53D20, 53D35.

In this paper, we classify Hamiltonian $S^1$-actions on compact, four dimensional
symplectic orbifolds that have isolated singular
points with cyclic orbifold structure groups, thus extending the
classification due to Karshon to the orbifold setting. To such a
space, we associated a combinatorial invariant, a labeled multigraph,
that determines the isomorphism type of the space. Moreover, we show
that any such space can be obtained by applying finitely many
equivariant weighted blow-ups to a minimal space, i.e., one on which
no equivariant weighted blow-down can be applied.

§ INTRODUCTION

In this paper we study Hamiltonian $S^1$-actions on compact, four dimensional
symplectic orbifolds that have isolated singular
points with cyclic orbifold structure groups. Intuitively, an orbifold is a space that is locally
modeled on a quotient of Euclidean space by an effective action of a finite group (see Definition
<ref>). First introduced by Satake
in [65], not only are they intrinsically interesting since
they are examples of `singular spaces', but in fact play an important
role in complex algebraic and symplectic geometry, and beyond. In the
former, they represent examples of `mild' singular behavior and have
been extensively studied in relation to desingularizations,
i.e., ways to resolve the singular locus to obtain a smooth complex
variety (see, for instance, [61]). In the latter, they arise naturally in the context of
symplectic reduction at a regular value of a moment map of a
Hamiltonian action. Thus, as stated in [43],

`we need to understand symplectic orbifolds even if we only
want to understand symplectic manifolds.'

In this direction, in recent years there has been plenty of interest in
desingularizing symplectic orbifolds to construct
interesting examples of symplectic manifolds (see,
among others, [18, 19]). Finally, symplectic
orbifolds also arise naturally when studying the so-called almost
regular (or Besse) contact manifolds, as the latter are principal
$S^1$-orbi-bundles over the former (see [38]).

Let $M$ be an orbifold of dimension $n$ and let $x \in M$ be a point. Associated to $x$
is its orbifold structure group $\Gamma$, which, using the above
intuitive definition, corresponds to the isotropy of the orbit of the
finite group action that $x$ represents (see Definition
<ref>). We say that $x$ is singular if $\Gamma$
is not trivial and that $M$ has isolated singular points if the
set of singular points of $M$ is discrete. In this paper, we are only
concerned with orbifolds with isolated singular points, the main
reason being that many of the subtleties of the differential topology
of orbifolds are made significantly simpler by imposing this
condition. For instance, for such orbifolds, the standard differential
topological notions of vector fields, differential forms, etc. can be
defined `naively'. Hence, the definitions of a symplectic manifold
and of a Hamiltonian action are the usual ones (see Definitions
<ref> and <ref>).

Let $(M,\omega)$ be a compact, four dimensional
symplectic orbifold that has isolated singular
points and suppose that $H \colon M \to \R$ is the moment map for an
effective Hamiltonian $S^1$-action on $M$. In this paper, we study
such objects under the extra condition that all orbifold structure groups be cyclic. This turns out to be a mild restriction, as
explained below. Let $x \in M$ be a singular
point with non-trivial orbifold structure group $\Gamma$. Since $M$ has only
isolated singular points, it follows that $x$ is a fixed point of the
action (see Corollary <ref>). Moreover, since $(M,\omega)$ is
symplectic and has dimension four, the group $\Gamma$ is (isomorphic
to) a subgroup of $\mathrm{U}(2)$ – see Theorem
<ref>. If $\Gamma$ is not abelian, by Lemma
<ref> and Remark
<ref>, it follows that $x$ is an
isolated fixed point that is a local (hence global) extremum. Otherwise, since the singular set of $M$ is
isolated, it follows that $\Gamma$ has to be cyclic (see Lemma
<ref>). In other words, by restricting
the orbifold structure groups to be cyclic, we are only excluding isolated
extrema with non-abelian structure groups. A Hamiltonian $S^1$-action
on $(M,\omega)$ with the above hypotheses is called a Hamiltonian
$S^1$-space and denoted by $(M,\omega,H)$.

The aim of this paper is to classify Hamiltonian $S^1$-spaces up to
the natural notion of isomorphism (see Definition
<ref>), thus generalizing the main results in
[34] to the setting of orbifolds. Given a Hamiltonian $S^1$-space $(M,\omega,H)$, we associate a
labeled multigraph to it that, when $M$ is a manifold, coincides with
the invariant introduced in <cit.>. However,
the invariant that we introduce is necessarily a
multigraph even when all fixed points of the action
are isolated and edges that are incident to a vertex need not have
coprime labels (see, for instance, Example <ref> and Figure <ref>).

(Q1) at (7,3);
(Q1) circle (4pt);
(Q2) at (7,0);
(Q2) circle (4pt);

(Q0) at (6,1.5);
(Q0) circle (4pt);

[label=left:$2$] (BC) at ($(6.9,1.5)$);
[label=right:$2$] (BC) at ($(7.1,1.5)$);

[label=below:${\frac{1}{4}(1,3), \color{red}{0}}$] (BC) at ($(Q2)$);
[label=above:${\frac{1}{4}(1,3), \color{red}{\frac{1}{4}}}$] (BC) at ($(Q1)$);
[label=left:${\frac{1}{2}(1,1),\color{red} \frac{1}{8}}\,\,$] (BC) at ($(Q0)$);

[very thick](7.1,3) – (7.1,0);
[very thick](6.9,3) – (6.9,0);

Example of a labeled multigraph of a Hamiltonian $S^1$-space
(see Example <ref>).

Moreover, there are more
labels than in the setting of manifolds, reflecting the richness of
the (local) orbifold structure. Our first main
result states that the labeled multigraph is a complete invariant up
to isomorphism (see Lemma
<ref> and Theorem <ref>).

Two Hamiltonian $S^1$-spaces are isomorphic if and only if the
associated labeled multigraphs are equal.

Intuitively, our strategy to prove Main Result <ref> is to use an equivariant
desingularization via weighted blow-ups to reduce the problem to
one between smooth manifolds, so that we can invoke <cit.>. We believe that the existence of an equivariant desingularization
(see Theorem <ref>) should be of interest in its own
right, especially in light of its relation to Hirzebruch-Jung
continued fractions and, therefore, to the well-known Hirzebruch-Jung
resolution of cyclic quotient singularities in complex algebraic
geometry (see [61]).

Our second main result is concerned with determining the so-called
minimal spaces, i.e., those Hamiltonian $S^1$-spaces on which no
equivariant weighted blow-down can be performed. Together with
equivariant weighted blow-ups, these can be thought of
as the fundamental building blocks of Hamiltonian $S^1$-spaces. More
precisely, the following holds (see Lemmas <ref>
and <ref>, and Theorems
<ref> and <ref>).

Any Hamiltonian $S^1$-space can be obtained by applying finitely
many equivariant weighted blow-ups to a minimal space.

We split the proof of Main Result <ref> in two cases, depending on
whether the given Hamiltonian $S^1$-space has at least one fixed orbi-surface (see Theorem
<ref>), or not (see Theorem <ref>). In each
case, our strategy for the proof is
the same and consists of two steps. First, we use the
localization formula for orbifolds (see Theorem
<ref>) to determine the possible labeled multigraph
of the minimal models (see Theorems <ref> and
<ref>). Second, we construct
Hamiltonian $S^1$-spaces that have as labeled multigraphs those
determined in the first step (see Sections <ref> and
<ref>). In all but two cases we construct the spaces explicitly; in the remaining two, we
obtain them from quotients of weighted projective spaces after
applying finitely many equivariant weighted blow-ups and
blow-downs. As a simple consequence of Main Result <ref>, we
exhibit minimal spaces with isolated fixed points for which
the $S^1$-action cannot be extended to a toric one (see Lemma
<ref>). These include an example studied in
[69]; moreover, this phenomenon should be contrasted with
<cit.> in the case of manifolds.

We view this paper as a step towards studying
`large' Hamiltonian actions on symplectic orbifolds, extending the
work in the completely integrable case carried out in
[43]. As such, there are several potential extensions
of the present paper that we intend to work on in the future. These
include removing the (easier) assumption on the orbifold structure
group and the (harder) assumption on the singular set of the orbifold,
as well as computing the equivariant cohomology of these
spaces. Further ahead, there may be scope to extend the classification
of tall complexity one Hamiltonian torus actions in
[34, 36, 37] to the realm of orbifolds.

§.§ Structure of the paper
In Section <ref>
we review standard differential topological
constructions for orbifolds, ranging from the definition and examples
of theses spaces
to maps between them, orbi-bundles, quotients, group actions and the study of
symplectic geometry for orbifolds. While most of the material in this
section is probably well-known to experts, we set the notation and
construct some fundamental examples that we use throughout the
paper. In Section <ref> we
define orbifolds using local charts (but see also
<ref>). We introduce the orbifold structure group of a
point on an orbifold. Depending on whether this group is trivial or
not, the point is regular or singular respectively. We discuss maps
between orbifolds, coming to the notion of good maps introduced in
<cit.>. We conclude this section by recalling the
classification of compact, orientable two dimensional orbifolds with
isolated singular points, which is a simple extension of the analogous
result for manifolds. In Section <ref> we discuss the
analog of fiber bundles in the setting of orbifolds, which are known
as orbi-bundles. We define these objects using local charts and
construct some examples of vector and principal orbi-bundles that are
used throughout the paper. Special attention is given to Seifert
fibrations, i.e., principal
$S^1$-orbi-bundles over compact, two dimensional orbifolds with
isolated singular points. In particular, we discuss the construction of the
associated Seifert invariant (see Definition
<ref>). To conclude this section, we consider
orbifold covering maps which generalize topological covering maps. In
Section <ref> we introduce suborbifolds, placing
particular emphasis on full (or strong) embedded suborbifolds (see Definition
<ref>). These can be thought of as being the
appropriate analog of embedded submanifolds in the setting of
orbifolds. For instance, such suborbifolds have a normal orbi-bundle
and there is a tubular neighborhood theorem (see Theorem
<ref>). We discuss group actions on orbifolds in
Section <ref>. To start, we define them and
construct several examples that are useful for our purposes. As in the
case of manifolds, there are linearization results near fixed points
(see Corollary <ref>). To conclude the
section, we state that finite quotients of orbifolds are again
orbifolds (see Corollary <ref>), and use this
result to describe cyclic quotients of weighted projective
planes. Once endowed with a suitable symplectic form and Hamiltonian
$S^1$-action, these underlie some of the minimal spaces that
we obtain. In Section <ref> we discuss aspects
of symplectic geometry on orbifolds, with a view to introduce Hamiltonian
group actions. These are very similar to the better known versions for
manifolds. We start by defining and giving examples of symplectic
orbifolds. The connection between symplectic forms and compatible
almost complex structures is discussed and then the Darboux theorem is
stated (see Theorem <ref>). We use this result
to classify open neighborhoods of isolated singular points with cyclic
orbifold structure group in a four
dimensional symplectic orbifold (see Corollary
<ref>). For our purposes, it is also useful to
recall the classification of compact symplectic surfaces with isolated
singular points (see Theorem
<ref>), as these appear in
our spaces as connected components of the fixed point set. We conclude
this section by defining Hamiltonian group actions, giving examples,
stating the equivariant Darboux and symplectic neighborhood theorems
(see Theorems <ref> and <ref>), and by stating
the connectedness result for fibers of the moment map of Hamiltonian
torus actions on a compact symplectic manifold (see Theorem

In Section <ref> we take a closer look at the local
behavior of Hamiltonian $S^1$-actions near fixed points. We start in
Section <ref> by discussing the orbi-weights of a
fixed point of a Hamiltonian $S^1$-action. First, we prove a general
result about finite extensions of tori (see Theorem
<ref>), which may be of independent interest and of use
in the study of Hamiltonian torus actions on orbifolds. The crucial
result that we need to define orbi-weights is Lemma
<ref>. In Section <ref> we prove
the local normal form for an isolated fixed point with cyclic orbifold
structure group of a Hamiltonian $S^1$-action on a four dimensional
symplectic orbifold (see Theorem <ref>). Moreover,
this section contains several technical results that we use in the
rest of the paper.

In Section <ref> we introduce the complete
combinatorial invariant of a Hamiltonian $S^1$-space, the
labeled multigraph. As a first step, in Section
<ref> we discuss properties of the connected
component of the fixed point set of such a space, as well as the
so-called isotropy orbi-spheres, which are the closure of a connected
component of the set of points with finite stabilizer. Then we proceed
to define the desired invariant in Section
<ref>. We also discuss some crucial examples
that, eventually, turn out to be minimal spaces (see Examples
<ref> and <ref>). Moreover, we prove the `easy'
direction of Main Result <ref> in Lemma
<ref>. As a first partial converse to this
result, in Section <ref> we show that the
labeled multigraph of a Hamiltonian $S^1$-space allows one to
reconstruct invariant open neighborhoods of all of the connected
components of the fixed point set (see Lemma
<ref> and Proposition
<ref>). In the proof of Proposition
<ref> we make use of the localization formula (see
Theorem <ref>) and so we also give
a short introduction to equivariant cohomology for orbifolds.

Section <ref> is the technical heart of
our paper, where we discuss the effects of equivariant weighted
blow-ups and blow-downs on the labeled multigraphs of Hamiltonian
$S^1$-spaces. In Section <ref> we deal with
equivariant weighted blow-ups. We start by recalling the original
(equivariant) definition of this operation introduced in [22, 49], as well as
establishing some basic properties (see, for instance,
<ref>). Then we turn to the two fundamental
technical results of the paper, Propositions <ref> and
<ref>, which describe the effects of certain equivariant
weighted blow-ups on the labeled multigraph of a Hamiltonian
$S^1$-space. Using Proposition <ref>, we prove the key
result that every Hamiltonian $S^1$-space may be desingularized after
finitely many equivariant
weighted blow-ups (see Theorem <ref>). In Section
<ref> we discuss equivariant weighted
blow-downs. This is a fundamental tool that we use to prove Main
Result <ref>. In particular, in this section we prove several
technical results that culminate in Corollary
<ref>, which gives a sufficient condition to
blow-down an $S^1$-invariant orbi-sphere.

We prove Main Results <ref> and <ref> in Section
<ref>. The `hard' part of Main Result <ref>
is proved in Section <ref>, while Section
<ref> deals with Main Result <ref>. To prove the
latter, we need to introduce chains of $S^1$-invariant orbi-spheres,
which are entirely analogous to those considered in <cit.>. Finally, we prove Main Result <ref> by showing a
sequence of results (see Lemma <ref>, Theorem
<ref>, Lemma <ref>, Theorem
<ref>, as well as Sections <ref> and
<ref>). We conclude this section by showing that certain minimal
models with only isolated fixed points are such that the $S^1$-action
cannot be extended to a toric one (see Lemma

In the final section of this paper, Section <ref>,
we prove some additional results regarding Hamiltonian $S^1$-spaces,
namely that they are Kähler (see Theorem <ref>), and that
some of them are birationally equivalent to a certain minimal model
(see Theorem <ref>), thus answering a question
posed to us by Margaret Symington.

§.§ Conventions

Given a positive integer $c > 1$, we denote the
cyclic group of order $c$ by $\Z_c$ and we identify it as the subgroup
of $S^1 = \{z \in \C \mid |z| = 1\}$ of $c$-th roots of unity. A
generator of $\Z_c$ is denoted by $\xi_c$.

§.§ Acknowledgments
We would like to thank Margaret
Symington for interesting discussions. L.G. was partially funded by
FCT/Portugal through UID/MAT/04459/2020 and PTDC/MATPUR/29447/2017. G.T.M.O. was supported by FCT/Portugal through PD/BD/128420/2017.
D.S. was partially supported by FAPERJ
grant JCNE E-26/202.913/2019 and by a CAPES/Alexander von Humboldt
Fellowship for Experienced Researchers
88881.512955/2020-01. D.S. would like to thank Instituto Superior
Técnico for the kind hospitality during several short visits. This
study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brazil
(CAPES) – Finance code 001.

§ A PRIMER ON (SYMPLECTIC) ORBIFOLDS, ORBI-BUNDLES AND GROUP ACTIONS

In this section we review some background material on (symplectic)
orbifolds and group actions (see for example
[2, 11, 30, 43, 70] for

§.§ Orbifolds

§.§.§ Definition and first examples

Throughout this section, we denote a topological vector space with
underlying set $M$ by $|M|$ and every subset of $|M|$ is endowed with
the relative topology.

Let $U \subseteq |M|$ be an open set. A local uniformizing chart (l.u.c. for short) on $|M|$ is a triple
$\oc$, where $\widehat{U}$ is a connected smooth manifold, $\Gamma
\subset \mathrm{Diff}(\widehat{U})$ is a finite group
acting effectively, and $\varphi :
\widehat{U} \to U$ is a  $\Gamma$-invariant continuous
map such that the map $\bar{\varphi} : \widehat{U}/\Gamma \to
U$ given by $\hat{\varphi}([\hat{x}]) = \varphi(\hat{x})$ is a homeomorphism. We say that $U$ is uniformized by $\oc$.

The above definition is equivalent to the standard one, in
which $\widehat{U}$ is assumed to be a connected open subset of
$\mathbb{R}^n$ (see <cit.>). Like charts for (topological) manifolds, l.u.c.'s need not be
unique. For this reason, it is useful to introduce the following

Given two l.u.c.'s $\oc$ and $\otheroc$ on $|M|$,

* an embedding

$$\lambda : \oc \hookrightarrow \otheroc$$

is a smooth open
embedding $\lambda : \widehat{U} \to \widehat{V}$ such that $\psi \circ
\lambda = \varphi$, and

* we say that $\oc$ and $\otheroc$ are isomorphic if
there exists an embedding $\lambda : \oc \hookrightarrow \otheroc$
that is a diffeomorphism.

An embedding $\lambda : \oc \hookrightarrow \otheroc$ of l.u.c.'s
induces an injective group homomorphism $\Gamma \to \Gamma'$ (which we
also denote by $\lambda$), satisfying the following property:

$$ \lambda(\gamma \cdot \hat{x}) = \lambda(\gamma) \cdot
\lambda(\hat{x}) \text{ for all } \hat{x} \in \widehat{U} \text{ and
for all } \gamma \in \Gamma. $$

Moroever, the image of $\lambda : \Gamma \to \Gamma'$ can be
characterized as follows:

\begin{equation}
\label{eq:9}
\gamma' \in \mathrm{im}\, \lambda \Leftrightarrow
\lambda(\widehat{U}) \cap \gamma' \cdot \lambda(\widehat{U}) \neq
\emptyset,
\end{equation}

(see <cit.>). In
particular, if $\lambda$ is an isomorphism, then $\lambda: \Gamma
\to \Gamma'$ is also an isomorphism. As a
consequence, the above notion of isomorphism is equivalent to the notion of
isomorphism of l.u.c.'s given in <cit.>.

Given a l.u.c. $\oc$ on $|M|$ and $\gamma_0 \in \Gamma$,
the map $\lambda_{\gamma_0} : \hat{U} \to \hat{U}$ given by $\hat{x}
\mapsto \gamma_0 \cdot \hat{x}$ is an automorphism of $\oc$. In fact,
any automorphism of $\oc$ is of this form. This is the content
of the following result, stated below without proof (see <cit.> and

Let $\oc$ be a l.u.c. on $|M|$. Given an automorphism $\lambda : \oc \to \oc$, there exists a unique $\gamma_0 \in \Gamma$ such that
$\lambda = \lambda_{\gamma_0}$.

The property of being uniformized behaves well under restriction to
connected subsets. More precisely, let $U \subseteq |M|$ be an open
set that is uniformized by the l.u.c. $\oc$ and let $U' \subseteq U$
be a connected subset. If we set $\widehat{U}'$ to be a connected
component of $\varphi^{-1}(U')$, $\Gamma'$ to be the subgroup that
fixes $\widehat{U}'$ and $\varphi':= \varphi|_{\widehat{U}'}$, then
$(\widehat{U}', \Gamma', \varphi')$ is a l.u.c. that uniformizes
$U'$. Moreover, the isomorphism class of
$(\widehat{U}', \Gamma', \varphi')$ does not depend on the choices
made (see <cit.>). We say that
$(\widehat{U}', \Gamma', \varphi')$ is the l.u.c. on
$U' \subseteq U$ induced by $\oc$.

Let $\lvert M \rvert$ be a topological space.

*  An orbifold atlas on $\lvert M \rvert$ is a collection
$\mathcal{A} = \{\oc\}$ of l.u.c.'s such that
$\{\varphi(\widehat{U})\}$ covers $\lvert M \rvert$ and
satisfies the following compatibility condition: For every $\oc,\otheroc
\in \mathcal{A}$ such that $U \cap V \neq \emptyset$ (where $U = \varphi(\widehat{U})$ and $V =
\psi(\widehat{V})$) and every $x \in U \cap V$, there exist an open
subset $W \subset U \cap V$ containing $x$, a l.u.c.
$\thirdoc \in \mathcal{A}$ with $W = \chi(\widehat{W})$ and embeddings
$\thirdoc \hookrightarrow \oc$ and $\thirdoc \hookrightarrow
\otheroc$.

*  An orbifold atlas $\mathcal{B}$ on $\lvert M \rvert$ refines the orbifold atlas $\mathcal{A}$ if every
l.u.c. in $\mathcal{B}$ admits an embedding in some
l.u.c. in $\mathcal{A}$.

*  Two orbifold atlases $\mathcal{A}$ and
$\mathcal{A}'$ of $\lvert M \rvert$ are equivalent if they admit a common

*  An (effective) orbifold is a paracompact, Hausdorff
topological space $\lvert M \rvert$ endowed with the equivalence class of an
orbifold atlas. To simplify notation, we denote an orbifold by $M$.

Using the above notions of isomorphism of l.u.c.'s and of induced
l.u.c.'s, it follows that Definition <ref> is
equivalent to <cit.>. In particular,
given an orbifold $M$ and a point $x \in M$, there exists an open
neighborhood $U$ of $x$ that is uniformized by some l.u.c. $\oc$;
moreover the isomorphism class of the germ of $\oc$ at $x$ is unique.

In analogy with the case of manifolds, if $M$ is an orbifold and
$|M|$ is connected, then there is a well-defined notion of dimension of $M$, which is given by the dimension of the domain
of any l.u.c. on $|M|$.

The `simplest' families of orbifolds are illustrated in the following
two examples.

Any (paracompact, Hausdorff) manifold is an orbifold: Given a smooth atlas
$\mathcal{A}$ on a
manifold $M$, we endow each
coordinate chart $(U,\varphi)$ with the action of the trivial
subgroup of $\mathrm{Diff}(U)$. $\Diamond$

[Orbi-vector spaces]
Let $\widehat{V}$ be a finite dimensional vector space over $\mathbb{K} \in
\{\R, \C\}$ and let $\Gamma < \mathrm{GL}(\widehat{V})$ be a finite
subgroup. The global quotient $\widehat{V}/\Gamma$ is an orbifold that is
known as an orbi-vector space. $\Diamond$

Example <ref> can be greatly generalized as follows. Let $G$ be a Lie group acting on (the left on) a manifold
$\widehat{M}$. We say that the action is  proper if the
map $\phi:G \times \widehat{M} \to \widehat{M} \times \widehat{M}$ given by $\phi(g,p)= g \cdot p$ is proper, and
that it is locally free if all stabilizers are finite. If the
$G$-action on $\widehat{M}$ is proper, then the quotient space $\widehat{M}/G$ is paracompact and Hausdorff (see <cit.>). If the action is also locally free, then the Slice Theorem
(see <cit.>),
implies that $\widehat{M}/G$ can naturally be endowed with
the structure of an orbifold (see <cit.>): A l.u.c. centered at a point $[p] \in \widehat{M}/G$ is given by the triple $(\widehat{U}, \Gamma, \varphi)$,

* $\widehat{U} \subseteq T_p\widehat{M}$ is an open neighborhood of
the origin that is invariant under the the linear action of the isotropy
group $G_p$,

* $\Gamma = \rho(G_p)$, where
$\rho : G_p \to \mathrm{GL}(T_pM)$ is the slice
representation[It is a consequence of the Slice Theorem
that $\rho$ is injective if the $G$-action is effective.], and

* $\varphi : \widehat{U} \to \widehat{M}/G$ is the
composite of the quotient map $\widehat{U} \to
\widehat{U}/\Gamma$ and the inclusion $
\widehat{U}/\Gamma \hookrightarrow \widehat{M}/G$ arising from the
Tube Theorem.

Following <cit.>, we call
$\widehat{M}/G$ a(n effective) quotient
orbifold; moreover, if $G$ is finite, we say that $
\widehat{M}/G$ is a(n effective) global quotient.

[Weighted projective spaces]
Fix an integer $n \geq 1$ and positive
integers $m_0,\dots, m_{n}$ satisfying $\gcd(m_0,\dots,m_n)=1$. Consider the smooth function $H :
\C^{n+1} \to \R$ given by

$$H(z_0,\dots,z_{n})=\frac{1}{2} \sum\limits_{j=0}^nm_j \lvert z_j \rvert ^2.$$

The level set $E:=H^{-1}(\frac{1}{2}\Pi_{j=0}^n \,\, m_j)$ is regular (and
diffeomorphic to a $2n+1$ dimensional sphere). Moreover it is invariant under the smooth $S^1$-action on $\C^{n+1}$ given by

\begin{equation}
\label{eq:8}
\begin{split}
\left(\lambda,(z_0,\dots,z_{n})\right)&\mapsto(\lambda^{m_0}z_0,\dots,\lambda^{m_n}z_n).
\end{split}
\end{equation}

The induced $S^1$-action on $E$ is proper,
effective and locally free. Hence, the quotient

\mathbb{C}P^n(m_0,\dots,m_n):= E / S^1  $$

is an
orbifold, known in the literature as a weighted
projective space of (complex) dimension $n$.

$\mathbb{C}P^1(1,a)$, also called a teardrop.

The following example was constructed by Singer, Talvacchia and
Watson in [69], and is carried out
throughout the text. The
effective action of
$\mathbb{Z}_4$  on $\widehat{M}:=\mathbb{C}P^1\times\mathbb{C}P^1$
uniquely determined by

\begin{equation}
\label{eq:29}
e^{\frac{\pi \imath}{2}} \cdot \left([z_0:z_1],[w_0:w_1]\right) = \left([-w_0:w_1],[z_0:z_1]\right),
\end{equation}

is proper and locally free. Hence, the
quotient $M:= \widehat{M}/\mathbb{Z}_4$ inherits the
structure of an orbifold. 	$\Diamond$

Let $c > 1$ be an integer and fix $1 \leq l < c$ such that
$\gcd(l,c)=1$. The effective action of $\mathbb{Z}_c$ on
$\mathbb{C}P^1\times\mathbb{C}P^1$ given by

\begin{equation}
\label{eq:30}
\lambda \cdot \left([z_0: z_1],[w_0 : w_1]\right)=\left([z_0: \lambda z_1], [ w_0:  \lambda^{-l} w_1]\right),
\end{equation}

is proper and locally free. Hence, the quotient $M_c:=
\mathbb{C}P^1\times\mathbb{C}P^1/\mathbb{Z}_c$ inherits the structure of an
orbifold. In order to stress the dependence on $c$, we denote the
elements of $M_c$ by $[[z_0:z_1],[w_0,w_1]]_c$. $\Diamond$

To conclude this section, we introduce the notion of orientation and
orientability for orbifolds. To this end, we fix once and for all the
standard orientation on $\R^n$.

Let $|M|$ be a topological space.

* A l.u.c. $\oc$ on $|M|$ is oriented if $\Gamma$ is a
subgroup of the group of orientation-preserving diffeomorphisms of

* An embedding $\lambda : \oc \to \otheroc$ between oriented l.u.c.'s on
$|M|$ is orientation-preserving if $\lambda : \widehat{U}
\to \widehat{V}$ is orientation-preserving.

* An orbifold atlas $\mathcal{A} =\{\oc\}$ on $|M|$ is oriented if
each l.u.c. $\oc$ in $\mathcal{A}$ is oriented and the embeddings
of part <ref> of Definition <ref> are

* An orbifold $M$ is orientable if it admits an oriented
orbifold atlas. In this case, an orientation on $M$ is a
choice of (equivalence class of) such an orbifold atlas.

* A manifold is orientable as an orbifold if and only if it is
orientable as a manifold.

* An orbi-vector space $\widehat{V}/\Gamma$ is orientable if and only if
$\Gamma$ is a subgroup of $\mathrm{SL}(\widehat{V})$.

* Let $\widehat{M}$ be an oriented manifold and let $G$ be a Lie
group acting on $\widehat{M}$ properly, in a locally free fashion,
and by orientation-preserving diffeomorphisms. The quotient orbifold
$\widehat{M}/G$ inherits a natural orientation.

If $M_1$ and $M_2$ are orbifolds, there is a natural equivalence
class of orbifold atlases on the product $|M_1| \times |M_2| =|M_1
\times M_2|$: If
$\mathcal{A}_1 = \{(\widehat{U}_1,\Gamma_1,\varphi_1)\}$ and
$\mathcal{A}_2 = \{(\widehat{U}_2,\Gamma_2,\varphi_2)\}$ are
orbifold atlases representing the orbifold structures on $|M_1|$ and
$|M_2|$ respectively, the orbifold structure on $|M_1 \times M_2|$
is represented by the `product' atlas

$$\mathcal{A}_1 \times
\mathcal{A}_2 = \{(\widehat{U}_1 \times \widehat{U}_2, \Gamma_1
\times \Gamma_2, \varphi_1 \times \varphi_2)\}.$$

The resulting orbifold is denoted by $M_1 \times M_2$.
Moreover, if both $M_1$ and $M_2$ are orientable, so is $M_1 \times

§.§.§ Orbifold structure group

Let $M$ be an orbifold, fix a point $x \in \lvert M \rvert$ and let $\oc$ be an
l.u.c. such that $x \in U = \varphi(\widehat{U})$. The stabilizers
of any two points $\hat{x},\hat{x}' \in \widehat{U}$ such that
$\varphi(\hat{x}) = x = \varphi(\hat{x}')$ are conjugate
as subgroups of $\Gamma$. We denote the underlying abstract group by $\Gamma_x\oc$. If $\otheroc$ is an orbifold
chart satisfying the above properties, then
$\Gamma_x \oc \cong \Gamma_x \otheroc$ – for a proof,
see the discussion preceding <cit.>.

Let $M$ be an orbifold. Given a point $x \in \lvert M
\rvert$, let $\oc$ be an
l.u.c. such that $x \in \varphi(\widehat{U})$. We call
$\Gamma_x \oc$  the orbifold structure group of
$\boldsymbol{x}$ and denote it  simply by $\Gamma_x$.

(Orbi-vector spaces, continued)
Let $\widehat{V}/\Gamma$ be an orbi-vector space in the sense of Example
<ref>. The
orbifold structure group of $[0] \in \widehat{V}/\Gamma$ is $\Gamma$. $\Diamond$

For $i=1,2$, let $M_i$ be an orbifold and let $x_i \in M_i$ be a point with
orbifold structure group $\Gamma_i$. The point $(x_1,x_2) \in M_1
\times M_2$ has orbifold structure group given by $\Gamma_1 \times \Gamma_2$.

By the Slice Theorem  (see <cit.>) and since l.u.c.'s behave well with respect
to restriction (see Remark <ref>), for every $x \in \lvert M \rvert$, there exists
a l.u.c. $\oc$ with the
following properties

* $\Gamma$ is the orbifold structure group of $x$ and $\Gamma <
\mathrm{O}(n)$,

* the open subset $\widehat{U}$ of $\R^n$ is
invariant under the standard linear action of $\mathrm{O}(n)$, and

* the point $0 \in \widehat{U}$ satisfies $\varphi(0) = x$.

In this case, we say that $\oc$ is a l.u.c. centered
at $\boldsymbol{x}$. If, in addition, $M$ is
oriented, then $\Gamma$ is a subgroup of $\mathrm{SO}(n)$.

Let $M$ be an effective orbifold. We say that a point $x \in |M|$ is
singular if $\Gamma_x \neq 1$, and regular otherwise.

Since the group action in any l.u.c. is assumed to be effective, it follows that the set of regular points of
an effective orbifold is necessarily open and dense.

Remark <ref> implies that any orbifold that has no
singular point can be viewed as a manifold, the smooth atlas being
given by (the equivalence class of) the union of the l.u.c.'s
centered at each point.

We say that an orbifold $M$ has isolated singular
points if the set of singular points is discrete.

[Weighted projective spaces, continued]
If, in Example <ref>, the positive integers
$m_0,\ldots, m_n$ are chosen to be pairwise
coprime, then the weighted
projective space $\mathbb{C}P^n(m_0,\dots,m_n)$ has isolated
singular points. These are $[(2\prod\limits_{j \neq 0} m_j)^{1/2}:0:\ldots:0] ,[0: (2\prod\limits_{j \neq 1} m_j)^{1/2}:\ldots:0], \ldots,
[0:0:\ldots: (2\prod\limits_{j \neq n} m_j)^{1/2}]$ with orbifold structure groups given by
$\mathbb{Z}_{m_0},\ldots, \mathbb{Z}_{m_n}$ respectively.  $\Diamond$

[Example <ref>, continued]
The orbifold of Example <ref> has isolated singular
points. These are $[[0:1],[0:1]], [[1:0],[1:0]]$ and $[[0:1],[1:0]]$
with orbifold structure groups given by $\mathbb{Z}_4$,
$\mathbb{Z}_4$ and $\mathbb{Z}_2$ respectively. 	$\Diamond$

[Example <ref>, continued]
The orbifold of Example <ref> has
isolated singular points. These are $[[1:0], [1:0]]_c, [[1:0], [0:1]]_c,
[[0:1], [1:0]]_c$ and $[[0:1], [0:1]]_c$ and all have orbifold structure
group given by $\mathbb{Z}_c$. $\Diamond$

§.§.§ Maps between orbifolds

In this section, we define a certain class of maps between
orbifolds. We start by considering what happens for fixed l.u.c.'s.

Let $|M_i|$ be a topological space and let $U_i \subseteq |M_i|$ be an
open set that is uniformized by the l.u.c. $(\widehat{U}_i,
\Gamma_i,\varphi_i)$ for $i=1,2$. Let $f: U_1 \to U_2$ be a
continuous map.

* A $\boldsymbol{C^{\infty}}$ lift of $f$ is a
$C^{\infty}$ map $\widehat{f} : \widehat{U}_1 \to \widehat{U}_2$
such that:

* $\varphi_2 \circ \widehat{f} = f \circ \varphi_1$, and

* for all $\gamma_1 \in \Gamma_1$, there exists a
$\gamma_2 \in \Gamma_2$ such that
$\widehat{f}(\gamma_1 \cdot \hat{x}) = \gamma_2 \cdot \widehat{f}(\hat{x})$
for all $\hat{x} \in \widehat{U}_1$.

* A good $\boldsymbol{C^{\infty}}$ lift of $f$ is a pair
$(\widehat{f},\Theta)$ such that:

* $\widehat{f} : \widehat{U}_1 \to \widehat{U}_2$
is a $C^{\infty}$ map with $\varphi_2 \circ \widehat{f} = f \circ \varphi_1$, and

* $\Theta : \Gamma_1 \to \Gamma_2$ is a homomorphism making
$\hat{f}$ is $\Theta$-equivariant.

It is immediate from Definition <ref> that the
$C^{\infty}$ map of a good $C^{\infty}$ lift is, in fact, a $C^{\infty}$ lift. However,
the converse need not hold (see <cit.>).

Just as l.u.c.'s are not unique, neither are (good) $C^{\infty}$ lifts of
continuous maps between l.u.c.'s (whenever they exist).

Let $|M_i|$ be a topological space and let $U_i \subseteq |M_i|$ be an
open set that is uniformized by the l.u.c. $(\widehat{U}_i,
\Gamma_i,\varphi_i)$ for $i=1,2$. Let $f: U_1 \to U_2$ be a
continuous map and let $\widehat{f}$ be a $C^{\infty}$ lift of $f$. If $\lambda:
(\widehat{U}'_1, \Gamma'_1, \varphi'_1) \hookrightarrow (\widehat{U}_1,
\Gamma_1,\varphi_1)$ (respectively $\lambda:
(\widehat{U}'_2, \Gamma'_2, \varphi'_2) \hookrightarrow (\widehat{U}_2,
\Gamma_2,\varphi_2)$) is an isomorphism in the sense of Definition
<ref>, it follows from Remark
<ref> that $\widehat{f} \circ \lambda$
(respectively $\lambda^{-1} \circ \widehat{f}$) is a
$C^{\infty}$ lift of $f$. Analogously, if $(\widehat{f},\Theta)$ is
a good $C^{\infty}$ lift of $f$, then by Remark
<ref> (and, in particular,
(<ref>)), $(\widehat{f} \circ \lambda, \Theta \circ \lambda)$
(respectively $(\lambda^{-1} \circ \widehat{f}, \lambda^{-1} \circ
\Theta)$) is a good $C^{\infty}$ lift of $f$.

In light of Remark <ref>, we introduce the following equivalence relation between (good)
$C^{\infty}$ lifts of
continuous maps between l.u.c.'s.

Let $|M_i|$ be a topological space and let $U_i \subseteq |M_i|$ be an
open set that is uniformized by the l.u.c.'s $(\widehat{U}_i,
\Gamma_i,\varphi_i)$ and $(\widehat{U}'_i,
\Gamma'_i,\varphi'_i)$ for $i=1,2$. Let $f :U_1 \to U_2$ be a
continuous map. We say that

* two $C^{\infty}$ lifts $\widehat{f} :
\widehat{U}_1 \to \widehat{U}_2$ and  $\widehat{f}' :
\widehat{U}'_1 \to \widehat{U}'_2$ of $f$ are isomorphic if there exist isomorphisms $\lambda_1 : (\widehat{U}_1,
\Gamma_1,\varphi_1) \to (\widehat{U}'_1,
\Gamma'_1,\varphi'_1)$ and $\lambda_2 : (\widehat{U}_2,
\Gamma_2,\varphi_2) \to (\widehat{U}'_2,
\Gamma'_2,\varphi'_2)$ of l.u.c.'s such that $\widehat{f}' \circ \lambda_1 =
\lambda_2 \circ \widehat{f}$, and

* two good  $C^{\infty}$ lifts $(\widehat{f}, \Theta: \Gamma_1
\to \Gamma_2)$ and $(\widehat{f}', \Theta': \Gamma'_1
\to \Gamma'_2)$ are isomorphic if there exist isomorphisms $\lambda_1 : (\widehat{U}_1,
\Gamma_1,\varphi_1) \to (\widehat{U}'_1,
\Gamma'_1,\varphi'_1)$ and $\lambda_2 : (\widehat{U}_2,
\Gamma_2,\varphi_2) \to (\widehat{U}'_2,
\Gamma'_2,\varphi'_2)$ of l.u.c.'s such that $\widehat{f}' \circ \lambda_1 =
\lambda_2 \circ \widehat{f}$ and $\Theta' \circ \lambda_1 =
\lambda_2 \circ \Theta$.

Whenever they exist, (good) $C^{\infty}$ lifts behave well with respect to
restriction to connected open subsets. More precisely, let $|M_i|$ be a topological space
and let $V_i \subseteq |M_i|$ be an open set that is uniformized by
the l.u.c. $(\widehat{V}_i, \Gamma'_i,\psi_i)$ for $i=1,2$. Let
$f: V_1 \to V_2$ be a continuous map that has a $C^{\infty}$ lift
$\widehat{f} : \widehat{V}_1 \to \widehat{V}_2$. Let $U_i \subseteq V_i$ be an
open set that is uniformized by $(\widehat{U}_i, \Gamma_i,
\varphi_i)$ for $i=1,2$ such that $f(U_1) \subseteq U_2$.  Suppose that we are given an embedding
$\lambda_1 : (\widehat{U}_1, \Gamma_1, \varphi_1) \hookrightarrow
(\hat{V}_1, \Gamma'_1,\psi_1)$. By construction,
$\psi_2(\hat{f} \circ \lambda_1(\widehat{U}_1)) \subseteq
U_2$. Therefore, there exists an
$\lambda_2 : (\widehat{U}_2,\Gamma_2,\varphi_2) \hookrightarrow
(\hat{V}_2, \Gamma'_2,\psi_2)$ such that
$\hat{f} \circ \lambda_1(\widehat{U}_1) \subseteq \lambda_2(\widehat{U}_2)$
(see <cit.>). Using Remark
<ref> (and, in particular,
(<ref>)), it follows that the map
$\lambda_2^{-1} \circ \widehat{f} \circ \lambda_1 : \widehat{U}_1 \to
\widehat{U}_2$ is a $C^{\infty}$ lift of the restriction of $f$ to
$U_1$. Analogously, if $(\widehat{f}, \Theta)$ is a good
$C^{\infty}$ lift of $f$, then $(\lambda_2^{-1} \circ \widehat{f}
\circ \lambda_1, \lambda_2^{-1} \circ \Theta \circ \lambda_1)$ is a
good $C^{\infty}$ lift of the restriction of $f$ to
$U_1$. We refer to the $C^{\infty}$ lift (respectively good $C^{\infty}$ lift) constructed above as the restriction of
$\widehat{f}$ (respectively $(\widehat{f}, \Theta)$) to $\widehat{U}_1$.

Following <cit.>, we break the definition of the
maps of our interest in two parts. First, we introduce $C^{\infty}$
lifts of maps between orbifolds.

Let $M_1, M_2$ be orbifolds and let $f : |M_1| \to |M_2|$ be a
continuous map. A $\boldsymbol{C}^{\infty}$ lift of $f$
consists of the following data:

*  orbifold atlases $\mathcal{A}_1 =
\{(\widehat{U}_1,\Gamma_1,\varphi_1)\}$ and $\mathcal{A}_2 =
\{(\widehat{U}_2,\Gamma_2,\varphi_2)\}$ such that, for all
$(\widehat{U}_1,\Gamma_1,\varphi_1) \in \mathcal{A}_1$, there
exists $(\widehat{U}_2,\Gamma_2,\varphi_2) \in \mathcal{A}_2$ with
$f(\varphi(\widehat{U}_1)) \subseteq \varphi_2(\widehat{U}_2)$,

* for all  $(\widehat{U}_1,\Gamma_1,\varphi_1) \in
\mathcal{A}_1$ and $(\widehat{U}_2,\Gamma_2,\varphi_2) \in
\mathcal{A}_2$ such that
$f(\varphi(\widehat{U}_1)) \subseteq \varphi_2(\widehat{U}_2)$, a
$C^{\infty}$ lift of the restriction of $f$ to $\varphi_1

satisfying the following compatibility condition:

Given $(\widehat{U}_1,\Gamma_1,\varphi_1), (\widehat{V}_1,\Gamma'_1,\psi_1)
\in \mathcal{A}_1$ with $U_1 \cap V_1 \neq \emptyset$ (where $U_1 = \varphi(\widehat{U_1})$ and $V_1 =
\psi(\widehat{V_1})$), and given a point $x \in U_1 \cap V_1$,
there exist

* an open subset $W_1 \subseteq U_1 \cap V_1$
uniformized by a l.u.c. $(\widehat{W}_1,\Gamma''_1,\chi_1) \in
\mathcal{A}_1$ containing $x$,

* an open subset $W_2 \subseteq |M_2|$ uniformized by
$(\widehat{W}_2, \Gamma_2'', \chi_2)$ such that
$f(W_1) \subseteq W_2$, and

* embeddings $(\widehat{W}_1,\Gamma''_1,\chi_1) \hookrightarrow
(\widehat{U}_1,\Gamma_1,\varphi_1)$ and
$(\widehat{W}_1,\Gamma''_1,\chi_1) \hookrightarrow

such that the $C^{\infty}$ lift of the restriction
of $f$ to $W_1$ is isomorphic to the restrictions of the
$C^{\infty}$ lifts of $f|_{U_1}$ and $f|_{V_1}$ to $W_1$.
We denote a $C^{\infty}$ lift of $f$ by the triple
$(\mathcal{A}_1,\mathcal{A}_2,\{\widehat{f}_{\widehat{U}_1} : \widehat{U}_1 \to
\widehat{U}_2\})$.

In order to make Definition <ref> independent of
choices, we introduce the following notion.

Let $M_1, M_2$ be orbifolds and let $f : |M_1| \to |M_2|$ be a
continuous map. We say that the $C^{\infty}$ lifts $(\mathcal{A}_1,\mathcal{A}_2,\{\widehat{f}_{\widehat{U}_1} : \widehat{U}_1 \to
\widehat{U}_2\})$ and $(\mathcal{A}'_1,\mathcal{A}'_2,\{\widehat{f}'_{\widehat{U}'_1} : \widehat{U}'_1 \to
\widehat{U}'_2\})$ of $f$ are equivalent if there exists a
$C^{\infty}$ lift $(\mathcal{B}_1,\mathcal{B}_2,\{\widehat{f}_{\widehat{V}_1} : \widehat{V}_1 \to
\widehat{V}_2\})$ such that

* the orbifold atlas $\mathcal{B}_i$ is a common refinement of
$\mathcal{A}_i$ and $\mathcal{A}_i'$ for $i=1,2$, and

* for all l.u.c.'s $(\widehat{V}_1,
\tilde{\Gamma}_1,\tilde{\varphi}_1)$ and for all embeddings $(\widehat{V}_1,
\tilde{\Gamma}_1,\tilde{\varphi}_1) \hookrightarrow
(\widehat{U}_1,\Gamma_1,\varphi_1)$ and $(\widehat{V}_1,
\tilde{\Gamma}_1,\tilde{\varphi}_1) \hookrightarrow
(\widehat{U}'_1,\Gamma'_1,\varphi'_1)$ , the $C^{\infty}$ lift
$\widehat{f}_{\widehat{V}_1}$ of the restriction of $f$ to $V_1 =
\tilde{\varphi}_1(\widehat{V}_1)$ is equivalent to the restrictions of
$\widehat{f}_{\widehat{U}_1}$ and $\widehat{f}'_{\widehat{U}'_1}$
to $V_1$.

Using Definitions <ref> and
<ref>, we can introduce smooth maps between

Let $M_1$ and $M_2$ be orbifolds. A $\boldsymbol{C^{\infty}}$
map between $M_1$ and $M_2$ is a continuous map $f : |M_1| \to
|M_2|$ together with an equivalence class of $C^{\infty}$ lifts of
$f$. For simplicity, we denote a $C^{\infty}$ map by $f : M_1 \to

Unfortunately, $C^{\infty}$ maps in the sense of Definition
<ref> do not have good geometric properties (for
instance, in general they do not pull vector bundles back to vector bundles). To
rectify this issue, following <cit.>, we consider those $C^{\infty}$ maps that have the
following type of representative in the equivalence class of $C^{\infty}$ lifts.

Let $M_1, M_2$ be orbifolds and let $f : |M_1| \to |M_2|$ be a
continuous map. A $C^{\infty}$ lift $(\mathcal{A}_1,\mathcal{A}_2,\{\widehat{f}_{\widehat{U}_1} : \widehat{U}_1 \to
\widehat{U}_2\})$ of $f$ is good if

* for all $(\widehat{U}_1,\Gamma_1,\varphi_1) \in
\mathcal{A}_1$, there exists a homomorphism $\Theta_{\widehat{U}_1} :
\Gamma_1 \to \Gamma_2$ such that $(\widehat{f}_{\widehat{U}_1},
\Theta_{\widehat{U}_1})$ is a good $C^{\infty}$ lift of the
restriction of $f$ to $U_1$,

* if $\lambda_1 :
(\widehat{U}'_1,\Gamma'_1,\varphi'_1) \hookrightarrow
(\widehat{U}_1,\Gamma_1,\varphi_1)$ is an embedding between
l.u.c.'s in $\mathcal{A}_1$, then there
exists an embedding $\lambda_2 : (\widehat{U}'_2,\Gamma'_2,\varphi'_2) \hookrightarrow
(\widehat{U}_2,\Gamma_2,\varphi_2)$ between the corresponding
l.u.c.'s in $\mathcal{A}_2$ such that the restriction of $(\widehat{f}_{\widehat{U}_1},
\Theta_{\widehat{U}_1})$ to $\widehat{U}'_1$ equals $(\widehat{f}_{\widehat{U}'_1},
\Theta_{\widehat{U}'_1})$.

Let $M_1$ and $M_2$ be orbifolds.

* A $C^{\infty}$ map $f : M_1 \to M_2$ is good if there
exists a good $C^{\infty}$ lift in the equivalence class of $f$.

* A diffeomorphism between $M_1$ and $M_2$ is a good
$C^{\infty}$ map $f : M_1 \to M_2$ that admits an inverse $g : M_2
\to M_1$ that is also a good $C^{\infty}$ map.

The above definition of good $C^{\infty}$ maps is somewhat
cumbersome and can be made more transparent using the approach to
orbifolds using Lie groupoids (see <cit.>
and references therein for more
details). Loosely speaking, an orbifold $M$ is an
equivalence class of certain Lie groupoids, where the topological
space $|M|$ is homeomorphic to the orbit space of any representative
in the equivalence class. (The equivalence relation is under the
so-called Morita equivalences, see <cit.>.) In the case of a global quotient
$\widehat{M}/G$, a Lie groupoid representing $\widehat{M}/G$ is the
action groupoid $G \ltimes \widehat{M}$. The orbifold structure
group at a point $x \in M$ corresponds to the isotropy Lie group of
the orbit representing $x$ with respect to any Lie groupoid
representing $M$.

From this perspective, if
$M_1,M_2$ are orbifolds, a
continuous map $f : |M_1| \to |M_2|$ is a good $C^{\infty}$ map if
and only if there exist Lie groupoids $\mathcal{G}_1$ and
$\mathcal{G}_2$ representing $M_1$ and $M_2$ respectively, and a Lie
groupoid homomorphism $\mathcal{F} : \mathcal{G}_1 \to
\mathcal{G}_2$, such that the induced map on the orbit spaces equals
$f$ – once the homeomorphisms identifying the orbit spaces of
$\mathcal{G}_1$ and of $\mathcal{G}_2$ with $|M_1|$ and $|M_2|$
respectively are
taken into account – (see <cit.>).

The perspective on orbifolds via Lie groupoids goes beyond the scope
of this paper. However, it is used sparingly to give
quick sketches of proofs (see, for instance, Lemma <ref>

In what follows we give families of $C^{\infty}$ maps between
orbifolds that are good. We start by introducing the following notion (see

A $C^{\infty}$ map $f : M_1 \to M_2$ between orbifolds is regular if the preimage of the subset of $M_2$ consisting of
regular points under $f$ is open, dense and connected in $M_1$.

The following result, stated below without proof, is proved in

Regular $C^{\infty}$ maps are good.

The next result provides another large class of good $C^{\infty}$

Let $\widehat{M}_i$ be a manifold and let $G_i$ be a Lie group
acting on $\widehat{M}_i$ so that the action is proper and locally
free for $i=1,2$. If $\widehat{f} : \widehat{M}_1 \to \widehat{M}_2$
is a smooth map and if $\Theta : G_1 \to G_2$ is a homomorphism of
Lie groups such that $\widehat{f}$ is $\Theta$-equivariant, then the
continuous map $f : \widehat{M}_1/G_1 \to \widehat{M}_2/G_2$ given
by $[\widehat{x}] \mapsto [\widehat{f}(\widehat{x})]$ is a good
$C^{\infty}$ map.

We give two strategies to prove the above result. The first is to
use the l.u.c.'s for quotients given in Section
<ref>, i.e., to check the result in `local
coordinates'. It can be checked directly that $\widehat{f}$ and
$\Theta$ induce a good $C^{\infty}$ lift of the restriction of $f$
to any open neighborhood that is uniformized by a l.u.c., and that
the compatibility condition of Definition <ref>

The second (more direct) strategy is to use the Lie groupoid approach to orbifolds
(see Remark <ref>). The map $\widehat{f}$ and the
homomorphism $\Theta$ can be used to construct a Lie groupoid
homomorphism $\mathcal{F} : G_1 \ltimes M_1 \to G_2 \ltimes M_2$
with the property that the induced map on orbit spaces coincides
with $f$. The result then follows at once from <cit.>.

In the following example, we illustrate an application of Lemma

[Weighted projective spaces, continued]
As in Example <ref>, we fix an integer $n \geq 1$ and positive
integers $m_0,\dots, m_{n}$ satisfying $\gcd(m_0,\dots,m_n)=1$. We
consider the smooth $\C^*$-action on $\C^{n+1} \smallsetminus
\{0\}$ given by

\begin{equation}
\label{eq:11}
\begin{split}
\C^*\times\mathbb{C}^{n+1} \smallsetminus \{0\}
&\rightarrow \mathbb{C}^{n+1} \smallsetminus \{0\}\\
\left(w,(z_0,\dots,z_{n})\right)&\mapsto(w^{m_0}z_0,\dots,w^{m_n}z_n),
\end{split}
\end{equation}

(cf. (<ref>)). This action is proper and locally free,
so that the quotient $(\C^{n+1} \smallsetminus
\{0\})/\C^*$ inherits the structure of an orbifold. We denote an
element in $(\C^{n+1} \smallsetminus
\{0\})/\C^*$ by $[z_0:\ldots :z_n]$. We remark that, just as for
projective spaces, we may exploit the equivalence relation to carry
out calculations in `local coordinates' that we refer to as homogeneous coordinates (see, for instance, Example
<ref> below).

Let $E \subset \C^{n+1} \smallsetminus \{0\}$ be as in Example
<ref>. If $\widehat{f} : E \to \C^{n+1}
\smallsetminus \{0\}$ and $\Theta : S^1 \to \C^*$ denote the
inclusions, then, by Lemma <ref>, the induced continuous map $f : \C P^n(m_0,\ldots,
m_n) \to  (\C^{n+1} \smallsetminus
\{0\})/\C^*$ is a good $C^\infty$ map. In fact, it is a
diffeomorphism and the inverse is given by the `standard' projection
maps $\C^{n+1}
\smallsetminus \{0\} \to E$ and $\C^* \to S^1$. Throughout
this paper, we use this diffeomorphism to identify $(\C^{n+1} \smallsetminus
\{0\})/\C^*$ with $\C P^n(m_0,\ldots,
m_n)$ tacitly, trusting that this does not cause confusion. $\Diamond$

To conclude this section, we give a sketch of a proof of the following
result regarding good $C^{\infty}$ maps.

Let $f : M_1 \to M_2$ be a good $C^{\infty}$ map. For any point $x
\in M_1$, there is an induced conjugacy class of homomorphisms
$\Gamma_x \to \Gamma_{f(x)}$.

As in the proof of Lemma <ref>, we give two
strategies. The first is to check the result directly as in
<cit.>. The second (more direct) strategy is to use the Lie groupoid approach to orbifolds
(see Remark <ref>): With the latter, the result is
immediate as a Lie groupoid morphism induces homomorphisms between
isotropy groups and Morita equivalences preserve the conjugacy class
of these homomorphisms.

§.§.§ Compact, connected, orientable orbi-surfaces with isolated
singular points

In this section, we state a classification result for a certain family
of two-dimensional orbifolds, which we call orbi-surfaces. This
result should be seen as the analog of the well-known classification of closed
two-dimensional manifolds for orbifolds.

Let $\Sigma$ be an orientable orbi-surface, let $x \in
\Sigma$ be a point, and let $\oc$ be a l.u.c. centered at $x$ as in Remark
<ref>. The orbifold structure group $\Gamma$ of $x$ is a
subgroup of $\mathrm{SO}(2)$. Hence, if $\Gamma$ is not trivial, then it is
a finite cyclic subgroup $\Z_c$ for some natural number $c \geq
2$. If, in addition, $\Sigma$ is compact and has only isolated singular
points, then it has finitely many such points. The following result,
stated below without proof, provides a complete classification of such orbi-surfaces.

*  If $\Sigma$ be a compact, connected, orientable orbi-surface with isolated
singular points, then there exists a non-negative integer $g$ such
that $|\Sigma|$ is homeomorphic to the compact orientable surface of
genus $g$.

*  For $i=1,2$, let $\Sigma_i$ be a compact, connected, orientable orbi-surface with isolated
singular points, let $g_i$ be the genus of $|\Sigma_i|$, let
$S_i$ be the set of singular points of $\Sigma_i$, and let $c_i : S_i
\to \mathbb{N}$ be the map that sends a singular point to the order of its orbifold structure group. The two orbifolds $\Sigma_1$ and $\Sigma_2$ are diffeomorphic if
and only if $g_1 = g_2$ and there exists a bijection $\sigma : S_1
\to S_2$ such that $c_2 \circ \sigma = c_1$.

*  For any non-negative integers $g, n$ and for any map $c :
\{1,\ldots, n\} \to \mathbb{N}$ such that $c(j) \geq 2$ for all
$j=1,\ldots, n$, there exists a compact, connected, orientable orbi-surface with isolated
singular points $\Sigma$ such that $|\Sigma|$ is homeomorphic to the compact orientable surface of
genus $g$, the set of singular points $S$ of $\Sigma$ has cardinality
$n$ and there exists a bijection $\sigma : S \to \{1,\ldots, n\}$
with the property that $c(\sigma(p))$ is the order of the orbifold
structure group of $p$, for any $p \in S$.

§.§ Orbi-bundles

§.§.§ Definition and Examples

In this section, we follow mainly [20, 22].

Let $|M_i|$ be a topological space for $i=1,2$, let $\mathrm{pr}: |M_1| \to |M_2|$ be a
surjective continuous map, let $F$ be a connected smooth manifold and let $G$
be a Lie group acting on $F$. We say that $\mathrm{pr}: |M_1| \to
|M_2|$ admits a local trivializing l.u.c.
over the open set $U_2 \subseteq |M_2|$ with generic fiber $F$ and
structure group $G$ if there exist

* a l.u.c. $(\widehat{U}_2, \Gamma_2,\varphi_2)$ uniformizing

* a  l.u.c. $(\widehat{U}_1, \Gamma_1,\varphi_1)$ uniformizing

* a smooth map $\rho : \Gamma_2 \times \widehat{U}_2\to G$, and

* a diffeomorphism $\lambda : \widehat{U}_2 \times F \to
\widehat{U}_1$,

such that

* the map

\begin{equation*}
\begin{split}
\Gamma_2 \times \widehat{U}_2 \times F &\to \widehat{U}_2
\times F \\
(\gamma,\widehat{x},y) &\mapsto (\gamma \cdot \widehat{x}, \rho(\gamma,\widehat{x})(y))
\end{split}
\end{equation*}

is an action of $\Gamma_2$ on $\widehat{U}_2 \times F$, where
$\cdot$ denotes the $\Gamma_2$-action on $\widehat{U}_2$,

* the open set $U_1$ is uniformized by the l.u.c.
$(\widehat{U}_2 \times F, \Gamma_2, \varphi_1 \circ \lambda)$, and

* the pair $(\widehat{\mathrm{pr}}_1,
\mathrm{id})$ is a good $C^{\infty}$ lift of $\mathrm{pr}$.

We denote a local trivializing l.u.c. for $\mathrm{pr}: |M_1| \to
|M_2|$ over $U_2$ by

$$ (\widehat{U}_1 \to \widehat{U}_2, \Gamma_2,
\varphi_1,\varphi_2,\rho, \lambda). $$

The conditions in Definition <ref> imply that $\lambda : (\widehat{U}_2 \times F,
\Gamma_2, \lambda \circ \varphi_1) \hookrightarrow (\widehat{U}_1,
\Gamma_1,\varphi_1)$ is an isomorphism of l.u.c.'s and that
$(\widehat{\mathrm{pr}}_1 \circ \lambda^{-1},\lambda^{-1})$ is a
good $C^{\infty}$ lift of $\mathrm{pr}$. This is summarized in the
following commutative diagram:

\begin{equation*}
\xymatrix{\widehat{U}_2 \times F \ar[r]_-{\simeq}^-{\lambda}
\ar[dr]_-{\widehat{\mathrm{pr}}_1} & \widehat{U}_1
\ar[r]^-{\varphi_1} \ar[d]^-{\widehat{\mathrm{pr}}_1 \circ
\lambda^{-1}} & U_1 \ar[d]^-{\mathrm{pr}} \\
& \widehat{U}_2 \ar[r]_-{\varphi_2}& U_2.}
\end{equation*}

As the terminology suggests, local trivializing l.u.c.'s should be
thought of as being analogous to l.u.c.'s and local trivializations for
fiber bundles with a given structure group. In this sense, the following notion is the
analog of Definition <ref>.

Let $|M_i|$ be a topological space for $i=1,2$, let $\mathrm{pr}: |M_1| \to |M_2|$ be a
surjective continuous map, let $F$ be a smooth manifold, let $G$ be
a Lie group acting effectively on $F$, and let
$U_2, V_2 \subseteq |M_2|$ be open sets. Given local trivializing
l.u.c.'s $(\widehat{U}_1 \to \widehat{U}_2, \Gamma_2,
\varphi_1,\varphi_2,\lambda)$ and $(\widehat{V}_1 \to \widehat{V}_2, \Gamma'_2,
\psi_1,\psi_2,\mu)$ for $\mathrm{pr}: |M_1| \to |M_2|$ with generic
fiber $F$ and structure group $G$ over $U_2$
and $V_2$ respectively, an embedding of $(\widehat{U}_1 \to \widehat{U}_2, \Gamma_2,
\varphi_1,\varphi_2,\lambda)$ into $(\widehat{V}_1 \to \widehat{V}_2, \Gamma'_2,
\psi_1,\psi_2,\mu)$ is an embedding $\Lambda : (\widehat{U}_2,
\Gamma_2, \varphi_2) \hookrightarrow  (\widehat{V}_2,
\Gamma'_2, \psi_2)$ together with a map $g: \widehat{U}_2 \to
G$ such that the map

\begin{equation*}
\begin{split}
\widehat{U}_2 \times F &\to \widehat{V}_2 \times F \\
(\widehat{x},y) &\mapsto (\Lambda(x), g(x)(y))
\end{split}
\end{equation*}

is an embedding between $(\widehat{U}_2 \times F, \Gamma_2,
\varphi_1 \circ \lambda)$ and $(\widehat{V}_2 \times F, \Gamma'_2,
\psi_1 \circ \lambda)$. We denote such an embedding by
$(\Lambda, g)$ and say that $(\Lambda, g)$ extends $\Lambda$.

We say that $(\widehat{U}_1 \to \widehat{U}_2, \Gamma_2,
\varphi_1,\varphi_2,\lambda)$ and $(\widehat{V}_1 \to \widehat{V}_2, \Gamma'_2,
\psi_1,\psi_2,\mu)$ are isomorphic if there exists an
embedding  $(\Lambda, g)$ of $(\widehat{U}_1 \to \widehat{U}_2, \Gamma_2,
\varphi_1,\varphi_2,\lambda)$ into $(\widehat{V}_1 \to \widehat{V}_2, \Gamma'_2,
\psi_1,\psi_2,\mu)$ such that $\Lambda: (\widehat{U}_2,
\Gamma_2, \varphi_2) \hookrightarrow  (\widehat{V}_2,
\Gamma'_2, \psi_2)$ is an isomorphism.

It can be checked that the analogs of Remarks
<ref> and <ref> hold for
local trivializing l.u.c.'s.

Let $|M_i|$ be a topological space for $i=1,2$,  let $\mathrm{pr} : |M_1| \to |M_2|$ be a surjective
continuous map, let $F$ be a smooth manifold, and let $G$ be a Lie
group acting on $F$. We say that $\mathrm{pr} : |M_1| \to |M_2|$ admits an orbi-bundle atlas
with generic fiber $F$ and structure group $G$ if there exists an orbifold atlas
$\mathcal{A}_2$ on $|M_2|$ such that

* for each $(\widehat{U}_2,\Gamma_2,\varphi_2) \in
\mathcal{A}_2$, the map $\mathrm{pr} : |M_1| \to |M_2|$ admits a
local trivializing l.u.c. over $U_2 = \varphi_2(\widehat{U}_2)$ with generic fiber $F$ and structure group $G$ of
the form $(\widehat{U}_1 \to \widehat{U}_2, \Gamma_2,
\varphi_1,\varphi_2,\lambda)$, and

* for every pair $(\widehat{U}_2,\Gamma_2,\varphi_2),
(\widehat{V}_2,\Gamma'_2,\psi_2) \in
\mathcal{A}_2$ with $U_2 \cap V_2 \neq \emptyset$, for
every $x \in U_2 \cap V_2$, and for every
$(\widehat{W}_2,\Gamma''_2,\chi_2) \in \mathcal{A}_2$ with $x \in
W_2$ and with embeddings $(\widehat{W}_2,\Gamma''_2,\chi_2)
\hookrightarrow (\widehat{U}_2,\Gamma_2,\varphi_2)$ and $(\widehat{W}_2,\Gamma''_2,\chi_2)
\hookrightarrow  (\widehat{V}_2,\Gamma'_2,\psi_2)$, there exist
embeddings of $(\widehat{W}_1 \to \widehat{W}_2, \Gamma''_2,
\chi_1,\chi_2,\nu)$ into $(\widehat{U}_1 \to \widehat{U}_2, \Gamma_2,
\varphi_1,\varphi_2,\lambda)$ and $(\widehat{V}_1 \to \widehat{V}_2, \Gamma'_2,
\psi_1,\psi_2,\mu)$ that extend the
above embeddings of l.u.c.'s.

Given an orbi-bundle atlas as in Definition
<ref>, the collection
$\mathcal{A}_1=\{(\widehat{U}_1, \Gamma_1, \varphi_1)\}$ defines an orbifold atlas
on $|M_1|$. Hence, a posteriori, $M_1$ is an orbifold. The
second bullet point in Definition <ref>
is a compatibility condition between $\mathcal{A}_1$ and $\mathcal{A}_2$. For this reason, we denote an orbi-bundle atlas
by $(\mathcal{A}_1,\mathcal{A}_2)$ and we refer to the local
trivializing l.u.c.'s of Definition
<ref> as lying in $(\mathcal{A}_1,\mathcal{A}_2)$.

Let $|M_i|$ be a topological space for $i=1,2$,  let $\mathrm{pr} : |M_1| \to |M_2|$ be a surjective
continuous map, let $F$ be a smooth manifold and let $G$ be a Lie
group acting on $F$. Let $(\mathcal{A}_1,\mathcal{A}_2)$ and
$(\mathcal{B}_1,\mathcal{B}_2)$ be orbi-bundle atlases for
$\mathrm{pr} : |M_1| \to |M_2|$ with generic fiber $F$ and structure
group $G$.

*  The orbi-bundle atlas
$(\mathcal{B}_1,\mathcal{B}_2)$ refines
$(\mathcal{A}_1,\mathcal{A}_2)$ if every local trivializing
l.u.c. lying in $(\mathcal{B}_1,\mathcal{B}_2)$ admits an
embedding into some local trivializing
l.u.c. lying in $(\mathcal{A}_1,\mathcal{A}_2)$.

*  The orbi-bundle atlases
$(\mathcal{A}_1,\mathcal{A}_2)$ and
$(\mathcal{B}_1,\mathcal{B}_2)$ are equivalent if they admit
a common refinement.

*  An orbi-bundle with generic fiber $F$ and
structure group $G$ is a
surjective continuous map $\mathrm{pr} : |M_1| \to |M_2|$ endowed
with an equivalence class of orbi-bundle
atlases with generic fiber $F$ and
structure group $G$. To simplify notation, we denote an
orbi-bundle by $\mathrm{pr} : M_1 \to M_2$.

By Remark <ref>, Definition
<ref> agrees with the notion of
orbi-bundles in <cit.> and <cit.>. Moreover, an orbi-bundle atlas can be understood in
terms of the analog of transition maps (see <cit.>): this comes from unraveling the compatibility
condition between the atlases $\mathcal{A}_1$ and $\mathcal{A}_2$ in
Definition <ref>.

If $\mathrm{pr} : M_1 \to M_2$ is an orbi-bundle, then $\mathrm{pr}$
is a good $C^{\infty}$ map (see <cit.>).

In analogy with the case of manifolds, if $F$ is a vector space and
$G$ is (a subgroup of) $\mathrm{GL}(F)$, we call an orbi-bundle with
generic fiber $F$ and structure group $G$ a vector orbi-bundle.

If $M$ is an orbifold, one can construct the tangent
orbi-bundle $TM \to M$ in a natural fashion (see, for instance,
<cit.>). The fiber $T_x M$ over $x \in
M$ can be identified with the following orbi-vector space: If $\oc$
is a l.u.c. centered at $x$ and $\widehat{U} \subseteq \R^n$ (see Remark <ref>), then $T_x
M \simeq \R^n/\Gamma$. If $f : M_1 \to M_2$ is a good $C^{\infty}$
map, then there is a well-defined notion of derivative of $f$
at $x \in M_1$, i.e., a $\Gamma$-equivariant linear map $d_x f : T_x
M_1 \to T_{f(x)} M_2$, where $\Gamma$ is the orbifold structure
group of $x$ (see <cit.>).

[Standard vector orbi-bundles
over orbifolds]
All the standard operations on
vector bundles over manifolds extend to vector orbi-bundles over
orbifolds (see <cit.>). In particular, given an
orbifold $M$, one can construct the cotangent
orbi-bundle $T^*M \to M$ and all tensor bundles over $M$.  $\Diamond$

The construction of orbifolds using locally free and proper actions
described in Section <ref> can be generalized to
construct orbi-bundles. For our purposes, we need only the following
special case: Let $\widehat{M}, F$ be smooth manifolds, let
$G$ be a Lie group acting on $F$ and let $H$ be a Lie group. We
suppose that

* $H$ acts on $\widehat{M}$,

* there exists a smooth map $\rho : H \times \widehat{M} \to G$
such that the map

\begin{equation*}
\begin{split}
H \times \widehat{M} \times F &\to \widehat{M} \times F \\
(h,\widehat{x},y) &\mapsto (h \cdot x, \rho(h,\widehat{x})(y))
\end{split}
\end{equation*}

defines an $H$-action on $\widehat{M} \times F$, where $\cdot$
denotes the $H$-action on $\widehat{M}$,

* the $H$-action on $\widehat{M}$ is locally free, and

* the $H$-actions on $\widehat{M}$ and on  $\widehat{M} \times F$
are proper.

Since the $H$-actions on $\widehat{M}$ and $\widehat{M} \times F$ are
locally free and proper, we obtain orbifolds $\widehat{M}/H$ and
$(\widehat{M} \times F)/H$. Moreover, there is a surjective continuous
map $(\widehat{M} \times F)/H \to \widehat{M}/H$ such that the
following diagram commutes

\begin{equation*}
\xymatrix{\widehat{M} \times F \ar[d] \ar[r] & (\widehat{M} \times
F)/H \ar[d] \\
\widehat{M} \ar[r] & \widehat{M}/H,}
\end{equation*}

where the horizontal arrows are the quotient maps and the vertical
arrow on the left is projection onto the first component. Using the
standard l.u.c.'s on $\widehat{M}/H$ and
$(\widehat{M} \times F)/H$ (described in Section <ref>), it can be checked directly that
$(\widehat{M} \times F)/H \to \widehat{M}/H$ is an orbi-bundle with
generic fiber $F$ and structure group $G$. The following example gives
a concrete application of this construction.

[Weighted projective spaces, continued]
Fix an integer $n \geq 1$ and positive
integers $m_0,\dots, m_{n}$ satisfying
$\gcd(m_0,\dots,m_n)=1$. Fix an integer $k$ and consider the
$\C^*$-action on $(\C^{n+1} \smallsetminus \{0\}) \times\mathbb{C}$ given by

\begin{align*}
\C^* \times (\C^{n+1} \smallsetminus \{0\})\times\mathbb{C}&\rightarrow (\C^{n+1} \smallsetminus \{0\})\times\mathbb{C}\\
\end{align*}

(cf. (<ref>)). Together with the $\C^*$-action on $\C^{n+1} \smallsetminus \{0\}$ given in Example
<ref>, these two $\C^*$-action satisfy the above
hypotheses, where $G = \C^* = \mathrm{GL}(\C)$. The resulting complex vector

$$(  (\C^{n+1} \smallsetminus \{0\}) \times \C)/\C^* \to
\mathbb{C}P^{n}(m_0,\dots,m_n)$$

of rank 1 is denoted by
$\mathcal{O}_{m_0,\dots,m_n}(k)$. Finally, we observe that it is
also possible to construct $\mathcal{O}_{m_0,\dots,m_n}(k)$ as an
appropriate $S^1$-quotient of the product $E \times \C$, where $E$
is as in Example <ref>. We leave the details of this
latter construction to the reader (see Example
<ref>). $\Diamond$

Given an orbi-bundle $\mathrm{pr} : M_1 \to M_2$, there is a natural
notion of a $C^{\infty}$ section of $\mathrm{pr} : M_1 \to
M_2$ (see <cit.>): This is a $C^{\infty}$ map
$s : M_2 \to M_1$ that is a set-theoretic section of $\mathrm{pr}$
such that the $C^{\infty}$ lifts of $s$ in local trivializing
l.u.c.'s for $\mathrm{pr} : M_1 \to
M_2$ are equivariant sections. By <cit.>,
any $C^{\infty}$ section of $\mathrm{pr} : M_1 \to M_2$ is good.

In particular, it makes sense to define tensors on an orbifold
$M$ as $C^{\infty}$ sections of the appropriate vector
orbi-bundle constructed in Remark <ref> (see,
for instance, <cit.>). Moreover, the de
Rham differential $d : \Omega^{\bullet}(M) \to \Omega^{\bullet
+1}(M)$ can be defined in the same way as for manifolds. Hence, it
also makes sense to consider closed and exact forms, as well as de
Rham cohomology (see <cit.> for an approach
involving Lie groupoids). Finally, integration of differential
forms can be defined using the standard formulas.

By Remark <ref>, we may consider a vector field $\xi$ on an
orbifold $M$. The naive definition of the flow of $\xi$ on $M$
makes sense and satisfies the standard properties (see
[31] for a more sophisticated approach using Lie
groupoids). If $M$ has isolated singular points, this is
particularly simple using l.u.c.'s and their compatibility. In particular, it can be shown that the standard Cartan
calculus holds for orbifolds.

To conclude this section, we introduce principal
$G$-orbi-bundles: These are orbi-bundles in the sense of Definition
<ref>, where $F = G$ and $G$ acts on
itself by left multiplication. In fact, just as for manifolds, it is
possible to go from orbi-bundles over $M$ with typical fiber $F$ and
structure group $G$ to principal $G$-orbi-bundles by means of the
standard construction and vice versa (see, for instance, <cit.>). In particular, this establishes the `standard' correspondence
between complex line orbi-bundles and principal $S^1$-orbi-bundles over orbisurfaces (see, for example, [20]).

§.§.§ Seifert manifolds

To start, we define certain 3-manifolds endowed with an
$S^1$-action, first introduced by Seifert in [67].

A Seifert manifold is a 3-dimensional compact, connected oriented manifold
$M$ endowed with an effective, locally free,
orientation-preserving $S^1$-action. We call the
orbit map $\pi: M\rightarrow M/S^1$ a Seifert

Given a Seifert fibration $\pi : M \to M/S^1$, the construction
illustrated in Section <ref> yields that
orbit space $\Sigma:=M/S^1$ is a compact, connected oriented

Seifert manifolds and fibrations have been classified by
Raymond [60]. For completeness, in what follows we
illustrate this classification (see also <cit.>
and [33, 56]). Let $\pi:M\rightarrow \Sigma$ be a
Seifert fibration. Since $S^1$ is abelian, it makes sense to consider
the stabilizer of an orbit $S^1\cdot x$, where $x \in M$, which we
denote by $\Gamma$. Since the $S^1$-action on $M$ is effective, the
only possible stabilizers are the trivial subgroup and the cyclic
subgroups $\Z_{\alpha}$ given by $\alpha$-th roots of unity, where
$\alpha > 1$ is an integer. In particular, every orbit is
diffeomorphic to $S^1$. We say that an orbit $S^1 \cdot x$ is principal (respectively exceptional) if its stabilizer is
trivial (respectively $\Z_{\alpha}$ for some $\alpha >1$).

The Slice
Theorem (see <cit.>) provides a description of $\pi : M \to \Sigma$ near any
orbit, which we illustrate below. To this end, we fix $x \in M$ and let
$D^2\subset \R^2 \simeq \mathbb{C}$ be an open disc centred at the
origin. Here, $\R^2$ is identified with the normal space to the orbit
$S^1 \cdot x$ at $x$. Since the $S^1$-action is
orientation-preserving, the image of the isotropy representation $\rho
(\Gamma)$ lies in $\mathrm{SO}(2) \simeq \mathrm{U}(1)$, so that
$\rho$ is completely determined by $\nu \in \Z$ such that $\rho
(\gamma) = \gamma^{\nu}$, for all $\gamma \in \Gamma
\subset S^1$. We consider the $\Gamma$-action on $S^1 \times D^2$
given by

\begin{equation}
\label{eq:10}
\gamma \cdot (\lambda, w) = (\lambda \gamma^{-1}, \gamma^{\nu}z).
\end{equation}

This action is free and proper so that the quotient $(S^1 \times
D^2)/\Gamma$ is a smooth manifold. We endow $(S^1 \times
D^2)/\Gamma$ with the $S^1$-action given by

By the Slice Theorem, an
$S^1$-invariant neighborhood of $S^1 \cdot x$ is $S^1$-equivariantly
isomorphic to $(S^1 \times
D^2)/\Gamma$. In particular, since the $S^1$-action on $M$ is
effective, it follows that $\nu \neq 0$ and, if $\Gamma = \Z_{\alpha}$
for some $\alpha >1$, then $\gcd(\alpha, \nu)=1$. In the latter case
(i.e., if the orbit is exceptional),
there exists $\gamma \in \Gamma$ such that $\gamma^{\nu} =
e^{\frac{2\pi i}{\alpha}}$ and $\beta \in \Z$ with $\gcd(\alpha ,
\beta)=1$ such that (<ref>) becomes

$$ e^{\frac{2\pi i}{\alpha}} \cdot (\lambda, w) = \left(\lambda
e^{-\frac{2\pi i \beta}{\alpha}}, e^{\frac{2\pi i}{\alpha}}w
\right). $$

If we choose $0 < \beta < \alpha$, we obtain the Seifert
invariant of an exceptional orbit, which we denote by $(\alpha,
\beta)$.

A closer look at the above argument yields the following extra

*  exceptional orbits are isolated in $\Sigma$, and

*  locally near an orbit with stabilizer $\Gamma$ (and isotropy
representation determined by the integer $\nu$), the Seifert
fibration $\pi : M \to \Sigma$ is isomorphic to the map $ (S^1 \times
D^2)/\Gamma \to D^2/\Gamma$ sending $[\lambda, w] \to [w]$, where
the $\Gamma$-action on $D^2$ is given by the second component of

In particular, since exceptional orbits are precisely the singular points
of the orbifold $\Sigma$, by property <ref> the orbifold structure on $B$ has only isolated
singular points. In particular, by Remark
<ref> and part <ref> of Theorem
<ref>, there exists a non-negative integer $g$
such that $|B|$ is homeomorphic to the compact orientable surface of
genus $g$. Finally, by property <ref>, $\pi : M \to
\Sigma$ is a principal $S^1$-orbi-bundle.

Let $S^1 \cdot x$ be an exceptional orbit with stabilizer $\Gamma$
and Seifert invariant
$(\alpha, \beta)$. It can be shown that the quotient $(S^1 \times
D^2)/\Gamma$ is, in fact, $S^1$-equivariantly
isomorphic to a solid torus $S^1 \times
D^2$ endowed with an $S^1$-action that is not trivial on both components – for
more details, see <cit.>. Moreover, this identification can be extended to an identification
between $(S^1 \times \overline{D}^2)/\Gamma$ and $S^1 \times
\overline{D}^2$. Under this identification, the exceptional orbit corresponds to $S^1
\times \{0\}$ and a principal orbit lying on the boundary $S^1
\times S^1$ intersects the `meridian' $\{1\} \times S^1$ in precisely
$\alpha$ points.

By Remark <ref>, there exists a positive integer
$n$ such that $\Sigma$ has precisely $n$ singular points, that we denote by
$[x_1],\dots,[x_n]$ for some $x_1,\ldots, x_n \in
M$. Let $(\alpha_i,\beta_i)$ denote the Seifert invariant of $S^1
\cdot x_i$ for $i=1,\ldots, n$. The remaining invariant of the Seifert
fibration $\pi : M \to \Sigma$ is encoded by how the neighborhoods of the
exceptional orbits are `glued in'. To make this precise, for each
$i=1,\ldots, n$, let $\Gamma_i$ be the stabilizer of $x_i$ and let $\mathcal{V}_i \subset M$ be an $S^1$-invariant
open neighborhood of $[x]$ that is $S^1$-equivariantly
diffeomorphic to $(S^1 \times D^2)/\Gamma_i$. Moreover, we fix a point
$x_0 \in M$ with trivial stabilizer and let $\mathcal{V}_0 \subset M$
be an open neighborhood of $[x_0]$ in $M$. (The reason for
this extra neighborhood becomes clear in what follows.) Finally, we choose
$\mathcal{V}_0, \mathcal{V}_1,\ldots , \mathcal{V}_n$ sufficiently
small so that their closures are pairwise disjoint. For $i=0,\ldots, n$, we set $D_i:= \pi
(\mathcal{V}_i) \subset \Sigma$; by construction, these are homeomorphic to
open discs centered at the respective orbits that have closures that are pairwise
disjoint. By Remark <ref>, the complement $\Sigma
\smallsetminus\bigcup_{i=0}^n \overline{D}_i$ is a surface with vanishing
second cohomology. Moreover, the restriction

\begin{equation}
\label{eq:12}
\pi : M \smallsetminus \bigcup_{i=0}^n \overline{\mathcal{V}}_i \to \Sigma
\smallsetminus\bigcup_{i=0}^n \overline{D}_i
\end{equation}

is a principal $S^1$-bundle. Hence, it is a trivial principal
$S^1$-bundle. In particular, by Remark
<ref>, in order to reconstruct the
manifold $M$ from the non-negative integers $g, n$ and the Seifert invariants
$(\alpha_1,\beta_1),\ldots, (\alpha_n,\beta_n)$, it suffices to
understand how the closed solid tori $\overline{\mathcal{V}}_0,
\ldots, \overline{\mathcal{V}}_n$ are glued to $M \smallsetminus
\bigcup_{i=0}^n \mathcal{V}_i$.

To understand the gluing maps, we start by observing that, since
(<ref>) is trivial, there exists a smooth section
$\sigma: \Sigma
\smallsetminus\bigcup_{i=0}^n \overline{D}_i \to
M \smallsetminus \bigcup_{i=0}^n \overline{\mathcal{V}}_i$. Moreover, $\sigma$ extends to
the closure $\Sigma
\smallsetminus\bigcup_{i=0}^n D_i $ of $\Sigma
\smallsetminus\bigcup_{i=0}^n \overline{D}_i$; by a slight abuse of
notation, we denote the extension also by $\sigma$. We set $\partial_i
\sigma:=
\sigma(\partial \overline{D}_i)$ and let $b_i \in \partial
\overline{\mathcal{V}}_i$ be a principal orbit (see Remark
<ref>), for $i=0,\ldots, n$. Since
$\sigma$ is a section, the homology classes of $\partial_i \sigma$ and $b_i$ form a basis of $H_1(\partial
\overline{\mathcal{V}}_i)$. A different basis of  $H_1(\partial
\overline{\mathcal{V}}_i)$ is given by the homology classes of a choice of meridian $a_i$ and
parallel $\ell_i$, which are the `standard' ones for
$\overline{\mathcal{V}}_0$ and the ones given by the images of $\{1\} \times S^1$ and $S^1
\times \{1\}$ under the identification of Remark
<ref> for $i=1,\ldots, n$. (In what
follows we abuse notation and denote a curve and its homology class by
the same symbol.) Since $\partial
\overline{\mathcal{V}}_i \simeq S^1 \times S^1$, the isotopy
class of the gluing map of $\overline{\mathcal{V}}_i$ is determined by
the induced automorphism on $H_1(\partial\overline{\mathcal{V}}_i)$, which, in turn, is codified by how the basis
$a_i, \ell_i$ is related to the basis $\partial_i \sigma$ and
$b_i$. It can be shown (see <cit.>), that there
exists $\beta_0 \in \Z$ such that the section $\sigma$ can be chosen to
satisfy[In some sense, the appearance of $\beta_0$ is so that
(<ref>) holds and can be thought of as a `balancing' constant. This justifies the removal of an extra solid
torus in the beginning of the discussion.]

\begin{equation}
\label{eq:13}
\begin{split}
a_0 &= \partial \sigma_0 - \beta_0 b \\
a_i& =\alpha_i\partial\sigma_i-\beta_ib_i\quad\text{for}\quad 1 \leq
i\le n.
\end{split}
\end{equation}

Moreover, together with the identification of Remark <ref>, (<ref>) determines how $a_i, \ell_i$ is related to the basis $\partial_i \sigma$ and
$b_i$ for $i=0,\ldots, n$.

Let $\pi : M \to \Sigma$ be a Seifert fibration. Its Seifert
invariant is given by

$$ (g;\beta_0, (\alpha_1,\beta_1),\dots,(\alpha_n,\beta_n)). $$

If two Seifert fibrations have equal Seifert invariants, then the
total spaces are diffeomorphic. Conversely, given any suitable
abstract data $
(g;\beta_0, (\alpha_1,\beta_1),\dots,(\alpha_n,\beta_n)) $, there
exists a Seifert fibration with the above as its Seifert invariant
(see <cit.>).

If a Seifert fibration $\pi : M \to \Sigma$ has Seifert invariant $
(g;\beta_0, (\alpha_1,\beta_1),\dots,(\alpha_n,\beta_n)) $, then the
compact, connected orientable orbi-surface $\Sigma$ only has isolated
singular points (see Remark <ref>). Hence, by
Theorem <ref>, $\Sigma$
is classified by $g$,
the non-negative integer $n$ and the pairs  $(\alpha_1,\beta_1),\dots,(\alpha_n,\beta_n)$.

Following <cit.>, we state a result that shows how the Seifert
invariant of a Seifert fibration $\pi : M \to \Sigma$ determines the Euler
class of $\pi : M \to \Sigma$. First, we need to define the latter
notion. Instead of using a conceptually clearer but more involved
approach (as in, e.g., [29]), we follow a simpler, ad
hoc path as
in <cit.> or <cit.>. Let $\pi : M
\to \Sigma$ be a Seifert fibration with Seifert invariant $(g;\beta_0,
(\alpha_1,\beta_1),\dots,(\alpha_n,\beta_n))$. Let $\alpha =
\mathrm{lcm}(\alpha_1,\ldots, \alpha_n)$, where $\alpha :=1$ if
$n=0$. We set $M':= M/\Z_{\alpha}$, where $\Z_{\alpha}$ acts on $M$ via
the inclusion in $S^1$. By <cit.>, the
quotient $M'$ can be endowed with a structure of smooth manifold such

* the quotient map $q: M \to M'$ is smooth,

* there is a free and proper smooth action of $S^1/\Z_{\alpha}$ on
$M'$, and

* the quotient map is equivariant with respect to the projection $S^1 \to

In the presence of exceptional orbits, the quotient map $q$ fails to
be a covering; in fact, it is a branched covering along the
exceptional orbits. (It is what is known as an orbifold covering
map, see Definition <ref> below.) Together with the second bullet point above, this
implies that the base space of the principal $S^1$-bundle $\pi' : M'
\to M'/(S^1/\Z_{\alpha})$ can be identified
with $|\Sigma|$. Abusing notation, we denote $|\Sigma|$ with the resulting
manifold structure by $B$. If $e' \in H^2(|\Sigma|;\Z) \simeq \Z$ denotes the Euler
class of $\pi' : M' \to \Sigma$, then we define the Euler class of
$\pi : M \to \Sigma$ to be $e:= e'/\alpha \in H^2(|\Sigma|;\Q) \simeq \Q$
(cf. <cit.>).

The Euler class $e$ of a Seifert fibration with Seifert invariant $ (g;\beta_0, (\alpha_1,\beta_1),\dots,(\alpha_n,\beta_n))$
is given by

\begin{equation}
\label{eq:23}
\end{equation}

The idea is to calculate $e$ using standard obstruction theory as in
the manifold case, using <cit.>. We fix the notation
as in the derivation of the Seifert invariant of a Seifert
fibration. Since all meridians are homologous to zero in $M$,
(<ref>) implies that

\begin{equation}
\label{eq:14}
\partial \sigma_0 = \beta_0 b_0 \text{ and } \partial \sigma_i =
\frac{\beta_i}{\alpha_i}b_i \text{ for } 1 \leq i \leq n.
\end{equation}

Any two principal orbits are homologous in $M$: To see this, first one can
prove it for any two principal orbits that are contained in an
$S^1$-invariant open neighborhood of a principal orbit and then use
the fact that the open subset of principal orbits is path-connected
in $M$. Hence, using (<ref>) and the fact that $\partial
\sigma = \sum\limits_{i=0}^n \partial \sigma_i$, we obtain that

$$ \partial \sigma = \left(\beta_0 + \sum\limits_{i=1}^n
\frac{\beta_i}{\alpha_i}\right) b_0. $$

Unraveling the obstruction-theoretic definition of $e$ given in
<cit.>, the desired result follows.

Under the correspondence between complex line orbi-bundles and principal
$S^1$-orbi-bundles over (compact, connected orientable)
orbi-surfaces (see Section <ref>), the Euler
class of a Seifert fibration is equal to the first Chern class
(or degree) of the corresponding complex line orbi-bundle. (For
our purposes, we take this to be a definition of the latter,
which can be alternatively defined intrinsically using the ideas in [29].)

The above discussion of Seifert manifolds can be generalized to Seifert orbifolds (see <cit.>): the former are 3-dimensional, compact connected oriented
orbifolds that are the total spaces of principal $S^1$-orbi-bundles
over compact, connected oriented orbi-surfaces that have only
isolated singular points. Such a principal $S^1$-orbi-bundle $\pi :
M \to \Sigma$ is called a Seifert orbi-fibration. Associated
to $\pi : M \to \Sigma$ is a Seifert invariant given as in
Definition <ref> and the Euler class of $\pi
: M \to \Sigma$ can be obtained from the Seifert invariant by

To conclude this section, we use Seifert manifolds to construct
certain four-dimensional orbifolds that play an important role in our

[From Seifert fibrations to projectivized plane bundles]
Let $\pi : M \to \Sigma$ be a Seifert fibration
and let $\mathrm{pr} : L \to \Sigma$ be the corresponding complex
line orbi-bundle. The projectivization $\mathbb{P}(L \oplus
\C) \to \Sigma$ is an orbifold that is the total space of an
orbi-bundle with generic fiber $\C P^1$. To see this, we consider
the manifold (!) $M \times \C P^1$ endowed with the following

\begin{equation}
\label{eq:16}
\lambda \cdot (x,[z_0:z_1]) = (\lambda x, [z_0:\lambda z_1]).
\end{equation}

Since the $S^1$-action on $M$ is locally free, so is this
$S^1$-action on $M \times \C P^1$. Hence, the quotient $(M \times \C
P^1)/S^1$ is an orbifold. In fact, since the exceptional orbits in
$M$ are isolated, it follows that $(M \times \C
P^1)/S^1$ has isolated singular points. The set of singular points

\begin{equation*}
\begin{split}
& \{[x,[0:1]] \in (M \times \C
P^1)/S^1 \mid [x] \text{ is singular in } \Sigma \} \\
\cup & \{[x,[1:0]] \in (M \times \C
P^1)/S^1 \mid [x] \text{ is singular in } \Sigma \}.
\end{split}
\end{equation*}

Moreover, since the above $S^1$-action preserves the projection $M
\times \C P^1 \to \Sigma$, there is a $C^{\infty}$ map $(M \times \C
P^1)/S^1 \to \Sigma$ that can be checked to be an orbi-bundle with
generic fiber $\C P^1$. It remains to show that the total space of
this orbi-bundle can be identified with  $\mathbb{P}(L \oplus
\C)$. To this end, we use the idea of <cit.>:
the map $(M \times \C
P^1)/S^1 \to \mathbb{P}(L \oplus
\C)$ that sends $[x,[z_0:z_1]]$ to $[xz_0:z_1]$ is the desired
identification. $\Diamond$

§.§.§ Orbifold covering maps

For our purposes, it is also useful to introduce orbi-bundles with
zero dimensional disconnected fibers. To this end, we follow
<cit.> and <cit.>.

Let $M_1$ and $M_2$ be orbifolds. A surjective continuous map
$\mathrm{pr}: |M_1| \to |M_2|$ is an orbifold covering map if,
for any $x \in M_2$, there exists an open neighborhood $U \subseteq
M_2$ of $x$ and a l.u.c. $\oc$ uniformizing $U$ such that, for any
connected component $V$ of $\mathrm{pr}^{-1}(U)$, there exist a
subgroup $\Gamma'$ of $\Gamma$ and a
l.u.c. $(\widehat{U},\Gamma', \psi)$ uniformizing $V$ with $\varphi
= \mathrm{pr} \circ \psi$. We denote such a map by $\mathrm{pr}: M_1 \to M_2$.

In other words, orbifold covering maps are locally modeled on taking
quotients of subgroups. Hence, in general, an orbifold covering
map fails to be a topological covering map, but it is one away from
singular points. Moreover, any orbifold covering map is automatically
a good $C^{\infty}$ map.

If $\mathrm{pr}: M_1 \to M_2$ is an orbifold covering map and the set
of singular points in $M_2$ has codimension at least two (as in the
case that is of most interest to us), then there exists $r \in
\Z_{\geq 0} \cup \{\infty\}$ such that, for any regular $x \in M_2$,
the cardinality of $\mathrm{pr}^{-1}(x)$ equals $r$. Following <cit.>, we call $r$ the
geometric degree of $\mathrm{pr}: M_1 \to M_2$.

[An orbifold covering map for the weighted projective line]
Let $p, q$ be positive integers that are coprime. We claim that the

\begin{equation}
\label{eq:19}
\begin{split}
\mathrm{pr}: \C P^1 & \to \C P^1(p,q) \\
[z_0:z_1] &\mapsto [z^p_0:z^q_1]
\end{split}
\end{equation}

is an orbifold covering map of geometric degree $pq$. To see this,
we observe that $\mathrm{pr}$ is induced by the smooth map

\begin{equation}
\label{eq:20}
\begin{split}
\widehat{\mathrm{pr}}: \C^2 \smallsetminus \{0\} & \to \C^2 \smallsetminus
\{0\} \\
(z_0,z_1) &\mapsto (z^p_0,z^q_1).
\end{split}
\end{equation}

This map of (<ref>) is surjective; hence so is $\mathrm{pr}$. Moreover, it
is a covering map when restricted to
$(\C^*)^2$. Hence, the restriction of $\mathrm{pr}$ to the
complement of the singular points in $\C P^1(p,q)$ is an (orbifold)
covering map. Finally, by considering local charts near $[1:0]$ and
$[0:1]$ in $\C P^1$ and the standard l.u.c.'s near $[1:0]$ and
$[0:1]$ in $\C P^1(p,q)$, it can be checked that $\mathrm{pr} : \C P^1 \to \C P^1(p,q)$
satisfies the condition of Definition
<ref> near the singular points of
P^1(p,q)$. Finally, the geometric degree of $\mathrm{pr} : \C P^1 \to \C
P^1(p,q)$ is $pq$.

To conclude this section, we use Example <ref>
to compute the degree of the complex line
orbi-bundle $\mathcal{O}_{p,q}(k) \to \C P^1(p,q)$ (see Example
<ref> and Remark <ref>). While this result
is known (see <cit.> and
<cit.>), we provide a proof for completeness.

Let $p,q$ be positive integers that are coprime and let $k$ be an
integer. The degree of the complex line orbi-bundle
$\mathcal{O}_{p,q}(k) \to \C P^1(p,q)$ equals $k/pq$.

Let $\mathcal{O}(k) \to \C P^1$ denote the complex line bundle of
degree $k$. We claim that the map

\begin{equation}
\label{eq:18}
\begin{split}
\mathrm{Pr}: \mathcal{O}(k) & \to \mathcal{O}_{p,q}(k) \\
[z_0,z_1,v] &\mapsto [z^p_0,z^q_1,v]
\end{split}
\end{equation}

is an orbifold covering map that makes the following diagram commute

\begin{equation}\label{eq:21}
\begin{tikzcd}
\mathcal{O}(k) \arrow[r,"\mathrm{Pr}"] \ar{d} &
\mathcal{O}_{p,q}(k)\arrow{d} \\
\C P^1 \arrow[r,"\mathrm{pr}"] & \C P^1(p,q),
\end{tikzcd}
\end{equation}

where $\mathrm{pr} : \C P^1 \to \C P^1(p,q)$ is the orbifold
covering map of Example <ref>. This follows
by the same argument as in Example <ref> once
we observe that $\mathrm{Pr}$ is induced by the smooth map

\begin{equation*}
\begin{split}
\widehat{\mathrm{Pr}}: (\C^2 \smallsetminus \{0\}) \times \C & \to (\C^2 \smallsetminus \{0\}) \times \C \\
(z_0,z_1,v) &\mapsto (z^p_0,z^q_1,v),
\end{split}
\end{equation*}

which makes the following diagram commute

\begin{equation*}
\begin{tikzcd}
(\C^2 \smallsetminus \{0\}) \times \C \arrow[r,"\widehat{\mathrm{Pr}}"] \ar{d} &
(\C^2 \smallsetminus \{0\}) \times \C\arrow{d} \\
\C^2 \smallsetminus \{0\} \arrow[r,"\widehat{\mathrm{pr}}"] & \C^2 \smallsetminus \{0\},
\end{tikzcd}
\end{equation*}

where $\widehat{\mathrm{pr}} : \C^2 \smallsetminus \{0\} \to \C^2
\smallsetminus \{0\}$ is given by (<ref>). Moreover, this
argument also implies that $\mathrm{pr}^*(\mathcal{O}_{p,q}(k))
\simeq \mathcal{O}_k$.

Let $S(\mathcal{O}_k) \to \C P^1$ and
$S(\mathcal{O}_{p,q}(k)) \to \C P^1(p,q)$ denote the principal
$S^1$-orbi-bundles associated to $\mathcal{O}(k) \to \C P^1$ and
$\mathcal{O}_{p,q}(k) \to \C P^1(p,q)$ respectively. Then
$\mathrm{Pr} : \mathcal{O}(k) \to \mathcal{O}_{p,q}(k)$ induces an
orbifold covering map $S(\mathcal{O}_k) \to S(\mathcal{O}_{p,q}(k))$
that we also denote by $\mathrm{Pr}$ by a slight abuse of
notation. By construction, the analog of the commutative diagram
(<ref>) holds replacing the top row with $\mathrm{Pr}:
S(\mathcal{O}_k) \to S(\mathcal{O}_{p,q}(k))$, i.e., $\mathrm{Pr}$
sends fibers to fibers. Moreover, since $\mathrm{pr}^*(\mathcal{O}_{p,q}(k))
\simeq \mathcal{O}_k$, the preimage of a fiber of
$S(\mathcal{O}_{p,q}(k)) \to \C P^1(p,q)$ consisting of regular
points is the disjoint union of $pq$ fibers of $S(\mathcal{O}_k) \to
\C P^1$. Since $\mathrm{pr} : \C P^1 \to \C P^1(p,q)$ has geometric
degree $pq$ and since $S(\mathcal{O}_k) \to \C P^1$ has Euler class
equal to $k$, by the covering formula of <cit.>, the Euler class of $S(\mathcal{O}_{p,q}(k)) \to \C
P^1(p,q)$ equals $k/pq$. The result follows from Remark <ref>.

Combining Lemma <ref>, Remark
<ref> and Lemma <ref>, we obtain
the following result immediately.

Let $p,q$ be positive integers that are coprime and let $k$ be an
integer. Let $S(\mathcal{O}_{p,q}(k)) \to \C P^1(p,q)$ denote the principal
$S^1$-orbi-bundle associated to $\mathcal{O}_{p,q}(k) \to \C
P^1(p,q)$ and let $(0;b_0, (p,l_p),(q,l_q))$ be its Seifert
invariant. Then

\begin{equation}
\label{eq:24}
\frac{k}{pq} = b_0 + \frac{l_p}{p} + \frac{l_q}{q}.
\end{equation}

§.§ Suborbifolds

In this section, we follow [8, 50, 73].

Let $|M|$ be a topological space, let $\mathcal{A} =
\{(\widehat{U}_{\alpha},\Gamma_{\alpha},\varphi_{\alpha})\}_{\alpha \in A}$ be
an orbifold atlas on $|M|$, and let $|N| \subseteq |M|$ be a subset
endowed with the subspace topology. A suborbifold atlas on
$|N|$ is a subset $B \subseteq A$ together with a family of
connected embedded submanifolds $\{\widehat{V}_{\beta} \subseteq
\widehat{U}_{\beta}\}_{\beta \in B}$ such that

*  for all $\beta \in B$,
$\varphi_{\beta}(\widehat{V}_{\beta})$ is a non-empty open subset
of $|N|$,

*  $\bigcup\limits_{\beta \in B}
\varphi_{\beta}(\widehat{V}_{\beta}) = |N|$, and

*  for all $\beta \in B$, there exists a subgroup
$\Delta_{\beta}$ of $\Gamma_{\beta}$ so that

* $\widehat{V}_{\beta}$ is $\Delta_{\beta}$-invariant, and

* if $\gamma \in \Gamma_\beta$ and $\widehat{x} \in
\widehat{V}_{\beta}$ are such that $\gamma \cdot \widehat{x} \in
\widehat{V}_{\beta}$, then there exists $\delta \in
\Delta_{\beta}$ with $\gamma \cdot \widehat{x} = \delta \cdot
\widehat{x}$.

If, in addition, $\Delta_{\beta} = \Gamma_{\beta}$ for all $\beta
\in B$, the suborbifold atlas is said to be full.

As proved in <cit.>, if $M$ is an orbifold
and a subset $|N| \subseteq |M|$ has a suborbifold atlas, then $|N|$
inherits the structure of an orbifold, which we denote by
$N$. Moreover, the notion of full suborbifold atlas corresponds to
that of suborbifold atlas introduced in <cit.>.

Let $M$ be an orbifold. A (full) suborbifold of $M$ is a subset $|N|$
that admits a (full) suborbifold atlas.

A full orbifold is also known as a strong suborbifold in the
literature (see,
e.g. <cit.>).

If $N$ is a full suborbifold of $M$, the tangent orbi-bundle to $N$ can
be seen as an orbi-subbundle of $TM$ and so the normal
orbi-bundle $\nu_N$ to $N$
in $M$ may be defined as usual (see the discussion following

The proof of the existence of a tubular neighborhood for submanifolds
works mutatis mutandis for full suborbifolds. Hence, the
following result holds.

Let $M$ be an orbifold and let $N$ be a full suborbifold. Then there
exist an open neighborhood $U \subseteq M$ of $N$, an open
neighborhood $V \subseteq \nu_N$ of the zero section and a
diffeomorphism $\varphi : U \to V$ that equals the zero section when
restricted to $N$.

[Weighted projective spaces, continued]
Fix an integer $n \geq 1$, positive
integers $m_0,\dots, m_{n-1}$ satisfying
$\gcd(m_0,\dots,m_{n-1})=1$, and a positive integer $m_n$. As in the
case of manifolds, the
weighted projective space $\C P^{n-1}(m_0,\ldots, m_{n-1})$ can be
viewed as a (full) suborbifold of $\C P^n(m_0,\ldots,m_{n-1},m_n)$ as
follows. We observe that the inclusion $\C^n \smallsetminus \{0\}
\hookrightarrow \C^{n+1} \smallsetminus \{0\}$ given by

$$(z_0,\ldots, z_{n-1}) \mapsto (z_0,\ldots, z_{n-1},0) $$

is $\C^*$-equivariant under the actions on $\C^n
\smallsetminus \{0\}$ and $\C^{n+1}
\smallsetminus \{0\}$ defined in Example
<ref>. Hence, there is an induced map
$\C P^{n-1}(m_0,\ldots, m_{n-1}) \to \C P^{n}(m_0,\ldots, m_{n-1},
m_n)$ that is, in fact, a topological embedding; the image of this
map is precisely

\begin{equation}
\label{eq:15}
\{[z_0:\ldots: z_{n-1}:z_n] \in \C P^{n}(m_0,\ldots, m_{n-1},
m_n) \mid z_n = 0\}.
\end{equation}

It can be checked, either directly, or using <cit.>, that the subset (<ref>) is a (full) suborbifold of $ \C P^{n}(m_0,\ldots, m_{n-1},
m_n)$ that is diffeomorphic to $\C
P^{n-1}(m_0,\ldots, m_{n-1})$. The normal orbi-bundle to $\C
P^{n-1}(m_0,\ldots, m_{n-1})$ in $ \C P^{n}(m_0,\ldots, m_{n-1},
m_n)$ can be identified with the complement of $[0:\ldots:0:1]$ in $ \C P^{n}(m_0,\ldots, m_{n-1},
m_n)$ and is isomorphic to $\mathcal{O}_{m_0,\dots,m_{n-1}}(m_n)$. $\Diamond$

Let $p,q,k$ be pairwise coprime positive integers. We can use Example <ref> to calculate the
Seifert invariant of $S(\mathcal{O}_{p,q}(k)) \to \C P^1(p,q)$,
since this is the principal $S^1$-orbi-bundle associated to the
normal bundle to $\C P^1(p,q)$ in $\C P^2(p,q,k)$. We know that the
Seifert invariant is of the form $(0;b_0, (p,l_p),(q,l_q))$.

First we determine $l_p$ and $l_q$. Since $p,q$ are coprime, by
Bézout's lemma, there exist unique integers $n_1,n_2$ with
$|n_1|<q$ and $|n_2|<p$ such that

\begin{equation}
\label{eq:25}
\end{equation}

Moreover, since
$p,q>0$, $n_1 n_2 < 0$. Without loss of generality we may assume
that $n_1<0$. If we set $n_1^\prime:=-n_1$, $\beta:=n_2k$ and
$\alpha:=n_1^\prime k$, then, by (<ref>),

\begin{align}\label{ex_seifertinv}
\end{align}

If we view $\C P^2(p,q,k)$ as $(\C^3 \smallsetminus \{0\})/\C^*$ as
in Example <ref>, then $l_p$ (respectively $l_q$)
is obtained by understanding the action of $\Z_p$ (respectively
$\Z_q$) near the point $(1,0,0)$ (respectively $(0,1,0)$). If
$\lambda \in \Z_p$ and $\mu \in \Z_q$, by (<ref>),

\begin{equation*}
\lambda \cdot (z_0,z_1,z_2) = (z_0, \lambda^q z_1, \lambda^{q\beta}
z_2) \text{ and } \mu \cdot (z_0,z_1,z_2) = (\mu^p  z_0, z_1,\mu^{-p\alpha}z_2)
\end{equation*}

for any $(z_0,z_1,z_2) \in \C^3 \smallsetminus \{0\}$. Let

\begin{equation}
\label{eq:32}
\beta^\prime\equiv\beta\mod p \quad \text{and} \quad -\alpha^\prime\equiv-\alpha\mod q
\end{equation}

be such that $0<\beta^\prime<p$ and $0<q-\alpha^\prime<q$ respectively.
Since the two  structure groups act with exponent $\beta^\prime$ and
$q-\alpha^\prime$ in the fiber direction of the normal bundle (given
by the $z_2$-coordinate), $l_p = \beta'$ and $l_q =

It remains to calculate $b_0$. To this end, by definition of
$\alpha'$ and $\beta'$ and by
(<ref>), there exists an integer $N$ such that

\begin{equation}\label{eq:degree}
\beta^\prime q-\alpha^\prime p=k-Npq.
\end{equation}

By Corollary <ref>,

$$ \frac{k}{pq} = b_0

Hence, by (<ref>), $b_0=N-1$, so that the Seifert
invariant is

$$(0;N-1, (p,\beta'),(q,q-\alpha^{\prime})), $$

$\alpha^{\prime}, \beta'$ are defined in (<ref>).

If a complex line bundle $L\rightarrow \mathbb{C}P^1/\mathbb{Z}_m$  is
trivial, then the Seifert invariant of the corresponding  Seifert
fibration satisfies the following a useful condition.

Let $L\rightarrow \mathbb{C}P^1/\mathbb{Z}_m$ be a complex line
orbi-bundle whose associated circle orbi-bundle has Seifert invariant $(0;b,(m,l),(m,\hat{l}))$. If the degree of $L$ is zero then $l+\hat{l}=m$.

If the degree of $L$ is zero, then, by Proposition <ref>, we have

$$0 = \deg(L)=b+\frac{l+\hat{l}}{m},$$

and so $l+\hat{l}=\alpha\cdot m$ for some positive integer $\alpha$ since $1\leq l,\hat{l}<m$.  If  $\alpha\geq 2$, then we have

$$l = \alpha\cdot m-\hat{l} \geq 2 m -\hat{l}> m $$

which is impossible. Hence, $\alpha=1$ and the result follows.

[Projectivized plane bundles, continued]
Let $\Sigma$ be a compact, connected orientable orbi-surface and let
$\mathrm{pr} : L \to \Sigma$ be a complex line orbi-bundle. As in
the case of manifolds, the
projectivization $\mathbb{P}(L \oplus \C) \to \Sigma$ constructed in
Example <ref> admits two `standard' $C^{\infty}$
sections, namely $x \mapsto [0:1]$ and $x \mapsto [1:0]$, where the
former lifts to a section of the rank-two complex orbi-bundle $L \oplus
\C \to \Sigma$ and the latter does not. The images of these two
sections are full suborbifolds of $\mathbb{P}(L \oplus \C)$ that are
diffeomorphic to $\Sigma$. $\Diamond$

§.§ Group actions on orbifolds

§.§.§ Definition and examples

Let $G$ be a Lie group and let $M$ be an orbifold. A (smooth)
action of $G$ on $M$ is a continuous action of $G$ on $|M|$ such
that the map

\begin{equation*}
\begin{split}
\phi:G\times M &\rightarrow M \\
(g,x) &\mapsto g \cdot x
\end{split}
\end{equation*}

is a good $C^{\infty}$ map.

As in the case of manifolds, given a $G$-action on an orbifold $M$,
any element $\xi$ of the Lie algebra $\mathfrak{g}$ of $G$ gives
rise to a vector field $\xi^M $ on $M$ via the usual formula:

$$ \xi^M(x) := \frac{d}{d t}\bigg\rvert_{t = 0} \phi(\exp(t\xi),x)
\quad \text{for any } x \in M, $$

where $\exp : \mathfrak{g} \to G$ denotes the exponential map.

[Orbi-vector spaces, continued]
Let $\widehat{V}/\Gamma$ be an orbi-vector space. We define
$\mathrm{GL}(\widehat{V}/\Gamma): = N(\Gamma)/\Gamma$, where $N(\Gamma) \leq
\mathrm{GL}(\widehat{V})$ is the normalizer of $\Gamma$ in
$\mathrm{GL}(\widehat{V})$. A representation of a group $G$ on $\widehat{V}/\Gamma$ is a homomorphism $\rho : G \to
\mathrm{GL}(\widehat{V}/\Gamma)$. If $G$ is a Lie group, a smooth action of
$G$ on $\widehat{V}/\Gamma$ is linear if there exists a representation
$\rho$ of $G$ on $\widehat{V}/\Gamma$ such that $\phi(g,x) = \rho(g)(x)$ for
all $g \in G$ and all $x \in \widehat{V}/\Gamma$. In analogy with <cit.>, given a linear action of $G$ on $\widehat{V}/\Gamma$,
there exist a Lie group $\widehat{G}$, a representation $\rho:
\widehat{G} \to N(\Gamma)$ and a short exact sequence of Lie groups

\begin{equation*}
\begin{tikzcd}
1\arrow{r} & \Gamma\arrow[r,hook,"\iota"]& \widehat{G}\arrow[r,"\pi"] & G\arrow{r} & 1,
\end{tikzcd}
\end{equation*}

such that the following diagram commutes

\begin{equation*}
\begin{tikzcd}
1\arrow{r} & \Gamma\arrow[r,hook,"\iota"] \arrow[d,equal]&
\widehat{G}\arrow[r,"\pi"] \arrow[d,"\widehat{\rho}"]
& G\arrow{r} \arrow[d,"\rho"] & 1 \\
1\arrow{r} & \Gamma\arrow[r,hook,"\iota"]& N(\Gamma)\arrow{r}
& \mathrm{GL}(V/\Gamma) \arrow{r} & 1.
\end{tikzcd}
\end{equation*}

Moreover, $\rho$ is faithful if and only if $\widehat{\rho}$ is
faithful. $\Diamond$

Many of the smooth actions that we consider in this paper arise from
the following construction: Let $H$ be a Lie group that acts properly
and in a locally free fashion on the manifold $\widehat{M}$. Let $G$
be a Lie group that acts on $\widehat{M}$ so that the $G$- and
$H$-actions commute. The action of $g \in G$ on $\widehat{x} \in
\widehat{M}$ is denoted by $g\cdot x$. Then there is a continuous
$G$-action on $|\widehat{M}/H|$ given by

\begin{equation*}
\begin{split}
\phi: G \times |\widehat{M}/H| & \to |\widehat{M}/H| \\
(g, [\widehat{x}]) & \mapsto [g \cdot \widehat{x}].
\end{split}
\end{equation*}

By Lemma <ref>, $\phi$ is a good $C^{\infty}$
map. We stress that the $G$-action on $\widehat{M}/H$ need not be
effective even if the $G$-action on $\widehat{M}$ is. The following
examples illustrate the above construction.

[Weighted projective spaces, continued]
Fix an integer $n \geq 1$, positive
integers $m_0,\dots, m_{n}$ satisfying
$\gcd(m_0,\dots,m_{n})=1$, and integers $a_0,\ldots, a_n$. The map

\begin{equation*}
\begin{split}
\phi: S^1 \times \C P(m_0,\ldots, m_n) &\to \C P(m_0,\ldots,
m_n) \\
(\lambda, [z_0:\ldots:z_n]) &\mapsto [\lambda^{a_0}z_0:\ldots: \lambda^{a_n}z_n]
\end{split}
\end{equation*}

is an $S^1$-action on $\C P(m_0,\ldots, m_n)$. $\Diamond$

[Example <ref>, continued]
We let $M =  (\C P^1 \times \C P^1)/\Z_4$ be the orbifold
constructed in Example <ref>. The map

\begin{equation*}
\begin{split}
\phi: S^1 \times M &\to M \\
\left(\lambda,\left[[z_0:z_1],[w_0:w_1]\right]\right)&\mapsto
\left[[\lambda^{1/2}z_0:z_1],[ \lambda^{1/2} w_0:w_1]\right]
\end{split}
\end{equation*}

is an $S^1$-action on $M$. We observe that the expression involving
square roots of elements of $S^1$ are well-defined because of the
definition of the $\Z_4$-action on $\C P^1 \times \C P^1$. $\Diamond$

[Example <ref>, continued]
Given an integer $c > 1$, let $M_c =  (\C P^1 \times \C P^1)/\Z_c$ be the orbifold constructed in Example
<ref> and let $m,n$ be positive integers. The map

\begin{equation*}
\begin{split}
\phi: S^1 \times M_c &\to M_c \\
(\lambda, [[z_0: z_1],[w_0: w_1]]_c) &\mapsto [[ z_0: \lambda^{ \frac{m}{c}} z_1], [ w_0: \lambda^{\frac{n}{c}} w_1]]_c
\end{split}
\end{equation*}

is an $S^1$-action on $M_c$, where, as in Example
<ref>, there is no ambiguity in the
expressions involving $c$-th roots because of the $\Z_c$ action on
$\C P^1 \times \C P^1$. $\Diamond$

[Projectivized plane bundles, continued]
Let $\pi : M \to \Sigma$ be a Seifert fibration. The map

\begin{equation*}
\begin{split}
\phi : S^1 \times ((M
\times \C P^1)/S^1) & \to (M
\times \C P^1)/S^1 \\
(\lambda, [x,[z_0:z_1]]) &\mapsto [x,[\lambda z_0:z_1]]
\end{split}
\end{equation*}

is an $S^1$-action on the orbifold $(M
\times \C P^1)/S^1$ constructed in Example <ref>
(see (<ref>) for the $S^1$-action on $M \times \C
P^1$). Let $\mathrm{pr} : L \to \Sigma$ be the complex line
orbi-bundle corresponding to $\pi : M \to \Sigma$. Using the
identification between $(M
\times \C P^1)/S^1$ and $\mathbb{P}(L \oplus \C)$ given in Example
<ref>, we obtain an $S^1$-action on the latter. Note that  if $[x]\in \Sigma$ is a singular point with orbifold
structure group $\Z_k$, then by (<ref>)

[x,[\xi_k z_0:z_1]] = [x,[z_0:\xi_k^{-1} z_1]] =  [\xi_k x,[z_0: z_1]] = [x,[z_0: z_1]],

and so each point in the fiber over $[x]$ has $\mathbb{Z}_k$ as stabilizer.

§.§.§ Linearization at a fixed point

Lemma <ref> has the
following immediate consequence that is important for our purposes.

Let $G$ act on an orbifold $M$. If $x, y \in M$ lie in the same
$G$-orbit, then $\Gamma_x \simeq \Gamma_y$. In particular, if $M$ has
isolated singular points and $G$ is
connected, then any singular point in $M$ is fixed by the $G$-action.

Let $G$ be a compact Lie group acting on an orbifold $M$. The local structure of a $G$-action in a neighborhood of a fixed point
$x \in M$ is relevant for the purposes of our paper. First, we observe
that there exists a l.u.c. $\oc$
centered at $x$ such that the open neighborhood $U$ of $x$ that is
uniformized by $\oc$ is $G$-invariant. We call such a l.u.c. $\boldsymbol{G}$-invariant. Moreover, the $G$-action on $U$
can be described entirely by an action of a `larger' Lie group on
$\widehat{U}$. More precisely, the following result holds (for a proof, see <cit.> or
<cit.>; cf. Example <ref>).

Let $G$ be a compact Lie group acting on an orbifold $M$, let $x
\in M$ be a fixed point and let $\oc$ be a $G$-invariant
l.u.c. centered at $x$. There exist a Lie group $\widehat{G}$, a
smooth action of $\widehat{G}$ on $\widehat{U}$ fixing $0$, and a
short exact sequence of Lie groups

\begin{equation}\label{exact_sequ}
\begin{tikzcd}
1\arrow{r} & \Gamma\arrow[r,hook,"\iota"]& \widehat{G}\arrow[r,"\pi"] & G\arrow{r} & 1
\end{tikzcd}
\end{equation}

such that the following diagram commutes

G×U [r,"ϕ"] [d, "(π,φ)"]
U [d, "φ"]

G×U[r, "ϕ"]

where $\widehat{\phi} : \widehat{G} \times \widehat{U} \to
\widehat{U}$ encodes the $\widehat{G}$-action. Moreover, $\Gamma$ commutes with every connected subgroup of $\widehat{G}$.

On the other hand, as in the case of manifolds, there is a
linear $G$-action on the tangent space $T_xM$ given by taking
derivatives of the $G$-action at $x$ (see Remark
<ref> and Example
<ref>). The next result explains
the relation between the $G$-action near $x$ and the linear $G$-action
on $T_xM$; it is the analog of the Bochner linearization theorem for
orbifolds (see <cit.>).

Let $G$ be a compact Lie group acting on an orbifold $M$. If $x$ is a
fixed point, then there exist $G$-invariant open neighborhoods $U
\subseteq M$ and $W \subseteq T_xM$ of $x$ and $[0]$ respectively,
and a $G$-equivariant diffeomorphism $f : U \to W$ that sends $x$ to $[0]$.

Let $\oc$ be a $G$-invariant l.u.c. centered at $x$ and let
$\widehat{G}$ be the Lie group extending $G$ given by Proposition
<ref>. Since $\widehat{G}$ is compact,
by the Slice Theorem (see <cit.>), there is no loss of generality in assuming that the
$\widehat{G}$-action on $\widehat{U}$ is the restriction of a linear
$\widehat{G}$-action. By Proposition
<ref>, the map $\overline{\varphi} :
\widehat{U}/\Gamma \to U$ is a $G$-equivariant
diffeomorphism, where $\widehat{U}/\Gamma \subseteq \R^n/\Gamma$ is
endowed with the restriction of a linear $\widehat{G}/\Gamma \simeq
G$-action. Since the latter
linear action corresponds to the linear $G$-action on $T_x M$ under
the isomorphism $T_x M \simeq \R^n/\Gamma$, the result follows.

We conclude this section by stating the following consequence of
Corollary <ref> (see <cit.> for a proof).

Let $G$ be a compact Lie group that acts effectively on an orbifold
$M$. Any connected component of the fixed point set of the action
that has positive dimension[This restriction is
necessary as we are working only with effective
orbifolds.] is
a full suborbifold of $M$.

§.§.§ Quotients of orbifolds by finite

In this subsection we show that the quotient of an orbifold by an
effective action of a finite group inherits the structure of an
orbifold (see Corollary <ref> below), and then we
illustrate this construction with several examples that are relevant
for our purposes.

Let $M$ be an orbifold of dimension $n$. The orthonormal frame orbi-bundle
$\mathrm{Fr}(M) \to M$ can be constructed as in <cit.>: The total space can be obtained by gluing together the
orthonormal frame bundles of l.u.c.'s for $M$. (Alternatively, one can
define a Riemannian metric for an orbifold, prove that any orbifold
admits such a metric and construct the orthonormal frame
orbi-bundle as in the case of manifolds. We leave the details to the reader.) This principal
$\mathrm{O}(n)$-orbi-bundle has the following important property (see
<cit.> for a proof).

Let $M$ be an orbifold of dimension $n$. The total space
$\mathrm{Fr}(M)$ of the
orthonormal frame orbi-bundle of $M$ is a smooth manifold endowed with an
effective, locally free $\mathrm{O}(n)$-action such that the
quotient orbifold $\mathrm{Fr}(M)/\mathrm{O}(n)$ is diffeomorphic to

As a consequence of Theorem <ref>, every orbifold $M$ can be
thought of as a quotient orbifold $\widehat{M}/H$, where $H$ is
compact. This fact has the following useful consequence, stated below
without proof (see <cit.>).

If $G$ is a finite group acting on an orbifold $M$, then the
quotient $M/G$ can be endowed with the structure of an orbifold such
that the quotient map $M \to M/G$ is an orbifold covering map.

[Cyclic quotients of weighted projective lines]
Let $p,q$ be positive integers that are pairwise coprime and let
$a,b$ be integers. We consider the following smooth action of
$\mathbb{Z}_m$  on $\mathbb{C}P^1(p,q)$:

\begin{equation}
\label{eq:26}
\begin{split}
\phi:\mathbb{Z}_m\times\mathbb{C}P^1(p,q)&\rightarrow \mathbb{C}P^1(p,q)\\
\end{split}
\end{equation}

By the weighted homogeneity of the coordinates in $\C P^1(p,q)$,

\begin{equation}
\label{eq:27}
\mu \cdot [z_0:z_1] =  [z_0: \mu^{\frac{bp-aq}{p}}z_1] =
\end{equation}

for any $\mu \in \Z_m$ and any $[z_0:z_1] \in \C P^1(p,q)$. We set
$k_1:=|aq-bp|$.  A necessary and sufficient condition for this action to be effective
and so that the set of points with non-trivial stabilizer is discrete
is that $\gcd(k_1,m) = 1$. In this case, the only points with
non-trivial stabilizers are $[1:0]$ and $[0:1]$. By Corollary
<ref>, the quotient $\C P^1(p,q)/\Z_m(a,b)$ inherits the
structure of an orbifold that has two singular points (corresponding
to $[1:0]$ and $[0:1]$), and with respect
to which the quotient map $\C P^1(p,q) \to \C P^1(p,q)/\Z_m(a,b)$ is
an orbifold covering map. This orbifold structure can also be obtained
as a quotient orbifold (see Example

We claim that the orbifold structure group of the singular point
corresponding to $[1:0]$ is $\Z_{mp}$. To see this, we consider the
standard $\Z_{mp}$-action on $\C$ and observe that an open
neighborhood of $[1:0] \in \C P^1(p,q)$ is diffeomorphic to an open
neighborhood of $[0] \in \C/\Z_p$, where $\Z_p$ acts on $\C$ via the
inclusion $\Z_p \hookrightarrow \Z_{mp}$. Since $\gcd(k_1,m) = 1$, the residual $\Z_{mp}/\Z_p
\simeq \Z_m$-action on $\C/\Z_p$ can be identified with the
$\Z_m$-action on $\C/\Z_p$ given by (<ref>). Analogously, the structure group of the singular point
corresponding to $[0:1]$ is $\Z_{mq}$. In particular, by Theorem
<ref>, if $a', b' \in \Z$ are also such that
$k'_1:=|a'q-b'p|$ is coprime with $m$, then $\C P^1(p,q)/\Z_m(a,b)$
and $\C P^1(p,q)/\Z_m(a',b')$ are diffeomorphic. Hence, we write
simply $\C P^1(p,q)/\Z_m$ for this quotient.

line bundles over cyclic quotients of weighted projective lines]
In analogy with Example <ref>, let $p,q,k$ be positive integers that are pairwise coprime and let
$a,b,c$ be integers. We consider the following smooth action of $\mathbb{Z}_m$  on $\mathbb{C}P^2(p,q,k)$:

\begin{equation}
\label{eq:17}
\begin{split}
\phi:\mathbb{Z}_m\times\mathbb{C}P^2(p,q,k)&\rightarrow \mathbb{C}P^2(p,q,k)\\
\end{split}
\end{equation}

By the weighted homogeneity of the coordinates in

\begin{equation}
\label{eq:22}
\begin{split}
\lambda \cdot [z_0:z_1:z_2] & = [z_0: \lambda^{\frac{bp-aq}{p}}z_1:
\lambda^{\frac{cp-ak}{p}}z_2] \\
& = [\lambda^{\frac{aq-bp}{q}}z_0: z_1:
\lambda^{\frac{cq-bk}{q}}z_2] \\
& = [\lambda^{\frac{ak-cp}{k}}z_0: \lambda^{\frac{bk-cq}{k}} z_1:z_2],
\end{split}
\end{equation}

for any $\lambda \in \Z_m$ and any
$[z_0:z_1:z_2] \in \mathbb{C}P^2(p,q,k)$.
We set $k_1:=|aq-bp|, k_2:=|bk-cq|$ and $k_3:=|cp-ak|$. A necessary and sufficient condition for this action to be effective
and so that the set of points with non-trivial stabilizer is discrete

\begin{equation*}
\gcd(k_1,m) = \gcd(k_2,m) = \gcd(k_3,m) = 1.
\end{equation*}

In this case, the points that have non-trivial stabilizers are
$[1:0:0]$, $[0:1:0]$ and $[0:0:1]$. By Corollary
<ref>, the quotient $\mathbb{C}P^2(p,q,k)/\Z_m(a,b,c)$
inherits the structure of an orbifold that has three singular points,
corresponding to $[1:0:0]$, $[0:1:0]$ and $[0:0:1]$. This
orbifold structure can also be realized as a quotient orbifold (see
Example <ref> below). By (<ref>),
the orbifold structure groups of the singular points
corresponding to $[1:0:0]$, $[0:1:0]$ and $[0:0:1]$ are
$\Z_{mp}$, $\Z_{mq}$ and $\Z_{mk}$.

In analogy with Example <ref>, the subset
of $\mathbb{C}P^2(p,q,k)/\Z_m(a,b,c)$ given by $\{[z_0:z_1:0] \in \C P^2(p,q,k)\}/\Z_m $
is a (full) suborbifold that can be identified with the orbifold
$\C P^1(p,q)/\Z_m = \C P^1(p,q)/\Z_m(a,b)$ of Example <ref>. The
normal orbi-bundle to $\C P^1(p,q)/\Z_m$ in $\mathbb{C}P^2(p,q,k)/\Z_m(a,b,c)$ is
precisely $\mathcal{O}_{p,q}(k)/\Z_m(a,b,c) \to \C P^1(p,q)/\Z_m$, where the
$\Z_m$-action on $\mathcal{O}_{p,q}(k)$ is given by a formula entirely
analogous to (<ref>). $\Diamond$

For our purposes, we need to compute the degree of the complex line
orbi-bundle $\mathcal{O}_{p,q}(k)/\Z_m(a,b,c) \to \C P^1(p,q)/\Z_m$
of Example <ref> and to relate it to the Seifert invariant of the corresponding circle
bundle. The first of these results can be proved with the techniques of Lemma
<ref> using the orbifold covering map $\C P^1(p,q) \to
\C P^1(p,q)/\Z_m$. Hence, we omit its proof.

Let $p,q,k$ be positive integers that are pairwise coprime, let $m$
be a positive integer, and let
$a,b,c$ be integers such that

$$ \gcd(|aq-bp|,m) = \gcd(|bk-cq|,m) = \gcd(|cp-ak|,m) = 1.$$

The degree of the complex line orbi-bundle

$$\mathcal{O}_{p,q}(k)/\Z_m(a,b,c) \to \C P^1(p,q)/\Z_m$$

is $k/pqm$.

Combining Lemma <ref>, Remark
<ref> and Lemma <ref>, the
following result holds.

Let $p,q,k$ be positive integers that are pairwise coprime, let $m$
be a positive integer, and let
$a,b,c$ be integers such that

$$ \gcd(|aq-bp|,m) = \gcd(|bk-cq|,m) = \gcd(|cp-ak|,m) = 1.$$

Let $S(\mathcal{O}_{p,q}(k)/\Z_m(a,b,c)) \to \C
P^1(p,q)/\Z_m$ denote the principal $S^1$-orbi-bundle
associated to $\mathcal{O}_{p,q}(k)/\Z_m(a,b,c) \to \C
P^1(p,q)/\Z_m$ and let

$$(0;b_0, (mp, l_{mp}),

be the Seifert invariant of $S(\mathcal{O}_{p,q}(k)/\Z_m(a,b,c)) \to \C
P^1(p,q)/\Z_m$. Then

\begin{equation}
\label{eq:28}
\frac{k}{pqm} = b_0 + \frac{l_{mp}}{mp} + \frac{l_{mq}}{mq}.
\end{equation}

§.§ Symplectic orbifolds and Hamiltonian

§.§.§ Definition and Examples

The definition of a symplectic form carries on from manifolds to

Let $M$ be an orbifold. A symplectic form on $M$ is a closed,
non-degenerate 2-form $\omega \in \Omega^2(M)$. A symplectic
orbifold is a pair $(M,\omega)$, where $\omega$ is a symplectic
form on $M$. A symplectomorphism between two symplectic
orbifolds $(M_1,\omega_1),(M_2,\omega_2)$ is a diffeomorphism
$\varphi:M\rightarrow M'$ that satisfies
$\varphi^*\omega'=\omega$. We denote a symplectomorphism by $\varphi
: (M_1,\omega_1) \to (M_2,\omega_2)$.

As for manifolds, the existence of a non-degenerate
2-form on a connected orbifold $M$ implies that the dimension of $M$
is even. The simplest examples of symplectic orbifolds are

[Symplectic orbi-vector spaces]
Let $(\widehat{V},\widehat{\omega})$ be a real symplectic vector space
and let $\mathrm{Sp}(\widehat{V},\widehat{\omega}) < \mathrm{GL}(V)$
be the subgroup of linear maps that are symplectomorphisms. If $\Gamma < \mathrm{Sp}(\widehat{V},\widehat{\omega})$ is a
finite subgroup, then the orbi-vector space $\widehat{V}/\Gamma$
inherits a symplectic form $\omega$. We call $(\widehat{V}/\Gamma,
\omega)$ a symplectic orbi-vector space.

Let $(\widehat{V},\widehat{\omega})$ be as in Example
<ref> and let $\widehat{J}$ be a $\Gamma$-invariant (almost) complex structure
on $\widehat{V}$ that is compatible with $\widehat{\omega}$. (Such a
structure exists since $\Gamma$ is compact.) This allows us to consider $\widehat{V}$ as a Hermitian vector
space. Since the unitary group
$\mathrm{U}(\widehat{V})$ is the maximal compact
subgroup of $\mathrm{Sp}(\widehat{V},\widehat{\omega})$ (see
<cit.>), $\Gamma <
\mathrm{U}(\widehat{V})$. In particular, the real orbi-vector
space $\widehat{V}/\Gamma$ can be endowed with the structure of a
Hermitian orbi-vector spaces.

There are two basic constructions of symplectic orbifolds starting
from symplectic manifolds, which should be thought of as the analogs
of global quotients and quotients in the symplectic category (see Section
<ref>). Let $(\widehat{M},\widehat{\omega})$ be
a symplectic manifold. First, we suppose that $\Gamma$ is a finite group that acts on
$\widehat{M}$ effectively and by symplectomorphisms. The global
quotient $M:= \widehat{M}/\Gamma$ inherits a symplectic form
$\omega$; this is clearly the analog of a global quotient. More generally, we suppose that $G$ is a Lie group that acts properly and
in a locally free fashion on $\widehat{M}$ so that the action is Hamiltonian, i.e., there exists a smooth $G$-equivariant map $\Phi : M \to
\mathfrak{g}^*$ to the dual of the Lie algebra of $G$, called moment map, such that

$$ \omega(\xi^M, \cdot) = d \langle \xi, \Phi \rangle \quad \text{ for
all } \xi \in \mathfrak{g}, $$

where $G$ acts on $\mathfrak{g}^*$ with the coadjoint action, $\xi^M
\in \mathfrak{X}(M)$ is the vector field determined by $\xi$ as in
Remark <ref>, and $\langle \cdot,\cdot \rangle$ is the natural pairing
$\mathfrak{g} \times \mathfrak{g}^* \to \R$. If $\alpha \in \mathfrak{g}^*$ is a
regular value that is fixed by the coadjoint action of $G$, then the
reduced space $M_{\alpha}:=\Phi^{-1}(\alpha)/G$ is an orbifold
that inherits a symplectic form $\omega_\alpha$ (see [46]
and <cit.>). This is the analog of a quotient
in symplectic geometry. The following examples are various instances of the above two constructions.

[Weighted projective spaces, continued]
Fix an integer $n \geq 1$ and positive
integers $m_0,\dots, m_{n}$ satisfying $\gcd(m_0,\dots,m_n)=1$. We
endow $\C^{n+1}$ with the standard symplectic form $\omega_0$. The
$S^1$-action on $\C^{n+1}$ given by (<ref>) is
Hamiltonian. Hence, $\C P^n(m_0,\ldots, m_n)$ inherits a symplectic
form $\omega$, which we call the standard
symplectic form on $\C P^n(m_0,\ldots, m_n)$. We observe that, if
$m_i = 1$ for all $i =0, \ldots, n$, then $\C P^n(m_0,\ldots, m_n)
\simeq \C P^n$ and $\omega$ is the Fubini-Study form.

[Example <ref>, continued]
We let $M =  (\C P^1 \times \C P^1)/\Z_4$ be the orbifold
constructed in Example <ref>. We endow $\C P^1 \times \C
P^1$ with the symplectic form $\widehat{\omega}$ given by taking the
product of the Fubini-Study form on each factor. Since the $\Z_4$-action
of (<ref>) is symplectic, $M$ inherits a symplectic form $\omega$.

[Example <ref>, continued]
Given an integer $c > 1$, let $M_c =  (\C P^1 \times \C P^1)/\Z_c$ be the orbifold constructed in Example
<ref>. We endow $\C P^1 \times \C
P^1$ with the same symplectic form $\widehat{\omega}$ as in Example
<ref>. Since the $\Z_c$-action of (<ref>) is
symplectic, $M_c$ inherits a symplectic form $\omega_c$.

[Cyclic quotients of weighted projective spaces, continued]
The orbifolds $\C P^1(p,q)/\Z_m = \C P^1(p,q)/\Z_m(a,b)$ and $\C
P^2(p,q,k)/\Z_m(a,b,c)$ of Examples <ref> and
<ref> admit symplectic forms. To see this, we
present them as quotient orbifolds. In what follows, we deal only
with $\C P^1(p,q)/\Z_m$, as the argument for $\C
P^2(p,q,k)/\Z_m(a,b,c)$ is entirely analogous. Moreover, we fix the
notation and conditions of Example <ref>.

We start by considering $\C^2$ endowed with the standard symplectic
form $\omega_0$. Let $\Z_m \times S^1$ act on $\C^2$ by

\begin{equation}
\label{eq:31}
\begin{split}
\mathbb{Z}_m\times S^1 \times \C^2&\rightarrow \C^2\\
((\lambda,\mu),(z_0,z_1))&\mapsto(\lambda^a\mu^p z_0,\lambda^b \mu^qz_1),
\end{split}
\end{equation}

(cf. (<ref>) and (<ref>)). This action is locally free
and Hamiltonian with moment map $H : \C^2 \to \R$ given

$$ H(z_0,z_1) = \frac{1}{2}\left(p|z_0|^2 +q|z_1|^2\right).$$

Since $pq \neq 0$, $pq$ is a regular value of $H$. Hence, the reduced space
$M_{pq}(a,b)$ is an orbifold that inherits a symplectic form
$\omega_{\mathrm{red}}$. The techniques used in Example
<ref> can be used to show that $M_{pq}(a,b)$
is diffeomorphic to $\C P^1(p,q)/\Z_m$. Throughout this paper,
we use this diffeomorphism to identify $M_{pq}(a,b)$ with $\C
P^1(p,q)/\Z_m$, trusting that this does not cause confusion.

Next we show that certain projectivized plane
bundles over orbi-surfaces admit a symplectic form (see Example

[Projectivized plane bundles, continued]
Let $\pi : M \to \Sigma$ be a Seifert fibration with Euler class
different from zero and such that $M$ is a manifold. Let
$\mathrm{pr} : L \to \Sigma$ be the corresponding complex line
orbi-bundle. We aim to construct a family of symplectic forms on the
total space of the projectivization $\mathbb{P}(L\oplus \C) \to
\Sigma$. To this end, we fix the identification between $ \mathbb{P}(L\oplus \C)$ and the
quotient $(M \times \C P^1)/S^1$ as in Example

Since the Euler class of $\pi : M \to \Sigma$ does not vanish, by
<cit.>, there exists an $S^1$-invariant 1-form $\lambda \in
\Omega^1(M)$ such that $\lambda(X^M) =1$, where $X^M \in
\mathfrak{X}(M)$ is the infinitesimal generator of the
$S^1$-action. (This can be thought of as a connection 1-form on the
Seifert fibration.) Moreover, in analogy with the case of principal
$S^1$-bundles over manifolds, since the Euler class of $\pi : M \to
\Sigma$ does not vanish, there exists a symplectic form
$\omega_\Sigma \in \Omega^2(\Sigma)$ such that $d\lambda = \pi^*
\omega_{\Sigma}$. (The 1-form $\lambda$ is a so-called Besse
contact form, see [38] for more details.)

We fix $s > 0$. We consider the Fubini-Study symplectic form $\omega_{\C P^1}$ on
$\C P^1$ and we let $h : \C P^1 \to \R$ be the smooth function given

$$[z_0:z_1] \mapsto \frac{|z_0|^2}{|z_0|^2+|z_1|^2} + \frac{1}{s}.$$

This is a moment map for the $S^1$-action on $(\C P^1, \omega_{\C P^1}$ given by
$\lambda \cdot [z_0:z_1] = [z_0:\lambda z_1]$.
We denote the projections from $M \times \C P^1$
to the first and second component by $\mathrm{pr}_1$ and
$\mathrm{pr}_2$ respectively and consider the
following 2-form on $M \times \C P^1$:

$$ s\left(\mathrm{pr}^*_2 \omega_{\C P^1} - d\left(\mathrm{pr}^*_2h\,
\mathrm{pr}^*_1 \lambda \right)\right).$$

By construction, this form is closed, has maximal rank equal to
four and its kernel is precisely the span of the infinitesimal
generator of the $S^1$-action of Example
<ref>. Hence, we obtain a symplectic form $\omega_s
\in \Omega^2((M \times \C P^1)/S^1)$ as desired.

We conclude this section with the following observation, which,
intuitively speaking, states that we may work with full
submanifolds of symplectic suborbifolds as in the case of manifolds.

By Remark <ref>, the tangent bundle of
a full suborbifold of $M$ can be seen as an orbi-subbundle of $TM$. Hence, the notion of Lagrangian/isotropic/symplectic full suborbifolds of a symplectic
orbifold $(M,\omega)$ can be defined as in the case of manifolds.

To conclude this section, we state the following important
property of symplectic orbifolds, which is the analog of a standard
result for symplectic manifolds (see <cit.> for a

Let $(M,\omega)$ be a symplectic orbifold. Then there exists a
compatible almost complex structure $J$ on $M$, i.e.,
$\omega(\cdot,J\cdot)$ is a Riemannian metric on $M$.

Since the proof of <cit.> is
entirely analogous to the case of symplectic manifolds, the space of
compatible almost complex structures on a given symplectic orbifold
is contractible.

Proposition <ref> has the following immediate consequence.

Let $(M,\omega)$ be a symplectic orbifold and let $N \subset M$ be a
full symplectic suborbifold. Then $TM$ admits the structure of a
complex vector orbi-bundle such that the restriction $TM|_N$ has
$TN$ as a complex vector orbi-subbundle. In particular, the normal
bundle to $N$ has the structure of a complex vector orbi-bundle.

We observe that, by Remark <ref>, the structures of
complex vector orbi-bundles of Corollary <ref> do
not depend on the choice of compatible almost complex structure up to isomorphism.

§.§.§ Darboux's theorem and its consequences in dimension four

A version of Darboux's theorem holds for symplectic orbifolds (see,
for instance, <cit.>), i.e., Example
<ref> is locally the only one up to symplectomorphism.

Let $(M,\omega)$ be a symplectic orbifold. If $x \in M$ is a
point with orbifold structure group $\Gamma$, then there exists a
l.u.c. $\oc$ centered at $x$ such that $\widehat{U}$ is an open
subset of $\R^{2n} \simeq \C^n$, $\Gamma < \mathrm{U}(n)$ and $\omega|_U$ is
as in Example <ref>.

With the notation of Theorem <ref>, we observe
that the orbi-vector space $T_x M$ can be identified with
$\R^{2n}/\Gamma$ and that it inherits a linear symplectic form
$\omega_x$. Hence, it is a symplectic orbi-vector space (see Example <ref>).

Since the fixed point set of any linear unitary action has real
codimension at least two, the following result is an immediate
consequence of Theorem <ref>.

Let $(M,\omega)$ be a symplectic orbifold. Then the singular set of
$M$ has codimension at least two.

Let $(M,\omega)$ be a symplectic orbifold. By Theorem
<ref>, the local invariants of a neighborhood of a point $x \in M$ with
structure group $\Gamma$ up to symplectomorphisms are precisely the invariants of a Hermitian
orbi-vector space $\C^n/\Gamma$ up to unitary isomorphisms. For this
reason, and since we are particularly interested in the case of
four-dimensional symplectic orbifolds with isolated singular points and abelian orbifold structure groups,
in what follows we study Hermitian
orbi-vector spaces
$\C^2/\Gamma$ where $\Gamma$ is abelian and $[(0,0)]$
is the only singular point.

Fix a faithful representation $\Gamma \hookrightarrow
\mathrm{U}(2)$. Since $\Gamma$
is finite and abelian, we may assume that the image of $\Gamma$ in $\mathrm{U}(2)$ is contained
in the maximal torus consisting of diagonal matrices $(S^1)^2 <
\mathrm{U}(2)$. Moreover, since the rank of that torus is two,
$\Gamma$ has at most two generators.

Suppose that $\Gamma$ is a finite abelian group and let $\Gamma \hookrightarrow
\mathrm{U}(2)$ be a faithful representation. If the quotient
$\C^2/\Gamma$ orbifold has isolated singular points, then $\Gamma$
is cyclic.

Suppose that there exist
integers $m_1,m_2 \geq 2$ with $\gcd(m_1,m_2) > 1$ such that $\Gamma
\simeq \Z_{m_1} \times \Z_{m_2}$. We may assume that there exist
$k_1,k_2,j_1,j_2 \in \Z$ such that the $\Gamma$-action on
$\C^2$ induced by the representation is given by

\begin{equation*}
\begin{split}
\Gamma \times \C^2 &\to \C^2 \\
((e^{\frac{2\pi i l_1}{m_1}}, e^{\frac{2\pi i
l_2}{m_2}}),(z_1,z_2)) &\mapsto (e^{\frac{2\pi i l_1k_1}{m_1}}e^{\frac{2\pi i
l_2k_2}{m_2}}z_1, e^{\frac{2\pi i l_1j_1}{m_1}}e^{\frac{2\pi i
\end{split}
\end{equation*}

A simple calculation shows that the above action is effective if and
only if

\begin{equation}
\label{eq:62}
\gcd(|k_2j_1 - k_1j_2|,m_1) = 1 = \gcd(|k_2j_1 - k_1j_2|,m_2).
\end{equation}

We set $\gcd(m_1,m_2) = c$ and write $m_i = m'_ic$ for $i=1,2$. We
set $l_1 = m'_1k_2$ and $l_2 = -m'_2k_1$. Then the subgroup of $\Gamma$ generated by $(e^{\frac{2\pi i l_1}{m_1}}, e^{\frac{2\pi i
l_2}{m_2}})$ fixes all points in the subspace $z_2 =
0$. Moreover, either $l_1$ is not divisible by $m_1$ or
$l_2$ is not divisible by $m_2$. Indeed, if otherwise, then $c > 1$ divides both $k_1$
and $k_2$, contradicting (<ref>). Hence, all points in the
subspace $z_2=0$ have non-trivial stabilizer.

By Lemma <ref>, we may assume that
$\Gamma$ is cyclic, i.e., there exists an integer $m \geq 2$ such that
$\Gamma \simeq \Z_m$. As in the proof of Lemma
given any representation $\Z_m \to \mathrm{U}(2)$, there exist integers $a,b \in \Z$ such that
the induced action on $\C^2$ is given by

\begin{equation}
\label{eq:5}
\begin{split}
\Z_m \times \C^2 &\to \C^2 \\
(e^{\frac{2\pi ik}{m}},(z_1,z_2)) &\mapsto (e^{\frac{2\pi i
ak}{m}}z_1, e^{\frac{2\pi ibk}{m}}z_2).
\end{split}
\end{equation}

A simple calculation yields the following result, which gives a
necessary and sufficient condition for the action of (<ref>) to be such that the origin is the only point
with non-trivial stabilizer.

Consider the $\Z_m$-action on $\C^2$ given by (<ref>). The
quotient $\C^2/\Z_m$ is an orbifold such that $[(0,0)]$ is the only
singular point if and only if

\begin{equation}
\label{eq:2}
\gcd(a,m) = 1 =\gcd(b,m).
\end{equation}

If (<ref>) holds, by composing with a suitable isomorphism
$\Z_m \to \Z_m$, without loss of generality we may assume that
either $a$ or $b$ in (<ref>) equals 1.

Remark <ref> motivates introducing the
following terminology that is also used in complex algebraic geometry
(see, e.g., [61]).

Let $m \geq 2$ be an integer and let $l \in \Z$ be such that $1\le
l<m$ and $\gcd(l,m)=1$. Let $\mathbb{C}^2/\mathbb{Z}_m$ be the
Hermitian orbi-vector space that is the quotient of the following
$\mathbb{Z}_m$-action on $\C^2$:

\begin{align} \label{eq:typesing}
\mathbb{Z}_m \times \mathbb{C}^2 & \to \mathbb{C}^2  \nonumber \\
(\xi_m , (z_1,z_2))  & \mapsto (\xi_m z_1,\xi_m^l z_2).
\end{align}

We call $\C^2/\Z_m$ the model for a cyclic
isolated singular point of type $\frac{1}{m}(1,l)$. The integers
$(1,l)$ are the weights of the $\mathbb{Z}_m$-action on

In Definition <ref>, we choose to make the
weight of the $\Z_m$-action of the first coordinate to be equal to one. The other choice
would yield an integer  $1\le l'<m$ such that $\gcd(l',m)=1$ and $l
\cdot l'= 1 \mod m$.

The following result states that the ambiguity of Remark
<ref> disappears when considering the unitary
isomorphism type of the corresponding Hermitian orbi-vector space. Its
proof is a straightforward calculation and is left to the reader.

There exists a unitary isomorphism between models for a cyclic
singular point of types $\frac{1}{m}(1,l)$ and $\frac{1}{m}(1,l')$ if and only if either $l = l'$ or
$l \cdot l' = 1 \mod m$.

As an immediate consequence of Theorem <ref>,
Lemma <ref> and Remark <ref>, we
obtain the following result.

Let $(M,\omega)$ be a four dimensional symplectic orbifold. Let $x
\in M$ be an isolated singular point with structure group $\Z_m$ for
some integer $m \geq 2$. Then there exist an integer $1 \leq l < m$
with $\gcd(l,m)=1$, an open neighborhood $U \subseteq M$ of $x$, an
open neighborhood $V \subseteq \C^2/\Z_m$ of $[0,0]$, and a symplectomorphism
$\varphi : (U,\omega) \to (V,\omega_0)$ that sends $x$ to $[0,0]$,
where $\C^2/\Z_m$ is the global quotient constructed from $l$ as in
Definition <ref> and $\omega_0$ is the
symplectic form on $\C^2/\Z_m$ that descends from the standard
symplectic form on $\C^2$.

Motivated by Definition <ref> and Corollary
<ref>, we introduce the following terminology.

Let $x$ a singular point in a symplectic orbifold of
dimension four. Let $\Z_m$ be the orbifold structure group of $x$
and let $1 \leq l < m$
with $\gcd(l,m)=1$ be the integer of Corollary
<ref>. We say that is a cyclic
isolated singular point of type $\frac{1}{m}(1,l)$.

Consider the weighted projective plane $\C P^2(3,5,7)$. The type of
the singular
points $[1:0:0]$, $[0:1:0]$ and $[0:0:1]$ is $\frac{1}{3}(1,2)$,
$\frac{1}{5}(1,4)$ and  $\frac{1}{7}(1,4)$ respectively.  $\Diamond$

The type of the three singular points in the quotient $\C
P^2(3,5,7)/\mathbb{Z}_{11}(1,2,10)$ are $\frac{1}{33}(1,23)$,
$\frac{1}{55}(1,19)$ and  $\frac{1}{77}(1,25)$.  $\Diamond$

(Example <ref>, continued)
Let $M:=(\mathbb{C}P^1\times\mathbb{C}P^1)/\mathbb{Z}_4$ be the
orbifold constructed in Example <ref>. It has three isolated
singular points, $[[0:1],[0:1]], [[1:0],[1:0]]$ and $[[0:1],[1:0]]$
with orbifold structure groups given by $\mathbb{Z}_4$,
$\mathbb{Z}_4$ and $\mathbb{Z}_2$ respectively. By diagonalizing the action of $\mathbb{Z}_4$ on $\C P^1\times \C P^1$ around  $\left([0:1],[0:1]\right)\quad \text{and} \quad
\left([1:0],[1:0]\right)$,
there exist coordinates around these points for which the $\mathbb{Z}_4$-action is given by

\begin{align*}
\mathbb{Z}_4 \times \mathbb{C}^2 & \to \mathbb{C}^2  \nonumber \\
(\xi_4 , (z,w))  & \mapsto (\xi_4 z,\xi_4^3 w).
\end{align*}

Similarly, there exist coordinates around $\left([0:1],[1:0]\right)\in \C P^1\times \C P^1$ for which the $\mathbb{Z}_2$-action is given by

\begin{align*}
\mathbb{Z}_2 \times \mathbb{C}^2 & \to \mathbb{C}^2  \nonumber \\
(\xi_2 , (z,w))  & \mapsto (\xi_2 z,\xi_2 w).
\end{align*}

(Note that $\left([1:0],[0:1]\right)$ and $\left([0:1],[1:0]\right)$ are in the same $\mathbb{Z}_4$-orbit).
We conclude that the type of $[[0:1],[0:1]], [[1:0],[1:0]]$ is
$\frac{1}{4}(1,3)$, while that of $[[0:1],[1:0]]$ is $\frac{1}{2}(1,1)$.