# Constant Scalar Curvature Sasaki Metrics and Projective Bundles

Charles P. Boyer  Charles P. Boyer, Department of Mathematics and Statistics,
University of New Mexico, Albuquerque, NM 87131<EMAIL_ADDRESS>and
Christina W. Tønnesen-Friedman  Christina W. Tønnesen-Friedman, Department of
Mathematics, Union College, Schenectady, New York 12308, USA
<EMAIL_ADDRESS>
###### Abstract.

In this paper we consider the Boothby-Wang construction over twist 1 stage 3
Bott orbifolds given in terms of the log pair $(S_{\bf n},\Delta_{\bf m})$. We
give explicit constant scalar curvature (CSC) Sasaki metrics either directly
from CSC Kähler orbifold metrics or by using the weighted extremal approach of
Apostolov and Calderbank. The Sasaki 7-manifolds (orbifolds) are finitely
covered by compact simply connected manifolds (orbifolds) with the rational
homology of the 2-fold connected sum of $S^{2}\times S^{5}$.

###### Key words and phrases:

Admissible construction, extremal Sasaki metrics, CSC Sasaki metrics

###### 2000 Mathematics Subject Classification:

Primary: 53C25; Secondary: 53C55

The authors were partially supported by grants from the Simons Foundation, CPB
by (#519432), and CWT-F by (#422410)

###### Contents

  1. 1 The Projective Bundles
  2. 2 The Orbifold Boothby-Wang Construction
  3. 3 The Topology of the Orbifolds
    1. 3.1 The Orbifold Cohomology Groups
    2. 3.2 The Cohomology of the Branched Covers
    3. 3.3 The Regular Case.
    4. 3.4 The General Case; Branched Covers
    5. 3.5 When $M^{7}$ is an $S^{3}_{\bf w}$-join
  4. 4 Admissible Projective Bundles
    1. 4.1 Orbifolds
    2. 4.2 Connection with $S^{3}_{\bf w}$-joins
    3. 4.3 Connection with Yamazaki’s Fiber joins
    4. 4.4 Ricci Solitons and Kähler-Einstein
      1. 4.4.1 Kähler-Einstein
    5. 4.5 Extremal and CSC Kähler metrics
    6. 4.6 Weighted Extremal Metrics

## Introduction

Arguably the foremost problem in Sasaki geometry is that of determining which
isotopy classes of Sasakian structures admit Sasaki metrics of constant scalar
curvature (CSCS). The Sasaki version of the Yau-Tian-Donaldson conjecture says
roughly that CSCS metrics correspond to the affine variety satisfying some
type of K-stability requirement. Collins and Székelyhidi [CS18] proved that a
CSC Sasaki metric implies the K-semistablility of corresponding affine Kähler
variety. Recently, following the important work in the Kähler category by Chen
and Cheng [CC21a, CC21b], He and Li [He18, HL21] have related the existence of
CSC Sasaki metrics to the reduced properness of the Mabuchi K-energy.

Complementary to these existence results, it is important to explore explicit
examples. Of particular interest to us are those Sasaki manifolds (orbifolds)
that admit a transverse admissible construction [ACGTF08a]. A special case of
such Sasaki orbifolds consist of principle $S^{1}$ orbibundles over Bott
manifolds (orbifolds) [GK94, BCTF19] to be described in the first Section. In
this paper we discuss two fairly explicit methods for constructing constant
scalar curvature Sasaki metrics on a class of 7-manifolds having the rational
cohomology of the 2-fold connected sum of $S^{2}\times S^{5}$. Specifically,
these $7$-manifolds, $M^{7}_{{\bf n},{\bf m}}$, arise by a Boothby-Wang
construction over a polarized admissible orbifold, $S_{{\bf n},{\bf m}}$,
given as a log pair $(S_{\bf n},\Delta_{\bf m})$, where $S_{\bf
n}={\mathbb{P}}\bigl{(}\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(n_{1},n_{2})\bigr{)}\longrightarrow{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$
and $\Delta_{\bf
m}=\Bigl{(}1-\frac{1}{m_{0}}\Bigr{)}e_{0}+\Bigl{(}1-\frac{1}{m_{\infty}}\Bigr{)}e_{\infty}$,
with $e_{0}$ and $e_{\infty}$ are the infinity sections of the bundle $S_{\bf
n}$. The first method is the standard Boothby-Wang construction on twist one
stage 3 Bott orbifolds (KS orbifolds). The second method applies the weighted
extremal approach of Apostolov and Calderbank [AC21]. Both methods employ the
admissible construction at the Kähler level (see [ACGTF08a]) either directly
or indirectly. Specifically, in Section 4, with the main work horses being
Proposition 4.16 and Proposition 4.14 (the latter with limitations as
mentioned in Remark 4.15), we prove Theorem 4.17 which together with Remark
4.18 give

Main Theorem. Let $M^{7}_{{\bf n},{\bf m}}$ be a Boothby-Wang constructed
manifold with maximal symmetry over a polarized KS orbifold. Then there always
exists a positive constant scalar curvature Sasaki metric in the Sasaki cone.

In terms of K-stability we have the following corollary of the Main Theorem
and Corollary 1.1 of [CS18]

Corollary. Let $M^{7}_{{\bf n},{\bf m}}$ be a Boothby-Wang constructed Sasaki
manifold over the KS orbifold $(S_{\bf n},\Delta_{\bf m})$. Then there exists
a Reeb vector field $\xi$ in the Sasaki cone such that the corresponding
polarized affine variety $(Y,\xi)$ is K-semistable, equivalently the Sasakian
structure $(\xi,\eta,\Phi,g)$ is K-semistable.

In the Gorenstein case, a much stronger statement is known, namely the Sasaki
version of the famous Chen-Donaldson-Sun [CDS15a, CDS15b, CDS15c] result by
Collins and Székelyhidi [CS19]. In particular, the recent classification of
all smooth Fano threefolds that admit Kähler-Einstein metrics [ACC+21] gives a
classification of all regular Sasaki-Einstein manifolds over smooth KS
threefolds.

It is interesting to compare the Sasaki-Einstein solutions obtained from the
Boothby-Wang construction from the orbifold Kähler-Einstein solutions of
Section 4.4 with those described in [BTF19]. Both types live on simply
connected 7-manifolds with the rational cohomology of the 2-fold connected sum
of $S^{2}\times S^{5}$ with torsion in $H^{4}$. However, they are of a
somewhat different nature. Those in [BTF19] have holomorphic twist 2, whereas,
those in this paper have holomorphic twist 1. Assuming they have the same
torsion group one may ask whether they are homotopy equivalent, have the same
diffeomorphism type, and/or belong to inequivalent CR structures, contact
structures, etc.

######  Acknowledgements.

The authors of this paper have benefitted from conversations with Vestislav
Apostolov, David Calderbank, Hongnian Huang, Eveline Legendre, and Carlos
Prieto.

##  1\. The Projective Bundles

We view our projective bundle as a special type of stage 3 Bott manifold,
namely the Bott tower $M_{3}(0,n_{1},n_{2})$ where
$n_{1},n_{2}\in{\mathbb{Z}}$. We recall the definition of a _stage $k$ Bott
manifold_ of the _Bott tower of height $n$_:

$M_{n}\xrightarrow{\pi_{n}}M_{n-1}\xrightarrow{\pi_{n-1}}\cdots\rightarrow
M_{2}\xrightarrow{\pi_{2}}M_{1}={\mathbb{C}}{\mathbb{P}}^{1}\xrightarrow{\pi_{1}}\\{pt\\}$

where $M_{k}$ is defined inductively as the total space of the projective
bundle

${\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\oplus
L_{k})\xrightarrow{\pi_{k}}M_{k-1}$

with fiber ${\mathbb{C}}{\mathbb{P}}^{1}$, and some holomorphic line bundle
$L_{k}$ on $M_{k-1}$.We refer to [GK94, BCTF19] for details. In this paper we
treat twist 1 stage 3 Bott manifolds $M_{3}(0,n_{1},n_{2})$ and orbifolds.
Explicitly these are ruled manifolds of the form

(1) $S_{\bf
n}={\mathbb{P}}\bigl{(}\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(n_{1},n_{2})\bigr{)}\longrightarrow{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$

with $n_{1},n_{2}\in{\mathbb{Z}}^{*}$. To this we add an orbifold structure
given in terms of the log pair $(S_{\bf n},\Delta_{\bf m})$ where

(2) $\Delta_{\bf
m}=\Bigl{(}1-\frac{1}{m_{0}}\Bigr{)}e_{0}+\Bigl{(}1-\frac{1}{m_{\infty}}\Bigr{)}e_{\infty}$

and $e_{0},e_{\infty}$ are defined by the zero and infinity sections of the
bundle. A motivation of our study comes from the 1986 paper of Koiso and
Sakane [KS86] where they study Kähler-Einstein metrics on projectivizations of
the form ${\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\oplus
L_{1}\otimes_{ext}L_{2})$ over $N\times N$ where $N$ has a Kähler-Einstein
metric. For this reason we refer to our log pairs $(S_{\bf n},\Delta_{\bf m})$
as KS orbifolds. The underlying complex manifolds are smooth toric varieties
whose Kähler geometry was studied in [BCTF19]. The positive integers
$m_{0},m_{\infty}$ are called ramification indices. For each choice of
orbifold Kähler form $\omega_{{\bf n},{\bf m}}$, defining an integer Kähler
class on $(S_{\bf n},\Delta_{\bf m})$ we have a principal $S^{1}$ orbibundle

$S^{1}\longrightarrow M^{7}\longrightarrow(S_{\bf n},\Delta_{\bf m})$

with an induced Sasakian structure ${\mathcal{S}}_{{\bf n},{\bf m}}$;
moreover, when the class $[\omega_{{\bf n},{\bf m}}]$ is primitive in
$H^{2}_{orb}((S_{\bf n},\Delta_{\bf m}),{\mathbb{Z}})$, $M^{7}$ is simply
connected. While there are only 2 equivalences classes of Fano KS manifolds,
it is clear from Proposition 1.1 below that log Fano KS orbifolds are
abundant.

In [BTF] we described a functor from the category of $S^{3}_{\bf w}$ Sasaki
joins to the category of Bott orbifolds. However, as noted there, this functor
is not surjective on objects. One purpose of this paper is to study this lack
of surjectivity in the particular case of twist 1 stage 3 Bott orbifolds. A
key notion of Sasaki joins is that of cone decomposability which can be
thought of as the Sasaki version of de Rham decomposability in the Kähler case
[BHLTF18]. Cone decomposability is an invariant of the underlying Sasaki CR
structure. We shall see how our KS orbifolds give rise to both cone
decomposable Sasakian structures as well as Sasakian structures that are
likely cone indecomposable. See Remark 3.10 below.

Note that the ordinary Bott manifold $M_{3}(0,n_{1},n_{2})$ corresponds to the
trivial orbifold $(S_{\bf n},\emptyset)$. We denote by $KS^{orb}$ the set of
manifolds with these orbifold structures, that is,

$KS^{orb}=\\{(S_{\bf n},\Delta_{\bf
m})~{}|~{}n_{1},n_{2}\in{\mathbb{Z}}^{*},~{}m_{0},m_{\infty}\in{\mathbb{Z}}^{+}\\}.$

Such orbifolds are the objects of a groupoid whose morphisms are
biholomorphisms that intertwine the orbifold structures. Actually we are
intererested in polarized KS orbifolds that are polarized by a (primitive)
orbifold Kähler class $[\omega_{{\bf n},{\bf m}}]\in H^{2}_{orb}(S_{\bf
n},{\mathbb{Z}})$. These form the objects of our groupoid
${\mathcal{K}}{\mathcal{S}}$ whose morphisms are biholomorphisms that
intertwine the orbifold structures and intertwine their orbifold Kähler
classes. We remark that since the objects of ${\mathcal{K}}{\mathcal{S}}$ are
themselves orbifolds, we are working with a 2-category. The action of the
Coexter group $\mathrm{Sym}_{2}\ltimes{\mathbb{Z}}_{2}^{3}$ on $KS_{orb}$ is
generated by the transposition $(S_{\bf n},\Delta_{\bf
m})\mapsto(S_{n_{2},n_{1}},\Delta_{\bf m})$ and the fiber inversion sending
$(S_{\bf n},\Delta_{\bf m})\mapsto(S_{-{\bf n}},\Delta_{m_{\infty},m_{0}})$.
Note the interchange of ramification indices $m_{0},m_{\infty}$ induced by the
fiber inversion map.

Of special interest is the full subgroupoid ${\mathcal{K}}{\mathcal{S}}^{mon}$
of monotone KS orbifolds which we now describe. Consider the orbifold
canonical divisor or dually the orbifold first Chern class. In particular,
when is the orbifold $(S_{\bf n},\Delta_{\bf m})$ log Fano, equivalently when
does $c_{1}^{orb}(S_{\bf n},\Delta_{\bf m})$ lie in the Kähler cone? From
Section 1.3 of [BTF19] we have

(3) $\displaystyle c_{1}^{orb}(S_{\bf n},\Delta_{\bf m})$ $\displaystyle=$
$\displaystyle\Bigl{(}2-\frac{n_{1}}{m_{\infty}}\Bigr{)}y_{1}+\Bigl{(}2-\frac{n_{2}}{m_{\infty}}\Bigr{)}y_{2}+\Bigl{(}\frac{1}{m_{0}}+\frac{1}{m_{\infty}}\Bigr{)}y_{3}$
$\displaystyle=$
$\displaystyle\Bigl{(}2+\frac{n_{1}}{m_{0}}\Bigr{)}x_{1}+\Bigl{(}2+\frac{n_{2}}{m_{0}}\Bigr{)}x_{2}+\Bigl{(}\frac{1}{m_{0}}+\frac{1}{m_{\infty}}\Bigr{)}x_{3}.$

with respect to the invariant bases $\\{x_{1},x_{2},x_{3}\\}$ and
$\\{y_{1},y_{2},y_{3}\\}$, respectively. For $i=1,2$ here $x_{i}=y_{i}$ is the
class of the Fubini-Study metric on the ith factor of the product
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$ pulled back
to $S_{\bf n}$, while $x_{3}(y_{3})$ is the Poincaré dual of
$e_{\infty}(e_{0})$, respectively. From this we arrive at a special case of
Lemma 1.2 of [BTF19]

######  Proposition 1.1.

The orbifold $(S_{\bf n},\Delta_{\bf m})$ in $KS^{orb}$ is log Fano if and
only if the inequalities

$\frac{n_{1}}{m_{\infty}}<2,\quad\frac{n_{2}}{m_{\infty}}<2,\quad-\frac{n_{1}}{m_{0}}<2,\quad-\frac{n_{2}}{m_{0}}<2$

hold.

We obtain a primitive Kähler class $\frac{c_{1}^{orb}(S_{\bf n},\Delta_{\bf
m})}{{\mathcal{I}}_{{\bf n},{\bf m}}}$ in $H^{2}_{orb}((S_{\bf n},\Delta_{\bf
m}),{\mathbb{Z}})$ with Fano index given by

(4) ${\mathcal{I}}_{{\bf n},{\bf
m}}=\gcd(2mv_{0}v_{\infty}-n_{1}v_{0},2mv_{0}v_{\infty}-n_{2}v_{0},v_{0}+v_{\infty})$

where ${\bf m}=(m_{0},m_{\infty})=m(v_{0},v_{\infty})=m{\bf v}$ and
$m=\gcd(m_{0},m_{\infty})$. Here we have used the identity
$x_{3}=y_{3}-n_{1}x_{1}-n_{2}x_{2}$. The orbifold $(S_{\bf n},\Delta_{\bf m})$
is log Fano if and only if

$c_{1}^{orb}(S_{\bf n},\Delta_{\bf
m})=\frac{1}{mv_{0}v_{\infty}}\Bigl{[}\Bigl{(}2mv_{0}v_{\infty}-n_{1}v_{0}\Bigr{)}y_{1}+\Bigl{(}2mv_{0}v_{\infty}-n_{2}v_{0}\Bigr{)}y_{2}+\Bigl{(}v_{0}+v_{\infty}\Bigr{)}y_{3}\Bigr{]}$

is positive.

The Poincaré dual to $c_{1}^{orb}(S_{\bf n},\Delta_{\bf m})$ is the orbifold
anti-canonical divisor which is an ample ${\mathbb{Q}}$-divisor on $S_{\bf n}$
when $c_{1}^{orb}(S_{\bf n},\Delta_{\bf m})$ is positive. The primitive class
$\frac{c_{1}^{orb}(S_{\bf n},\Delta_{\bf m})}{{\mathcal{I}}_{{\bf n},{\bf
m}}}$ can be represented by an orbifold Kähler form $\omega_{{\bf n},{\bf m}}$
on $S_{\bf n}$, or equivalently an orbifold Kähler metric $g_{{\bf n},{\bf
m}}$.

##  2\. The Orbifold Boothby-Wang Construction

The well known Boothby-Wang Theorem [BW58] that associates regular contact
structures to the total space of $S^{1}$ bundles over symplectic manifolds
generalizes easily to the orbifold category. See for example Theorem 7.1.3 of
[BG08]. Here as in this theorem we are interested in Sasakian structures over
Kählerian structures. Given an integral Kähler class $[\omega_{{\bf n},{\bf
m}}]\in H^{1,1}_{orb}((S_{\bf n},\Delta_{\bf m}),{\mathbb{Z}})$ we shall
always choose a Kähler form $\omega_{{\bf n},{\bf m}}$ with maximal symmetry.
In this case the total space $M$ of the corresponding principal $S^{1}$
orbibundle has a maximal family ${\mathfrak{t}}^{+}$ of Sasakian structures,
called the Sasaki cone. Moreover, choosing a connection 1-form $\eta$ on $M$
such that $d\eta=\omega_{{\bf n},{\bf m}}$ determines a natural Sasakian
structure ${\mathcal{S}}_{{\bf n},{\bf m}}=(\xi,\eta,\Phi,g)$ in
${\mathfrak{t}}^{+}$. We refer to this construction as the orbifold Boothby-
Wang construction. Generally, unlike $(S_{\bf n},\Delta_{\bf m})$, the total
space $M$ has an orbifold structure whose underlying topological space is
singular with finite cyclic singularities. The general problem of finding
conditions when $M$ is smooth is quite subtle [Qui71, Kol07, KL21]: however,
Kegel and Lange show that $M$ is a smooth manifold if and only if
$H^{r}_{orb}(M,{\mathbb{Z}})=0$ for all $r>2n+1$. More precisely $M$ is smooth
if and only if multiplication by the Euler class
$H^{r}_{orb}({\mathcal{Z}},{\mathbb{Z}})\raise 4.0pt\hbox{$\cup_{e}\atop\hbox
to36.0pt{\rightarrowfill}$}H^{r+2}_{orb}({\mathcal{Z}},{\mathbb{Z}})$ in the
Gysin sequence of the orbibundle $M\hbox
to24.0pt{\rightarrowfill}{\mathcal{Z}}$ is an isomorphism for all $r>2n+1$. As
we shall see in Section 3.5 precise conditions can be obtained in certain
special cases (see also [BG08]).

The question arises as to when there exists a constant scalar curvature Sasaki
metric in ${\mathfrak{t}}^{+}$ and how many. In the Gorenstein case this was
answered by Futaki, Ono, and Wang [FOW09], but the general CSC case remains
open. Some partial results have been obtained by Legendre [Leg11], and by the
authors [BTF16, BTF21a] and references therein.

######  Remark 2.1.

In categorical language we work with the groupoid
${\mathcal{S}}{\mathcal{K}}{\mathcal{S}}$ whose objects $\\{M^{7}_{{\bf
n},{\bf m}}\\}$ are simply connected quasiregular Sasaki 7-orbifolds whose
projective algebraic base is a Koiso-Sakane orbifold, and whose morphisms are
orbifold biholomorphisms. Here the objects are classes of Sasakian or Koiso-
Sakane structures, thus again, they are 2-categories. We are mainly interested
in certain subgroupoids, namely, the full subgroupoid
${\mathcal{S}}{\mathcal{K}}{\mathcal{S}}^{F}$ of positive Sasakian structures
whose base Koiso-Sakane orbifold is log Fano. Actually, we consider the full
subgroupoid ${\mathcal{S}}{\mathcal{K}}{\mathcal{S}}^{MF}$ whose Kähler form
$\omega_{{\bf n},{\bf m}}$ lies in $\frac{c_{1}^{orb}(S_{\bf n},\Delta_{\bf
m})}{{\mathcal{I}}_{{\bf n},{\bf m}}}$, that is the Koiso-Sakane base orbifold
is $(S_{\bf n},\Delta_{\bf m})$ is monotone log Fano. We also define the
subgroupoids ${\mathcal{S}}{\mathcal{K}}{\mathcal{S}}_{\bf
m},{\mathcal{S}}{\mathcal{K}}{\mathcal{S}}^{F}_{\bf
m},{\mathcal{S}}{\mathcal{K}}{\mathcal{S}}^{MF}_{\bf m}$ consisting of the
corresponding objects with fixed ${\bf m}$. The objects of the groupoid
${\mathcal{S}}{\mathcal{K}}{\mathcal{S}}_{\bf m}$ are Sasaki classes on the
7-manifolds $M^{7}_{\bf m}$ whose cohomology we describe in the next section.
The morphisms are those induced by diffeomorphisms that intertwine the
corresponding Sasakian structures.

##  3\. The Topology of the Orbifolds

In this section we describe both the orbifold topology of $(S_{\bf
n},\Delta_{\bf m})$ and topology of the total space of the principal $S^{1}$
orbibundles over the log Fano Koiso-Sakane orbifolds $(S_{\bf n},\Delta_{\bf
m})$. First we recall the cohomology ring of the complex manifold $S_{\bf n}$
which is well known and can be found in [BCTF19] and references therein.

(5) $H^{*}(S_{\bf
n},{\mathbb{Z}})={\mathbb{Z}}[y_{1},y_{2},y_{3}]/\bigl{(}y_{1}^{2},y_{2}^{2},y_{3}(-n_{1}y_{1}-n_{2}y_{2}+y_{3})\bigr{)}$

We remark that the addition of orbifold structures on the invariant divisors
does not effect the cohomology $H^{*}(S_{\bf n},{\mathbb{Z}})$; however, as we
shall see it strongly effects the cohomology of the Sasakian 7-manifolds.

###  3.1. The Orbifold Cohomology Groups

Next we compute the orbifold cohomology groups.

######  Lemma 3.1.

$H^{r}_{orb}((S_{\bf n},\Delta_{\bf
m}),{\mathbb{Z}})=\begin{cases}{\mathbb{Z}}&~{}~{}\text{if $r=0$}\\\
{\mathbb{Z}}^{3}&~{}~{}\text{if $r=2$}\\\
{\mathbb{Z}}^{3}\oplus{\mathbb{Z}}_{m^{0}}^{2}\oplus{\mathbb{Z}}_{m^{\infty}}^{2}&~{}~{}\text{if
$r=4$}\\\
{\mathbb{Z}}\oplus{\mathbb{Z}}_{m^{0}}^{3}\oplus{\mathbb{Z}}_{m^{\infty}}^{3}&~{}~{}\text{if
$r=6$}\\\
{\mathbb{Z}}_{m^{0}}^{3}\oplus{\mathbb{Z}}_{m^{\infty}}^{3}&~{}~{}\text{if
$r=8,10,\cdots$}\\\ 0&~{}~{}\text{if $r$ is odd.}\end{cases}$

###### Proof.

Using Lemma 4.3.7 of [BG08] we compute the Leray spectral sequence of the
classifying map

$p:\mathsf{B}(S_{\bf n},\Delta_{\bf m})\hbox
to30.0pt{\rightarrowfill}~{}S_{\bf n}.$

The Leray sheaf of $p$ is the derived functor sheaf $R^{s}p_{*}{\mathbb{Z}}$,
that is, the sheaf associated to the presheaf $U\mapsto
H^{s}(p^{-1}(U),{\mathbb{Z}})$. For $s>0$ the stalks of
$R^{s}p_{*}{\mathbb{Z}}$ at points of $U$ vanish if $U$ lies in the regular
locus of $(S_{\bf n},\Delta_{\bf m})$ which is the complement of the union of
the zero $e_{0}$ and infinity $e_{\infty}$ sections of the natural projection
$S_{\bf n}\hbox
to19.20007pt{\rightarrowfill}{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$.
At points of $e_{0}$ and $e_{\infty}$ the fibers of $p$ are the Eilenberg-
MacLane spaces $K({\mathbb{Z}}_{m^{0}},1)$ and
$K({\mathbb{Z}}_{m^{\infty}},1)$, respectively. So at points of
$e_{0}(e_{\infty})$ the stalks are the group cohomology
$H^{s}({\mathbb{Z}}_{m^{0}},{\mathbb{Z}})\bigl{(}H^{s}({\mathbb{Z}}_{m^{\infty}},{\mathbb{Z}})\bigr{)}$.
This is ${\mathbb{Z}}$ for $s=0$ and
${\mathbb{Z}}_{m^{0}}({\mathbb{Z}}_{m^{\infty}})$ at points of
$e_{0}(e_{\infty})$ when $s>0$ is even; it vanishes when $s$ is odd. The
$E_{2}$ term of the Leray spectral sequence of the map $p$ is

$E_{2}^{r,s}=H^{r}(S_{\bf n},R^{s}p_{*}{\mathbb{Z}})$

and by Leray’s theorem this converges to the orbifold cohomology
$H^{r+s}_{orb}((S_{\bf n},\Delta_{\bf m}),{\mathbb{Z}})$. Now
$E_{2}^{r,0}=H^{r}(S_{\bf n},{\mathbb{Z}})$ and $E_{2}^{r,s}=0$ for $r$ or $s$
odd. For $r=0$ the only continuous section of $R^{s}p_{*}{\mathbb{Z}}$ is the
0 section which implies that $E_{2}^{0,s}=0$ for all $s$. Now we have
$E_{2}^{2r,2s}=0$ for $r>2$ and

$\displaystyle E_{2}^{2,2s}$ $\displaystyle=$ $\displaystyle H^{2}(S_{\bf
n},R^{2s}p)=H^{2}(e_{0},{\mathbb{Z}}_{m^{0}})\oplus
H^{2}(e_{\infty},{\mathbb{Z}}_{m^{\infty}})={\mathbb{Z}}_{m^{0}}^{2}\oplus{\mathbb{Z}}_{m^{\infty}}^{2},$
$\displaystyle E_{2}^{4,2s}$ $\displaystyle=$ $\displaystyle H^{4}(S_{\bf
n},R^{2s}p)=H^{4}(e_{0},{\mathbb{Z}}_{m^{0}})\oplus
H^{4}(e_{\infty},{\mathbb{Z}}_{m^{\infty}})={\mathbb{Z}}_{m^{0}}\oplus{\mathbb{Z}}_{m^{\infty}}.$

One easily sees that this spectral sequence collapses whose limit is the
orbifold cohomology $H^{r}_{orb}((S_{\bf n},\Delta_{\bf m}),{\mathbb{Z}})$
which implies the result. ∎

In addition Lemma 3.1 implies

######  Lemma 3.2.

$\pi_{1}^{orb}(S_{\bf n},\Delta_{\bf
m})=\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}.$

###### Proof.

From the homotopy sequence of the orbibundle

${\mathbb{C}}{\mathbb{P}}^{1}[{\bf v}]/{\mathbb{Z}}_{m}\longrightarrow(S_{\bf
n},\Delta_{\bf
m})\longrightarrow{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$

one easily sees that $\pi_{1}^{orb}(S_{\bf n},\Delta_{\bf m})$ is Abelian. But
this implies

$\pi_{1}^{orb}(S_{\bf n},\Delta_{\bf m})\approx H_{1}^{orb}(S_{\bf
n},\Delta_{\bf m}),{\mathbb{Z}}),$

and this vanishes by Lemma 3.1 and universal coefficients. ∎

###  3.2. The Cohomology of the Branched Covers

We consider the total space $M^{7}_{{\bf n},{\bf m}}$ of an $S^{1}$ orbibundle
over a KS orbifold $(S_{\bf n},\Delta_{\bf m})$ with an orbifold Kähler form
$\omega_{{\bf n},{\bf m}}$. Note that in our case $S_{\bf n}$ is a smooth
projective variety so $M^{7}_{{\bf n},{\bf m}}$ is a Seifert bundle as defined
in the foundational paper [OW75] of Orlik and Wagreich (see also the more
recent unpublished description by Kollár [Kol04]). A fundamental result of
Orlik and Wagreich says that every ${\mathbb{C}}^{*}$ Seifert bundle is a
branched cover over a principal ${\mathbb{C}}^{*}$ bundle. In our case this
branching occurs precisely along the orbifold divisor $\Delta_{\bf m}$. The
goal is to understand the integral cohomology of $M^{7}_{{\bf n},{\bf m}}$.
However, before treating the orbifold case, we need to consider the regular
case.

###  3.3. The Regular Case.

For simplicity we treat only the case where the Kähler class is primitive,
i.e.

(6) $[\omega_{\bf n}]=c_{1}y_{1}+c_{2}y_{2}+c_{3}y_{3}~{}\text{with
$c_{i}\in{\mathbb{Z}}^{+}$},\qquad\gcd(c_{1},c_{2},c_{3})=1.$

From Section 3.3 of [BCTF19] this is a Kähler class for all positive $c_{i}$
if $n_{1}n_{2}>0$, whereas, if $n_{1}n_{2}<0$, it is a Kähler class for
$c_{1}>0,~{}c_{2}>-n_{2}c_{3},~{}c_{3}>0$.

One sees from the long exact homotopy sequence of the fibration

(7) $S^{1}\longrightarrow M^{7}_{\bf n}\longrightarrow S_{\bf n}$

that

######  Lemma 3.3.

Let $M^{7}_{\bf n}$ denote the total space of the principal $S^{1}$ bundle
over a KS manifold $S_{\bf n}$ defined by a primitive Kähler class
$[\omega_{\bf n}]\in H^{2}(S_{\bf n},{\mathbb{Z}})$. Then $M^{7}_{\bf n}$ has
the rational cohomology of the 2-fold connected sum $2\\#(S^{2}\times S^{5})$,
and we have

  1. (1)

$\pi_{1}(M^{7}_{\bf n})=\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}$,

  2. (2)

$\pi_{2}(M^{7}_{\bf n})={\mathbb{Z}}^{2}$.

Thus, by the Hurewicz Theorem we also have $H_{2}(M^{7}_{\bf
n},{\mathbb{Z}})={\mathbb{Z}}^{2}$ which implies that $H^{3}(M^{7}_{\bf
n},{\mathbb{Z}})_{tor}=0$. One can also see from the Leray-Serre spectral
sequence of the fibration (7) that $M^{7}_{\bf n}$ has the rational cohomology
of the 2-fold connected sum $2\\#(S^{2}\times S^{5})$, and we also have
$H^{3}(M^{7}_{\bf n},{\mathbb{Z}})=0$. We now show that all such simply
connected spaces have nonzero torsion in $H^{4}$. There will be zero torsion
in $H^{4}$ if and only if the differential $d_{2}:E^{2,1}_{2}\longrightarrow
E^{4,0}_{2}$ is invertible over ${\mathbb{Z}}$. This differential is
represented by the matrix

(8) $C=\begin{pmatrix}c_{2}&c_{3}&0\\\ c_{1}&0&c_{3}\\\
0&c_{1}+n_{1}c_{3}&c_{2}+n_{2}c_{3}\end{pmatrix}$

with respect to the basis $\\{\alpha\otimes y_{1},\alpha\otimes
y_{2},\alpha\otimes y_{3}\\}$ of $E^{2,1}_{2}$ and the basis
$\\{y_{1}y_{2},y_{1}y_{3},y_{2}y_{3}\\}$ of $E^{4,0}_{2}$. But then we have

(9) $-\det C=c_{3}[c_{2}(c_{1}+n_{1}c_{3})+c_{1}(c_{2}+n_{2}c_{3})]$

which implies that the order of $H^{4}(M^{7}_{\bf n},{\mathbb{Z}})$ is
$c_{3}[c_{2}(c_{1}+n_{1}c_{3})+c_{1}(c_{2}+n_{2}c_{3})]$. This is always
greater than $1$, so $H^{4}(M^{7}_{\bf n},{\mathbb{Z}})_{tor}\neq 0$.
Summarizing we have

######  Theorem 3.4.

Let $M^{7}_{\bf n}$ denote the total space of the principal $S^{1}$ bundle
over a KS manifold $S_{\bf n}$ defined by a primitive Kähler class
$[\omega_{\bf n}]=c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3}.$ Then

$H^{r}(M^{7}_{{\bf n}},{\mathbb{Z}})=\begin{cases}{\mathbb{Z}}&~{}~{}\text{if
$r=0,7$}\\\ {\mathbb{Z}}^{2}&~{}~{}\text{if $r=2,5$}\\\ G_{reg}&~{}~{}\text{if
$r=4$}\\\ 0&~{}~{}\text{otherwise.}\end{cases}$

where $G_{reg}$ is an Abelian group of order
$c_{3}[c_{2}(c_{1}+n_{1}c_{3})+c_{1}(c_{2}+n_{2}c_{3})]$. Moreover, $G_{reg}$
is never the identity.

######  Example 3.1.

The simplist case is the standard product
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$
with Kähler class

$[\omega]=c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3},\qquad\gcd(c_{1},c_{2},c_{3})=1$

and $n_{1}=n_{2}=0$ which gives $|G_{reg}|=2c_{1}c_{2}c_{3}$. Taking
$c_{1}=c_{2}=c_{3}=1$ gives the homogeneous space

$M^{7}=\bigl{(}SU(2)\times SU(2)\times SU(2)\bigr{)}/U(1)\times U(1)$

with its Sasaki-Einstein metric (see for example [BFGK91]). Here
$G_{reg}={\mathbb{Z}}_{2}$.

######  Example 3.2.

Monotone Fano case. Up to equivalence there are two monotone Fano cases
polarized by the first Chern class, namely, ${\bf n}=(1,-1)$ and ${\bf
n}=(1,1)$. From Equations (3) and (6) we have $c_{1}=1,c_{2}=3,c_{3}=2$ when
${\bf n}=(1,-1)$, and $c_{1}=1,c_{2}=1,c_{3}=2$ when ${\bf n}=(1,1)$. These
give $|G_{reg}|=20$ and $|G_{reg}|=12$, respectively. Note that $M^{7}_{1,-1}$
is conjecturely111See Remark 3.10 below cone indecomposable; whereas,
$M^{7}_{1,1}$ is cone decomposible. The latter is described by Theorem 2.5 in
[BTF] with ${\bf l}_{1}=(1,1),{\bf w}_{1}=(1,1),{\bf l}_{2}=(1,2)$, and ${\bf
w}_{2}=(3,1)$.

###  3.4. The General Case; Branched Covers

Returning to the orbifold case, we consider the Leray spectral sequence of the
quotient map $M^{7}_{{\bf n},{\bf m}}\raise
4.0pt\hbox{$\pi\atop\longrightarrow$}(S_{\bf n},\Delta_{\bf m})$ viewed as an
$S^{1}$-Seifert bundle over $(S_{\bf n},\Delta_{\bf m})$. Since the fibers are
$S^{1}$ we only have the $R^{1}\pi_{*}{\mathbb{Z}}_{M}$ direct image sheaf,
the $E_{2}$ term of the Leray spectral sequence

(10) $E^{p,q}_{2}=H^{p}(S_{\bf n},R^{q}\pi_{*}{\mathbb{Z}})$

satisfies $E^{p,q}_{2}=0$ when $q\neq 0,1$, and this converges to
$H^{p+q}(M^{7}_{{\bf n},{\bf m}},{\mathbb{Z}})$. So the only differentials are
those induced by

(11) $d_{2}:E^{0,1}_{2}=H^{0}(S_{\bf
n},R^{1}\pi_{*}{\mathbb{Z}}_{M})\longrightarrow E^{2,0}_{2}=H^{2}(S_{\bf
n},{\mathbb{Z}})\approx{\mathbb{Z}}^{3}.$

We note that the sheaf injection

(12) $R^{1}\pi_{*}{\mathbb{Z}}_{M}\longrightarrow{\mathbb{Z}}_{S_{\bf n}}$

is multiplication by $1$ on the regular locus, multiplication by $m_{0}$ on
$e_{0}$ and by $m_{\infty}$ on $e_{\infty}$. So the image of the map (11) is
the order $\mu={\rm lcm}(m_{0},m_{\infty})$ times a primitive orbifold Kähler
class $[\omega_{{\bf n},{\bf m}}]\in H^{2}_{orb}((S_{\bf n},\Delta_{\bf
m}),{\mathbb{Z}})$. To compute the cohomology of such spaces we consider the
${\mathbb{C}}^{*}$ Seifert bundle over $(S_{\bf n},\Delta_{\bf m})$ which is
the cone $C(M^{7}_{{\bf n},{\bf m}})=Y_{{\bf n},{\bf m}}=M^{7}_{{\bf n},{\bf
m}}\times{\mathbb{R}}^{+}$ so that $Y_{{\bf n},{\bf m}}|_{r=1}=M^{7}_{{\bf
n},{\bf m}}$. Now $Y_{{\bf n},{\bf m}}$ is a branched cover of $Y_{{\bf
n},{\bf m}}/{\mathbb{Z}}_{\mu}$ with branching locus $\Delta_{\bf m}$. This
follows from [Kol04] by identifying the orbifold class $[\omega_{{\bf n},{\bf
m}}]$ with the rational first Chern class $c_{1}(Y/X)$ in [Kol04] with
$X=S_{\bf n}$. Now generally the total space $M^{7}_{{\bf n},{\bf m}}$ is a
Sasaki orbifold whose underlying topological space is a compactly generated
Hausdorff space which allows us to apply the theory of ramified covers [Smi83,
AP06]. The space $M^{7}_{{\bf n},{\bf m}}$ is simply connected, and a
$\mu$-fold cover of the total space of the ordinary $S^{1}$ bundle defined by
the primitive integral class $\mu[\omega_{{\bf n},{\bf m}}]$. Thus, we have
the commutative diagram of Seifert orbibundles:

(13) $\begin{matrix}S^{1}&\hbox to24.0pt{\rightarrowfill}&M^{7}_{{\bf n},{\bf
m}}&\raise 4.0pt\hbox{$f\atop\hbox to24.0pt{\rightarrowfill}$}&(S_{\bf
n},\Delta_{\bf m})\\\
\phantom{\hbox{$\scriptstyle{}$}}\left\downarrow\vbox{\vskip
15.0pt\hbox{$\scriptstyle{}$}}\right.&&\phantom{\hbox{$\scriptstyle{\pi}$}}\left\downarrow\vbox{\vskip
15.0pt\hbox{$\scriptstyle{\pi}$}}\right.&&\phantom{\hbox{$\scriptstyle{}$}}\left\downarrow\vbox{\vskip
15.0pt\hbox{$\scriptstyle{}$}}\right.\\\ S^{1}/{\mathbb{Z}}_{\mu}&\hbox
to24.0pt{\rightarrowfill}&M^{7}_{{\bf n},{\bf m}}/{\mathbb{Z}}_{\mu}&\raise
4.0pt\hbox{$f_{\mu}\atop\hbox to24.0pt{\rightarrowfill}$}&(S_{\bf
n},\emptyset).\end{matrix}$

Note that since $M^{7}_{{\bf n},{\bf m}}/{\mathbb{Z}}_{\mu}$ is the total
space of a principal $S^{1}$ bundle over the simply connected smooth
projective algebraic variety $S_{\bf n}$ defined by a primitive integral class
$\mu[\omega_{{\bf n},{\bf m}}]$, it is a simply connected smooth 7-manifold.
Moreover, the covering map $\pi$ induces a map of the corresponding Leray
spectral sequences, and since the fiber is an $S^{1}$ the only nonzero higher
direct image sheaf is $R^{1}\pi_{*}{\mathbb{Z}}$. This gives the natural
isomorphism $R^{1}\pi_{*}{\mathbb{Q}}_{M}\approx{\mathbb{Q}}_{S_{\bf n}}$
which then implies

######  Lemma 3.5.

There is an isomorphism $H^{*}(M^{7}_{{\bf n},{\bf m}},{\mathbb{Q}})\approx
H^{*}(M^{7}_{{\bf n},{\bf m}}/{\mathbb{Z}}_{\mu},{\mathbb{Q}})$.

However, we would also like information about integral cohomology groups. For
this we consider the transfer homomorphism for ramified covers [Smi83].
Following [AP06] we apply this to our branched cover $\pi:M^{7}_{{\bf n},{\bf
m}}\hbox to24.0pt{\rightarrowfill}M^{7}_{{\bf n},{\bf m}}/{\mathbb{Z}}_{\mu}$
. Theorem 5.4 of [AP06] says that the transfer homomorphism

$\tau:H^{*}(M^{7}_{{\bf n},{\bf m}},{\mathbb{Z}})\hbox
to24.0pt{\rightarrowfill}H^{*}(M^{7}_{{\bf n},{\bf
m}}/{\mathbb{Z}}_{\mu},{\mathbb{Z}})$

induces multiplication by $\mu$ in $H^{*}(M^{7}_{{\bf n},{\bf
m}}/{\mathbb{Z}}_{\mu},{\mathbb{Z}})$. We set $G_{reg}=H^{4}(M^{7}_{{\bf
n},{\bf m}}/{\mathbb{Z}}_{\mu},{\mathbb{Z}})$ and apply this to $*=4$. If
$\gcd(|G|,\mu)=1$ we see that

$\tau\circ\pi^{*}:G_{reg}=H^{4}(M^{7}_{{\bf n},{\bf
m}}/{\mathbb{Z}}_{\mu},{\mathbb{Z}})\hbox
to24.0pt{\rightarrowfill}G_{reg}=H^{4}(M^{7}_{{\bf n},{\bf
m}}/{\mathbb{Z}}_{\mu},{\mathbb{Z}})$

is an isomorphism which implies that $\pi^{*}:H^{4}(M^{7}_{{\bf n},{\bf
m}}/{\mathbb{Z}}_{\mu},{\mathbb{Z}})\hbox
to24.0pt{\rightarrowfill}H^{4}(M^{7}_{{\bf n},{\bf m}},{\mathbb{Z}})$ is
injective. Thus, $\pi^{*}(G_{reg})$ is a subgroup of $H^{4}(M^{7}_{{\bf
n},{\bf m}},{\mathbb{Z}})$ in this case. When $\gcd(|G_{reg}|,\mu)\neq 1$ the
homomorphism $\tau\circ\pi^{*}$ has a non-trivial kernel. This kernel consists
of all the prime factors
${\mathbb{Z}}_{p_{i}^{r_{1}}}\times\cdots\times{\mathbb{Z}}_{p_{k}^{r_{k}}}$
of $G_{reg}$ such that $p_{i}$ is a prime in $\mu$. Denote this group by
$G_{reg}^{\mu}$ In this case $H^{4}(M^{7}_{{\bf n},{\bf m}},{\mathbb{Z}})$
contains the subgroup of $G_{reg}$ containing only those prime factors whose
primes $p$ are not in $\mu$, that is the factor group $G_{reg}/G_{reg}^{\mu}$.
Summarizing we have

######  Lemma 3.6.

$H^{4}(M^{7}_{{\bf n},{\bf m}},{\mathbb{Z}})$ contains $G_{reg}/G_{reg}^{\mu}$

We also have

######  Lemma 3.7.

$H^{3}(M^{7}_{{\bf n},{\bf m}},{\mathbb{Z}})=0.$

###### Proof.

The only non-zero term of total degree 3 in (10) is

$E^{2,1}_{2}=H^{2}(S_{\bf n},R^{1}\pi_{*}{\mathbb{Z}})$

where we have a differential

$d_{2}:E^{2,1}_{2}=H^{2}(S_{\bf n},R^{1}\pi_{*}{\mathbb{Z}})\hbox
to24.0pt{\rightarrowfill}E^{4,0}_{2}=H^{4}(S_{\bf
n},{\mathbb{Z}})={\mathbb{Z}}^{3}.$

This differential gives torsion in $H^{4}(M^{7}_{{\bf n},{\bf
m}},{\mathbb{Z}})$ implying that $H^{4}(M^{7}_{{\bf n},{\bf m}},{\mathbb{Q}})$
vanishes which in turn implies that $E^{2,1}_{r}=0$ for $r>2$ which proves the
lemma. ∎

######  Theorem 3.8.

Let $M^{7}_{{\bf n},{\bf m}}$ denote the total space of the principal $S^{1}$
orbibundle over a KS orbifold $(S_{\bf n},\Delta_{\bf m})$ defined by a
primitive orbifold Kähler class $[\omega_{{\bf n},{\bf
m}}]=c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3}$ such that $\mu[\omega_{{\bf n},{\bf
m}}]$ is a primitive class in $H^{2}(S_{\bf n},{\mathbb{Z}})$. Then

$H^{r}(M^{7}_{{\bf n},{\bf
m}},{\mathbb{Z}})=\begin{cases}{\mathbb{Z}}&~{}~{}\text{if $r=0,7$}\\\
{\mathbb{Z}}^{2}&~{}~{}\text{if $r=2,5$}\\\ G&~{}~{}\text{if $r=4$}\\\
0&~{}~{}\text{otherwise.}\end{cases}$

where $G$ is an Abelian group that contains $G_{reg}/G^{\mu}_{reg}$. In
particular, if $\gcd(|G_{reg}|,\mu)=1$ then $G$ contains $G_{reg}$.

###### Proof.

Since $[\omega_{{\bf n},{\bf m}}]$ is a primitive orbifold class and $(S_{\bf
n},\Delta_{\bf m})$ has $\pi_{1}^{orb}(S_{\bf n},\Delta_{\bf
m})=\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}$, then the orbifold
$M^{7}_{{\bf n},{\bf m}}$ has $\pi_{1}^{orb}(M^{7}_{{\bf n},{\bf
m}})=\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}$ by the long exact
homotopy sequence of the classifying spaces. The remainder of the theorem then
follows from lemmas 3.5, 3.6, 3.7. ∎

######  Remark 3.9.

It follows from Theorem 3.4 that $G_{reg}$ is never the identity; although, we
can have $G_{reg}=G_{reg}^{\mu}$ so that
$G_{reg}/G^{\mu}=\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}$ in this case.
However, this does not imply that $H^{4}(M^{7}_{{\bf n},{\bf
m}},{\mathbb{Z}})$ is the identity.

######  Remark 3.10.

Let $M$ be a Boothby-Wang Sasaki manifold over
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$. The set of
KS orbifolds ${\mathcal{K}}{\mathcal{S}}_{0}$ splits into a disjoint union two
types; those that have a polarization such that the corresponding Boothby-Wang
constructed Sasaki orbifold is represented by a quasi-regular ray in the ${\bf
w}$-cone of a join of the form $M\star_{{\bf l}}S^{3}_{{\bf w}}$, and those
that do not have such a polarization. The former are characterized by the
condition $n_{1}n_{2}>0$ (see Theorem 4.3 below) and the corresponding Sasaki
manifolds are called cone decomposable (see Definitions 3.1 and 4.1 in
[BHLTF18]). It remains an open question whether a Boothby-Wang constructed
Sasaki orbifold over the latter type, with $n_{1}n_{2}<0$, is necessarily cone
indecomposable or simply not represented by a quasi-regular ray in the ${\bf
w}$-cone of a join of the form $M\star_{{\bf l}}S^{3}_{{\bf w}}$.

###  3.5. When $M^{7}$ is an $S^{3}_{\bf w}$-join

Generally, we are interested in how $G$ depends on ${\bf m}$ and ${\bf n}$.
Unfortunately, Theorem 3.8 does not give us much useful information about this
dependence. However, we can determine this dependence in a particular cone
decomposable case, namely when the space $M^{7}_{{\bf n},{\bf m}}$ can be
represented as a join of the form $M_{{\bf l},{\bf w}}=M\star_{\bf
l}S^{3}_{\bf w}$ where $M$ is a principal $S^{1}$ bundle over
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$. Let $M^{5}$
denote the total space of the Boothby-Wang bundle over
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$ determined by
the Kähler class $k_{1}y_{1}+k_{2}y_{2}$ with
$k_{1},k_{2}\in{\mathbb{Z}}^{+}$. When $k_{1}$ and $k_{2}$ are relatively
prime, $M^{5}$ is diffeomorphic to $S^{2}\times S^{3}$. More generally
$S^{2}\times S^{3}$ is an $k$-fold cover of $M^{5}$, namely we have
$M^{5}\approx(S^{2}\times S^{3})/{\mathbb{Z}}_{k}$ where
$k=\gcd(k_{1},k_{2})$. Moreover, if
$\gcd(l_{\infty},w_{0})=\gcd(l_{\infty},w_{\infty})=1$, $M_{{\bf l},{\bf w}}$
will be smooth, and we have an $S^{1}$ bundle

$S^{1}\hbox to31.20007pt{\rightarrowfill}M^{5}\times S^{3}_{\bf w}\hbox
to31.20007pt{\rightarrowfill}M^{5}\star_{\bf l}S^{3}_{\bf w}$

which gives the homotopy exact sequence

(14) $0\hbox to24.0pt{\rightarrowfill}\pi_{2}(M^{5})\hbox
to24.0pt{\rightarrowfill}\pi_{2}(M^{5}\star_{\bf l}S^{3}_{\bf w})\hbox
to24.0pt{\rightarrowfill}{\mathbb{Z}}\hbox
to24.0pt{\rightarrowfill}{\mathbb{Z}}_{k}\hbox
to24.0pt{\rightarrowfill}\pi_{1}(M^{5}\star_{\bf l}S^{3}_{\bf w})\hbox
to24.0pt{\rightarrowfill}1.$

So when $k=1$ so $M^{5}=S^{2}\times S^{3}$, the join $M^{7}_{{\bf l},{\bf w}}$
will also be simply connected with $\pi_{2}(M^{7}_{{\bf l},{\bf
w}})={\mathbb{Z}}^{2}$. More generally, the topological analysis proceeds as
in Section 4 of [BTF16], see also [BTF19]. We consider the commutative diagram
of fibrations

(15) $\begin{matrix}(S^{2}\times S^{3})/{\mathbb{Z}}_{k}\times S^{3}_{\bf
w}&\hbox to31.20007pt{\rightarrowfill}&M_{{\bf l},{\bf w}}&\hbox
to31.20007pt{\rightarrowfill}&\mathsf{B}S^{1}\\\
\phantom{\hbox{$\scriptstyle{=}$}}\left\downarrow\vbox{\vskip
15.0pt\hbox{$\scriptstyle{=}$}}\right.&&\phantom{\hbox{$\scriptstyle{}$}}\left\downarrow\vbox{\vskip
15.0pt\hbox{$\scriptstyle{}$}}\right.&&\phantom{\hbox{$\scriptstyle{\psi}$}}\left\downarrow\vbox{\vskip
15.0pt\hbox{$\scriptstyle{\psi}$}}\right.\\\ (S^{2}\times
S^{3})/{\mathbb{Z}}_{k}\times S^{3}_{\bf w}&\hbox
to31.20007pt{\rightarrowfill}&{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}\times\mathsf{B}{\mathbb{C}}{\mathbb{P}}^{1}[{\bf
w}]&\hbox
to31.20007pt{\rightarrowfill}&\mathsf{B}S^{1}\times\mathsf{B}S^{1}\,\end{matrix}\qquad\qquad$

where $\mathsf{B}G$ is the classifying space of a group $G$ or Haefliger’s
classifying space [Hae84] of an orbifold if $G$ is an orbifold. Note that
$H^{2}((S^{2}\times
S^{3})/{\mathbb{Z}}_{k},{\mathbb{Z}})={\mathbb{Z}}\oplus{\mathbb{Z}}_{k}$. The
diagram gives a map of spectral sequences, and we note that the lower
fibration is a product of well understood fibrations. The $E_{2}$ term of the
top fibration is

(16) $E_{2}^{p,q}=H^{p}(\mathsf{B}S^{1},H^{q}(S^{2}\times S^{3}\times
S^{3}_{\bf
w},{\mathbb{Z}}))\approx{\mathbb{Z}}[s]\otimes\Lambda[\alpha,\beta,\gamma],$

where $\alpha$ is a 2-class, $\beta,\gamma$ are 3-classes, and $s_{1},s_{2}$
are the positive generators of $H^{*}(\mathsf{B}S^{1},{\mathbb{Z}})$. By the
Leray-Serre Theorem this converges to $H^{p+q}(M_{{\bf l},{\bf
w}},{\mathbb{Z}})$. The non-vanishing differentials of the bottom product
fibration are, first $d_{4}(\beta)=s_{1}^{2}$ and second
$d_{4}(\gamma)=w_{0}w_{\infty}s_{2}^{2}$ and those induced by naturality. As
described in [BTF16] we also have $\psi^{*}s_{1}=l_{\infty}s$ and
$\psi^{*}s_{2}=-l_{0}s$. So by naturality the differentials of the top
fibration are $d_{4}(\beta)=l_{\infty}^{2}s^{2}$ and
$d_{4}(\gamma)=w_{0}w_{\infty}l_{0}^{2}s^{2}$. Now $M_{{\bf l},{\bf w}}$ is
smooth if and only if $\gcd(l_{\infty},w_{0}w_{\infty})=1$. This gives
$H^{4}(M_{{\bf l},{\bf
w}},{\mathbb{Z}})={\mathbb{Z}}_{l_{\infty}^{2}}\times{\mathbb{Z}}_{w_{0}w_{\infty}l_{0}^{2}}={\mathbb{Z}}_{w_{0}w_{\infty}l_{0}^{2}l_{\infty}^{2}}$.

##  4\. Admissible Projective Bundles

Admissible projective bundles were described in general [ACGTF08a]. Here we
will restrict ourselves to a specific type of these, namely projective bundles
of the form

$S_{\bf n}={\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(n_{1},n_{2}))\hbox
to21.60004pt{\rightarrowfill}{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$

that satisfy the following conditions:

  * •

For $i=1,2$ let $(\pm\omega_{i},\pm g_{i})$ be Kähler metrics with constant
scalar curvature $\pm 2s_{i}=\pm 4/n_{i}$ on ${\mathbb{C}}{\mathbb{P}}^{1}$.
[It is assumed that $n_{i}\neq 0$.] The $\pm$ means that either $+g_{i}$ or
$-g_{i}$ is positive definite.

  * •

${\mathcal{O}}(n_{1},n_{2})\hbox
to21.60004pt{\rightarrowfill}{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$
is the holomorphic line bundle over
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$ which satisfy

$c_{1}({\mathcal{O}}(n_{1},n_{2}))=\frac{1}{2\pi}[\omega_{1}+\omega_{2}]=n_{1}x_{1}+n_{2}x_{2}$

On admissible projective bundles such as the ones defined above we can now
construct the admissible metrics, [ACGTF08a]. Here we recover the main points
of this construction.

Consider the standard circle action on $S_{\bf
n}={\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(n_{1},n_{2}))\hbox
to21.60004pt{\rightarrowfill}{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$.
It extends to a holomorphic ${\mathbb{C}}^{*}$ action. The open and dense set
$S_{\bf n}^{0}$ of stable points with respect to the latter action has the
structure of a principal circle bundle over the stable quotient. The hermitian
norm on the fibers induces via a Legendre transform a function
${\mathfrak{z}}:S_{\bf n}^{0}\rightarrow(-1,1)$ whose extension to $S_{\bf n}$
consists of the critical manifolds
$e_{0}:={\mathfrak{z}}^{-1}(1)={\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\oplus 0)$ and
$e_{\infty}:={\mathfrak{z}}^{-1}(-1)={\mathbb{P}}(0\oplus{\mathcal{O}}(n_{1},n_{2}))$.
Letting $\theta$ be a connection one form for the Hermitian metric on $S_{\bf
n}^{0}$, with curvature $d\theta=\omega_{1}+\omega_{2}$, an admissible Kähler
metric and form are given (up to scale) by the respective formulas

(17)
$g=\frac{1+r_{1}{\mathfrak{z}}}{r_{1}}g_{1}+\frac{1+r_{2}{\mathfrak{z}}}{r_{2}}g_{2}+\frac{d{\mathfrak{z}}^{2}}{\Theta({\mathfrak{z}})}+\Theta({\mathfrak{z}})\theta^{2},\quad\omega=\frac{1+r_{1}{\mathfrak{z}}}{r_{1}}\omega_{1}+\frac{1+r_{2}{\mathfrak{z}}}{r_{2}}\omega_{2}+d{\mathfrak{z}}\wedge\theta,$

valid on $S_{\bf n}^{0}$. Here $\Theta$ is a smooth function with domain
containing $(-1,1)$ and $r_{i}$, $i=1,2$ are real numbers of the same sign as
$g_{i}$ and satisfying $0<|r_{i}|<1$. The complex structure yielding this
Kähler structure is given by the pullback of the base complex structure along
with the requirement $Jd{\mathfrak{z}}=\Theta\theta$. The function
${\mathfrak{z}}$ is hamiltonian with $K=J\,grad\,{\mathfrak{z}}$ a Killing
vector field. In fact, ${\mathfrak{z}}$ is the moment map on $S_{\bf n}$ for
the circle action, decomposing $M$ into the free orbits $S_{\bf
n}^{0}={\mathfrak{z}}^{-1}((-1,1))$ and the special orbits
${\mathfrak{z}}^{-1}(\pm 1)$. Finally, $\theta$ satisfies $\theta(K)=1$.

Now $g$ is a (positive definite) Kähler metric which extends smoothly to all
of $S_{\bf n}$ if and only if $\Theta$ satisfies the following positivity and
boundary conditions

(18) $\displaystyle(i)\
\Theta({\mathfrak{z}})>0,\quad-1<{\mathfrak{z}}<1,\quad(ii)\ \Theta(\pm
1)=0,\quad(iii)\ \Theta^{\prime}(\pm 1)=\mp 2.$

The Kähler class $\Omega_{{\bf r}}=[\omega]$ of an admissible metric as in
(17) is also called admissible and is uniquely determined by the parameters
$r_{1},r_{2}$, once the data associated with $M$ (i.e. $s_{i}=2/n_{i}$,
$g_{i}$ etc.) is fixed. Indeed, we have

$\Omega_{{\bf
r}}=\frac{[\omega_{1}]}{r_{1}}+\frac{[\omega_{2}]}{r_{2}}+2\pi\Xi,$

where $\Xi$ is the Poincare dual of $e_{0}+e_{\infty}$. For a more thorough
description of $\Xi$, please consult Section 1.3 of [ACGTF08a]. Note that on
$S_{\bf n}$ any Kähler class is admissible up to scale. Using that
$y_{3}-x_{3}=n_{1}x_{1}+n_{2}x_{2}$, we can also write

$\begin{array}[]{ccl}\Omega_{{\bf
r}}/2\pi&=&\frac{n_{1}}{r_{1}}x_{1}+\frac{n_{2}}{r_{2}}x_{2}+x_{3}+y_{3}\\\
\\\
&=&\frac{n_{1}(1+r_{1})}{r_{1}}x_{1}+\frac{n_{2}(1+r_{2})}{r_{2}}x_{2}+2x_{3}\\\
\\\
&=&\frac{n_{1}(1-r_{1})}{r_{1}}x_{1}+\frac{n_{2}(1-r_{2})}{r_{2}}x_{2}+2y_{3}.\end{array}$

Define a function $F({\mathfrak{z}})$ by the formula
$\Theta({\mathfrak{z}})=F({\mathfrak{z}})/p_{c}({\mathfrak{z}})$, where
$p_{c}({\mathfrak{z}})=(1+r_{1}{\mathfrak{z}})(1+r_{2}{\mathfrak{z}})$. Since
$p_{c}({\mathfrak{z}})$ is positive for $-1\leq{\mathfrak{z}}\leq 1$,
conditions (18) are equivalent to the following conditions on
$F({\mathfrak{z}})$.

(19) $\displaystyle(i)\
F({\mathfrak{z}})>0,\quad-1<{\mathfrak{z}}<1,\quad(ii)\ F(\pm 1)=0,\quad(iii)\
F^{\prime}(\pm 1)=\mp 2p_{c}(\pm 1).$

###  4.1. Orbifolds

Now we allow our admissible metrics to compactify as orbifold metrics on the
log pair

$(S_{\bf n},\Delta_{\bf
m})=\left({\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(n_{1},n_{2}),\Delta_{\bf m}\right)\hbox
to21.60004pt{\rightarrowfill}{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1},$

where

$\Delta_{\bf
m}=(1-1/m_{0})D_{1}+(1-1/m_{\infty})D_{2}=(1-1/m_{0})e_{0}+(1-1/m_{\infty})e_{\infty},$

and $m_{0},m_{\infty}\in{\mathbb{Z}}^{+}$. Then (18) generalizes to

(20)
$\begin{array}[]{cl}(i)&\Theta({\mathfrak{z}})>0,\quad-1<{\mathfrak{z}}<1,\\\
(ii)&\Theta(\pm 1)=0,\\\
(iii)&\Theta^{\prime}(-1)=2/m_{\infty},\quad\text{and}\quad\Theta^{\prime}(1)=-2/m_{0}.\end{array}$

and, with $\Theta({\mathfrak{z}})=F({\mathfrak{z}})/p_{c}({\mathfrak{z}})$ as
above, we get that this is equivalent to

(21) $\begin{array}[]{cl}(i)&F({\mathfrak{z}})>0,\quad-1<{\mathfrak{z}}<1,\\\
(ii)&F(\pm 1)=0,\\\
(iii)&F^{\prime}(-1)=2p_{c}(-1)/m_{\infty},\quad\text{and}\quad
F^{\prime}(1)=-2p_{c}(1)/m_{0}.\end{array}$

Note that this does not change the expression for $\Omega_{r}$ above whereas
from (3) we already know that the adjusted Chern class is

$\begin{array}[]{ccl}c_{1}^{orb}(S_{\bf n},\Delta_{\bf
m})&=&\Bigl{(}2-\frac{n_{1}}{m_{\infty}}\Bigr{)}x_{1}+\Bigl{(}2-\frac{n_{2}}{m_{\infty}}\Bigr{)}x_{2}+\Bigl{(}\frac{1}{m_{0}}+\frac{1}{m_{\infty}}\Bigr{)}y_{3}\\\
&=&\Bigl{(}2+\frac{n_{1}}{m_{0}}\Bigr{)}x_{1}+\Bigl{(}2+\frac{n_{2}}{m_{0}}\Bigr{)}x_{2}+\Bigl{(}\frac{1}{m_{0}}+\frac{1}{m_{\infty}}\Bigr{)}x_{3}.\end{array}.$

###  4.2. Connection with $S^{3}_{\bf w}$-joins

Consider $(S_{\bf n},\Delta_{\bf
m})=\left({\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(n_{1},n_{2}),\Delta_{\bf m}\right)\hbox
to21.60004pt{\rightarrowfill}{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1},$
polarized with a primitive orbifold Kähler class $[\omega_{{\bf n},{\bf m}}]$.
As mentioned above, this will be the rescale of some admissible Kähler class
$\Omega_{\bf r}$ determined by ${\bf r}=(r_{1},r_{2})$. If $n_{1}n_{2}>0$, we
will say that $[\omega_{{\bf n},{\bf m}}]$ is diagonally admissible if and
only if $r_{1}=r_{2}$. Note that if $n_{1}n_{2}<0$, it is never possible to
have $r_{1}=r_{2}$, since the sign of $r_{i}$ is equal to the sign of $n_{i}$.
Note that diagonally admissible is equivalent to being admissible (up to
scale) as defined (more narrowly) in Section 2.5.1 of [BTF21a] with
$N={\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$ and
$[\omega_{N}]=\frac{n_{1}}{n}x_{1}+\frac{n_{2}}{n}x_{2}$, where $n=(\text{sign
of}\,n_{i})\gcd(|n_{1}|,|n_{2}|)$.

Indeed, suppose $n_{1}n_{2}>0$ and $r_{1}=r_{2}=r$ in the above setting. Then
the admissible metric simplifies to

(22)
$g=\frac{1+r{\mathfrak{z}}}{r}g_{N_{n}}+\frac{d{\mathfrak{z}}^{2}}{\Theta({\mathfrak{z}})}+\Theta({\mathfrak{z}})\theta^{2},\quad\omega=\frac{1+r{\mathfrak{z}}}{r}\omega_{N_{n}}+d{\mathfrak{z}}\wedge\theta,$

with admissible Kähler class

$\begin{array}[]{ccl}\Omega_{{\bf
r}}/2\pi&=&\frac{n}{r}\left(\frac{n_{1}}{n}x_{1}+\frac{n_{2}}{n}x_{2}\right)+x_{3}+y_{3}\\\
\\\
&=&\frac{n(1+r)}{r}\left(\frac{n_{1}}{n}x_{1}+\frac{n_{2}}{n}x_{2}\right)+2x_{3}\\\
\\\
&=&\frac{n(1-r)}{r}\left(\frac{n_{1}}{n}x_{1}+\frac{n_{2}}{n}x_{2}\right)+2y_{3},\end{array}$

where $g_{N_{n}}=n_{1}g_{1}+n_{2}g_{2}$. As such, we can view this metric as
an admissible metric on $S_{\bf
n}={\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold 1$}}\oplus L_{n})\hbox
to21.60004pt{\rightarrowfill}N$, where
$N={\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$ and

  * •

$(\pm\omega_{N_{n}},\pm g_{N_{n}})$ is a Kähler metric with constant scalar
curvature $\pm 4s=\pm 4(\frac{1}{n_{1}}+\frac{1}{n_{2}})$. [The $\pm$ still
means that either $+g_{N_{n}}$ or $-g_{N_{n}}$ is positive definite.]

  * •

$L_{n}\hbox to21.60004pt{\rightarrowfill}N$ is the holomorphic line bundle
over $N={\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$ which
satisfy

$c_{1}(L_{n})=\frac{1}{2\pi}[\omega_{N_{n}}]=n\left(\frac{n_{1}}{n}x_{1}+\frac{n_{2}}{n}x_{2}\right).$

This special case, is exactly of the type considered in e.g. Section 5 of
[BTF16] or more generally (including cyclic orbifolds $N$) in Section 2.5 of
[BTF21a]. Note that the base metric $(\pm\omega_{N_{n}},\pm g_{N_{n}})$ is
only Kähler-Einstein if also $n_{1}=n_{2}$.

Now consider the natural Boothby-Wang constructed Sasaki structure on the
$S^{1}$-bundle $M\rightarrow N$ defined by the primitive
$[\omega_{N}]=\left(\frac{n_{1}}{n}x_{1}+\frac{n_{2}}{n}x_{2}\right)$.
Following Proposition 4.22 of [BHLTF18] (with the amendment of Lemma 2.12 and
Corollary 2.14 of [BTF21a]) we then realize the following connection with
$S^{3}_{\bf w}$-joins.

######  Proposition 4.1.

For $n_{1}n_{2}>0$, $r_{1}=r_{2}=r$ rational, $n=(\text{sign
of}\,n_{i})\gcd(|n_{1}|,|n_{2}|)$, and $\gcd(m_{1},m_{2},|n|)=1$, there is a
choice of co-prime $w_{0}$, $w_{\infty}\in{\mathbb{Z}}^{+}$ and co-prime
$l_{0},l_{\infty}$ such that, when we form the $S^{3}_{\bf w}$-join
$M_{l_{0},l_{\infty},{\bf
w}}:=M\star_{l_{0},l_{\infty}}S^{3}_{w_{0},w_{\infty}}=M\star_{{\bf
l}}S^{3}_{{\bf w}}$, the quasi-regular quotient of $M_{l_{0},l_{\infty},{\bf
w}}$ by the flow of the Reeb vector field $\xi_{\bf v}$ determined by
$(v_{0},v_{\infty})$ in the ${\bf w}$-cone (where $m=\gcd(m_{0},m_{\infty})$
and ${\bf m}=(m_{0},m_{\infty})=m{\bf v}=m(v_{0},v_{\infty})$), is the log
pair $(S_{\bf n},\Delta_{\bf m})$ with induced (transverse) Kähler class
$m\gcd({\mathfrak{s}},w_{0}v_{\infty})[\omega_{{\bf n},{\bf m}}]$ where
${\mathfrak{s}}=\gcd(l_{\infty},|w_{0}v_{\infty}-w_{\infty}v_{0}|)$ and
$[\omega_{{\bf n},{\bf m}}]$ is a orbifold primitive class that is an
appropriate rescale of the admissible class $\Omega_{\bf r}$ on $(S_{\bf
n},\Delta_{\bf m})$. The join is smooth if and only if
$\gcd(w_{0},l_{\infty})=\gcd(w_{\infty},l_{\infty})=1$. In particular, the
join is smooth if $m_{0}=m_{\infty}=1$.

###### Proof.

We follow the proof of Proposition 4.22 in [BHLTF18] (including the paragraphs
leading up to the proposition), to identify the join by first picking ${\bf
w}=(w_{0},w_{\infty})$ to be the unique positive, integer, and co-prime
solution of

$r=\frac{w_{0}m_{\infty}-w_{\infty}m_{0}}{w_{0}m_{\infty}+w_{\infty}m_{0}}$

and then, as the next step, picking the pair $(l_{0},l_{\infty})$ as the
unique, positive integers, and co-prime solution of

$l_{\infty}n=l_{0}(w_{0}m_{\infty}-w_{\infty}m_{0}).$

Proposition 4.22 in [BHLTF18] was slightly misleading in implying that the
quasi-regular quotient of $\xi_{\bf v}$ would always produce a primitive
orbifold Kähler class, but, with Lemma 2.12 and Corollary 2.14 of [BTF21a] in
hand, we can say that the transverse Kähler class is
$m\gcd({\mathfrak{s}}\Upsilon_{N},w_{0}v_{\infty}l_{0})[\omega_{{\bf n},{\bf
m}}]$ where $[\omega_{{\bf n},{\bf m}}]$ is a orbifold primitive class that is
an appropriate rescale of the admissible class $\Omega_{\bf r}$ on $(S_{\bf
n},\Delta_{\bf m})$ and $\Upsilon_{N}$ is the orbifold order of $N$. Since
$\Upsilon_{N}=1$ and $\gcd({\mathfrak{s}},l_{0})=1$ (recall
$\gcd(l_{0},l_{\infty})=1$) we get the desired Kähler class.

The rest of the claims follow straight from the proof Proposition 4.22 in
[BHLTF18]. In particular, note that $m=l_{\infty}/{\mathfrak{s}}$, as it
should be according to Theorem 3.8 in [BTF16]. ∎

######  Remark 4.2.

Note that when the join is smooth, it is easy to see that
$\gcd({\mathfrak{s}},w_{0}v_{\infty})=1$ and so the transverse Kähler class is
$m[\omega_{{\bf n},{\bf m}}]$. In particular, if $m_{0}=m_{\infty}=1$, we do
have a primitive (and admissible) transverse Kähler class.

Combining Theorem 2.7 and Lemma 2.12 from [BTF21a] for the“only if ” and
Proposition 4.1 for the “if”, we arrive at the following theorem.

######  Theorem 4.3.

Assume $\gcd(m_{1},m_{2},|n_{1}|,|n_{2}|)=1$ and $[\omega_{{\bf n},{\bf m}}]$
is a orbifold primitive class on $(S_{\bf n},\Delta_{\bf m})$. There exist a
constant $k\in{\mathbb{Z}}^{+}$ such that the polarized orbifold $(S_{\bf
n},\Delta_{\bf m},k[\omega_{{\bf n},{\bf m}}])$ is the quotient with respect
to (the canonical Reeb vector field in) a quasi-regular ray in the ${\bf
w}$-cone of a (possibly non-smooth) join of the form $M\star_{{\bf
l}}S^{3}_{{\bf w}}$, where $M$ is a Boothby-Wang constructed Sasaki manifold
over ${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$, if and
only if $n_{1}n_{2}>0$ and $[\omega_{{\bf n},{\bf m}}]$ is a diagonally
admissible Kähler class.

Suppose now that $n_{1}n_{2}>0$, we are in the log Fano case as in Proposition
1.1, and the polarization is chosen such that $c_{1}^{orb}(S_{\bf
n},\Delta_{\bf m})={\mathcal{I}}_{{\bf n},{\bf m}}[\omega_{{\bf n},{\bf m}}]$.
Since $c_{1}^{orb}(S_{\bf n},\Delta_{\bf
m})=\Bigl{(}2-\frac{n_{1}}{m_{\infty}}\Bigr{)}x_{1}+\Bigl{(}2-\frac{n_{2}}{m_{\infty}}\Bigr{)}x_{2}+\Bigl{(}\frac{1}{m_{0}}+\frac{1}{m_{\infty}}\Bigr{)}y_{3}$
and a diagonally admissible Kähler class is a rescale of $\Omega_{\bf
r}=\frac{(1-r)}{r}\left(n_{1}x_{1}+n_{2}x_{2}\right)+2y_{3}$ for some
$0<|r|<1$ with $rn_{i}>0$, we see that $[\omega_{{\bf n},{\bf
m}}]=c_{1}^{orb}(S_{\bf n},\Delta_{\bf m})/{\mathcal{I}}_{{\bf n},{\bf m}}$ is
diagonally admissible if and only if

$2-\frac{n_{1}}{m_{\infty}}=\left(\frac{1}{m_{0}}+\frac{1}{m_{\infty}}\right)\frac{n_{1}(1-r)}{2r}\quad\text{and}\quad
2-\frac{n_{2}}{m_{\infty}}=\left(\frac{1}{m_{0}}+\frac{1}{m_{\infty}}\right)\frac{n_{2}(1-r)}{2r},$

i.e.,

$n_{1}\left(\frac{1}{m_{\infty}}+\left(\frac{1}{m_{0}}+\frac{1}{m_{\infty}}\right)\frac{(1-r)}{2r}\right)=2=n_{2}\left(\frac{1}{m_{\infty}}+\left(\frac{1}{m_{0}}+\frac{1}{m_{\infty}}\right)\frac{(1-r)}{2r}\right).$

This clearly implies that we must have $n_{1}=n_{2}$. On the other hand, if
$n_{1}=n_{2}$ (and still assuming log Fano) we can solve for an appropriate
$r$. In conclusion,

######  Corollary 4.4.

In the log Fano case, there exists a constant $k\in{\mathbb{Z}}^{+}$ such that
the polarized orbifold $(S_{\bf n},\Delta_{\bf m},kc_{1}^{orb}(S_{\bf
n},\Delta_{\bf m}))$ is the quotient with respect to a quasi-regular ray in
the ${\bf w}$-cone of a (possibly non-smooth) join of the form $M\star_{{\bf
l}}S^{3}_{{\bf w}}$, where $M$ is a Boothby-Wang constructed Sasaki manifold
over ${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$, if and
only if $n_{1}=n_{2}$.

Returning to Proposition 4.1 and Theorem 4.3, we note that in the case where
$(S_{\bf n},\Delta_{\bf m},[\omega_{{\bf n},{\bf m}}])$ is the quotient with
respect to a quasi-regular ray in the ${\bf w}$-cone of a join of the form
$M\star_{{\bf l}}S^{3}_{{\bf w}}$ (so assuming $k=1$ in Theorem 4.3, i.e.,
assuming the transverse Kähler class is primitive) we must have that $m=1$ and
$\gcd({\mathfrak{s}},w_{0}v_{\infty})=1$. The first equality implies that
$l_{\infty}={\mathfrak{s}}=\gcd(l_{\infty},|w_{0}v_{\infty}-w_{\infty}v_{0}|)$
and hence $l_{\infty}$ is a factor of $|w_{0}v_{\infty}-w_{\infty}v_{0}|$.
Since we also have that $\gcd(l_{\infty},w_{0}v_{\infty})=1$, we must have
$\gcd(l_{\infty},w_{\infty}v_{0})=1$ and in particular
$\gcd(w_{0},l_{\infty})=\gcd(w_{\infty},l_{\infty})=1$. This means that
$M\star_{{\bf l}}S^{3}_{{\bf w}}$ is smooth. We therefore have the following
companion to Theorem 4.3.

######  Proposition 4.5.

Consider the polarized orbifold $(S_{\bf n},\Delta_{\bf m},[\omega_{{\bf
n},{\bf m}}])$, where $[\omega_{{\bf n},{\bf m}}]$ is a primitive integer
orbifold Kähler class and $\gcd(m_{0},m_{\infty},|n_{1}|,|n_{2}|)=1$.

If $(S_{\bf n},\Delta_{\bf m},[\omega_{{\bf n},{\bf m}}])$ is the quotient
with respect to (the canonical Reeb vector field in) a quasi-regular ray in
the ${\bf w}$-cone of a join of the form $M\star_{{\bf l}}S^{3}_{{\bf w}}$,
then this join is smooth. Moreover, in this case $\gcd(m_{0},m_{\infty})=1$.

###  4.3. Connection with Yamazaki’s Fiber joins

In [Yam99] T. Yamazaki introduced the fiber join for $K$-contact structures
and in [BTF21b] this was extended to Sasaki structures. In the smooth case
($m_{0}=m_{\infty}=1$) of KS orbifolds, it follows from Section 5.3 of
[BTF21b] that $S_{\bf n}$ polarized by a primitive Kähler class, which in turn
is an appropriate rescale of $\Omega_{\bf r}$, is the quotient of the regular
ray in the ${\mathfrak{t}}^{+}_{sphr}$ cone of a Yamazaki fiber join over
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$, if an only
if there exist $k_{1}^{1},k_{1}^{2},k_{2}^{1},k_{2}^{2}\in{\mathbb{Z}}^{+}$
such that

$\displaystyle k_{1}^{1}-k_{2}^{1}$ $\displaystyle=$ $\displaystyle
n_{1}\qquad k_{1}^{2}-k_{2}^{2}=n_{2}$ (23)
$\displaystyle\frac{k_{1}^{1}-k_{2}^{1}}{k_{1}^{1}+k_{2}^{1}}$
$\displaystyle=$ $\displaystyle
r_{1}\qquad\frac{k_{1}^{2}-k_{2}^{2}}{k_{1}^{2}+k_{2}^{2}}=r_{2}.$

In that case, the corresponding Yamazaki fiber join is given as follows:222We
refer to [BTF21b] for details and a more general description of the Yamazaki
fiber joins.

Let
$\pi_{j}:{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}\rightarrow{\mathbb{C}}{\mathbb{P}}^{1}$
denote the natural projection to the $j^{t}$ factor of the product
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}\rightarrow{\mathbb{C}}{\mathbb{P}}^{1}$
and let ${\mathcal{K}}$ denote the canonical bundle on
${\mathbb{C}}{\mathbb{P}}^{1}$. Let $L_{i}$ be a holomorphic line bundle over
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}\rightarrow{\mathbb{C}}{\mathbb{P}}^{1}$
given by
$L_{i}=k^{1}_{i}\pi_{1}^{*}{\mathcal{K}}^{\frac{-1}{2}}+k^{2}_{i}\pi_{2}^{*}{\mathcal{K}}^{\frac{-1}{2}}$,
for $k^{j}_{i}\in{\mathbb{Z}}^{+}$. The choice of $L_{1}$ and $L_{2}$ can be
given by the matrix $K=\begin{pmatrix}k^{1}_{1}&k^{2}_{1}\\\
k^{1}_{2}&k^{2}_{2}\end{pmatrix}$. Note that $c_{1}(L_{i})$ are both in the
Kähler cone of
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$ and hence
$L_{i}$ are positive line bundles over
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$. Each
$c_{1}(L_{i})$ also defines a principal $S^{1}$-bundle over
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$,
$M_{i}\rightarrow{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$,
and we identify $L_{i}$ with $M_{i}\times_{S^{1}}{\mathbb{C}}$. Then
$M_{i}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$
has a natural Sasaki structure defined by the Boothby-Wang construction.
Consider now $L_{1}^{*}\oplus L_{2}^{*}$ and equip each $L_{i}^{*}$ with a
Hermitian metric giving us a norm $d_{i}:L_{i}^{*}\to{\mathbb{R}}^{\geq 0}$.
Then, the fiber join, $M_{K}=M_{1}\star_{f}M_{2}$ is defined as the
$S^{3}$-bundle over $S$ whose fibers are given by $d_{1}^{2}+d_{2}^{2}=1$. Now
$M_{K}$ has a natural $CR$-structure $({\mathcal{D}},J)$ with a family,
${\mathfrak{t}}^{+}_{sphr}$, of compatible Sasaki structures
${\mathcal{S}}_{\bf a}=(\xi_{\bf a},\eta_{\bf a},\Phi_{\bf a},g_{\bf a})$,
where ${\bf a}=(a_{1},a_{2})\in({\mathbb{R}}^{+})^{2}$ and
$(a_{1},a_{2})=(1,1)$ corresponds to the regular Sasaki structure in
${\mathfrak{t}}^{+}_{sph}({\mathcal{D}},J)$. Note that
${\mathfrak{t}}^{+}_{sph}$ is a proper subcone of the unreduced Sasaki cone
${\mathfrak{t}}^{+}$ of $(M_{K},{\mathcal{D}},J)$.

######  Remark 4.6.

In [BTF21b] we did not consider the more general question of determining the
quotients of quasi-regular Sasaki structures in ${\mathfrak{t}}^{+}_{sph}$ and
the transverse Kähler class. We conjecture that those would indeed be certain
polarized KS orbifolds and will explore this in future studies.

###  4.4. Ricci Solitons and Kähler-Einstein

It is well-known that there exists a unique Kähler-Ricci soliton on any toric
compact Fano complex orbifold [Don08, TZ00, SZ12, WZ04, Zhu00]. The existence
proofs by Wang-Zhu [WZ04], Shi-Zhu [SZ12] and Donaldson [Don08] all use a
continuity method and thus do not provide an explicit expression of the
Kähler-Ricci soliton. It is therefore interesting to explore cases where
explicit descriptions of Kähler-Ricci solitons are possible. Explicit examples
can be found in e.g.[DW11, LTF13].

Here we want to explore the Ricci Soliton and more specifically the Kähler-
Einstein equations under the constraint of the orbifold endpoint conditions
(21). This will yield explicit examples on Fano $(S_{\bf n},\Delta_{\bf m})$
and is essentially a mild orbifold extension of the work by Koiso and Sakane
[KS86, Koi90]. We will follow in their footsteps using the notation of Section
3 of [ACGTF08b].

The admissible metrics are Ricci solitons with
$V=(\frac{c}{2})\,grad_{g}{\mathfrak{z}}$ if and only if

(24) $\rho-\lambda\omega={\mathcal{L}}_{V}\omega,$

where the Ricci form is

$\rho=s_{1}\omega_{1}+s_{2}\omega_{2}-\frac{1}{2}dd^{c}\log
F=\left(s_{1}-\frac{1}{2}\frac{F^{\prime}({\mathfrak{z}})}{p_{c}({\mathfrak{z}})}\right)\omega_{1}+\left(s_{2}-\frac{1}{2}\frac{F^{\prime}({\mathfrak{z}})}{p_{c}({\mathfrak{z}})}\right)\omega_{2}-\frac{1}{2}\left(\frac{F^{\prime}}{p_{c}}\right)^{\prime}({\mathfrak{z}})\,d{\mathfrak{z}}\wedge\theta$

and $\lambda,c\in{\mathbb{R}}$. Obviously, $c=0$ corresponds to Kähler-
Einstein metrics. Since
${\mathcal{L}}_{V}\omega=(\frac{c}{2})\,dd^{c}{\mathfrak{z}}$ and
$s_{i}=2/n_{i}$, equation (24) becomes a pair of ODEs

(25)
$\begin{array}[]{ccl}\frac{F^{\prime}({\mathfrak{z}})}{p_{c}({\mathfrak{z}})}+c\frac{F({\mathfrak{z}})}{p_{c}({\mathfrak{z}})}&=&4/n_{1}-2\lambda({\mathfrak{z}}+\frac{1}{r_{1}})\\\
\\\
\frac{F^{\prime}({\mathfrak{z}})}{p_{c}({\mathfrak{z}})}+c\frac{F({\mathfrak{z}})}{p_{c}({\mathfrak{z}})}&=&4/n_{2}-2\lambda({\mathfrak{z}}+\frac{1}{r_{2}}),\\\
\end{array}$

Note that (25) implies that $n_{1}=n_{2}$ if and only if $r_{1}=r_{2}$.

Note that (ii) and (iii) of (21) together with (25) implies the necessary
conditions

(26) $2\lambda=\frac{1}{m_{0}}+\frac{1}{m_{\infty}}$

and

(27)
$r_{1}=\frac{\frac{1}{m_{0}}+\frac{1}{m_{\infty}}}{\frac{4}{n_{1}}+\frac{1}{m_{0}}-\frac{1}{m_{\infty}}},\quad\quad
r_{2}=\frac{\frac{1}{m_{0}}+\frac{1}{m_{\infty}}}{\frac{4}{n_{2}}+\frac{1}{m_{0}}-\frac{1}{m_{\infty}}}.$

On the other hand, assuming (26) and (27), the ODEs of (25) are equivalent to
the single ODE

(28)
$\begin{array}[]{ccl}\frac{F^{\prime}({\mathfrak{z}})}{p_{c}({\mathfrak{z}})}+c\frac{F({\mathfrak{z}})}{p_{c}({\mathfrak{z}})}&=&(\frac{1}{m_{\infty}}-\frac{1}{m_{0}})-(\frac{1}{m_{0}}+\frac{1}{m_{\infty}}){\mathfrak{z}}\end{array}$

Further, (28) and (ii) of (21) together imply (iii) of (21). In summary, we
will get an admissible Ricci soliton solution as in (24) exactly when (26) and
(27) are satisfied and (28) has a solution $F({\mathfrak{z}})$ that satisfies
(i) and (ii) of (21).

Notice that since we require $0<|r_{i}|<1$ and $r_{i}n_{i}>0$, (27) has an
appropriate solution $(r_{1},r_{2})$ iff

$\frac{n_{1}}{m_{\infty}}<2,\quad\frac{n_{2}}{m_{\infty}}<2,\quad-\frac{n_{1}}{m_{0}}<2,\quad-\frac{n_{2}}{m_{0}}<2.$

This is exactly the Fano condition in Proposition 1.1. In turn, under this
condition, (26) and (27) correspond exactly to $c_{1}^{orb}(S_{\bf
n},\Delta_{\bf m})=\lambda\Omega_{{\bf r}}/2\pi$.

Similarly to the smooth case, we shall see that in the Fano case, there is
indeed always an admissible Ricci soliton in the appropriate Kähler class:

Using the integrating factor method, observe that

(29)
$\begin{array}[]{ccl}F({\mathfrak{z}})&=&e^{-c\,{\mathfrak{z}}}\int_{-1}^{{\mathfrak{z}}}e^{c\,t}\bigl{(}(\frac{1}{m_{\infty}}-\frac{1}{m_{0}})-(\frac{1}{m_{0}}+\frac{1}{m_{\infty}})t\bigr{)}p_{c}(t)dt\\\
\\\
&=&e^{-c\,{\mathfrak{z}}}\int_{-1}^{{\mathfrak{z}}}e^{c\,t}(t-t_{0})g(t)dt\end{array}$

solves (28) and (ii) of (21) iff $G(c)=0$, where

(30) $G(k)=e^{k\,t_{0}}\int_{-1}^{1}e^{k\,(t-t_{0})}(t-t_{0})g(t)dt,$

$t_{0}=\frac{m_{0}-m_{\infty}}{m_{0}+m_{\infty}}$, and
$g(t)=-(\frac{1}{m_{0}}+\frac{1}{m_{\infty}})p_{c}(t)$.

Note that $t_{0}\in(-1,1)$ and $g(t)<0$ for $t\in[-1,1]$. Thus
$e^{-kt_{0}}G(k)$ is a strictly decreasing function of $k$ tending to
$\mp\infty$ as $k\to\pm\infty$, and hence has a unique zero $c$ (consistent
with the uniqueness of Ricci solitons).

To check (i) of (21) we consider another auxiliary function

$h({\mathfrak{z}})=e^{c\,{\mathfrak{z}}}F({\mathfrak{z}})=\int_{-1}^{{\mathfrak{z}}}e^{c\,t}(t-t_{0})g(t)dt.$

Note that the sign of $h({\mathfrak{z}})$ equals the sign of
$F({\mathfrak{z}})$. In particular, $h(\pm 1)=0$, and (due to (iii) of (21))
$h$ is positive to the immediate right of ${\mathfrak{z}}=-1$ and immediate
left of ${\mathfrak{z}}=+1$. Now, since $h^{\prime}$ clearly has exactly one
zero (namely $t_{0}$) in $(-1,1)$, it is positive on $(-1,1)$. Therefore (i)
of (21) is also satisfied.

Combining the arguments above with Proposition 1.1 we conclude the following.

######  Proposition 4.7.

For $(S_{\bf n},\Delta_{\bf m})$ the following conditions are equivalent:

  * •

The inequalities
$\frac{n_{1}}{m_{\infty}}<2,\,\frac{n_{2}}{m_{\infty}}<2,\,-\frac{n_{1}}{m_{0}}<2,\,-\frac{n_{2}}{m_{0}}<2$
are satisfied;

  * •

$(S_{\bf n},\Delta_{\bf m})$ is log Fano;

  * •

There exist a Kähler-Ricci soliton on $(S_{\bf n},\Delta_{\bf m})$.

In this case, the Kähler-Ricci soliton $(g,\omega)$ is admissible and
satisfies $\rho={\mathcal{L}}_{V}\omega,$ with
$\lambda=\frac{1}{2m_{0}}+\frac{1}{2m_{\infty}}$ and
$V=(\frac{c}{2})\,grad_{g}{\mathfrak{z}}$ for a suitable real constant $c$.
The Kähler-Ricci soliton is Kähler-Einstein iff this $c$ is equal to zero.

Of course this can also be seen as an orbifold extension of Theorem 3.1 in
[ACGTF08a] (which in turn is essentially due to Koiso [Koi90]) in the case of
$S_{\bf n}={\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(n_{1},n_{2}))\hbox
to21.60004pt{\rightarrowfill}{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$.

######  Remark 4.8.

Note that, assuming we are in the Fano case, $c$ in the above proposition is a
non-zero multiple of the Futaki invariant, ${\mathcal{F}}(V)$, of
$c_{1}^{orb}(S_{\bf n},\Delta_{\bf m})$ applied to $V$. This underscores the
well-known fact that the existence of a Kähler-Ricci Soliton with respect to a
non-trivial holomorphic vector field is an obstruction to the existence of a
Kähler-Einstein metric.

####  4.4.1. Kähler-Einstein

Now we turn our attention to Kähler-Einstein examples on $(S_{\bf
n},\Delta_{\bf m})$. From the work above we see that under the assumption that
the log Fano condition and (27) hold, $\Omega_{\bf r}$ contains an admissible
Kähler-Einstein metric with
$\rho=(\frac{1}{2m_{0}}+\frac{1}{2m_{\infty}})\omega$ if and only if $G$
defined in (30) satisfies that $G(0)=0$, i.e.,

(31)
$\int_{-1}^{1}\bigl{(}(\frac{1}{m_{\infty}}-\frac{1}{m_{0}})-(\frac{1}{m_{0}}+\frac{1}{m_{\infty}})t\bigr{)}(1+r_{1}t)(1+r_{2}t)dt=0.$

Carrying out the integration in (31) and substituting (27), we have the
following

######  Proposition 4.9.

The Bott orbifold $(S_{\bf n},\Delta_{\bf m})$ admits a Kähler-Einstein metric
(which happens to be admissible) iff the following two conditions are
satisfied

  1. (1)

$(S_{\bf n},\Delta_{\bf m})$ is log Fano (i.e.,
$\frac{n_{1}}{m_{\infty}}<2,\,\frac{n_{2}}{m_{\infty}}<2,\,-\frac{n_{1}}{m_{0}}<2,\,-\frac{n_{2}}{m_{0}}<2$)

  2. (2)

$24(m_{0}^{3}m_{\infty}^{2}-m_{0}^{2}m_{\infty}^{3})-8(n_{1}+n_{2})(m_{0}^{3}m_{\infty}-m_{0}^{2}m_{\infty}^{2}+m_{0}m_{\infty}^{3})+3n_{1}n_{2}(m_{0}^{3}-m_{0}^{2}m_{\infty}+m_{0}m_{\infty}^{2}-m_{\infty}^{3})=0$

######  Example 4.1.

If we assume $n_{1}=1$ and $n_{2}=2$, then the equation in Proposition 4.9
rewrites to

$6(2m_{0}m_{\infty}-m_{0}+m_{\infty})(m_{0}^{2}(-1+2m_{\infty})-m_{\infty}^{2}(1+2m_{0}))=0.$

Clearly $2m_{0}m_{\infty}-m_{0}+m_{\infty}=0$ has no positive integer
solutions $(m_{0},m_{\infty})$. Likewise, assume by contradiction that
$(m_{0},m_{\infty})$ are positive integer solutions of

$m_{0}^{2}(-1+2m_{\infty})-m_{\infty}^{2}(1+2m_{0})=0,$

i.e.,

$m_{0}^{2}(-1+2m_{\infty})=m_{\infty}^{2}(1+2m_{0}).$

Since $gcd(2m_{i}\pm 1,m_{i})=1$ for $i=0,\infty$ we see that this would imply
that $m_{0}=m_{\infty}$ (since they would have to have the same prime
factorization). But $m_{0}=m_{\infty}$ will clearly never solve the equation.

######  Remark 4.10.

The diophantine nature of the equation in Proposition 4.9 makes it hard to
spot solutions other than the classic smooth Koiso-Sakane example (see Example
4.2 below). Further, Example 4.1 and Proposition 4.9 tells us that there exist
at least one pair ${\bf n}=(n_{1},n_{2})$ such that for all pairs ${\bf
m}=(m_{0},m_{\infty})\in{\mathbb{Z}}^{+}\times{\mathbb{Z}}^{+}$, the Bott
orbifold $(S_{\bf n},\Delta_{\bf m})$ admits no Kähler-Einstein metric. For
these reasons we will instead view a choice of
$(r_{1},r_{2})\in{\mathbb{Q}}^{2}$ such that $0<|r_{i}|<1$ as a pair of
parameters that determine a unique KE example on the appropriate corresponding
KS orbifold.

From this point of view, note that (31) is equivalent to the equation

(32)
$\frac{m_{\infty}}{m_{0}}=\frac{3+r_{1}r_{2}-r_{1}-r_{2}}{3+r_{1}r_{2}+r_{1}+r_{2}}.$

and, assuming (32), equation (27) is equivalent to

(33)
$\frac{n_{1}}{m_{0}}=\frac{2r_{1}(3+r_{1}r_{2}-r_{1}-r_{2})}{3+2r_{1}r_{2}+r_{1}^{2}},\quad\frac{n_{2}}{m_{0}}=\frac{2r_{2}(3+r_{1}r_{2}-r_{1}-r_{2})}{3+2r_{1}r_{2}+r_{2}^{2}}.$

We notice that if we pick a rational pair $(r_{1},r_{2})$ such that
$0<|r_{i}|<1$, then there is a unique quadruple of appropriate integers
$(n_{1},n_{2},m_{0},m_{\infty})$, solving (32) and (33), such that $n_{i}$ has
the same sign as $r_{i}$, $m_{i}>0$, and
$\gcd(|n_{1}|,|n_{2}|,m_{0},m_{\infty})=1$. This yields a KE example on the
corresponding KS orbifold $(S_{\bf n},\Delta_{\bf m})\hbox
to21.60004pt{\rightarrowfill}{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$
with fibers
${\mathbb{C}}{\mathbb{P}}^{1}[v_{0},v_{\infty})]/{\mathbb{Z}}_{m}$. Here we
have used our earlier notation; $m=\gcd(m_{0},m_{\infty})$ and ${\bf
m}=(m_{0},m_{\infty})=m{\bf v}=m(v_{0},v_{\infty})$.

######  Example 4.2.

Assume $r_{1}=r$ and $r_{2}=-r$ with $r\in(0,1)\cap{\mathbb{Q}}$. Then (32)
and (33) yields

$\frac{m_{\infty}}{m_{0}}=1,\quad\frac{n_{1}}{m_{0}}=2r,\quad\frac{n_{2}}{m_{0}}=-2r.$

When $r=1/2$ this yields the Koiso-Sakane smooth Kähler-Einstein metric on
${\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(1,-1))\rightarrow{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$
([KS86]). In general, we write $r=p/q$ in reduced form with co-prime integers
$0<p<q$. Then

$(n_{1},n_{2},m_{0},m_{\infty})=\left\\{\begin{array}[]{cl}(2p,-2p,q,q)&\text{if}\,\,q\,\,\text{is
odd}\\\ \\\ (p,-p,q/2,q/2)&\text{if}\,\,q\,\,\text{is even}\end{array}\right.$

This gives us Kähler-Einstein metrics on
${\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(2p,-2p))$ with fibers
${\mathbb{C}}{\mathbb{P}}^{1}/{\mathbb{Z}}_{q}$ and on
${\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(p,-p))$ with fibers
${\mathbb{C}}{\mathbb{P}}^{1}/{\mathbb{Z}}_{q/2}$, respectively. Using (4) we
see that the index is equal to $2$ if $q$ is odd and equal to $1$ if $q$ is
even.

Below we will explore more examples beyond the Koiso-Sakane examples. We set
$r_{i}=p_{i}/q_{i}$ with $q_{i}\in{\mathbb{Z}}^{\geq 2}$ and
$p_{i}\in{\mathbb{Z}}$ such that $0<|p_{i}|<q_{i}$ and
$\gcd(|p_{i}|,q_{i})=1$. Further, without loss we can assume that $r_{1}$,
hence $p_{1}$, is positive. We then observe that (32) and (33) are equivalent
to

(34)
$\begin{array}[]{ccl}\frac{m_{\infty}}{m_{0}}&=&\frac{3q_{1}q_{2}+p_{1}p_{2}-p_{1}q_{2}-p_{2}q_{1}}{3q_{1}q_{2}+p_{1}p_{2}+p_{1}q_{2}+p_{2}q_{1}}\\\
\\\
\frac{n_{1}}{m_{0}}&=&\frac{2p_{1}(3q_{1}q_{2}+p_{1}p_{2}-p_{1}q_{2}-p_{2}q_{1})}{3q_{1}^{2}q_{2}+2p_{1}p_{2}q_{1}+p_{1}^{2}q_{2}}\\\
\\\
\frac{n_{2}}{m_{0}}&=&\frac{2p_{2}(3q_{1}q_{2}+p_{1}p_{2}-p_{1}q_{2}-p_{2}q_{1})}{3q_{1}q_{2}^{2}+2p_{1}p_{2}q_{2}+p_{2}^{2}q_{1}}.\end{array}$

Note that, due to $0<|p_{i}|<q_{i}$, the denominator of each the right hand
fractions in (34) are all positive. Further, the numerator of the first two
right hand fractions are positive while the numerator of the last right hand
fraction has the same sign as $p_{2}$. Thus if we set

$k=\gcd\left(\begin{array}[]{r}(3q_{1}q_{2}+p_{1}p_{2}+p_{1}q_{2}+p_{2}q_{1})\left(3q_{1}^{2}q_{2}+2p_{1}p_{2}q_{1}+p_{1}^{2}q_{2}\right)\left(3q_{1}q_{2}^{2}+2p_{1}p_{2}q_{2}+p_{2}^{2}q_{1}\right),\\\
(3q_{1}q_{2}+p_{1}p_{2}-p_{1}q_{2}-p_{2}q_{1})\left(3q_{1}^{2}q_{2}+2p_{1}p_{2}q_{1}+p_{1}^{2}q_{2}\right)\left(3q_{1}q_{2}^{2}+2p_{1}p_{2}q_{2}+p_{2}^{2}q_{1}\right),\\\
2p_{1}(3q_{1}q_{2}+p_{1}p_{2}-p_{1}q_{2}-p_{2}q_{1})(3q_{1}q_{2}+p_{1}p_{2}+p_{1}q_{2}+p_{2}q_{1})\left(3q_{1}q_{2}^{2}+2p_{1}p_{2}q_{2}+p_{2}^{2}q_{1}\right),\\\
2|p_{2}|(3q_{1}q_{2}+p_{1}p_{2}-p_{1}q_{2}-p_{2}q_{1})(3q_{1}q_{2}+p_{1}p_{2}+p_{1}q_{2}+p_{2}q_{1})\left(3q_{1}^{2}q_{2}+2p_{1}p_{2}q_{1}+p_{1}^{2}q_{2}\right)\end{array}\right)$

we have an appropriate solution $(n_{1},n_{2},m_{0},m_{\infty})$ to (34) given
by

(35)
$\begin{array}[]{ccl}n_{1}&=&\frac{2p_{1}(3q_{1}q_{2}+p_{1}p_{2}-p_{1}q_{2}-p_{2}q_{1})(3q_{1}q_{2}+p_{1}p_{2}+p_{1}q_{2}+p_{2}q_{1})\left(3q_{1}q_{2}^{2}+2p_{1}p_{2}q_{2}+p_{2}^{2}q_{1}\right)}{k}\\\
n_{2}&=&\frac{2p_{2}(3q_{1}q_{2}+p_{1}p_{2}-p_{1}q_{2}-p_{2}q_{1})(3q_{1}q_{2}+p_{1}p_{2}+p_{1}q_{2}+p_{2}q_{1})\left(3q_{1}^{2}q_{2}+2p_{1}p_{2}q_{1}+p_{1}^{2}q_{2}\right)}{k}\\\
m_{0}&=&\frac{(3q_{1}q_{2}+p_{1}p_{2}+p_{1}q_{2}+p_{2}q_{1})\left(3q_{1}^{2}q_{2}+2p_{1}p_{2}q_{1}+p_{1}^{2}q_{2}\right)\left(3q_{1}q_{2}^{2}+2p_{1}p_{2}q_{2}+p_{2}^{2}q_{1}\right)}{k}\\\
m_{\infty}&=&\frac{(3q_{1}q_{2}+p_{1}p_{2}-p_{1}q_{2}-p_{2}q_{1})\left(3q_{1}^{2}q_{2}+2p_{1}p_{2}q_{1}+p_{1}^{2}q_{2}\right)\left(3q_{1}q_{2}^{2}+2p_{1}p_{2}q_{2}+p_{2}^{2}q_{1}\right)}{k}.\end{array}$

Each of these solutions then yields a KE example on the KS orbifold $(S_{\bf
n},\Delta_{\bf
m})=({\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(n_{1},n_{2})),\Delta_{\bf m})$ with fibers
${\mathbb{C}}{\mathbb{P}}^{1}[m_{0}/\gcd(m_{0},m_{\infty}),m_{\infty}/\gcd(m_{0},m_{\infty})]/{\mathbb{Z}}_{\gcd(m_{0},m_{\infty})}$.

######  Proposition 4.11.

There exists a four-parameter family of KS orbifolds with KE orbifold metrics.
The parameters $(p_{1},p_{2},q_{1},q_{2})$ are integers such that
$0<p_{1}<q_{1}$, $0<|p_{2}|<q_{2}$, and $\gcd(|p_{i}|,q_{i})=1$.

######  Remark 4.12.

One might ask the following question. Suppose we have a co-prime quadruple
$(n_{1},n_{2},m_{0},m_{\infty})$ satisfying the conditions in Proposition 4.9,
that is, $(S_{\bf n},\Delta_{\bf m})$ admits a KE metric with ${\bf
n}=(n_{1},n_{2})$ and ${\bf m}=(m_{0},m_{\infty})$. Fixing ${\bf
n}=(n_{1},n_{2})$, does there exist another pair $\tilde{{\bf
m}}=(\tilde{m}^{0},\tilde{m}^{\infty})$ with $\tilde{{\bf m}}\neq{\bf m}$,
such that $(n_{1},n_{2},\tilde{m}^{0},\tilde{m}^{\infty})$ is coprime and also
satisfies the conditions in Proposition 4.9. In other words, if ${\bf n}$ has
an appropriate choice of ${\bf m}$ such that $(S_{\bf n},\Delta_{\bf m})$
admits a KE metric, is this ${\bf m}$ then unique?

######  Example 4.3.

If we pick $p_{1}=p_{2}=1$, $q_{1}=2$, and $q_{2}=q>2$, then (35) becomes

$\begin{array}[]{ccccl}n_{1}&=&\frac{4(5q-1)(7q+3)\left(3q^{2}+q+1\right)}{k}&=&\frac{2(5q-1)(7q+3)\left(3q^{2}+q+1\right)}{\hat{k}}\\\
n_{2}&=&\frac{2(5q-1)(7q+3)\left(13q+4\right)}{k}&=&\frac{(5q-1)(7q+3)\left(13q+4\right)}{\hat{k}}\\\
m_{0}&=&\frac{2(7q+3)\left(13q+4\right)\left(3q^{2}+q+1\right)}{k}&=&\frac{(7q+3)\left(13q+4\right)\left(3q^{2}+q+1\right)}{\hat{k}}\\\
m_{\infty}&=&\frac{2(5q-1)\left(13q+4\right)\left(3q^{2}+q+1\right)}{k}&=&\frac{(5q-1)\left(13q+4\right)\left(3q^{2}+q+1\right)}{\hat{k}},\end{array}$

where

$\hat{k}=\gcd\left(\begin{array}[]{r}2(5q-1)(7q+3)\left(3q^{2}+q+1\right),\\\
(5q-1)(7q+3)\left(13q+4\right),\\\
(7q+3)\left(13q+4\right)\left(3q^{2}+q+1\right),\\\
(5q-1)\left(13q+4\right)\left(3q^{2}+q+1\right)\end{array}\right).$

Likewise, if we pick $p_{1}=1$, $p_{2}=-1$, $q_{1}=2$, and $q_{2}=q>2$, then
(35) becomes

$\begin{array}[]{ccccl}n_{1}&=&\frac{4(5q+1)(7q-3)\left(3q^{2}-q+1\right)}{k}&=&\frac{2(5q+1)(7q-3)\left(3q^{2}-q+1\right)}{\hat{k}}\\\
n_{2}&=&\frac{-2(5q+1)(7q-3)\left(13q-4\right)}{k}&=&\frac{-(5q+1)(7q-3)\left(13q-4\right)}{\hat{k}}\\\
m_{0}&=&\frac{2(7q-3)\left(13q-4\right)\left(3q^{2}-q+1\right)}{k}&=&\frac{(7q-3)\left(13q-4\right)\left(3q^{2}-q+1\right)}{\hat{k}}\\\
m_{\infty}&=&\frac{2(5q+1)\left(13q-4\right)\left(3q^{2}-q+1\right)}{k}&=&\frac{(5q+1)\left(13q-4\right)\left(3q^{2}-q+1\right)}{\hat{k}},\end{array}$

where

$\hat{k}=\gcd\left(\begin{array}[]{r}2(5q+1)(7q-3)\left(3q^{2}-q+1\right),\\\
(5q+1)(7q-3)\left(13q-4\right),\\\
(7q-3)\left(13q-4\right)\left(3q^{2}-q+1\right),\\\
(5q+1)\left(13q-4\right)\left(3q^{2}-q+1\right)\end{array}\right).$

The appendix contains a table representing a sample family of these solutions
as well as the classic smooth Koiso-Sakane example from Example 4.2. Using
(4), we calculated the (orbifold) index ${\mathcal{I}}_{{\bf n},{\bf m}}$ of
each $(S_{\bf n},\Delta_{\bf m})$.

###  4.5. Extremal and CSC Kähler metrics

More generally, the admissible metrics are extremal, as defined by Calabi
[Cal82], if and only if the scalar curvature of $g$, which is given as a
function of ${\mathfrak{z}}$ by

(36)
$Scal(g)=\frac{2s_{1}r_{1}}{1+r_{1}{\mathfrak{z}}}+\frac{2s_{2}r_{2}}{1+r_{2}{\mathfrak{z}}}-\frac{F^{\prime\prime}({\mathfrak{z}})}{p_{c}({\mathfrak{z}})},$

is a holomorphic potential, i.e., a linear affine function of
${\mathfrak{z}}$. Following the arguments in [ACGTF08a] and considering the
orbifold case at hand, it is easy to see that the proof of Proposition 11
together with Section 2.2 in [ACGTF08a] adapts to give us the following
result.

######  Proposition 4.13.

Any Kähler class $\Omega_{\bf r}$ on $(S_{\bf n},\Delta_{\bf m})$ admits an
admissible extremal metric with scalar curvature equal to an affine linear
function of ${\mathfrak{z}}$. Moreover, this metric is (positive) CSC (i.e.,
the function is constant) if and only if
$\alpha_{0}\beta_{1}-\alpha_{1}\beta_{0}=0$, where

(37) $\begin{split}\alpha_{r}=&\int_{-1}^{1}t^{r}p_{c}(t)dt\\\
\beta_{r}=&\int_{-1}^{1}\Big{(}r_{1}s_{1}(1+r_{2}t)+r_{2}s_{2}(1+r_{1}t)\Big{)}t^{r}dt\\\
&+(-1)^{r}p_{c}(-1)/m_{\infty}+p_{c}(1)/m_{0}.\end{split}$

In the smooth case, this has been thoroughly explored in [Gua95], [Hwa94], and
[HS02], as well as Section 3.4 of [ACGTF08a]. Here we will just mention that
with $s_{1}=2/n_{1}$ and $s_{2}=2/n_{2}$, the equation
$\alpha_{0}\beta_{1}-\alpha_{1}\beta_{0}=0$ is equivalent to
$f(r_{1},r_{2})=0$ where

(38)
$\begin{split}f(r_{1},r_{2})=&9(m_{0}-m_{\infty})n_{1}n_{2}-6(m_{0}+m_{\infty})n_{1}n_{2}(r_{1}+r_{2})+6(m_{0}-m_{\infty})n_{1}n_{2}r_{1}r_{2}\\\
&+3n_{2}(4m_{0}m_{\infty}-n_{1}(m_{0}-m_{\infty}))r_{1}^{2}+3n_{1}(4m_{0}m_{\infty}-n_{2}(m_{0}-m_{\infty}))r_{2}^{2}\\\
&-(4m_{0}m_{\infty}(n_{1}+n_{2})-3(m_{0}-m_{\infty})n_{1}n_{2})r_{1}^{2}r_{2}^{2}.\end{split}$

######  Proposition 4.14.

For any value of $n_{1},n_{2}\neq 0$, there exist a choice of $m_{0}$ and
$m_{\infty}$ such that $(S_{\bf n},\Delta_{\bf m})$ admits Kähler classes
$\Omega_{\bf r}$ with admissible constant scalar curvature Kähler metrics.

###### Proof.

In the case of $n_{1}n_{2}<0$ we can choose $m_{0}=m_{\infty}$ and the
existence of CSC Kähler classes follows from [Hwa94] (see also [ACGTF08a] for
details using the present notation). In the case of $n_{1}n_{2}>0$, notice
that $f(0,0)=9(m_{0}-m_{\infty})n_{1}n_{2}$ while
$f(1,1)=8m_{\infty}(m_{0}(n_{1}+n_{2})-3n_{1}n_{2})$. So, if we choose $m_{0}$
and $m_{\infty}$ such that $m_{\infty}>m_{0}>\frac{3n_{1}n_{2}}{n_{1}+n_{2}}$,
we have $f(0,0)<0$ and $f(1,1)>0$. This means that any continuous curve going
from $(0,0)$ to $(1,1)$ in the square $0<r_{1},r_{2}<1$ must contain at least
one point $(r_{1},r_{2})$ where $f$ vanishes. This ensures the existence of
solutions $0<r_{1},r_{2}<1$ to the equation $f(r_{1},r_{2})=0$. In turn, this
implies the existence of classes $\Omega_{\bf r}$ with admissible constant
scalar curvature Kähler metrics. ∎

######  Example 4.4.

When $m_{0}=m_{\infty}=1$, $n_{1}=1$, and $n_{2}=-1$, we have
$f(r_{1},r_{2})=-12(-1+r_{1}-r_{2})(r_{1}+r_{2})$, reconfirming the smooth CSC
Kähler metrics on
${\mathbb{P}}(\mathchoice{\scalebox{1.16}{$\displaystyle\mathbbold
1$}}{\scalebox{1.16}{$\textstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptstyle\mathbbold
1$}}{\scalebox{1.16}{$\scriptscriptstyle\mathbbold
1$}}\oplus{\mathcal{O}}(1,-1)\hbox
to21.60004pt{\rightarrowfill}{\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$
as stated in Theorem 9 of [ACGTF08a].

######  Remark 4.15.

Note that in general there is no guarantee the $CSC$ Kähler classes
$\Omega_{\bf r}$ from Proposition 4.14 are rational (i.e. both $r_{1}$ and
$r_{2}$ are rational) and thus useful from the Sasakian geometry point of
view. With the constraint that $r_{1},r_{2}$ are rational, the equation
$f(r_{1},r_{2})=0$ is diophantine in nature. Suppose $f(r_{1},r_{2})\neq 0$
for some rational class $\Omega_{\bf r}$ and assume we have a Boothby-Wang
constructed Sasaki manifold defined by an integer orbifold Kähler class
obtained by an appropriate rescale of $\Omega_{r}$. We shall see in the next
subsection that then we will have a CSC Sasaki metric somewhere else in the
Sasaki cone of this Sasaki maniifold. The key to seeing this is to use the
connection discovered by Apostolov and Calderbank in [AC21] between the so-
called weighted extremal metrics and extremal metrics appearing as transverse
structures for different rays in the same Sasaki cone.

###  4.6. Weighted Extremal Metrics

Let $(M,g,\omega)$ be a Kähler orbifold of complex dimension $m$, $f$ a
positive Killing potential on $M$, and (weight) $p\in{\mathbb{R}}$. Then the
$(f,p)$-Scalar curvature of $g$ is given by

(39) $Scal_{f,p}(g)=f^{2}Scal(g)-2(p-1)f\Delta_{g}f-p(p-1)|df|^{2}_{g},$

If $Scal_{f,p}(g)$ is a Killing potential, $g$ is said to be a
$(f,p)$-extremal Kähler metric. The case $p=2m$ has been studied by several
people and is interesting due to the fact that $Scal_{f,2m}(g)$ computes the
scalar curvature of the Hermitian metric $h=f^{-2}g$.

However, the case of interest to us here is when $p=m+2$. This case is related
to the study of extremal Sasaki metrics [AC21, ACL21, AJL21]. Indeed, if we
assume that the Kähler class $[\omega/2\pi]$ is an integer orbifold class
giving a Boothby-Wang constructed (smooth) Sasaki manifold over
$(M,g,\omega)$, then $\frac{Scal_{f,m+2}(g)}{f}$ is equal to the transverse
scalar curvature of a certain Sasaki structure (determined by $f$) in the
Sasaki cone. More precisely, if $\chi$ is the Reeb vector field of the Sasaki
structure coming directly from the Boothby-Wang construction over
$(M,g,\omega)$ and $f$ is viewed as a pull-back to the Sasaki manifold, then
(mod ${\mathcal{D}}$) $\xi:=f\chi$ is a Reeb vector field in the Sasaki cone
giving a new Sasaki structure. While the pull-back from $M$ of $Scal(g)$ is
the Tanaka-Webster scalar curvature of the Tanaka-Webster connection induced
by $\chi$, the expression $\frac{Scal_{f,m+2}(g)}{f}$ pulls back from $M$ to
be the Tanaka-Webster scalar curvature of the Tanaka-Webster connection
induced by $\xi$. The latter is then also identified with the transverse
scalar curvature of the Sasaki structure defined by $\xi$. This fact is seen
from the details of the proof of Lemma 3 in [AC21]. As also follows from
[AC21] (see their Theorem 1), the Sasaki structure determined by $f$ is
extremal if and only if $g$ is $(f,m+2)$-extremal.

Now, returning to our orbifolds at hand we have that $m=3$ and so $m+2=5$.
Note that $|{\mathfrak{z}}+b|$ for $b\in{\mathbb{R}}$ such that $|b|>1$
defines a Killing potential on $(S_{\bf n},\Delta_{\bf m})$.

It is not hard to check that Section 2 and the existence result in the second
half of Theorem 3.1. of [AMTF] adapts to get us the next proposition. Note in
particular, that whether we use $f={\mathfrak{z}}+b$ (when ($b>1$) or
$f=-({\mathfrak{z}}+b)$ (when $b<-1$), the formula for $Scal_{f,p}$ in (39)
will give us the same result, i.e., the right hand side of (19) in [AMTF]. In
general the assumption of $b>1$ in [AMTF] is merely practical and all the
arguments are easily adapted to include the $b<-1$ case as well. Further, the
root counting argument in the proof of Theorem 3.1.in [AMTF] is not affected
by our mild orbifold conditions.

######  Proposition 4.16.

Let $b\in{\mathbb{R}}$ such that $|b|>1$. Any Kähler class $\Omega_{\bf r}$ on
$(S_{\bf n},\Delta_{\bf m})$ admits a $(|{\mathfrak{z}}+b|,5)$-extremal Kähler
metric with weighted scalar curvature
$Scal_{|{\mathfrak{z}}+b|,5}=A_{1}{\mathfrak{z}}+A_{2}$ for constants
$A_{1},A_{2}$ given as the unique solutions of the following linear system

(40)
$\begin{array}[]{ccc}\alpha_{1,-6}A_{1}+\alpha_{0,-6}A_{2}=2\beta_{0,-4}\\\
\\\ \alpha_{2,-6}A_{1}+\alpha_{1,-6}A_{2}=2\beta_{1,-4},\end{array}$

where

(41) $\begin{split}\alpha_{r,-6}=&\int_{-1}^{1}(t+b)^{-6}t^{r}p_{c}(t)dt\\\
\beta_{r,-4}=&\int_{-1}^{1}\Big{(}\frac{2r_{1}}{n_{1}}(1+r_{2}t)+\frac{2r_{2}}{n_{2}}(1+r_{1}t)\Big{)}t^{r}(t+b)^{-4}dt\\\
&+(-1)^{r}(b-1)^{-4}p_{c}(-1)/m_{\infty}+(1+b)^{-4}p_{c}(1)/m_{0}.\end{split}$

Moreover, assuming that $\Omega_{\bf r}$ is rational and we can form a
Boothby-Wang constructed Sasaki manifold with respect to an appropriate
rescale of $\Omega_{\bf r}$, the extremal Sasaki structure determined by
$f=|{\mathfrak{z}}+b|$ has constant scalar curvature if and only if
$Scal_{|{\mathfrak{z}}+b|,5}=A_{1}{\mathfrak{z}}+A_{2}$ is a constant multiple
of $|{\mathfrak{z}}+b|$, i.e., if and only if $A_{1}b-A_{2}=0$.

Suppose a rational $\Omega_{\bf r}$ is given on a specific $(S_{\bf
n},\Delta_{\bf m})$. Solving the linear system for $A_{1}$ and $A_{2}$ in
Proposition 4.16 (note that $\alpha_{1,-6}^{2}-\alpha_{0,-6}\alpha_{2,-6}\neq
0$) and simplifying a bit, the equation $A_{1}b-A_{2}=0$ may be re-written as
$h(b)=0$ where

(42)
$h(b)=(b^{2}-1)^{7}(b(\alpha_{1,-6}\beta_{0,-4}-\alpha_{0,-6}\beta_{1,-4})-(\alpha_{1,-6}\beta_{1,-4}-\alpha_{2,-6}\beta_{0,-4})).$

Using (41) we see that $h(b)$ simplifies as a polynomial of degree $5$ in the
variable $b$ with leading coefficient equal to
$\frac{2f(r_{1},r_{2})}{9m_{0}m_{\infty}n_{1}n_{2}}$, where $f(r_{1},r_{2})$
is given by (38). Since further $\lim_{b\rightarrow\pm 1^{\mp}}h(b)>0$, we
conclude that unless $f(r_{1},r_{2})=0$ (in which case $\Omega_{\bf r}$ itself
has an admissible CSC representative), there must exist at least one value
$b\in(-\infty,-1)\cup(1,+\infty)$ such that $h(b)=0$, hence $A_{1}b-A_{2}=0$,
and therefore, for this $b$ value, the extremal Sasaki structure determined by
$f=|{\mathfrak{z}}+b|$ has constant scalar curvature. If $b$ is an irrational
number, then the corresponding Sasaki structure is irregular.

######  Theorem 4.17.

Suppose $\Omega_{\bf r}$ is a rational admissible Kähler class on a KS
orbifold of the form $(S_{\bf n},\Delta_{\bf m})$. Let ${\mathcal{S}}$ be a
Boothby-Wang constructed Sasaki manifold given by an appropriate rescale of
$\Omega_{\bf r}$. Then the corresponding Sasaki cone will always have a
(possibly irregular) CSC-ray (up to isotopy).

######  Remark 4.18.

It is not too difficult (using (36) and (21)) to produce some admissible
Kähler metric of positive scalar curvature in $\Omega_{\bf r}$. Using the
corresponding Kähler form of this metric for the Boothby-Wang construction, we
can use Lemma 5.2 in [BHL18]) to conclude that the CSC-ray alluded to in the
theorem above has positive transverse scalar curvature. This together with
Theorem 4.17 proves the main theorem in the introduction.

######  Remark 4.19.

The observations in this section - in particular, Proposition 4.16 and Theorem
4.17 \- can be generalized to no-blown-down admissible manifolds in general
[BHLTF]. Indeed, in [BHLTF] we use a more general version of Proposition 4.16
to answer the open question in Problem 6.1.2 of [BHLTF21] by providing a
counter example where the Sasaki cone contains some extremal Sasaki
structures, but no CSC Sasaki metric at all.

######  Example 4.5.

Suppose $m_{1}=m_{2}=1$ and suppose $n_{1}>4$. Now let
$r_{1}=\frac{5n_{1}^{2}-4}{n_{1}\left(n_{1}^{2}+4\right)}$, $r_{2}=2/n_{1}$,
and $n_{2}$ be equal to any positive integer. Note that since $n_{1}>4$ we
have ensured that $r_{i}\in(0,1)$ for $i=1,2$ and $r_{1}\neq r_{2}$. Then
$\Omega_{\bf r}$ is a rational (non-diagonally) admissible Kähler class on the
smooth KS manifold $(S_{\bf n})$. Note that since $\Omega_{\bf r}$ is non-
diagonally admissible, we know from Theorem 4.3 that $S_{\bf n}$ (with any
appropriate rescale of $\Omega_{\bf r}$) is not the quotient of a regular ray
in the ${\bf w}$-cone of the type of $S^{3}_{\bf w}$-join described in Theorem
4.3. Further, it can be shown (as we will do near the end of this example)
that $S_{\bf n}$ (with any appropriate rescale of $\Omega_{\bf r}$) is also
not the quotient of the regular ray in the ${\mathfrak{t}}^{+}_{sphr}$ cone of
a Yamazaki fiber join over
${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$. The Boothby-
Wang constructed Sasaki manifold over $S_{\bf n}$, given by an appropriate
rescale of $\Omega_{\bf r}$ is thus in a certain sense a truly new example.

By Theorem 8 and Proposition 11 of [ACGTF08a] we know that the Kähler class
$\Omega_{\bf r}$, while having an extremal admissible Kähler metric
representative, does not admit a CSC Kähler metric. [This fact can also easily
be checked by seeing that $f(r_{1},r_{2})$ from (38) cannot be zero in this
case.] We now calculate that with the given choices of $({\bf m},{\bf n},{\bf
r})$, we have that $h(b)$ from (42) is equal to

$h(b)=\frac{4(n_{1}-2b)p(b)}{9n_{1}^{5}\left(n_{1}^{2}+4\right)^{2}n_{2}}$

where $p(b)$ is given by

$\begin{array}[]{ccl}p(b)&=&3n_{1}^{8}n_{2}-4n_{1}^{7}-13n_{1}^{6}n_{2}+268n_{1}^{5}+1252n_{1}^{4}n_{2}-544n_{1}^{3}-1456n_{1}^{2}n_{2}+192n_{1}+192n_{2}\\\
\\\
&-&4n_{1}\left(9n_{1}^{6}n_{2}+20n_{1}^{5}+482n_{1}^{4}n_{2}+64n_{1}^{3}-464n_{1}^{2}n_{2}-64n_{1}+224n_{2}\right)b\\\
\\\
&+&16\left(n_{1}^{7}+79n_{1}^{6}n_{2}-17n_{1}^{5}-35n_{1}^{4}n_{2}+56n_{1}^{3}+32n_{1}^{2}n_{2}-16n_{1}-16n_{2}\right)b^{2}\\\
\\\
&-&4n_{1}\left(63n_{1}^{6}n_{2}-20n_{1}^{5}+106n_{1}^{4}n_{2}-64n_{1}^{3}-16n_{1}^{2}n_{2}+64n_{1}-32n_{2}\right)b^{3}\\\
\\\
&+&(21n_{1}^{8}n_{2}-12n_{1}^{7}+21n_{1}^{6}n_{2}+4n_{1}^{5}+268n_{1}^{4}n_{2}-352n_{1}^{3}-208n_{1}^{2}n_{2}+64n_{1}+64n_{2})b^{4}.\end{array}$

Clearly, $b=n_{1}/2$ is a rational root of $h(b)$. Note that since $n_{1}>4$
and $n_{2}>0$, we also have that

  * •

The $b^{4}$ coefficient of $p(b)$ is positive

  * •

$p(\pm 1)>0$

  * •

$p^{\prime}(-1)<0$ and $p^{\prime}(1)>0$

  * •

$f(b):=p^{\prime\prime}(b)$ is a concave up 2nd order polynomial with $f(\pm
1)>0$, $f^{\prime}(-1)<0$, and $f^{\prime}(1)>0$. Thus any roots of $f(b)$
would be inside the interval $(-1,1)$.

Putting this together we have that $p(b)$ is positive and decreasing at
$b=-1$, positive and increasing at $b=1$, and concave up for $|b|\geq 1$.
Hence $p(b)$ has no roots in $(-\infty,-1)\cup(1,+\infty)$ and hence
$b=n_{1}/2$ is the ONLY root of $h(b)$.

All in all we conclude that while the original regular ray coming from the
Boothby-Wang constructed Sasaki manifold over $S_{\bf n}$, given by an
appropriate rescale of $\Omega_{\bf r}$ is not CSC, the CSC ray alluded to in
Theorem 4.17 is quasi-regular. It would be interesting to explore what the
transverse Kähler orbifold is for this ray. Conjecturely, it may be a KS
orbifold.

Finally, let us finish this example by confirming that $S_{\bf n}$ (with any
appropriate rescale of $\Omega_{\bf r}$) is also not the quotient of the
regular ray in the ${\mathfrak{t}}^{+}_{sphr}$ cone of a Yamazaki fiber join
over ${\mathbb{C}}{\mathbb{P}}^{1}\times{\mathbb{C}}{\mathbb{P}}^{1}$:

For $S_{\bf n}$ (with an appropriate rescale of $\Omega_{\bf r}$) to be such a
quotient, it follows from Section 4.3 and (4.3) that there should exist
$k_{1}^{1},k_{1}^{2},k_{2}^{1},k_{2}^{2}\in{\mathbb{Z}}^{+}$ such that

$k_{1}^{1}-k_{2}^{1}=n_{1},\quad
k_{1}^{2}-k_{2}^{2}=n_{2},\quad\frac{k_{1}^{1}-k_{2}^{1}}{k_{1}^{1}+k_{2}^{1}}=\frac{5n_{1}^{2}-4}{n_{1}\left(n_{1}^{2}+4\right)},\quad\frac{k_{1}^{2}-k_{2}^{2}}{k_{1}^{2}+k_{2}^{2}}=2/n_{1}.$

This would place further restrictions on $n_{1}$ and $n_{2}$ and, in
particular, we would need $n_{1}$ to satisfy that
$\frac{n_{1}^{2}\left(n_{1}^{2}+4\right)}{5n_{1}^{2}-4}$ is a positive integer
(that positive integer is $k_{1}^{1}+k_{2}^{1}$). By the following elementary
lemma this is not possible and so $S_{\bf n}$ (with any appropriate rescale of
$\Omega_{\bf r}$) is not the quotient of the regular ray in the
${\mathfrak{t}}^{+}_{sphr}$ cone of a Yamazaki fiber join.

######  Lemma 4.20.

For all integers $x\geq 3$, $\frac{x^{2}\left(x^{2}+4\right)}{5x^{2}-4}$ is
NOT an integer.

###### Proof.

Assume for contradiction that there exists an integer $x\geq 3$ such that
$\frac{x^{2}\left(x^{2}+4\right)}{5x^{2}-4}$ is an integer. Let $d>1$ be a
prime divisor of $5x^{2}-4$. Then, due to our assumption, $d\mid x^{2}$ or
$d\mid(x^{2}+4)$. If $d\mid x^{2}$, then, since also $d\mid(5x^{2}-4)$, we
have that $d\mid 4$ and so $d=2$. If $d\mid(x^{2}+4)$, then, since again
$d\mid(5x^{2}-4)$, we have that $d\mid(5(x^{2}+4)-(5x^{2}-4))$, i.e., $d\mid
24$, and so $d=2$ or $d=3$.

By considering the three possible cases $x=3k$, $x=3k+1$, and $x=3k+2$ with
$k\in{\mathbb{Z}}$, we realize that $3\nmid(5x^{2}-4)$. Therefore, $d=2$ and
$5x^{2}-4$ is even. This implies that $x^{2}$ is even and hence $x$ is even.
In particular, $x\neq 3$ and moving forward we may assume $x\geq 4$.

Since $5x^{2}-4$ is even, we have that $(5x^{2}-4)=2^{i}$ for some non-
negative integer $i$. We also know that $5x^{2}-4\geq 76$ (since $x\geq 4$),
so $i>6$.

Now, writing $x=2k$ for some $k\in{\mathbb{Z}}^{+}$ we see that
$(5x^{2}-4)=2^{i}$ implies that $20k^{2}-4=2^{i}$ and hence
$5k^{2}=2^{i-2}+1$. Thus $5k^{2}$ is odd, which gives us that $k^{2}$ is odd
and hence $k$ is odd. Let us write $k=2l+1$ for some $l\in{\mathbb{Z}}^{+}$.
So $x=2(2l+1)$. Then

$\begin{array}[]{ccl}x^{2}\left(x^{2}+4\right)&=&2^{2}(4l^{2}+4l+1)(2^{2}(4l^{2}+4l+1)+4)\\\
\\\ &=&2^{2}(4l^{2}+4l+1)(2^{2}(4l^{2}+4l+2))\\\ \\\
&=&2^{2}(4l^{2}+4l+1)(2^{3}(2l^{2}+2l+1))\\\ \\\
&=&2^{5}(2(2l^{2}+2l)+1)(2(l^{2}+l)+1).\end{array}$

Since $(2(2l^{2}+2l)+1)(2(l^{2}+l)+1)$ is odd, we realize that
$\frac{x^{2}\left(x^{2}+4\right)}{5x^{2}-4}=\frac{(2(2l^{2}+2l)+1)(2(l^{2}+l)+1)}{2^{i-5}}$.
Since $i>6$, this cannot possible be an integer and hence we have arrived at a
contraction. This completes the proof of the lemma. ∎

As an immediate consequence of Theorem 4.17 we get the following special case
of the existence result for toric Sasaki-Einstein metrics due to Futaki, Ono,
and Wang [FOW09].

######  Corollary 4.21.

Suppose a KS orbifold of the form $(S_{\bf n},\Delta_{\bf m})$ is log Fano and
consider a Boothby-Wang constructed Sasaki manifold over $(S_{\bf
n},\Delta_{\bf m})$ given by some Kähler form representing $c_{1}^{orb}(S_{\bf
n},\Delta_{\bf m})/{\mathcal{I}}_{{\bf n},{\bf m}}$. Then the corresponding
Sasaki cone will always have a (possibly irregular) Sasaki-Einstein structure
(up to isotopy).

###### Proof.

First, we note that $c_{1}({\mathcal{D}})=0$ and so the CSC Sasaki metrics
from Theorem 4.17 above are now $\eta$-Einstein. Second, we see from
Proposition 5.3 in [BHL18]) that since the basic first Chern class of the
initial Sasaki metric is positive (as a pullback of the positive class
$c_{1}^{orb}(S_{\bf n},\Delta_{\bf m})/{\mathcal{I}}_{{\bf n},{\bf m}}$), the
average transverse scalar curvature of any Sasaki structure in the Sasaki cone
must be positive. In particular, the transverse (constant) scalar curvature of
any $\eta$-Einstein structure in the cone must be positive. This means that
any $\eta$-Einstein ray admits a Sasaki-Einstein structure. ∎

######  Remark 4.22.

Examples similar in spirit to Corollary 4.21, but also including non-toric
cases have been given by Mabuchi and Nakagawa [MN13]. See also [AJL21] and
references therein.

## Appendix

$p_{1}$ | $q_{1}$ | $p_{2}$ | $q_{2}$ | $n_{1}$ | $n_{2}$ | $m_{0}$ | $m_{\infty}$ | $m$ | $v_{0}$ | $v_{\infty}$ | ${\mathcal{I}}_{{\bf n},{\bf m}}$
---|---|---|---|---|---|---|---|---|---|---|---
1 | 2 | -1 | 15 | 5124072 | -740316 | 6438801 | 4797538 | 126251 | 51 | 38 | 89
1 | 2 | -1 | 14 | 775675 | -120061 | 972325 | 726685 | 10235 | 95 | 71 | 83
1 | 2 | -1 | 13 | 48 | -8 | 60 | 45 | 15 | 4 | 3 | 7
1 | 2 | -1 | 12 | 2080161 | -375516 | 2591676 | 1951756 | 31996 | 81 | 61 | 71
1 | 2 | -1 | 11 | 1462832 | -288008 | 1815479 | 1373876 | 49067 | 37 | 28 | 65
1 | 2 | -1 | 10 | 110483 | -23919 | 136479 | 103887 | 2037 | 67 | 51 | 59
1 | 2 | -1 | 9 | 129720 | -31188 | 159330 | 122153 | 5311 | 30 | 23 | 53
1 | 2 | -1 | 8 | 80401 | -21730 | 98050 | 75850 | 1850 | 53 | 41 | 47
1 | 2 | -1 | 7 | 25944 | -8004 | 31349 | 24534 | 1363 | 23 | 18 | 41
1 | 2 | -1 | 6 | 124527 | -44733 | 148629 | 118141 | 3811 | 39 | 31 | 35
1 | 2 | -1 | 5 | 59072 | -25376 | 69296 | 56303 | 4331 | 16 | 13 | 29
1 | 2 | -1 | 4 | 525 | -280 | 600 | 504 | 24 | 25 | 21 | 23
1 | 2 | -1 | 3 | 1440 | -1008 | 1575 | 1400 | 175 | 9 | 8 | 17
1 | 2 | -1 | 2 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1
1 | 2 | 1 | 2 | 51 | 51 | 85 | 45 | 5 | 17 | 9 | 13
1 | 2 | 1 | 3 | 10416 | 7224 | 15996 | 9331 | 1333 | 12 | 7 | 19
1 | 2 | 1 | 4 | 31217 | 16492 | 46004 | 28196 | 1484 | 31 | 19 | 25
1 | 2 | 1 | 5 | 8208 | 3496 | 11799 | 7452 | 621 | 19 | 12 | 31
1 | 2 | 1 | 6 | 30015 | 10701 | 42435 | 27347 | 943 | 45 | 29 | 37
1 | 2 | 1 | 7 | 54808 | 16796 | 76570 | 50065 | 2945 | 26 | 17 | 43
1 | 2 | 1 | 8 | 51389 | 13806 | 71154 | 47034 | 1206 | 59 | 39 | 49
1 | 2 | 1 | 9 | 552 | 132 | 759 | 506 | 253 | 3 | 2 | 5
1 | 2 | 1 | 10 | 1112447 | 239659 | 1521101 | 1021013 | 20837 | 73 | 49 | 61
1 | 2 | 1 | 11 | 36000 | 7056 | 49000 | 33075 | 1225 | 40 | 27 | 67
1 | 2 | 1 | 12 | 456837 | 82128 | 619440 | 420080 | 7120 | 87 | 59 | 73
1 | 2 | 1 | 13 | 3134336 | 520384 | 4236251 | 2884256 | 90133 | 47 | 32 | 79
1 | 2 | 1 | 14 | 466923 | 72013 | 629331 | 429939 | 6231 | 101 | 69 | 85
1 | 2 | 1 | 15 | 5522472 | 795204 | 7425486 | 5087833 | 137509 | 54 | 37 | 91

TABLE: This gives a sample of Kähler-Einstein orbifold solutions. See Example
4.3.

## References

  * [AC21] Vestislav Apostolov and David M. J. Calderbank, _The CR geometry of weighted extremal Kähler and Sasaki metrics_ , Math. Ann. 379 (2021), no. 3-4, 1047–1088. MR 4238260
  * [ACC+21] Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Kento Fujita, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Süss, and Nivedita Viswanathan, _The Calabi problem for Fano threefolds_ , 2021\.
  * [ACGTF08a] Vestislav Apostolov, David M. J. Calderbank, Paul Gauduchon, and Christina W. Tønnesen-Friedman, _Hamiltonian 2-forms in Kähler geometry. III. Extremal metrics and stability_ , Invent. Math. 173 (2008), no. 3, 547–601. MR MR2425136 (2009m:32043)
  * [ACGTF08b] by same author, _Hamiltonian 2-forms in Kähler geometry. IV. Weakly Bochner-flat Kähler manifolds_ , Comm. Anal. Geom. 16 (2008), no. 1, 91–126. MR 2411469 (2010c:32043)
  * [ACL21] Vestislav Apostolov, David M. J. Calderbank, and Eveline Legendre, _Weighted K-stability of polarized varieties and extremality of Sasaki manifolds_ , Adv. Math. 391 (2021), Paper No. 107969, 63. MR 4300921
  * [AJL21] Vestislav Apostolov, Simon Jubert, and Abdellah Lahdili, _Weighted $K$-Stability and coercivity with applications to Extremal Kähler and Sasaki Metrics_, arXiv: 2104.09709 (2021).
  * [AMTF] Vestislav Apostolov, Gideon Maschler, and Christina W. Tønnesen-Friedman, _Weighted Extremal Kähler metrics and the Einstein-Maxwell geometry of Projective Bundles_ , arXiv:1808.02813 (to appear in Comm. Anal. Geom.).
  * [AP06] Marcelo A. Aguilar and Carlos Prieto, _Transfers for ramified covering maps in homology and cohomology_ , Int. J. Math. Math. Sci. (2006), Art. ID 94651, 28. MR 2251716
  * [BCTF19] Charles P. Boyer, David M. J. Calderbank, and Christina W. Tønnesen-Friedman, _The Kähler geometry of Bott manifolds_ , Adv. Math. 350 (2019), 1–62. MR 3945589
  * [BFGK91] H. Baum, T. Friedrich, R. Grunewald, and I. Kath, _Twistors and Killing spinors on Riemannian manifolds_ , Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 124, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1991, With German, French and Russian summaries. MR 94a:53077
  * [BG08] Charles P. Boyer and Krzysztof Galicki, _Sasakian geometry_ , Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008. MR MR2382957 (2009c:53058)
  * [BHL18] Charles Boyer, Hongnian Huang, and Eveline Legendre, _An application of the Duistermaat–Heckman theorem and its extensions in Sasaki geometry_ , Geom. Topol. 22 (2018), no. 7, 4205–4234. MR 3890775
  * [BHLTF] Charles P. Boyer, Hongnian Huang, Eveline Legendre, and Christina W. Tønnesen-Friedman, _Work in progress_.
  * [BHLTF18] Charles P. Boyer, Hongnian Huang, Eveline Legendre, and Christina W. Tønnesen-Friedman, _Reducibility in Sasakian geometry_ , Trans. Amer. Math. Soc. 370 (2018), no. 10, 6825–6869. MR 3841834
  * [BHLTF21] by same author, _Some open problems in Sasaki geometry_ , Differential Geometry in the Large, Cambridge University Press, Cambridge, UK, 2021, pp. 143–168.
  * [BTF] Charles P. Boyer and Christina W. Tønnesen-Friedman, _Iterated ${S}^{3}$ Sasaki joins and Bott orbifolds_, arXiv:2006.06596, to appear in Annales Scientifiques de la Faculté de Toulouse.
  * [BTF16] by same author, _The Sasaki join, Hamiltonian 2-forms, and constant scalar curvature_ , J. Geom. Anal. 26 (2016), no. 2, 1023–1060. MR 3472828
  * [BTF19] by same author, _Sasaki-Einstein metrics on a class of 7-manifolds_ , J. Geom. Phys. 140 (2019), 111–124. MR 3923473
  * [BTF21a] by same author, _The ${S}^{3}$ Sasaki join construction_, arXiv:1911.11031, to appear in J. Math. Soc. Japan (2021).
  * [BTF21b] by same author, _Sasakian geometry on sphere bundles_ , Differential Geom. Appl. 77 (2021), 101765, 37. MR 4253896
  * [BW58] W. M. Boothby and H. C. Wang, _On contact manifolds_ , Ann. of Math. (2) 68 (1958), 721–734. MR 22 #3015
  * [Cal82] E. Calabi, _Extremal Kähler metrics_ , Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259–290. MR 83i:53088
  * [CC21a] Xiuxiong Chen and Jingrui Cheng, _On the constant scalar curvature Kähler metrics (I)—A priori estimates_ , J. Amer. Math. Soc. 34 (2021), no. 4, 909–936. MR 4301557
  * [CC21b] by same author, _On the constant scalar curvature Kähler metrics (II)—Existence results_ , J. Amer. Math. Soc. 34 (2021), no. 4, 937–1009. MR 4301558
  * [CDS15a] Xiuxiong Chen, Simon Donaldson, and Song Sun, _Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities_ , J. Amer. Math. Soc. 28 (2015), no. 1, 183–197. MR 3264766
  * [CDS15b] by same author, _Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than $2\pi$_, J. Amer. Math. Soc. 28 (2015), no. 1, 199–234. MR 3264767
  * [CDS15c] by same author, _Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches $2\pi$ and completion of the main proof_, J. Amer. Math. Soc. 28 (2015), no. 1, 235–278. MR 3264768
  * [CS18] Tristan C. Collins and Gábor Székelyhidi, _K-semistability for irregular Sasakian manifolds_ , J. Differential Geom. 109 (2018), no. 1, 81–109. MR 3798716
  * [CS19] Tristan C. Collins and Gábor Székelyhidi, _Sasaki-Einstein metrics and K-stability_ , Geom. Topol. 23 (2019), no. 3, 1339–1413. MR 3956894
  * [Don08] S. K. Donaldson, _Extremal metrics on toric surfaces: a continuity method_ , J. Differential Geom. 79 (2008), no. 3, 389–432. MR MR2433928 (2009j:58018)
  * [DW11] Andrew S. Dancer and McKenzie Y. Wang, _On Ricci solitons of cohomogeneity one_ , Ann. Global Anal. Geom. 39 (2011), no. 3, 259–292. MR 2769300
  * [FOW09] Akito Futaki, Hajime Ono, and Guofang Wang, _Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds_ , J. Differential Geom. 83 (2009), no. 3, 585–635. MR MR2581358
  * [GK94] Michael Grossberg and Yael Karshon, _Bott towers, complete integrability, and the extended character of representations_ , Duke Math. J. 76 (1994), no. 1, 23–58. MR 1301185 (96i:22030)
  * [Gua95] Daniel Guan, _Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends_ , Trans. Amer. Math. Soc. 347 (1995), no. 6, 2255–2262. MR 1285992 (96a:58059)
  * [Hae84] A. Haefliger, _Groupoïdes d’holonomie et classifiants_ , Astérisque (1984), no. 116, 70–97, Transversal structure of foliations (Toulouse, 1982). MR 86c:57026a
  * [He18] Weiyong He, _Scalar curvature and properness on Sasaki manifolds_ , arXiv:1802.03841 (2018).
  * [HL21] Weiyong He and Jun Li, _Geometric pluripotential theory on Sasaki manifolds_ , J. Geom. Anal. 31 (2021), no. 2, 1093–1179. MR 4215260
  * [HS02] Andrew D. Hwang and Michael A. Singer, _A momentum construction for circle-invariant Kähler metrics_ , Trans. Amer. Math. Soc. 354 (2002), no. 6, 2285–2325 (electronic). MR 1885653 (2002m:53057)
  * [Hwa94] Andrew D. Hwang, _On existence of Kähler metrics with constant scalar curvature_ , Osaka J. Math. 31 (1994), no. 3, 561–595. MR 1309403 (96a:53061)
  * [KL21] Marc Kegel and Christian Lange, _A Boothby-Wang theorem for Besse contact manifolds_ , Arnold Math. J. 7 (2021), no. 2, 225–241. MR 4260074
  * [Koi90] Norihito Koiso, _On rotationally symmetric Hamilton’s equation for Kähler-Einstein metrics_ , Recent topics in differential and analytic geometry, Adv. Stud. Pure Math., vol. 18, Academic Press, Boston, MA, 1990, pp. 327–337. MR 1145263 (93d:53057)
  * [Kol04] J. Kollár, _Seifert $G_{m}$-bundles_, preprint; arXiv:math.AG/0404386 (2004).
  * [Kol07] by same author, _Einstein metrics on connected sums of $S^{2}\times S^{3}$_, J. Differential Geom. 77 (2007), no. 2, 259–272. MR 2269781
  * [KS86] Norihito Koiso and Yusuke Sakane, _Nonhomogeneous Kähler-Einstein metrics on compact complex manifolds_ , Curvature and topology of Riemannian manifolds (Katata, 1985), Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986, pp. 165–179. MR 859583 (88c:53047)
  * [Leg11] Eveline Legendre, _Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics_ , Compos. Math. 147 (2011), no. 5, 1613–1634. MR 2834736
  * [LTF13] Eveline Legendre and Christina W. Tønnesen-Friedman, _Toric generalized Kähler-Ricci solitons with Hamiltonian 2-form_ , Math. Z. 274 (2013), no. 3-4, 1177–1209. MR 3078263
  * [MN13] Toshiki Mabuchi and Yasuhiro Nakagawa, _New examples of Sasaki-Einstein manifolds_ , Tohoku Math. J. (2) 65 (2013), no. 2, 243–252. MR 3079287
  * [OW75] Peter Orlik and Philip Wagreich, _Seifert $n$-manifolds_, Invent. Math. 28 (1975), 137–159. MR 361150
  * [Qui71] Daniel Quillen, _The spectrum of an equivariant cohomology ring. I, II_ , Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694
  * [Smi83] Larry Smith, _Transfer and ramified coverings_ , Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 3, 485–493. MR 698352
  * [SZ12] Yalong Shi and Xiaohua Zhu, _Kähler-Ricci solitons on toric Fano orbifolds_ , Math. Z. 271 (2012), no. 3-4, 1241–1251. MR 3665915
  * [TZ00] Gang Tian and Xiaohua Zhu, _Uniqueness of Kähler-Ricci solitons_ , Acta Math. 184 (2000), no. 2, 271–305. MR 1768112
  * [WZ04] Xu-Jia Wang and Xiaohua Zhu, _Kähler-Ricci solitons on toric manifolds with positive first Chern class_ , Adv. Math. 188 (2004), no. 1, 87–103. MR MR2084775 (2005d:53074)
  * [Yam99] T. Yamazaki, _A construction of $K$-contact manifolds by a fiber join_, Tohoku Math. J. (2) 51 (1999), no. 4, 433–446. MR 2001e:53094
  * [Zhu00] Xiaohua Zhu, _Kähler-Ricci soliton typed equations on compact complex manifolds with $C_{1}(M)>0$_, J. Geom. Anal. 10 (2000), no. 4, 759–774. MR 1817785