\in
B}\right)$ and $F_{*}=\left(F,\\{\Psi_{x}\\}_{x\in B},\\{\theta_{x}\\}_{x\in
B}\right)$, a morphism $E_{*}\longrightarrow F_{*}$ is defined to be a pair
$\left(g,\\{\sigma_{x}\\}_{x\in B}\right)$ where $g\colon E\longrightarrow F$
is a homomorphism of vector bundles on $X$, and for each $x\in B$,
$\sigma_{x}\colon E_{x}\otimes_{\mathcal{O}_{X,x}}R_{x}\longrightarrow
F_{x}\otimes_{\mathcal{O}_{X,x}}R_{x}$ is a $G_{x}$-equivariant (with respect
to the $\Phi_{x}$ and $\Psi_{x}$-actions) homomorphism of $R_{x}$-modules such
that the following diagram is commutative.

${E_{x}\otimes_{\mathcal{O}_{X,x}}P(x)}$${F_{x}\otimes_{\mathcal{O}_{X,x}}P(x)}$${E_{x}\otimes_{\mathcal{O}_{X,x}}P(x)}$${F_{x}\otimes_{\mathcal{O}_{X,x}}P(x)}$$\scriptstyle{g_{x}\otimes\text{\rm
Id}_{P(x)}}$$\scriptstyle{\eta_{x}}$$\scriptstyle{\theta_{x}}$$\scriptstyle{\sigma_{x}^{0}}$

Here $\sigma_{x}^{0}$ is the homomorphism induced by $\sigma_{x}$.

We denote the category of vector bundles on a formal orbifold $(X,P)$ by
$\text{\rm Vect}(X,P)$.

To simplify the notation, we write $E_{*}=\left(E,\Phi,\eta\right)$ when the
context is clear.

######  Example 3.11.

For any formal orbifold curve $(X,P)$, we have the trivial vector bundle
$\mathcal{O}_{(X,P)}\in\text{\rm Vect}(X,P)$. The underlying vector bundle on
$X$ is the structure sheaf $\mathcal{O}_{X}$, and for any $x\in B$, the action
maps are given by

$\Phi_{x}=\phi_{x}\colon G_{x}\longrightarrow\text{\rm Aut}_{\text{\rm
Ab}}(\mathcal{O}_{X,x}\otimes_{\mathcal{O}_{X,x}}R_{x})=\text{\rm
Aut}_{\text{\rm Ab}}(R_{x}),$

and so $\eta_{x}$ is the identity map.

######  Example 3.12.

Suppose that $\text{\rm char}(k)=0$. Let $(X,P)$ be a formal orbifold curve.
So $(X,P)$ is determined by a finite set $B$ of closed points and for each
$x\in B$, a positive integer $n_{x}$. By [6, Proposition 5.15], $\text{\rm
Vect}(X,P)$ is equivalent to the category of parabolic vector bundles on $X$
with respect to the divisor $\sum_{\begin{subarray}{c}x\in B\end{subarray}}x$
and over each $x\in B$, the weights are of the form $a/n_{x}$, $0\leq
a<n_{x}$.

######  Definition 3.13.

Let $E_{*}=\left(E,\Phi,\eta\right)\in\text{\rm Vect}(X,P)$. A vector bundle
$F_{*}=\left(F,\Psi,\theta\right)\in\text{\rm Vect}(X,P)$ is called a sub-
bundle of $E_{*}$ if there is a morphism $\left(g,\sigma\right)\colon
F_{*}\longrightarrow E_{*}$ such that $g\colon F\longrightarrow E$ is an
injective homomorphism making $F$ into a sub-bundle of $E$ (so the quotient
$E/g(F)$ is a vector bundle) and for each $x\in\text{\rm Supp}(P)$, the
$R_{x}$-module homomorphism $\sigma_{x}\colon
F_{x}\otimes_{\mathcal{O}_{X,x}}R_{x}\longrightarrow
E_{x}\otimes_{\mathcal{O}_{X,x}}R_{x}$ is a $G_{x}$-equivariant monomorphism.

######  Remark 3.14.

For a sub-bundle $F_{*}\subset E_{*}$ in $\text{\rm Vect}(X,P)$ as above, for
any $g\in G_{x}$ and $f\in F_{x}\otimes_{\mathcal{O}_{X,x}}R_{x}$, we have
$\sigma_{x}(\Psi_{x}(g)(f))=\Phi_{x}(g)(\sigma_{x}(f))$. Thus for any $g,\,f$,
and $r\in R_{x}$, we have the following.

$\begin{array}[]{rcl}\Phi_{x}(g)(r\cdot\sigma_{x}(f))&=&\phi_{x}(g)(r)\cdot\Phi_{x}(g)(\sigma_{x}(f))\\\
&=&\phi_{x}(g)(r)\cdot\sigma_{x}(\Psi_{x}(g)(f))\\\
&=&\sigma_{x}(\Psi_{x}(g)(r.f)).\end{array}$

Thus the action morphisms $\Psi_{x}$, and hence the $G_{x}$-equivariant
isomorphisms $\theta_{x}$ are restriction of $\Phi_{x}$ and $\eta_{x}$,
respectively.

We now summarize some of the important operations on $\text{\rm Vect}(X,P)$
following [6].

  1. (a)

([6, Theorem 4.12]) Let $(X,P)$ be a formal orbifold curve and
$g\colon(Z,O)\longrightarrow(X,P)$ be a $G$-Galois étale cover for a finite
group $G$. There is an equivalence of categories

$T_{(X,P)}^{Z}\colon\text{\rm
Vect}(X,P)\overset{\sim}{\longrightarrow}\text{\rm Vect}_{G}(Z),$

of vector bundles on $(X,P)$ and the $G$-equivariant vector bundles on $Z$.
The quasi-inverse to $T_{(X,P)}^{Z}$ is denotes by $S_{(X,P)}^{Z}$ which is
constructed using the invariant pushforward functor $g_{*}^{G}$.

  2. (b)

([6, Section 5]) Let $X$ be a smooth $k$-curve, $P^{\prime}\geq P$ be two
branch data on $X$. The identity map on $X$ induces a morphism
$\iota\colon(X,P^{\prime})\longrightarrow(X,P)$. We have an embedding ([6,
Corollary 5.10])

$\iota^{*}\colon\text{\rm Vect}(X,P)\longrightarrow\text{\rm
Vect}(X,P^{\prime}),$

where $\iota^{*}$ is given as follows. Let
$E_{*}=\left(E,\Phi,\eta\right)\in\text{\rm Vect}(X,P)$. Take the underlying
vector bundle of $\iota^{*}E_{*}$ to be $E$. For closed points $x\in X$, let
$R_{x}$ and $R^{\prime}_{x}$ be the integral closure of
$\widehat{\mathcal{O}}_{X,x}$ in $P(x)$ and $P^{\prime}(x)$, respectively. Set
$G^{\prime}_{x}=\text{\rm Gal}\left(P^{\prime}(x)/K_{X,x}\right)$, and let
$\phi^{\prime}\colon G^{\prime}_{x}\longrightarrow\text{\rm Aut}_{\text{\rm
Ring}}\left(R^{\prime}_{x}\right)$ be the action morphism. We have

$E_{x}\otimes_{\mathcal{O}_{X,x}}R^{\prime}_{x}\cong(E_{x}\otimes_{\mathcal{O}_{X,x}}R_{x})\otimes_{R_{x}}R^{\prime}_{x},$
$\text{and
}\,E_{x}\otimes_{\mathcal{O}_{X,x}}P^{\prime}(x)\cong(E_{x}\otimes_{\mathcal{O}_{X,x}}P(x))\otimes_{P(x)}P^{\prime}(x).$

Define $\iota^{*}\Phi_{x}\colon G^{\prime}_{x}\longrightarrow\text{\rm
Aut}_{\text{\rm Ab}}(E_{x}\otimes_{\mathcal{O}_{X,x}}R^{\prime}_{x})$ by

$\iota^{*}\Phi_{x}(g^{\prime})(e\otimes
r^{\prime})=\Phi_{x}(g)(e)\otimes_{R_{x}}\phi^{\prime}_{x}(r^{\prime}),$

for any $g^{\prime}\in G^{\prime}_{x}$, $e\in
E_{x}\otimes_{\mathcal{O}_{X,x}}R_{x}$ and $r^{\prime}\in R^{\prime}_{x}$,
where $g$ is the image of $g^{\prime}$ under the surjection
$G^{\prime}_{x}\twoheadrightarrow G_{x}$. Also define
$\iota^{*}\eta_{x}=\eta_{x}\otimes_{P(x)}\text{\rm Id}_{P^{\prime}(x)}$. Then
$\iota^{*}E_{*}\coloneqq\left(E,\iota^{*}\Phi,\iota^{*}\eta\right)\in\text{\rm
Vect}(X,P^{\prime})$.

  3. (c)

(Pullback; [6, Section 5.1]) For a finite cover $f\colon Y\longrightarrow X$
of smooth connected projective $k$-curves and a geometric branch data $P\geq
B_{f}$ on $X$, one can define the pullback morphism

$f^{*}\colon\text{\rm Vect}(X,P)\longrightarrow\text{\rm Vect}(Y,f^{*}P)$

as follows. Since $P\geq B_{f}$, for any closed point $y\in Y$, we have
$f^{*}P(y)=P(x)\cdot K_{Y,y}=P(x)$ as extensions of $K_{X,x}$, where $x=f(y)$.
So $f$ is an essentially étale cover and by [5, Lemma 2.12], the morphism
$f\colon(Y,f^{*}P)\longrightarrow(X,P)$ of formal orbifold curves is an étale
map. Let $g\colon Z\longrightarrow(X,P)$ be a $G$-Galois étale map of
connected projective formal orbifold curves such that $g$ factors as

${Z}$${(Y,f^{*}P)}$${(X,P)}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{f}$

Here $h\colon Z\longrightarrow(Y,f^{*}P)$ is a Galois étale cover with group
$H$, say (necessarily, $H\leq G$). For any $E_{*}\in\text{\rm Vect}(X,P)$,
define

(3.1) $f^{*}(E_{*})\coloneqq S^{Z}_{(Y,f^{*}P)}\circ T_{(X,P)}^{Z}(E_{*}).$

It follows that the definition is independent of the choice of $g\colon
Z\longrightarrow(X,P)$, up to an isomorphism. Also observe that
$g^{*}E_{*}=T_{(X,P)}^{Z}(E_{*})\in\text{\rm Vect}_{G}(Z)$.

  4. (d)

When $f\colon Y\rightarrow X$ and $P$ are as above, one can also define the
pushforward functor ([6, Definition 5.26])

$f_{*}\colon\text{\rm Vect}(Y,f^{*}P)\longrightarrow\text{\rm Vect}(X,P),$

and the tensor ([6, Definition 2.20]) $E_{*}\otimes F_{*}\in\text{\rm
Vect}(X,P)$ for $E_{*},\,F_{*}\in\text{\rm Vect}(X,P)$. Under this definition,
for any geometric formal orbifold curve $(X,P)$ with $g\colon
Z\longrightarrow(X,P)$ a $G$-Galois étale cover of connected formal orbifold
curves, the functor $S_{(X,P)}^{Z}$ producing the equivalence $\text{\rm
Vect}_{G}(Z)\overset{\sim}{\longrightarrow}\text{\rm Vect}(X,P)$ is none other
than the pushforward $g_{*}$.

  5. (e)

([6, Proposition 5.28]) As before, let $f\colon Y\longrightarrow X$ be a
finite cover of smooth connected projective $k$-curves, and let $P\geq B_{f}$
be a geometric branch data on $X$. For any $E_{*}\in\text{\rm Vect}(Y,f^{*}P)$
and $F_{*}\in\text{\rm Vect}(X,P)$, have the following natural isomorphisms.

(3.2) $\begin{array}[]{cc}\operatorname*{Hom}_{\text{\rm
Vect}(Y,f^{*}P)}(f^{*}F_{*},E_{*})\cong\operatorname*{Hom}_{\text{\rm
Vect}(X,P)}(F_{*},f^{*}E_{*}),\\\ f_{*}(f^{*}F_{*}\otimes E_{*})\cong
F_{*}\otimes f_{*}E_{*}.\end{array}$

For the purpose of our paper, we will work with the pullback of vector bundles
on formal orbifold curves that are not necessarily geometric, which we now
define. Suppose that $f\colon Y\rightarrow X$ is a morphism of smooth
connected projective $k$-curves. Consider the (not necessarily étale) morphism
$f\colon(Y,f^{*}P)\longrightarrow(X,P)$ of formal orbifold curves induced by
$f$. Let $E_{*}=(E,\Phi,\eta)\in\text{\rm Vect}(X,P)$. We will construct a
vector bundle $f^{*}E_{*}$ on $(Y,f^{*}P)$.

For any closed point $y\in Y$, $x=f(y)$, we have $f^{*}P(y)=P(x)\cdot K_{X,x}$
as extensions of $K_{X,x}$. Let $R_{x}$ and $R_{y}$ be the integral closure of
$\widehat{\mathcal{O}}_{X,x}$ in $P(x)$ and in $f^{*}P(y)$, respectively. So
$R_{y}$ is also the integral closure of $\widehat{\mathcal{O}}_{Y,y}$ in
$f^{*}P(y)$, and $\text{\rm Gal}\left(f^{*}P(y)/K_{Y,y}\right)\subset\text{\rm
Gal}\left(P(x)/K_{x,x}\right)$. We have the following isomorphisms of
$\widehat{\mathcal{O}}_{Y,y}$-modules.

(3.3)
$\begin{array}[]{rcl}(f^{*}E)_{y}\otimes_{\mathcal{O}_{Y,y}}R_{y}&\cong&(E_{x}\otimes_{\mathcal{O}_{X,x}}\mathcal{O}_{Y,y})\otimes_{\mathcal{O}_{Y,y}}R_{y}\\\
&\cong&E_{x}\otimes_{\mathcal{O}_{X,x}}R_{y}\\\
&\cong&(E_{x}\otimes_{\mathcal{O}_{X,x}}R_{x})\otimes_{R_{x}}R_{y}.\end{array}$

Similarly,

(3.4)
$(f^{*}E)_{y}\otimes_{\mathcal{O}_{Y,y}}f^{*}P(y)\cong(E_{x}\otimes_{\mathcal{O}_{X,x}}P(x))\otimes_{P(x)}f^{*}P(y).$

Now we define
$f^{*}(E_{*})=\left(f^{*}E,\Psi=f^{*}\Phi,\theta=f^{*}\eta\right)$ as follows.

For any $g\in\text{\rm Gal}\left(f^{*}P(y)/K_{Y,y}\right)$, $v\in
E_{x}\otimes_{\mathcal{O}_{X,x}}R_{x}$, $r\in R_{y}$, set

(3.5) $\Psi_{y}(g)(v\otimes
r)\coloneqq\Phi_{x}(g)(v)\otimes_{R_{x}}\phi^{\prime}_{y}(g)(r),$

where $\phi^{\prime}_{y}\colon\text{\rm
Gal}\left(f^{*}P(y)/K_{Y,y}\right)\longrightarrow\text{\rm Aut}_{\text{\rm
Ring}}(R_{y})$ is the action map. Also set
$\theta_{y}=\eta_{x}\otimes_{P(x)}\text{\rm Id}_{f^{*}P(y)}$.

From the above definition, we see that $\Psi=f^{*}\Phi$ satisfies the required
properties as in Definition 3.10 (i), and we need to check Definition 3.10
(ii):

$\text{for any }g\in\text{\rm
Gal}\left(f^{*}P(y)/K_{Y,y}\right),\,\theta_{y}\circ(\text{\rm
Id}_{(f^{*}E)_{y}}\otimes_{\mathcal{O}_{Y,y}}{\phi^{\prime}}^{0}_{y}(g))=\Psi^{0}_{y}(g)\circ\theta_{y}.$

By the isomorphism in (3.4), any arbitrary element of
$(f^{*}E)_{y}\otimes_{\mathcal{O}_{Y,y}}(f^{*}P)(y)$ is a finite sum of
elements of the form $(e\otimes 1)\otimes s$ for some $e\in E_{x}$, $1\in
P(x)$ and $s\in(f^{*}P)(y)$. Then

(3.6) $\begin{array}[]{rcl}\theta_{y}\circ(\text{\rm
Id}_{(f^{*}E)_{y}}\otimes_{\mathcal{O}_{Y,y}}{\phi^{\prime}}^{0}_{y}(g))((e\otimes
1)\otimes s)&=&\theta_{y}\left((e\otimes
1)\otimes{\phi^{\prime}}^{0}_{y}(g)(s)\right)\\\ &=&\eta_{x}(e\otimes
1)\otimes{\phi^{\prime}}^{0}_{y}(g)(s)\\\
&=&\left[\eta_{x}\circ\left(\text{\rm
Id}_{E_{x}}\otimes\phi_{x}(g)\right)(e\otimes
1)\right]\otimes{\phi^{\prime}}^{0}_{y}(g)(s)\\\
&=&\left(\Phi^{0}_{x}(g)\circ\eta_{x}\right)(e\otimes
1)\otimes{\phi^{\prime}}^{0}_{y}(g)(s)\\\
&=&\Psi^{0}_{y}(g)\left(\eta_{x}(e\otimes 1)\otimes s\right)\\\
&=&\Psi^{0}_{y}(g)\circ\theta_{y}\left((e\otimes 1)\otimes
s\right).\end{array}$

This shows that $f^{*}(E_{*})$ is a vector bundle on $(Y,f^{*}P)$.

######  Definition 3.15.

Let $f\colon(Y,Q)\longrightarrow(X,P)$ be a morphism of formal orbifold curves
where $f\colon Y\longrightarrow X$ is a finite cover of smooth projective
connected $k$-curves. We necessarily have $Q\geq f^{*}P$, and the morphism $f$
factors as the composition

$(Y,Q)\overset{\jmath}{\longrightarrow}(Y,f^{*}P)\overset{f^{\prime}}{\longrightarrow}(X,P).$

Define the pullback functor $f^{*}\colon\text{\rm
Vect}(X,P)\longrightarrow\text{\rm Vect}(Y,Q)$ as

$f^{*}(E)\coloneqq\jmath^{*}\circ(f^{\prime})^{*},$

where $(f^{\prime})^{*}$ is defined as above and $\jmath^{*}$ is defined in
(b).

In the following, we establish that the above definition coincide with the
earlier definition – 3.1 (3.1) of pullback in the geometric set up.

######  Proposition 3.16.

Let $f\colon Y\longrightarrow X$ be a finite cover of smooth projective
connected $k$-curves. Let $P\geq B_{f}$ be a geometric branch data on $X$.
Suppose that $g\colon Z\longrightarrow(X,P)$ be a $G$-Galois étale morphism of
smooth connected formal orbifolds that factors as $g=f\circ h$, where $h\colon
Z\longrightarrow(Y,f^{*}P)$ is an $H$-Galois étale covering map (see Diagram
3.3). Then we have

$S^{Z}_{(Y,f^{*}P)}\circ T^{Z}_{(X,P)}=f^{*},$

where $f^{*}$ is defined as in Definition 3.15.

###### Proof.

For any closed point $y\in Y$, $x=f(y)$, we have $f^{*}P(y)=P(x)$ as extension
of $K_{X,x}$. Let $R_{x}$ be the integral closure of $K_{X,x}$ in $P(x)$,
which is also same as the integral closure of $\widehat{\mathcal{O}}_{Y,y}$ in
$P(x)$. So the action map $\phi^{\prime}_{y}\colon\text{\rm
Gal}\left(P(x)/K_{Y,y}\right)\longrightarrow\text{\rm Aut}_{\text{\rm
Ring}}(R_{x})$ is the restriction of the action map $\phi_{x}\colon\text{\rm
Gal}\left(P(x)/K_{X,x}\right)\longrightarrow\text{\rm Aut}_{\text{\rm
Ring}}(R_{x})$. Let $E_{*}=\left(E,\Phi,\eta\right)\in\text{\rm Vect}(X,P)$
and $f^{*}(E_{*})=(f^{*}E,f^{*}\Phi,f^{*}\eta)$. Then by (3.5), we have

$(f^{*}\Phi)_{y}(g)(e)=\Phi_{x}(g)(e)$

for any $g\in\text{\rm Gal}\left(P(x)/K_{Y,y}\right)$ and
$e\in(f^{*}E)_{y}\otimes_{\mathcal{O}_{Y,y}}R_{x}\cong
E_{x}\otimes_{\mathcal{O}_{X,x}}R_{x}$. In other words, the $\text{\rm
Gal}\left(P(x)/K_{Y,y}\right)$-action $(f^{*}\Phi)_{y}$ is the restriction of
the $\text{\rm Gal}\left(P(x)/K_{X,x}\right)$-action $\Phi_{x}$. Since
$K_{Z,z}\cong P(x)$ as extensions of $K_{X,x}$ for some point $z\in Z$ with
$h(z)=y$, the statement follows. ∎

We will use the following result later.

######  Lemma 3.17.

Let $f\colon Y\longrightarrow X$ be a finite cover of smooth connected
projective $k$-curves. Let $P\geq B_{f}$ be a geometric branch data on $X$.
Consider the following cartesian square.

${(Y,f^{*}P)}$${(X,P)}$${(Y,O)}$${(X,O)}$$\scriptstyle{f}$$\scriptstyle{\jmath}$$\scriptstyle{\iota}$$\scriptstyle{f}$

Then we have the following.

(3.7) $\iota^{*}f_{*}\mathcal{O}_{(Y,O)}=f_{*}\mathcal{O}_{(Y,f^{*}P)}.$

###### Proof.

Since $P\geq B_{f}$, the morphism $f\colon(Y,f^{*}P)\longrightarrow(X,P)$ is
an étale map. Let $g\colon Z\longrightarrow(X,P)$ be a $G$-Galois étale map of
connected projective formal orbifold curves such that $g$ factors as $g=f\circ
h$, where $h\colon Z\longrightarrow(Y,f^{*}P)$ is an $H$-Galois étale covering
map. Set $\widetilde{Z}$ as the normalization of the fiber product
$Y\times_{X}Z$. Since $P\geq B_{f}$, we also have
$(Y,f^{*}P)\times_{(X,P)}(Z,O)\cong(\widetilde{Z},O)$ as formal orbifold
curves. Let $p_{1}\colon\widetilde{Z}\longrightarrow(Y,f^{*}P)$ and
$p_{2}\colon\widetilde{Z}\longrightarrow(X,P)$ be the projection morphisms. So
$p_{1}$ is a $G$-Galois étale cover and $p_{2}$ is a finite étale morphism of
formal orbifold curves. To summarize, we have the following commutative
diagram where the squares are cartesian.

${\tilde{Z}}$${Z}$${(Y,f^{*}P)}$${(X,P)}$${Y}$${X}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{\jmath}$$\scriptstyle{f}$$\scriptstyle{\iota}$$\scriptstyle{f}$

Here $\jmath$ and $\iota$ are the morphisms induced by $\text{\rm Id}_{Y}$ and
$\text{\rm Id}_{X}$, respectively.

Using the definition of the pushforward ([6, Definition 5.26])
$f_{*}=S^{Z}_{(X,P)}\circ(p_{2})_{*}\circ T^{\tilde{Z}}_{(Y,f^{*}P)}$ and the
observation that $T^{Z}_{(Y,f^{*}P)}=h^{*}$, $T^{Z}_{(X,P)}=g^{*}$, we have

$\begin{array}[]{rcl}f_{*}\mathcal{O}_{(Y,f^{*}P)}&=&S^{Z}_{(X,P)}\circ(p_{2})_{*}\circ
T^{\widetilde{Z}}_{(Y,f^{*}P)}\left(\mathcal{O}_{(Y,f^{*}P)}\right)\\\
&=&S^{Z}_{(X,P)}\circ(p_{2})_{*}\left(\mathcal{O}_{\widetilde{Z}}\right)\\\
&=&S^{Z}_{(X,P)}\circ(p_{2})_{*}\circ(p_{1}^{*}\jmath^{*})\left(\mathcal{O}_{Y}\right)\\\
&=&S^{Z}_{(X,P)}\circ g^{*}\iota^{*}\left(f_{*}\mathcal{O}_{Y}\right)\hskip
28.45274pt(\text{\rm by finite base change of schemes})\\\
&=&S^{Z}_{(X,P)}\circ
T^{Z}_{(X,P)}\circ\iota^{*}\left(f_{*}\mathcal{O}_{Y}\right)\\\
&=&\iota^{*}\left(f_{*}\mathcal{O}_{Y}\right).\end{array}$

∎

###  3.4. Genuinely Ramified Morphisms

We recall the definition and properties of genuinely ramified morphisms of
smooth projective connected $k$-curves. For any finite cover $f\colon
Y\longrightarrow X$ of smooth projective connected $k$-curves, we have
$\mathcal{O}_{X}\subseteq f_{*}\mathcal{O}_{Y}$. Moreover, the maximal
destabilizing sub-sheaf $(f_{*}\mathcal{O}_{Y})_{1}$ is a semistable sub-
bundle of degree $0$ containing $\mathcal{O}_{X}$, and it is also a sheaf of
$\mathcal{O}_{X}$-algebras ([2, Equation (2.7), Lemma 2.4]).

######  Definition 3.18 ([2, Definition 2.5, Proposition 2.6, Lemma 3.1]).

A finite cover $f\colon Y\longrightarrow X$ of smooth projective connected
$k$-curves is said to be genuinely ramified if $f$ satisfies any of the
following equivalent conditions.

  1. (1)

The maximal destabilizing sub-bundle of $f_{*}\mathcal{O}_{Y}$ is
$\mathcal{O}_{X}$.

  2. (2)

The map $f$ does not factor through any non-trivial étale cover of $Y$ (in
particular, $f$ is not an étale map).

  3. (3)

The homomorphism between étale fundamental groups

$f_{*}\colon\pi^{\text{\'{e}t}}_{1}(Y)\longrightarrow\pi_{1}^{\text{\'{e}t}}(X)$

induced by $f$ is a surjection.

  4. (4)

$\text{\rm dim}_{k}H^{0}(Y,f^{*}f_{*}\mathcal{O}_{Y})=1$.

  5. (5)

The fibre product $Y\times_{X}Y$ is connected.

In [2, Corollary 2.7] it is shown that for any finite cover $f\colon
Y\longrightarrow X$ of smooth projective connected $k$-curves, the following
hold. The maximal destabilizing sub-bundle $(f_{*}\mathcal{O}_{Y})_{1}\subset
f_{*}\mathcal{O}_{Y}$ is a sheaf of $\mathcal{O}_{X}$-algebras inducing a
finite étale cover

$\hat{X}=\underline{\text{\rm
Spec}}\left((f_{*}\mathcal{O}_{Y})_{1}\right)\longrightarrow X$

of degree given by the rank of the bundle $(f_{*}\mathcal{O}_{Y})_{1}$. It is
the maximal étale cover of $X$ via which the map $f$ factors. Moreover, the
induced finite cover $Y\longrightarrow\hat{X}$ is genuinely ramified.

The main result of [2] gives another important criterion of a map to be
genuinely ramified in terms of the stability of vector bundles under pullback
maps.

######  Theorem 3.19 ([2, Theorem 5.3]).

Let $f\colon Y\longrightarrow X$ be a finite cover of smooth projective
connected $k$-curves. The map $f$ is genuinely ramified if and only if the
vector bundle $f^{*}E$ is stable on $Y$ for every stable vector bundle $E$ on
$X$.

##  4\. Formal orbifold curves: a stacky view

Throughout this section, let $k$ be an algebraically closed field. We will see
that a formal orbifold curve can be viewed as a Deligne-Mumford stack which is
a smooth, proper orbifold curve. This view lets us define concepts like the
degree of a vector bundle on a formal orbifold curve, and a stability
condition.

The definition of a Deligne-Mumford stack is technical which we avoid. Our
main reference are [7] and [9, Appendix A]. We mention a few important
properties of a Deligne-Mumford stack $\mathfrak{X}$ (which will be assumed to
be smooth, separated and of finite type over $k$). We also note that a
representable morphism $\mathfrak{Y}\longrightarrow\mathfrak{X}$ in our paper
is a morphism representable by a scheme222This is done to avoid the language
of algebraic spaces, taken in sense of [9, Definition 7.11], i.e. for any
scheme $Z$ and a morphism $Z\longrightarrow\mathfrak{X}$ of stacks, the fibre
product $\mathfrak{Y}\times_{\mathfrak{X}}Z$ is a scheme. A representable
morphism $\mathfrak{Y}\longrightarrow\mathfrak{X}$ is said to be unramified if
for any scheme $Z$ and a morphism $Z\longrightarrow\mathfrak{X}$ of stacks,
the morphism $\mathfrak{Y}\times_{\mathfrak{X}}Z\longrightarrow Z$ is a
formally unramified morphism of schemes.

  1. (1)

The diagonal morphism

$\Delta_{\mathfrak{X}}\colon\mathfrak{X}\longrightarrow\mathfrak{X}\times_{\text{\rm
Spec}(k)}\mathfrak{X}$

is a representable, unramified morphism (see [9, Proposition 7.15], [7,
Theorem 8.3.3]). We assume that the Deligne-Mumford stack $\mathfrak{X}$ is
separated, i.e. for any morphism
$Y\longrightarrow\mathfrak{X}\times_{\text{\rm Spec}(k)}\mathfrak{X}$ of
stacks where $Y$ is a scheme, the morphism

(4.1) $\mathfrak{X}\times_{\mathfrak{X}\times_{\text{\rm
Spec}(k)}\mathfrak{X}}Y\longrightarrow Y$

of schemes is a proper morphism.

  2. (2)

There exists an étale surjective morphism $Z\longrightarrow\mathfrak{X}$ from
a scheme $Z$. The smoothness and the finite type over the field $k$
assumptions on $\mathfrak{X}$ implies that we can find a scheme $Z$ as above,
which is smooth, of finite type over $k$. We say that
$Z\longrightarrow\mathfrak{X}$ is an atlas.

  3. (3)

When the diagonal morphism $\Delta_{\mathfrak{X}}$ is also finite, i.e. the
morphism of schemes in (4.1) is finite for any morphism
$Y\longrightarrow\mathfrak{X}$ where $Y$ is a scheme, the Deligne-Mumford
stack admits a Coarse moduli space $\pi\colon\mathfrak{X}\longrightarrow X$
([7, Theorem 11.1.2]). If the stack $\mathfrak{X}$ is one-dimensional (admits
an atlas from a smooth $k$-curve), the moduli space $X$ is also a $k$-curve
together with the following properties.

    1. (a)

The morphism $\pi$ is initial among all morphisms from $\mathfrak{X}$ to
schemes.

    2. (b)

$\pi$ induces a bijective correspondence between the $k$-points of $X$ and the
isomorphism classes of $k$-points of $\mathfrak{X}$.

Moreover, $\pi$ is a proper morphism of stacks, $X$ is separated and of finite
type over $k$. We say that $\mathfrak{X}$ is proper if $X$ is a proper scheme
([7, Exercise 11.E]).

######  Example 4.1.

One important example of a Deligne-Mumford stack is a quotient stack. Let $Y$
be a quasi-projective $k$-variety. Let $G$ be a finite group acting on $Y$
such that there is a quotient scheme $X=Y/G$. We can assign a Deligne-Mumford
stack $[Y/G]$ to this data (see [7, Example 8.1.12]). The $k$-points of
$[Y/G]$ (i.e., the isomorphism classes of objects over $k$) and the closed
points of $X$ are both canonically identified with the $G$-orbits of the
closed points of $Y$. For each such point $x\in[Y/G](k)$, we obtain a
stabilizer group $G_{x}$, which is the group of automorphisms lying over
$\text{\rm Id}_{\text{\rm Spec}\left(k\right)}$. More precisely, the fiber
product
$[Y/G]\,\times_{\Delta,\,\left([Y/G]\times[Y/G]\right),\,(x,x)}\,\text{\rm
Spec}\left(k\right)$ is a constant $k$-group scheme associated to the finite
group $G_{x}$. A point $x\in[Y/G](k)$ is called a stacky point if the
stabilizer group $G_{x}$ is trivial. The canonical morphism
$Y\longrightarrow[Y/G]$ is an atlas, and so $[Y/G]$ is a smooth (respectively,
proper) Deligne-Mumford stack if and only if $Y$ is smooth (respectively,
proper). Moreover, the stack $[Y/G]$ admits a Coarse moduli scheme
$[Y/G]\longrightarrow X$.

When $Y$ is a smooth variety, the finite surjective morphism $Y\longrightarrow
X$ is étale over a dense open subset $U\subset X$, and for any point
$x\in[Y/G](k)$ whose image is a closed point in $U$, the stabilizer group
$G_{x}$ is trivial. It follows that $[Y/G]\times_{X}U\cong U$.

We now define an orbifold curve (our definition is a special case of the
general definition).

######  Definition 4.2.

A Deligne-Mumford stack $\mathfrak{X}$ is said to be an orbifold curve if it
satisfies the following properties.

  1. (1)

$\mathfrak{X}$ admits an étale surjective morphism from a smooth (not
necessarily connected) $k$-curve;

  2. (2)

the diagonal $\Delta_{\mathfrak{X}}$ is a finite morphism of stacks;

  3. (3)

$\mathfrak{X}$ admits an irreducible $k$-curve as its Coarse moduli space;

  4. (4)

there exists an open subset $U\subset X$ such that the induced map
$\mathfrak{X}\times_{X}U\longrightarrow U$ is an isomorphism.

When $Y\longrightarrow X$ is a $G$-Galois cover of smooth $k$-curves for some
finite group $G$ and $X$ is connected, the quotient stack $[Y/G]$ in Example
4.1 is an orbifold curve.

Let $\mathfrak{X}$ be an orbifold curve. For any point $x\in\mathfrak{X}(k)$,
the fiber product
$\mathfrak{X}\times_{\Delta_{\mathfrak{X}},\mathfrak{X}\times_{k}\mathfrak{X},(x,x)}\text{\rm
Spec}(k)=\underline{\text{\rm Isom}}(x,x)$ is a constant group $k$-scheme
associated to a finite group $G_{x}$ (up to a canonical isomorphism). We say
that $G_{x}$ is the stabilizer group at $x$. So a closed point in
$\mathfrak{X}$ is identified with a closed point $x$ of $X$ together with the
group $G_{x}$ that acts as the group of automorphisms on $x$. It follows that
$G_{x}$ is the trivial group if and only if $x$ lies in the open sub-scheme of
$\mathfrak{X}$.

######  Proposition 4.3.

Let $\mathfrak{X}$ be an orbifold curve and
$\pi\colon\mathfrak{X}\longrightarrow X$ be a Coarse moduli $k$-curve. Let
$U\subset X$ be the maximal open subset such that
$\mathfrak{X}\times_{X}U\cong U$. Set $B=X-U$. For each point $x\in B$, let
$G_{x}$ be the stabilizer group at $x$. Then for each point $x\in B$, the
following hold.

There is an open subset $U_{x}\subset X$ together with a $\Gamma_{x}$-Galois
cover $V_{x}\longrightarrow U_{x}$ (for some finite group $\Gamma_{x}$) that
is étale away from $x$, $U_{x}\cap B=\\{x\\}$, the inertia groups at points in
$V_{x}$ lying above $x$ are conjugates of $G_{x}$ in $\Gamma_{x}$ such that

$\mathfrak{X}\times_{X}U_{x}\cong\left[V_{x}/\Gamma_{x}\right].$

###### Proof.

Let $V\longrightarrow\mathfrak{X}$ be an atlas where $V$ is a smooth
$k$-curve. The induced morphism $g\colon V\longrightarrow X$ is a finite
surjective morphism of $k$-curves, étale over $U$. In particular, $g$ is a
finite cover. By the maximality of $U$, the branched locus of $g$ is $B$. Let
$x\in B$. Then

$V\times_{X}\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X,x}\right)=\sqcup_{\begin{subarray}{c}1\leq
i\leq n\end{subarray}}\text{Spec}\left(S_{i}\right),$

a finite disjoint union, where each $S_{i}$ is a complete discrete valuation
domain and the maps $\text{\rm Spec}\left(S_{i}\right)\longrightarrow\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X,x}\right)$ are finite morphisms of regular
schemes. Set $\mathfrak{X}_{x}=\mathfrak{X}\times_{X}\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X,x}\right)$. Let $W_{1}=\text{\rm
Spec}\left(S_{1}\right)$.

We first show that $W_{1}$ is equipped with a $G_{x}$-action such that
$\mathfrak{X}_{x}\cong\left[W_{1}/G_{x}\right]$ (the same holds for each $i$,
$1\leq i\leq n$).

Both the projection maps
${W_{1}\times_{\mathfrak{X}_{x}}W_{1}}$${W_{1}}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$
are étale. So via the first projection map $p_{1}$, each connected component
of $W_{1}\times_{\mathfrak{X}_{x}}W_{1}$ is identified with $W_{1}$. Since

$\left(W_{1}\times_{\mathfrak{X}_{x}}W_{1}\right)\times_{\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X,x}\right)}\text{\rm
Spec}\left(k\right)\cong\text{\text@underline{\rm Isom}}(x,x)\cong G_{x},$

we have an isomorphism $W_{1}\times_{\mathfrak{X}_{x}}W_{1}\cong W_{1}\times
G_{x}$, where the composition
$W_{1}\times_{\mathfrak{X}_{x}}W_{1}\overset{\sim}{\longrightarrow}W_{1}\times
G_{x}\longrightarrow W_{1}$ is $p_{1}$. We also obtain a morphism

$\rho_{x,1}\colon W_{1}\times
G_{x}\overset{\sim}{\longrightarrow}W_{1}\times_{\mathfrak{X}_{x}}W_{1}\overset{p_{2}}{\longrightarrow}W_{1}.$

As in the proof of [7, Theorem 11.3.1], it follows that $\rho_{x,1}$ equips
$W_{1}$ with a $G_{x}$-action such that
$\mathfrak{X}_{x}\cong\left[W_{1}/G_{x}\right]$.

Since $g$ is a finite morphism of smooth curves, there is an open irreducible
smooth subset $V_{x}\subset V$ containing $W_{1}$ such that the map $g$
restricts to a finite surjective cover $g\colon V_{1}\longrightarrow
U_{x}=g(V_{1})$ of smooth connected $k$-curves, étale away from $x$,
$U_{x}\cap B=\\{x\\}$, and there a point $v_{1}\in V_{1}$ in the fiber over
$x$ with $\widehat{\mathcal{O}}_{V_{1},v_{1}}=S_{1}$.

Let $g_{x}\colon V_{x}\longrightarrow U_{x}$ be the Galois closure of the
cover $V_{1}\longrightarrow U_{x}$, with Galois group $\Gamma_{x}$, say. For
each point $v\in V_{1}$ above $x$, we have
$\widehat{\mathcal{O}}_{V_{1},v}\cong S_{i}$ for some $i$, and so
$K_{V_{1},v}\cong K_{V_{1},v_{1}}$ as $G_{x}$-Galois extensions of $K_{X,x}$.
Also since $V_{1}\longrightarrow U_{x}$ is étale away from $x$, the induced
map $V_{x}\longrightarrow V_{1}$ is a Galois étale cover. So the
$\Gamma_{x}$-Galois cover $g_{x}$ is a finite surjective cover of smooth
connected $k$-curves, étale away from $x$, and the inertia groups at the
points in the fiber $g_{x}^{-1}(x)$ are conjugate to $G_{x}$ in $\Gamma_{x}$.
Then the induced map $V_{x}\longrightarrow\mathfrak{X}\times_{X}U_{x}$ is an
étale surjective morphism,

$\left[\left(V_{x}-g_{x}^{-1}(x)\right)/\Gamma_{x}\right]\cong
U_{x}-\\{x\\}=U\cap U_{x}\cong\mathfrak{X}\times_{X}(U_{x}\cap U),$
$\left[\left(\sqcup_{\begin{subarray}{c}g_{x}(v)=x\end{subarray}}\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{V_{x},v}\right)\right)/\Gamma_{x}\right]\cong\left[\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{V_{1},v_{1}}\right)/G_{x}\right]\cong\mathfrak{X}_{x}.$

This implies

$\mathfrak{X}\times_{X}U_{x}\cong\left[V_{x}/\Gamma_{x}\right].$

∎

######  Notation 4.4.

Let $\mathfrak{X}$ be any orbifold curve. For a point $x\in B$ as in
Proposition 4.3, and the $\Gamma_{x}$-Galois cover $V_{x}\longrightarrow
U_{x}$, we have

$\widehat{\mathcal{O}}_{V_{x},v}\cong\widehat{\mathcal{O}}_{V_{x},v^{\prime}}$

for any two points $v,\,v^{\prime}\in V_{x}$ over $x$, as $G_{x}$-Galois
extensions of $\widehat{\mathcal{O}}_{X,x}$-algebras. Moreover,

$\mathfrak{X}\times_{X}\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X,x}\right)\cong\left[\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{V_{x},v}\right)/G_{x}\right]\cong\left[\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{V_{x},v^{\prime}}\right)/G_{x}\right].$

We fix a point $v\in V_{x}$ above $x$ and set

$R_{x}\coloneqq\widehat{\mathcal{O}}_{V,v}.$

To simplify the notation, to any orbifold curve $\mathfrak{X}$ with $B=X-U$ as
before, to each point $x\in B$ we associate the tuple

$(U_{x},V_{x},R_{x}).$

######  Remark 4.5.

Let $\mathfrak{X}$ be an orbifold curve. For $x\in B$ as in Proposition 4.3 we
have a tuple $(U_{x},V_{x},R_{x})$. For a different choice of
$(U^{\prime}_{x},V^{\prime}_{x},R^{\prime}_{x})$, by shrinking $U_{x}$ and
$U^{\prime}_{x}$, and taking a $\Gamma^{\prime\prime}_{x}$-Galois cover
$V^{\prime\prime}_{x}\longrightarrow U^{\prime\prime}_{x}$ dominating the
$\Gamma_{x}$ and $\Gamma^{\prime}_{x}$-Galois covers, we can find another
tuple $(U^{\prime\prime}_{x},V^{\prime\prime}_{x},R^{\prime\prime}_{x})$ such
that

$R_{x}\cong R^{\prime\prime}_{x}\cong R^{\prime}_{x}$

as $G_{x}$-Galois extensions of $\widehat{\mathcal{O}}_{X,x}$. So for
different choices of tuples which arise from the choices made in the proof of
Proposition 4.3, the complete discrete valuation domains are all
$G_{x}$-equivariantly isomorphic.

Let $\mathfrak{X}$ be a proper orbifold curve and $X$ be its Coarse moduli
scheme. In particular, $X$ is a smooth ([7, Exercise 11.H]) connected
projective $k$-curve. We construct a connected formal orbifold curve
$\lambda(\mathfrak{X})$ as follows.

Let $U\subset X$ be the maximal open subset such that
$\mathfrak{X}\times_{X}U\cong U$. Set $B\coloneqq X-U$. Then $B$ is a finite
set of closed points of $X$. The set $B$ is empty if and only if
$\mathfrak{X}\cong X$, in which case we set
$\lambda(\mathfrak{X})=(X,\mathcal{O})$ where $\mathcal{O}$ is the trivial
branch data on $X$. By Notation 4.4, we obtain a tuple $(U_{x},V_{x},R_{x})$
associated to $\mathfrak{X}$. More precisely, for each point $x\in B$, we have
an open connected subset $U_{x}\subset X$ and a $\Gamma_{x}$-Galois cover
$g_{x}\colon V_{x}\longrightarrow U_{x}$ of smooth connected $k$-curves, étale
away from $x$, and for a point $v_{x}\in V_{x}$ over $x$,
$R_{x}=\widehat{\mathcal{O}}_{V_{x},v_{x}}$ is a $G_{x}$-Galois
$\widehat{\mathcal{O}}_{X,x}$-algebra which induces an isomorphism

$\mathfrak{X}\times_{X}\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X,x}\right)\cong\left[\text{\rm
Spec}\left(R_{x}\right)/G_{x}\right].$

For any closed point $x\in X$, consider the $G_{x}$-Galois extensions of
$K_{X,x}$ given by

$P(x)=\begin{cases}QF(R_{x})=K_{V_{x},v_{x}},&\text{if }x\in B\\\
K_{X,x},&\text{if }x\in U.\end{cases}$

By Remark 4.5, the $G_{x}$-Galois extension over each $K_{X,x}$ is independent
(up to isomorphism) of the choices of $(U_{x},V_{x},R_{x})$. We define
$\lambda(\mathfrak{X})$ to be the connected formal orbifold curve $(X,P)$.
Note that $\text{\rm Supp}(P)=B$.

Now we construct $\lambda(f)$ for any finite covering map
$f\colon\mathfrak{Y}\longrightarrow\mathfrak{X}$ of proper orbifold curve (a
finite covering map is a finite representable morphism such that the morphism
induced on the Coarse moduli spaces is a finite cover). Suppose that
$\lambda(\mathfrak{Y})=(Y,Q)$ and $\lambda(\mathfrak{X})=(X,P)$. For a point
$y\in\mathfrak{Y}(k)$ with image $x\in\mathfrak{X}(k)$, let $G^{\prime}_{y}$
and $G_{x}$ denote the stabilizer groups at $y$ and at $x$, respectively. Then
$f$ induces a finite cover $f\colon Y\longrightarrow X$ of smooth projective
connected $k$-curves. Moreover, for each closed point $y\in Y$, we obtain
tuples $(U^{\prime}_{y},V^{\prime}_{y},R^{\prime}_{y})$ and
$(U_{x},V_{x},R_{x})$ with $x=f(y)$ that are compatible with the given
morphism $f$ of orbifold curves. More precisely, $f$ restricts to a finite
surjective cover $f\colon U^{\prime}_{y}\longrightarrow U_{x}$ and there is a
finite surjective cover $h\colon V^{\prime}_{y}\longrightarrow V_{x}$ making
the following diagram commute.

(4.2)
${V^{\prime}_{y}}$${V_{x}}$${U^{\prime}_{y}}$${U_{x}}$${Y}$${X}$$\scriptstyle{h}$$\scriptstyle{g^{\prime}_{y}}$$\scriptstyle{g_{x}}$$\scriptstyle{f}$$\scriptstyle{\delta^{\prime}_{y}}$$\scriptstyle{\delta_{x}}$$\scriptstyle{f}$

The above diagram induces the following commutative diagram.

${\text{\rm Spec}\left(R^{\prime}_{y}\right)}$${\text{\rm
Spec}\left(R_{x}\right)}$${\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{Y,y}\right)}$${\text{\rm
Spec}\left(\widehat{\mathcal{O}}_{X,x}\right)}$$\scriptstyle{h_{y}}$$\scriptstyle{f_{y}}$

So for every closed point $y\in Y$, we have

$Q(y)\supset P(x)\cdot K_{Y,y}\supset P(x),$

and $f\colon\mathfrak{Y}\longrightarrow\mathfrak{X}$ defines a morphism
$\lambda(f)\colon(Y,Q)\longrightarrow(X,P)$ of connected formal orbifold
curves (up to isomorphism).

The above construction produces a faithful functor $\lambda$ of categories
from the isomorphism classes of proper orbifold curves together with finite
morphisms to the category of isomorphism classes of connected formal orbifold
curves.

######  Theorem 4.6.

The functor $\lambda$ defines an equivalence of categories

$\begin{pmatrix}\text{Isomorphism classes of }\\\ \text{proper orbifold
curves}\end{pmatrix}\overset{\sim}{\longrightarrow}\begin{pmatrix}\text{Isomorphism
classes of }\\\ \text{connected projective formal orbifold
curves}\end{pmatrix}.$

We will denote the quasi-inverse of $\lambda$ by $\alpha$.

###### Proof.

Suppose that $\mathfrak{Y}$ and $\mathfrak{X}$ be two proper orbifold curves.
Let $\lambda(\mathfrak{Y})=(Y,Q)$ and $\lambda(\mathfrak{X})=(X,P)$. We first
show that

$\text{\rm Hom}_{k-\text{\rm
stacks}}\left(\mathfrak{Y},\mathfrak{X}\right)=\text{\rm
Mor}\left((Y,Q),\,(X,P)\right).$

Since $\lambda$ is faithful, we need to show that for any morphism
$f\colon(Y,Q)\longrightarrow(X,P)$, there is a finite covering morphism
$h\colon\mathfrak{Y}\longrightarrow\mathfrak{X}$ such that $\lambda(h)=f$. We
have an induced finite cover $f\colon Y\longrightarrow X$ of smooth projective
connected $k$-curves. Also $f\left(Y-\text{\rm Supp}(Q)\right)\subset
X-\text{\rm Supp}(P)$. We take $h=f$ on $Y-\text{\rm
Supp}(Q)\cong\mathfrak{Y}\times_{Y}(Y-\text{\rm Supp}(Q))$. So it is enough to
construct $h$ étale locally on $\mathfrak{Y}$ and $\mathfrak{X}$ for each
point $y\in\text{\rm Supp}(Q)$.

Let $y\in\text{\rm Supp}(Q)$, $x=f(y)$ with respective stabilizer groups
$G^{\prime}_{y}$ at $y$ and $G_{x}$ at $x$. Consider tuples
$(U^{\prime}_{y},V^{\prime}_{y},R^{\prime}_{y})$ and $(U_{x},V_{x},R_{x})$ as
in Notation 4.4. We have the inclusion $Q(y)\supset P(x)\cdot K_{Y,y}\supset
K_{X,x}$. Since $R^{\prime}_{y}$ is the integral closure of
$\widehat{\mathcal{O}}_{Y,y}$ in $Q(y)$ and $R_{x}$ is the integral closure of
$\widehat{\mathcal{O}}_{X,x}$ in $P(x)$, we obtain

$R^{\prime}_{y}\supset R_{x}$

as $\widehat{\mathcal{O}}_{X,x}$-algebras, and a morphism

$\mathfrak{Y}\times_{Y}\widehat{\mathcal{O}}_{Y,y}\cong\left[R^{\prime}_{y}/G^{\prime}_{y}\right]\overset{h_{y}}{\longrightarrow}\left[R_{x}/G_{x}\right]\cong\mathfrak{X}\times_{X}\widehat{\mathcal{O}}_{X,x}.$

By shrinking around $y$ and $x$, we assume that $U^{\prime}_{y}\longrightarrow
Y$ and $U_{x}\longrightarrow X$ are étale neighborhoods of $y$ and $x$,
respectively, and the following hold. $\Gamma^{\prime}_{y}=G^{\prime}_{y}$,
$\Gamma_{x}=G_{x}$, the morphism $f$ induces maps $f\colon
U^{\prime}_{y}\longrightarrow U_{x}$ and $h\colon
V^{\prime}_{y}\longrightarrow V_{x}$ so that the diagram (4.2) commutes, and
that the inclusion $R^{\prime}_{y}\supset R_{x}$ of
$\widehat{\mathcal{O}}_{X,x}$-algebras is obtained via the pullback of $h$ to
$\text{\rm Spec}\left(\widehat{\mathcal{O}}_{X,x}\right)$. This produces our
required étale local morphism

$\mathfrak{Y}\times_{Y}U^{\prime}_{y}\cong\left[V^{\prime}_{y}/G^{\prime}_{y}\right]\overset{\alpha(f)}{\longrightarrow}\left[V_{x}/G_{x}\right]\cong\mathfrak{X}\times_{X}U_{x}.$

Now we show that $\lambda$ is essentially surjective. Let $(X,P)$ be a
connected formal orbifold curve. For any closed point $x\in X$, set
$G_{x}\coloneqq\text{\rm Gal}\left(P(x)/K_{X,x}\right)$. Let $B$ be the
support of the branch data $P$, and set $U=X-B$. Let $x\in B$. Identifying the
complete local ring $\widehat{\mathcal{O}}_{X,x}$ at $x$ with
$\widehat{\mathcal{O}}_{\mathbb{P}^{1},\infty}$, we have a $G_{x}$-Galois
field extension $P(x)\supset K_{\mathbb{P}^{1},\infty}$. Let $\phi_{x}\colon
W_{x}\longrightarrow\mathbb{P}^{1}$ be the Harbater-Katz-Gabber cover (the
Kummer cover if $\text{char}(k)=0$; [4, Theorem 1.4.1]) corresponding to the
extension $P(x)/K_{\mathbb{P}^{1},\infty}$. So $\phi_{x}$ is a $G_{x}$-Galois
cover of smooth projective connected $k$-curves, étale away from $\infty$,
tamely ramified over $0$, and over the unique point $w_{x}\in W_{x}$ over
$\infty$, we have

$K_{W_{x},w_{x}}\cong P(x)$

as $G_{x}$-Galois extensions of $K_{\mathbb{P}^{1},\infty}$. By the Noether
Normalization Lemma ([3, Corollary 16.18]), there exists a finite cover
$g_{x}\colon X\longrightarrow\mathbb{P}^{1}$ of smooth connected projective
$k$-curves that is étale at $x$. Consider the component $V_{x}$ in the
normalization of $X\times_{g_{x},\mathbb{P}^{1},\phi_{x}}W_{x}$. So we obtain
a Galois cover $\theta_{x}\colon V_{x}\longrightarrow X$ with the Galois group
contained in $G_{x}$. Since $g_{x}$ is étale at the point $x$, the cover
$\theta_{x}$ is $G_{x}$-Galois, and at the unique point $v_{x}\in V_{x}$ over
$x$, we have $K_{V_{x},v_{x}}\cong P(x)$ as $G_{x}$-Galois extensions of
$K_{X,x}$. Consider the open subset $U_{x}\subset X$ by removing the branched
points of $\theta_{x}$, except for the point $x$, i.e.
$U_{x}=X-\text{BL}(\theta_{x})\sqcup\\{x\\}$. Set
$Z_{x}\coloneqq\theta_{x}^{-1}(U_{x})$.

So for each $x\in B$, we have constructed an open subset $U_{x}\subset X$ such
that $x^{\prime}\not\in U_{x}$ for $x^{\prime}\in B-\\{x\\}$, together with a
$G_{x}$-Galois cover $V_{x}\longrightarrow U_{x}$ of smooth connected
$k$-curves, étale away from $x$, and $P(x)/K_{X,x}$ occurs as the local Galois
extension near $x$. We obtain a stack $\mathfrak{X}$ by gluing the quotients
stacks $[Z_{x}/G_{x}]$, $x\in B$, with the curve $U$. From the above
construction, $\lambda(\mathfrak{X})$ is isomorphic to the formal orbifold
curve $(X,P)$. ∎

######  Remark 4.7.

We remark some important properties of the functor $\lambda$ which will be
used frequently for the rest of this section. For any proper orbifold curve
$\mathfrak{X}$ with Coarse moduli $k$-curve $X$ and
$\lambda(\mathfrak{X})=(X,P)$, we have the following.

  1. (1)

The support $\text{\rm Supp}(P)$ of the branch data is given by $X-U$, where
$U$ is the maximal open subset of $X$ satisfying $\mathfrak{X}\times_{X}U$.

  2. (2)

For any closed point $x\in X$, we have

$G_{x}=\text{\rm Gal}\left(P(x)/K_{X,x}\right)$

where $G_{x}$ is the stabilizer group at $x$. By (1), $x\in X(k)$ corresponds
to a stacky point in $\mathfrak{X}(k)$ if and only if $G_{x}$ is a non-trivial
group if and only if $x\in\text{\rm Supp}(P)$.

  3. (3)

For any closed point $x\in X$, let $R_{x}$ denote the integral closure of
$\widehat{\mathscr{O}}_{X,x}$ in $P(x)$. Then we have the isomorphism

$\mathfrak{X}\times_{X}\text{\rm
Spec}\left(\widehat{\mathscr{O}}_{X,x}\right)\cong\left[\text{\rm
Spec}\left(R_{x}\right)/G_{x}\right].$

Under the above equivalence, we can view a connected formal orbifold curve as
a proper orbifold curve. We have a well developed theory of quasi-coherent
sheaves on an orbifold curve. We will show that the vector bundles on a proper
orbifold curves $\mathfrak{X}$ are the same as the vector bundles on
$(X,P)=\lambda(\mathfrak{X})$ as defined in Definition 3.10. We start by
recalling the definition of a quasi-coherent sheaf on an orbifold curve (as a
smooth Deligne-Mumford stack).

######  Definition 4.8 ([9, Definition 7.18]).

Let $\mathfrak{X}$ be an orbifold $k$-curve (see Definition 4.2). A quasi-
coherent sheaf $E$ on $\mathfrak{X}$ is defined by the following set of data.

  1. (1)

For each atlas $Z\longrightarrow\mathfrak{X}$, a quasi-coherent sheaf $E_{Z}$
on the smooth scheme $Z$.

  2. (2)

For atlases $Z_{1}\longrightarrow\mathfrak{X}$ and
$Z_{2}\longrightarrow\mathfrak{X}$, and each commutative diagram

${Z_{1}}$${Z_{2}}$${\mathfrak{X}}$$\scriptstyle{\phi}$

of stacks, an isomorphism $\rho_{\phi}\colon
E_{Z_{1}}\overset{\sim}{\longrightarrow}\phi^{*}E_{Z_{2}}$.

  3. (3)

(cocyle condition) For any three atlases $Z_{1}$, $Z_{2}$, $Z_{3}$ of
$\mathfrak{X}$, and any commutative diagram

${Z_{1}}$${Z_{2}}$${Z_{3}}$${\mathfrak{X}}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$

of stacks, we have

$\alpha_{\psi\circ\phi}=\alpha_{\phi}\circ\phi^{*}\alpha_{\psi}\colon
E_{Z_{1}}\overset{\sim}{\longrightarrow}(\psi\circ\phi)^{*}E_{Z_{3}}=\phi^{*}(\psi^{*}E_{Z_{3}}).$

A coherent sheaf on $\mathfrak{X}$ is a quasi-coherent sheaf $E$ on
$\mathfrak{X}$ such that for any atlas $Z\longrightarrow\mathfrak{X}$, the
sheaf $E_{Z}$ is a coherent sheaf on $Z$. If moreover, each $E_{Z}$ is a
vector bundle on $Z$, we say that $E$ is a vector bundle on $\mathfrak{X}$.

The structure sheaf $\mathcal{O}_{\mathfrak{X}}$ on an orbifold curve
$\mathfrak{X}$ is the quasi-coherent sheaf defined by associating the
structure sheaf $\mathcal{O}_{Z}$ for every atlas $Z$ of $\mathfrak{X}$. It
should be noted that a quasi-coherent sheaf in the above sense is actually a
quasi-coherent sheaf of $\mathcal{O}_{\mathfrak{X}}$-modules as in [7,
Definition 9.1.14, Proposition 9.1.15].

######  Theorem 4.9.

Let $\mathfrak{X}$ be a proper orbifold curve. Consider the formal orbifold
curve $(X,P)=\lambda(\mathfrak{X})$, where $\lambda$ is the categorical
equivalence defined in Theorem 4.6. We have an equivalence of categories

$\lambda_{\text{\rm Vect},\mathfrak{X}}\colon\text{\rm
Vect}(\mathfrak{X})\overset{\sim}{\longrightarrow}\text{\rm Vect}(X,P)$

with a quasi-inverse $\alpha_{\text{\rm Vect},\mathfrak{X}}$.

###### Proof.

The smooth projective connected $k$-curve $X$ is the Coarse moduli space for
$\mathfrak{X}$. Let $U\subset X$ be maximal open such that
$\mathfrak{X}\times_{X}U\cong U$. Set $B=X-U$. By Remark 4.7, $B=\text{\rm
Supp}(P)$, and for $x\in B$, the group $\text{\rm
Gal}\left(P(x)/K_{X,x}\right)$ is the stabilizer group $G_{x}$ at $x$, and we
have

(4.3) $\mathfrak{X}\times_{X}\widehat{\mathcal{O}}_{X,x}\cong\left[\text{\rm
Spec}\left(R_{x}\right)/G_{x}\right]$

where $R_{x}$ is the integral closure of $\widehat{\mathcal{O}}_{X,x}$ in
$P(x)$.

Let $E\in\text{\rm Vect}(\mathfrak{X})$. Then $E$ restricts to a vector bundle
$E|_{U}$ on $U$. Let $x\in B$. By Notation 4.4, We have the tuple
$(U_{x},V_{x},R_{x})$ where $U_{x}\longrightarrow X$ is an open connected
subset, $U\cap U_{x}=\\{x\\}$, $V_{x}\longrightarrow U_{x}$ is a finite
surjective $\Gamma_{x}$-Galois cover that is étale away from $x$, and
$\mathfrak{X}\times_{X}U_{x}\cong\left[V_{x}/\Gamma_{x}\right]$ which in turn
induces the isomorphism in Equation (4.3) above. So $E$ restricts to a vector
bundle $E^{(x)}$ on the quotient sub-stack $\mathfrak{X}\times_{X}U_{x}$.

By our construction of $\lambda$, we have

$\lambda\left(\left[V_{x}/\Gamma_{x}\right]\right)=(U_{x},P|_{U_{x}})$

where $P|_{U_{x}}$ is the restriction of branch data on the connected curve
$U_{x}$. We have the following equivalences of categories

$\text{\rm
Vect}\left(\left[V_{x}/\Gamma_{x}\right]\right)\overset{\sim}{\longrightarrow}\text{\rm
Vect}_{\Gamma_{x}}(V_{x})\overset{\sim}{\longrightarrow}\text{\rm
Vect}(U_{x},P|_{U_{x}}),$

where the first equivalence is [9, Example 7.21] and the second one is [6,
Theorem 4.12]333Although the statement is for smooth projective curves,
projectivity is not used in the proof of the result.. So the vector bundle
$E^{(x)}$ on $\mathfrak{X}\times_{X}U_{x}$ under the above equivalence
corresponds to a bundle $E^{(x)}_{*}\in\text{\rm Vect}(U_{x},P|_{U_{x}})$. Let
$E_{0}^{(x)}$ be the underlying vector bundle on $U_{x}$. Also notice that
since $g_{x}$ is étale over $U_{x}-\\{x\\}$, the restriction of $E^{(x)}_{0}$
to $U_{x}-\\{x\\}=U_{x}\cap U$ is isomorphic to the restriction of $E|_{U}$ to
$U_{x}\cap U$. We obtain a bundle $E_{0}$ on X via gluing. Set
$\lambda_{\text{\rm Vect}}(E)$ to be the bundle $E_{0}$ together with the
action and equivariant isomorphism maps obtained from $E^{(x)}_{*}$.

We note that $\lambda_{\text{\rm Vect}}$ is functorial and up to an
isomorphism of the vector bundles in $\text{\rm Vect}(X,P)$, it is independent
of the choice of atlases for the open sub-stacks $\mathfrak{X}\times_{X}U$.

To construct the quasi-inverse functor $\alpha_{\text{\rm Vect}}$, start with
a bundle $E_{*}\in\text{\rm Vect}(X,P)$. Let $E_{0}$ be the underlying vector
bundle on $X$. Via restriction, for each $x\in B$, we obtain the vector bundle
$E^{(x)}_{*}\in\text{\rm Vect}(U_{x},P|_{U_{x}})$. Again using the previous
equivalences, we obtain a vector bundle $E_{U_{x}}$ on the open sub-stack
$\mathfrak{X}\times_{X}U_{x}$. Then the restriction of $E_{U_{x}}$ to the
curve $U\cap U_{x}$ is isomorphic to $E_{0}|_{U\cap U_{x}}$. Now we glue
together the bundles $E_{U_{x}}$ to obtain $\alpha_{\text{\rm Vect}}(E_{*})$.
The construction is again functorial and independent of the choices on local
atlases, up to an isomorphism. ∎

######  Remark 4.10.

  1. (1)

When $\mathfrak{X}=[Z/G]$ corresponding to a $G$-Galois cover $g\colon
Z\longrightarrow X$ of smooth projective connected $k$-curves, we have
$\lambda([Z/G])=(X,B_{g})$ and the equivalence $\lambda_{\text{\rm
Vect},\mathfrak{X}}$ is the equivalence

$\text{\rm Vect}\left([Z/G]\right)\overset{\sim}{\longrightarrow}\text{\rm
Vect}_{G}(Z)\overset{g_{*}}{\longrightarrow}\text{\rm Vect}(X,B_{g}).$

  2. (2)

Under $\lambda_{\text{\rm Vect},\mathfrak{X}}$, the trivial vector bundle
$\mathcal{O}_{\lambda(\mathfrak{X})}$ on $\lambda(\mathfrak{X})$ corresponds
to the structure sheaf $\mathcal{O}_{\mathfrak{X}}$.

######  Remark 4.11.

For any smooth curves $X$ (not necessarily connected or projective) We can
define a quasi-coherent sheaf on a formal orbifold curve $(X,P)$ and morphisms
of quasi-coherent sheaves similar to Definition 3.10 by replacing the
underlying vector bundle by a quasi-coherent sheaf of
$\mathcal{O}_{X}$-modules. One can see that the equivalence [6, Theorem 4.12]
holds. As in Theorem 4.9, we can then establish that there is an equivalence
of categories

$\text{\rm Qcoh}(\mathfrak{X})\overset{\sim}{\longrightarrow}\text{\rm
Qcoh}(\lambda(X,P)),$

for any proper orbifold curve $\mathfrak{X}$, where $\text{\rm Qcoh}(X,P)$
denote the category of quasi-coherent sheaves on an orbifold curve $(X,P)$,
and $\text{\rm Qcoh}(\mathfrak{X})$ is the category of the quasi-coherent
sheaves on $\mathfrak{X}$. Moreover, the equivalence restricts to the
equivalence of Theorem 4.9.

##  5\. Stability conditions for vector bundles on formal orbifold curves

###  5.1. Degree of a line bundle on an orbifold curve

Let $\mathfrak{X}$ be a proper orbifold curve with coarse moduli space $X$. We
make the following definitions with an analogy as in scheme theory.

  * •

The (Weil) divisor group of $\mathfrak{X}$, denoted by
$\text{Div}\left(\mathfrak{X}\right)$, is the free abelian group generated by
the closed points of $\mathfrak{X}$.

  * •

A principal (Weil) divisor on $\mathfrak{X}$ is a divisor div$(f)$ associated
to a rational section $f$ of $\mathcal{O}_{\mathfrak{X}}$ given by

div$(f)\coloneqq\sum\limits_{x}\nu_{x}(f)x,$

where $\nu_{x}(f)$ is the valuation of $f$ in the local ring
$\mathcal{O}_{X,x}$. Two divisors $D,D^{\prime}\in\text{\rm Div}\mathfrak{X}$
are said to be linearly equivalent, denoted by $D\sim D^{\prime}$ if
$D=D^{\prime}+$ div$(f)$ for some rational section
$f:\mathfrak{X}\longrightarrow\mathbb{P}^{1}_{k}$ of $\mathfrak{X}$.

  * •

The degree of a divisor $D=\sum\limits_{x}n_{x}x\in\text{\rm
Div}\mathfrak{X}$, denoted by $\text{deg}(D)$ is the sum

$\text{deg}(D)=\sum\limits_{x}n_{x}\text{deg}(x)=\sum\limits_{x}n_{x}\frac{1}{\lvert
G_{x}\rvert},$

where $G_{x}$ is the stabilizer at $x$.

  * •

For a closed point $x\in\mathfrak{X}$, we denote the ideal sheaf at $x$ by
$\mathcal{O}_{\mathfrak{X}}(-x)$. Defining as usual

$\mathcal{O}_{\mathfrak{X}}\left(x\right)=\mathcal{O}_{\mathfrak{X}}\left(-x\right)^{\vee}=\operatorname*{Hom}\left(\mathcal{O}_{\mathfrak{X}}\left(-x\right),\mathcal{O}_{\mathfrak{X}}\right),$

and then extending this definition linearly to any (Weil) divisor on
$\mathfrak{X}$, we obtain a line bundle $\mathcal{O}_{\mathfrak{X}}(D)$
associated to a (Weil) divisor $D$.

######  Lemma 5.1.

[10, Remark 5.4.4] Let $\mathfrak{X}$ be a proper orbifold curve over $k$ with
Coarse space $\pi\colon\mathfrak{X}\longrightarrow X$. Then for any non-
constant map $f\colon\mathfrak{X}\longrightarrow\mathbb{P}_{k}^{1}$,
div$(f)=\pi^{*}$div$(f^{\prime})$, where
$f^{\prime}:X\longrightarrow\mathbb{P}^{1}_{k}$ is the unique map making the
following diagram commutative :

${X}$${\mathfrak{X}}$${\mathbb{P}^{1}_{k}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{f}$

######  Lemma 5.2.

[10, Lemma 5.4.5] Every line bundle $L$ on $\mathfrak{X}$ is of the form
$\mathcal{O}_{\mathfrak{X}}(D)$ for some divisor
$D\in\text{Div}(\mathfrak{X})$. Moreover,
$\mathcal{O}_{\mathfrak{X}}(D)\cong\mathcal{O}_{\mathfrak{X}}(D^{\prime})$ if
and only if $D\sim D^{\prime}$ in $\text{\rm Div}(\mathfrak{X})$.

######  Definition 5.3.

Let $L$ be a line bundle on a proper orbifold curve $\mathfrak{X}$. Then
$L\cong\mathcal{O}_{\mathfrak{X}}(D)$ where $D$ is a divisor on
$\mathfrak{X}$, up to a numerical equivalence (Lemma 5.2). Define the degree
of $L$ to be

$\text{\rm deg}(L)\coloneqq\text{\rm deg}(D).$

###  5.2. Degree, slope and (semi)stability

In this section, we introduce the notion of stability and semistability for
any vector bundle on a formal orbifold curve. For this, we will use the
equivalence of categories from Theorem 4.6 and Theorem 4.9 as well as the
notion of degree of a line bundle on an orbifold curve (Definition 5.3).
Throughout the section, any formal orbifold curve will be assumed to be
connected and projective.

Let $(X,P)$ be a connected projective formal orbifold curve such that
$(X,P)=\lambda(\mathfrak{X})$ for a proper orbifold curve $\mathfrak{X}$,
where $\lambda$ is the categorical equivalence as in Theorem 4.6. We will now
define the degree of a vector bundle in $\text{\rm Vect}(X,P)$ using the
equivalence of categories in Theorem 4.9.

Let $E_{*}=(E,\Phi,\eta)\in\text{\rm Vect}(X,P)$ be a vector bundle on
$(X,P)$. Then the rank of the bundle $E_{*}$ is defined to be the rank of the
underlying vector bundle $E$ on $X$. Let $n$ be the rank of $E_{*}$. Then we
define its determinant bundle

$\text{\rm
det}\left(E_{*}\right)\coloneqq(\wedge^{n}E,\wedge^{n}\Phi,\wedge^{n}\eta)\in\text{\rm
Vect}(X,P).$

Since $\wedge$ commutes with $\otimes$, we have a line bundle $\text{\rm
det}\left(E_{*}\right)\in\text{\rm Vect}(X,P)$ whose underlying vector bundle
is the usual determinant line bundle $\text{\rm det}(E)$ on $X$.

We define the degree of $E_{*}$ with respect to $P$ to be

$\text{\rm deg}_{P}(E_{*})\coloneqq\text{\rm deg}_{P}\left(\text{\rm
det}\left(E_{*}\right)\right)\coloneqq\text{\rm deg}(\alpha_{\text{\rm
Vect},\mathfrak{X}}\left(\text{\rm det}\left(E_{*}\right)\right),$

where $\alpha_{\text{\rm Vect},\mathfrak{X}}$ is the functor defined in
Theorem 4.9 and $\text{\rm deg}(\alpha_{\text{\rm
Vect},\mathfrak{X}}\left(\text{\rm det}\left(E_{*}\right)\right)$ is defined
in Definition 5.3.

The $P$-slope of $E_{*}$, denoted by $\mu_{P}(E_{*})$, is the fraction

$\mu_{P}(E_{*})=\frac{\text{deg}_{P}(E_{*})}{\text{rank}(E_{*})}.$

######  Definition 5.4.

A vector bundle $E_{*}\in\text{\rm Vect}(X,P)$ is called $P$-(semi)stable if
for all its sub-bundle $0\neq F_{*}\subset E_{*}$ in $\text{\rm Vect}(X,P)$,
we have

$\mu_{P}(F_{*})\hskip 5.69046pt(\leq)\hskip 5.69046pt\mu_{P}(E_{*}).\,$

A $P$-polystable bundle on $(X,P)$ bundle that is a finite sum of $P$-stable
vector bundles in $\text{\rm Vect}(X,P)$ having the same $P$-slope.

In the following two lemmas we see how the degrees with respect to branch data
behave for the pullback under a morphism of formal orbifold curves.

######  Lemma 5.5.

Let $P^{\prime}\geq P$ be two branch data on a smooth projective connected
$k$-curve $X$. The identity map on $X$ induces
$\iota\colon(X,P^{\prime})\longrightarrow(X,P)$. Consider the embedding
$\iota^{*}:\text{\rm Vect}(X,P)\longrightarrow\text{\rm Vect}(X,P^{\prime})$.
Then $\text{\rm deg}_{P^{\prime}}(\iota^{*}E_{*})=\text{\rm deg}_{P}(E_{*})$
for any $E_{*}\in\text{\rm Vect}(X,P)$.

###### Proof.

Let $\mathfrak{X}$ and $\mathfrak{X}^{\prime}$ be the orbifold curves such
that $\lambda(\mathfrak{X})=(X,P)$ and
$\lambda(\mathfrak{X}^{\prime})=(X,P^{\prime})$. From the definition, it
follows that for any $F_{*}\in\text{\rm Vect}(X,P)$, we have

$\iota^{*}\text{\rm det}(F_{*})=\text{\rm det}(\iota^{*}E_{*}).$

We defined the $P$-degree of any bundle $F_{*}\in\text{\rm Vect}(X,P)$ as the
degree of the line bundle $\alpha_{\text{\rm Vect}}(\text{\rm det}(F_{*}))$ on
$\mathfrak{X}$, which in turn is defined to be the degree of a (Weil) divisor
associated to it. So it is enough to show that for any closed point
$x\in\mathfrak{X}$, the degree of the point $x$ considered as a divisor on
$\mathfrak{X}$ and the divisor $\alpha(\iota)^{*}x$ on $\mathfrak{X}^{\prime}$
have the same degree. Note that

$\alpha(\iota)^{*}x=[P^{\prime}(x):P(x)]x.$

Now note that $\text{\rm
deg}_{\mathfrak{X}^{\prime}}(\alpha(\iota)^{*}x)=\frac{\left[P^{\prime}(x):P(x)\right]}{\left[P^{\prime}(x):K_{X,x}\right]}=\frac{\left[P^{\prime}(x):P(x)\right]}{\left[P^{\prime}(x):P(x)\right]\left[P(x):K_{X,x}\right]}=\text{\rm
deg}_{\mathfrak{X}}(x)$. ∎

######  Lemma 5.6.

Let $f\colon(Y,Q)\longrightarrow(X,P)$ be a morphism of formal orbifold
curves. Then for any $E_{*}\in\text{\rm Vect}(X,P)$, we have the following.

$\text{\rm deg}_{Q}(f^{*}E_{*})=\text{\rm deg}(f)\hskip 2.84526pt\text{\rm
deg}_{P}(E_{*}).$

###### Proof.

The map $f$ factors as
$(Y,Q)\overset{\jmath}{\longrightarrow}(Y,f^{*}P)\overset{f}{\longrightarrow}(X,P).$
By Lemma 5.5, we have $\text{\rm deg}_{Q}(\jmath^{*}f^{*}E_{*})=\text{\rm
deg}_{f^{*}P}(f^{*}E_{*})$. So it is enough to show that

$\text{\rm deg}_{f^{*}P}(f^{*}E_{*})=\text{\rm deg}(f)\hskip
2.84526pt\text{\rm deg}_{P}(E_{*}).$

Let $\alpha(X,P)=\mathfrak{X}$ and $\alpha(Y,f^{*}P)=\mathfrak{X}^{\prime}$ be
the orbifold curves as in Theorem 4.6. Using the definition, it is again
enough to show that for any closed point $x\in\mathfrak{X}$, the degree of the
divisor $\alpha(f)^{*}x$ in $\mathfrak{X}^{\prime}$ is equal to the degree of
the point $x\in\mathfrak{X}$ considered as a divisor, multiplied by $\text{\rm
deg}(f)$. We have

$\alpha(f)^{*}(x)=\sum\limits_{f(y)=x}[f^{*}P(y):P(x)]y.$

So

$\text{\rm
deg}_{\mathfrak{X}^{\prime}}(\alpha(f)^{*}x)=\sum\limits_{f(y)=x}\frac{\left[f^{*}P(y)\hskip
2.84526pt:\hskip 2.84526ptP(x)\right]}{\left[P(x)\cdot K_{Y,y}\hskip
2.84526pt:\hskip
2.84526ptK_{Y,y}\right]}=\sum\limits_{f(y)=x}\frac{\left[P(x)\cdot
K_{Y,y}\hskip 2.84526pt:\hskip 2.84526ptP(x)\right]}{\left[P(x)\cdot
K_{Y,y}\hskip 2.84526pt:\hskip 2.84526ptK_{X,x}\right]}\frac{\left[P(x)\cdot
K_{Y,y}\hskip 2.84526pt:\hskip
2.84526ptK_{X,x}\right]}{\left[P(x)K_{Y,y}\hskip 2.84526pt:\hskip
2.84526ptK_{Y,y}\right]}.$

$=\sum\limits_{f(y)=x}\frac{\left[K_{Y,y}\hskip 2.84526pt:\hskip
2.84526ptK_{X,x}\right]}{\left[P(x)\hskip 2.84526pt:\hskip
2.84526ptK_{X,x}\right]}=\text{\rm deg}(f)\hskip 2.84526pt\text{\rm
deg}_{\mathfrak{X}}(x).$ This completes the proof. ∎

As an immediate consequence of the above results, the following shows that
under the equivalence of vector bundles on a geometric formal orbifold curve
with the equivariant vector bundles, the above defined stability is the same
as the equivariant stability.

######  Corollary 5.7.

Let $g\colon Z\longrightarrow(X,P)$ be a $G$-Galois étale map of connected
projective formal orbifold curves. Then a vector bundle $E_{*}\in\text{\rm
Vect}(X,P)$ is $P$-(semi)stable (respectively, $P$-polystable) if and only if
$g^{*}E_{*}\in\text{\rm Vect}_{G}(Z)$ is $G$-(semi)stable (respectively,
$G$-polystable).

###### Proof.

First suppose that $E_{*}\in\text{\rm Vect}(X,P)$ is $P$-(semi)stable. Suppose
that $W\subseteq V\coloneqq g^{*}E_{*}$ be a $G$-equivariant sub-bundle in
$\text{\rm Vect}_{G}(Z)$, $0<\text{\rm rank}(W)<\text{\rm rank}(V)=\text{\rm
rank}(E_{*})$. Then we have a sub-bundle $S^{Z}_{(X,P)}(W)=g_{*}W\subseteq
E_{*}$, $0<\text{\rm rank}(g_{*}W)<\text{\rm rank}(E_{*})$. By Lemma 5.6, we
have

$\mu(W)=|G|\mu_{P}(g_{*}W)\hskip 28.45274pt\text{and}\hskip
28.45274pt\mu(V)=|G|\mu_{P}(E_{*}).$

Since $E_{*}$ is $P$-(semi)stable, we have $\mu_{P}(g_{*}W)\hskip
5.69046pt(\leq)\hskip 5.69046pt\mu_{P}(E_{*})$. This implies $\mu(W)\hskip
5.69046pt(\leq)\hskip 5.69046pt\mu(V)$. So $V=g^{*}E_{*}$ is also
$G$-(semistable).

To see that converse, suppose that $V=g^{*}E_{*}$ is a $G$-(semi)stable bundle
on $Z$. Let $F_{*}\subseteq g_{*}V=E_{*}$ be a sub-bundle, $0<\text{\rm
rank}(F_{*})<\text{\rm rank}(E_{*})$. Then we obtain a $G$-equivariant sub-
bundle $g^{*}F_{*}\subseteq V$. Using the same slope condition as above, we
again conclude that $\mu_{P}(F_{*})\hskip 5.69046pt(\leq)\hskip
5.69046pt\mu_{P}(E_{*})$. This shows that $E_{*}$ is $P$-(semi)stable.

The statement about polystability follows the same way by observing that
$\oplus$ is preserved under the equivalence $g^{*}$ (with quasi-inverse
$g_{*}$). ∎

In view of the above, for a geometric orbifold curve, we define the Harder-
Narasimhan filtration and the socle as follows.

######  Definition 5.8.

Let $(X,P)$ be a geometric formal orbifold curve. Let $g\colon
Z\longrightarrow(X,P)$ be a $G$-Galois étale cover for a finite group $G$. We
have an equivalence of categories $g^{*}=T^{Z}_{(X,P)}\colon\text{\rm
Vect}(X,P)\longrightarrow\text{\rm Vect}_{G}(Z)$ with a quasi-inverse
$g_{*}=S^{Z}_{(X,P)}$.555Here the maps $g^{*}$ and $g_{*}$ are in the sense of
formal orbifold curves.

For any vector bundle $E_{*}\in\text{\rm Vect}(X,P)$, the $P$-Harder-
Narasimhan filtration of $E_{*}$ is defined to be the image of the Harder
Narasimhan filtration of $T^{Y}_{(X,P)}(E_{*})$ under the functor
$S^{Z}_{(X,P)}$. We also define

$\mu_{P,max}(E_{*})\coloneqq\mu_{P}\left(g_{*}(g^{*}E_{*})_{1}\right),$

where $(g^{*}E_{*})_{1}$ is the unique $G$-equivariant maximal destabilizing
sub-bundle of $g^{*}E_{*}$.

Since the Harder-Narasimhan filtration of a vector bundle on $Z$ and the socle
of a $G$-semistable vector bundle on $Z$ are both invariant under any
automorphism of $G$, the filtration and the socle are $G$-equivariant, and the
definition make sense. Note that a vector bundle $E_{*}\in\text{\rm
Vect}(X,P)$ is $P$-semistable if and only if
$E_{*}=g_{*}\left(g^{*}E_{*}\right)_{1}$, and this is equivalent to
$\mu_{P}(E_{*})=\mu_{P,max}(E_{*})$. Similarly, for a $P$-semistable vector
bundle $E_{*}\in\text{\rm Vect}(X,P)$, the bundle $E_{*}$ is $P$-polystable if
and only if the $P$-socle of $E_{*}$ is the same as $E_{*}$.

The following result shows that under a morphism of formal orbifolds induced
by the identity map of the underlying curve, the stability conditions are
always preserved. When dealing with morphisms of formal orbifold curves, it is
always easier to assume that the formal orbifold curves are geometric. On the
other hand, we know that for any formal orbifold curve $(X,P)$, there is a
geometric branch data $P^{\prime}$ on $X$ such that $P^{\prime}\geq P$. This
observation together with the following result helps us to translate a problem
into one only involving geometric formal orbifold curves.

######  Lemma 5.9.

Let $X$ be a smooth projective connected $k$-curves. Let $P^{\prime}\geq P$ be
two branch data on $X$. Consider the embedding $\iota^{*}\colon\text{\rm
Vect}(X,P)\longrightarrow\text{\rm Vect}(X,P^{\prime})$ induced by the
identity map $\text{\rm Id}_{X}$. Then a vector bundle $E_{*}\in\text{\rm
Vect}(X,P)$ is $P$-(semi)stable (respectively, $P$-polystable) if and only if
$\iota^{*}(E_{*})$ is $P^{\prime}$-(semi)stable (respectively,
$P^{\prime}$-polystable).

###### Proof.

Suppose that $\iota^{*}E_{*}\in\text{\rm Vect}(X,P^{\prime})$ is
$P^{\prime}$-(semi)stable for some vector bundle $E_{*}\in\text{\rm
Vect}(X,P)$. Consider a nonzero proper sub-bundle $F_{*}\subset E_{*}$. Then
$i^{*}F_{*}\subset i^{*}E_{*}$ is a nonzero proper sub-bundle of $i^{*}E_{*}$.
Since $i^{*}E_{*}$ is $P^{\prime}$-(semi)stable, by Lemma 5.5,

$\mu_{P}(F_{*})=\mu_{P^{\prime}}(i^{*}F_{*})\hskip 5.69046pt(\leq)\hskip
5.69046pt\mu_{P^{\prime}}(i^{*}E_{*})=\mu_{P}(E_{*}).$

This shows that $E_{*}$ is $P$-(semi)stable.

Conversely, let $E_{*}$ is $P$-(semi)stable. We will show that $i^{*}E_{*}$ is
$P^{\prime}$-(semi)stable. Let $F^{\prime}_{*}\subset i^{*}E_{*}$ be a nonzero
proper sub-bundle of $i^{*}E_{*}$. Then there exists a nonzero proper sub-
bundle $F_{*}\subset E_{*}$ such that $i^{*}F_{*}=F^{\prime}_{*}$ (see Remark
3.14). Thus

$\mu_{P}(F_{*})=\mu_{P^{\prime}}(F^{\prime}_{*})\hskip 5.69046pt(\leq)\hskip
5.69046pt\mu_{P^{\prime}}(i^{*}E_{*})=\mu_{P}(E_{*}).$

This shows that $i^{*}E_{*}$ is $P^{\prime}$-(semi)stable.

Note that $\iota^{*}\left(\oplus_{\begin{subarray}{c}1\leq i\leq
n\end{subarray}}E_{i,*}\right)=\oplus_{\begin{subarray}{c}1\leq i\leq
n\end{subarray}}\iota^{*}E_{i,*}$ for $E_{i,*}\in\text{\rm Vect}(X,P)$. So for
a vector bundle $E_{*}=\oplus_{\begin{subarray}{c}1\leq i\leq
n\end{subarray}}E_{i,*}\in\text{\rm Vect}(X,P)$, $E_{*}$ is $P$-polystable if
and only if $\iota^{*}E_{*}$ is a $P^{\prime}$-semistable which is the a
direct sum of $P^{\prime}$-stable bundles $\iota^{*}(E_{i,*})$, i.e.
$\iota^{*}E_{*}$ is $P^{\prime}$-polystable. ∎

The above lemma together with Lemma 5.5 and Lemma 5.6 have the following
consequence when $\text{\rm char}(k)=0$. In particular, the slopes $\mu_{P}$
in this case are the same as parabolic slopes, and the stability conditions
are the same as parabolic stability conditions.

######  Remark 5.10.

Suppose that $\text{\rm char}(k)=0$. Let $(X,P)$ be a connected projective
formal orbifold curve, determined by a finite set $B\subset X$ of closed
points and a positive integers $n_{x}$ for each $x\in B$. Let
$D=\sum_{\begin{subarray}{c}x\in B\end{subarray}}x\in\text{\rm Div}(X)$. As
noted in Example 3.12, there is an equivalence of categories

(5.1) $\text{\rm Vect}(X,P)\overset{\sim}{\longrightarrow}\text{\rm
Vect}_{\text{\rm par, rat}}\text{\rm Vect}(X,D)$

where $\text{\rm Vect}_{\text{\rm par, rat}}\text{\rm Vect}(X,D)$ is the
category of parabolic vector bundles on $X$ with respect to the divisor $D$
and over each $x\in B$, the weights are of the form $a/n_{x}$, $0\leq
a<n_{x}$.

There exists a connected $G$-Galois cover $g\colon Z\longrightarrow X$ of
smooth projective connected $k$-curves that is branched over the set $B$, and
for each point $x\in B$, the integer $n_{x}$ divides the ramification index at
any point $z\in g^{-1}(x)$. In other words, $B_{g}\geq P$ and $g$ induces
$Z\overset{g^{\prime}}{\longrightarrow}(X,B_{g})\overset{\iota}{\longrightarrow}(X,P),$
where $g^{\prime}$ is a $G$-Galois étale map and $\iota$ is induced from
$\text{\rm Id}_{X}$. By [1], [8], for each parabolic vector bundle
$V_{*}\in\text{\rm Vect}_{\text{\rm par, rat}}\text{\rm Vect}(X,D)$, there is
a unique $G$-bundle $\hat{V}\in\text{\rm Vect}_{G}(Z)$, and

$\mu(\hat{V})=|G|\mu_{\text{\rm para}}(V_{*}).$

Moreover, $V$ is $G$-(semi)stable (respectively, $G$-polystable) if and only
if $V_{*}$ is parabolic (semi)stable (respectively, parabolic polystable).

Let $E_{*}\in\text{\rm Vect}(X,P)$ be the vector bundle corresponding to
$V_{*}$. By Lemma 5.6 and Lemma 5.5, we have

$|G|\mu_{P}(E_{*})=|G|\mu_{B_{g}}(\iota^{*}E_{*})=\mu(g^{*}E_{*}).$

Moreover, by Lemma 5.9, $E_{*}$ is $P$-(semi)stable if and only if
$\iota^{*}E_{*}$ is $B_{g}$-(semi)stable. So by Corollary 5.7, $E_{*}$ is
$P$-(semi)stable or $P$-polystable if and only if $g^{*}E_{*}$ is
$G$-(semi)stable or $G$-polystable.

From the above discussion, we see that under the equivalence (5.1), the
parabolic slope is the same as $P$-slope, and parabolic (semi)stability or
parabolic polystability are the same as $P$-(semi)stability.

We conclude this section by the following results which are classical for
curves.

######  Proposition 5.11.

  1. (1)

Let $f\colon(Y,Q)\longrightarrow(X,P)$ be a morphism of connected projective
formal orbifold curves. Let $E_{*}\in\text{\rm Vect}(X,P)$. Then $E_{*}$ is
$P$-semistable (respectively, $P$-polystable) if and only if $f^{*}E_{*}$ is
$Q$-semistable (respectively, $Q$-polystable).

  2. (2)

Let $f\colon(Y,Q)\longrightarrow(X,P)$ be a morphism of connected projective
orbifold curves. Let $E_{*}\in\text{\rm Vect}(X,P)$ be such that
$f^{*}E_{*}\in\text{\rm Vect}(Y,Q)$ is $f^{*}P$-stable. Then $E_{*}$ is a
$P$-stable vector bundle on $(X,P)$.

  3. (3)

Let $(X,P)$ be a geometric projective connected orbifold curve. Let
$E_{*},\,F_{*}\in\text{\rm Vect}(X,P)$ where $E_{*}$ is $P$-semistable. If
there is a non-zero homomorphism $\beta\in\operatorname*{Hom}_{\text{\rm
Vect}(X,P)}\left(E_{*},F_{*}\right)$, then we have

$\mu_{P}(E_{*})\leq\mu_{P,max}(F_{*}).$

  4. (4)

Let $(X,P)$ be a geometric connected projective formal orbifold curve. If
$E_{*}\in\text{\rm Vect}(X,P)$ is $P$-semistable and $L_{*}\in\text{\rm
Vect}(X,P)$ is a line bundle, the tensor product $E_{*}\otimes L_{*}$ is also
$P$-semistable.

###### Proof.

  1. (1)

We prove 1 and 2.

As usual, we have $Q\geq f^{*}P$ on $Y$. Also choose a geometric branch data
$P^{\prime}$ on $X$ such that $P^{\prime}\geq P,\,B_{f}$. Then the morphism
$(Y,f^{*}P^{\prime})\longrightarrow(X,P^{\prime})$ is an étale map. Also
suppose that $g\colon Z\longrightarrow(X,P^{\prime})$ be a $G$-Galois étale
morphism that factors as $g=f_{2}\circ h$ where $h\colon
Z\longrightarrow(Y,f^{*}P^{\prime})$ is $H$-Galois étale (necessarily, $H\leq
G$). Then we have the following diagram with the square being cartesian.

(5.2)
${Z}$${(Y,f^{*}P^{\prime})}$${(X,P^{\prime})}$${(Y,Q)}$${(Y,f^{*}P)}$${(X,P)}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{f_{2}}$$\scriptstyle{\jmath_{1}}$${\square}$$\scriptstyle{\iota}$$\scriptstyle{\jmath}$$\scriptstyle{f_{1}}$

Here $f_{2}=f_{1}=f$ as covers of underlying curves, $\jmath$, $\jmath_{1}$
are induced by $\text{\rm Id}_{Y}$ and $\iota$ is induced by $\text{\rm
Id}_{X}$.

Now let $E_{*}\in\text{\rm Vect}(X,P)$. By functoriality of the pullback, we
have
$g^{*}\iota^{*}E_{*}=h^{*}(f_{2}^{*}\iota^{*})=h^{*}(\jmath_{1}^{*}f_{1}^{*})$.
Thus we have

$\begin{array}[]{rcl}&&f^{*}E_{*}\text{ is }Q-\text{semistable}\\\ &&\hskip
28.45274pt\big{\Updownarrow}(\text{Lemma}~{}\ref{lem_iota_stability_equivalence})\\\
E_{*}\text{ is }P-\text{semistable}&&f_{1}^{*}E_{*}\text{ is }f^{*}P\\\
(\text{Lemma}~{}\ref{lem_iota_stability_equivalence})\hskip
5.69046pt\big{\Updownarrow}&&\hskip 28.45274pt\big{\Updownarrow}\hskip
5.69046pt(\text{Lemma}~{}\ref{lem_iota_stability_equivalence})\\\
\iota^{*}E_{*}\text{ is
}P^{\prime}-\text{semistable}&&\jmath_{1}^{*}f_{1}^{*}E_{*}\text{ is
}f^{*}P^{\prime}-\text{semistable}\\\
(\text{Corollary}~{}\ref{cor_geom_semi/poly_same})\hskip
5.69046pt\big{\Updownarrow}&(\text{Proposition}~{}\ref{prop_G_stability})&\hskip
28.45274pt\big{\Updownarrow}\hskip
5.69046pt(\text{Corollary}~{}\ref{cor_geom_semi/poly_same})\\\
g^{*}\iota^{*}E_{*}\text{ is
}G-\text{semistable}&\Leftrightarrow&h^{*}\jmath_{1}^{*}f_{1}^{*}E_{*}\text{
is }H-\text{semistable}\end{array}$

Replacing ‘semistable’ by ‘polystable’, we obtain the complete statement of 1.
To see 2, note that all other implications hold, except for the bottom one; by
Proposition 3.9, if the $G$-equivariant bundle
$h^{*}\jmath_{1}^{*}f_{1}^{*}E_{*}=g^{*}\iota^{*}E_{*}$ on $Z$ is $H$-stable,
then it is also $G$-stable.

  2. (2)

As before, invoking Lemma 5.5, we may assume that $Q=f^{*}P$ and $P\geq B_{f}$
is a geometric branch data. Also choose a $G$-Galois étale cover $g\colon
Z\longrightarrow(X,P)$ which factors as the composition of an $H$-Galois étale
cover $h\colon Z\longrightarrow(Y,f^{*}P)$ followed by the étale cover
$f\colon(Y,f^{*}P)\longrightarrow(X,P)$. Since $f^{*}E_{*}$ is
$f^{*}P$-stable, $E_{*}$ is $P$-semistable by (1).

Suppose that $0\neq S_{*}\subseteq E_{*}$ be a sub-bundle in $\text{\rm
Vect}(X,P)$ with $\mu_{P}(S_{*})=\mu_{P}(E_{*})$. Since the pullback map
$f^{*}=h_{*}\circ g^{*}$ is the composition of two functors which are
equivalence of categories, we obtain

$0\neq f^{*}S_{*}\subseteq f^{*}E_{*},$

a sub-bundle of $f^{*}E_{*}\in\text{\rm Vect}(Y,f^{*}P)$, and
$\mu_{f^{*}P}(f^{*}S_{*})=\mu_{f^{*}P}(f^{*}E_{*})$ (Lemma 5.6). So we have
$f^{*}S_{*}=f^{*}E_{*}$. Thus

$S_{*}=g_{*}\circ h^{*}\left(f^{*}S_{*}\right)=g_{*}\circ
h^{*}\left(f^{*}E_{*}\right)=E_{*}.$

  3. (3)

Let $g\colon Z\longrightarrow(X,P)$ be a $G$-Galois étale morphism of
connected projective orbifold curves. We obtain a non-zero $G$-invariant
homomorphism

$g^{*}\beta\colon g^{*}F_{*}\longrightarrow g^{*}E_{*}$

of $G$-equivariant vector bundles on $Z$. By Lemma 5.6,
$\mu(g^{*}E_{*})=|G|\,\mu_{P}(E_{*})$ and
$\mu(g^{*}F_{*})=|G|\,\mu_{P}(F_{*})$. By Corollary 5.7, $g^{*}E_{*}$ is also
$G$-semistable. Thus we obtain

$\mu(g^{*}E_{*})\leq\mu_{\text{max}}(g^{*}F_{*})$

and the statement now follows from the above slope-relations.

  4. (4)

Let $g\colon Z\longrightarrow(X,P)$ be a $G$-Galois étale map of connected
projective orbifold curves. By Corollary 5.7, $g^{*}E_{*}$ is $G$-semistable.
So tensoring $g^{*}E_{*}$ with the $G$-equivariant line bundle $g^{*}L_{*}$,
we see that $g^{*}E_{*}\otimes g^{*}L_{*}\in\text{\rm Vect}_{G}(Z)$ is
$G$-semistable. As the tensor product operation is preserved under this
equivalence $g^{*}$ of categories ([6, Proposition 5.21]),

$g^{*}E_{*}\otimes g^{*}L_{*}=g^{*}(E_{*}\otimes L_{*}),$

and the statement now follows from Corollary 5.7.

∎

##  6\. Stability of pullback under genuinely ramified map

Throughout this section, all formal orbifold curves considered are connected
and projective, unless otherwise specified. We will show (Theorem 6.4, Theorem
6.5) that the morphisms of orbifold curves under which the pullback of every
stable vector bundle remains stable are precisely the morphisms whose
underlying finite cover of curves is a genuinely ramified map. We first prove
several results that are needed for our main result.

######  Lemma 6.1.

Let $f\colon Y\longrightarrow X$ be a $G$-Galois genuinely ramified map of
smooth projective $k$-curves and $P$ be a branch data on $X$. Let
$f\colon(Y,f^{*}P)\longrightarrow(X,P)$ be the induced morphism of formal
orbifolds curves. Then in $\text{\rm Vect}(Y,f^{*}P)$ we have

$f^{*}\left(\left(f_{*}\mathcal{O}_{(Y,f^{*}P)}\right)/\mathcal{O}_{(X,P)}\right)\subset\bigoplus_{\begin{subarray}{c}1\leq
i\leq|G|-1\end{subarray}}M_{i}$

where each $M_{i}\in\text{\rm Vect}(Y,f^{*}P)$ is a line bundle of negative
$f^{*}p$-degree.

###### Proof.

As in the proof of [2, Proposition 3.5], there is an injective homomorphism
([2, Equation 3.17])

(6.1) $\Psi\colon
f^{*}\left(\left(f_{*}\mathcal{O}_{Y}\right)/\mathcal{O}_{X}\right)\longrightarrow\mathcal{O}_{Y}^{\oplus|G|-1}$

which is an isomorphism on the étale locus of $f$, and there are points ([2,
Equation 3.18]) $z_{i}\in Y$, $1\leq i\leq|G|-1$, such that for each
projection map
$p_{i}\colon\mathcal{O}_{Y}^{\oplus|G|-1}\longrightarrow\mathcal{O}_{Y}$ to
the $i^{\text{th}}$ factor ($1\leq i\leq|G|-1$), the composition morphism
$p_{i}\circ\Psi$ vanishes at the point $z_{i}$. This implies

$p_{i}\circ\Psi\left(f^{*}\left(\left(f_{*}\mathcal{O}_{Y}\right)/\mathcal{O}_{X}\right)\right)\subset
L_{i}\coloneqq\mathcal{O}_{Y}(-z_{i})\subset\mathcal{O}_{Y},$

and
$f^{*}\left(\left(f_{*}\mathcal{O}_{Y}\right)/\mathcal{O}_{X}\right)\subset\bigoplus_{\begin{subarray}{c}1\leq
i\leq|G|\end{subarray}}L_{i}$.

Set $\jmath\colon(Y,f^{*}P)\longrightarrow(Y,O)$ and
$\iota\colon(X,P)\longrightarrow(X,O)$ as the morphisms induced by the
respective identity maps. By Lemma 3.17, we have

$\iota^{*}f_{*}\mathcal{O}_{Y}=f_{*}\jmath^{*}\mathcal{O}_{Y}=f_{*}\mathcal{O}_{(Y,f^{*}P)}.$

This implies

$\jmath^{*}f^{*}f_{*}\mathcal{O}_{Y}=f^{*}\iota^{*}f_{*}\mathcal{O}_{Y}=f^{*}f_{*}\mathcal{O}_{(Y,f^{*}P)}.$

Thus we have
$\jmath^{*}\left(\left(f^{*}f_{*}\mathcal{O}_{Y}\right)/\mathcal{O}_{X}\right)=f^{*}\left(\left(f_{*}\mathcal{O}_{(Y,f^{*}P)}\right)/\mathcal{O}_{(X,P)}\right)$.
Using this and applying $\jmath^{*}$ to the composite morphism

$f^{*}\left(\left(f_{*}\mathcal{O}_{Y}\right)/\mathcal{O}_{X}\right)\overset{\Psi}{\longrightarrow}\mathcal{O}_{Y}^{\oplus|G|-1}\overset{p_{i}}{\longrightarrow}\mathcal{O}_{Y},$

for each $1\leq i\leq|G|-1$, we obtain the following morphism

$f^{*}\left(\left(f_{*}\mathcal{O}_{(Y,f^{*}P)}\right)/\mathcal{O}_{(X,P)}\right)\overset{\jmath^{*}\Psi}{\longrightarrow}\mathcal{O}_{(Y,f^{*}P)}^{\oplus|G|-1}\overset{\jmath^{*}p_{i}}{\longrightarrow}\mathcal{O}_{(Y,f^{*}P)}.$

Since for each $i$, the morphism $p_{i}\circ\Psi$ vanishes at the point
$z_{i}$, the morphism $\jmath^{*}p_{i}\circ\jmath^{*}\Psi$ vanishes at the
divisor $n_{i}z_{i}$ where
$n_{i}=|\text{Gal}\left((f^{*}P)(z_{i})/K_{Y,z_{i}}\right)|$, and hence we
have

$\jmath^{*}p_{i}\circ\jmath^{*}\Psi\left(f^{*}\left(\left(f_{*}\mathcal{O}_{(Y,f^{*}P)}\right)/\mathcal{O}_{(X,P)}\right)\right)\subset
M_{i}\coloneqq\mathcal{O}_{(Y,f^{*}P)}(-n_{i}z_{i})\subset\mathcal{O}_{(Y,f^{*}P)}.$

This shows that

$f^{*}\left(\left(f_{*}\mathcal{O}_{(Y,f^{*}P)}\right)/\mathcal{O}_{(X,P)}\right)\subset\bigoplus_{\begin{subarray}{c}1\leq
i\leq|G|-1\end{subarray}}M_{i}.$

∎

######  Lemma 6.2.

Let $f\colon Y\longrightarrow X$ be a $G$-Galois genuinely ramified map of
smooth projective $k$-curves and $P$ be a branch data on $X$. Let
$f\colon(Y,f^{*}P)\longrightarrow(X,P)$ be the induced morphism of formal
orbifolds curves. Then for any $P$-semistable vector bundle $F_{*}\in\text{\rm
Vect}(X,P)$, we have

$\mu_{P,\max}\left(F_{*}\otimes_{\mathcal{O}_{(X,P)}}f_{*}\mathcal{O}_{(Y,f^{*}P)}/\mathcal{O}_{(X,P)}\right)<\mu_{P}(F_{*}).$

###### Proof.

Since $F_{*}$ is $P$-semistable, the vector bundle $f^{*}P\in\text{\rm
Vect}(Y,f^{*}P)$ is $f^{*}P$-semistable by Proposition 5.11 1. We have

(6.2) $\begin{array}[]{cc}(\text{\rm
deg}(f))\,\mu_{P,\max}\left(F_{*}\otimes_{\mathcal{O}_{(X,P)}}f_{*}\mathcal{O}_{(Y,f^{*}P)}/\mathcal{O}_{(X,P)}\right)&=\,\mu_{f^{*}P,\max}\left(f^{*}\left(F_{*}\otimes_{\mathcal{O}_{(X,P)}}f_{*}\mathcal{O}_{(Y,f^{*}P)}/\mathcal{O}_{(X,P)}\right)\right)\end{array}$
$\begin{array}[]{c}=\mu_{f^{*}P,\max}\left(f^{*}F_{*}\otimes_{\mathcal{O}_{(Y,f^{*}P)}}f^{*}\left(\left(f_{*}\mathcal{O}_{(Y,f^{*}P)}\right)/\mathcal{O}_{(X,P)}\right)\right)\end{array}$

By Lemma 6.1, we have

$f^{*}\left(\left(f_{*}\mathcal{O}_{(Y,f^{*}P)}\right)/\mathcal{O}_{(X,P)}\right)\subset\bigoplus_{\begin{subarray}{c}1\leq
i\leq|G|-1\end{subarray}}M_{i}$

where each $M_{i}\in\text{\rm Vect}(Y,f^{*}P)$ is such that
$\mu_{f^{*}P}(M_{i})<0$. So the last slope in Equation (6.2) is

$\begin{array}[]{cc}\leq\mu_{f^{*}P,\max}\left(f^{*}F_{*}\otimes_{\mathcal{O}_{(Y,f^{*}P)}}\bigoplus_{\begin{subarray}{c}1\leq
i\leq|G|-1\end{subarray}}M_{i}\right)&\leq\max_{\begin{subarray}{c}1\leq
i\leq|G|-1\end{subarray}}\left\\{\mu_{f^{*}P}\left(f^{*}F_{*}\otimes_{\mathcal{O}_{(Y,f^{*}P)}}M_{i}\right)\right\\}.\end{array}$

Here we have used that each
$f^{*}F_{*}\otimes_{\mathcal{O}_{(Y,f^{*}P)}}M_{i}$ is $f^{*}P$-semistable
(Proposition 5.11 4) since $f^{*}F_{*}$ is $f^{*}P$-semistable. Since $M_{i}$
is a line bundle on $(Y,f^{*}P)$ of negative $f^{*}P$-degree, we have

$(\text{\rm
deg}(f))\,\mu_{P,\max}\left(F_{*}\otimes_{\mathcal{O}_{(X,P)}}f_{*}\mathcal{O}_{(Y,f^{*}P)}/\mathcal{O}_{(X,P)}\right)<\mu_{f^{*}P}\left(f^{*}F_{*}\right)=(\text{\rm
deg}(f))\,\mu_{P}(F_{*}).$

Thus we have

$\mu_{P,\max}\left(F_{*}\otimes_{\mathcal{O}_{(X,P)}}f_{*}\mathcal{O}_{(Y,f^{*}P)}/\mathcal{O}_{(X,P)}\right)<\mu_{P}(F_{*}).$

∎

######  Lemma 6.3.

Let $f\colon Y\longrightarrow X$ be a finite cover of smooth projective
$k$-curves and $P$ be a branch data on $X$. Let
$f\colon(Y,f^{*}P)\longrightarrow(X,P)$ be the induced morphism of formal
orbifolds curves. Then for any two $E_{*},\,F_{*}\in\text{\rm Vect}(X,P)$, we
have

$\text{\rm Hom}_{\text{\rm Vect}(X,P)}\left(E_{*},F_{*}\right)\subset\text{\rm
Hom}_{\text{\rm Vect}(Y,f^{*}P)}\left(f^{*}E_{*},f^{*}F_{*}\right).$

Moreover, if $f$ is a Galois genuinely ramified morphism and $E_{*}$, $F_{*}$
are $P$-semistable with $\mu_{P}(E_{*})=\mu_{P}(F_{*})$, we have

$\text{\rm Hom}_{\text{\rm
Vect}(Y,f^{*}P)}\left(f^{*}E_{*},f^{*}F_{*}\right)=\text{\rm Hom}_{\text{\rm
Vect}(X,P)}\left(E_{*},F_{*}\right).$

###### Proof.

By (3.2), we have

(6.3) $\begin{array}[]{rcl}\operatorname*{Hom}_{\text{\rm
Vect}(Y,f^{*}P)}\left(f^{*}E_{*},f^{*}F_{*}\right)&=&\operatorname*{Hom}_{\text{\rm
Vect}(X,P)}\left(E_{*},f_{*}f^{*}F_{*}\right)\\\
&=&\operatorname*{Hom}_{\text{\rm
Vect}(X,P)}\left(E_{*},F_{*}\otimes_{\mathcal{O}_{(X,P)}}f_{*}\mathcal{O}_{(Y,f^{*}P)}\right).\end{array}$

The first statement follows immediately from this.

Now suppose that $f$ is $G$-Galois genuinely ramified, and
$\mu_{P}(E_{*})=\mu_{P}(F_{*})$ where $E_{*},\,F_{*}$ are $P$-semistable
bundles. By Lemma 6.2, we have

$\mu_{P,\max}\left(F_{*}\otimes_{\mathcal{O}_{(X,P)}}f_{*}\mathcal{O}_{(Y,f^{*}P)}/\mathcal{O}_{(X,P)}\right)<\mu_{P}(F_{*})=\mu_{P}(E_{*}).$

By Proposition 5.11 (3), $\text{ \rm Hom}_{\text{\rm
Vect}(X,P)}\left(E_{*},F_{*}\otimes_{\mathcal{O}_{(X,P)}}f_{*}\mathcal{O}_{(Y,f^{*}P)}/\mathcal{O}_{X,P}\right)$
is trivial. Therefore the result follows from the following exact sequence

$0\longrightarrow\text{\rm Hom}_{\text{\rm
Vect}(X,P)}\left(E_{*},F_{*}\right)\longrightarrow\text{\rm Hom}_{\text{\rm
Vect}(X,P)}\left(E_{*},F_{*}\otimes_{\mathcal{O}_{(X,P)}}f_{*}\mathcal{O}_{(Y,f^{*}P)}\right)\longrightarrow$

$\longrightarrow\text{\rm Hom}_{\text{\rm
Vect}(X,P)}\left(E_{*},F_{*}\otimes_{\mathcal{O}_{(X,P)}}(f_{*}\mathcal{O}_{(Y,f^{*}P)}/\mathcal{O}_{(X,P)})\right)\longrightarrow
0.$

∎

Now we come to our main theorem.

######  Theorem 6.4.

Let $f\colon(Y,Q)\longrightarrow(X,P)$ be a morphism of connected projective
formal orbifold curves. Suppose that the underlying map $f\colon
Y\longrightarrow X$ is genuinely ramified. Then for every $P$-stable vector
bundle $E_{*}$ on $(X,P)$, the pullback bundle $f^{*}E_{*}$ on $(Y,Q)$ is
$Q$-stable.

###### Proof.

Suppose that $E_{*}\in\text{\rm Vect}(X,P)$ is $P$-stable. By [5, Lemma 2.12],
we have $Q\geq f^{*}P$, and so $f\colon(Y,Q)\longrightarrow(X,P)$ factors as

$(Y,Q)\overset{\jmath}{\longrightarrow}(Y,f^{*}P)\overset{f}{\longrightarrow}(X,P).$

By Proposition 5.11 (1), it is enough to prove that the bundle
$f^{*}E_{*}\in\text{\rm Vect}(Y,f^{*}P)$ is $f^{*}P$-stable. Further repeated
application of the same proposition, we may assume that $P$ is a geometric
branch data such that $P\geq B_{f}$. By Proposition 5.11 (1), the vector
bundle $f^{*}E_{*}$ on $(Y,f^{*}P)$ is $f^{*}P$-polystable.

Let
$\bar{f}\colon\bar{Y}\overset{\hat{f}}{\longrightarrow}Y\overset{f}{\longrightarrow}X$
be the Galois closure of $f$ with Galois group $\bar{G}$. Then
$\hat{f}\colon\bar{Y}\longrightarrow Y$ is also Galois. The map $\bar{f}$
factors through the maximal finite étale sub-cover

$g\colon\hat{X}\coloneqq\text{Spec}\left((\bar{f}_{*}\mathcal{O}_{\bar{Y}})_{1}\right)\longrightarrow
X$

where $(\bar{f}_{*}\mathcal{O}_{\bar{Y}})_{1}$ is the maximal destabilizing
sub-bundle of $\bar{f}_{*}\mathcal{O}_{Y}$. We see that $\bar{f}$ is genuinely
ramified if and only if $\hat{X}=X$ and $g=\text{\rm Id}_{X}$. By the
maximality, $g$ is a Galois cover with group $G$, say. By [2, Corollary 2.7],
the induced map

$\hat{g}\colon\bar{Y}\longrightarrow\hat{X}$

is a Galois genuinely ramified morphism. We have the following commutative
diagram.

${\bar{Y}}$${\hat{Y}\coloneqq
Y\times_{X}\hat{X}}$${\hat{X}}$${Y}$${X}$$\scriptstyle{\hat{g}}$$\scriptstyle{\hat{f}}$$\scriptstyle{\delta}$${\square}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$$\scriptstyle{g}$$\scriptstyle{f}$

Since $g$ is a $G$-Galois étale map and $f$ is genuinely ramified, from the
above cartesian square we see that the first projection
$p_{1}\colon\hat{Y}\longrightarrow Y$ is a $G$-Galois étale cover and the
second projection $p_{2}\colon\hat{Y}\longrightarrow\hat{X}$ is a finite
genuinely ramified morphism.

Now let $0\neq S_{*}\subseteq f^{*}E_{*}$ be a $f^{*}P$-stable sub-bundle of
$f^{*}E_{*}$ such that $\mu_{f^{*}P}(S_{*})=\mu_{f^{*}P}(f^{*}E_{*})$. We will
show that $f^{*}E_{*}=S_{*}$. To show this, we will first construct a sub-
bundle $V_{*}\subseteq g^{*}E_{*}$ having the same $\mu_{g^{*}P}$ such that
$p_{2}^{*}V_{*}=p_{1}^{*}S_{*}$. Using the fact that
$p_{1}^{*}S_{*}\subseteq(f\circ p_{1})^{*}E_{*}$ is $G$-invariant, we will
descend the bundle $V_{*}$ to a sub-bundle of $E_{*}$ on $(X,P)$ having the
same $\mu_{P}$, and this will conclude the proof.

First, taking pullback under the Galois morphism
$\hat{f}\colon(\bar{Y},\bar{f}^{*}P)\longrightarrow(Y,f^{*}P)$, we obtain a
sub-bundle $\hat{f}^{*}S_{*}\subseteq\hat{f}^{*}f^{*}E_{*}=\bar{f}^{*}E_{*}$
on $(\bar{Y},\bar{f}^{*}P)$. By Lemma 5.6, we have
$\mu_{\bar{f}^{*}P}(\hat{f}^{*}S_{*})=\mu_{\bar{f}^{*}P}(\bar{f}^{*}E_{*})$.
Since $S_{*}$ is a $f^{*}P$-stable bundle on $(Y,f^{*}P)$ and $E_{*}$ is a
$P$-stable bundle on $(X,P)$, both $\hat{f}^{*}S_{*}$ and $\bar{f}^{*}E_{*}$
are $\bar{f}^{*}P$-polystable by Proposition 5.11 (1). Define the right ideal
$\Theta_{\bar{Y}}$ of the associative algebra $\text{\rm End}_{\text{\rm
Vect}(\bar{Y},\bar{f}^{*}P)}(\bar{f}^{*}E)$ by

$\Theta_{\bar{Y}}\coloneqq\\{\gamma\in\text{\rm End}_{\text{\rm
Vect}(\bar{Y},\bar{f}^{*}P)}(\bar{f}^{*}E)\hskip 5.69046pt|\hskip
5.69046pt\gamma(\bar{f}^{*}E)\subset\hat{f}^{*}S_{*}\\}.$

Since $\hat{f}^{*}S_{*}$ is a direct summand of $\bar{f}^{*}E_{*}$, the bundle
$\hat{f}^{*}S_{*}$ coincides with the vector sub-bundle of $\bar{f}^{*}E_{*}$
generated by the images of the endomorphisms in $\Theta_{\bar{Y}}$.

Similarly, via a pullback under the $G$-Galois étale cover
$p_{1}\colon(\hat{Y},(f\circ p_{1})^{*}P)\longrightarrow(Y,f^{*}P)$, we obtain
a sub-bundle $p_{1}^{*}S_{*}\subseteq p_{1}^{*}f^{*}E_{*}$ on
$(\hat{Y},(f\circ p_{1})^{*}P)$, which are also $(f\circ
p_{1})^{*}P$-polystable, $\mu_{(f\circ
p_{1})^{*}P}(p_{1}^{*}S_{*})=\mu_{(f\circ p_{1})^{*}P}(f\circ
p_{1})^{*}E_{*})$. Further, $p_{1}^{*}S_{*}$ is generated by the images of
endomorphisms in the right ideal $\Theta_{\hat{Y}}\subset\text{\rm
End}_{\text{\rm Vect}(\hat{Y},(f\circ p_{1})^{*}P)}((f\circ p_{1})^{*}E_{*})$
defined as

$\Theta_{\hat{Y}}\coloneqq\\{\gamma^{\prime}\in\text{\rm End}_{\text{\rm
Vect}(\hat{Y},(f\circ p_{1})^{*}P)}(f\circ p_{1})^{*}E_{*})\hskip
5.69046pt|\hskip 5.69046pt\gamma^{\prime}((f\circ p_{1})^{*}E_{*})\subset
p_{1}^{*}S_{*}\\}.$

Applying Lemma 6.3 to the genuinely ramified Galois map
$\hat{g}\colon\bar{Y}\longrightarrow\hat{X}$, we obtain

(6.4) $\text{\rm End}_{\text{\rm Vect}(\hat{X},g^{*}P)}(g^{*}E_{*})=\text{\rm
End}_{\text{\rm Vect}(\bar{Y},\bar{f}^{*}P)}(\bar{f}^{*}E).$

Since under the above identification, $\gamma\in\text{\rm End}_{\text{\rm
Vect}(\hat{X},g^{*}P)}(g^{*}E_{*})$ is mapped to the endomorphism
$\hat{g}^{*}\gamma\in\text{\rm End}_{\text{\rm
Vect}(\bar{Y},\bar{f}^{*}P)}(\bar{f}^{*}E)$, the associative algebra
structures are preserved. Let the right ideal
$\Theta_{\hat{X}}\subset\text{\rm End}_{\text{\rm
Vect}(\hat{X},g^{*}P)}(g^{*}E_{*})$ be the image of $\Theta_{\bar{Y}}$. Since
$g^{*}E_{*}$ is a $g^{*}P$-polystable bundle on $(\hat{X},g^{*}P)$, the image
of any endomorphism of it is a sub-bundle. Let $V_{*}$ be the sub-bundle of
$g^{*}E_{*}$ generated by the images $\gamma(g^{*}E_{*})$ for
$\gamma\in\Theta_{\hat{X}}$. Then we have $\hat{g}^{*}V_{*}=\hat{f}^{*}S_{*}$.
Moreover, $V_{*}$ is again a $g^{*}P$-polystable bundle on $(\hat{X},g^{*}P)$,
and by Lemma 5.6, we have

$\mu_{g^{*}P}\left(V_{*}\right)=\mu_{g^{*}P}\left(g^{*}E_{*}\right).$

Now applying the first part of Lemma 6.3 to the finite covers $p_{2}$ and
$\delta$, we obtain the following.

$\text{\rm End}_{\text{\rm
Vect}(\hat{X},g^{*}P)}(g^{*}E_{*})\subseteq\text{\rm End}_{\text{\rm
Vect}(\hat{Y},(g\circ p_{2})^{*}P)}((g\circ p_{2})^{*}E_{*})\subseteq\text{\rm
End}_{\text{\rm Vect}(\bar{Y},\bar{f}^{*}P)}(\bar{f}^{*}E).$

By (6.4), both the above containment are equality, and the associative algebra
structures are preserved. Thus the ideal $\Theta_{\hat{X}}$ maps onto the
ideal $\Theta_{\hat{Y}}$ which maps onto the ideal $\Theta_{\bar{f}}$. We have
seen that $V_{*}$ is generated by the images of the endomorphisms in
$\Theta_{\hat{X}}$, and $p_{1}^{*}S_{*}$ is generated by the images of the
endomorphisms in $\Theta_{\hat{Y}}$. Thus we have

(6.5) $p_{1}^{*}S_{*}=p_{2}^{*}V_{*}.$

Since $p_{1}$ is a $G$-Galois étale cover, the injective morphism
$p_{1}^{*}S\hookrightarrow p_{1}^{*}f^{*}E$ underlying the sub-bundle
$p_{1}^{*}S_{*}\subseteq p_{1}^{*}f^{*}E_{*}$ on $(\hat{Y},(f\circ
p_{1})^{*}P)$ is $G$-equivariant. From (6.5), the injective morphism
$V\hookrightarrow g^{*}E$ underlying the sub-bundle $V_{*}\subseteq
g^{*}E_{*}$ is also preserved under the $G$-action. By Galois étale descent
under the map $g$, there is a sub-bundle $W\subset E$ on $X$ such that
$g^{*}W=V$. Let $W_{*}\subseteq E_{*}$ be the sub-bundle on $(X,P)$ with
underlying bundle $W$ on $X$ and equipped with the induced action maps from
$E_{*}$. Thus $g^{*}W_{*}=V_{*}$, and $f^{*}W_{*}=S_{*}$. Again by Lemma 5.6,
we have

$\mu_{P}(W_{*})=\mu_{P}(E_{*}).$

Since $E_{*}$ was assumed to be $P$-stable, we obtain $W_{*}=E_{*}$, and
consequently, $S_{*}=f^{*}E_{*}$. ∎

We also establish the converse to the above theorem, namely, if the cover
$f\colon Y\longrightarrow X$ is not genuinely ramified, and $P$ is any branch
data on $X$, the pullback of a $P$-stable bundle need not be $f^{*}P$-stable.

######  Theorem 6.5.

Let $f\colon(Y,Q)\longrightarrow(X,P)$ be a morphism of connected projective
orbifold curves such that the underlying finite cover $f\colon
Y\longrightarrow X$ is not genuinely ramified. Then there is a $P$-stable
vector bundle $E_{*}$ on $(X,P)$ whose pullback $f^{*}E_{*}\in\text{\rm
Vect}(Y,Q)$ is not $Q$-stable.

###### Proof.

First suppose that $f\colon Y\longrightarrow X$ is a finite étale cover. By
[2, Proposition 5.2], there is a stable vector bundle $E$ on $X$ such that the
vector bundle $f^{*}E$ is non stable on $Y$. By Lemma 5.5, the vector bundle
$\iota^{*}E\in\text{\rm Vect}(X,P)$ is $P$-stable, where
$\iota\colon(X,P)\longrightarrow X$ is the morphism induced by $\text{\rm
Id}_{X}$. By the functoriality of the pullback morphism, we have
$\jmath^{*}f^{*}E=f^{*}\iota^{*}E$ in $\text{\rm Vect}(Y,f^{*}P)$, where
$\jmath\colon(Y,f^{*}P)\longrightarrow Y$ is the morphism induced by
$\text{\rm Id}_{Y}$. Since $f^{*}E\in\text{\rm Vect}(Y)$ is not stable, by
Lemma 5.5, $\jmath^{*}f^{*}E\in\text{\rm Vect}(Y,f^{*}P)$ is not
$f^{*}P$-stable, and again applying the same lemma, the final pullback to
$\text{\rm Vect}(Y,Q)$ is not $Q$-stable.

Now if $f\colon Y\longrightarrow X$ is an arbitrary finite cover that is not
genuinely ramified, the map $f$ factors as

$f\colon
Y\overset{f_{1}}{\longrightarrow}X_{1}\overset{f_{2}}{\longrightarrow}X,$

where $f_{2}$ is the maximal étale finite sub-cover given by the spectrum of
the maximal destabilizing sub-sheaf of $f_{*}\mathcal{O}_{Y}$, and $f_{2}$ is
a genuinely ramified map. By the previous paragraph, choose a stable vector
bundle $E_{*}$ on $(X,P)$ such that $f_{1}^{*}E_{*}\in\text{\rm
Vect}(X_{1},f_{1}^{*}P)$ which is not $f_{1}^{*}P$-stable. By Proposition 5.11
2, $f^{*}E_{*}\in\text{\rm Vect}(Y,f^{*}P)$ is not $f^{*}P$-stable, and
applying Lemma 5.5, the pullback of $E_{*}$ in $\text{\rm Vect}(Y,Q)$ is not
$Q$-stable. ∎

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