# A Gromov-Witten approach to $G$-equivariant birational invariants

Leonardo F. Cavenaghi  College of Arts and Sciences, Department of
Mathematics, University of Miami, Ungar Bldg, 1365 Memorial Dr 515, Coral
Gables, FL 33146. & Instituto de Matemática, Estatística e Computação
Científica da Universidade Estadual de Campinas (Unicamp), Cidade
Universitária, Campinas - SP, 13083-856<EMAIL_ADDRESS>,  Lino
Grama  Instituto de Matemática, Estatística e Computação Científica (IMECC) da
Universidade Estadual de Campinas (Unicamp), Cidade Universitária, Campinas -
SP, 13083-856<EMAIL_ADDRESS>and  Ludmil Katzarkov  College of Arts and
Sciences, Department of Mathematics, University of Miami, Ungar Bldg, 1365
Memorial Dr 515, Coral Gables, FL 33146. & Institute of the Mathematical
Sciences of the Americas (IMSA), 1365 Memorial Drive, Ungar 515, Coral Gables,
FL 33146<EMAIL_ADDRESS>
###### Abstract.

In [12], we initiated a program linking birational invariants with smooth ones
and offering new interpretations of classical invariants, such as the
Kervaire-Milnor invariants [34, 47]. Here, we rely on the profound geometric
reasoning provided by Lupercio and Uribe in [43, 44] to establish a connection
between Chen-Ruan cohomology and several $G$-birational invariants introduced
in the pioneering works [36, 38, 39], along with presenting applications.
Combined with the theory of atoms by Katzarkov, Kontsevich, Pantev, and Yu
([33]), the proposal in this paper program will lead to a theory of
equivariant atoms, [14].

###### Contents

  1. 1 Introduction
  2. 2 A minimum on Chen-Ruan cohomology
  3. 3 $G$-birational obstruction via the Chen-Ruan cohomology of Deligne-Mumford stacks
  4. 4 Discrete torsion, Chen-Ruan cohomology, and non-$G$-birational varieties
  5. 5 Some conjectures and further preliminary results
  6. A Algebraic group actions: biregular vs birational actions

##  1\. Introduction

This paper continues the research line initiated in [12] to produce birational
invariants out of higher homotopy invariants [47, 34]. Here, we indirectly
connect Gromov-Witten invariants with $G$-birational invariants. The final
goal of the presented approach will culminate in an equivariant theory of
atoms [14] relying upon [33].

Let $Y$ be a smooth projective variety of dimension $d\geq 2$. Given a finite
group $G$ acting effectively and regularly on $Y$, we can construct the orbit
space $X=Y/G$. If the $G$-action is _generically free_ , then $X$ naturally
becomes a global quotient orbifold, as illustrated in Example 2.9.

When applying standard equivariant cohomology to study $X$, information about
different isotropy subgroups is lost due to the finite nature of $G$, as its
infinitesimal data from Lie algebras is not preserved [51]. To address this,
_orbifold cohomology_ (Chen-Ruan cohomology) was developed [16], rooted in
Gromov-Witten theory (see Example 2.20). Fundamentally, orbifold cohomology
should not rely on ordinary cohomology but rather on the degree-zero part of
quantum cohomology [35].

This paper obtains a Chen-Ruan obstruction (Theorem A) to $G$-birationality
(Definition 3.1). Pick two smooth projective varieties $Y,\leavevmode\nobreak\
Y^{\prime}$ with effective generically free actions by the same finite group
$G$. We assume that these actions are by birational transformations. Moreover,
we assume that these actions _extend_ to $G$-biregular actions on
$Y,\leavevmode\nobreak\ Y^{\prime}$ (Definition A.10). In Appendix A, we add
details on this kind of action and obstructions. Essentially, we assume the
following:

######  Remark 1.1.

In this paper, whenever we consider a regular $G$-action on irreducible
algebraic varieties $Y$, we mean a $G$-regular action extended from a
$G$-birational action.

  1. (1)

Given an irreducible algebraic variety $Y$ with a $G$-birational action, there
always exists a $G$-equivariant resolution of singularity $\widetilde{Y}$ that
makes the $G$-action regular, see [8, Theorem 2.17]. Here, we assume that the
action extends to a regular action. In other words, we can think that the
$G$-equivariant singularity resolution $\widetilde{Y}$ is isomorphic to the
original variety $Y$.

  2. (2)

Requiring that the $G$-action extends makes the corresponding stack $[Y/G]$ be
normal.

∎

As we learn from [40, 39], the global quotient orbifolds
$Y/G,\leavevmode\nobreak\ Y^{\prime}/G$ with generically trivial stabilizers
can be seen as Deligne-Mumford stacks (DM stacks)
$\mathcal{X}=[Y/G],\leavevmode\nobreak\ \mathcal{X}^{\prime}:=[Y^{\prime}/G]$.
To them, the notion of birationality (as DM stacks) is in hand, Definition
3.4. Here, we prove:

######  Theorem A (=Theorems 3.5,4.19).

Let $G$ be a finite group acting regularly and generically free on a smooth
irreducible projective variety $Y$ of dimension $d\geq 2$ and let $X:=Y/G$ be
the resulting global quotient orbifold. Pick a linear $G$-action on
$\mathbb{P}^{d}$. If the Chen-Ruan cohomology rings of $X$ and
$\mathbb{P}^{d}/G$ are not isomorphic, i.e.,

$\mathrm{H}^{*}_{CR}(X=Y/G)\not\cong\mathrm{H}^{*}_{CR}(\mathbb{P}^{d}/G),$

then $Y\curvearrowleft G$ is not $G$-birational to
${\mathbb{P}^{d}}\curvearrowleft G$.

######  Remark 1.2.

Let $V$ be a complex vector space of dimension $d+1$ and consider a group
homomorphism $G\rightarrow\mathrm{GL}_{d+1}(V)$. We see $\mathbb{P}^{d}$ as
the space of lines through the origin in $V$. By a linear $G$-action on
$\mathbb{P}^{d}$ one means the restriction of $\rho$ to automorphisms of
$\mathbb{P}^{d}$, meaning, a group homomorphism
$G\rightarrow\mathrm{PGL}_{d+1}(V)$. ∎

As an application of the former, we prove:

######  Theorem B (=Corollary3.7).

Let $G=C_{4}$ (the cyclic group of order $4$) act

  1. (a)

linearly in $\mathbb{P}^{4}$, that is, as a homomorphism
$\rho:C_{4}\rightarrow\mathrm{PGL}_{5}(\mathbb{C})\cong\mathrm{Aut}(\mathbb{P}^{4})$
given in homogeneous coordiantes by

$\rho(g)\cdot[z_{0}:z_{1}:z_{2}:z_{3}:z_{4}]=[\rho_{0}(g)z_{0}:\rho_{1}(g)z_{1}:\rho_{2}(g)z_{2}:\rho_{3}(g)z_{3}:\rho_{4}(g)z_{4}],\leavevmode\nobreak\
\rho_{i}:G\rightarrow\mathrm{GL}(\mathbb{C})$

  2. (b)

and cyclic permuting coordinates on
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Then $\mathbb{P}^{4}$ with the linear $C_{4}$-action is not
$C_{4}$-birationally equivalent to
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
with the $C_{4}$-cyclic permuting action, i.e.,

$\mathbb{P}^{4}\not\sim_{C_{4}}\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}.$

Let $X(1,1,1,1)$ be the blowup of
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ on an elliptic curve.
Assume that

  1. (a)

$C_{4}$ acts in $X(1,1,1,1)$ via restricting the cyclic coordinate permutation
in $\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}$

  2. (b)

and $C_{4}$ acts linearly in $\mathbb{P}^{3}$, that is, as a homomorphism
$\rho:C_{4}\rightarrow\mathrm{PGL}_{4}(\mathbb{C})\cong\mathrm{Aut}(\mathbb{P}^{3})$.

Then $\mathbb{P}^{3}$ with the linear $C_{4}$-action is not
$C_{4}$-birationally equivalent to $X(1,1,1,1)$, i.e.,

$X(1,1,1,1)\not\sim_{C_{4}}\mathbb{P}^{3}.$

######  Remark 1.3.

  1. (1)

Observe that in Theorem B, the $C_{4}$-action in $\mathbb{P}^{4}$, once
linear, is already regular. The same happens for the $\mathrm{C}_{4}$-action
in $\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}$ since it is transitive in
$\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}$. Therefore, both actions
reduce (uniquely) to birational actions on $\mathbb{C}^{4}$. Moreover, we
remark that the cyclic permuting action in
$\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}$ is not induced by any finite
subgroup of $\mathrm{PGL}_{5}(\mathbb{C})$.

  2. (2)

In contrast to Theorem B, we observe that
$\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}$ with the $C_{4}$-cyclic
permuting action is $C_{4}$-rational. We can induce this action via
stereographic projection in $\mathbb{C}^{4}$ to see this. Moreover, this
action admits a unique extension to $\mathbb{P}^{4}$.

∎

From the perspective of equivariant birational geometry, one learns from [15]
that the hardest problem is to decide when a nodal cubic threefold (i.e., a
cubic threefold with ordinary double points) with a regular group action by a
finite group $G$ is _linearizable_. As a corollary of Theorem A, we can
recover Example 2.7 in [15]. We leave open the question of whether the theory
developed here can be related to Proposition 2.6 in the same reference.

######  Theorem C (=Theorem4.24).

Let $X\subset\mathbb{P}^{4}$ be a 2-nodal cubic given by

$x_{1}x_{2}x_{3}+x_{1}(x^{2}_{4}+x^{2}_{5})+x_{2}(x^{2}_{4}-x^{2}_{5})+x_{3}^{3}=0,$

with a $G\cong C_{4}$-action generated by

$\iota:(x_{1},x_{2},x_{3},x_{4},x_{5})\to(x_{2},x_{1},x_{3},x_{4},\zeta_{4}x_{5}).$

Pick any $G\cong C_{4}$-linear action in $\mathbb{P}^{3}$. Then

$\mathrm{H}_{CR}^{*}(X/C_{4})\not\cong\mathrm{H}^{*}_{CR}(\mathbb{P}^{3}/C_{4}).$

Consequently, $X$ is not $C_{4}$-birational to $\mathbb{P}^{3}$ with any
$C_{4}$-linear action.

Following [1, Chapter 4.5] choosing flat line bundles (with properties) over
twisted sectors may twist the Chen-Ruan cohomology. Special forms of these
twists are those induced by _discrete torsion_ (Definition 4.10).

In the literature, these twisting processes appear relevant in different
situations. There are two methods to remove singularities: resolution and
deformation. A smooth manifold $Y$ obtained from an orbifold $X$ via a
sequence of resolutions and deformations is called a desingularization of $X$.
In string theory, people usually require all the resolutions to be crepant
(even though these may fail to exist in dimensions higher than $3$). Because
of this, it is also interesting to allow the desingularization to be an
orbifold. In both cases, it is conjectured that the Chen–Ruan cohomology of
$X$ should correspond to that of the crepant resolution.

Suppose that $f:\widetilde{X}\to X$ is an orbifold universal cover, and let
$G=\pi^{\mathrm{orb}}_{1}(X)$ be the orbifold fundamental group ([1,
Definition 1.50]). Then $G$ acts on $\widetilde{X}$ such that
$X=\widetilde{X}/G$. Any non-identity element $g\in G$ acts on $\widetilde{X}$
as an orbifold morphism. The orbifold fixed point set $\widetilde{X}^{g}$ of
$\widetilde{X}$ under $g$ is a smooth suborbifold of $\widetilde{X}$.
Moreover, $X_{(g)}=\widetilde{X}^{g}/C(g)$ is a _twisted sector_ (Definition
2.25) of $X$, where $C(g)$ is the centralizer of $g$ in $G$. Importantly, not
all twisted sectors of $X$ necessarily arise as this. We call them dormant
sectors.

Let $e:X_{(g)}\to X$ be the evaluation map (Definition 2.27). We can view
$\widetilde{X}\to X$ as an orbifold principal $G$-bundle over $X$. Hence, we
can pull back to get an orbifold principal $G$-bundle $Z=e^{*}\widetilde{X}\to
X_{(g)}$ over $X_{(g)}$. Then $X_{(g)}$ is dormant if, and only if, the
$G$-action on $Z$ has no kernel. Moreover, $Z$ is a $G$-invariant suborbifold
of $\widetilde{X}$. We term $Z$ a $\pi^{\text{orb}}_{1}(X)$-effective
suborbifold.

As we explain in Example 4.3, a discrete torsion $\alpha$ is a 2-cocycle,
i.e., $\alpha\in Z^{2}(G;\mathrm{U}(1))$. For each $g\in G$, the cocycle
$\alpha$ defines a function $\gamma^{\alpha}_{g}:G\to\mathrm{U}(1)$ by
$\gamma^{\alpha}_{g}(h)=\alpha(g,h)\alpha(ghg^{-1},g)^{-1}$. We can use
$\gamma^{\alpha}_{g}$ to define a flat complex orbifold line bundle

$L_{(g),\alpha}=\widetilde{X}^{g}\times_{\gamma^{\alpha}_{g}}\mathbb{C}$

over $X_{(g)}$. We always assign a trivial line bundle for a dormant or non-
twisted sector. Let $\mathcal{L}^{\alpha}=\\{L_{(g),\alpha}\\}$. The
$k$-twisted Chen-Ruan cohomology group of $X$ is given by

$\mathrm{H}^{k}_{CR,\alpha}(X;\mathcal{L})=\oplus_{g\in(T_{1})}\mathrm{H}^{k-2\iota_{(g)}}(X_{(g)};L_{(g),\alpha}),$

where $\iota_{(g)}$ is termed _defree shifting_ (Definition 2.28). As per [43,
44], the previous construction can be extrapolated.

  1. (1)

Each orbifold $X$ admits a groupoid description
$\mathcal{G}=(\mathcal{G}_{0},\mathcal{G}_{1},s,t)$ (Example 2.12). To each of
such orbifold $\mathcal{G}$, one can obtain the _intertia orbifold
$\wedge\mathcal{G}$_ (Definition 2.19). The Chen-Ruan cohomology of $X$ is the
(de Rham) cohomology of $\wedge\mathcal{G}$.

  2. (2)

Let $X$ be an orbifold and $\mathcal{G}$ denote its groupoid description.
There is a group of _distinguished_ line bundles over the $\mathcal{G}_{1}$
consisting of _gerbes_ (Definition 4.1) over $X$. These are classified by
$\mathrm{H}^{3}(BX;\mathbb{Z})$, where $BX$ is the classifying space of $X$.
_Gerbes with connections_ over $X$ are classified by the enhancement of
$\mathrm{H}^{3}(BX;\mathbb{Z})$ given by the third Deligne cohomology group
$\mathrm{H}^{3}(X;\mathbb{Z}(3)_{D}^{\infty})$, Theorem 5.5.

  3. (3)

Each gerbe $\mathcal{L}$ over $X$ gives rise to flat line bundles
$L_{(g)}\rightarrow X_{(g)}$ over each _twisted sector_ $X_{(g)}$ (Definition
2.25). The $k$-th Chen-Ruan cohomology group of $X$ admits the following
description in terms of this line bundles family

$\mathrm{H}^{k}(X;\mathcal{L}):=\oplus_{g\in
T_{1}}\mathrm{H}^{k-2\iota_{(g)}}(X_{(g)};L_{(g)}),$

where $\iota_{(g)}$ is the _degree shifting_ associated with $g\in G$
(Definition 2.28).

Gathering the former information, we _describe the $G$-birational types of
$Y\curvearrowleft G$ by classifying the orbifold quotients $X/G$ regarding
their gerbes and their manifestations at the Chen-Ruan cohomology level._

We continue to list our results. In Section 5, we extrapolate the connections
here proposed promoting twisted K-theoretical interpretations of birational
invariants, besides proposing a motivic interpretation of birational
invariants, among other connections, such as related to $\mathrm{tmf}$, [28,
27, 46, 45, 3, 4]. We are mind-guided by the problems and conjectures
presented in [12]. In Section 5.4, we demonstrate a possible application in
studying the $G$-nonrationality of Johnson-Kollár Fano orbifolds appearing in
[32] \- a major area of applications of our invariants.

Our next result provides an intriguing connection between gerbes and the
theory of topological modular forms. In this paper, we only present how it is
obtained. Applications and a more detailed account shall be fully developed in
[13, 14].

######  Theorem D (=Theorem3.5).

Fix an orbifold $X$. Then, every gerbe $\mathcal{L}$ over $X$ gives rise to a
class in the generalized $\mathrm{tmf}$-cohomology of $X$.

As we learn from [55, Section 4.4], a $G$-birational invariant commonly
appearing in the literature is the cohomology group
$\mathrm{H}^{1}(G;\mathrm{Pic}(Y))$ where $\mathrm{Pic}(Y)$ is the Picard
group of the underlined variety $Y$ ( [55]). Each gerbe over an orbifold $X$
can be represented as a line bundle over $\mathcal{G}_{1}$, where
$\mathcal{G}=\mathcal{G}_{1}\rightarrow\mathcal{G}_{0}$ is a groupoid
description of $X$ (see Definition 4.1). Because of that, we can consider the
Picard group $\mathrm{Pic}_{\mathrm{gerbes}}(\mathcal{G}_{1})$ consisting of
the line bundles over $\mathcal{G}_{1}$ induced by gerbes over $X$ (Definition
4.14). We prove:

######  Theorem E (=Theorem 4.16).

Let $Y\curvearrowleft G,\leavevmode\nobreak\ Y^{\prime}\curvearrowleft G$ be
irreducible smooth projective varieties $Y,\leavevmode\nobreak\ Y^{\prime}$ of
the same dimension $d\geq 2$ and $G$ be a finite group acting regularly and
generically free on $Y$, $Y^{\prime}$. Let $X:=Y/G,\leavevmode\nobreak\
X^{\prime}:=Y^{\prime}/G$ be the resulting global quotient orbifolds with
groupoid representations
$\mathcal{G}=(\mathcal{G}_{0},\mathcal{G}_{1},s,t),\leavevmode\nobreak\
\mathcal{G}^{\prime}=(\mathcal{G}_{0}^{\prime},\mathcal{G}_{1}^{\prime},s^{\prime},t^{\prime})$.
Assume that the stacks $\mathcal{X},\leavevmode\nobreak\ \mathcal{X}^{\prime}$
for $X,\leavevmode\nobreak\ X^{\prime}$ are birational. Then

  1. (1)

$\mathrm{Pic}_{\mathrm{gerbes}}(\mathcal{G}_{1})\cong\mathrm{Pic}_{\mathrm{gerbes}}(\mathcal{G}^{\prime}_{1}).$

  2. (2)

hence, for a normal DM stack coming from a global quotient orbifold $X$, we
have that $\mathrm{H}^{1}(G;\mathrm{Pic}_{\mathrm{gerbes}}(\mathcal{G}_{1}))$
is a birational invariant.

The generic cubic is non-rational, [33]. On the other hand, Pfaffian cubics
are known to be rational and contain a $K3$ in its semi-orthogonal
decomposition. Here, we recover the result in [9] showing that the Pfaffian
cubic are G-non-birational to $\mathbb{P}^{4}$ by the Chen-Ruan cohomology
argument.

######  Theorem F (=Corollary 4.20, Theorem 16, p.22 in [9]).

Let $X\subset\mathbb{P}^{5}$ be a smooth Pfaffian cubic with an effective
regular action of the Frobenius group
$G=\mathrm{AGL}_{1}(\mathbb{F}_{7})=\mathbb{F}_{7}\rtimes\mathbb{F}_{6}=C_{7}\rtimes
C_{6}$. Then there is no regular $G$-linear action on $\mathbb{P}^{4}$ making
it $G$-birational to $X$.

### Terminology and conventions

  * •

Throughout this manuscript, we assume all considered varieties are defined
over $\mathbb{C}$.

  * •

We let $Y/G$ to stand to an orbifold and $[Y/G]$ to the corresponding Deligne-
Mumford stack. For general stacks one uses the convention letters
$\mathcal{X},\leavevmode\nobreak\ \mathcal{X}^{\prime}$.

  * •

When $G$ is a finite group, the superscript $\cdot^{G}$ denotes the fixed
point set. In Theorem 3.5, these may appear in two contexts: as the fixed
point set for the $G$-action on a certain variety $Y$ or at the cohomology
level. For example, if $\mathrm{H}^{*}(\cdot)$ represents the cohomology ring
of $\cdot$, then $\mathrm{H}^{*}(\cdot)^{G}$ or $(\mathrm{H}^{*}(\cdot))^{G}$
denote the fixed elements in this ring under the induced $G$-action on the
cohomology level.

##  2\. A minimum on Chen-Ruan cohomology

This section provides the minimum on orbifolds and Chen-Ruan cohomology
seeking self-containing. We suggest [43, 44, 16, 1] for detailed expositions.

###  2.1. A quick recall of orbifolds

######  Definition 2.1 (Uniformizing systems).

An $n$-dimension _uniformizing system_ for a connected topological space $U$
is a triple $(V,G,\pi)$ where $V$ is a connected $n$-dimension manifold and

  1. (a)

$G$ is a finite group acting on $V$ as diffeomorphisms;

  2. (b)

$\pi:V\rightarrow U$ is a continuous map which descends to a homeomorphism
$\pi:V/G\rightarrow U$.

######  Definition 2.2 (Isomorphic uniformizing systems).

Two uniformizing systems $(V_{1},G_{1},\pi_{1}),\leavevmode\nobreak\
(V_{2},G_{2},\pi_{2})$ are _isomorphic_ if there exists a pair of functions
$(\lambda,\phi)$ such that

  1. (a)

$\phi:V_{1}\rightarrow V_{2}$ is a diffeomorphism;

  2. (b)

$\lambda:G_{1}\rightarrow G_{2}$ is an isomorphism;

  3. (c)

$\phi$ is $\lambda$-equivariant and the following is a commutative diagram

${V_{1}}$${V_{2}}$${U}$${U}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\phi}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\mathrm{Id}}$

Let $U^{\prime}$ be an embedded connected open set
$i:U^{\prime}\hookrightarrow U$. Let $(V^{\prime},G^{\prime},\pi^{\prime})$ a
uniformizing system of $U^{\prime}$.

######  Definition 2.3 (Induced uniformizing system).

We say that $(V^{\prime},G^{\prime},\pi^{\prime})$ is _induced from a
uniformizing system_ $(V,G,\pi)$ in $U$ if one can find

  1. (a)

a monomorphism $\lambda:G^{\prime}\rightarrow G$ inducing an isomorphism
$\lambda:\ker\leavevmode\nobreak\ G^{\prime}\rightarrow\ker G$ where $\ker
G^{\prime}:=\\{g^{\prime}\in
G^{\prime}:g^{\prime}v^{\prime}=v^{\prime},\leavevmode\nobreak\ \forall
v^{\prime}\in V^{\prime}\\},\leavevmode\nobreak\ \ker G:=\\{g\in
G:gv=v,\leavevmode\nobreak\ \forall v\in V\\}$ are the ineffective kernels for
the $G^{\prime},\leavevmode\nobreak\ G$-actions, respectively;

  2. (b)

a $\lambda$-equivariant open embedding $\phi:V^{\prime}\rightarrow V$
commuting the following

${V^{\prime}}$${V}$${U^{\prime}}$${U}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\pi}$

######  Definition 2.4 (Isomorphic injections).

Two injections
$(\phi_{i},\lambda_{i}):(V^{\prime}_{i},G^{\prime}_{i},\pi^{\prime}_{i})\rightarrow(V,G,\pi)$
are _isomorphic_ if one can find

  1. (a)

an isomorphism
$(\psi,\tau):(V^{\prime}_{1},G^{\prime}_{1},\pi^{\prime}_{1})\rightarrow(V^{\prime}_{2},G^{\prime}_{2},\pi^{\prime}_{2})$
and

  2. (b)

an automorphism
$(\widetilde{\psi},\widetilde{\tau}):(V,G,\pi)\rightarrow(V,G,\pi)$ such that

$(\widetilde{\psi},\widetilde{\tau})\circ(\phi_{1},\lambda_{1})=(\phi_{2},\lambda_{2})\circ(\psi,\tau).$

######  Definition 2.5 (Germ of uniformizing system at a point).

Let $U$ be a connected topological space and $p\in U$. Assume that one can
find two open neighborhoods $U_{1},\leavevmode\nobreak\ U_{2}$ of $p$ with
uniformizing systems $(V_{1},G_{1},\pi_{1}),\leavevmode\nobreak\
(V_{2},G_{2},\pi_{2})$. We say that the former are _equivalent at $p$_ if one
can find $U_{3}\subset U_{1}\cap U_{2}$ and a uniformizing system
$(V_{3},G_{3},\pi_{3})$ isomorphic to the restrictions to $U_{3}$ of the
uniformizing systems $(V_{1},G_{1},\pi_{1})\leavevmode\nobreak\
(V_{2},G_{2},\pi_{2})$.

The _germ_ of a uniformizing system $(V,G,\pi)$ at $p$ is the set of
uniformizing systems of neighborhoods of $p$ which are equivalent to
$(V,G,\pi)$ _at_ $p$.

######  Definition 2.6 (Orbifold structure).

Let $X$ be a normal topological space. An $n$-dimension _orbifold structure_
on $X$ is a collection $\left\\{(V_{p},G_{p},\pi_{p})\right\\}_{p\in X}$ of
_germs_ of $n$-dimension uniformizing systems for each $p\in X$. That is, for
each $p\in X$ there exists $X\supset U\ni p$ such that for any $q\in U$ the
following uniformizing systems are equivalent,
$(V_{q},G_{q},\pi_{q}),\leavevmode\nobreak\ (V_{p},G_{p},\pi_{p})$.

######  Definition 2.7 ($n$-dimension orbifold).

A normal topological space $X$ is said to be an $n$-dimension orbifold if it
carries a $n$-dimension orbifold structure.

Fix $p\in X$. For any $q\in U_{p}$ the groups $G_{q},\leavevmode\nobreak\
G_{p}$ are isomorphic.

######  Definition 2.8.

We denote by $G_{p}$ the _isotropy_ or _local group_ at $p$. An orbifold is
said to be _reduced_ if each $G_{p}$ acts effectively on $X$ for every $p\in
X$. In other words, a reduced orbifold satisfies $\ker G_{p}={e}$ where $e$
denotes the unit element in $G_{p}$ at each $p$.

######  Example 2.9 (Global quotient orbifolds).

Let $G$ be a finite group acting smoothly and effectively on a smooth manifold
$Y$. Let $X=Y/G$ stand to the orbit space for this action. Then
$\\{(X,G,\pi)\\}$ is a _single_ collection of the uniformizing system (meaning
it is a uniformizing system for each of the points on $X$), where
$\pi:Y\rightarrow X$ is the _orbit map_.  $\blacksquare$

Not every orbifold is a global orbifold; for example:

######  Example 2.10 (Orbifolds as bases for hypersurface singularities
fibrations).

Pick positive integers $(w_{1},\ldots,w_{m})$. We consider the following
$\mathbb{C}^{*}$ action on $\mathbb{C}^{m}$

$\lambda\cdot(z_{1},\ldots,z_{m}):=(\lambda^{w_{1}/W}z_{1},\ldots,\lambda^{w_{m}/W}z_{m}),\leavevmode\nobreak\
\lambda\in\mathbb{C}^{*}$

where $W=\gcd(w_{1},\ldots,w_{m})$. The orbit space
$\mathbb{C}^{m}\setminus\\{0\\}/\mathbb{C}^{*}$ is denoted as
$\mathbb{P}(w_{1},\ldots,w_{m})$ and is named _weighted projective space_ with
weights $(w_{1},\ldots,w_{m})$.

Let $f:\mathbb{C}^{m}\rightarrow\mathbb{C}$ be a _weighted homogeneous
polynomial of degree $d$_, that is,

$f(\lambda\cdot(z_{1},\ldots,z_{m}))=f(\lambda^{w_{1}/W}z_{1},\ldots,\lambda^{w_{m}/W}z_{m})=\lambda^{d}f(z_{1},\ldots,z_{m}),\leavevmode\nobreak\
\forall\lambda\in\mathbb{C}^{*}.$

Assume that $0$ is the unique singularity of $f$ in an open ball centered in
$0$. Let $Y=f^{-1}(0)$. Then $\mathbb{C}^{*}$ defines an action in $Y$ whose
quotient $Y\setminus\\{0\\}/\mathbb{C}^{*}:=X$ is an orbifold hypersurface in
$\mathbb{P}(w_{1},\ldots,w_{m})$.  $\blacksquare$

###  2.2. Groupoids

In this section, we revisit the definition of a groupoid and some related
concepts. For a detailed exposition, see [1, 48]. In the next section, we
recall the concept of Chen-Ruan cohomology, obtained as the singular
cohomology of the inertia orbifold (Definition 2.30). We begin by stating the
main differences between groupoids and stacks and then proceed to a
“computational” treatment of groupoids.

####  2.2.1. Orbifolds as stacks?

According to [41, p.2], there are two complementary ways of thinking of
orbifolds as a 2-category. Both are guided by the idea that to think of
orbifolds as groupoids, one must treat them as a 2-category. Therefore, one
needs to consider maps between groupoids and maps between maps. Summarily,

  1. (i)

either one uses _bibundles as maps_ (Definition 3.25 in [41]), or

  2. (ii)

embed Lie groupoids into the 2-category of stacks.

If one follows (i), we say that a smooth orbifold is a Lie groupoid which is
_Morita equivalent_ (Definition in 1.43 in [1] or Definition 2.16) in the
category ($\mathrm{Bi}$111i.e., the weak 2-category with objects Lie
groupoids, 1-arrows bibundles and 2-arrows equivariant maps between bibundles)
to a proper étale Lie groupoid. We term such groupoids orbifold groupoids. A
map between two orbifolds is named a _smooth bibundle_. A downgrade point is
that the composition of bibundles is not strictly associative. Another is that
one loses the reasoning that groupoids are atlases on orbifolds (recall
Example 2.12).

The approach proposed in (ii) passes through embedding the weak 2-category
$\mathrm{Bi}$ into the 2-category of stacks $\mathrm{St}$, which is a _strict
2-category_. Because of this, the composition of 1-arrows is associative, and
groupoids can be seen as “coordinates” on a corresponding stack. Different
choices of coordinates define Morita equivalent groupoids. Morita equivalent
groupoids define isomorphic stacks.

####  2.2.2. Some generalities on groupoids

######  Definition 2.11 (Groupoids).

A _groupoid_ , shortly denoted as
$\mathcal{G}_{1}\rightrightarrows\mathcal{G}_{0}$ or
$\mathcal{G}=\mathcal{G}_{1}\rightrightarrows\mathcal{G}_{0}$, is a quadruple
$\mathcal{G}=(\mathcal{G}_{1},\mathcal{G}_{0},s,t)$ where

  1. (1)

$\mathcal{G}_{1},\leavevmode\nobreak\ \mathcal{G}_{0}$ are objects in a
category, where $\mathcal{G}_{1}$ is named _arrow set_ and $\mathcal{G}_{0}$
is named _object set_

  2. (2)

$s,\leavevmode\nobreak\ t:\mathcal{G}_{1}\rightarrow U$ are morphisms

  3. (3)

let
$\mathcal{G}_{1}{}_{t}\times_{s}\mathcal{G}_{1}:=\\{(u,v)\in\mathcal{G}_{1}:s(v)=t(u)\\}$.
One can find a multiplication map
$m:\mathcal{G}_{1}{}_{t}\times_{s}\mathcal{G}_{1}=\\{(u,v)\in\mathcal{G}_{1}:s(v)=t(u)\\}\rightarrow\mathcal{G}_{1}$
and _inverse_ and _identity_ maps, respectively

$i:\mathcal{G}_{1}\rightarrow\mathcal{G}_{1}$
$e:\mathcal{G}_{0}\rightarrow\mathcal{G}_{1}$

fully characterized by the compositions

$\displaystyle s\circ e=\mathrm{Id}_{\mathcal{G}_{0}}=t\circ e;\quad
s\circ\pi_{1}=s\circ m,\quad t\circ\pi_{2}=t\circ m;$ $\displaystyle
m\circ(m\times\mathrm{Id}_{\mathcal{G}_{1}})=m\circ(\mathrm{Id}_{\mathcal{G}_{1}}\times
m);$ $\displaystyle
m\circ(es,\mathrm{Id}_{\mathcal{G}_{1}})=\mathrm{Id}_{\mathcal{G}_{1}},\quad
m\circ(\mathrm{Id}_{\mathcal{G}_{1}},et)=\mathrm{Id}_{\mathcal{G}_{1}};$
$\displaystyle i\circ i=\mathrm{Id}_{\mathcal{G}_{1}},\quad s\circ i=t,\quad
t\circ i=s;$ $\displaystyle m\circ(\mathrm{Id}_{\mathcal{G}_{1}},i)=e\circ
s,\quad m\circ(i,\mathrm{Id}_{\mathcal{G}_{1}})=e\circ t.$

where
$\pi_{i}:\mathcal{G}_{1}{}_{t}\times_{s}\mathcal{G}_{1}\rightarrow\mathcal{G}_{1}$
are the factor projections, $i=1,2,$ and
$\mathrm{Id}_{\mathcal{G}_{0}}:\mathcal{G}_{0}\rightarrow\mathcal{G}_{0},\leavevmode\nobreak\
\mathrm{Id}_{\mathcal{G}_{1}}:\mathcal{G}_{1}\rightarrow\mathcal{G}_{1}$ are
the identity morphisms.

######  Example 2.12 (Orbifolds as groupoids).

Let $X$ be an orbifold and $\\{(V_{p},G_{p},\pi_{p})\\}_{p\in X}$ its orbifold
structure. The groupoid $\mathcal{G}_{1}\rightrightarrows\mathcal{G}_{0}$
associated with $X$ will be defined as follows. The set of objects is given by
$\mathcal{G}_{0}:=\bigsqcup_{p\in X}V_{p}$. An arrow
$g:(v_{1},V_{1})\to(v_{2},V_{2})\in\mathcal{G}_{1}$ with $v_{i}\in
V_{i},i=1,2$, will be a equivalence class of triples
$g=[\lambda_{1},w,\lambda_{2}]$ where $w\in W$ for another uniformizing system
$(W,H,\rho)$, and the $\lambda_{i}$’s are injections
$(\lambda_{i},\phi_{i}):(W,H,\rho)\to(V_{i},G_{i},\pi_{i})$ with
$\lambda_{i}(w)=v_{i},i=1,2$.

The maps $s,t,e,i,m$ are given as:

$s([\lambda_{1},w,\lambda_{2}])=(\lambda_{1}(w),V_{1}),\ \ \ \ \
t([\lambda_{1},w,\lambda_{2}])=(\lambda_{2}(w),V_{2})\ \ \ \
e(x,V)=[id_{V},x,id_{V}]$
$i([\lambda_{1},w,\lambda_{2}])=[\lambda_{2},w,\lambda_{1}]\ \ \ \
m([[\lambda_{1},w,\lambda_{2}],[\mu_{1},z,\mu_{2}])=[\lambda_{1}\circ\nu_{1},y,\mu_{2}\circ\nu_{2}]$

where $h=[\nu_{1},y,\nu_{2}]$ is an arrow joining $w$ and $z$.

It can be given a topology to $\mathcal{G}_{1}$ so that $s,t$ will be étale
maps, making it into a proper, étale, differentiable groupoid, [48, Theorem
4.1].  $\blacksquare$

######  Example 2.13 (Action groupoid and Orbit Space).

Let $G$ be a group acting on the left on a set $Y$. We make a groupoid out of
it in the following manner. The objects shall be elements of $Y$, and the
arrows shall be pairs $(x,g)$ with source $s((x,g))=x$ and target
$t((x,g))=gx$. The identity $e(x)=(x,1)$ and the inverse map
$i((x,g))=(gx,g^{-1})$.

Let $\mathcal{G}=\mathcal{G}_{1}\rightrightarrows\mathcal{G}_{0}$ be a
groupoid. Put

$\sim:=\\{(x,y)\in\mathcal{G}_{0}\times\mathcal{G}_{0}:\exists\gamma\in\mathcal{G}_{1}\leavevmode\nobreak\
\text{such that}\leavevmode\nobreak\
x\stackrel{{\scriptstyle\gamma}}{{\rightarrow}}y\\}.$

Then $\sim$ is an equivalence relation. We denote the quotient
$\mathcal{G}_{0}/\sim$ by $\mathcal{G}_{0}/\mathcal{G}_{1}$ and name it _the
orbit space of the groupoid $\mathcal{G}$_.

If $\mathcal{G}$ stands for an action groupoid, the orbit space
$\mathcal{G}_{1}/\mathcal{G}_{0}$ is also called _coarse moduli space_.
$\blacksquare$

######  Definition 2.14.

Let $\mathcal{G}$ be a groupoid description of an orbifold. A left
$\mathcal{G}$-space is a manifold $E$ equipped with an action by $G$. Such an
action is given by two maps: an anchor $\pi:E\to\mathcal{G}_{0}$, and an
action $\mu:\mathcal{G}_{1}\times_{\mathcal{G}_{0}}E\to E$. The latter map is
defined on pairs $(g,e)$ with $\pi(e)=s(g)$, and written $\mu(g,e)=g\cdot e$.
It satisfies the usual identities for an action:

$\pi(g\cdot e)=t(g),$ $1_{x}\cdot e=e,$

and

$g\cdot(h\cdot e)=(gh)\cdot e,$

for $x\xrightarrow{h}y\xrightarrow{g}z$ in $G_{1}$ and $e\in E$ with
$\pi(e)=x$.

######  Definition 2.15.

A homomorphism $\phi:\mathcal{G}^{\prime}\rightarrow\mathcal{G}$ between Lie
groupoids is termed an _equivalence_ if

  * (i)

the map

$t\circ\pi_{1}:\mathcal{G}_{1}\,{}_{s}\\!\times_{\phi}{\mathcal{G}^{\prime}}_{0}\rightarrow\mathcal{G}_{0}$

defined on the fibered manifold $\\{(g,y)\mid
g\in\mathcal{G}_{1},y\in{\mathcal{G}^{\prime}}_{0},s(g)=\phi(y)\\}$ is a
surjective submersion;

  * (ii)

the square

$\begin{array}[]{ccc}{\mathcal{G}^{\prime}}_{1}&\xrightarrow{\phi}&\mathcal{G}_{1}\\\
\downarrow_{(s,t)}&&\downarrow_{(s,t)}\\\
{\mathcal{G}^{\prime}}_{0}\times{\mathcal{G}^{\prime}}_{0}&\xrightarrow{\phi\times\phi}&{\mathcal{G}^{\prime}}_{0}\times{\mathcal{G}^{\prime}}_{0}\end{array}$

is a fibered product of manifolds.

######  Definition 2.16 (Morita equivalence).

Two Lie groupoids $\mathcal{G},\leavevmode\nobreak\ \mathcal{G}^{\prime}$ are
said to be _Morita equivalent_ if there exists a third groupoid
$\widetilde{\mathcal{G}}$ and two equivalences $\sigma,\leavevmode\nobreak\
\tau$ fitting the diagram

$\mathcal{G}\stackrel{{\scriptstyle\sigma}}{{\longleftarrow}}\widetilde{\mathcal{G}}\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}\mathcal{G}^{\prime}.$

######  Definition 2.17.

A $\mathcal{G}$-vector bundle over an orbifold groupoid $\mathcal{G}$ is a
$\mathcal{G}$-space $E$ for which $\pi:E\to\mathcal{G}_{0}$ is a vector
bundle, such that the action of $\mathcal{G}$ on $E$ is fiberwise linear. Any
arrow $g:x\to y$ induces a linear isomorphism $g:E_{x}\to E_{y}$. In
particular, $E_{x}$ is a linear representation of the isotropy group $G_{x}$
for each $x\in\mathcal{G}_{0}$.

Many properties of Lie groupoids are preserved under Morita equivalence, for
instance, _properness condition_ and _Hausdorff assumption_. However, the
property of being étale need not be preserved. More geometric preserved
properties are described next.

######  Proposition 2.18.

Let $\mathcal{G}=\mathcal{G}_{1}\rightrightarrows\mathcal{G}_{0}$ and
$\mathcal{H}=\mathcal{H}_{1}\rightrightarrows\mathcal{H}_{0}$ be two Morita
equivalent groupoids induced by orbifolds. Then, their transverse geometry is
preserved, meaning that

  1. (i)

there exists a homeomorphism between the orbit spaces (Example 2.13)
$\mathcal{G}_{0}/\mathcal{G}_{1}$ and $\mathcal{H}_{0}/\mathcal{H}_{1}$

  2. (ii)

for each point $x\in\mathcal{G}_{0}$ there is $y\in\mathcal{H}_{0}$ and an
isomorphism between the isotropy groups $G_{x}\cong H_{y}$

  3. (iii)

for each point $x\in\mathcal{G}_{0}$ there is $y\in\mathcal{H}_{0}$ and an
isomorphism $\mathcal{N}_{x}\cong\mathcal{N}_{y}$ between the normal
representations of the isotropy groups.

Moreover, the differentiable cohomologies of $\mathcal{G}$ and $\mathcal{H}$
are isomorphic.

We finish this section by presenting an important groupoid playing a major
role in the definition of Chen-Ruan cohomology. Let $X$ be an orbifold and let
$\mathcal{G}=(\mathcal{G}_{0},\mathcal{G}_{1},s,t)$ be its groupoid
representation.

######  Definition 2.19.

The inertia orbifold $\wedge\mathcal{G}$ is characterized by

  1. (i)

objects set is $(\wedge\mathcal{G})_{0}:=\\{r\in\mathcal{G}_{1}:s(r)=t(r)\\}$

  2. (ii)

arrows set is
$(\wedge\mathcal{G})_{1}:=\\{\lambda\in\mathcal{G}_{1}:r_{1}\stackrel{{\scriptstyle\lambda}}{{\rightarrow}}r_{2}\Leftrightarrow
m(\lambda,r_{2})=m(r_{1},\lambda)\\}$.

As one shall see in the next section, the Chen-Ruan cohomology is nothing but
the singular cohomology of the inertia orbifold.

###  2.3. Chen-Ruan cohomology for Abelian orbifolds

Let $\mathcal{G}=(\mathcal{G}_{0},\mathcal{G}_{1},s,t)$ stands for the
groupoid representation of an orbifold $X$. Herein, we assume all considered
orbifolds $X$ are almost complex. That is, one assumes that there is an almost
complex structure on the tangent bundle $T\mathcal{G}_{0}$, which is invariant
under the $\mathcal{G}$-action. We focus on Abelian orbifolds $X$, where each
local group $G_{p}$ in a chosen uniformizing system is Abelian. This class
encompasses toric varieties and complete intersections of toric varieties.

Example 2.20 below encapsulates the _core_ information regarding the concepts
related to Chen-Ruan cohomology that need to be recalled. For a more detailed
account of it, check [16].

######  Example 2.20 (The Gromov-Witten flavor in Chen-Ruan cohomology).

Given an orbifold $X$, let $\mathcal{G}$ stand to its groupoid description.
Let $\wedge\mathcal{G}$ be its inertia orbifold, or _intertia stack_.

Let $C$ be a genus-zero orbifold complex curve with marked $r$-points
$\\{x_{1},\ldots,x_{r}\\}$, that is, $C$ is a complex curve with $r$-non-
trivial isotropies on uniformizing systems indexed by these points. Let
$\overline{\mathcal{M}}_{0,r}(X,\beta)$ be the moduli space of degree-zero
maps $f:C\rightarrow X$ where $\beta=f_{\ast}([C])\in\mathrm{H}_{2}(X)$ for
$[C]$ the fundamental class. Recall that it carries an orbifold structure; see
[7, 21]. Below, we indiscriminately abuse notation and let
$\overline{\mathcal{M}}_{0,r}(X,\beta)$ denote the groupoid obtained from its
orbifold description.

Each groupoid is a _small category_ , so a groupoid morphism ([1, Definition
2.39, p.48]) is a category functor. Below, we consider the family
($i$-indexed) of _evaluation_ orbifold morphism (compare with Definition 2.27)

$e_{i}:\overline{\mathcal{M}}_{0,r}(X,\beta)\rightarrow\wedge\mathcal{G}$
$(f:C\rightarrow
X;x_{1},\ldots,x_{r})\mapsto(f(x_{i}),\lambda_{x_{i}}(1_{x_{i}}))$

where $1_{x_{i}}$ is the canonical generator of $G_{x_{i}}$.

Assume that $r=3$ and that all maps $f$ are degree-zero. Each map
$f:C\rightarrow X$ factors through maps $f:C\rightarrow BG_{p}\subset X$ where
$p\in X$ is an element in the image of $f$ and $BG_{p}$ is the classifying
space of bundles with structure group $G_{p}$, the isotropy at $p$. According
to [1, Chapter 2.3] one gets a $G_{p}$-principal bundle over
$C\setminus\\{x_{1},x_{2},x_{3}\\}$. Moreover, these are classified by the
$G_{p}$-conjugacy classes of group homomorphisms ([1, Theorem 2.36, p.47])

$\pi_{1}(C\setminus\\{x_{1},x_{2},x_{3}\\})\rightarrow G_{p}.$

Thus, they are fully determined by $\\{\lambda_{i}(1_{x_{i}})\\}_{i=1,2,3}$
constrained as $\prod_{i=1}^{3}\lambda_{i}(1_{x_{i}})=1$.

We conclude that the object set of $\overline{\mathcal{M}}_{0,3}(X,0)$ can be
described as

$\left(\overline{\mathcal{M}}_{0,3}(X,0)\right)_{0}=\\{(p,(g_{1},g_{2},g_{3})):g_{i}\in
G_{p},\leavevmode\nobreak\ g_{1}g_{2}g_{3}=1\\}$

with arrows

$\left(\overline{\mathcal{M}}_{0,3}(X,0)\right)_{1}=\\{(p,(g_{1},g_{2},g_{3}))\mapsto(hp,(hg_{1}h^{-1},hg_{2}h^{-1},hg_{3}h^{-1})),\leavevmode\nobreak\
h\in G_{p}\\}.$

Once $(\wedge\mathcal{G})_{0}=\\{(p,g):p\in X,\leavevmode\nobreak\ g\in
G_{p}\\}$ with arrows $\\{(p,g)\mapsto(hp,hgh^{-1}))\\}$ we re-read

$e_{i}:(p,(g_{1},g_{2},g_{3}))\mapsto(p,g_{i}).$

$\blacksquare$

Let $p\in X$ and $(V_{p},\pi,G_{p})$ be an uniformizing system at $U_{p}\ni
p$, that is, $\pi(V_{p})=U_{p}$ and $U_{p}\cong V_{p}/G_{p}$. Let $k>1$ be an
integer.

######  Definition 2.21.

Let $(\mathbf{g}):=[(g_{1},\ldots,g_{k})],\leavevmode\nobreak\
(g_{1},\ldots,g_{k})\in G_{p}^{k}$ be the conjugacy class of
$\mathbf{g}=(g_{1},\ldots,g_{k})$ in $G_{p}^{k}$. The _multi-sector_
$\widetilde{\Sigma_{k}X}$ is defined as

$\widetilde{\Sigma_{k}X}=\\{(p,({\bf g})):p\in X,\leavevmode\nobreak\ {\bf
g}\in G_{p}^{k}\\}.$

######  Remark 2.22.

When $k=1$, we may omit the subscript in $\widetilde{\Sigma_{k}X}$. ∎

Pick $q\in U_{p}$. Up to conjugation, $G_{p}$ and $G_{q}$ are isomorphic.
Hence, for any choice ${\bf g}\in G_{q}^{k}$ the conjugacy class $({\bf
g})_{G_{p}}$ is well-defined. Therefore, we let $({\bf g})_{G_{q}}\cong({\bf
g})_{G_{p}}$ for any $q\in U_{p},$ and any ${\bf g}\in G_{q}^{k}$. Set
$T_{k}:=\cup_{q\in U_{p}}\cup_{({\bf g})_{q}\in G_{q}}{({\bf g})_{q}}/\sim$
where $\sim$ is the former equivalence relation.

######  Lemma 2.23 (Chapter 4 in [1]).

Let $X_{({\bf g})}:=\\{(p,({\bf g^{\prime}})_{G_{p}}):{\bf g^{\prime}}\in
G_{p}^{k}\leavevmode\nobreak\ \&\leavevmode\nobreak\ ({\bf
g^{\prime}})_{G_{p}}\in({\bf g})\\}$. Then,

$\widetilde{\Sigma_{k}X}=\sqcup_{(\mathbf{g})\in T_{k}}X_{(\bf g)}$

######  Example 2.24.

Let $X=Y/G$ where $|G|<\infty$. For any point $p\in X$, one has that
$G_{p}\cong G$. Pick any $g\in G$ and let $(g)$ be the conjugacy class of $g$
in $G$. Then the sector $\widetilde{\Sigma X}$ is locally seen as
$\widetilde{\Sigma X}\stackrel{{\scriptstyle\text{locally}}}{{=}}Y^{(g)}/C(g)$
where $Y^{(g)}:=\\{y\in Y:g^{\prime}y=yg^{\prime}\leavevmode\nobreak\ \forall
g^{\prime}\in(g)\\}$ and $C(g)$ is the centralizer of $g$ in $G$. Moreover,

$\widetilde{\Sigma X}=\cup_{(g)}Y^{(g)}/C(g).$

$\blacksquare$

######  Definition 2.25 (Twisted sector).

Let $X_{({g})}:=\\{(p,({g^{\prime}})_{G_{p}}):{g^{\prime}}\in
G_{p}\leavevmode\nobreak\ \&\leavevmode\nobreak\
({g^{\prime}})_{G_{p}}\in({g})\\}$.

  1. (a)

if $g=1$ we name $X_{(g)}$ by _untwisted sector_

  2. (b)

for each $g\neq 1$ the sets $X_{(g)}$ are named _twisted sectors_

######  Example 2.26 (Twisted sectors for global quotients).

Let $X=Y/G$ with $|G|<\infty$ and pick $g\neq 1$. We have that
$X_{(g)}:=\\{(p,({g^{\prime}})_{G_{p}}:{g^{\prime}}\in
G_{p}\leavevmode\nobreak\ \&({g^{\prime}})_{G_{p}}\in({g})\\}\cong
Y^{(g)}/C(g)$.

For $g=1$ though we have that $Y^{(g)}=Y$ and $C(g)=G$. Therefore,
$X_{(1)}\cong Y/G=X$.  $\blacksquare$

######  Definition 2.27 (Natural maps between multi-sectors).

Pick positive integers $k,\leavevmode\nobreak\ l$. We let

  1. (a)

$e_{i_{1}\ldots
i_{l}}:\widetilde{\Sigma_{k}X}\rightarrow\widetilde{\Sigma_{l}X}$ be the
_evaluation map_ defined as

$e_{i_{1}\ldots
i_{l}}:(x,(g_{1},\ldots,g_{k}))\mapsto(x,(g_{i_{1}},\ldots,g_{i_{l}})).$

  2. (b)

the _inversion map_
$I:\widetilde{\Sigma_{k}X}\rightarrow\widetilde{\Sigma_{k}X}$ be defined as

$I:(x,(g_{1},\ldots,g_{k}))\rightarrow(x,(g_{1}^{{}_{1}},\ldots,g_{k}^{-1})).$

Recall that $\mathcal{G}=(\mathcal{G}_{0},\mathcal{G}_{1},s,t)$ represents the
groupoid of the orbifold $X$. Let
$e:\mathcal{G}_{0}\rightarrow\mathcal{G}_{1}$ be the identity map. It defines
an embedding, which we also denote by
$e:\wedge\mathcal{G}\rightarrow\mathcal{G}$, given by $e(g):=s(g)=t(g)$. Thus,
we can pull back the groupoid bundle $T\mathcal{G}$ to $\wedge\mathcal{G}$ via
$e^{*}$.

Consider $g\in(\wedge\mathcal{G})_{0}$ such that $s(g)=t(g)=x$. Then $g$ acts
on $e^{*}(T_{x}\mathcal{G}_{0})$ via a group representation
$\rho(g):T_{x}\mathcal{G}_{0}\rightarrow T_{x}\mathcal{G}_{0}$. Collect its
eigenvalues $\left\\{e^{\tfrac{2\pi\mathbf{i}m_{j}}{m}}\right\\}$, where
$m=\mathrm{order}_{G}(g)$ and $m_{j}\leq m_{j+1}\leq\ldots$. Let $\\{E_{j}\\}$
be the corresponding eigenspaces. We have
$e^{*}(T_{x}\mathcal{G}_{0})=\oplus_{j}E_{j}$. Assume that $\dim E_{j}=1$.

Let $v\in\mathcal{G}_{1}$ be an arrow such that $s(v)=x$. Viewed as an arrow
in $(\wedge\mathcal{G})_{1}$, $v$ connects $g$ with $vgv^{-1}$. The
differential of the local diffeomorphism associated with $v$ maps
$(E_{j})_{g}$ to an eigenspace with the same eigenvalue. This map preserves
the splitting into one-dimensional eigenspaces given by $\\{E_{j}\\}$ for each
$v$. Thus, each $E_{j}$ forms a line bundle over $\wedge\mathcal{G}$.

Let
$\theta_{j}\in\mathrm{H}^{1}(E_{j},E_{j}\setminus\wedge\mathcal{G};\mathbb{Z})$
be the Thom class of $E_{j}\rightarrow\mathcal{\wedge}G$.

######  Definition 2.28.

The formal product

$\mathrm{t}(g):=\prod_{j}\theta_{j}^{\tfrac{m_{j}}{m}}$

is named _twisted factor_ associated with the twisted sector $X_{(g)}$. The
_degree_ $\mathrm{deg}(t(g))$ of the twisted factor $t(g)$ is given by

$\mathrm{deg}(t(g)):=2\sum_{j}\frac{m_{j}}{m}=:2\iota_{(g)}.$

Each element in $\\{\iota_{(g)}\\}_{g\in G}$ is named a _degree-shifting_.

######  Remark 2.29.

For Definition 2.28, we are adopting the conventions $\theta_{j}^{0}=1$ and
$\theta_{j}^{1}=\theta_{j}$. ∎

Following [43], one gets that the inertia orbifold represents the twisted
sectors of $X$. Therefore, for any (invariant) form
$\omega\in\Omega^{*}(X_{(g)})$, consider the formal product $\omega t(g)$. We
name it a _twisted form_ (or formal form) associated with $X_{(g)}$. We define
the de Rham complex of twisted forms by setting

$\Omega^{p}_{CR}(\mathcal{G})=\left\\{\omega_{1}t(g_{1})+\cdots+\omega_{k}t(g_{k})\mid\sum_{i}\deg(\omega_{i})\deg(t(g_{i}))=p\right\\}.$

The coboundary operator $\mathrm{d}$ is given by the formula

$\mathrm{d}(\omega_{i}t(g_{i}))=\mathrm{d}(\omega_{i})t(g_{i}).$

It can be checked that $\\{\Omega^{*}_{CR}(\mathcal{G}),\mathrm{d}\\}$ is a
chain complex. Let

$\mathrm{H}^{*}(\\{\Omega^{*}_{CR}(\mathcal{G}),\mathrm{d}\\})$

to be its associated cohomology. One can easily verify that there are
homomorphisms.

$\mathrm{H}^{*}(X_{(g)};\mathbb{R})\to\mathrm{H}^{*+2\iota_{(g)}}(\\{\Omega^{*}_{CR}(\mathcal{G}),\mathrm{d}\\}).$

Define the wedge product formally by setting

(2.1)
$\omega_{1}t(g_{1})\wedge\omega_{2}t(g_{2})=\omega_{1}\wedge\omega_{2}t(g_{1})t(g_{2}).$

Lemma 5.2 in [1] ensures that $\omega_{1}\wedge\omega_{2}t(g_{1})t(g_{2})$ can
be identified with an element of $\Omega^{*}_{CR}(\mathcal{G})$. Thus, this
operation defines an associative ring structure on
$\mathrm{H}^{*}(\\{\Omega^{*}_{CR}(\mathcal{G}),\mathrm{d}\\})$.

######  Definition 2.30 (Chen-Ruan cohomology ring).

Let $\mathcal{G}$ be the groupoid representation of $X$ represented by the
orbifold $\mathcal{G}$. The Chen-Ruan cohomology ring $\mathrm{H}^{*}_{CR}(X)$
of $X$ is defined as the ring

$\mathrm{H}^{*}_{CR}(X):=\mathrm{H}^{*}(\\{\Omega^{*}_{CR}(\mathcal{G}),\mathrm{d}\\})$

with product induced by the wedge product given in Equation (2.1).

##  3\. $G$-birational obstruction via the Chen-Ruan cohomology of Deligne-
Mumford stacks

######  Definition 3.1.

Let $Y,\leavevmode\nobreak\ Y^{\prime}$ be smooth projective varieties of
dimension $d\geq 2$ with biregular $G$-actions. One says that
$Y,\leavevmode\nobreak\ Y^{\prime}$ are $G$-equivariantly birationally
equivalent, writing $Y\sim_{G}Y^{\prime}$, if there exists a $G$-equivariant
birational morphisms

$\phi:Y-\\!\\!\rightarrow Y^{\prime}.$

This section proposes understanding obstruction to $G$-birationality by
examining the resulting orbifolds) and understanding non-birationality for
Deligne-Mumford stacks (DM stack), [39]. A standard example of a DM stack is
the stack $[Y/G]$ obtained from the global quotient orbifold determined by an
algebraic variety $Y$ with a generically free regular action of a finite group
$G$. This is the category where an object is a $G$-torsor $E\to T$ (over an
arbitrary $\mathbb{C}$-scheme $T$) with a $G$-equivariant morphism $E\to Y$,
and a morphism is a $G$-equivariant isomorphism of torsors (over a morphism of
$\mathbb{C}$-schemes), compatible with the morphisms to $Y$. The category
$[Y/G](\mathbb{C})$ of $\mathbb{C}$-points of $[Y/G]$ has isomorphism classes
of objects in bijective correspondence with the $G$-orbits of $Y(\mathbb{C})$.
If $Y=\text{Spec}(\mathbb{C})$ then $[Y/G]$ coincides with $BG$, the category
of $G$-torsors (over $\mathbb{C}$-schemes) and $G$-equivariant morphisms (over
morphisms of $\mathbb{C}$-schemes).

The stack $[Y/G]$ has a coarse moduli space $(Y/G)_{\mathrm{coarse}}$
consisting of the topological quotient space (orbit space). When $Y$ is
nonsingular and the $G$-action is effective, then $Y/G$ is an orbifold.

######  Definition 3.2 ([40]).

Assume that $Y/G$ is a global quotient orbifold from a regular generically
free $G$-action on a smooth projective variety $Y$. One says that

  1. (a)

$Y/G$ is a _projective orbifold_ if the coarse moduli space
$(Y/G)_{\mathrm{coarse}}$ is a projective scheme. In this case, one says that
$[Y/G]$ is a projective stack

  2. (b)

$Y/G$ is a _quasi-projective orbifold_ if the coarse module space
$(Y/G)_{\mathrm{coarse}}$ is a quasi-projective scheme. In this case, one says
that $[Y/G]$ is a quasi-projective stack.

######  Remark 3.3.

  1. (1)

For a quasi-projective Deligne-Mumford stack $\mathcal{X}$, there always
exists a linear algebraic group $G$ acting on a smooth quasi-projective scheme
$Y$ such that $\mathcal{X}\cong[Y/G]$. However, there is no reason for $G$ to
be finite.

  2. (2)

When DM stack given as $[Y/G]$ for a finite group $G$ it is automatically
_separated_. This means that the diagonal morphism
$\Delta:[Y/G]\rightarrow[Y/G]\times_{T}[Y/G]$ is an immersion where $T$ is an
arbitrary $\mathbb{C}$-scheme.

∎

######  Definition 3.4 (Birational equivalence of smooth stacks, Definitions
4.1, 4.21+Remark 4.22 in [39]).

Let $\mathcal{X},\leavevmode\nobreak\ {\mathcal{X}}^{\prime}$ be two smooth
stacks. One says that $\mathcal{X}$ and ${\mathcal{X}}^{\prime}$ are
birationally equivalent if there exists a smooth quasi-projective stack
$\widetilde{\mathcal{X}}$ and projective birational morphisms
$\sigma,\leavevmode\nobreak\ \tau$ encompassing the diagram below

(3.1)
${\widetilde{\mathcal{X}}}$${\mathcal{X}}$${{\mathcal{X}}^{\prime}}$$\scriptstyle{\sigma}$$\scriptstyle{\tau}$

By a birational map between stacks, one means an isomorphism between open
dense substacks.

Theorem 3.5 below is a simplified version of Theorem A since one assumes $G$
to be Abelian. The complete proof of Theorem A shall be presented in the next
section.

######  Theorem 3.5.

Let $G$ be a finite Abelian group acting regularly and generically free on a
smooth irreducible projective variety $Y$ of dimension $d\geq 2$ and let
$X:=Y/G$ be the resulting orbifold. Pick an effective linear $G$-action on
$\mathbb{P}^{d}$. If the Chen-Ruan cohomology rings of $X$ and
$\mathbb{P}^{d}/G$ are not isomorphic, i.e.,

$\mathrm{H}^{*}_{CR}(X=Y/G)\not\cong\mathrm{H}^{*}_{CR}(\mathbb{P}^{d}/G),$

then $Y\curvearrowleft G$ is not $G$-birational to
${\mathbb{P}^{d}}\curvearrowleft G$.

The proof of Theorem 3.5 is done by contrapositive. One assumes that
$\mathbb{P}^{d}\curvearrowleft G,\leavevmode\nobreak\ Y\curvearrowleft G$ are
$G$-birational, passes to the resulting stacks $[\mathbb{P}^{d}/G]$ and
$[Y/G]$ and shows that (because they are birational) there exists an
isomorphism between the corresponding Chen-Ruan cohomology rings of the
underlined orbifolds. A major observation is that, in general, dealing with
the orbifold $X$ instead of $Y$ with the $G$-action could lead to information
loss, as we disregard the group action. Even the orbifold description of $X$
carrying the isotropy information (via uniformizing systems), the Chen-Ruan
cohomology encodes the needed ingredients to obstruct birationality for DM
stacks.

One needs the following auxiliary result to prove Theorem 3.5.

######  Proposition 3.6.

Assume that $\mathcal{X},\leavevmode\nobreak\ \mathcal{X}^{\prime}$ are
birational quasi-projective normal Deligne-Mumford stacks. Then

  1. (i)

there is a linear algebraic group $G$ and quasi-projective schemes
$Y,\leavevmode\nobreak\ Y^{\prime}$ such that

$\mathcal{X}\cong[Y/G],\leavevmode\nobreak\
\mathcal{X}^{\prime}\cong[Y^{\prime}/G]$

  2. (ii)

if $Y/G,\leavevmode\nobreak\ Y^{\prime}/G$ are Abelian orbifolds, then

$\mathrm{H}_{CR}^{*}(Y/G)\cong\mathrm{H}_{CR}^{*}(Y^{\prime}/G).$

###### Proof.

The proof of item $(i)$ follows from Item 1 in Remark 3.3 because, being
$\mathcal{X},\leavevmode\nobreak\ \mathcal{X}^{\prime}$ quasi-projective,
there are algebraic linear groups $G,\leavevmode\nobreak\ G^{\prime}$ and
quasi-projective schemes $Y,\leavevmode\nobreak\ Y^{\prime}$ such that
$\mathcal{X}\cong[Y/G]$ and
$\mathcal{X}^{\prime}\cong[Y^{\prime}/G^{\prime}]$. However, since
$\mathcal{X},\leavevmode\nobreak\ \mathcal{X}^{\prime}$ are birational, it
holds that $G=G^{\prime}$.

We proceed to prove item $(ii)$. Observe that we can obtain the inertia stacks
$\mathcal{I}_{\mathcal{X}},\leavevmode\nobreak\
\mathcal{I}_{\mathcal{X}^{\prime}}$ described as the stacks associated with
the inertia orbifolds for the groupoid realizations of
$Y/G,\leavevmode\nobreak\ Y^{\prime}/G$ as the fibered products

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Both $\mathcal{I}_{\mathcal{X}},\mathcal{I}_{\mathcal{X}^{\prime}}$ are
separated quasi-projective normal stacks. Because
$\mathcal{X},\leavevmode\nobreak\ \mathcal{X}^{\prime}$ are birational, so are
$\mathcal{I}_{\mathcal{X}},\leavevmode\nobreak\
\mathcal{I}_{\mathcal{X}^{\prime}}$ by a projective birational morphism
$\phi$. Proposition 4.2 in [39] teaches us that there exists a maximal open
substack $\mathcal{U}\subset\mathcal{X}$ of codimension $\geq 2$ making
$\mathcal{I}_{\mathcal{\mathcal{missing}}U}$ and
$\mathcal{I}_{\phi^{-1}(\mathcal{U})}$ isomorphic as stacks. Because
$\mathcal{X},\leavevmode\nobreak\ \mathcal{X}^{\prime}$ are quasi-projective
stacks one has that $\mathcal{U},\leavevmode\nobreak\ \phi^{-1}(\mathcal{U})$
are quasi-projective stacks. Thus, $\mathcal{U},\leavevmode\nobreak\
\phi^{-1}(\mathcal{U})$ are isomorphic to stacks from global quotient of
$G$-actions on certain quasi-projective schemes. We abuse notation and denote
these global quotients by $U/G$ and $\phi^{-1}(U)/G$, respectively.

Now treat the stacks $\mathcal{I}_{\mathcal{U}}$ and
$\mathcal{I}_{\phi^{-1}(\mathcal{U})}$ by the groupoid schemes (by looking at
the groupoid descriptions of $Y/G$ and $Y^{\prime}/G$). We have a Morita
equivalence between these groupoids. Thus, Proposition 2.18 ensures that any
data used to define the Chen-Ruan cohomology of the orbifolds
$U/G,\leavevmode\nobreak\ \phi^{-1}(U)/G$ coincide (recall from Section 2.3
that these are essentially line bundles associated with eigenspaces for
actions on the inertia orbifolds). Therefore,

$\mathrm{H}^{*}(\mathcal{I}_{\mathcal{U}})\cong\mathrm{H}^{*}(\mathcal{I}_{\phi^{-1}(\mathcal{U})}).$

Using that $\mathcal{U}$ is of codimension $\geq 2$ in $\mathcal{X}$, we can
apply a Hartogs’ extension-type theorem ([23, Lemma 3.5] or [22, Remarque
19.9.9]) to conclude that these quantities are extended to
$Y/G,\leavevmode\nobreak\ Y^{\prime}/G$ ensuring then

$\mathrm{H}^{*}(\mathcal{I}_{X})\cong\mathrm{H}^{*}(\mathcal{I}_{\mathcal{X}^{\prime}}).$

∎

###### Proof of Theorem 3.5.

Assume that $\mathbb{P}^{d}\curvearrowleft G,\leavevmode\nobreak\
Y\curvearrowleft G$ are $G$-birational. Then $[\mathbb{P}^{d}/G]$ and $[Y/G]$
are birational as Deligne-Mumford stacks (Definition 4.21+Example 4.24 in
[39]). Once these orbifolds satisfy the hypothesis in Proposition 3.6, the
proof is done. Observe that the remaining step is to verify that the quotient
stacks are normal. However, this holds because we assume that the $G$-actions
are birational actions extending to regular actions (see Remark 1.1). ∎

Next, we prove Theorem B. It will follow as a corollary of Theorem A. Below,
we re-state it for readers’ convenience.

######  Corollary 3.7.

Let $G=C_{4}$ (the cyclic group of order $4$) act

  1. (a)

linearly in $\mathbb{P}^{4}$, that is, as a homomorphism
$\rho:C_{4}\rightarrow\mathrm{PGL}_{5}(\mathbb{C})\cong\mathrm{Aut}(\mathbb{P}^{4})$
given in homogeneous coordiantes by

$\rho(g)\cdot[z_{0}:z_{1}:z_{2}:z_{3}:z_{4}]=[\rho_{0}(g)z_{0}:\rho_{1}(g)z_{1}:\rho_{2}(g)z_{2}:\rho_{3}(g)z_{3}:\rho_{4}(g)z_{4}],\leavevmode\nobreak\
\rho_{i}:G\rightarrow\mathrm{GL}(\mathbb{C})$

  2. (b)

and cyclic permuting coordinates on
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Then $\mathbb{P}^{4}$ with the linear $C_{4}$-action is not
$C_{4}$-birationally equivalent to
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
with the $C_{4}$-cyclic permuting action, i.e.,

$\mathbb{P}^{4}\not\sim_{C_{4}}\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}.$

Let $X(1,1,1,1)$ be the blowup of
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ on an elliptic curve.
Assume that

  1. (a)

$C_{4}$ acts in $X(1,1,1,1)$ via restricting the cyclic coordinate permutation
in $\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}$

  2. (b)

and $C_{4}$ acts linearly in $\mathbb{P}^{3}$, that is, as a homomorphism
$\rho:C_{4}\rightarrow\mathrm{PGL}_{4}(\mathbb{C})\cong\mathrm{Aut}(\mathbb{P}^{3})$.

Then $\mathbb{P}^{3}$ with the linear $C_{4}$-action is not
$C_{4}$-birationally equivalent to $X(1,1,1,1)$, i.e.,

$X(1,1,1,1)\not\sim_{C_{4}}\mathbb{P}^{3}.$

###### Proof.

According to Theorem A, one only needs to compare the corresponding Chen-Ruan
cohomology rings.

For the cyclic permuting action in
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$
one oberves that for each $g\neq 1\in C_{4}$ we have that
$(\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1})^{g}=\mathbb{P}^{1}$.
Therefore, using formulae given Remark 4.18 in [1], one gets that

$\mathrm{H}^{*}_{CR}(\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}/C_{4})=(4\mathrm{H}^{*}(\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}))^{C_{4}}.$

On the other hand, for any $C_{4}$-linear action in $\mathbb{P}^{4}$, we have
a more involved analysis. The group $C_{4}$ is the cyclic group of order 4,
represented by $\langle g\rangle$ where $g^{4}=1$. The action is given by the
functions $\rho_{i}:C_{4}\to\mathbb{C}^{*}$. Since $C_{4}$ has 4 elements (1,
$g$, $g^{2}$, $g^{3}$), for each $i$, $\rho_{i}(g)$ is a fourth root of unity.
The fourth roots of unity are $1$, $\omega$, $\omega^{2}$, and $\omega^{3}$,
where $\omega=e^{2\pi i/4}=\mathbf{i}$.

We can write the action of the generator $g$ in homogeneous coordinates as:

$g\cdot[z_{0}:z_{1}:z_{2}:z_{3}:z_{4}]=[\rho_{0}(g)z_{0}:\rho_{1}(g)z_{1}:\rho_{2}(g)z_{2}:\rho_{3}(g)z_{3}:\rho_{4}(g)z_{4}].$

We do not go through all the possible actions, but we show below how, for a
generic choice of linear action, the Chen-Ruan cohomology ring shall not be
isomorphic to that of the $C_{4}$-cyclic permuting action in
$\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}$.

  1. (i)

If $\rho_{i}(g)=1$ for all $i$, the action is trivial; every point is fixed.
Therefore:

$(\mathbb{P}^{4})^{g}=\mathbb{P}^{4}$

  2. (ii)

Assume that $\rho_{0}(g)=1$, $\rho_{1}(g)=\mathbf{i}$, $\rho_{2}(g)=-1$,
$\rho_{3}(g)=-\mathbf{i}$, $\rho_{4}(g)=1$. Then

$g\cdot[z_{0}:z_{1}:z_{2}:z_{3}:z_{4}]=[z_{0}:\mathbf{i}z_{1}:-z_{2}:-\mathbf{i}z_{3}:z_{4}]$

The fixed points satisfy:

$[z_{0}:z_{1}:z_{2}:z_{3}:z_{4}]=[z_{0}:\mathbf{i}z_{1}:-z_{2}:-\mathbf{i}z_{3}:z_{4}]$

Thus, the fixed points are of the form:

$[z_{0}:0:0:0:z_{4}]$

Therefore:

$(\mathbb{P}^{4})^{g}=\\{[z_{0}:0:0:0:z_{4}]\mid z_{0},z_{4}\in\mathbb{C},\
(z_{0},z_{4})\neq(0,0)\\}\cong\mathbb{P}^{1}$

Going case-by-case, such as before, one can rely on the formulae given Remark
4.18 in [1] to conclude that the Chen-Ruan cohomology rings for
$\mathbb{P}^{4}/C_{4}$ and
$\mathbb{P}^{1}\times\ldots\times\mathbb{P}^{1}/C_{4}$ are not isomorphic.
Theorem 3.5 concludes the proof. The second part of the statement follows
similarly. ∎

##  4\. Discrete torsion, Chen-Ruan cohomology, and non-$G$-birational
varieties

Let $X$ be a (not necessarily a global quotient) orbifold. Let $\mathcal{G}$
denote its groupoid description and $\wedge\mathcal{G}$ its inertia orbifold.
It is noteworthy, as per [43, 44], that the notion of _gerbes_ ([26]) extends
to orbifolds.

######  Definition 4.1.

A gerbe over an orbifold
$\mathcal{G}=(\mathcal{G}_{1}\rightrightarrows\mathcal{G}_{0})$ is a complex
line bundle $\mathcal{L}$ over $\mathcal{G}_{1}$ satisfying the following
conditions:

  * •

$I^{*}\mathcal{L}\cong\mathcal{L}^{-1}$

  * •

$\pi_{1}^{*}\mathcal{L}\otimes\pi_{2}^{*}\mathcal{L}\otimes
m^{*}i^{*}\mathcal{L}\overset{\theta}{\cong}1$

  * •

$\theta:\mathcal{G}_{1}{}_{t}\times_{s}\mathcal{G}_{1}\rightarrow\mathrm{U}(1)$
is a 2-cocycle

where
$\pi_{1},\pi_{2}:\mathcal{G}_{1}{}_{t}\times_{s}\mathcal{G}_{1}\rightarrow\mathcal{G}_{1}$
are the projections on the first and the second coordinates, and $\theta$ is a
trivialization of the line bundle.

The next proposition plays a major role if one aims to describe analogous
_Azumaya algebras_ (Definition 4.12) in terms of gerbes.

######  Proposition 4.2.

Gerbes $\mathcal{L}$ over a groupoid $\mathcal{G}$ are in bijective
correspondence with central extension of groupoids

$1\rightarrow\overline{\mathrm{U}(1)}\rightarrow\tilde{\mathcal{G}}\rightarrow\mathcal{G}\rightarrow
1$

where $\overline{\mathrm{U}(1)}$ is the groupoid
$\star\times\mathrm{U}(1)\rightarrow\star$ with the trivial maps $s,t$ and
$i(g)=g^{-1}$ and $m(h,g)=hg$.

######  Example 4.3 (Gerbes over a discrete group $G$).

Gerbes over a discrete group are in bijective correspondence with the two
cocycles $Z^{2}(G;\mathrm{U}(1))$. That is, let $\overline{G}$ denotes the
groupoid $\star\times G\rightrightarrows\star$ with the trivial maps $s,t$ and
$i(g)=g^{-1}$ and $m(h,g)=hg$.

A gerbe over $\overline{G}$ is a line bundle $\mathcal{L}$ over $G$ such that,
if we call $\mathcal{L}_{g}$ the fiber at $g$,
$\mathcal{L}_{g}^{-1}\stackrel{{\scriptstyle\beta}}{{\cong}}\mathcal{L}_{g^{-1}}$
and $\mathcal{L}_{g}\mathcal{L}_{h}=\mathcal{L}_{gh}$. Thus, for each $g,h\in
G$ there is a a trivialization $\beta_{g,h}\in\mathrm{U}(1)$ satisfying
$\beta_{g,h}\beta_{gh,k}=\beta_{g,hk}\beta_{h,k}$ Hence, $\beta:G\times
G\rightarrow\mathrm{U}(1)$ satisfies the cocycle condition, being then a two-
cocycle. Analogously, it is possible to construct a gerbe over $\overline{G}$
once given a two-cocycle.  $\blacksquare$

We relate gerbes over orbifolds with _Inner Local Systems_.

######  Definition 4.4 (Inner Local Systems).

Let $X$ be an orbifold. An _inner local system_
$\mathcal{L}=\left\\{L_{(g)}\right\\}_{(g)\in T_{1}}$ is an assignment of a
flat complex line orbi-bundle $L_{(g)}\rightarrow X_{(g)}$ to each twisted
sector $X_{(g)}$ satisfying the following compatibility conditions

  1. (a)

$L_{(1)}$ is a trivial orbifold line bundle with chosen trivialization

  2. (b)

$\mathrm{I}^{*}L_{(g^{-1})}=L_{(g)}$, where $I$ is the inversion map
(Definition 2.27)

  3. (c)

over each $X_{({\bf g})}$ with ${\bf
g}=(g_{1},g_{2},g_{2}),\leavevmode\nobreak\ g_{1}g_{2}g_{3}=1$

$e^{*}_{1}L_{(g_{1})}\otimes e^{*}_{2}L_{(g_{2})}\otimes
e^{*}_{3}L_{(g_{3})}=1$

######  Example 4.5 (Continuation of Example 2.20).

For each $i$ we have the evaluation morphisms
$e_{i}:\overline{\mathcal{M}}_{0,3}(X,0)\rightarrow\wedge X$ mapping
$(x,(g_{1},g_{2},g_{3}))\mapsto(x,g_{i})$. Consider to each $X_{(g_{i})}$ a
flat line bundle $L_{(g_{i})}$. We make the pullbacks
$\\{e_{i}^{*}(L_{(g_{i})})\\}$ and make a line bundle

$L_{123}:=e_{1}^{*}L_{(g_{1})}\otimes e_{1}^{*}L_{(g_{2})}\otimes
e_{1}^{*}L_{(g_{3})}\rightarrow\overline{\mathcal{M}}_{0,3}(X,0).$

We can associate to $L_{123}$ its first Chern class
$c_{1}(L_{123})\in\mathrm{H}^{2}(\overline{\mathcal{M}}_{0,3}(X,0);\mathbb{Z})$.
It is well-known that $L_{123}$ is trivial if, and only if,
$c_{1}(L_{123})=0$. Hence, an inner local system always gives rise to flat
line bundles over $\overline{\mathcal{M}}_{0,3}(X,0)$.  $\blacksquare$

######  Example 4.6 (Inner local systems on global quotient orbifolds).

Assume that $X=Y/G$ where $|G|<\infty$. For $g=1$, we have nothing but a
trivial line bundle $L_{(1)}=X\times\mathbb{C}$ over $X_{(1)}=X=Y/G$. In the
remaining cases, we have a flat line bundle $L_{(g)}$ over $Y^{(g)}/C(g)$ with
typical fiber $\mathbb{C}$. We can relate that with line bundles over
$Y^{(g)}$ in the following way. For each $(g)$ consider the projection
$\pi^{(g)}:Y^{(g)}\rightarrow Y^{(g)}/C(g)$. We can then pullback $L_{(g)}$ to
$Y^{(g)}$ defining a line bundle ${\pi^{(g)}}^{*}L_{(g)}\rightarrow Y^{(g)}$.
Not every line bundle over $Y^{(g)}$ comes in this way. However, Lemma 4.7
provides a criterion for that.

######  Lemma 4.7.

A line bundle $L^{Y}_{(g)}$ carrying a linear $C(g)$-action defines a line
bundle $L^{(g)}$ over $Y^{(g)}/C(g)$ if, and only if, for every $y\in Y^{(g)}$
the isotropy group $C(g)_{y}$ acts trivially on the fiber
${L^{Y}_{(g)}}_{y}\cong\mathbb{C}$. Consequently, the collection of inner
local systems $\mathcal{L}=\\{L_{(g)}\\}_{(g)\in T_{1}}$ over $X_{(g)}$ is is
bijective correspondence with flat line bundles $\\{L^{Y}_{(g)}\\}$ over
$Y^{(g)}$ with $C(g)$-linear actions with

  1. (a)

trivial $C(g)_{y}$-actions in
${L^{Y}_{(g)}}_{y}\cong\mathbb{C},\leavevmode\nobreak\ y\in Y^{(g)}$

  2. (b)

$L^{Y}_{(1)}=Y\times\mathbb{C}$ is the trivial bundle over $Y$

  3. (c)

${\pi^{(g)}}^{*}I\circ\pi^{(g)}_{\ast}L^{Y}_{(g)}\cong L^{Y}_{(g^{-1})}$

  4. (d)

${\left(\pi^{(g_{1})}\otimes\pi^{(g_{2})}\otimes\pi^{(g_{3})}\right)}^{*}\left(e_{1}^{*}\pi^{(g_{1})}_{\ast}L^{Y}_{(g_{1})}\otimes
e_{2}^{*}\pi^{(g_{2})}_{\ast}L^{Y}_{(g_{2})}\otimes
e_{3}^{*}\pi^{(g_{3})}_{\ast}L^{Y}_{(g_{3})}\right)=1$

###### Proof.

The only non-trivial claim is about the characterization of which bundles
descend to $Y^{(g)}/C(g)$ for each $g$. It is proven, however, in Lemma 2.2 or
Theorem 2.3 in [20]. ∎

$\blacksquare$

The relation between gerbes and inner local systems is recorded in the
following result:

######  Theorem 4.8 (Corollary 6.4.3 in [43]).

A gerbe over an orbifold $X$ determines an inner local system over the twisted
sectors $\widetilde{\Sigma X}$.

Because of Theorem 4.8, herein we think on gerbes over orbifolds in terms of
their defined inner local systems $\mathcal{L}=\\{L_{(g)}\\}$. Following [44],
one can use these line bundle families to introduce twisted in the Chen-Ruan
cohomology of an orbifold. To know, one can consider

(4.1)
$\mathrm{H}^{k}_{CR}(X;\mathcal{L})=\oplus_{g\in(T_{1})}\mathrm{H}^{k-2\iota_{(g)}}(X_{(g)};L_{(g)}).$

Gerbes over an orbifold $X$ are classified by the third integral cohomology
$\mathrm{H}^{3}(BX;\mathbb{Z})$ where $BX$ is the classifying space of the
orbifold $X$, see [1, Chapter 1.4]. An enhancement comes if one considers
instead of gerbes, _gerbes with connections_ over $X$. These form a group and
are classified by the Deligne cohomology group
$\mathrm{H}^{3}(X,{\mathbb{Z}(3)}^{\infty}_{D})$, see [43, Proposition 6.3.3,
p.473]. A related concept to the former is that of _discrete torsion_
(Definition 4.10). It plays a meaningful role in Chen-Ruan cohomology.
Discrete torsion and $B$-fields (as in [19, 18]) can be seen as elements in
$\mathrm{H}^{3}(X,{\mathbb{Z}(3)}^{\infty}_{D})$.

Assume that $X$ is a global quotient orbifold. The choice of discrete torsion
is a subgroup of the equivalence classes of gerbes with connection over the
$X$ ([43, Theorem 4.2.1]).

######  Example 4.9.

Consider a global quotient orbifold $X=Y/G$ with a choice of inner local
system $\\{L_{(g)}\\}$. We have the following decomposition of the twisted
sectors

$\widetilde{\Sigma X}=\cup_{(g)\in T_{1}}X_{(g)}\cong\cup_{(g)\in
T_{1}}Y^{(g)}/C(g),$

along with line bundles $\\{L_{(g)}\\}$ over each $Y^{(g)}/C(g)$.

Following Example 2.20 in [1, p.42], let $Z$ be a universal cover of $Y$. Then
$Z\rightarrow Y\rightarrow X$ is the orbifold universal cover of $X$. One gets
an exact sequence ([1, Definition 2.16, p. 40]).

$1\rightarrow\pi_{1}(Y)\rightarrow\pi_{1}^{\text{orb}}(X)\rightarrow
G\rightarrow 1.$

The former induces cohomology-level map

$\phi:\mathrm{H}^{2}(G;\mathbb{C}^{*})\rightarrow\mathrm{H}^{2}(\pi_{1}^{\text{orb}}(X);\mathbb{C}^{*}).$

######  Definition 4.10.

We name the classes
$\alpha\in\mathrm{H}^{2}(\pi_{1}^{\text{orb}}(X);\mathbb{C}^{*})$ as _discrete
torsion elements_.

Hence, elements in $\mathrm{H}^{2}(G;\mathbb{C}^{*})$ induce (via the image of
$\phi$) discrete torsion elements in $X$.

Expanding upon Example 4.3, pick a discrete torsion element
$\alpha\in\mathrm{H}^{2}(G;\mathbb{C}^{*})$. The element $\alpha$ is a gerbe
over the discrete group $G$. Recalling that each $\alpha$ can be identified
with a two cocycle, let us show that $\alpha$ induces a local inner system
$\mathcal{L}_{\alpha}:=\\{L_{(g),\alpha}\\}$ on $X$.

Think about $\alpha$ as a map $\alpha:G\times G\to\mathrm{U}(1)$ satisfying
$\alpha_{1,g}=\alpha_{g,1}=1$ and
$\alpha_{g,hk}\alpha_{h,k}=\alpha_{g,h}\alpha_{gh,k}$ for any $g,h,k\in G$.
Consider the quantity $\gamma(\alpha)_{g,h}:=\alpha_{g,h}\alpha^{-1}_{h,g}$.
It induces a representation of $C(g)$ (the centralizer of $g$ in $G$) in
$\mathrm{U}(1)$

$L^{\alpha}_{g}:=\gamma(\alpha)_{g,-}:C(g)\to\mathrm{U}(1).$

For each conjugacy class $(g)$ we consider
$L_{(g),\alpha}:=Y^{g}\times_{L_{g}^{\alpha}}\mathbb{C}$. This constitutes our
desired local inner system. Recalling from [16] that the Chen-Ruan cohomology
of $X=Y/G$ is given by

$\mathrm{H}^{k}_{CR}(X)=\bigoplus_{(g)\in
T_{1}}\left(\mathrm{H}^{k-2\iota_{(g)}}(Y^{g};\mathbb{C})\right)^{C(g)}$

let us understand the effect of the twist by discrete torsion on such a
cohomology.

Th inner local system $\mathcal{L}_{\alpha}$ induces a twisted action of
$C(g)$ on the cohomology of the fixed point set
$\mathrm{H}^{*}(Y^{g},\mathbb{C})$ by $\beta\mapsto
L_{g}^{\alpha}(h)h^{*}\beta$ for $h\in C(g)$. Let
$\mathrm{H}^{*}(Y^{g},\mathbb{C})^{C^{\alpha}(g)}$ be the invariant subspace
under this twisted action. Then

(4.2) $\mathrm{H}^{k}_{CR,\alpha}(X;\mathbb{C})=\bigoplus_{(g)\in
T_{1}}\left(\mathrm{H}^{k-2\iota_{(g)}}(Y^{g};\mathbb{C})\right)^{C^{\alpha}(g)}.$

$\blacksquare$

Let $G$ be a finite group. Recall that:

######  Definition 4.11.

  1. (1)

A _projective representation_ of a group $G$ on a vector space $V$ over a
field $k$ is a group homomorphism from $G$ to the projective linear group

$\mathrm{PGL}_{k}(V):=\mathrm{GL}(V)/k^{*}$

where $\mathrm{GL}(V)$ is the general linear group of invertible linear
transformations of $V$ over $k$, and $k^{*}$ is the normal subgroup whose
elements are nonzero scalar multiples of the identity.

  2. (2)

A _linear representation_ of a group $G$ on a vector space $V$ over a field
$k$ is a group homomorphism from $G$ to $\mathrm{GL}(V)$.

Following, [30, Section 3], let

  1. (i)

$RG$: monoid whose underlying set consists of isomorphism classes of linear
representations of $G$

  2. (ii)

$PG$: monoid whose underlying set consists of isomorphism classes of
projective representations of $G$.

Above, the group structure is induced by the tensor product.

There is a short exact sequence ([30, Equation 3.1, p.298])

$RG\rightarrow PG\rightarrow\mathrm{H}^{2}(G;\mathbb{C}^{*})\rightarrow 0.$

Moreover,

  1. (a’)

$\mathrm{H}^{2}(G;\mathbb{C}^{*})$ is a finite group ([30, 3.2, p.298])

  2. (b’)

$\mathrm{H}^{2}(G;\mathbb{C}^{*})\cong\mathrm{H}^{3}(G;\mathbb{Z})$ ([30,
Theorem 3.6, p.298]).

######  Definition 4.12.

We name $\mathrm{H}^{2}(G;\mathbb{C}^{*})$ as _the Schur multiplier_ of the
finite group $G$. To each nonzero element
$\alpha\in\mathrm{H}^{2}(G;\mathbb{C}^{*})$, we term an Azumaya
algebra.222This terminology may not be very usual. Nevertheless, we chose that
in relation to [30].

Expanding upon Example 4.9, we define.

######  Definition 4.13.

Let $X=Y/G$ be a global quotient orbifold. Recall that there exists a map
$\phi:\mathrm{H}^{2}(G;\mathbb{C}^{*})\rightarrow\mathrm{H}^{2}(\pi^{\mathrm{orb}}(X);\mathbb{C}^{*})$.
An _Azumaya algebra over_ $X$ is a non-trivial element in the image of $\phi$.

The next section discusses certain subgroups of the Picard group on orbifolds
and their possible relations with $G$-birational invariants.

###  4.1. Gerbes, discrete torsion, and birational invariants

Let $\mathcal{G}=(\mathcal{G}_{0},\mathcal{G}_{1},s,t)$ be the groupoid
representation of an orbifold $X$. Herein, we refer to the translating Table 1
below, built based upon Theorem 4.8.

Table 1. Orbifold vs. Groupoid quantities Orbifold $X$ | Groupoid representation $\mathcal{G}=(\mathcal{G}_{0},\mathcal{G}_{1},s,t)$
---|---
Twisted sectors $\widetilde{\Sigma X}=\bigcup_{(g)\in T_{1}}X_{(g)}$ | Inertial groupoid $\wedge\mathcal{G}$
Inner local systems $\mathcal{L}=\\{L_{(g)}\\}$ | (Gerbes) Line bundles $\mathcal{L}$ over $\mathcal{G}_{1}$ with properties

######  Definition 4.14.

We denote by

  1. (1)

$\mathrm{Pic}_{\mathrm{gerbes}}(\mathcal{G}_{1})$ the Picard group of line
bundles over $\mathcal{G}_{1}$ coming from gerbes over $X$.

  2. (2)

$\mathrm{Pic}_{\mathrm{torsion}}(\mathcal{G}_{1})$ the Picard group of line
bundles over $\mathcal{G}_{1}$ coming from discrete torsion elements
$\alpha\in\mathrm{H}^{2}({\pi_{1}}^{\mathrm{orb}}(X);\mathbb{C}^{*})$.

In [55, Section 4.4, p.47] it is presented a necessary condition for
birational equivalence.

######  Theorem 4.15.

Let $X,\leavevmode\nobreak\ Y$ be nonsingular projective algebraic varieties
over a field $k$ of characteristic zero. Let $L/k$ be a normal extension with
Galois group $\Pi$. If $X,\leavevmode\nobreak\ Y$ are birationally equivalent
over $k$, the $\Pi$-modules

$\mathrm{Pic}(X\otimes_{k}L),\leavevmode\nobreak\ \mathrm{Pic}(Y\otimes_{k}L)$

are similar. That is, there exists $\Pi$-modules $S_{1},\leavevmode\nobreak\
S_{2}$ such that

$\mathrm{Pic}(X\otimes_{k}L)\oplus S_{1}\cong\mathrm{Pic}(Y\otimes_{k}L)\oplus
S_{2}.$

Here, we obtain its analogous in the context of orbifolds (for birationality
as DM stacks), Theorem 4.16. We observe that the proof of the former theorem
is based on the following facts:

  1. (i)

if $X,\leavevmode\nobreak\ Y$ are birational over $k$, there exists a
nonsingular projective variety $Z$ over $k$ and and birational morphisms

$X\longleftarrow Z\longrightarrow Y$

  2. (ii)

Each birational morphism induces an exact sequence

$0\rightarrow\mathrm{Pic}(Y)\rightarrow\mathrm{Pic}(Z)\rightarrow S_{Y}$
$0\rightarrow\mathrm{Pic}(X)\rightarrow\mathrm{Pic}(Z)\rightarrow S_{X}$

  3. (iii)

it can be shown that

$\mathrm{Pic}(Z)\cong\mathrm{Pic}(X)\oplus S_{X},\leavevmode\nobreak\
\mathrm{Pic}(Z)\cong\mathrm{Pic}(Y)\oplus S_{Y}.$

######  Theorem 4.16.

Let $X,\leavevmode\nobreak\ X^{\prime}$ be two global quotient orbifolds with
projective and normal coarse moduli spaces. Let
$\mathcal{G},\leavevmode\nobreak\ \mathcal{G}^{\prime}$ extend to their
groupoid representations. Assume that the stacks
$\mathcal{X},\leavevmode\nobreak\ \mathcal{X}^{\prime}$ for
$X,\leavevmode\nobreak\ X^{\prime}$ are birational. Then

$\mathrm{Pic}_{\mathrm{gerbes}}(\mathcal{G}_{1})\cong\mathrm{Pic}_{\mathrm{gerbes}}(\mathcal{G}^{\prime}_{1}).$

###### Proof.

If $\mathcal{X},\leavevmode\nobreak\ \mathcal{X}^{\prime}$ are birational as
Deligne-Mumford stacks, one gets a smooth projective stack $\mathcal{Z}$ and
projective birational morphisms
$\mathcal{X}\longleftarrow\mathcal{Z}\longrightarrow\mathcal{X}^{\prime}$.
Thus, if $\mathcal{G},\leavevmode\nobreak\ \mathcal{G}^{\prime}$ denote the
groupoid descriptions of $X,\leavevmode\nobreak\ X^{\prime}$, there exists
open dense substacks $\mathcal{U}\subset\mathcal{X},\leavevmode\nobreak\
\mathcal{U}^{\prime}\subset\mathcal{X}^{\prime}$ such that

$\mathcal{I}_{\mathcal{U}},\leavevmode\nobreak\
\mathcal{I}^{\prime}_{\mathcal{U}^{\prime}}$

are isomorphic stacks. Thus, the restriction inertia orbifolds
$\wedge\mathcal{G}|_{\mathcal{U}},\leavevmode\nobreak\
\wedge\mathcal{G}|_{\mathcal{U}^{\prime}}$ are Morita equivalent.

Proposition 2.18 ensures that any line bundle $\mathcal{L}$ over
$\mathcal{G}_{\mathcal{U}}$ can be pushforwarded to
$\mathcal{G}^{\prime}_{\mathcal{U}^{\prime}}$, moreover, if it satisfies the
requirements in Definition 4.1 over $\mathcal{G}_{1}|_{\mathcal{U}}$, it does
satisfy on $\mathcal{G}_{1}|_{\mathcal{U}^{\prime}}$. Via a Hartogs’
extension-type theorem ([23, Lemma 3.5] or [22, Remarque 19.9.9]), one can
assume that these line bundles extend over, concluding the proof.∎

We proceed to extend Theorem 3.5 to global quotient orbifolds for non-Abelian
finite groups. We need to introduce more terminology. We follow [43]. The idea
is to describe the Chen-Ruan cohomology of a global quotient orbifold via
Chen-Ruan twisted cohomology.

Let $G$ be a finite group and $\overline{G}$ be the groupoid
$G\times\star\rightrightarrows\star$. Then the loop groupoid
$\mathrm{L}\overline{G}$ of $\overline{G}$ is Morita equivalent to the
groupoid $[\mathrm{Hom}(\mathbb{Z},G)/G]$ where $G$ acts on
$\mathrm{Hom}(\mathbb{Z},G)$ by conjugation ([43, Lemma 4.2.1]). Therefore,
$\mathrm{L}\overline{G}$ can be seen as the groupoid $G\times
G\rightrightarrows G$ where $s(g,h)=g$ and $t(g,h)=h^{-1}gh$. The groupoids
below are Morita equivalent, where the right-hand side runs over the conjugacy
classes of elements in $G$ and $C(g)$ is the centralizer of $g$.

$G\times G\rightrightarrows G\cong\bigsqcup_{\left(g\right)}\left(\star\times
C(g)\rightrightarrows\star\right)$

The groupoid $\bigsqcup_{\left(g\right)}\left(\star\times
C(g)\rightrightarrows\star\right)$ is also a subgroupoid of
$\mathrm{L}\overline{G}=\wedge\overline{G}$:

$\bigsqcup_{\left(g\right)}\left(\star\times
C(g)\rightrightarrows\star\right)\hookrightarrow G\times G\rightrightarrows
G.$

Consider now a gerbe over the groupoid $\overline{G}$. Example 4.3 teaches us
that it can be thought of as a line bundle $L$ over $G$ and a 2-cocycle
$\beta:G\times G\to\mathrm{U}(1)$ such that $L^{-1}_{g}\cong L_{g^{-1}}$ and
$L_{g}L_{h}\stackrel{{\scriptstyle\beta(g,h)}}{{\cong}}L_{gh}$. This can be
used to define a line bundle on the inertia groupoid.

######  Lemma 4.17 (Lemma 4.3.1 in [43]).

Let $\wedge\overline{G}=G\times G\rightrightarrows G$ be the inertia groupoid
of $\overline{G}$. There is a morphism of groupoids
$\rho:\wedge\overline{G}\to\overline{\mathrm{U}}(1)$ which produces a line
bundle on $\wedge\overline{G}$ coinciding with a gerbe over $\overline{G}$.

Finally, using the subgroupoid representation

$\sqcup_{(g)}\overline{C(g)}\hookrightarrow\wedge\overline{G}$

we can consider the map $\rho$ restricted to the centralizers $C(g)$ to
obtain:

######  Proposition 4.18 (Proposition 4.3.2 in [43]).

The map $\rho(g,\cdot):C(g)\to\mathrm{U}(1)$ is a representation that defines
a line bundle on the groupoid $\overline{C}(g)$.

Expanding upon Example 4.9, we can produce a twisted Chen-Ruan cohomology
description of the Chen-Ruan cohomology of any global quotient orbifold $Y/G$.
In the possession of this information, adapting the proof of Proposition 3.6
ensures the following:

######  Theorem 4.19.

Let $G$ be a finite group acting regularly and generically free on a smooth
irreducible projective variety $Y$ of dimension $d\geq 2$ and let $X:=Y/G$ be
the resulting orbifold. Pick an effective linear $G$-action on
$\mathbb{P}^{d}$. If the Chen-Ruan cohomology rings of $X$ and
$\mathbb{P}^{d}/G$ are not isomorphic, i.e.,

$\mathrm{H}^{*}_{CR}(X=Y/G)\not\cong\mathrm{H}^{*}_{CR}(\mathbb{P}^{d}/G),$

then $Y\curvearrowleft G$ is not $G$-birational to
${\mathbb{P}^{d}}\curvearrowleft G$.

We pass to an application. Following [9, Example 14, p.20], let
$X\subset\mathbb{P}^{5}$ be a hypersurface admitting an effective action of
the Frobenious group
$G=\mathrm{AGL}_{1}(\mathbb{F}_{7})=\mathbb{F}_{7}\rtimes\mathbb{F}_{6}=C_{7}\rtimes
C_{6}$. Then $X$ is a Pfaffian cubic fourfold

$x_{1}^{2}x_{2}+x_{2}^{2}x_{3}+x_{3}^{2}x_{4}+x_{4}^{2}x_{5}+x_{5}^{2}x_{6}+x_{1}x_{6}^{2}+\lambda^{2}(x_{1}x_{3}x_{5}+x_{2}x_{4}x_{6}).$

We have that $X$ is smooth for $\lambda\neq 0,\xi,\sqrt{3}\xi$ for $\xi$ a
$6$-th root of unity. In the possession of the former, we now prove:

######  Corollary 4.20.

Let $X\subset\mathbb{P}^{5}$ be a smooth Pfaffian cubic with an effective
regular action of the Frobenius group
$G=\mathrm{AGL}_{1}(\mathbb{F}_{7})=\mathbb{F}_{7}\rtimes\mathbb{F}_{6}=C_{7}\rtimes
C_{6}$. Then there is no regular $G$-linear action on $\mathbb{P}^{4}$ making
it $G$-birational to $X$.

###### Proof.

Pick any regular $G$-linear action on $\mathbb{P}^{4}$. We thus have a
homomorphism
$\rho:G\rightarrow\mathrm{Aut}(\mathbb{P}^{4})=\mathrm{PGL}_{5}(\mathbb{C})$
induced by a representation
$\widetilde{\rho}:G\rightarrow\mathrm{GL}_{5}(\mathbb{C})$. However, as we
learn from [9, Example 14, p.20], $\rho$ have a non-trivial ineffective
kernel, say $H$. Using that

$\mathrm{H}^{*}_{CR}(\mathbb{P}^{4}/G)=\oplus_{(g):g\in
G}\left(\mathrm{H}^{*}((\mathbb{P}^{4})^{g})\right)^{C(g)}$

we see that $\mathrm{H}^{*}_{CR}(\mathbb{P}^{4}/G)$ contains $|H|+1$ copies of
$\mathrm{H}^{*}(\mathbb{P}^{4})$. If $X$ with the $G$-action were
$G$-birational to $\mathbb{P}^{4}$ with a given linear $G$-action, because of
Theorem A we would have that the Chen-Ruan cohomology ring of $X$ would
contain $|H|+1$ isomorphic copies of $\mathrm{H}^{*}(\mathbb{P}^{4})$. As

$\mathrm{H}^{*}_{CR}(X/G)=\oplus_{(g):g\in
G\setminus\\{1\\}}\left(\mathrm{H}^{*}(X^{g})\right)^{C(g)}\oplus\mathrm{H}^{*}(X),$

and each $X^{g}$ is embedded in $X$, this is a contradiction. ∎

######  Remark 4.21.

In [9] it is shown that despite the $G$-equivariant SOD of $X$ with this
action contains a component equivariantly isomorphic to $\mathrm{D}^{b}(K3)$,
$X\curvearrowleft G$ is not $G$-birational to $\mathbb{P}^{4}$ with any linear
$G$-action. The way that we prove Corollary 3.7, however, points out that this
equivariant SOD is not related to a SOD for the stack $\mathcal{X}=[X/G]$. In
other words, following [33], we conclude that the atoms for the SOD of $X/G$
are essentially different from the atoms for $X$. This further justifies
developing the theory of equivariant atoms in [14]. ∎

Recall that a nodal cubic threefold is a cubic with ordinary double points. In
[15], it is proved the following:

######  Proposition 4.22.

Let $X$ be a nodal cubic threefold with a regular action by a finite group
$G$. Assume that there is an element $\tau\in G$ such that

  1. (1)

the $\tau$-fixed locus contains a cubic surface $S\subset X$,

  2. (2)

the subgroup $Y\subseteq G$ preserving $S$ acts generically nontrivially on
$S$ and contains an element fixing a curve of genus $\geq 1$.

Then, the $G$-action on $X$ is not $G$-birational to $\mathbb{P}^{3}$ with any
linear $G$-action.

An essential ingredient in the proof is the fact
$\mathrm{H}^{1}(Y/H,\mathrm{Pic}(S))\neq 0$ where $H=\langle\tau\rangle$. As a
corollary, authors obtain

######  Theorem 4.23.

Let $X\subset\mathbb{P}^{4}$ be a 2-nodal cubic given by

$x_{1}x_{2}x_{3}+x_{1}(x^{2}_{4}+x^{2}_{5})+x_{2}(x^{2}_{4}-x^{2}_{5})+x_{3}^{3}=0,$

with a $G\cong C_{4}$-action generated by

$\iota:(x_{1},x_{2},x_{3},x_{4},x_{5})\to(x_{2},x_{1},x_{3},x_{4},\zeta_{4}x_{5}).$

Then $X$ with the former action is not $G$-birational to $\mathbb{P}^{3}$ with
any $G\cong C_{4}$-linear action.

With this paper’s techniques, we can show the following.

######  Theorem 4.24.

Let $X\subset\mathbb{P}^{4}$ be a 2-nodal cubic given by

$x_{1}x_{2}x_{3}+x_{1}(x^{2}_{4}+x^{2}_{5})+x_{2}(x^{2}_{4}-x^{2}_{5})+x_{3}^{3}=0,$

with a $G\cong C_{4}$-action generated by

$\iota:(x_{1},x_{2},x_{3},x_{4},x_{5})\to(x_{2},x_{1},x_{3},x_{4},\zeta_{4}x_{5}).$

Pick any $G\cong C_{4}$-linear action in $\mathbb{P}^{3}$. Then

$\mathrm{H}_{CR}^{*}(X/C_{4})\not\cong\mathrm{H}^{*}_{CR}(\mathbb{P}^{3}/C_{4}).$

Consequently, $X$ is not $C_{4}$-birational to $\mathbb{P}^{3}$ with any
$C_{4}$-linear action.

###### Proof.

The proof follows along the lines of Theorem B combined, and it is just a
matter of computing the fixed point set for each element for the
$C_{4}$-actions.∎

In the next section, we propose connecting Chen-Ruan cohomology quantities on
orbifolds with certain combinatorial quantities on projective smooth varieties
with regular group actions.

###  4.2. A Gromov-Witten reading of $\beta$-classes in $\mathcal{B}_{d}(G)$

In the pioneering work [36], M. Kontsevich, V. Pestun, and Y. Tschinkel
introduced new invariants in equivariant birational geometry represented by
specific classes in a group $\mathcal{B}_{d}(G)$. Although we prove no
concrete result here, we propose an analogy, summarized by the “translation”
Table 2 below, to connect geometric quantities on global quotient orbifolds
with the $\beta$-classes defining quantities’.

Let $G$ be a finite group acting generically free and birationally on a
(complex) smooth irreducible projective variety $Y$ of dimension $d\geq 2$.
After a $G$-equivariant singularity resolution, one can assume the $G$-action
is regular. Let

$Y^{G}=\left\\{y\in Y:gy=y\leavevmode\nobreak\ \forall g\in
G\right\\}=\cap_{g\in G}Y^{g},$

where $Y^{g}=\\{y\in Y:gy=y\\}$, be the $G$-action fixed point locus.

There is a collection of indexes $\mathcal{A}$ such that

$Y^{G}=\sqcup_{\alpha\in\mathcal{A}}F_{\alpha}$

where $\\{F_{\alpha}\\}$ are closed smooth irreducible subvarieties of $Y$.
Let $d_{\alpha}:=\dim F_{\alpha}\leq d-1$. Pick _any_ $y_{\alpha}\in
F_{\alpha}$ and let $G$-act on $T_{y_{\alpha}}Y$ as the isotropy
representation $\rho:G\rightarrow\mathrm{Aut}(T_{y_{\alpha}}Y)$, i.e., the
linearization of the $G$-action on $Y$ at $y_{\alpha}$. It decomposes as

$T_{y_{\alpha}}Y=\oplus_{i=1}^{d}E_{a_{i,\alpha}}$

where $E_{a_{i,\alpha}}$ are one-dimensional (over $\mathbb{C}$) eigenspaces
with characters $\\{a_{i,\alpha}\\}$. Moreover,

$\sum_{i=1}^{d}\mathbb{Z}a_{i,\alpha}=\mathrm{Hom}(G,\mathbb{C}^{\times}).$

Let $\mathcal{B}_{d}(G)$ be the group collecting

  1. (a)

for each $\alpha$ the symbols $[a_{1,\alpha},\ldots,a_{d,\alpha}]$

  2. (b)

subjected to the relations

    1. (i)

the symbols are invariant under permutation

    2. (ii)

for every positive integer $2\leq k\leq d$

$[a_{1,\alpha},\ldots,a_{d,\alpha}]=\sum_{1\leq i\leq k,\leavevmode\nobreak\
a_{i,\alpha}\neq a_{i^{\prime},\alpha},\leavevmode\nobreak\ \forall
i^{\prime}<i}[a_{1,\alpha}-a_{i,\alpha},\ldots,a_{i,\alpha},\ldots,a_{k,\alpha}-a_{i,\alpha},a_{k+1,\alpha},\ldots,a_{d,\alpha}]$

In [52, 39, 38], the authors explore the $\beta$-classes above to study
$G$-birationality problems.

######  Theorem 4.25 (Theorem 3 in [36]).

For any smooth projective irreducible variety $Y$ of dimension $d\geq 2$ with
a generically free action by a finite group $G$, the class
$\beta(Y):=\sum_{\alpha}[a_{1,\alpha},\ldots,a_{d,\alpha}]\in\mathcal{B}_{d}(G)$
is an equivariant birational invariant. Consequently, given two
$X,\leavevmode\nobreak\ Y$ as formerly, if $\beta(X)\neq\beta(Y)$ then
$X\not\sim_{G}Y$.

We believe one can read the information above described by looking at the
orbifold quotient $Y/G$. In the table below, the quantities on the right are
an orbifold manifestation of those on the left. Assume that $G$ is an Abelian
orbifold so that the Chen-Ruan cohomology description known as “de Rham model”
is in hand (recall Section 2.3). We propose the following conjecture.

######  Conjecture 4.26.

Let $Y,\leavevmode\nobreak\ Y^{\prime}$ be smooth irreducible projective
varieties with regular generically free group actions by a finite Abelian
group $G$. Consider the resulting global quotient orbifolds
$X:=Y/G,\leavevmode\nobreak\ X^{\prime}:=Y^{\prime}/G$. For each global
quotient orbifolds $X,\leavevmode\nobreak\ X^{\prime}$, collect the twisted
factors $\\{\mathrm{t}(g)\\},\\{\mathrm{t^{\prime}}(g)\\}$. If they do not
coincide as sets, $\beta(X)\neq\beta(X^{\prime})$.

Table 2. Translating concepts between $\mathcal{B}_{d}(G)$ and Chen-Ruan cohomology Projective varieties with $G$-free birational actions $Y\curvearrowleft G$ | Global quotient orbifolds $X=Y/G$
---|---
Fixed point set $Y^{G}$ | Twisted sectors $\widetilde{\Sigma X}=\bigcup_{(g)\in T_{1}}X_{(g)}$ or inertia $\wedge\mathcal{G}$
Eigenspaces for linearized $G$-actions $T_{y}Y\curvearrowleft G$, $y\in Y^{G}$ | Eigenspaces $\\{E_{j}\\}$ yielding $e^{*}(T_{x}\mathcal{G}_{0})=\oplus_{j}E_{j}$
Symbols in $\mathcal{B}_{d}(G)$ | Eigenvalues $\\{e^{2\pi i\frac{m_{j}}{m}}:g\in G,m=\mathrm{order}(g)\\}$
$\beta$-class in $\mathcal{B}_{d}(G)$ | Twisted factors $\mathrm{t}(g)$

##  5\. Some conjectures and further preliminary results

Below, we furnish a possible _twisted K-theory approach_ to $G$-birationality.
First, we provide a _motivic_ interpretation of $G$-birationality.

###  5.1. A motivic interpretation of $G$-birationality

Following [42], the cohomological McKay-Reid correspondence states that for a
global quotient orbifold $X=Y/G$ having a smooth crepant resolution, there is
a correspondence between the cohomology generators of the resolution and the
conjugacy classes of the group $G$.

######  Theorem 5.1 (Theorem 1 in [42]).

Let $X,\leavevmode\nobreak\ X^{\prime}$ to (complete) global quotient
orbifolds. Assume there are common birational singularity resolutions
$\phi:\widetilde{X}\rightarrow X,\leavevmode\nobreak\
\psi:\widetilde{X}\rightarrow X$ such that
$\phi^{*}(K_{X})\cong\psi^{*}(K_{\widetilde{X}})$. Then, their orbifold Hodge
numbers, Hodge structures, and orbifold Euler characteristics coincide.

Theorem 5.1 is proved using a motivic measure formula, Equation (5.1) below.

As one learns from [16], it is expected that “orbifold quantum cohomology”,
meaning Chen-Ruan cohomology, is isomorphic to the ordinary quantum cohomology
of the orbifold singularity crepant resolutions. We conjecture the following.

######  Conjecture 5.2.

The motivic measure below is a $G$-birational invariant

$\mu_{\mathrm{orb}}(\mathrm{L}(X))=\sum_{(g)\in
T_{1}}\left[X_{(g)}/X\right]\mathbb{L}^{\iota_{(g^{-1})}},$

where $\mathbb{L}=[\mathbb{A}^{1}]$ is the _Tate motive_ and $\mathrm{L}(X)$
is the space of arcs in $X$ whose $\mathbb{C}$-points correspond to formal
arcs $\mathrm{Spec}\mathbb{C}[[z]]\rightarrow X$. In other words, let a finite
group $G$ act regularly generically free on smooth projective irreducible
varieties $Y,\leavevmode\nobreak\ Y^{\prime}$ of dimension $d\geq 2$. Let
$X,\leavevmode\nobreak\ X^{\prime}$ be the resulting global quotient orbifolds
(obtained as orbit spaces). If

(5.1)
$\mu_{\mathrm{orb}}(\mathrm{L}(X))\neq\mu_{\mathrm{orb}}(\mathrm{L}(X^{\prime}))$

then $Y\not\sim_{G}Y^{\prime}$.

###  5.2. Twisted K-theory

Herein, let $X$ be an orbifold. This section proposes some connections between
birational invariants and twisted K-theory. Recall Table 1. Gerbes with
connections correspond with $\mathcal{L}$ _twisted bundles_ over $X$:

######  Definition 5.3 ($\mathcal{L}$ twisted bundles over orbifolds,
Definition 7.2.1+Proposition 7.2.2 in [43]).

An $n$-dimension _twisted line bundle_ over $X$ is a vector bundle
$E\rightarrow\mathcal{G}_{0}$ with an isomorphism

$\mathcal{L}\otimes t^{*}E\cong s^{*}E.$

######  Definition 5.4 (Twisted K-theory, Definition 7.2.3 in [43]).

Let $X$ be an orbifold and $\mathcal{L}$ a gerbe over $X$. The Grothendieck
group generated by the isomorphism classes of $\mathcal{L}$ twisted bundles
over $X$ is the $\mathcal{L}$ twisted K-theory
${}^{\mathcal{L}}K_{\mathrm{grp}}(X)$.

######  Theorem 5.5 (Proposition 6.3.3 in [44]).

Let $X$ be an orbifold. The gerbes $\mathcal{L}$ over $X$ are classified by
$\mathrm{H}^{3}(BX;\mathbb{Z})$. Gerbes with connections, and hence,
$\mathcal{L}$ twisted bundles over $X$, are classified by the enhancement
$\mathrm{H}^{3}(X;\mathbb{Z}(3)^{\infty}_{D})$ of
$\mathrm{H}^{3}(BX;\mathbb{Z})$.

Given an inner local system $\mathcal{L}_{\alpha}$ coming from a discrete
torsion $\alpha\in\mathrm{H}^{2}(\pi_{1}^{\mathrm{orb}}(X);\mathbb{C}^{*})$ we
let

$\mathrm{H}^{*}_{CR,\alpha}(X;\mathbb{C}):=\mathrm{H}^{*}_{CR}(X;\mathcal{L}_{\alpha}).$

Theorem 7.2.16 in [44] ensures that

(5.2)
${}^{\mathcal{L}_{\alpha}}K_{\mathrm{grp}}(X)\otimes{\mathbb{C}}\cong\mathrm{H}^{*}_{CR;\alpha}(X;\mathbb{C}).$

######  Question 1.

Let $X$ be a global quotient orbifold $X$. Let $\\{\mathcal{L}\\}$ collect all
the gerbes with connections over $X$. Are
$\\{{}^{\mathcal{L}_{\alpha}}K_{\mathrm{grp}}(X)\\}$ birational invariants for
Deligne-Mumford stacks? Can this provide easier computation methods (relying
upon K-theory methods) to decide birationality?

Equation (4.2) in Example 4.9 presents a twisting of the Chen-Ruan cohomology
given by discrete torsion. Conjecture 5.6 proposes a motivic measure
interpretation of the Chen-Ruan cohomology.

######  Conjecture 5.6.

The quantity given by Equation (5.3) below

(5.3) $\mu_{\mathrm{orb},\alpha}(\mathrm{L}(X))=\sum_{(g)\in
T_{1}}\left[\left(Y^{g}/C^{\alpha}(g)\right)/X\right]\mathbb{L}^{\iota_{(g^{-1})}}$

is

  1. (1)

a motivic measure

  2. (2)

a $G$-birational invariant

where $\mathbb{L}=[\mathbb{A}^{1}]$ is the _Tate motive_ and $\mathrm{L}(X)$
is the space of arcs in $X$ whose $\mathbb{C}$-points correspond to formal
arcs $\mathrm{Spec}\mathbb{C}[[z]]\rightarrow X$.

###  5.3. Birational invariants for orbifolds via twisted K-theory

Throughout this manuscript, we have mostly considered global quotient
orbifolds. One of the key reasons for that is that the $\beta$-classes
(belonging to $\mathcal{B}_{d}(G)$) are related to finite group actions on
smooth irreducible projective varieties $Y$. However, there is no reason why
we do not try to approach birational invariants for orbifolds $X$ such as the
one presented in Example 2.10.

In [12, Conjectures F, G], we have proposed a connection between smooth
invariants ([47, 34]) related to Homotopy Hopf manifolds, i.e.,
$\Sigma^{7}\times\mathrm{S}^{1}$, where $\Sigma^{7}$ is a $7$-dimension
homotopy sphere, and birational invariants, on certain orbi-fanos coming from
$\mathrm{T}^{2}$-fibrations. Expanding on Example 2.10, pick
$f_{k}(u,v,z_{0},z_{1},z_{2}):=u^{6k-1}+v^{3}+z_{0}^{2}+z_{1}^{2}+z_{2}^{2},\leavevmode\nobreak\
k\in\\{1,\ldots,28\\}$. Then $(0,\ldots,0)$ is the only singularity in any
arbitrarily small ball around the origin in $\mathbb{C}^{5}$ and
$f^{-1}(0)\cap\mathrm{S}^{9}$ inherits a $\mathrm{S}^{1}$-action from the
$\mathbb{C}^{*}$-action defined in Example 2.10. The orbit spaces
$f^{-1}(0)\cap\mathrm{S}^{9}/\mathrm{S}^{1}=:V_{k}$ are Fano orbifold
hypersurfaces in a weighted projective space, [12, Lemma 5.4].

Using that for each $k$ the manifolds $f_{k}^{-1}(0)\cap\mathrm{S}^{9}$
correspond to one of the diffeomorphism classes of smooth structures of seven
dimension spheres ([11]), it is natural to ask if for each $k\neq
k^{\prime}\in\\{1,\ldots,28\\}$, $V_{k}$ is not birational to
$V_{k^{\prime}}$. Now that we have argued how gerbes can be associated with
birational invariants, we propose the following approach to decide this
question.

Consider the third Deligne cohomology
$\mathrm{H}^{3}(V_{k};\mathbb{Z}(3)_{D}^{\infty})$ of each $V_{k}$. This
cohomology group parameterizes all the gerbes with connections over $V_{k}$.
As we explained, these gerbes with connections give rise to the Grothendieck
group of isomorphism classes of twisted bundles over $V_{k}$. Given that among
the $\mathrm{T}^{2}$-bundles over $V_{k}$, for different $k$, different
diffeomorphism types total spaces exist, it is reasonable to expect the
existence of different isomorphism classes of twisted bundles over two
$V_{k},\leavevmode\nobreak\ V_{k^{\prime}},\leavevmode\nobreak\ k\neq
k^{\prime}$. In other words, non-isomorphic twisted K-theories.

######  Conjecture 5.7.

For each $k\in\\{1,\ldots,28\\}$ there exist gerbes with connections
$\mathcal{L}_{k}$ over the Fano orbifolds $V_{k}$ have non-such that, for each
$k\neq k^{\prime}\in\\{1,\ldots,28\\}$ it holds that

${}^{\mathcal{L}_{k}}K_{\mathrm{grp}}(V_{k})\not\cong{}^{\mathcal{L}_{k^{\prime}}}K_{\mathrm{grp}}(V_{k^{\prime}}).$

The string orientation of $\mathrm{tmf}$ is the universal orientation for
$\mathrm{tmf}$-cohomology given by a homomorphism

$\sigma:M\mathrm{String}\rightarrow\mathrm{tmf}$

from the String structure-Thom spectrum to the $\mathrm{tmf}$-spectrum, [45].

On the other hand, the String is a $K(\mathbb{Z},2)$-fibration over Spin. This
induces a map

$\pi:K(\mathbb{Z},3)\to M\mathrm{String}.$

The composition $\sigma\circ\pi$ yeilds a map
$K(\mathbb{Z},3)\rightarrow\mathrm{tmf}$. Let $\mathbb{S}(K(\mathbb{Z},3))$ be
the $\infty$-group $\infty$-ring of $K(\mathbb{Z},3)$, [4]. Following [3,
Section 8], there is a canonical homomorphism of $E$-$\infty$-rings

$\mathbb{S}(K(\mathbb{Z},3))\rightarrow\mathrm{tmf}$

in other words, the following holds.

######  Theorem 5.8.

Fix an orbifold $X$. Then, every gerbe $\mathcal{L}$ over $X$ gives rise to a
class in the generalized $\mathrm{tmf}$-cohomology of $X$.

We conjecture the following.

######  Conjecture 5.9.

The induced classes in $\mathrm{tmf}$-generalized cohomology by gerbes over
orbifolds $X$ are birational invariants.

In the next section, we will explain how one can combine our techniques with
that in [38, 39] to obtain birational invariants for _divisorial orbifolds_
(Definition 5.10).

###  5.4. Birational invariants for divisorial orbifolds

Here, we explain our ideas for studying the non-rationality of divisorial
orbifolds. Examples of divisorial orbifolds are Fano hypersurface
singularities (Example 2.10), [12, 32, 10].

######  Definition 5.10 (Definition 5.1 in [39]).

Let $X$ be an orbifold (seen as a Deligne-Mumford stack) obtained as the
quotient by an algebraic linear group acting generically free and regularly on
some projective scheme $Y$. Let $\mathcal{X}$ stand to its DM stack. We say
that $\mathcal{X}$ is _divisorial_ if it admits a representable morphism to a
stack of the form ${B{\mathbb{G}_{m}}^{r}}$ for some positive integer $r$.

We give a more geometric account of Definition 5.10:

######  Lemma 5.11 (Equivalent definition for Divisorial orbifolds, [39]).

An orbifold $X$ as in Definition 5.10 is divisorial if, and only if, there is
a finite collection of line bundles $\\{L_{j}\\}_{j=1,\ldots,r}$ over $X$
inducing faithful representations of the isotropy groups of $X$.

######  Example 5.12.

The orbifolds appearing as orbit spaces for $\mathbb{C}^{*}$-actions on
hypersurface singularities in Example 2.10 are divisorial orbifolds.

Theorem 4.1 in [37] provides the necessary data for defining a birational
invariant for divisorial Deligne-Mumford stacks. Notably, these data also
appear in Chen-Ruan cohomology. For instance, see Section 4.2 for comparison.

######  Proposition 5.13 (Theorem 4.1 in [37]).

Let $X$ be a divisorial orbifold. The following data furnishes a birational
invariant for $X$.

  1. (1)

A stratification333Check [2] for more information regarding stratification and
tubular neighborhood on orbit spaces. of $X=\sqcup X_{\sim}$ regarding
stabilizer (or isotropy) types.

  2. (2)

Tubular neighborhoods of the $X_{\sim}$.

With the former information, one can work to expand Theorem A to obtain a re-
reading of Theorem 8.1 in [39]. We can use Proposition 3.6 to conclude the
following.

######  Theorem 5.14.

Let $X,\leavevmode\nobreak\ X^{\prime}$ be two projective divisorial orbifolds
of the same dimension $d\geq 2$. If the Chen-Ruan cohomology rings of
$X,\leavevmode\nobreak\ X^{\prime}$ are not isomorphic, then
$X,\leavevmode\nobreak\ X^{\prime}$ are not birational.

Theorem 5.14 furnishes a strong criterium to decide the non-rationality of the
Fano orbifolds discussed in [12, 10, 32]. The proof and details on what is
next shall appear in [14]. For instance, the combination of the Chen-Ruan
computations for the orbi-Fanos $V_{k}$ appearing in [12] with Theorem 5.14
imply the following:

######  Theorem 5.15.

For each $k\neq k^{\prime}\in\\{1,\ldots,28\\}$ the orbi-Fanos
$V_{k},\leavevmode\nobreak\ V_{k^{\prime}}$ obtained as
$f_{k}^{-1}(0)\cap\mathrm{S}^{9}/\mathrm{S}^{1}$ for
$f_{k}=u^{6k-1}+v^{3}+z_{0}^{2}+z_{1}^{2}+z_{2}^{2}$ are non birational.

######  Remark 5.16.

Observe that in Theorem 5.15 we are not talking about the hypersurface
$f_{k}^{-1}(0)\subset\mathbb{C}^{5}$. Instead, we are dealing with the orbi-
Fanos $V_{k}$ that are base for the $\mathrm{S}^{1}$-fibration

$\mathrm{S}^{1}\hookrightarrow f^{-1}(0)\cap\mathrm{S}^{9}\rightarrow V_{k}.$

Therefore, we are referring to the rationality of $V_{k}$ as orbi-Fano.
However, we believe that understanding the $\mathrm{tmf}$-classes obtained
from gerbes over each $V_{k}$ (Theorem D) could lead to stable $G$-rationality
obstructions for $f_{k}^{-1}(0)$ with $\mathbb{C}^{*}$-actions (check the
Example 2.10). ∎

Observe that any weighted projective space is rational. The reason for that is
that if $\mathbb{P}(w_{1},\ldots,w_{m})$ is a weighted projective space with
weights $(w_{1},\ldots,w_{m})$ (recall Example 2.10) then it is rational. It
contains a copy of $(\mathbb{C}^{*})^{m}$. Indeed, denote the elements of
$(\mathbb{C}^{*})^{m}$ by $(q_{1},\ldots,q_{m})$. We can define a
$\mathrm{T}:=(\mathbb{C}^{*})^{m}/\\{(q,\ldots,q)\\}$-action in
$\mathbb{P}(w_{0},\ldots,w_{m})$ via

$(q_{1},\ldots,q_{m})\cdot[z_{1}:\ldots:z_{m}]:=[q^{w_{1}}z_{1}:\ldots:q^{w_{m}}z_{m}].$

The orbit through $[1:\ldots:1]$ is precisely
$\mathrm{T}/\mathbb{Z}_{w_{1}}\times\ldots\times\mathbb{Z}_{w_{m}}$, which is
isomorphic to $(\mathbb{C}^{*})^{m}$. Since this orbit is dense, one has that
both $\mathbb{P}(w_{1},\ldots,w_{m})$ and $\mathbb{P}^{m}$ are birational. If
$m=1$, as one learns that the birational group of weighted projective spaces
is the same as its group of automorphisms (recall Example A.11), one gets the
following:

######  Proposition 5.17.

Two weighted projective lines
$\mathbb{P}^{1}(w_{1},w_{2}),\leavevmode\nobreak\
\mathbb{P}^{1}(w^{\prime}_{1},w^{\prime}_{2})$ are birational if, and only if
_the normalizations_ of $(w_{1},w_{2})$ and $(w^{\prime}_{1},w^{\prime}_{2})$
coincide (up to order). That is, if
$\gcd(w_{1},w_{2})=\gcd(w^{\prime}_{1},w^{\prime}_{2})=1$ then
$(w_{1}^{\prime},w_{2}^{\prime})$ is a permutation of $(w_{1},w_{2})$.

###### Proof.

Since any birational equivalence between $\mathbb{P}^{1}(w_{1},w_{2})$ and
$\mathbb{P}^{1}(w_{1}^{\prime},w_{2}^{\prime})$ is a biregular equivalence
(isomorphism as algebraic varieties), one applies Theorem 1 in [6].∎

Proposition 5.17 combined with Theorem 5.14 leads to the following:

######  Proposition 5.18.

Two weighted projective lines
$\mathbb{P}^{1}(w_{1},w_{2}),\leavevmode\nobreak\
\mathbb{P}^{1}(w^{\prime}_{1},w^{\prime}_{2})$ are isomorphic only if their
Chen-Ruan cohomology ring is isomorphic.

######  Remark 5.19.

According to Theorem 2.1 in [6], Proposition 5.18 is non-longer true if one
changes the Chen-Ruan cohomology to cohomology. ∎

As we learn from [1, Example 5.6, p.112], the twisted sectors on weighted
projective spaces are smaller dimension weighted projective spaces. We wonder
if we can adapt Theorem 5.14 to prove the following.

######  Conjecture 5.20.

Let $\mathbb{P}(w_{1},\ldots,w_{m})$,
$\mathbb{P}(w^{\prime}_{1},\ldots,w^{\prime}_{m})$ be two weighted projective
spaces with $\gcd(w_{i},w_{j})=\gcd(w^{\prime}_{i},w^{\prime}_{j})=1$ for any
$i,j\in\\{1,\ldots,m\\}$ with $w_{i}\neq w_{j}$, $w^{\prime}_{i}\neq
w^{\prime}_{j}$. Then $\mathbb{P}(w_{1},\ldots,w_{m})$ is birational to
$\mathbb{P}(w^{\prime}_{1},\ldots,w^{\prime}_{m})$ if and only if they share
birational twisted sectors.

Assuming that Conjecture 5.20 holds, we can obtain the following application.
A log del Pezzo surface is a projective surface with quotient singularities
with ample anticanonical class. Theorem 8 in [31] provides the existence of
anticanonically embedded quasi smooth log del Pezzo surface $X_{d}$ of degree
$d$ in $\mathbb{P}(a_{0},a_{1},a_{2},a_{3})$ for $d,\leavevmode\nobreak\
\\{a_{i}\\}$ either coming in series

(5.4) $d=8k+4\quad\&\quad(a_{0},a_{1},a_{2},a_{3})=(2,2k+1,2k+1,4k+1)$

or sporadically. With the here-developed technology, we have:

######  Theorem 5.21.

Let $X_{d}$ be an anticanonically embedded log del Pezzo surface of degree $d$
in

$X_{d}\subset\mathbb{P}(a_{0},a_{1},a_{2},a_{3})$

for $d,\leavevmode\nobreak\ \\{a_{i}\\}$ as in Equation (5.4), or
sporadically, as in Theorem 8 in [31], except for

$(a_{0},a_{1},a_{2},a_{3})=(1,1,1,1,),\leavevmode\nobreak\
(1,1,1,2),\leavevmode\nobreak\ (1,1,2,3).$

Then, $X_{d}$ is rational.

###### Proof.

Since we are pending on Conjecture 5.20, we only present a sketch. According
to [31, p. 71], except for the cases

$(a_{0},a_{1},a_{2},a_{3})=(1,1,1,1,),\leavevmode\nobreak\
(1,1,1,2),\leavevmode\nobreak\ (1,1,2,3)$

$X_{d}$ is not smooth and passes through some of the vertices
$(x_{i}=x_{j}=x_{k}=0)$ for $i,\leavevmode\nobreak\ j,\leavevmode\nobreak\
k\in\\{0,1,2,3\\}$. We claim that any two contain a smooth projective line and
are birational. The proof of this fact passes through assuming Conjecture
5.20. We can show that any two of $X_{d}$ are birational because they share
common intersections with the twisted sectors of
$\mathbb{P}(a_{0},a_{1},a_{2},a_{3})$. The common shared line $\mathbb{P}^{1}$
is the intersection of $X_{d}$ with one of these twisted sectors.

Thus, it suffices to prove that one $X_{d}$ is rational to conclude what is
desired. However, since any of contains $\mathbb{P}^{1}$, each $X_{d}$ is
uniruled. Because $X_{d}$ is a surface, one concludes that it is rationally
connected and hence rational. ∎

In [32, Theorem 2.2] is presented a family of anticanonically embedded Fano
threefolds in certain weighted projective spaces

$X_{2k(b_{1}+b_{2}+b_{3})}\subset\mathbb{P}(2,kb_{1},kb_{2},kb_{3},k(b_{1}+b_{2}+b_{3})-1,\leavevmode\nobreak\
k\equiv 1\pmod{2},$

where $b_{1},b_{2},b_{3}$ are given in Remark 2.3 in [32]. In [14], we aim to
explore the non-rationality of members of the above family and approach
Conjecture 5.20.

In the next section, we discuss in more detail how we can combine our here-
presented methods to the ones in [53, 54, 25] to work with stable
$G$-rationality. A complete account shall appear in [14].

###  5.5. Stable G-rationality

In [25], the authors provide a sufficient condition for the non-stable
rationality of certain conic bundles over rational surfaces. Let $Y$ be a
smooth projective variety. From [25, Proposition 7], if $Y$ is stably
rational, then $Y$ admits a _Chow decomposition of the diagonal_ [5].
Additionally, the group $\mathrm{CH}_{0}(Y)$ is a stable birational invariant
[54, Lemma 2.11].

Let $X$ be an orbifold. To the proof of the main result in [25] authors rely
upon the Brauer group

$\mathrm{Br}(X):=\mathrm{H}^{2}(X;\mathbb{C}^{*})_{\mathrm{tors}}$

For smooth projective varieties $Y$, the Brauer group is a stable birational
invariant, [54, Lemma 2.9]. If $Y$ _rationally connected_ over $\mathbb{C}$,
then $\mathrm{Br}(Y)\cong\mathrm{H}^{3}(Y;\mathbb{Z})$. Thus, the Brauer group
is also a birational invariant. Considering Example 4.9, we pose the following
questions and conjecture.

######  Question 2.

Let $X=Y/G$ be a global quotient orbifold. Is the quantity

(5.5) $\mu_{\mathrm{orb},\alpha}(\mathrm{L}(X))=\sum_{(g)\in
T_{1}}\left[\left(Y^{g}/C^{\alpha}(g)\right)/X\right]\mathbb{L}^{\iota_{(g^{-1})}}$

  1. (1)

a motivic measure

  2. (2)

a _stable_ $G$-birational invariant

where $\mathbb{L}=[\mathbb{A}^{1}]$ is the _Tate motive_ and $\mathrm{L}(X)$
is the space of arcs in $X$ whose $\mathbb{C}$-points correspond to formal
arcs $\mathrm{Spec}\mathbb{C}[[z]]\rightarrow X$? In other words, let a finite
group $G$ act generically free on smooth projective irreducible varieties
$Y,\leavevmode\nobreak\ Y^{\prime}$ of dimension $d\geq 2$. Let
$X,\leavevmode\nobreak\ X^{\prime}$ be the resulting global quotient orbifolds
(obtained as orbit spaces). If _for some_ $N\geq 1$ there exist
$\alpha\in\mathrm{H}^{2}(\pi_{1}^{\text{orb}}(X\times\mathbb{P}^{N});\mathbb{C}^{*}),\leavevmode\nobreak\
\alpha^{\prime}\in\mathrm{H}^{2}(\pi_{1}^{\text{orb}}(X^{\prime}\times\mathbb{P}^{N});\mathbb{C}^{*})$
such that

(5.6)
$\mu_{\mathrm{orb},{\alpha}}(\mathrm{L}(X\times\mathbb{P}^{N}))\neq\mu_{\mathrm{orb},{\alpha^{\prime}}}(\mathrm{L}(X^{\prime}\times\mathbb{P}^{N}))$

then

$Y\times\mathbb{P}^{N}\not\sim_{G}Y^{\prime}\times\mathbb{P}^{N}?$

######  Question 3.

Can one classify all the possible $G$-actions on $Y\times\mathbb{P}^{N}$ that
are trivial on the second factor and whose quotient is isomorphic as orbifold
to $X\times\mathbb{P}^{N}$?

######  Question 4.

To the former classification, can one relate the gerbes over
$X\times\mathbb{P}^{N}$ with that on $X$?

Taking into account Theorem 4.16, let us consider the Picard group
$\mathrm{Pic}_{\mathrm{torsion}}(\mathcal{G}_{1})$ (Definition 4.14), we ask:

######  Question 5.

Is $\mathrm{H}^{1}(G;\mathrm{Pic}_{\mathrm{torsion}}(\mathcal{G}_{1}))$ a
stable $G$-birational invariant?

Summarizing, we conjecture the following.

######  Conjecture 5.22.

Let $X=Y/G$ be a global quotient orbifold. Then

  1. (1)

The images
$\mathrm{H}^{2}(G;\mathbb{C}^{*})\rightarrow\mathrm{H}^{2}(\pi_{1}^{\text{orb}}(X);\mathbb{C}^{*})$
is a stable birational invariant for $X$ just as the Brauer group
$\mathrm{Br}(Y)$ is a stable birational invariant of $Y$.

  2. (2)

If $Y$ is rationally connected, then the $G$-rational types of $Y$ are
classified by the gerbes with connections over $X$, thus, can be ready from
$\mathrm{H}^{3}(BX;\mathbb{Z})$. For the $G$-stable birational types of $Y$,
these can be classified by the images
$\mathrm{H}^{2}(G;\mathbb{C}^{*})\rightarrow\mathrm{H}^{2}(\pi_{1}^{\text{orb}}(X);\mathbb{C}^{*})$.

The former questions and conjectures will be treated in [13, 14]. When
completed, this program shall give a new read of the new approach in
birational geometry initiated in [53, 54].

## Acknowledgments

The authors are grateful to Yuri Tschinkel and Ivan Cheltsov for critical
reading of the first versions of this paper. This forced the authors to
clarify many points, contributing immensely to the exposition. The authors are
also thankful to Maxim Kontsevich for discussing several points related to the
paper, which helped us to clarify many aspects of our writing and proposed
connections.

The São Paulo Research Foundation FAPESP supports L. F. C grants 2022/09603-9
and 2023/14316-1. L. G’s research is partially supported by São Paulo Research
Foundation FAPESP grants 2018/13481-0 and 2023/13131-8. L. K. is supported by
The Basic Research Program of the National Research University Higher School
of Economics, the Simons Investigators Award of the Simons Collaboration on
Homological Mirror Symmetry, and partially supported by the National Science
Fund of Bulgaria, National Scientific Program "Excellent Research and People
for the Development of European Science" (VIHREN), Project No. KP-06-DV-7.

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