# Foliation Divisorial Contraction by the Sasaki-Ricci Flow on Sasakian Five-
Manifolds

∗Shu-Cheng Chang  Department of Mathematics, National Taiwan University,
Taipei, Taiwan<EMAIL_ADDRESS>,  ∗∗Chien Lin  Mathematical Science
Research Center, Chongqing University of Technology, 400054, Chongqing, P.R.
China<EMAIL_ADDRESS>and  ∗∗∗Chin-Tung Wu  Department of Applied
Mathematics, National Pingtung University, Pingtung 90003, Taiwan
<EMAIL_ADDRESS>
###### Abstract.

Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian $5$-manifold
with finite cyclic quotient foliation singularities of type
$\frac{1}{r}(1,a).$ First, we derive the foliation minimal model program by
applying the resolution of cyclic quotient foliation singularities. Secondly,
based on the study of local model of resolution of foliation singularities, we
prove the foliation canonical surgical contraction or the foliation extremal
ray contraction under the Sasaki-Ricci flow. As a consequence, we provea
Sasaki analogue ofanalytic minimal model program with the Kaehler-Ricci flow
due to Song-Tian and Song-Weinkove.

###### Key words and phrases:

Foliation singularities, Foliation canonical surgical contraction, Foliation
extremal ray contraction, Sasaki-Ricci flow, Transverse Mori program,
Foliation minimal model program.

###### 1991 Mathematics Subject Classification:

Primary 53E50, 53C25; Secondary 53C12, 14E30.

∗Shu-Cheng Chang and ∗∗∗Chin-Tung Wu are partially supported in part by the
MOST of Taiwan.

###### Contents

  1. 1 Introduction
  2. 2 Sasakian Geometry
    1. 2.1 Sasakian Structure and Orbifold Structure of Its Leave Space
    2. 2.2 The Foliated Normal Coordinate
    3. 2.3 Basic Cohomology and Type II Deformation in a Sasakian Manifold
  3. 3 Blow-up and Resolution of Foliation Singularities
    1. 3.1 Foliation Singularities
    2. 3.2 Local Model of $\frac{1}{k}(1,1)$-type Foliation Singularities
      1. 3.2.1 The Coordinates on $\mathcal{S}(-k)$
      2. 3.2.2 Distance to the Zero Section
      3. 3.2.3 Distance to the Singular Fiber
      4. 3.2.4 Foliation Blow-up and Sasaki Castelnuovo’s Contraction Theorem
    3. 3.3 Reid’s Model and Resolution of $\frac{1}{r}(1,a)$-type Foliation Singularities
  4. 4 The Sasaki-Ricci Flow Through Singularities
    1. 4.1 The Sasaki-Ricci Flow
    2. 4.2 Basic Cohomology Characterization of the Maximal-Time Solution
    3. 4.3 The Weak Sasaki–Ricci Flow
    4. 4.4 The Sasaki-Ricci Flow Through Divisorial Contractions
  5. 5 Foliation MMP with Scaling via the Sasaki-Ricci Flow
    1. 5.1 Canonical Surgical Contraction for Floating Foliation $(-1)$-Curves
      1. 5.1.1 Key Estimates for the Sasaki-Ricci Flow
      2. 5.1.2 Gromov-Hausdorff Convergence as $t\rightarrow T_{0}^{-}$
      3. 5.1.3 Higher Order Estimates for $\omega(t)$ as $t\rightarrow T_{0}^{-}$
      4. 5.1.4 Continuing the Sasaki-Ricci Flow at $T_{0}$
      5. 5.1.5 Gromov-Hausdorff Convergence as $t\rightarrow T_{0}^{+}$
    2. 5.2 Analytic Foliation Minimal Model Program with Scaling
  6. A  Foliation Minimal Model Program
    1. A.1 Basic Holomorphic Line Bundles, Basic Divisors on Sasakian Manifolds
    2. A.2 Minimal Model Program on Compact Quasi-Regular Sasakian $5$-Manifolds

##  1\. Introduction

Sasakian geometry is very rich as the odd-dimensional analogous of Kähler
geometry. A Sasaki-Einstein $(2n+1)$-manifold is to say that its Kaehler cone
is a Calabi-Yau $(n+1)$-fold. Let $(M,\eta,\xi,\Phi,g)$ be a compact Sasakian
$(2n+1)$-manifold. If the orbits of the Reeb vector field $\xi$ are all
closed, and hence circles, then integrates to an isometric $U(1)$ action on
$(M,g)$. Since it is nowhere zero this action is locally free; that is, the
isotropy group of every point in $M$ is finite. If the $U(1$) action is in
fact free then the Sasakian structure is said to be regular. Otherwise, it is
said to be quasi-regular. Thus the space of leaves of the canonical
$U(1)$-fibration will have orbifold singularities (cf section $3$). In
general, it is said to be irregular if the orbits of are not all closed.

In particular, Sasaki-Einstein $5$-manifolds provide interesting examples of
the AdS/CFT correspondence. On the other hand, The class of simply connected,
closed, oriented, smooth, $5$-manifolds is classifiable under diffeomorphism
due to Smale-Barden ([S], [B]). Then, in this paper, it is our goal to focus
on a classification of compact quasi-regular Sasakian $5$-manifolds according
to the global properties of the Reeb $U(1)$-fibration.

Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian $5$-manifold.
Then by the first structure theorem, $M$ is a principal $S^{1}$-orbibundle
($V$-bundle) over an orbifold $\mathbf{Z}=(Z\emph{,}\Delta)$ which is also a
$Q$-factorial, polarized, normal projective orbifold surface such that there
is an orbifold Riemannian submersion

$\pi:(M,g,\omega)\rightarrow(Z,h,\omega_{h})$

and

(1.1)
$K_{M}^{T}=\pi^{\ast}(K_{Z}^{orb})=\pi^{\ast}(\varphi^{\ast}(K_{Z}+[\Delta])).$

If the orbifold structure of the leave space $Z$ is well-formed (c.f. section
$3$), then the orbifold canonical divisor $K_{\emph{Z}}^{orb}$ and canonical
divisor $K_{Z}$ are the same and thus

$K_{M}^{T}=\pi^{\ast}(\varphi^{\ast}(K_{Z})).$

In a such case, there is the Sasaki analogue of Mori’s minimal model program
with respect to $K_{Z}$ in a such compact quasi-regular Sasakian $5$-manifold.
More precisely, one can ask the following so-called foliation minimal model
program :

Is there a foliation $(-1)$-curve ? One of Mori program ([KMM], [M], [KM]) is
to replace this criterion with the one dictated by the transverse canonical
divisor $K_{M}^{T}$ of $M$ : Does the transverse canonical divisor have
nonnegative intersection

$K_{M}^{T}\cdot V\geq 0$

with any invariant submanifold $V$ on $M$ which is a Sasakian $3$-submanifold.
In other words,

$\mathrm{is}\ \ \ K_{M}^{T}\ \ \ \text{\ }\mathrm{nef\ }?$

If $K_{M}^{T}$ is not nef, there is an extremal transverse contraction map
which turns out not only to generalize the Sasaki analogue of Castelnuovo’s
contractibility criterion but also to provide decisive information on the
global structures of the end results of the foliation minimal model program.
Then an end result of foliation MMP starting from $M$ is a transverse Mori
fiber space if and only if there exists a nonempty open set $U\subset M$ such
that for any $S^{1}$-fibre in $U$, there is an irreducible invariant
$3$-submanifold $V$ passing through such a fibre $S^{1}$ with $\
K_{M}^{T}\cdot V<0.$ We refre to Proposition 11. Moreover, the Mori’s minimal
model program in birational geometry can be viewed as the complex analogue of
Thurston’s geometrization conjecture which was proved via Hamilton’s Ricci
flow with surgeries on $3$-dimensional Riemannian manifolds by Perelman ([P1],
[P2], [P3]). Likewise, there is a conjecture picture by Song-Tian ([ST]) that
the Kaehler-Ricci flow should carry out an analytic minimal model program with
scaling on projective varieties. Recently, Song and Weinkove ([SW1])
established the above conjecture on a projective algebraic surface.

The Sasaki–Ricci flow is introduced by Smoczyk–Wang–Zhang ([SWZ]) to study the
existence of Sasaki $\eta$-Einstein metrics on Sasakian manifolds. They showed
that the flow has the longtime solution and asymptotic converges to a Sasaki
$\eta$-Einstein metric when the basic first Chern class is negative or null.
It can be viewed as a Sasaki analogue of Cao’s result ([Cao]) for the
Kaehler–Ricci flow.

In view of the previous discusses, it is natural to conjecture that the
Sasaki-Ricci flow will carry out an analytic foliation minimal model program
with scaling on quasi-regular Sasakian $5$-manifolds as well. In this paper,
we deal with the case of its orbifold structure $(Z,\Delta)$ of the leave
space $Z$ is well-formed which has the codimension two fixed point set of
every non-trivial isotropy subgroup with no branch divisors
($\Delta=\emptyset$).

In section $3,$ we first derive the following result concering its foliation
cyclic quotient singularities on a compact quasi-regular Sasakian $5$-manifold
:

######  Theorem 1.

Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian $5$-manifold and
$Z$ be its leave space of the characteristic foliation.  Then $Z$ is a
$Q$-factorial normal projective algebraic orbifold surface satisfying

  1. (1)

if its leave space $(Z,\emptyset)$ has at least codimension two fixed point
set of every non-trivial isotropy subgroup. That is to say $Z$ is well-formed,
then $Z$ has isolated singularies of a finite cyclic quotient of
$\mathbf{C}^{2}$ and the action is

$\mu_{Z_{r}}:(z_{1},z_{2})\rightarrow(\zeta^{a}z_{1},\zeta^{b}z_{2}),$

where $\zeta$ is a primitive $r$-th root of unity. We denote the cyclic
quotient singularity by $\frac{1}{r}(a,b)$ with $(a,r)=1=(b,r)$. In
particular, the action can be rescaled so that every cyclic quotient
singularity corresponds to a $\frac{1}{r}(1,a)$-type singularity with
$(r,a)=1,\zeta=e^{\frac{2\pi i}{r}}$. In particular, it is klt (Kawamata log
terminal) singularities. Moreover, the corresponding singularities in
$(M,\eta,\xi,\Phi,g)$ is called foliation cyclic quotient singularities of
type

$\frac{1}{r}(1,a)$

at a singular fibre $\mathbf{S}_{p}^{1}$ in $M$.

  2. (2)

if its leave space $(Z,\Delta)$ has the codimension one fixed point set of
some non-trivial isotropy subgroup. Then the action is

$\mu_{Z_{r}}:(z_{1},z_{2})\rightarrow(e^{\frac{2\pi
a_{1}i}{r_{1}}}z_{1},e^{\frac{2\pi a_{2}i}{r_{2}}}z_{2}),$

for some positive integers $r_{1,}r_{2}$ whose least common multipler is $r$,
and $a_{i},i=1,2$ are integers coprime to $r_{i},i=1,2$. Then the foliation
singular set contains some $3$-dimensional submanifolds of $M.$ More
precisely, the corresponding singularities in $(M,\eta,\xi,\Phi,g)$ is called
the Hopf $S^{1}$-orbibundle over a Riemann surface $\Sigma_{h}.$

In particular, it is a foliation $(-1)$-curve (Theorem 5) at the regular point
if $r=a=1$ as in $(1)$ of Theorem 1. Then we have the following definition
([Cu]):

######  Definition 1.

Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian $5$-manifold
with foliation singularities of type $\frac{1}{r}(1,a).$ A foliation
$(-1)$-curve $V$ in $M$ is said to be floating if $V$ is entirely contained in
the smooth locus of $M$ with respective to the foliation $\mathcal{F}_{\xi}$.
Then $M$ is said to be minimal if it has no floating foliation $(-1)$-curves.

Secondly, a detail work on the local model of resolution of foliation cyclic
quotient singularities and a Castelnuovo’s contraction theorem, we have the
following Sasaki analogue of Castelnuovo’s contraction Theorem on Sasakian
$5$-Manifolds as in section $3:$

######  Theorem 2.

(Sasaki Castelnuovo’s Contraction Theorem) Let $(M,\eta,\xi,\Phi,g)$ be a
compact quasi-regular Sasakian $5$-manifold with foliation singularities of
type $\frac{1}{r}(1,a)$ and $V$ a floating foliation $(-1)$-curve. Then there
exists a transverse birational morphism $f:M\rightarrow N$ that contracts
$V\in S^{3}\times D^{2}\subset M$ to a regular fibre $S_{\mathbf{o}}^{1}\in
S^{1}\times D^{4}\subset N$ and it is an isomorphism outside of $V$. $N$ is
again a compact quasi-regular Sasakian $5$-manifold. In particular, every
proper transverse birational morphism between compact regular Sasakian
$5$-manifolds can be factored in a sequence of contractions of foliation
$(-1)$-curves.

In section $4,$ we work on some basic facts of the Sasaki analogue of the
Kaehler-Ricci flow through singularities due to Song-Tian ([ST]). In
particular, we have the following definition for the foliation canonical
surgical contraction on a compact quasi-regular Sasakian $5$-manifold.

######  Definition 2.

We say that the solution $g(t)$ of the Sasaki-Ricci flow on a compact Sasakian
$5$-manifold $M$ performs a floating foliation canonical surgical contraction
if the following holds: There exist distinct floating foliation $(-1)$-curves
$V_{1},...,V_{k}$ of $M$, a compact Sasakian $5$-manifold $N$ and a divisorial
contraction $\psi:M\rightarrow N$ with $\psi(V_{i})=S_{i}^{1}\subset N$ and
$\psi:M\backslash\cup_{i=1}^{k}V_{i}\rightarrow
N\backslash\\{S_{1}^{1},...,S_{k}^{1}\\}$ a basic transverse biholomorphism
onto $N\backslash\\{S_{1}^{1},...,S_{k}^{1}\\}$ such that :

  1. (1)

The metrics $g(t)$ converge to a smooth Sasakain metric $g_{T}$ on
$M\backslash\cup_{i=1}^{k}V_{i},$ as $t\rightarrow T^{-}$, smoothly on compact
subsets of $M\backslash\cup_{i=1}^{k}V_{i}.$

  2. (2)

$(M,g(t))$ converges to a unique compact metric space $(N,d_{T})$ in the
Gromov-Hausdorff sense as $t\rightarrow T^{-}$. In particular, $(N,d_{T})$ is
homeomorphic to the Sasakain $5$-manifold $N$. Here $d_{T}$ is defined to be
the metric on $N$ by extending $(\psi^{-1})^{\ast}g_{T}$ to be zero on
$\\{S_{1}^{1},...,S_{k}^{1}\\}$.

  3. (3)

There exists a unique maximal smooth solution $g(t)$ of the Sasaki-Ricci flow
on $N$ for $t\in(T,T_{N})$ with $T<T_{N}\leq\infty$, such that $g(t)$
converges to $(\psi^{-1})^{\ast}g_{T}$ as $t\rightarrow T^{+}$ smoothly on
compact subsets of $N\backslash\\{S_{1}^{1},...,S_{k}^{1}\\}.$

  4. (4)

$(N,g(t))$ converges to $(N,d_{T})$ in the Gromov-Hausdorff sense as
$t\rightarrow T^{+}$.

In the final section, we will apply the results of local model of resolution
of foliation singularities as in section $3$ to prove the foliation canonical
surgical contraction on a compact quasi-regular Sasakian $5$-manifold $M.$

Along the lines of the arguments in [SW2], we first deal with the case of
floating foliation canonical surgical contraction :

######  Theorem 3.

Let $g(t)$ be a smooth solution of the Sasaki-Ricci flow on a compact quasi-
regular Sasakian $5$-manifold $M$ with the foliation singularitie of type
$\frac{1}{r}(1,a)$ for $t\in[0,T)$ and assume $T<\infty$. Suppose there exists
a blow-down map $\psi:M\rightarrow N$ contracting disjoint floating foliation
$(-1)$-curves $V_{1},...,V_{k}$ on $M$ with $\psi(V_{i})=S_{i}^{1}\subset N$,
for a smooth compact Sasakian $5$-manifold $(N,\omega_{N})$ such that the
limiting transverse Kaehler class satisfies

(1.2) $[\omega_{0}]_{B}-Tc_{1}^{B}(M)=[\psi^{\ast}\omega_{N}]_{B}.$

Then the Sasaki-Ricci flow $g(t)$ performs a foliation canonical surgical
contraction with respect to the data $V_{1},...,V_{k}$, $N$ and $\psi$.

By applying Theorem 3 to the minimal resolution of $\frac{1}{r}(1,a)$-type
foliation singularities of $M$ as in Theorem 6, we have

######  Corollary 1.

Let $g(t)$ be a smooth solution of the Sasaki-Ricci flow on a compact regular
Sasakian $5$-manifold $M$ for $t\in[0,T)$ and assume $T<\infty$. Suppose there
exists a divisorial contraction $\psi:M\rightarrow N$ contracting disjoint
foliation $(-1)$-curves $V_{1},...,V_{k}$ on $M$ with
$\psi(V_{i})=S_{i}^{1}\subset N$, for a smooth compact Sasakian $5$-manifold
$(N,\omega_{N})$ such that the limiting transverse Kaehler class satisfies

$[\omega_{0}]_{B}-Tc_{1}^{B}(M)=[\psi^{\ast}\omega_{N}]_{B}.$

Then the Sasaki-Ricci flow $g(t)$ performs a foliation canonical surgical
contraction with respect to the data $V_{1},...,V_{k}$, $N$ and $\psi$.

As a consequence of Theorem 3 and Proposition 11, we have our main results in
the paper on the analytic f oliation minimal model program with scaling in a
compact quasi-regular Sasakian $5$-Mmanifold.

######  Theorem 4.

Let $(M,\xi,\omega_{0})$ be a compact quasi-regular Sasakian $5$-manifold with
the foliation singularitie of type $\frac{1}{r}(1,a).$ Then there exists a
unique maximal Sasaki-Ricci flow $\omega(t)$ on $M_{0},M_{1},...,M_{k}$
starting at $(M,\omega_{0})$ with floating foliation canonical surgical
contractions of a finite number of disjoint floating foliation $(-1)$-curves
or foliation extremal ray contractions of foliation $K_{M}^{T}$-negative
curves $\psi_{i}:M_{i-1}\rightarrow M_{i}$. In addition, we have

  1. (1)

Either $T_{k}<\infty$ and the flow $\omega(t)$ collapses in the sense that

$Vol_{\xi}(M_{k},\omega(t))\rightarrow 0$

$\mathrm{as}\ \ \ t\rightarrow T_{k}^{-}.$

    1. (a)

there exists a contraction

$\varphi:M_{k}\rightarrow pt,$

then $K_{M}^{T}<0$ and thus $M_{k}$ is transverse minimal Fano and the
foliation space $M_{k}/\mathcal{F}_{\xi}$ is minimal log del Pezzo surface of
at worst $\frac{1}{r}(1,a)$-type singularities and Picard number one, or

    2. (b)

there exists a fibration

$\varphi:M_{k}\rightarrow\Sigma_{h},$

then $M_{k}$ is an $S^{1}$-orbibundle of a rule surface over Riemann surfaces
$\Sigma_{h}$ of genus $h$. Or

  2. (2)

$T_{k}=\infty$ and $M_{k}$ is nef : $\psi_{i}=\psi_{k}$, and $M_{k}$ has at
worst foliation cyclic quotient singularities (orbifold singularities) and has
no foliation $K_{M}^{T}$-negative curves.

######  Corollary 2.

Let $(M,\xi,\omega_{0})$ be a compact regular Sasakian $5$-manifold with a
smooth transverse Kaehler metric $\omega_{0}$. Then there exists a unique
maximal Sasaki-Ricci flow $\omega(t)$ on $M_{0},M_{1},...,M_{k}$ with
foliation canonical surgical contractions starting at $(M,\omega_{0})$.
Moreover, each foliation canonical surgical contraction corresponds to a
divisorial contraction $\psi_{i}:M_{i-1}\rightarrow M_{i}$ of a finite number
of disjoint foliation $(-1)$-curves on $M_{i}$. In addition, we have

  1. (1)

Either $T_{k}<\infty$ and the flow $\omega(t)$ collapses in the sense that

$Vol_{\xi}(M_{k},\omega(t))\rightarrow 0,$

$\mathrm{as}\ \ \ t\rightarrow T_{k}^{-}.$Then $M_{k}$ is transverse
birational to a regular Sasakian sphere $\mathbf{S}^{5}$ or
$(\mathbf{S}^{2}\mathbf{\times S}^{3})$ or
$\mathbf{S}^{2}\widetilde{\times}\mathbf{S}^{3}$or $\Sigma_{h}\mathbf{\times
S}^{3}$ or $\Sigma_{h}\widetilde{\times}\mathbf{S}^{3}.$

  2. (2)

Or $T_{k}=\infty$ and $M_{k}$ has no foliation $(-1)$-curves.

In this paper, we assume that $M$ is a compact quasi-regular transverse
Sasakian manifold and the space $Z$ of leaves is well-formed which means its
orbifold singular locus and algebro-geometric singular locus coincide,
equivalently $Z$ has no branch divisors. However when its leave space $Z$ is
not well-formed, the orbifold structure $(Z,\Delta)$ of the leave space $Z$
has the codimension one fixed point set of some non-trivial isotropy subgroup.
Then the corresponding singularities in $(M,\eta,\xi,\Phi,g)$ is the Hopf
$S^{1}$-orbibundle over a Riemann surface $\Sigma_{h}.$ The orbifold canonical
divisor $K_{\emph{Z}}^{orb}$ and canonical divisor $K_{Z}$ are related by

$K_{\emph{Z}}^{orb}=\varphi^{\ast}(K_{Z}+[\Delta])$

and then

$K_{M}^{T}=\pi^{\ast}(K_{\emph{Z}}^{orb}).$

There is a Sasaki analogue of Log minimal model program ([KM]) with respect to
$K_{Z}+[\Delta].$ Thus we conjecture that there is a Sasaki analogue of
analytic Log minimal model program with respect to $K_{Z}+[\Delta]$ via the
conical Sasaki-Ricci flow which is the odd dimensional counterpart of the
conical Kähler-Ricci flow ([LZ], [Shen]). We hope to address this issue in the
near future.

Acknowledgement. The authors would like to thank Yi-Sheng Wang for sharing his
ideas on the construction of foliation singularities as in the section three.

##  2\. Sasakian Geometry

In this section, we will give some preliminaries on structures theorems for
Sasakian structures and the orbifold structure of its leave space, the
foliation normal local coordinates, the basic cohomology and its Type II
deformation . We refer to [BG], [FOW], and references therein for some
details.

###  2.1. Sasakian Structure and Orbifold Structure of Its Leave Space

Let $(M,g,\nabla)$ be a Riemannian $(2n+1)$-manifold. $(M,g)$ is called Sasaki
if the cone

$(C(M),\overline{g}):=(\mathbf{R}^{+}\times M\mathbf{,\ dr^{2}+r}^{2}g)$

such that $(C(M),\overline{g},J,\overline{\omega})$ is Kaehler with

$\overline{\omega}=\frac{1}{2}i\partial\overline{\partial}r^{2}.$

The function $\frac{1}{2}r^{2}$ is hence a global Kaehler potential for the
cone metric. As $\\{r=1\\}=\\{1\\}\times M\subset C(M)$, we may define

$\overline{\xi}=J(r\frac{\partial}{\partial r})$

and the Reeb vector field $\xi$ on $M$

$\xi=J(\frac{\partial}{\partial r}).$

Also

$\overline{\eta}(\cdot)=\frac{1}{2}\overline{g}(\xi,\cdot)$

and the contact $1$-form $\eta$ on $TM$

$\eta(\cdot)=g(\xi,\cdot)$

Then $\xi$ is the killing vector field with unit length such that

$\eta(\xi)=1\ \text{{and}}\ \ d\eta(\xi,X)=0.$

The tensor field of $type(1,1)$, defined by $\Phi(Y)=\nabla_{Y}\xi$, satisfies
the condition

$(\nabla_{X}\Phi)(Y)=g(\xi,Y)X-g(X,Y)\xi$

for any pair of vector fields $X$ and $Y$ on $M$. Then such a triple
$(\eta,\xi,\Phi)$ is called a Sasakian structure on a Sasakian manifold
$(M,g).$ Note that the Riemannian curvature satisfying the following

$R(X,\xi)Y=g(\xi,Y)X-g(X,Y)\xi$

for any pair of vector fields $X$ and $Y$ on $M$. In particular, the sectional
curvature of every section containing $\xi$ equals one.

The first structure theorem on Sasakian manifolds states that

######  Proposition 1.

([Ru], [Sp]) Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian
manifold of dimension $2n+1$ and $Z$ denote the space of leaves of the
characteristic foliation $\mathcal{F}_{\xi}$ (just as topological space). Then

(i) $Z$ carries the structure of a Hodge orbifold $\mathcal{Z=}(Z,\Delta)$
with an orbifold Kaehler metric $h$ and Kaehler form $\omega$ which defines an
integral class $[p^{\ast}\omega]$ in $H_{orb}^{2}(Z,\mathbf{Z)}$ in such a way
that $\pi:$ $(M,g)\rightarrow(Z,h)$ is an orbifold Riemannian submersion, and
a principal $S^{1}$-orbibundle ($V$-bundle) over $Z.$ Furthermore,it satisfies
$\frac{1}{2}d\eta=\pi^{\ast}(\omega).$The fibers of $\pi$ are geodesics.

(ii) $Z$ is also a $Q$-factorial, polarized, normal projective algebraic
variety.

(iii) $(M,\xi,g)$ is Sasaki-Einstein if and only if $(Z,h)$ is Kaehler-
Einstein with scalar curvature $4n(n+1).$

(iv) If $(M,\eta,\xi,\Phi,g)$ is regular then the orbifold structure is
trivial and $\pi$ is a principal circle bundle over a smooth projective
algebraic variety.

(v) As real cohomology classes, there is a relation between the basic Chern
class and orbifold chern class

$c_{k}^{B}(M):=c_{k}(\emph{F}_{\xi})=\pi^{\ast}c_{k}^{orb}(\mathbf{Z}).$

Conversely, let $\pi$: $M\rightarrow Z$ be a $U(1)$-orbibundle over a compact
Hodge orbifold $(Z,h)$ whose first Chern class is an integral class defined by
$[\omega_{Z}]$, and $\eta$ be a $1$-form with
$\frac{1}{2}d\eta=\pi^{\ast}\omega_{Z}$. Then
$(M,\pi^{\ast}h+\eta\otimes\eta)$ is a Sasakian manifold if all the local
uniformizing groups inject into the structure group $U(1).$

On the other hand, the second structure theorem on Sasakian manifolds states
that

######  Proposition 2.

([Ru]) Let $(M,g)$ be a compact Sasakian manifold of dimension $2n+1$. Any
Sasakian structure $(\xi,\eta,\Phi,g)$ on $M$ is either quasi-regular or there
is a sequence of quasi-regular Sasakian structures
$(M,\xi_{i},\eta_{i},\Phi_{i},g_{i})$ converging in the compact-open
$C^{\infty}$-topology to $(\xi,\eta,\Phi,g).$ In particular, if $M$ admits an
irregular Sasakian structure, it admits many locally free circle actions.

###  2.2. The Foliated Normal Coordinate

Let $(M,\eta,\xi,\Phi,g)$ be a compact Sasakian $(2n+1)$-manifold with
$g(\xi,\xi)=1$ and the integral curves of $\xi$ are geodesics. For any point
$p\in M$, we can construct local coordinates in a neighbourhood of $p$ which
are simultaneously foliated and Riemann normal coordinates ([GKN]). That is,
we can find Riemann normal coordinates
$\\{x,z^{1},z^{2},\cdot\cdot\cdot,z^{n}\\}$ on a neighbourhood $U$ of $p$,
such that $\frac{\partial}{\partial x}=\xi$ on $U$. Let
$\\{U_{\alpha}\\}_{\alpha\in A}$ be an open covering of the Sasakian manifold
and

$\pi_{\alpha}:U_{\alpha}\rightarrow V_{\alpha}\subset\mathbb{C}^{n}$

submersion such that
$\pi_{\alpha}\circ\pi_{\beta}^{-1}:\pi_{\beta}(U_{\alpha}\cap
U_{\beta})\rightarrow\pi_{\alpha}(U_{\alpha}\cap U_{\beta})$ is biholomorphic.
On each $V_{\alpha},$ there is a canonical isomorphism

$d\pi_{\alpha}:D_{p}\rightarrow T_{\pi_{\alpha}(p)}V_{\alpha}$

for any $p\in U_{\alpha},$ where $D=\ker\xi\subset TM.$ Since $\xi$ generates
isometries, the restriction of the Sasakian metric $g$ to $D$ gives a well-
defined Hermitian metric $g_{\alpha}^{T}$ on $V_{\alpha}.$ This Hermitian
metric in fact is Kähler. More precisely, let
$z^{1},z^{2},\cdot\cdot\cdot,z^{n}$ be the local holomorphic coordinates on
$V_{\alpha}$. We pull back these to $U_{\alpha}$ and still write the same. Let
$x$ be the coordinate along the leaves with $\xi=\frac{\partial}{\partial x}.$
Then we have the foliation local coordinate
$\\{x,z^{1},z^{2},\cdot\cdot\cdot,z^{n}\\}$ on $U_{\alpha}\ $and
$(D\otimes\mathbb{C})$ is spanned by the form

$Z_{\alpha}=\left(\frac{\partial}{\partial
z^{\alpha}}-\theta\left(\frac{\partial}{\partial
z^{\alpha}}\right)\frac{\partial}{\partial x}\right),\ \ \
\alpha=1,2,\cdot\cdot\cdot,n.$

Moreover

$\Phi=i\left(\frac{\partial}{\partial z^{j}}+ih_{j}\frac{\partial}{\partial
x}\right)\otimes dz^{j}+conj$

and

$\ \eta=dx-ih_{j}dz^{j}+ih_{\overline{j}}d\overline{z}^{j}.$

Here $h$ is basic : $\frac{\partial h}{\partial x}=0$ and
$h_{j}=\frac{\partial h}{\partial
z^{j}},h_{j\overline{l}}=\frac{\partial^{2}h}{\partial
z^{j}\partial\overline{z}^{l}}$ with

(2.1) $\text{{The normal coordinate :}
}h_{j}(p)=0,h_{j\overline{l}}(p)=\delta_{j}^{l},dh_{j\overline{l}}(p)=0.$

A frame

$\\{\frac{\partial}{\partial x},Z_{j}=\left(\frac{\partial}{\partial
z^{j}}+ih_{j}\frac{\partial}{\partial x}\right),\ \ \
j=1,2,\cdot\cdot\cdot,n\\}$

and the dual

$\\{\eta,dz^{j},\ j=1,2,\cdot\cdot\cdot,n\\}$

with

$[Z_{i},Z_{j}]=[\xi,Z_{j}]=0.$

Since $i(\xi)d\eta=0,$

$d\eta(Z_{\alpha},\overline{Z_{\beta}})=d\eta(\frac{\partial}{\partial
z^{\alpha}},\frac{\partial}{\overline{\partial}z^{\beta}}).$

Then the Kähler $2$-form $\omega_{\alpha}^{T}$ of the Hermitian metric
$g_{\alpha}^{T}$ on $V_{\alpha},$ which is the same as the restriction of the
Levi form $d\eta$ to $\widetilde{D_{\alpha}^{n}}$, the slice $\\{x=$
constant$\\}$ in $U_{\alpha},$ is closed. The collection of Kähler metrics
$\\{g_{\alpha}^{T}\\}$ on $\\{V_{\alpha}\\}$ is so-called a transverse Kähler
metric. We often refer to $d\eta$ as the Kähler form of the transverse Kähler
metric $g^{T}$ in the leaf space $\widetilde{D^{n}}.$

The Kaehler form $d\eta$ on $D$ and the Kaehler metric $g^{T}$ is define such
that

$g=g^{T}+\eta\otimes\eta$

and

$g_{i\overline{j}}^{T}=g^{T}(\frac{\partial}{\partial
z^{i}},\frac{\partial}{\partial\overline{z}^{j}})=d\eta(\frac{\partial}{\partial
z^{i}},\Phi\frac{\partial}{\partial\overline{z}^{j}})=2h_{i\overline{j}}.$

In terms of the normal coordinate, we have:

$g^{T}=g_{i\overline{j}}^{T}dz^{i}d\overline{z}^{j},\omega=d\eta=2ih_{i\overline{j}}dz^{i}\wedge
d\overline{z}^{j}.$

The transverse Ricci curvature $Ric^{T}$ of the Levi-Civita connection
$\nabla^{T}$ associated to $g^{T}$ is

$Ric^{T}=Ric+2g^{T}$

and

$R^{T}=R+2n.$

The transverse Ricci form $\rho^{T}$

$\rho^{T}=Ric^{T}(J\cdot,\cdot)=-iR_{i\overline{j}}^{T}dz^{i}\wedge
d\overline{z}^{j}$

with

$R_{i\overline{j}}^{T}=-\frac{\partial_{2}}{\partial
z^{i}\partial\overline{z}^{j}}\log\det(g_{\alpha\overline{\beta}}^{T})$

and it is a closed basic $(1,1)$-form

$\rho^{T}=\rho+2d\eta.$

###  2.3. Basic Cohomology and Type II Deformation in a Sasakian Manifold

######  Definition 3.

Let $(M,\eta,\xi,\Phi,g)$ be a Sasakian $(2n+1)$-manifold. Define

A $p$-form $\gamma$ is called basic if

$i(\xi)\gamma=0$

and

$\mathcal{L}_{\xi}\gamma=0.$

Let $\Lambda_{B}^{p}$ be the sheaf of germs of basic $p$-forms and
$\Omega_{B}^{p}$ be the set of all global sections of $\Lambda_{B}^{p}$. It is
easy to check that $d\gamma$ is basic if $\gamma$ is basic. Set
$d_{B}=d|_{\Omega_{B}^{p}}.$ Then

$d_{B}:\Omega_{B}^{p}\rightarrow\Omega_{B}^{p+1}.$

We then have the well-defined operators

$d_{B}:=\partial_{B}+\overline{\partial}_{B}$

with

$\partial_{B}:\Lambda_{B}^{p,q}\rightarrow\Lambda_{B}^{p+1,q}$

and

$\overline{\partial}_{B}:\Lambda_{B}^{p,q}\rightarrow\Lambda_{B}^{p,q+1}.$

Then for $d_{B}^{c}:=\frac{i}{2}(\overline{\partial}_{B}-\partial_{B}),$ we
have

$d_{B}d_{B}^{c}=i\partial_{B}\overline{\partial}_{B},d_{B}^{2}=(d_{B}^{c})^{2}=0.$

The basic Laplacian is defined by

$\Delta_{B}:=d_{B}d_{B}^{\ast}+d_{B}^{\ast}d_{B}.$

Then we have the basic de Rham complex $($ $\Omega_{B}^{\ast},d_{B})$ and the
basic Dolbeault complex $($ $\Omega_{B}^{p,\ast},\overline{\partial}_{B})$ and
its cohomology group $H_{B}^{\ast}(M,\mathbf{R})$ ([EKA]]).

######  Definition 4.

(i) We define the basic Cohomology of the foliation $F_{\xi}$by

$H_{B}^{\ast}(F_{\xi}):=H_{B}^{\ast}(M,\mathbf{R}).$

Then by transverse Hodge decomposition and transverse Serre duality

$H_{B}^{p,q}(F_{\xi})\simeq H_{B}^{q,p}(F_{\xi})$

and the cohomology of the leaf space $Z=M/U(1)$ to this basic cohomology of
the foliation :

$H_{orb}^{\ast}(Z,\mathbf{R})=H_{B}^{\ast}(F_{\xi}):=H_{B}^{\ast}(M,\mathbf{R}).$

(ii) ([34], Theorem 7.5.17) : Define the basic first Chern class
$c_{1}^{B}(M)$ by

$c_{1}^{B}=[\rho^{T}]_{B}$

and the transverse Einstein (Sasaki $\eta$-Einstein) equation up to a
$D$-homothetic deformation

$[\rho^{T}]_{B}=\varkappa[d\eta]_{B},\varkappa=-1,0,1.$

Basic $k$-th Chern class $c_{k}^{B}(M)$ is represented by a closed basic
$(k,k)$-form $\gamma_{k}$ which is determined by the formula :

$\det(\mathbf{I}_{n}-\frac{1}{2\pi i}\Omega^{T})=1+\gamma_{1}+...+\gamma_{k}.$

Here $\Omega^{T}$ is the curvature $2$-form of type basic $(1,1)$ with respect
to the transverse connection $\nabla^{T}$.

######  Example 1.

Let $(M,\eta,\xi,\Phi,g)$ be a compact Sasakian $(2n+1)$-manifold. If $g^{T}$
is a transverse Kaehler metric on $M,$ then
$h_{\alpha}=\det((g_{i\overline{j}}^{\alpha})^{T})$ on $U_{\alpha}$ defines a
basic Hermitian metric on the canonical bundle $K_{M}^{T}$. The inverse
$(K_{M}^{T})^{-1}$ of $K_{M}^{T}$ is sometimes called the anti-canonical
bundle. Its basic first Chern class $c_{1}^{B}((K_{M}^{T})^{-1})$ is called
the basic first Chern class of $M$ and is often denoted by $c_{1}^{B}(M).$Then
it follows from the previous result that $c_{1}^{B}(M)$ that
$c_{1}^{B}(M)=[Ric^{T}(\omega)]_{B}$ for any transversal Kaehler metric
$\omega$ on a Sasakian manifold $M$.

######  Definition 5.

We define Type II deformations of Sasakian structures $(M,\eta,\xi,\Phi,g)$ by
fixing the $\xi$ and varying $\eta$. That is, for $\varphi\in\Omega_{B}^{0}$,
define

$\widetilde{\eta}=\eta+d_{B}^{c}\varphi,$

then

$d\widetilde{\eta}=d\eta+d_{B}d_{B}^{c}\varphi=d\eta+i\partial_{B}\overline{\partial}_{B}\varphi$

and

$\widetilde{\omega}=\omega+i\partial_{B}\overline{\partial}_{B}\varphi.$

Note that we have the same transversal holomorphic foliation ( $\xi$ is fixed)
but with the new Kaehler structure on the Kaehler cone $C(M)$ and new contact
bundle $\widetilde{D}$ :
$\widetilde{\omega}=dd^{c}\widetilde{r},\widetilde{r}=re^{\varphi}$. The same
holomorphic structure : $r\frac{\partial}{\partial
r}=\widetilde{r}\frac{\partial}{\partial\widetilde{r}};\xi=J($
$r\frac{\partial}{\partial r})$ and $\xi+ir\frac{\partial}{\partial
r}=\xi-i\Phi(\xi)$ is the holomorphic vector field on $C(M).$ Moreover, we
have

$\begin{array}[c]{ccc}\widetilde{\Phi}&=&\Phi-\xi\otimes(d_{B}^{c}\varphi)\circ\Phi,\\\
\widetilde{g}&=&d\widetilde{\eta}\circ(Id\otimes\widetilde{\Phi})+\widetilde{\eta}\otimes\widetilde{\eta}.\end{array}$

and

$\mathcal{L}_{\xi}\widetilde{\Phi}=\mathcal{L}_{\xi}\Phi=0.$

##  3\. Blow-up and Resolution of Foliation Singularities

###  3.1. Foliation Singularities

In this section, parts of notions are due to the papers of Boyer-Galicki
([BG]) and Wang ([W]). Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular
Sasakian manifold of dimension five and $Z$ denote the space of leaves of the
characteristic foliation $\mathcal{F}_{\xi}$. It follows from the first
srtucture theorem as in Proposition 1 that $Z$ is a compact normal, orbifold
surface locally given by charts written as quotients of smooth coordinate
charts. That is, $Z$ can be covered by open charts $Z=\cup U_{i}.$ The
orbifold charts on $(Z,U_{i},\varphi_{i})$ is defined by the local
uniformizing systems $(\widetilde{U_{i}},G_{i},\varphi_{i})$ centered at the
point $z_{i}$, where $G_{{}_{i}}$ is the local uniformizing finite group
acting on a smooth complex space $\widetilde{U_{i}}$ such that

$\varphi_{i}:\widetilde{U_{i}}\rightarrow U_{i}=\widetilde{U_{i}}/G_{{}_{i}}$

is the biholomorphic map. A point $x$ of complex orbifold $Z$ whose isotropy
subgroup $\Gamma_{x}\neq Id$ is called a singular point. Those points with
$\Gamma_{x}=Id$ are called regular points. The set of singular points is
called the orbifold singular locus or orbifold singular set, and is denoted by
$\Sigma^{orb}(Z)$. Let $\Gamma\subset GL(2,\mathbf{C})$ be a finite subgroup.
Then the quotient space $\mathbf{C}^{2}/\Gamma$ is smooth if and only if
$\Gamma$ is a reflection group which fixes a hyperplane in $\mathbf{C}^{2}$.
By the first structure theorem, the underlying complex orbifold space
$\emph{Z}=(Z\emph{,U}_{i}\emph{)}$ is a normal, orbifold variety with the
algebro-geometric singular set $\Sigma(Z)$. Then
$\Sigma(Z)\subset\Sigma^{orb}(Z)$ and it follows that
$\Sigma(Z)=\Sigma^{orb}(Z)$ if and only if none of the local uniformizing
groups $\Gamma_{i}$ of the orbifold $\emph{Z}=(Z\emph{,U}_{i}\emph{)}$ contain
a refletion. Moreover if some $\Gamma_{i}$ contains a reflection, then the
reflection fixes a hyperplane in $\widetilde{\emph{U}_{i}}$ giving rise to a
ramification divisor on $\widetilde{\emph{U}_{i}}$ and a branch divisor on
$Z.$ Then we have the following definition regarding orbifold singular locus
and algebro-geometric singular locus.

######  Definition 6.

(i) The branch divisor $\Delta$ of an orbifold $\mathbf{Z}=(Z\emph{,}\Delta)$
is a $Q$-divisor on $Z$ of the form

$\Delta=\sum_{\alpha}(1-\frac{1}{m_{\alpha}})D_{\alpha},$

where the sum is taken over all Weil divisors $D_{\alpha}$ that lie in the
orbifold singular locus $\Sigma^{orb}(Z)$, and $m_{\alpha}$ is the $gcd$ of
the orders of the local uniformizing groups taken over all points of
$D_{\alpha}$ and is called the ramification index of $D_{\alpha}.$

(ii) The orbifold structure $\mathbf{Z}=(Z\emph{,}\Delta)$ is called well-
formed if the fixed point set of every non-trivial isotropy subgroup has
codimension at least two. Then $Z$ is well-formed if and only if its orbifold
singular locus and algebro-geometric singular locus coincide, equivalently $Z$
has no branch divisors.

######  Example 2.

Fo instance, the weighted projective $\mathbf{CP}(1;4;6)$ has a branch divisor
$\frac{1}{2}D_{0}=\\{z_{0}=0\\}.$ But $\mathbf{CP}(1;2;3)$ is a unramified
well-formed orbifold with two singular points, $(0;1;0)$ with local
uniformizing group the cyclic group $\mathbf{Z}_{2}$, and $(0;0;1)$ with local
uniformizing group $\mathbf{Z}_{3}$.

Let $(M,\xi,\alpha_{\xi})$ be a compact quasi-regular Sasakian $5$-manifold
and the leave space $Z:=M/S^{1}$ be an orbifold Kaehler surface. Denote by the
quotient map

$\pi:M\rightarrow Z$

as before, and call such a $5$-manifold $(M,\xi)$ an $S^{1}$-orbibundle. More
precisely, $M$ admits a locally free, effective $S^{1}$-action

$\alpha_{\xi}:S^{1}\times M\rightarrow M$

such that $\alpha_{\xi}(t)$ is orientation-preserving, for every $t\in S^{1}$.
Since $M$ is compact, $\alpha_{\xi}$ is proper and the isotropy group
$\Gamma_{p}$ of every point $p\in M$ is finite.

######  Definition 7.

The principal orbit type $M_{\operatorname{reg}}$ corresponds to points in
$(M,\xi,\alpha_{\xi})$ with the trivial isotropy group and
$M_{\operatorname{reg}}\rightarrow M_{\operatorname{reg}}/S^{1}$ is a
principle $S^{1}$-bundle. Furthermore, the orbit $\mathbf{S}_{p}^{1}$ of a
point $p\in M$ is called a regular fiber if $p\in M_{\operatorname{reg}}$, and
a singular fiber otherwise. In this case,
$M_{\mathrm{sing}}/S^{1}\simeq\Sigma^{orb}(Z).$

The Proof of Theorem 1 :

###### Proof.

Since each orbit ${\mathsf{S_{p}^{1}}}$ in an $S^{1}$-orbibundle
$(M,\xi,\alpha_{\xi})$ admits an $S^{1}$-invariant neighborhood $U_{p}$ such
that it is $S^{1}$-equivalent to a standard $S^{1}$-structure
$V_{p}:=S^{1}\times\mathbb{C}^{2}/\mu_{p}$, where $\mu_{p}$ is a linear
representation of $\Gamma_{p}$ of $p$ on the vector space $\mathbb{C}^{2}$,
and the $S^{1}$-action on $\mathbb{C}^{2}$ is induced by the Lie group
structure of $S^{1}$. Since $\alpha$ is effective and $\mu_{p}$ is faithful.
If in addition $\mathbb{C}^{2}$ is endowed with a complex structure, one can
enhance the notion of a standard $S^{1}$-structure, and define the standard
complex $S^{1}$-structure which is the standard $S^{1}$-structure
$S^{1}\times\mathbb{C}^{2}/\mu_{p}$ with $\mu_{p}$ complex linear. Note that,
up to conjugation, a representation $\mu_{p}$ of $\mathbb{Z}_{r}$ on
$\mathbb{C}^{2}$ is conjugate to

$\displaystyle\mathbb{Z}_{r}$ $\displaystyle\rightarrow\mathbb{C}^{2}$
$\displaystyle\gamma$ $\displaystyle\mapsto\begin{bmatrix}e^{2\pi
i\frac{a_{1}}{r_{1}}}&0\\\ 0&e^{2\pi i\frac{a_{2}}{r_{2}}},\end{bmatrix}$

for some positive integers $r_{1},r_{2}$ whose least common multipler is $r$,
and $a_{i},i=1,2$ are integers coprime to $r_{i},i=1,2$. If $r_{1},r_{2}<r$,
then $(M,\xi,\alpha_{\xi})$ has a non-discrete set of fibers; more precisely,
$M_{\mathrm{sing}}$ contains some $3$-dimensional submanifolds of $M$. In
particular, in the case where $M_{\mathrm{sing}}$ is discrete, we have

$r_{1}=r_{2}=r.$

Up to change of generator, the representation $\mu_{p}$ can be normalized as
follows:

(3.1) $\displaystyle\mathbb{Z}_{r}$ $\displaystyle\rightarrow\mathbb{C}^{2}$
$\displaystyle\gamma$ $\displaystyle\mapsto\begin{bmatrix}e^{2\pi
i\frac{1}{r}}&0\\\ 0&e^{2\pi i\frac{a}{r}},\end{bmatrix}$

for some $a$ coprime to $r$. Then Theorem 1 follows easily. ∎

Now we come out with the following definition.

######  Definition 8.

Given $p\in M_{\mathrm{sing}}$, the singular fiber $\mathbf{S}_{p}^{1}$ is a
foliation singularity of type $\frac{1}{r}(1,a)$ at $p$ with $r>0,a$ two
integers and $a$ coprime with $r$, if there exists an isomorphism
$\Gamma_{p}\simeq\mathbb{Z}_{r}=<\gamma>$ such that $\mu_{p}$ can be
identified with the representation (3.1), up to conjugation. This corresponds
to the orbifold structure $\mathbf{Z}$ is well-formed where the fixed point
set of every non-trivial isotropy subgroup has codimension at least two.

Since $M-M_{\operatorname{reg}}$ is a union of smooth submanifolds of $M$. The
present paper concerns mainly the case where
$M_{\mathrm{sing}}=M-M_{\operatorname{reg}}$ is a discrete, and hence finite
set.

###  3.2. Local Model of $\frac{1}{k}(1,1)$-type Foliation Singularities

Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian $5$-manifold and
$\mathbf{S}_{p}^{1}$ be a discrete singular fiber of type $\frac{1}{k}(1,a)$
at $p\in M.$ Then there is an $\mathbf{S}^{1}$-equivariant neighborhood
$U_{p}$ of $\mathbf{S}_{p}^{1}$ and $\mathbf{S}^{1}$-equivalent to a standard
complex $\mathbf{S}^{1}$-structure
$\mathbf{S}^{1}\times\mathbf{C}^{2}/\mathbf{Z}_{r}.$ In this subsection, we
examine the $S^{1}$-structure of a neighborhood of a singular fiber of
$\frac{1}{k}(1,a)$ type. Then, by the slice theorem as discussed in the
previous subsection, there is an $S^{1}$-equivariant neighborhood $U_{p}$ of
${\mathsf{S_{p}^{1},}}$ $S^{1}$-equivalent to a standard complex
$S^{1}$-structure $S^{1}\times\mathbb{C}^{2}/\mathbb{Z}_{r}$. The
$S^{1}$-action on $S^{1}\times_{\mathbb{Z}_{r}}\mathbb{C}^{2}$ is induced by
the $S^{1}$-action on $S^{1}\times\mathbb{C}^{2}$ given by

$\displaystyle S^{1}\times S^{1}\times\mathbb{C}^{2}$
$\displaystyle\rightarrow S^{1}\times\mathbb{C}^{2}$
$\displaystyle(t,w,z_{1},z_{2})$ $\displaystyle\mapsto(tw,z_{1},z_{2}).$

Now we consider the local diffeomorphism

$\displaystyle\pi:S^{1}\times\mathbb{C}^{2}$ $\displaystyle\rightarrow
S^{1}\times\mathbb{C}^{2}$ $\displaystyle(w,z_{1},z_{2})$
$\displaystyle\mapsto(w^{k},w^{-1}z_{1},w^{-a}z_{2})$

which descends to a diffeomorphism from
$S^{1}\times\mathbb{C}^{2}/\mathbb{Z}_{r}$ to $S^{1}\times\mathbb{C}^{2}$. Via
the diffeomorphism, one can identify the neighborhood $U_{p}$ with
$S^{1}\times\mathbb{C}^{2}$ with the $S^{1}$-action on
$S^{1}\times\mathbb{C}^{2}$ given by

(3.2) $\displaystyle S^{1}\times S^{1}\times\mathbb{C}^{2}$
$\displaystyle\rightarrow S^{1}\times\mathbb{C}^{2}$
$\displaystyle(t,u,v_{1},v_{2})$
$\displaystyle\mapsto(t^{k}u,t^{-1}v_{1},t^{-a}v_{2}).$

The above identification also yields an $S^{1}$-equivalence between a closed
tubular neighborhood of ${\mathsf{S_{p}^{1}}}$ and $S^{1}\times D^{4}$, where
$D^{4}$ is the $4$-ball in $\mathbb{C}^{2}$, and the $S^{1}$-action on
$S^{1}\times D^{4}$ is given by

(3.3) $\displaystyle S^{1}\times S^{1}\times D^{4}$ $\displaystyle\rightarrow
S^{1}\times D^{4}$ $\displaystyle(t,u,v_{1},v_{2})$
$\displaystyle\mapsto(t^{k}u,t^{-1}v_{1},t^{-a}v_{2}).$

Now we are ready to consider the resolution of a singular fiber
${\mathsf{S_{\mathbf{p}}^{1}}}$ of type $\frac{1}{k}(1,1)$ at $p\in M$.
Consider the space ${\mathcal{S}(-k)}$ given by

$\\{\big{(}(z_{1},z_{2}),(u_{1},u_{2})\big{)}\in\mathbb{C}^{2}\times S^{3}\mid
z_{1}u_{2}^{k}=z_{2}u_{1}^{k}\\},$

where $S^{3}$ is the standard sphere in $\mathbb{C}^{2}$ and oriented using
inward normal. ${\mathcal{S}(-k)}$ admits a natural free $S^{1}$-action given
by the restriction of

(3.4) $\displaystyle\alpha_{k}:S^{1}\times\mathbb{C}^{2}\times S^{3}$
$\displaystyle\rightarrow\mathbb{C}^{2}\times S^{3}$
$\displaystyle(t,z_{1},z_{2},u_{1},u_{2})$
$\displaystyle\mapsto(z_{1},z_{2},t^{-1}u_{1},t^{-1}u_{2}).$

${\mathcal{S}(-k)}$ can be identified with $\mathbb{C}\times S^{3}$ via the
diffeomorphism

(3.5) $\displaystyle\mathbb{C}\times S^{3}$
$\displaystyle\rightarrow{\mathcal{S}(-k)}$ $\displaystyle(z,(w_{1},w_{2}))$
$\displaystyle\mapsto(zw_{1}^{k},zw_{2}^{k},w_{1},w_{2})$

which induces a free $S^{1}$-action $\beta_{k}$ on $\mathbb{C}\times S^{3}$:

(3.6) $\displaystyle\beta_{k}$ $\displaystyle:S^{1}\times\mathbb{C}\times
S^{3}\rightarrow\mathbb{C}\times S^{3}$ $\displaystyle(t,z,w_{1},w_{2})$
$\displaystyle\mapsto(t^{k}z,t^{-1}w_{1},t^{-1}w_{2}).$

We remark that the quotient of the embedding
${\mathcal{S}(-k)}\subset\mathbb{C}^{2}\times S^{3}$ by $\alpha_{0}$ is the
embedding ${\mathcal{O}(-k)}\subset\mathbb{C}^{2}\times\mathbf{C}P^{1}$, and
furthermore, there is a pullback diagram

(3.7)
$\begin{array}[c]{ccccc}\overline{M}\supset\mathcal{S}(-k)\thickapprox\mathbf{C}\times\mathbf{S}^{3}&\overset{\pi_{2}}{\rightarrow}&\mathbf{S}^{3}&\overset{}{\longrightarrow}&M\supset\mathbf{S}_{\mathbf{p}}^{1}\\\
\downarrow&&\downarrow-H&&\downarrow\pi_{M}\\\ Z\supset
O(-k)&\overset{\pi_{d}}{\rightarrow}&\mathbf{CP}^{1}&\overset{}{\longrightarrow}&Z\ni\pi_{M}(\mathbf{p}).\end{array}$

Here $\pi_{2}$ is the projection onto the second factor, and $\mathsf{H}$ is
the Hopf fibration. Note that $D^{2}\times\mathbf{S}^{3}\simeq\mathbf{C\times
S}^{3}$ can be identified with the subspace of
$\mathbf{C}^{2}\times\mathbf{S}^{3}$ given by

$\\{(z_{1},z_{2}),(u_{1},u_{2})|\text{ }z_{1}u_{2}^{k}-z_{2}u_{1}^{k}=0\\}$

which is the coordinate for the canonical line bundle $\mathcal{S}(-k)$ over
$\mathbf{S}^{3}$and $\overline{H}$ can be realized by projecting down the
first three coordinates to with the subspace of
$\mathbf{C}^{2}\times\mathbf{CP}^{1}$ given by

$\\{(z_{1},z_{2},[l_{1},l_{2}])|\text{ }z_{1}l_{2}^{k}-z_{2}l_{1}^{k}=0\\}$

which is the coordinate for the canonical line bundle $\mathcal{O}(-k)$ over
$\mathbf{CP}^{1}.$

Now, to resolve the singular fiber ${\mathsf{S_{\mathbf{p}}^{1}}}$ amounts to
replacing $U$ with ${\mathcal{S}(-k)}$. To see how this is done, we observe
that there is an $S^{1}$-equivariant embedding

(3.8) $\displaystyle\Psi:S^{1}\times(\mathbb{C}^{2}-\\{0,0\\})$
$\displaystyle\xrightarrow{\iota}{\mathcal{S}(-k)}\subset\mathbb{C}^{2}\times
S^{3}$ $\displaystyle(s,z_{1},z_{2})$
$\displaystyle\mapsto(sz_{1}^{k},sz_{2}^{k},\frac{z_{1}}{\sqrt{|z_{1}|^{2}+|z_{2}|^{2}}},\frac{z_{2}}{\sqrt{|z_{1}|^{2}+|z_{2}|^{2}}}).$

The complement of the image in $\tilde{U}$ is precisely the $3$-sphere
$\\{0,0\\}\times S^{3}$ where $\alpha_{0}$ restricts to give the negative Hopf
fibration $-\mathsf{H}$.

We define $(M^{\prime},\alpha^{\prime})$ to be the new $S^{1}$-orbibundle
given by gluing $M-{\mathsf{S_{\mathbf{p}}^{1}}}$ and ${\mathcal{S}(-k)}$ via
the identification between $U-{\mathsf{S_{\mathbf{p}}^{1}}}$ and
${\mathcal{S}(-k)}-\\{0,0\\}\times S^{3}$ given in above with
$\alpha^{\prime}$ induced by $\alpha_{\xi}$ and $\alpha_{k}$. In other words,
we remove the singular fiber by replacing it with a negative Hopf fibration.
It follows that

######  Lemma 1.

$(M^{\prime},\alpha^{\prime})$ is again a compact quasi-regular Sasakian
$5$-manifold.

####  3.2.1. The Coordinates on $\mathcal{S}(-k)$

Let $U_{i},i=1,2$ be two copies of
$\mathbf{C}\times\mathbf{C}\times\mathbf{S}^{1}$ denoted by
$(x_{i},v_{i},s_{i})$ the coordinate of a point in $U_{i}$. Consider the
embedding

$\begin{array}[c]{ccl}U_{1}&\overset{}{\rightarrow}&\mathcal{S}(-k)\\\
(x_{1},v_{1},s_{1})&\mapsto&(v_{1},v_{1}x_{1}^{k},(\frac{s_{1}}{\sqrt{1+|x_{1}|^{2}}},\frac{s_{1}x_{1}}{\sqrt{1+|x_{1}|^{2}}}))\end{array}$

and

$\begin{array}[c]{ccl}U_{2}&\overset{}{\rightarrow}&\mathcal{S}(-k)\\\
(x_{2},v_{2},s_{2})&\mapsto&(v_{2}x_{2}^{k},v_{2},(\frac{s_{2}x_{2}}{\sqrt{1+|x_{2}|^{2}}},\frac{s_{2}}{\sqrt{1+|x_{2}|^{2}}}))\end{array}$

which induce coordinate charts covering $\mathcal{S}(-k)$ with the transition
function given by

$x_{2}=x_{1}^{-1},v_{2}=v_{1}x_{1}^{k},s_{2}=s_{1}\frac{x_{1}}{x_{2}}.$

Note that the quotients $\overline{U}_{i}$ of $U_{i},i=1,2,$ give us the
standard coordinates of $\mathcal{O}(-k),$ namely,

$\begin{array}[c]{ccl}\overline{U}_{1}&\overset{}{\rightarrow}&\mathcal{O}(-k)\\\
(x_{1},v_{1})&\mapsto&(v_{1},v_{1}x_{1}^{k},[1,x_{1}])\end{array}$

and

$\begin{array}[c]{ccl}\overline{U}_{i2}&\overset{}{\rightarrow}&\mathcal{O}(-k)\\\
(x_{2},v_{2})&\mapsto&(v_{2}x_{2}^{k},v_{2},[x_{2},1]),\end{array}$

where $[l_{1},l_{2}]$ is the homogeneous coordinate of a point in
$\mathbf{CP}^{1}$. In particular, we have
$x_{1}=\frac{l_{2}}{l_{1}};x_{2}=\frac{l_{1}}{l_{2}}$, and $v_{i},i=1,2$
correspond to vectors in a fiber of $\mathcal{O}(-k).$

####  3.2.2. Distance to the Zero Section

Let $\mathbf{S}_{\mathbf{0}}^{3}$ be the zero section of

$\pi_{2}:\mathcal{S}(-k)\subset\mathbf{C}^{2}\mathbf{\times
S}^{3}\rightarrow\mathbf{S}^{3}$

and endow $\mathcal{S}(-k)$ with the metric induced by the standard metric on
$\mathbf{C}^{2}\mathbf{\times S}^{3}.$ Then the distance square
$dist^{2}(y,\mathbf{S}^{3})$ between a point $y\in\mathcal{S}(-k)$ and
$\mathbf{S}^{3}$ expressed in terms of the coordinates charts is

$dist^{2}(y,\mathbf{S}_{0}^{3})=(1+|x_{i}|^{2k})|v_{i}|^{2},$

where $y=(x_{i},v_{i},s_{i})\in U_{i},i=1,2.$

Now we define the section $s$ of $[V\mathcal{]}$ over $\mathcal{S}(-k)$ by

$s_{i}:U_{i}\rightarrow\mathbf{C,}\text{ \ \ }s_{i}=v_{i}.$

Note that $s$ is vanishing along the exceptional basic divisor $V$ and the
hermitian metric $h$ on $[V]$ such that

$h_{i}=(\frac{|l_{1}|^{2k}+|l_{2}|^{2k}}{|l_{i}|^{2k}})=(1+|x_{i}|^{2k})$

on $U_{i}.$ If furthermore the composition

$\begin{array}[c]{ccc}\mathcal{S}(-k)\thickapprox\mathbf{C}\times\mathbf{S}^{3}&\overset{\pi_{2}}{\rightarrow}&\mathbf{S}^{3}\\\
&&\downarrow-H\\\ &&\mathbf{CP}^{1}\end{array}$

send $y$ to $[l_{1},l_{2}],$ then

(3.9)
$dist^{2}(y,\mathbf{S}^{3})=(\frac{|l_{1}|^{2k}+|l_{2}|^{2k}}{|l_{i}|^{2k}})|v_{i}|^{2}:=|s|_{h}^{2}$

where $y\in U_{i}.$

####  3.2.3. Distance to the Singular Fiber

The standard metric on $\mathbf{C}^{2}\mathbf{\times S}^{3}$ also induces a
metric on $U-\mathbf{S}_{\mathbf{p}}^{1}=$
$\mathbf{S}^{1}\times(\mathbf{C}^{2}-\\{(0,0)\\})$ via

$\begin{array}[c]{ccl}\mathbf{S}^{1}\times(\mathbf{C}^{2}-\\{(0,0)\\})&\overset{i}{\rightarrow}&\mathcal{S}(-k)=\mathbf{C}\times\mathbf{S}^{3}\subset\mathbf{C}^{2}\times\mathbf{S}^{3}.\end{array}$

In particular, given a point $x=(s,z_{1},z_{2})$, the image $\Psi\circ\tau(t)$
of the ray $\tau(t)=(s,tz_{1},tz_{2}),0\leq t\leq 1$ is a geodesic

$(sz_{1}^{k}t^{k},sz_{2}^{k}t^{k},\frac{z_{1}}{\sqrt{|z_{1}|^{2}+|z_{2}|^{2}}},\frac{z_{2}}{\sqrt{|z_{1}|^{2}+|z_{2}|^{2}}})\subset\mathbf{C}^{2}\times\mathbf{S}^{3}$

away from zero, so the distance square to the singular fiber is

(3.10)
$dist^{2}(x,\mathbf{S}_{p}^{1})=|z_{1}|^{2k}+|z_{2}|^{2k}:=r_{\mathbf{S}_{\mathbf{p}}^{1}}^{2k}.$

Furthermore, $\Psi\circ\tau(t)$ is a geodesic from $\Psi(x)\in\mathcal{S}(-k)$
to the zero section $\mathbf{S}_{\mathbf{0}}^{3}$. Then

(3.11)
$\Psi^{\ast}(dist^{2}(\Psi(x),\mathbf{S}^{3}))=dist^{2}(x,\mathbf{S}_{p}^{1}).$

####  3.2.4. Foliation Blow-up and Sasaki Castelnuovo’s Contraction Theorem

As in (3.7) with $k=1,$ we pull back a negative Hopf fibration $-H$ along the
canonical disk bundle $\pi_{d}:O(-1)\rightarrow\mathbf{CP}^{1}$, we get a
$\mathbf{S}^{1}$-bundle $\mathcal{S}(-1)$ over $O(-1)$, which is necessarily
diffeomorphic to $\mathbf{S}^{3}\times D^{2}$, since the only $D^{2}$-bundle
over $\mathbf{S}^{3}$ is the trivial one :

(3.12)
$\begin{array}[c]{ccccc}\overline{M}\supset\mathcal{S}(-1)=\mathbf{S}^{3}\times
D^{2}&\rightarrow&\mathbf{S}^{3}&\overset{\psi|_{V}}{\longrightarrow}&\mathbf{S}_{p}^{1}\subset\mathbf{S}^{1}\times
D^{4}\subset M\\\ \downarrow&&&&\downarrow\pi\\\ \overline{Z}\supset
O(-1)&\rightarrow&\mathbf{CP}^{1}&\overset{\phi|_{E}}{\longrightarrow}&z_{p}=\pi(p)\in
Z.\end{array}$

Here $\phi:\overline{Z}\rightarrow Z$ is a blow-up of $Z$ at a nonsingular
point $z_{p}=\pi(p)\in Z.$

As the first consequence of previous subsections (cf Lemma 1, etc), it follows
that we have the following foliation blow-up along a regular fibre
$\mathbf{S}_{p}^{1}$ of type $\frac{1}{k}(1,1)$ with $k=1:$

######  Theorem 5.

Let $(M,\eta,\xi,\Phi,g)$ be a compact regular Sasakian $5$-manifold and $Z$
denote the space of leaves of the characteristic foliation which is a smooth
projective surface with the regular $\mathbf{S}^{1}$-principal bundle over
$M$. Then

  1. (1)

There is a compact regular Sasakian $5$-manifold $\overline{M}$ which is
obtained by gluing

$M-\overset{\circ}{\mathcal{N}}({\mathsf{S_{\mathbf{p}}^{1}}})$

and $\mathbf{S}^{3}\times D^{2}$ along their boundary via an orientation-
preserving diffeomorphism

$f:\mathbf{S}^{3}\times\mathbf{S}^{1}\simeq\partial(\mathbf{S}^{3}\times
D^{2})\rightarrow-\partial(M-\overset{\circ}{\mathcal{N}}(\mathbf{S}_{p}^{1}))\simeq\mathbf{S}^{3}\times\mathbf{S}^{1}.$

In other words, we have the blow-up map $\psi$

$\psi:\overline{M}\rightarrow M$

and $\overline{M}$ is called the foliation blowing up along a regular fibre
$\mathbf{S}_{p}^{1}$ at a point $p\in M$

$\overline{M}=(M-\overset{\circ}{\mathcal{N}}(\mathbf{S}_{p}^{1}))\cup_{f}\mathbf{S}^{3}\times
D^{2}$

  2. (2)

For the irreducible transverse exceptional divisor $V\in Exc(\psi)$ and the
irreducible exceptional divisor $E\in Exc(\phi)$, the foliation $(-1)$-curve
$V$ of such a regular fibre $\mathbf{S}_{p}^{1}$ at a point $p\in M$ is a
regular $\mathbf{S}^{1}$-principal bundle over a compact Riemann sphere
$\mathbf{S}^{2}$. Moreover, $E$ is biholomorphic to $\mathbf{CP}^{1}$ and $V$
is transverse biholomorphic to $\mathbf{S}^{3}.$ Furthermore it follows from
(A.2) and (A.1) that

$E\cdot E=-1$

and then

$V\cdot V=-1.$

  3. (3)

For the foliation $(-1)$-curve blow-up map $\psi:\overline{M}\rightarrow M$,
it follows from (3.9), (3.10) and (3.11) that

(3.13) $\psi^{\ast}r_{\mathbf{S}_{p}^{1}}^{2}=|s|_{h}^{2}.$

In general, for the foliation $(-k)$-curve contraction map, we have

(3.14) $\psi^{\ast}r_{\mathbf{S}_{p}^{1}}^{2k}=|s|_{h}^{2}.$

In particular, let $X$ and $Y$ be smooth projective surfaces with the
Castelnuovo’s Contraction map $\phi:X\rightarrow Y,$ we have the following
lifting transverse contraction morphism $\psi:M\rightarrow N$ via the regular
$S^{1}$-principal bundles $\pi_{M}$ and $\pi_{N}$ as in

$\begin{array}[c]{ccl}M\supset
V&\overset{\psi}{\longrightarrow}&N\supset\mathbf{S}_{p}^{1}\\\
\downarrow\pi_{M}&\circlearrowright&\downarrow\pi_{N}\\\ X\supset
E&\overset{\phi}{\longrightarrow}&Y\ni\pi_{N}(p).\end{array}$

Then, as the consequence of Theorem 5, we have the Sasaki analogue of
Castelnuovo’s contraction Theorem 2 on Sasakian Five-Manifolds.

######  Remark 1.

In general for a Sasakian $3$-manifold, it is an $S^{1}$-Seifert $3$-manifold
and the Euler characteristic $e$ of the Seifert bundle is nonzero if it is
closed. For a Seifert bundle, the geometry is determined by the Euler
characteristic $\chi$ of the base $2$-orbifold ([Gei]) :

$\begin{tabular}[c]{llll}&$\chi>0$&$\chi=0$&$\chi<0$\\\
$e=0$&$S^{2}\times\mathbb{E}$&$\mathbb{E}^{3}=\mathbb{E}^{2}\times\mathbb{E}$&$H^{2}\times\mathbb{E}$\\\
$e\neq 0$&$S^{3}$&$Nil$&$\widetilde{SL(2,\mathbb{R)}}$\end{tabular}\ \ \ \ \ \
\ \ \ \ \ \ \ \ .\ \ $

######  Example 3.

As an example, $\mathbf{S}^{5}$ is equipped with a free
$\mathbf{S}^{1}$-action given by the Hopf bration:

$\begin{array}[c]{ccl}\mathbf{S}^{5}&\overset{H}{\longrightarrow}&\mathbf{CP}^{2}.\end{array}$

Then the blowing up along a regular fibre $\mathbf{S}_{p}^{1}$ in
$\mathbf{S}^{5}$ via the the gluing map gives us the $\mathbf{S}^{1}$-bundle

$\begin{array}[c]{ccl}\mathbf{S}^{5}\\#(\mathbf{S}^{3}\mathbf{\times
S}^{2})&\overset{\pi}{\longrightarrow}&\mathbf{CP}^{2}\\#\overline{\mathbf{CP}^{2}}\end{array}$

or

$\begin{array}[c]{ccl}\mathbf{S}^{5}\\#(\mathbf{S}^{3}\widetilde{\times}\mathbf{S}^{2})&\overset{\pi}{\longrightarrow}&\mathbf{CP}^{2}\\#\overline{\mathbf{CP}^{2}}.\end{array}$

In the case $N$ is simply-connected, the blowing up along a regular fibre
$\mathbf{S}_{p}^{1}$ is either $N\\#(\mathbf{S}^{3}\mathbf{\times S}^{2})$ or
$N\\#(\mathbf{S}^{3}\widetilde{\times}\mathbf{S}^{2})$, twisted
$\mathbf{S}^{3}$-bundle over $\mathbf{S}^{2}$, a non-spin manifold. More
precisely, since

$\pi_{1}(Diff(\mathbf{S}^{3}))\simeq\pi_{1}(SO(4))\simeq\mathbf{Z}_{2},$

the blowing up along $\mathbf{S}_{p}^{1}$ can produce at most two
$\mathbf{S}^{1}$-manifolds, depending on the gluing map. As an example,
$\mathbf{S}^{5}$ is equipped with a free $\mathbf{S}^{1}$-action given by the
negative Hopf fibration. Then blowing up along $\mathbf{S}_{p}^{1}$ via $f=id$
gives us the $\mathbf{S}^{1}$-bundle

$\begin{array}[c]{ccc}M\\#(\mathbf{S}^{3}\mathbf{\times
S}^{2})&\rightarrow&\mathbf{S}^{5}\\\ \downarrow&&\downarrow-H\\\
\mathbf{CP}^{2}\\#\overline{\mathbf{CP}^{2}}&\overset{}{\rightarrow}&\mathbf{CP}^{2}.\end{array}$

On the other hand, if we take the gluing map $f$ to be

$(t,[\begin{array}[c]{cccc}\cos t&-\sin t&0&0\\\ \sin t&\cos t&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{array}]),$

then

$\begin{array}[c]{ccc}M\\#(\mathbf{S}^{3}\widetilde{\times}\mathbf{S}^{2})&\rightarrow&\mathbf{S}^{5}\\\
\downarrow&&\downarrow-H\\\
\mathbf{CP}^{2}\\#\overline{\mathbf{CP}^{2}}&\overset{}{\rightarrow}&\mathbf{CP}^{2}.\end{array}$

###  3.3. Reid’s Model and Resolution of $\frac{1}{r}(1,a)$-type Foliation
Singularities

In general, for a foliation singularity of type $\frac{1}{r}(1,a),$ we can
reduce to a foliation singularity of type $\frac{1}{k}(1,1)$ in this
subsection. More precisely, given $p\in M_{\mathrm{sing}}$, the singular fiber
${\mathsf{S_{p}^{1}}}$ is a foliation singularity of type $\frac{1}{r}(1,a)$
at $p$, one can measure its complexity by the Hirzebruch-Jung continued
fraction.

######  Definition 9.

Given $p\in M_{\mathrm{sing}}$, the singular fiber ${\mathsf{S_{p}^{1}}}$ is
foliation singularity of type $\frac{1}{r}(1,a)$. The Hirzebruch Jung
continued fraction $[b_{1},\dots,b_{l}]$ of $\frac{r}{a}$ is defined by

$\frac{r}{a}=b_{1}-\frac{1}{b_{2}-\frac{1}{b_{3}-\dots}}.$

Then we define the length of the singular fiber to be $l$. Note that the
fraction can be calculated by the recursive formula:

$\displaystyle r$ $\displaystyle=ab_{1}-a_{1},$ $\displaystyle a$
$\displaystyle=a_{1}b_{2}-a_{2},$ $\displaystyle a_{i}$
$\displaystyle=a_{i+1}b_{i+2}-a_{i+2},$ (3.15) $\displaystyle\dots.$

In the paper of [W, section $2$], Wang presents a construction that reduce a
foliation singularity of type $\frac{1}{r}(1,a)$ to a foliation singularity of
type $\frac{1}{a}(1,a_{1})$ with $r=ab_{1}-a_{1}.$ More precisely, we consider
the five manifold ${\mathbf{C}}\times{\mathbf{S}^{3}}$ equipped with the
${\mathbf{S}^{1}}$-action

$\displaystyle\delta_{a}:{\mathbf{S}^{1}}\times({\mathbf{C}}\times{\mathbf{S}^{3}})$
$\displaystyle\rightarrow{\mathbf{C}}\times{\mathbf{S}^{3}}$
$\displaystyle(x,y_{1},y_{2})$
$\displaystyle\mapsto(t^{r}x,t^{-1}y_{1},t^{-a}y_{2}),$

where
$(x,y_{1},y_{2})\in{\mathbf{C}}\times{\mathbf{S}^{3}}\subset{\mathbf{C}}^{3}$.
$({\mathbf{C}}\times{\mathbf{S}^{3}},\delta_{a})$ has a singular fiber of type
$\frac{1}{a}(1,a_{1})$ with $\frac{a}{a_{1}}=[b_{2},\dots,b_{l}]$ and
$\frac{r}{a}=[b_{1},b_{2},\dots,b_{l}]$.

Now, given a Sasakian manifold $(M,\alpha),$ $p\in M_{\mathrm{sing}}$, the
singular fiber ${\mathsf{S_{\mathbf{p}}^{1}}}$ is of type $\frac{1}{r}(1,a)$
of lenth $l$. Then there exists an ${\mathbf{S}^{1}}$-equivariant neighborhood
$\overset{\circ}{\mathcal{N}}({\mathsf{S_{\mathbf{p}}^{1}}})$ of
${\mathsf{S_{\mathbf{p}}^{1}}}$ such that $\overset{\circ}{\mathcal{N}}$ is
${\mathbf{S}^{1}}$-equivariant to $(N,\nu)$ with
$N\simeq{\mathbf{S}^{1}}\times{\mathbf{C}}^{2}$ and $\nu$ given by

(3.16) $\begin{array}[c]{ccc}{\mathbf{S}^{1}}\times N&\rightarrow&N\\\
(t,u,v_{1},v_{2})&\mapsto&(t^{r}u,t^{-1}v_{1},t^{-a}v_{2}).\end{array}$

For the ${\mathbf{S}^{1}}$-equivariant embedding

$\displaystyle\Psi:(\overset{\circ}{\mathcal{N}}({\mathsf{S_{\mathbf{p}}^{1}}})-{\mathsf{S_{\mathbf{p}}^{1}}})\simeq(N-{\mathbf{S}^{1}}\times(0,0))$
$\displaystyle\rightarrow{\mathbf{C}}\times{\mathbf{S}^{3}}$ (3.17)
$\displaystyle(u,v_{1},v_{2})$
$\displaystyle\mapsto(u|v|,\frac{v_{1}}{|v|},\frac{v_{2}}{|v|}),$

where $v=(v_{1},v_{2})$, we glue the manifold

(3.18)
$(M^{\prime},\alpha^{\prime})=(M-\overset{\circ}{\mathcal{N}}({\mathsf{S_{\mathbf{p}}^{1}}}),\alpha)\underset{\Psi}{\cup}({\mathbf{C}}\times{\mathbf{S}^{3}},\delta_{a}).$

Then resulting new Sasakian manifold $(M^{\prime},\alpha^{\prime})$ has all
singular fibers the same as $(M,\alpha)$ except the singular fiber
${\mathsf{S_{\mathbf{p}}^{1}}}$ is now replaced with a (singular) fiber of
smaller length. Moreover, we have

$M^{\prime}\simeq M\\#{\mathbf{S}^{3}}\times{\mathbf{S}^{2}}$

topologically.

Furthermore, it follows from Reid’s model ([R]) on Kaehler surfaces that the
gluing map $\Psi$ descends to the gluing maps $(2.9)$ and $(2.10)$ of the
paper [W, section $2.2$]. Then we have the following the minimal resolution of
the general $\frac{1}{r}(1,a)$-type foliation singularities of $M$ which is
compatibable with the minimal resolution in case of Kaehler surfaces.

######  Theorem 6.

Let $(M,\xi,g)$ be a compact quasi-regular Sasakian $5$-manifold and $Z$
denote the space of leaves of the characteristic foliation which is a normal
projective orbifold surface of $\frac{1}{r}(1,a)$-type singularities. Let
$\phi:\overline{Z}\rightarrow Z$ be a resolution of singularities of $Z$ for a
nonsigular projective variety $\overline{Z}$ Then there exist a lifting
$\psi:\overline{M}\rightarrow M$ and an
$\overline{\pi}:\overline{M}^{5}\rightarrow\overline{Z}$ the regular
$S^{1}$-principal bundle over $\overline{Z}$ such that the following diagram
is commutative :

(3.19) $\begin{array}[c]{lcl}\overline{M}^{5}\supset
V_{i}&\overset{\psi}{\longrightarrow}&M^{5}\supset\mathbf{S}_{p}^{1}\\\
\downarrow\overline{\pi}&\circlearrowright&\downarrow\pi\\\
\overline{Z}\supset
E_{i}&\overset{\phi}{\longrightarrow}&Z\ni\pi(p),\end{array}$

where the irreducible transverse exceptional divisors $V_{i}\in Exc(\psi)$ and
the irreducible exceptional divisors $E_{i}\in Exc(\phi).$ $\psi$ is defined
to be the resolution of $\frac{1}{r}(1,a)$-type foliation singularities from
$\overline{M}$ to$M.$ Moreover, the Hirzebruch Jung continued fraction

$\frac{r}{a}=[b_{1},\cdot\cdot\cdot,b_{l}]$

gives the information on the foliation resolution
$\psi:\overline{M}\rightarrow M$ of $M$. The exceptional foliation
$(-b_{i})$-curves form a chain of $\\{V_{1},\cdot\cdot\cdot,V_{l}\\}$ such
that each $V_{i}$ has self intersection

$V_{i}^{2}=-b_{i}$

for every $i=1,...,l$ and $V_{i}$ intersects another foliation curve $V_{j}$
transversely only if $j=i-1$ or $j=i+1.$ In particular, for a foliation cyclic
quotient singularity of type $\frac{1}{k}(1,1),$ we have $\frac{k}{1}=[k]:$
the foliation $(-k)$-curve.

###### Proof.

Note that, for the quasi-regular Sasakian structure on $M$ with foliation
singularities of type $\frac{1}{r}(1,a)$ in which the well-formed leave
(foliation) space $Z$ has orbifold singularities as same as the algebraic
singularities

$\pi^{\ast}(K_{Z})=K_{M}^{T};\text{ \ \
}\overline{\pi}^{\ast}(K_{\overline{Z}})=K_{\overline{M}}^{T}$

and

$\psi^{\ast}\circ\pi^{\ast}=\overline{\pi}^{\ast}\circ\phi^{\ast}.$

It follows from (3.21) that

$\overline{\pi}^{\ast}(K_{\overline{Z}})=\overline{\pi}^{\ast}(\phi^{\ast}(K_{Z}))+\sum_{i}a_{i}\overline{\pi}^{\ast}([E_{i}]),$

where the sum is over the irreducible exceptional divisors $E_{i}\in
Exc(\phi)\subset\overline{Z}$, $\pi(p)\in Z$ such that

$\overline{\pi}(V_{i})=E_{i}$

with $\mathbf{S}_{p}^{1}\subset M,$ $p\in M.$

(i) Foliation singularity of type $\frac{1}{k}(1,1):$ It is the most simple
case. For a resolution of singularities of type $\frac{1}{k}(1,1)$ in $Z$

$\phi:\overline{Z}\rightarrow Z$

with a nonsigular projective surface $\overline{Z}$ and $E$ is the exceptional
curve of such resolution. Then, over the singular point
$\pi(\mathbf{S}_{p}^{1})$, the exceptional $(-k)$-curve $E$ has self
intersection

$E^{2}=-k.$

Then, from the previous section construction for $\mathcal{S}(-k)$ in
$\overline{M}^{5}$ as in subsection $3.3,$ it follows that there exists a
lifting $\psi:\overline{M}\rightarrow M$ as in Theorem 5. Thus there is an
$\overline{\pi}:\overline{M}^{5}\rightarrow\overline{Z}$ the regular
$S^{1}$-principal bundle over $\overline{Z}$ such that the following diagram
is commutative

(3.20)
$\begin{array}[c]{ccccc}\overline{M}^{5}\supset\mathcal{S}(-k)\thickapprox\mathbf{C}\times\mathbf{S}^{3}&\overset{\pi_{2}}{\rightarrow}&\mathbf{S}^{3}\thickapprox
V&\overset{\psi|_{V}}{\longrightarrow}&\mathbf{S}_{p}^{1}\in M\\\
\downarrow&&&&\downarrow\pi_{M}\\\ \overline{Z}\supset
O(-k)&\overset{\pi_{d}}{\rightarrow}&\mathbf{CP}^{1}\thickapprox
E&\overset{\phi|_{E}}{\longrightarrow}&\pi_{M}(p)\in Z.\end{array}$

Hence the exceptional foliation $(-k)$-curve $V$ has self intersection

$V^{2}=-k.$

(ii) Foliation singularity of type $\frac{1}{r}(1,a):$ The similar situation
as in (i). Let

$\phi:\overline{Z}\rightarrow Z$

be a resolution of singularities of type $\frac{1}{r}(1,a)$ in $Z$ with a
nonsigular projective surface $\overline{Z}$ and $E_{i}$ be the exceptional
curves of such resolution. Then it follows from M. Reid ([R]) thar the
Hirzebruch Jung continued fraction

$\frac{r}{a}=[b_{1},\cdot\cdot\cdot,b_{l}]$

gives the information on the resolution $\phi:\overline{Z}\rightarrow Z$. More
precisely, over the singular point $\pi(\mathbf{S}_{p}^{1})$, the exceptional
curves form a chain of $\\{E_{1},\cdot\cdot\cdot,E_{l}\\}$ such that each
$E_{i}$ has self intersection

$E_{i}^{2}=-b_{i}$

for every $i=1,...,l$ and the $(-b_{i})$-curve $E_{i}$ intersects another
$(-b_{j})$-curve $E_{j}$ transversely only if $j=i-1$ or $j=i+1.$

Again we consider the resolution of foliation singularities of type
$\frac{1}{r}(1,a)$

$\psi:\overline{M}\rightarrow M$

in $M$ with a regular Sasakian $5$-manifold $\overline{M}$. Let $V_{i}$ be the
exceptional foliation curves of $\psi$ lifting from $E_{i}$ via the diagram
(3.20) such that

$\pi(V_{i})=E_{i}$

Then, by the previous construction (3.18) of foliation singularity of type
$\frac{1}{r}(1,a)$ over the singular fiber $\mathbf{S}_{p}^{1}$ as in
subsection $3.4.$, the exceptional foliation curves form a chain of
$\\{V_{1},\cdot\cdot\cdot,V_{l}\\}$ such that each $V_{i}$ has self
intersection

$V_{i}^{2}=-b_{i}$

for every $i=1,...,l$ and the foliation $(-b_{i})$-curve $V_{i}$ intersects
another foliation $(-b_{j})$-curve $V_{j}$ transversely only if $j=i-1$ or
$j=i+1.$ We refer to (A.2), (A.1) and [CHLW, $(6.3)$] for some details. ∎

Here we come out some definition on the $\frac{1}{r}(1,a)$-type foliation
singularities of $M:$

######  Definition 10.

Let $(M,\xi,g)$ be a compact quasi-regular Sasakian $5$-manifold and $Z$
denote the space of leaves of the characteristic foliation which is a normal
projective orbifold surface of $\frac{1}{r}(1,a)$-type singularities. Let
$\psi:\overline{M}\rightarrow M$ be a minimal resolution of
$\frac{1}{r}(1,a)$-type foliation singularities from $\overline{M}$ to $M.$ If

(3.21) $K_{\overline{M}}^{T}=\psi^{\ast}(K_{M}^{T})+\sum_{i}a_{i}[V_{i}]_{B},$

where the sum is over the irreducible transverse exceptional divisors
$V_{i}\in Exc(\psi)$ over $\mathbf{S}_{p}^{1}\subset M$ and the $a_{i}$ are
rational numbers, called the discrepancies. Then the $\frac{1}{r}(1,a)$-type
foliation singularities of $M$ are called :

  1. (1)

terminal if $a_{i}>0$ for all $i.$

  2. (2)

canonical if $a_{i}\geq 0$ for all $i.$

  3. (3)

log terminal if $a_{i}>-1$ for all $i.$

  4. (4)

log canonical if $a_{i}\geq-1$ for all $i$.

##  4\. The Sasaki-Ricci Flow Through Singularities

In this section, we warm up some basic facts of the Sasaki analogue of the
Kaehler-Ricci flow through singularities due to Song-Tian ([ST]).

###  4.1. The Sasaki-Ricci Flow

By a $\partial_{B}\overline{\partial}_{B}$-Lemma ([EKA]]) in the basic Hodge
decomposition, there is a basic function $F:M\rightarrow\mathbb{R}$ such that

$\rho^{T}(x,t)-\varkappa
d\eta(x,t)=d_{B}d_{B}^{c}F=i\partial_{B}\overline{\partial}_{B}F.$

We focus on finding a new $\eta$-Einstein Sasakian structure
$(M,\xi,\widetilde{\eta},\widetilde{\Phi},\widetilde{g})$ with

$\widetilde{\eta}=\eta+d_{B}^{c}\varphi,\varphi\in\Omega_{B}^{0}$

and

$\widetilde{g}^{T}=(g_{i\overline{j}}^{T}+\varphi_{i\overline{j}})dz^{i}d\overline{z}^{j}=2i(h_{i\overline{j}}+\frac{1}{2}\varphi_{i\overline{j}})dz^{i}\wedge
d\overline{z}^{j}$

such that

$\widetilde{\rho}^{T}=\varkappa d\widetilde{\eta}.$

Hence

$\widetilde{\rho}^{T}-\rho^{T}=\kappa d_{B}d_{B}^{c}\varphi-d_{B}d_{B}^{c}F$

and it follows that

(4.1)
$\frac{\det(g_{\alpha\overline{\beta}}^{T}+\varphi_{\alpha\overline{\beta}})}{\det(g_{\alpha\overline{\beta}}^{T})}=e^{-\kappa\varphi+F}.$

This is a Sasakian analogue of the Monge-Ampere equation for the orbifold
version of Calabi-Yau Theorem (([EKA]]).

Now we consider the Sasaki-Ricci flow on $M\times[0,T)$

(4.2) $\frac{d}{dt}g^{T}(x,t)=-(Ric^{T}(x,t)-\varkappa g^{T}(x,t))$

or

$\frac{d}{dt}d\eta(x,t)=-(\rho^{T}(x,t)-\varkappa d\eta(x,t)).$

It is equivalent to consider

(4.3)
$\frac{d}{dt}\varphi=\log\det(g_{\alpha\overline{\beta}}^{T}+\varphi_{\alpha\overline{\beta}})-\log\det(g_{\alpha\overline{\beta}}^{T})+\kappa\varphi-F.$

Note that,for any two Saskian structures with the fixed Reeb vector field
$\xi,$ we have

$Vol(M,g)=Vol(M,g^{\prime})$

and

$\widetilde{\omega}^{n}\wedge\eta=i^{n}\det(g_{\alpha\overline{\beta}}^{T}+\varphi_{\alpha\overline{\beta}})dz^{1}\wedge
d\overline{z}^{1}\wedge...\wedge dz^{n}\wedge d\overline{z}^{n}\wedge dx.$

With all the above discussiones, let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-
regular Sasakian manifold of dimension $2n+1$ and $Z$ denote the space of
leaves of the characteristic foliation $\mathcal{F}_{\xi}$. There is an
orbifold Riemannian submersion, and a principal $S^{1}$-orbibundle
($V$-bundle) $\pi:(M,g,\omega)\rightarrow(Z,h,\omega_{h})$ with
$\omega=\pi^{\ast}(\omega_{h}).$ Now for the natual projection

(4.4) $\Pi:(C(M),\overline{g},J,\overline{\omega})\rightarrow(Z,h,\omega_{h})$

with $\Pi|_{(M,g,\omega)}=\pi,$ then we have the volume form of the Kaehler
cone metric on the cone $C(M):$

$\overline{\omega}^{n+1}=r^{2n+1}(\Pi^{\ast}\omega_{h})^{n}\wedge
dr\wedge\overline{\eta}$

and the volume form of the Sasaki metric on $M:$

(4.5) $i_{\frac{\partial}{\partial
r}}\overline{\omega}^{n+1}=(\Pi^{\ast}\omega_{h})^{n}\wedge\eta.$

###  4.2. Basic Cohomology Characterization of the Maximal-Time Solution

Let $(M,\xi_{0},\eta_{0},\Phi_{0},g_{0},\omega_{0})$ be a compact quasi-
regular Sasakian $(2n+1)$-manifold and its leave space $Z$ of the
characteristic foliation be well-formed. We consider a solution
$\omega=\omega(t)$ of the Sasaki-Ricci flow

(4.6) $\begin{array}[c]{c}\frac{\partial}{\partial
t}\omega(t)=-\mathrm{Ric}^{T}(\omega(t)),\text{
}\omega(0)=\omega_{0}.\end{array}$

As long as the solution exists, the cohomology class $[\omega(t)]_{B}$ evolves
by

$\begin{array}[c]{c}\frac{\partial}{\partial
t}\left[\omega(t)\right]_{B}=-c_{1}^{B}(M),\text{
}\left[\omega(0)\right]_{B}=\left[\omega_{0}\right]_{B},\end{array}$

and solving this ordinary differential equation gives

$\begin{array}[c]{c}\left[\omega(t)\right]_{B}=\left[\omega_{0}\right]_{B}-tc_{1}^{B}(M).\end{array}$

We see that a necessary condition for the Sasaki-Ricci flow to exist for $t>0$
such that

$\left[\omega_{0}\right]_{B}-tc_{1}^{B}(M)>0.$

This necessary condition is in fact sufficient. In fact we define

$\begin{array}[c]{c}T_{0}:=\sup\\{t>0|\text{
}\left[\omega_{0}\right]_{B}-tc_{1}^{B}(M)>0\\}.\end{array}$

That is to say that

(4.7) $\left[\omega_{0}\right]_{B}-T_{0}c_{1}^{B}(M)\in\overline{C_{M}^{B}}$

which is a nef class.

For a representative $\chi\in-C_{1}^{B}(M)$, we can fix the adapted measure
$\Omega$ on the leave space $Z$ and then a volume form $\Omega\wedge\eta_{0}$
on $(M,\xi_{0},\eta_{0},\Phi_{0},g_{0},\omega_{0})$ such that

(4.8) $\Omega\wedge\eta_{0}=(\sqrt{-1})^{n}F(z_{1},...,z_{n})dz_{1}\wedge
d\overline{z}_{1}\wedge...\wedge dz_{n}\wedge d\overline{z}_{n}\wedge dx$

with

$\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log F=-Ric^{T}(\Omega)=\chi$

and

$\int_{M}\Omega\wedge\eta_{0}=\int_{M}\omega_{0}^{n}\wedge\eta_{0}.$

We choose a reference (transverse) Kaehler metric

$\widehat{\omega}_{t}:=\omega_{0}+t\chi.$

Then the corresponding transverse parabolic Monge-Ampere equation for the
basic function $\varphi(x,t)$ to (4.6) on $M\times[0,T_{0})$ is

(4.9) $\left\\{\begin{array}[c]{rll}\frac{\partial}{\partial
t}\varphi(x,t)&=&\log\frac{(\widehat{\omega}_{t}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi)^{n}\wedge\eta_{0}}{\Omega\wedge\eta_{0}},\\\
\widehat{\omega}_{t}&=&\omega_{0}+t\chi.\\\
\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log\Omega&=&\chi,\\\
\widehat{\omega}_{t}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi&>&0,\\\
\varphi(0)&=&0.\end{array}\right.$

Based on [SW1], [T], [CHLW] and references therein as in the Kaehler case, we
have the following cohomological characterization for the maximal solution of
the Sasaki-Ricci flow :

######  Theorem 7.

There exists a unique maximal solution $\omega(t)$ of the Sasaki-Ricci flow
(4.6) on $M\times[0,T_{0})$ for $t\in[0,T_{o})$.

###  4.3. The Weak Sasaki–Ricci Flow

Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian $5$-manifold
with finite cyclic quotient foliation singularities of type
$\frac{1}{r}(1,a).$ Such singularities are rather mild and they do not become
worse after divisorial contractions are performed in the foliation minimal
model program we considered. More precisely, $\overline{M}\rightarrow M$ be a
resolution of foliation singularity, then the pullback of any volume measure
$\Omega\wedge\eta$ on $M$ is $L^{p}$-integrable on the nonsingular model
$\overline{M}$ for some $p>1$.

######  Definition 11.

Let $(M,\eta,\xi,\Phi,g)$ be a compact quasi-regular Sasakian $5$-manifold
with finite cyclic quotient foliation singularities of type $\frac{1}{r}(1,a)$
and $H^{T}$ be a big and semi-ample $Q$-divisor with a basic transverse
birationalmorphism $\Psi_{mH^{T}}:M\rightarrow\mathbf{CP}^{N_{m}}$ induced by
the linear system $|mH^{T}|$ for some $m>>0$. Let
$\overline{\omega}=\frac{1}{m}\Psi^{\ast}(\omega_{FS})\in[H^{T}]$ and an
volume measure $\Omega\wedge\eta$ on $M$, where $\omega_{FS}$ is the Fubini-
Study metric on $\mathbf{CP}^{N_{m}}$ and define for $p\in(1,\infty]$,
$\varphi$ is the basic function

$\mathcal{PSH}_{p}(M,\overline{\omega},\Omega\wedge\eta)=\\{\varphi\in\mathcal{PSH}(M,\overline{\omega})\cap
L^{\infty}|\frac{(\overline{\omega}+i\partial_{B}\overline{\partial}_{B}\varphi)^{n}}{\Omega\wedge\eta}\in
L^{p}(M,\Omega\wedge\eta)\\}$

and

$K_{H_{T_{0},,}^{T}p}(M)=\\{\overline{\omega}+i\partial_{B}\overline{\partial}_{B}\varphi|\varphi\in\mathcal{PSH}_{p}(M,\overline{\omega},\Omega\wedge\eta)\\}.$

Now we can define the weak Sasaki-Ricci flow on a compact quasi-regular
Sasakian $5$-manifold for which its leave space $Z$ is normal projective
surface with mild singularities.

######  Definition 12.

Let $(M,\xi,\eta_{0})$ be a compact quasi-regular Sasakian $5$-manifold and
its leave space $Z$ be normal projective variety with finite cyclic quotient
singularities of type $\frac{1}{r}(1,a)$ which are klt singularities and
$H^{T}$ be a big and semi-ample $Q$-divisor so that $H^{T}+tK_{M}^{T}$ is
ample for a small $t\in Q^{+}$ such that

$T_{0}=\sup\\{t>0\mathbf{\ |\ }H^{T}+tK_{M}^{T}\text{ {nef }}\\}>0.$

A family of closed basic semi-positive $(1,1)$-currents $\omega(t,\cdot)$ on
$M$ for $t\in[0,T_{0})$ are said to be a solution of the weak Sasaki–Ricci
flow starting $\omega_{0}\in K_{H_{,,}^{T}p}(M)$ for some $p>1$ if the
following conditions hold

  1. (1)

$\omega\in C^{\infty}((0,T_{0})\times M_{\operatorname{reg}})$ and $\varphi\in
L^{\infty}([0,T]\times M),$

  2. (2)

$\omega(t,\cdot)$ satisfies

(4.10) $\left\\{\begin{array}[c]{lcc}\frac{\partial}{\partial
t}\omega=-Ric^{T}(\omega)&\mathrm{on}&(0,T_{0})\times
M_{\operatorname{reg}},\\\
\omega(0)=\omega_{0}&\mathrm{on}&M.\end{array}\right.$

Let $(M,\xi,\eta_{0})$ be a compact quasi-regular Sasakian $5$-manifold and
the space of leaves $Z$ be normal projective surface with klt singularities
and $H^{T}$ be a big and semi-ample $Q$-divisor. There exists a basic
transverse holomorphic map $\Psi:M\rightarrow(\mathbf{CP}^{N},\omega_{FS})$
defined by the basic transverse holomorphic section
$\\{s_{0},s_{1},...s_{N}\\}$ of $H^{0}(M.(K_{M}^{T})^{m})$ which is
$S^{1}$-equivariant with respect to the weighted $\mathbf{C}^{\ast}$action in
$\mathbf{C}^{N+1}$ with $N=\dim H^{0}(M.(K_{M}^{T})^{m})-1$ for a large
positive integer $m$ and
$\widehat{\omega}_{\infty}=\frac{1}{m}\Psi^{\ast}(\omega_{FS})\in[H_{0}^{T}]$
and an adapted measure $\Omega_{Z}$ on $Z$ as in [ST, Theorem 4.3] with
$\Omega=\pi^{\ast}(\Omega_{Z})$, where $\omega_{FS}$ is the Fubini-Study
metric on $\mathbf{CP}^{N_{m}}.$ We also denote

$\chi=\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log\Omega\in-c_{1}^{B}(M)$

and

$\widehat{\omega}_{t}=\widehat{\omega}_{\infty}+t\chi.$

The Sasaki-Ricci flow is equivalent to the following Monge-Ampere flow for the
basic function $\varphi(x,t)$ :

(4.11) $\left\\{\begin{array}[c]{lll}\frac{\partial}{\partial
t}\varphi(x,t)&=&\log\frac{(\widehat{\omega}_{t}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi)^{n}\wedge\eta_{0}}{\Omega\wedge\eta_{0}},\\\
\widehat{\omega}_{t}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi&>&0,\\\
\varphi(0)&=&\varphi_{0}.\end{array}\right.$

In order to define the Monge-Ampere flow (4.11) on $M,$ one might want to lift
the equation to the regular Sasakian manifold $\overline{M}$ and its foliation
space $\overline{Z}$ is a smooth projective surfaces. However,
$\widehat{\omega}_{Z}$ is not Kaehler on $\overline{Z}$ and $\Omega_{Z}$ in
general vanishes or blow up along $E_{i}\in Exc(f)$, so $\widehat{\omega}$ is
not transverse Kaehler on $\overline{M}$ and $\Omega$ in general vanishes or
blow up along $V_{i}\in Exc(\overline{f}).$ Then the lifted equation is
degenerate near the exceptional locus $Exc(\overline{f})$. we have to perturb
the Monge–Ampere equation (4.11) to (4.12) and obtain uniform estimates so
that the equation descend to $M$.

######  Theorem 8.

Let $(M,\xi,\eta_{0})$ be a compact quasi-regular Sasakian $5$-manifold and
its leave space $Z$ be normal projective variety with finite cyclic quotient
singularities of type $\frac{1}{r}(1,a)$ which are klt singularities and
$\varphi_{0}\in\mathcal{PSH}_{p}(M,\widehat{\omega},\Omega)$ for some $p>1$.
Define the minimal resolution of foliation singularities of $M$ to be
$\overline{f}:\overline{M}\rightarrow M$. Then the Monge-Ampere flow on
$\overline{M}$ defined by

(4.12) $\left\\{\begin{array}[c]{lll}\frac{\partial}{\partial
t}\overline{\varphi}(x,t)&=&\log\frac{(\overline{f}^{\ast}\widehat{\omega}_{t}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\overline{\varphi})^{n}\wedge\eta_{0}}{\overline{f}^{\ast}\Omega\wedge\eta_{0}},\\\
\overline{f}^{\ast}\widehat{\omega}_{t}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\overline{\varphi}&>&0,\\\
\overline{\varphi}(0)&=&\overline{f}^{\ast}\varphi_{0}.\end{array}\right.$

has a unique solution

$\overline{\varphi}\in C^{\infty}((0,T_{0})\times\overline{M}\backslash V)\cap
C^{0}([0,T_{0})\times\overline{M}\backslash V)$

such that

$\overline{\varphi}\in
L^{\infty}(\overline{M})\cap\mathcal{PSH}(\overline{M},\overline{f}^{\ast}\omega_{t}),$

for all $t\in[0,T_{0}).$ Moreover, $\overline{\varphi}$ is constant along each
fibre of $\overline{f}$, and so $\overline{\varphi}$ descends to a unique
solution

$\varphi\in C^{\infty}((0,T_{0})\times M_{\operatorname{reg}})\cap
C^{0}([0,T_{0})\times M_{\operatorname{reg}})$

of the Monge–Ampere flow (4.11) such that

$\varphi\in C^{0}(M)\cap\mathcal{PSH}(M,\omega_{t})$

for each $t\in[0,T_{0}).$

###### Proof.

As notions in Proposition 1, for a transverse Kaehler metric $\omega_{Z}$ on
the foliation space $Z=M/\mathcal{F}_{\xi}$ such that

(i) $\omega_{Z}\in C^{\infty}((0,T_{0})\times Z_{\text{{reg}}})$ and
$\varphi\in L^{\infty}([0,T]\times Z),$

(ii) $\omega_{Z}(t,\cdot)$ satisfies

(4.13) $\left\\{\begin{array}[c]{lcl}\frac{\partial}{\partial
t}\omega_{Z}=-Ric(\omega_{Z})&\mathrm{on}&(0,T_{0})\times
Z_{\operatorname{reg}},\\\
\pi^{\ast}(\omega_{Z}(0))=\omega_{0}&\mathrm{on}&Z.\end{array}\right.$

It follows from Proposition 1 that

$\pi:(M,\xi,\eta,g)\rightarrow(Z,h,\omega_{h})$

is an orbifold Riemannian submersion and a principal $S^{1}$-orbibundle
($V$-bundle) over $Z$ with

$\frac{1}{2}d\eta=\pi^{\ast}(\omega).$

Then the basic transverse Kaehler form

$\omega=\pi^{\ast}(\omega_{Z})$

satisfies the flow (4.10).

Let $f:\overline{Z}\rightarrow Z$ be a minimal resolution of singularities of
$Z$ for a nonsigular projective variety $\overline{Z}$ and
$\overline{\pi}:\overline{M}\rightarrow\overline{Z}$ is the regular
$S^{1}$-principal bundle over $\overline{Z}$ and regular Sasakian manifold
$\overline{M}.$ The minimal resolution of foliation singularities of $M$ to be
$\overline{f}:\overline{M}\rightarrow M$ such that the following diagram is
commutative :

(4.14) $\begin{array}[c]{ccc}\overline{M}\supset
V_{i}&\overset{\overline{f}}{\longrightarrow}&M\supset S_{p}^{1}\\\
\downarrow\overline{\pi}&\circlearrowright&\downarrow\pi\\\
\overline{Z}\supset E_{i}&\overset{f}{\longrightarrow}&Z\ni\pi(p).\end{array}$

Then we have

$K_{\overline{M}}^{T}=\overline{f}^{\ast}(K_{M}^{T})+\sum_{i}a_{i}V_{i},$

where $V_{i}\in Exc(\overline{f})\subset\overline{M}$ over
$\mathbf{S}_{p}^{1}\subset M$ and $a_{i}>-1$ for all $i.$

As in [ST, Theorem 4.2], they lifted (4.13) to the minimal resolution
$\overline{Z}$ of singularities of $Z$ to have the desired estimates. Here we
have to lift the Monge–Ampere equation (4.11) to (4.12). Thus we obtain the
estimates via the diagram (4.14). ∎

As a consequence, we have the following existence theorem for the solution of
the weak Sasaki-Ricci flow (4.10).

######  Corollary 3.

([ST, Theorem 4.3]) Let $(M,\xi,\eta_{0})$ be a compact quasi-regular Sasakian
$5$-manifold and its leave space $Z$ be normal projective surface with finite
cyclic quotient orbifold singularities of type $\frac{1}{r}(1,a)$ which are
klt singularities. Let $H^{T}$ be an ample $Q$-divisor such that

$T_{0}=\sup\\{t>0\mathbf{\ |\ }H^{T}+tK_{M}^{T}\text{ {nef }}\\}>0.$

If $\omega_{0}\in K_{H_{,,}^{T}p}(M)$ for some $p>1$, then there exists a
unique solution $\omega(t,\cdot)$ of the weak Sasaki–Ricci flow (4.10) for
$t\in[0,T_{0})$. Moreover, $\omega(t,\cdot)$ is a smooth transverse orbifold
Kaehler-metric on $M$ for $t\in(0,T_{0})$ and so the weak Sasaki-Ricci flow
becomes the smooth orbifold Sasaki-Ricci flow on $M$ immediately when $t>0$.

###  4.4. The Sasaki-Ricci Flow Through Divisorial Contractions

Now based on the proofs of Theorem 8, Corollary 3 and [ST, Theorem 5.3], the
weak Sasaki-Ricci flow (4.10) can be continued through divisorial contractions
on a compact quasi-regular Sasakian $5$-manifold $M$.

######  Proposition 3.

Let $(M,\xi,\eta_{0})$ be a compact quasi-regular Sasakian $5$-manifold and
its leave space $Z$ be normal projective surface with finite cyclic quotient
orbifold singularities of type $\frac{1}{r}(1,a)$ which are klt singularities.
Let $H^{T}$ be an ample $Q$-divisor such that

$T_{0}=\sup\\{t>0\mathbf{\ |\ }H^{T}+tK_{M}^{T}\text{ {nef }}\\}>0.$

is the first singular time and $L_{T_{0}}^{T}:=H^{T}+tK_{M}^{T}$ is the semi-
ample basic divisorwhich induces a divisoral contraction

$\psi_{(L_{T_{0}}^{T})^{m}}:M\rightarrow
N\subset\mathcal{P(}H_{B}^{0}(M,(L_{T_{0}}^{T})^{m})),$

for some $m>>1.$ If $\omega_{0}\in K_{H^{T},p}(M)$ for some $p>1$, Let
$\omega(t,\cdot)$ be the unique solution $\omega(t,\cdot)$ of the weak
Sasaki–Ricci flow (4.10) for $t\in[0,T_{0})$If $\omega_{0}\in K_{H^{T},p}(M)$
for some $p>1$, starting with $\omega_{0}\in K_{H^{T},p}(M)$ for some $p>1$.
Then there esists $\omega_{N}(t,\cdot)$ such that

  1. (1)

$\omega_{N}\in K_{L_{T_{0}}^{T},p^{\prime}}(N)\cap
C^{\infty}(N_{\operatorname{reg}}\backslash\psi(Exc(\psi))$

for some $p^{\prime}>1$.

  2. (2)

$\omega(t,\cdot)\rightarrow\psi^{\ast}\omega_{N}$

in $C^{\infty}(M_{\operatorname{reg}}\backslash Exc(\psi))$-topology as
$t\rightarrow T_{0}^{-}.$

  3. (3)

There exists a unique solution $\omega(t,\cdot)$ of the weak Sasaki-Ricci flow
on $N$ starting with $\omega_{N}$ at $T_{0}$ for $t\in(T_{0},T_{N})$ with
$T_{0}<T_{N}\leq\infty$, such that

$\omega(t,\cdot)\rightarrow\omega_{N}$

in $C^{\infty}(M_{\operatorname{reg}}\backslash Exc(\psi))$-topology as
$t\rightarrow T_{0}^{+}.$

Therefore the weak Sasaki-Ricci flow can be uniquely continued on $N$ starting
with $\omega_{N}$ at $T_{0}.$

In particular, we have Definition 2 for the floating foliation canonical
surgical contraction on a compact quasi-regular Sasakian $5$-manifold.

##  5\. Foliation MMP with Scaling via the Sasaki-Ricci Flow

Since the Reeb vector field and the transverse holomorphic structure are both
invariant, all the quantities in this section are only involved with the
transverse Kähler structure $\omega(t)$ and basic tensors. Hence, under the
Sasaki-Ricci flow, when one applies for instance, the maximal principle and
foliation resolutions etc, the expressions involved behave essentially the
same as the Kähler-Ricci flow.

###  5.1. Canonical Surgical Contraction for Floating Foliation $(-1)$-Curves

In this subsection, primarily along the lines of the arguments in [SW2], we
are ready to give the proof of Theorem 3. We write the Sasaki-Ricci flow (4.6)
as a transverse parabolic complex Monge-Ampére equation. First, using the
assumption (4.2), define a family of reference transverse Kähler metrics
$\widehat{\omega}_{t}$ for $t\in[0,T_{0})$ by

(5.1)
$\begin{array}[c]{c}\widehat{\omega}_{t}=\frac{1}{T_{0}}((T_{0}-t)\omega_{0}-t\psi^{\ast}\omega_{N})\in[\omega(t)]_{B}=[\omega_{0}]_{B}-tc_{1}^{B}(M).\end{array}$

We can fix the adapted measure $\Omega$ on the leave space $Z$ and then a
volume form $\Omega\wedge\eta_{0}$ on $M$ such that

(5.2)
$\begin{array}[c]{c}\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log\Omega=\frac{\partial}{\partial
t}\widehat{\omega}_{t}=-\frac{1}{T_{0}}(\omega_{0}+\psi^{\ast}\omega_{N})\in
c_{1}^{B}(M),\text{ }\int_{M}\Omega\wedge\eta_{0}=1.\end{array}$

Then the corresponding transverse parabolic Monge-Ampére equation for the
basic function $\varphi=\varphi(t)$ to (4.9) on $M\times[0,T_{0})$ is

(5.3) $\begin{array}[c]{c}\frac{\partial}{\partial
t}\varphi=\log\frac{(\widehat{\omega}_{t}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi)^{2}\wedge\eta_{0}}{\Omega\wedge\eta_{0}},\text{
}\varphi(0)=0.\end{array}$

####  5.1.1. Key Estimates for the Sasaki-Ricci Flow

In the following, by using the Sasaki analogue of Koaira Lemma ([CHLW,
Proposition 3.3]) under assumption (1.2) and (A.3), we prove the main
estimates for the Sasaki-Ricci flow under the assumptions of Theorem 3. In
particular, the estimates of the first two lemmas are essentially contained in
[CHLW].

######  Lemma 2.

There is a uniform constant $C$ depending only on $(M,\omega_{0})$ such that
the solution $\varphi=\varphi(t)$ of (5.3) satisfies, for $t\in[0,T_{0}),$

$\left\|\varphi\right\|_{L^{\infty}}\leq C,\text{
}\varphi^{\prime}=\partial\varphi/\partial t\leq C\text{ }\mathrm{and}\text{
}\omega^{2}\wedge\eta_{0}\leq C\Omega\wedge\eta_{0}.$

As $t\rightarrow T_{0}^{-}$, $\varphi(t)$ converges pointwise on $M$ to a
bounded basic function $\varphi_{T}$ satisfying

$\begin{array}[c]{c}\omega_{T_{0}}:=\widehat{\omega}_{T_{0}}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi_{T_{0}}\geq
0,\end{array}$

and $\omega(t)$ converges weakly in the sense of currents to the closed
positive $(1,1)$ current $\omega_{T_{0}}$.

We have the $C^{\infty}$ estimates for the solution $\omega(t)$ of the Sasaki-
Ricci flow on compact subsets of $M\backslash V$ as following.

######  Lemma 3.

With the assumptions of Theorem 3, the solution $\omega=\omega(t)$ of the
Sasaki-Ricci flow (4.6) satisfies

(i) There exists a uniform constant $c>0$ such that

(5.4) $\begin{array}[c]{c}\omega\geq c\psi^{\ast}\omega_{N}.\end{array}$

(ii) For every compact subset $K\subset M\backslash V$, there exist constants
$C_{K,k}$ for $k=0,1,2,...,$ such that

(5.5) $\begin{array}[c]{c}\left\|\omega(t)\right\|_{C^{k}(K,\omega_{0})}\leq
C_{K,k}.\end{array}$

(iii) The closed $(1,1)$ current $\omega_{T_{0}}$, given in Lemma 2, is a
smooth transverse Kähler metric on $M\backslash V$.

(iv) As $t\rightarrow T_{0}^{-}$, the metrics $\omega(t)$ converge to
$\omega_{T_{0}}$ in $C^{\infty}$ on compact subsets of $M\backslash V$.

Then it follows from Lemma 3 that part $(1)$ in the Definition 2 of foliation
canonical surgical contraction holds.

We now compare $\psi^{\ast}\omega_{N}$ with a fixed transverse Kähler metric
$\omega_{0}$ on $M$ by using the description of the neighborhood of $V$ in the
blow-up map

$\begin{array}[c]{c}\psi:M\rightarrow N.\end{array}$

As the notion in section $2,$ for $p=(x_{i},v_{i},s_{i})\in
U_{i}\subset\mathbb{C}\times\mathbb{C}\times S^{1},$ $i=1,2,$ the composition

$\begin{array}[c]{ccc}\mathcal{S}(-1)\thickapprox\mathbb{C}\times
S^{3}&\overset{\pi_{2}}{\rightarrow}&S^{3}\\\ &&\downarrow H\\\
&&CP^{1}\end{array}$

send $p$ to $[l_{1},l_{2}].$ we have $x_{1}=\frac{l_{2}}{l_{1}};$
$x_{2}=\frac{l_{1}}{l_{2}}$, and $v_{i},$ $i=1,2$ correspond to vectors in a
fiber of $\mathcal{O}(-1).$ Now $s$ is the section of $[V\mathcal{]}$ over
$\mathcal{S}(-1)$ given by

$\begin{array}[c]{c}s_{i}:U_{i}\rightarrow\mathbb{C}\mathbf{,}\text{\
}s_{i}=v_{i}\end{array}$

and $h$ is the Hermitian metric on $[V]$ such that

$\begin{array}[c]{c}h_{i}=(\frac{|l_{1}|^{2}+|l_{2}|^{2}}{|l_{i}|^{2}})=(1+|x_{i}|^{2})\end{array}$

on $U_{i}.$ Then we have

(5.6)
$\begin{array}[c]{c}\mathrm{dist}^{2}(p,S_{0}^{3})=(\frac{|l_{1}|^{2}+|l_{2}|^{2}}{|l_{i}|^{2}})|v_{i}|^{2}:=|s|_{h}^{2}\end{array}$

On the other hand, the distance square to the singular fiber $S_{0}^{1}$ is

(5.7)
$\begin{array}[c]{c}\mathrm{dist}^{2}(p,S_{0}^{1})=|z_{1}|^{2}+|z_{2}|^{2}:=r_{S^{1}}^{2}.\end{array}$

If $\psi:\mathbb{C}^{1}\times S^{3}\rightarrow S^{1}\times\mathbb{C}^{2}$ is
the composition of

$\begin{array}[c]{ccccc}S(-k)\thickapprox\mathbb{C}\times
S^{3}&\overset{\pi_{2}}{\rightarrow}&S^{3}&\overset{}{\longrightarrow}&S^{1}\times\mathbb{C}^{2}\supset
S^{1}\times D^{4},\end{array}$

it follows from (5.6) and (5.7) that the function $|s|_{h}^{2}$ on
$\psi^{-1}(S^{1}\times D_{1/2}^{4})$ is given by

(5.8) $\begin{array}[c]{c}\psi^{\ast}r_{S^{1}}^{2}=|s|_{h}^{2}.\end{array}$

and then write

$|s|_{h}^{2}(x)=r_{S^{1}}^{2}=|z_{1}|^{2}+|z_{2}|^{2}$

for $\psi(x)=(s,z_{1},z_{2}),$ $s\in S^{1}.$ Hence the transverse Ricci
curvature of $h$ is given by

$\begin{array}[c]{c}R_{h}^{T}=-\frac{\sqrt{-1}}{2\pi}\partial_{B}\overline{\partial}_{B}\log(|z_{1}|^{2}+|z_{2}|^{2})\end{array}$

on $\psi^{-1}(S^{1}\times D_{1/2}^{4}\backslash\\{0\\}).$

We have the following lemmas.

######  Lemma 4.

For sufficiently small $\varepsilon_{0}>0$,

$\omega_{M}:=\psi^{\ast}\omega_{N}-\varepsilon_{0}R_{h}^{T}$

is a transverse Kähler form on $M$. Furthermore, in term of Sasaki normal
coordinate (2.1)

(5.9)
$\begin{array}[c]{c}\omega_{M}:=\psi^{\ast}\omega_{N}+\frac{\sqrt{-1}}{2\pi}\frac{\varepsilon_{0}}{r_{S^{1}}^{2}}\sum_{i,j=1}^{2}(\delta_{ij}-\frac{\overline{z}_{i}z_{j}}{r_{S^{1}}^{2}})dz_{i}\wedge
d\overline{z}_{j}\end{array}$

on $\psi^{-1}(S^{1}\times D_{1/2}^{4}\backslash\\{0\\}).$

######  Lemma 5.

There exist positive constants $C_{1},$ $C_{2}$ such that

(5.10)
$\begin{array}[c]{c}\psi^{\ast}\omega_{N}\leq\omega_{M}\leq\frac{C_{1}}{|s|_{h}^{2}}\psi^{\ast}\omega_{N}\end{array}$

and

(5.11)
$\begin{array}[c]{c}\frac{1}{C_{2}}\psi^{\ast}\omega_{N}\leq\omega_{0}\leq\frac{C_{2}}{|s|_{h}^{2}}\psi^{\ast}\omega_{N}.\end{array}$

###### Proof.

We observe that $\omega_{M}$ and $\omega_{0}$ are uniformly equivalent
transverse Kähler metrics on $M$. Hence (5.11) follows easily from (5.10).
Thus it suffices for us to prove (5.10) in $\psi^{-1}(S^{1}\times
D_{1/2}^{4}\backslash\\{0\\})$. By using (5.9), the first inequality of (5.10)
follows from the fact that if $X^{i}$ is any $T^{1,0}$ vector then by the
Cauchy-Schwarz inequality,

$\begin{array}[c]{c}\sum_{i,j=1}^{2}\frac{\overline{z}_{i}z_{j}}{r_{S^{1}}^{2}}X^{i}\overline{X^{j}}=\left(\sum_{i=1}^{2}\frac{\overline{z}_{i}}{r_{S^{1}}}X^{i}\right)\left(\sum_{j=1}^{2}\frac{z_{j}}{r_{S^{1}}}\overline{X^{j}}\right)\leq\left|X\right|^{2}=\sum_{i,j=1}^{2}\delta_{ij}X^{i}\overline{X^{j}}.\end{array}$

The second inequality of (5.10) follows from the fact that
$\overline{z}_{i}z_{j}$ is semi-positive definite. ∎

######  Lemma 6.

There exists $\delta>0$ and a uniform constant $C$ such that for
$\omega=\omega(t)$ a solution of the Sasaki-Ricci flow (4.6) satisfies

$\begin{array}[c]{c}\omega\leq\frac{C}{|s|_{h}^{2}}\psi^{\ast}\omega_{N}\text{
{and} }\omega\leq\frac{C}{|s|_{h}^{2(1-\delta)}}\omega_{0}.\end{array}$

###### Proof.

From Lemma 5 we have

(5.12)
$\begin{array}[c]{c}|s|_{h}^{2}\mathrm{tr}_{\psi^{\ast}\omega_{N}}\omega\leq
C\mathrm{tr}_{\omega_{0}}\omega.\end{array}$

We fix $0<\varepsilon\leq 1$ and will apply the maximum principle to the
quantity

$\begin{array}[c]{c}Q_{\varepsilon}=\log\mathrm{tr}_{\omega_{0}}\omega+A\log(|s|_{h}^{2+2\varepsilon}\mathrm{tr}_{\psi^{\ast}\omega_{N}}\omega)-A^{2}\varphi,\end{array}$

on $M\backslash V$ where $A$ is a constant to be determined later. It follows
that at any fixed time $t$, $Q_{\varepsilon}(x,t)$ tends to negative infinity
as $x\in M$ tends to $V$. Suppose there exists $(x_{0},t_{0})\in M\backslash
V\times(0,T_{0})$ with $\sup_{M\backslash
V\times(0,t_{0}]}Q_{\varepsilon}=Q_{\varepsilon}(x_{0},t_{0})$. For any fixed
transverse Kähler metric $\widetilde{\omega}$ on $M$, if $\omega=\omega(t)$
solves the Sasaki-Ricci flow, we have the following estimate like the Kähler-
Ricci flow

(5.13) $\begin{array}[c]{ll}&(\frac{\partial}{\partial
t}-\Delta_{B})\log\mathrm{tr}_{\widetilde{\omega}}\omega\\\
=&-\frac{1}{\mathrm{tr}_{\widetilde{\omega}}\omega}\left(g^{Ti\overline{j}}\widetilde{R}_{i\overline{j}}^{T\text{
\
}k\overline{l}}g_{k\overline{l}}^{T}+g^{Ti\overline{j}}\widetilde{g}^{Tk\overline{l}}g^{Tp\overline{q}}\widetilde{\bigtriangledown}_{i}^{T}g_{k\overline{q}}^{T}\widetilde{\bigtriangledown}_{\overline{j}}^{T}g_{p\overline{l}}^{T}-\frac{\left|\bigtriangledown^{T}\mathrm{tr}_{\widetilde{\omega}}\omega\right|^{2}}{\mathrm{tr}_{\widetilde{\omega}}\omega}\right)\\\
\leq&\widetilde{C}\mathrm{tr}_{\omega}\widetilde{\omega},\end{array}$

for $\widetilde{C}$ depending only on the lower bound of the transverse
bisectional curvature of $\widetilde{g}^{T}.$ At $(x_{0},t_{0})$ we have

(5.14) $\begin{array}[c]{lll}0\leq(\frac{\partial}{\partial
t}-\Delta_{B})Q_{\varepsilon}&\leq&C_{1}\mathrm{tr}_{\omega}\omega_{0}-A\mathrm{tr}_{\omega}(A\widehat{\omega}_{t_{0}}-(1+\varepsilon)R_{h}^{T}-C_{2}\psi^{\ast}\omega_{N})\\\
&&-A^{2}\log\frac{\omega^{2}\wedge\eta_{0}}{\Omega\wedge\eta_{0}}+16A,\end{array}$

for some uniform constants $C_{1},$ $C_{2}$. Here we apply (5.13) with
$\widetilde{\omega}=\omega_{0}$ and then
$\widetilde{\omega}=\psi^{\ast}\omega_{N}$ at the point $(x_{0},t_{0})$. From
Lemma 4 and the definition of $\widehat{\omega}_{t}$ we can choose $A$
sufficiently large and independent of $\varepsilon$, $t_{0}$ so that

$\begin{array}[c]{c}A(A\widehat{\omega}_{t_{0}}-(1+\varepsilon)R_{h}^{T}-C_{2}\psi^{\ast}\omega_{N})\geq(C_{1}+1)\omega_{0}.\end{array}$

Then we have

$\begin{array}[c]{l}(A^{2}\log\frac{\omega^{2}\wedge\eta_{0}}{\Omega\wedge\eta_{0}}+\mathrm{tr}_{\omega}\omega_{0})(x_{0},t_{0})\leq
16A,\end{array}$

which implies that

$\begin{array}[c]{l}(\mathrm{tr}_{\omega}\omega_{0})(x_{0},t_{0})\leq
C.\end{array}$

Since the volume form $\omega^{2}\wedge\eta_{0}$ is uniformly bounded from
above by Lemma 2,

$\begin{array}[c]{c}(\mathrm{tr}_{\omega_{0}}\omega)(x_{0},t_{0})\leq(\mathrm{tr}_{\omega}\omega_{0})(x_{0},t_{0})\left(\frac{\omega^{2}\wedge\eta_{0}}{\omega_{0}^{2}\wedge\eta_{0}}\right)(x_{0},t_{0})\leq
C.\end{array}$

From (5.12), we obtain

$\begin{array}[c]{c}(|s|_{h}^{2}\mathrm{tr}_{\psi^{\ast}\omega_{N}}\omega)(x_{0},t_{0})\leq
C.\end{array}$

Since $\varphi$ is uniformly bounded, we can get

$Q_{\varepsilon}\leq C,$

for $C$ independent of $\varepsilon$. Letting $\varepsilon\rightarrow 0$ and
applying (5.12) again we get

$\begin{array}[c]{c}\omega\leq\frac{C}{|s|_{h}^{2}}\psi^{\ast}\omega_{N}.\end{array}$

Using the inequality $\mathrm{tr}_{\omega_{0}}\omega\leq
C\mathrm{tr}_{\psi^{\ast}\omega_{N}}\omega$ in Lemma 5, we also have

$\begin{array}[c]{l}\log(|s|_{h}^{2A}(\mathrm{tr}_{\omega_{0}}\omega)^{A+1})\leq
C,\end{array}$

this yields that
$\mathrm{tr}_{\omega_{0}}\omega\leq\frac{C}{|s|_{h}^{2(1-\delta)}}$ for
$\delta=\frac{1}{A+1}>0,$ and also
$\omega\leq\frac{C}{|s|_{h}^{2(1-\delta)}}\omega_{0}.$ ∎

For $(s,z_{1},z_{2})\in S^{1}\times D^{4},$ we consider the transverse
holomorphic vector field

$\begin{array}[c]{c}\sum_{i=1}^{2}z_{i}\frac{\partial}{\partial
z_{i}}\end{array}$

on $S^{1}\times D^{4},$ which defines a transverse holomorphic vector field
$X$ on $\psi^{-1}(S^{1}\times D^{4})\subset M$ via $\psi$ and we extend $X$ to
be a smooth $T^{1,0}$ vector field on $M$. We then have the following lemma.

######  Lemma 7.

For $\omega=\omega(t)$ be a solution of the Sasaki-Ricci flow, we have the
estimate

(5.15) $\begin{array}[c]{c}|X|_{\omega}^{2}\leq C|s|_{h}.\end{array}$

for a uniform constant $C$. Locally, in $S^{1}\times
D_{1/2}^{4}\backslash\\{0\\}$ we have

(5.16) $\begin{array}[c]{c}|W|_{g^{T}}^{2}\leq\frac{C}{r_{S^{1}}},\end{array}$

for $W=\sum_{i=1}^{2}\frac{1}{r_{S^{1}}}(x_{i}\frac{\partial}{\partial
x_{i}}+y_{i}\frac{\partial}{\partial y_{i}})$ the unit transverse radial
vector field with respect to $g_{Eucl}^{T}$, where
$z_{i}=x_{i}+\sqrt{-1}y_{i}$.

###### Proof.

By the expression (5.9) of $\omega_{M},$ we have

$|X|_{\omega_{M}}^{2}=|X|_{\psi^{\ast}\omega_{N}}^{2}$

in $S^{1}\times D_{1/2}^{4}.$ It follows that $|X|_{\omega_{0}}^{2}$ is
uniformly equivalent to $|s|_{h}^{2}=r_{S^{1}}^{2}$ in $S^{1}\times
D_{1/2}^{4}$. Hence there exists a positive constant $C(t)$ depending on $t$
such that

(5.17) $\begin{array}[c]{c}\frac{1}{C(t)}|s|_{h}^{2}\leq|X|_{\omega}^{2}\leq
C(t)|s|_{h}^{2}.\end{array}$

Now define a transverse Kähler metric $\widetilde{\omega}_{N}$ on $N$ by
$\widetilde{\omega}_{N}=\omega_{Eucl}$ on $S^{1}\times D^{4}$, and extending
in an arbitrary way to be a smooth transverse Kähler metric on $N$. For small
$\varepsilon>0$, we apply the maximum principle to the quantity

$\begin{array}[c]{c}Q_{\varepsilon}=\log(|X|_{\omega}^{2+2\varepsilon}\mathrm{tr}_{\psi^{\ast}\widetilde{\omega}_{N}}\omega)-t.\end{array}$

For any fixed $t$, using (5.12), (5.17) and $\widetilde{\omega}_{N}$ is
uniformly equivalent to $\omega_{N}$, we know that
$(|X|_{\omega}^{2+2\varepsilon}\mathrm{tr}_{\psi^{\ast}\widetilde{\omega}_{N}}\omega)(x,t)$
tends to zero as $x$ tends to $V$ and thus $Q_{\varepsilon}(x,t)$ tends to
negative infinity. Since $Q_{\varepsilon}$ is uniformly bounded from above on
the complement of $S^{1}\times D_{1/2}^{4}\backslash\\{0\\}$, thus
$Q_{\varepsilon}$ may attain its maximum at a point in $S^{1}\times
D_{1/2}^{4}\backslash\\{0\\}$. So we assume that at some point
$(x_{0},t_{0})\in S^{1}\times D_{1/2}^{4}\backslash\\{0\\}\times(0,T)$ with
$\sup_{(M\backslash
V)\times(0,t_{0}]}Q_{\varepsilon}=Q_{\varepsilon}(x_{0},t_{0})$. By (5.13), we
have

(5.18) $\begin{array}[c]{c}(\frac{\partial}{\partial
t}-\Delta_{B})\mathrm{tr}_{\psi^{\ast}\widetilde{\omega}_{N}}\omega\leq
0,\end{array}$

since the transverse bisectional curvature of $\widetilde{\omega}_{N}$ is
zero. Using the Cauchy-Schwarz inequality to get

(5.19) $\begin{array}[c]{c}(\frac{\partial}{\partial
t}-\Delta_{B})\log|X|_{\omega}^{2}=\frac{1}{|X|_{\omega}^{2}}\left(-g^{Ti\overline{j}}g_{k\overline{l}}^{T}(\partial_{i}^{k}X)(\overline{\partial_{j}^{l}X})+\frac{\left|\bigtriangledown^{T}|X|_{\omega}^{2}\right|_{\omega}^{2}}{|X|_{\omega}^{2}}\right)\leq
0.\end{array}$

All these (5.18) and (5.19) yield that

$\begin{array}[c]{c}0\leq(\frac{\partial}{\partial
t}-\Delta_{B})Q_{\varepsilon}(x_{0},t_{0})<0.\end{array}$

This implies $Q_{\varepsilon}$ is uniformly bounded from above. Letting
$\varepsilon\rightarrow 0$ we obtain

$|X|_{\omega}^{2}\mathrm{tr}_{\psi^{\ast}\omega_{N}}\omega\leq C,$

for some uniform constant $C$. By Lemma 5 we have
$|X|_{\omega}^{2}\mathrm{tr}_{\omega_{0}}\omega\leq C$, and since
$|X|_{\omega}^{2}\leq|X|_{\omega_{0}}^{2}\mathrm{tr}_{\omega_{0}}\omega,$ this
gives $|X|_{\omega}^{4}\leq C|X|_{\omega_{0}}^{2}$ and (5.15) follows from the
fact that $|X|_{\omega_{0}}^{2}$ is uniformly equivalent to $|s|_{h}^{2}$ in
$S^{1}\times D_{1/2}^{4}$.

Since $2\operatorname{Re}(\frac{1}{r_{S^{1}}}X)=W$ in $S^{1}\times
D_{1/2}^{4}\backslash\\{0\\}$ and hence (5.16) follows from (5.15). ∎

From Lemma 7 we have the bound of the lengths of spherical and transverse
radial paths in $S^{1}\times D_{1/2}^{4}\backslash\\{0\\}.$ For
$0<r_{S^{1}}<1/2$, the diameter of the product sphere $S^{1}\times
S_{r_{S^{1}}}^{3}$ in $S^{1}\times D^{4}$ with the metric induced from
$\omega$ is uniformly bounded from above, independent of $r_{S^{1}}$. For any
$(s_{0},z_{1},z_{2})\in S^{1}\times D_{1/2}^{4}\backslash\\{0\\}$, the length
of a transverse radial path $\gamma(u)=(s_{0},uz_{1},uz_{2})$ for $u\in(0,1]$
with respect to $\omega$ is uniformly bounded from above by $Cr_{S^{1}}$ with
$r_{S^{1}}=\sqrt{|z_{1}|^{2}+|z_{2}|^{2}}$. Hence the diameter of $S^{1}\times
D_{1/2}^{4}\backslash\\{0\\}$ with respect to $\omega$ is uniformly bounded
from above and

(5.20) $\mathrm{diam}_{g^{T}(t)}M\leq C.$

We then obtain a diameter bound of $M$ with respect to $g^{T}(t)$.

####  5.1.2. Gromov-Hausdorff Convergence as $t\rightarrow T_{0}^{-}$

In this subsection, we will derive the part $(2)$ in the Definition 2 of
floating foliation canonical surgical contraction under the assumption (1.2).
For the simplicity, we assume that there is only one exceptional foliation
$(-1)$-curve $V$.

More precisely, we will show that $(M,g^{T}(t))$ converges to
$(N,d_{T_{0}}^{T})$ in the sense of Gromov-Hausdorff as $t\rightarrow
T_{0}^{-}$ and $\omega(t)$ converges in $C^{\infty}$ on any compact subset of
$M\backslash V$ to a smooth transverse Kähler metric $\omega_{T}$ on
$M\backslash V$. We will first define a compact metric space $(N,d_{T_{0}})$
and show that $(M,g(t))$ converges in the Gromov-Hausdorff sense to this
metric space.

Recall that $(M,\eta,\xi,\Phi,g)$ is a compact quasi-regular Sasakian
manifold, then by thefirst structure theorem on Sasakian manifolds, $M$ is a
principal $S^{1}$-orbibundle ($V$-bundle) over $Z$ which is also a
$Q$-factorial, polarized, normal projective orbifold such that there is an
orbifold Riemannian submersion
$\pi_{M}:(M,g,\omega)\rightarrow(Z,h,\omega_{h})$ with

$\begin{array}[c]{c}g=g^{T}+\eta\otimes\eta\end{array}$

such that

$\begin{array}[c]{c}g^{T}=\pi_{M}{}^{\ast}(h);\text{ \
}\frac{1}{2}d\eta=\pi_{M}{}^{\ast}(\omega_{h}).\end{array}$

The orbit $\xi_{x}$ is compact for any $x\in M,$ we then define the transverse
distance function as

$\begin{array}[c]{c}d^{T}(x_{1},x_{2})\triangleq
d_{g}(\xi_{x_{1}},\xi_{x_{2}}),\end{array}$

where $d_{g}$ is the distance function defined by the Sasaki metric $g.$ Then

$\begin{array}[c]{c}d_{M}^{T}(x_{1},x_{2})=d_{h}(\pi_{M}(x_{1}),\pi_{M}(x_{2})).\end{array}$

We define a transverse ball $B_{\xi,g}(x,r)$ as follows:

$\begin{array}[c]{c}B_{\xi,g}(x,r)=\left\\{\widetilde{x}:d_{M}^{T}(x,\widetilde{x})<r\right\\}=\left\\{y:d_{h}(\pi_{M}(x),\pi_{M}(\widetilde{x}))<r\right\\}.\end{array}$

Since $M\backslash V$ can be identified with $N\backslash S_{y_{0}}^{1}$ via
the map $\psi$. Abusing notation, we denote $g_{T_{0}}$ for the smooth Kähler
metric $(\psi^{-1})^{\ast}g_{T_{0}}$ on $N\backslash S_{y_{0}}^{1}$ and extend
$g_{T_{0}}$ to a nonnegative $(1,1$)-tensor $\overline{g}_{T_{0}}$ on the
whole space $N$ by setting
$\overline{g}_{T_{0}}|_{S_{y_{0}}^{1}}(\cdot,\cdot)=0.$ Since $N$ is a
$S^{1}$-bundle over $Y,$ there is a Riemannian submersion

(5.21)
$\pi_{N}:(N,\overline{g}_{T_{0}},\omega_{T_{0}})\rightarrow(Y,k_{T_{0}},\omega_{k})$

such that

$\pi_{N}^{\ast}(\omega_{k})=\omega_{T_{0}}.$

We define a distance function $d_{T_{0}}^{T}$ on $N$ as follows: For
$\overline{g}_{T_{0}}^{T}:=\overline{g}_{T_{0}}(\pi_{N}(\cdot),\pi_{N}(\cdot)),$

$\begin{array}[c]{c}d_{T_{0}}^{T}(y_{1},y_{2})=\inf\int_{0}^{1}\sqrt{\overline{g}_{T_{0}}^{T}(\gamma^{\prime}(s),\gamma^{\prime}(s))}ds\end{array}$

where the infimum is taken over all piecewise smooth paths
$\gamma:[0,1]\rightarrow N$ with $\gamma(0)=y_{1}$ and $\gamma(1)=y_{2}$ such
that

$\begin{array}[c]{c}d_{T_{0}}^{T}(y_{1},y_{2})=d_{\overline{g}_{T_{0}}}(\pi_{N}(y_{1}),\pi_{N}(y_{2})).\end{array}$

Then

$\begin{array}[c]{c}d_{T_{0}}^{T}=\pi_{N}^{\ast}(d_{k_{T_{0}}}),\end{array}$

where $d_{k_{T_{0}}}$ is the distance with respect to $k_{T_{0}}$ on $Y$ as in
(5.21).

Adapting from section $3$, it follows from the previous set-up that there is a
commutative diagram

(5.22) $\begin{array}[c]{cclcl}S(-1)\thickapprox\mathbb{C}\times
S^{3}&\overset{\pi_{2}}{\rightarrow}&S^{3}\subset(M,g^{T},\omega,d^{T})&\overset{}{\longrightarrow}&S_{y_{0}}^{1}\subset(N,\overline{g}_{T_{0}},\omega_{T},d_{T_{0}}^{T})\\\
\downarrow\overline{H}&&\downarrow\pi_{M}&&\downarrow\pi_{N}\\\
O(-1)&\overset{\pi_{d}}{\rightarrow}&CP^{1}\subset(Z,h,\omega_{h},d_{h})&\overset{}{\longrightarrow}&\pi_{N}(y_{0})\in(Y,k_{T_{0}},d_{k_{T_{0}}}).\end{array}$

Here $d_{h}$ is the distance with respect to $h$ on $Z$. Note that

(5.23) $d^{T}=\pi_{M}^{\ast}(d_{h})\text{ \ {and} \
}\omega=\pi_{M}^{\ast}(\omega_{h}).$

Hence, in view of the commutative diagram (5.22), all the estimates as in
[SW2] can be adapted on $Z,$ $Y$ and then on $M,$ $N$ via (5.23). More
precisely, it follows from [SW2, Lemma 3.2] that

######  Lemma 8.

There exists a uniform constant $C$ such that for any points $x_{1},x_{2}\in
V\subset(M,g^{T},\omega,d^{T})$

$\begin{array}[c]{c}d_{\omega}^{T}(x_{1},x_{2})\leq
C(T-t)^{\frac{1}{3}}\end{array}$

for any $t\in[0,T_{0})$.

Note that when $r$ small enough, $B_{\xi,\overline{g}_{T_{0}}}(y,r)$ is a
trivial $S^{1}$-bundle over the geodesic ball $B_{k}(\pi_{N}(y),r)$. Then it
follows from [SW2, Lemma 3.3], Lemma 8 and (5.20) that

######  Lemma 9.

For any $\varepsilon>0,$ there exist $\delta$ and $T\in[0,T_{0})$ such that

$\begin{array}[c]{c}\mathrm{diam}_{d_{T_{0}}^{T}}B_{\xi,\overline{g}_{T_{0}}}(y,\delta)<\varepsilon\end{array}$

and

$\begin{array}[c]{c}\mathrm{diam}_{\omega(t)}\psi^{-1}(B_{\xi,\overline{g}_{T_{0}}}(y,\delta))<\varepsilon\end{array}$

for all $t\in[0,T_{0}).$

Combining Lemma 9 and [CHLW, Theorem 6.2], it follows that

######  Proposition 4.

$(M,g^{T}(t))$ converges to $(N,d_{T_{0}}^{T})$ and $(M,g(t))$ converges to
$(N,d_{T_{0}})$ in the sense of Gromov-Hausdorff as $t\rightarrow T_{0}^{-}.$
In particular, $(N,d_{T_{0}})$ is compact metric space homeomorphic to the
Sasakian $5$-manifold $N$.

This establishes the part $(2)$ in the Definition 2 of floating foliation
canonical surgical contraction under the assumption (1.2).

####  5.1.3. Higher Order Estimates for $\omega(t)$ as $t\rightarrow
T_{0}^{-}$

Under the assumption of Theorem 3, we know that $(M,g(t))$ converges to
$(N,d_{T_{0}})$ in the sense of Gromov-Hausdorff as $t\rightarrow T_{0}^{-}$
and $\omega(t)$ converges in $C^{\infty}$ on any compact subset of
$M\backslash\cup_{i=1}^{k}V_{i}$ to a smooth transverse Kähler metric
$\omega_{T}$ on $M\backslash\cup_{i=1}^{k}V_{i}$. In particular, we already
have $C^{\infty}$ estimates for $\omega(t)$ away from the floating foliation
$(-1)$-curves $V_{i}$ as $t$ approaches $T_{0}^{-}$. However, to understand
how the Sasaki-Ricci flow can be continued past the singular time we need more
precise estimates.

We assume as before that there is only one exceptional foliation $(-1)$-curve
$V$. From Lemmas 3, 5 and 6, there exist positive constants $\delta$ and $C$
such that

(5.24)
$\begin{array}[c]{c}\frac{|s|_{h}^{2}}{C}\omega_{0}\leq\omega(t)\leq\frac{C}{|s|_{h}^{2(1-\delta)}}\omega_{0}.\end{array}$

For simple we assume that $|s|_{h}\leq 1$ on $M$. Following the same method in
[PSS], we define an endomorphism $H=H(t)$ of the transverse tangent bundle by
$H_{\ell}^{i}=g_{0}^{Ti\overline{j}}g_{\ell\overline{j}}^{T}$ and consider the
quantity $S=S(t)$ given by

$\begin{array}[c]{c}S=\left|\nabla^{T}\log H\right|^{2},\end{array}$

here the transverse covariant derivative $\nabla^{T}$ and the norm $|\cdot|$
are taken with respect to the evolving metric $g^{T}$.

######  Proposition 5.

There exist positive constants $\alpha$ and $C$ such that for $t\in[0,T)$,

(5.25) $\begin{array}[c]{c}S\leq\frac{C}{|s|_{h}^{2\alpha}}.\end{array}$

###### Proof.

Following the same computation as in [PSS], we have

(5.26) $\begin{array}[c]{l}(\frac{\partial}{\partial t}-\Delta_{B})S\\\
\leq-|\nabla^{T}\nabla^{T}\log H|^{2}-|\overline{\nabla}^{T}\nabla^{T}\log
H|^{2}+CS+C|\nabla^{T}Rm^{T}(g_{0}^{T})|^{2},\end{array}$

where $Rm^{T}(g_{0}^{T})$ is the transverse Riemannian curvature of
$g_{0}^{T}$. Using (5.24)

(5.27) $\begin{array}[c]{c}|\nabla^{T}Rm^{T}(g_{0}^{T})|^{2}\leq
C|s|_{h}^{-K}(S+1),\end{array}$

for a positive constant $K$. By the Cauchy-Schwarz inequality we have

(5.28) $|\nabla^{T}S|\leq S^{1/2}(|\nabla^{T}\nabla^{T}\log
H|+|\overline{\nabla}^{T}\nabla^{T}\log H|),$

and the inequalities

(5.29) $|\nabla^{T}|s|_{h}^{4K}|\leq C|s|_{h}^{3K},\text{
}|\Delta_{B}|s|_{h}^{4K}|\leq C|s|_{h}^{3K},$

where we are choosing $K$ sufficiently large. Combining (5.26), (5.27), (5.28)
and (5.29), we obtain

(5.30) $\begin{array}[c]{ll}&(\frac{\partial}{\partial
t}-\Delta_{B})(|s|_{h}^{4K}S)\\\ =&|s|_{h}^{4K}(\frac{\partial}{\partial
t}-\Delta_{B})S-2\operatorname{Re}(\nabla^{T}|s|_{h}^{4K}\cdot\overline{\nabla}^{T}S)-(\Delta_{B}|s|_{h}^{4K})S\\\
\leq&-|s|_{h}^{4K}(|\nabla^{T}\nabla^{T}\log
H|^{2}+|\overline{\nabla}^{T}\nabla^{T}\log H|^{2})+C|s|_{h}^{2K}S\\\
&+C|s|_{h}^{3K}S^{1/2}(|\nabla^{T}\nabla^{T}\log
H|+|\overline{\nabla}^{T}\nabla^{T}\log H|)+C\\\
\leq&C(1+|s|_{h}^{2K}S).\end{array}$

Moreover, we know that $\mathrm{tr}_{\omega_{0}}\omega$ satisfies

$\begin{array}[c]{lll}(\frac{\partial}{\partial
t}-\Delta_{B})\mathrm{tr}_{\omega_{0}}\omega&=&-g^{Ti\overline{j}}R_{i\overline{j}}^{T\text{
\
}k\overline{l}}(g_{0}^{T})g_{k\overline{l}}^{T}-g_{0}^{Tk\overline{l}}g^{Ti\overline{j}}g^{Tp\overline{q}}\overset{0}{\nabla_{i}^{T}}g_{k\overline{q}}^{T}\overset{0}{\nabla_{\overline{j}}^{T}}g_{p\overline{l}}^{T}\\\
&\leq&C|s|_{h}^{-K}-\frac{1}{C}|s|_{h}^{K}S-\frac{1}{2}g_{0}^{Tk\overline{l}}g^{Ti\overline{j}}g^{Tp\overline{q}}\overset{0}{\nabla_{i}^{T}}g_{k\overline{q}}^{T}\overset{0}{\nabla_{\overline{j}}^{T}}g_{p\overline{l}}^{T},\end{array}$

where we used (5.24) again. Here$\overset{0}{\text{ }\nabla^{T}}$ denotes the
transverse covariant derivative with respect to $g_{0}^{T}$. Since

(5.31)
$\begin{array}[c]{c}|\nabla^{T}\mathrm{tr}_{\omega_{0}}\omega|^{2}\leq(\mathrm{tr}_{\omega_{0}}\omega)g_{0}^{Tk\overline{l}}g^{Ti\overline{j}}g^{Tp\overline{q}}\overset{0}{\nabla_{i}^{T}}g_{k\overline{q}}^{T}\overset{0}{\nabla_{\overline{j}}^{T}}g_{p\overline{l}}^{T},\end{array}$

we have

(5.32) $\begin{array}[c]{ll}&(\frac{\partial}{\partial
t}-\Delta_{B})(|s|_{h}^{K}\mathrm{tr}_{\omega_{0}}\omega)\\\
\leq&-\frac{1}{C}|s|_{h}^{2K}S+C-2\operatorname{Re}(\nabla^{T}|s|_{h}^{K}\cdot\overline{\nabla}^{T}\mathrm{tr}_{\omega_{0}}\omega)\\\
&-\frac{1}{2}|s|_{h}^{K}g_{0}^{Tk\overline{l}}g^{Ti\overline{j}}g^{Tp\overline{q}}\overset{0}{\nabla_{i}^{T}}g_{k\overline{q}}^{T}\overset{0}{\nabla_{\overline{j}}^{T}}g_{p\overline{l}}^{T}\\\
\leq&-\frac{1}{C}|s|_{h}^{2K}S+C,\end{array}$

where

$\begin{array}[c]{lll}2|\operatorname{Re}(\nabla^{T}|s|_{h}^{K}\cdot\overline{\nabla}^{T}\mathrm{tr}_{\omega_{0}}\omega)|&\leq&C+\frac{1}{C}|\nabla^{T}|s|_{h}^{K}|^{2}|\nabla^{T}\mathrm{tr}_{\omega_{0}}\omega|^{2}\\\
&\leq&C+\frac{1}{2}|s|_{h}^{K}g_{0}^{Tk\overline{l}}g^{Ti\overline{j}}g^{Tp\overline{q}}\overset{0}{\nabla_{i}^{T}}g_{k\overline{q}}^{T}\overset{0}{\nabla_{\overline{j}}^{T}}g_{p\overline{l}}^{T},\end{array}$

which follows from (5.24) and (5.31).

By applying the maximum principle to
$Q=|s|_{h}^{4K}S+A|s|_{h}^{K}\mathrm{tr}_{\omega_{0}}\omega-Bt,$ for constants
$A$ and $B$ be choosing sufficiently large, from (5.30) and (5.32) we obtain

$\begin{array}[c]{c}(\frac{\partial}{\partial t}-\Delta_{B})Q<0.\end{array}$

This gives a uniform upper bound for $Q$ and (5.25) follows. ∎

We have the estimates of curvature of $g^{T}$ and all its higher order
estimates as in [SW2].

######  Proposition 6.

For each integer $m\geq 0$ there exist $C_{m},$ $\alpha_{m}>0$ such that for
$t\in[0,T_{0})$,

(5.33)
$\begin{array}[c]{c}|(\nabla_{\mathbb{R}}^{T})^{m}Rm^{T}(g^{T}(t))|\leq\frac{C_{m}}{|s|_{h}^{2\alpha_{m}}}\text{
\ {and \
}}|(\overset{0}{\nabla_{\mathbb{R}}^{T}})^{m}g^{T}(t)|_{g_{0}^{T}}\leq\frac{C_{m}}{|s|_{h}^{2\alpha_{m}}},\end{array}$

where $\nabla_{\mathbb{R}}^{T}=\frac{1}{2}(\nabla^{T}+\overline{\nabla}^{T})$
and $\overset{0}{\nabla_{\mathbb{R}}^{T}}$ denote the real covariant
derivative of $g^{T}$ and $g_{0}^{T},$ respectively.

####  5.1.4. Continuing the Sasaki-Ricci Flow at $T_{0}$

Now we will show how to continue the Sasaki-Ricci flow past time $T_{0}$ on
the manifold $N$ by following the methods in [SW2]. First we know that
$(M,g^{T}(t))$ converges to $(N,d_{T_{0}}^{T})$ in the sense of Gromov-
Hausdorff as $t\rightarrow T_{0}^{-}$. We explain how one can continue the
Sasaki-Ricci flow through the singularity. We replace $M$ with the manifold
$N$ at the singular time $T_{0}$. Again, we assume for simplicity that we have
only one exceptional floating foliation $(-1)$-curve $V$.

Write $\widehat{\omega}_{T_{0}}=\psi^{\ast}\omega_{N}$, where $\omega_{N}$ is
the smooth transverse Kähler metric on $N$. From Lemma 2, there is a closed
positive $(1,1)$ current $\omega_{T_{0}}$ and a bounded basic function
$\varphi_{T_{0}}$ with

(5.34)
$\begin{array}[c]{c}\omega_{T_{0}}=\widehat{\omega}_{T_{0}}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi_{T_{0}}\geq
0.\end{array}$

Moreover from Lemma 3, $\varphi(t)$ converges to $\varphi_{T_{0}}$ pointwise
on $M$ and smoothly on any compact subset of $M\backslash V$. Since
$\varphi_{T_{0}}$ is constant on $V,$ there exists a bounded function
$\phi_{T_{0}}$ on $N$ and is smooth on $N\backslash S^{1}$ such that
$\psi^{\ast}\phi_{T_{0}}=\varphi_{T_{0}}.$ We then define a closed positive
$(1,1)$ current $\omega^{\prime}$ on $N$ by

(5.35)
$\begin{array}[c]{c}\omega^{\prime}=\omega_{N}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\phi_{T_{0}}\geq
0.\end{array}$

This implies $\omega^{\prime}$ is smooth on $N\backslash S^{1}$ which
satisfies $\omega_{T_{0}}=\psi^{\ast}\omega^{\prime}$. It follows from [Kol1],
[Kol3] and [SW2, Lemma 5.1 and Lemma 5.2] that there exists $p>1$ such that

(5.36) $\frac{\omega^{\prime
n}\wedge\eta_{N}}{\omega_{N}^{n}\wedge\eta_{N}}\in L^{p}(N)$

and then $\phi_{T_{0}}$ is a bounded basic continuous function on $N$, which
is smooth on $N\backslash S^{1}$, such that
$\varphi_{T_{0}}=\psi^{\ast}\phi_{T_{0}}$.

Then, by using the method as in section $4$ and [ST], we construct a solution
of the Sasaki-Ricci flow on $N$ starting at $\omega^{\prime}$. First one fix a
smooth closed $(1,1)$ form $\chi\in c_{1}^{B}(N)$. There exists
$T^{\prime}>T_{0},$ this $T^{\prime}$ is strictly less than the maximal time
$T_{N},$ such that the closed $(1,1)$ form

$\begin{array}[c]{c}\widehat{\omega}_{t,N}:=\omega_{N}+(t-T)\chi\end{array}$

is a transverse Kähler metric for $t\in[T_{0},T^{\prime}]$. We use the metrics
$\widehat{\omega}_{t,N}$ as our reference metrics for the Sasaki-Ricci flow as
it continues on $N$. Fix a smooth adapted measure $\Omega_{N}$ and then a
volume form $\Omega_{N}\wedge\eta_{N}$ on $N$ satisfying

$\begin{array}[c]{c}\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log\Omega_{N}=\frac{\partial}{\partial
t}\widehat{\omega}_{t,N}=\chi\in c_{1}^{B}(N),\end{array}$

for $t\in[T_{0},T^{\prime}]$.

Next by using the method as in section $4$ and [ST], we will now construct a
family of basic functions $\phi_{T_{0},\varepsilon}$ on $N$ which converge to
$\phi_{T_{0}}$. For $\varepsilon>0$ sufficiently small and $K$ fixed be
sufficiently large, define a family of volume forms
$\Omega_{\varepsilon}\wedge\eta_{N}$ on $N$ by

$\begin{array}[c]{c}\Omega_{\varepsilon}\wedge\eta_{N}=(\psi|_{M\backslash
V}^{-1})^{\ast}(\frac{|s|_{h}^{K}\omega^{2}(T-\varepsilon)}{|s|_{h}^{K}+\varepsilon})\wedge\eta_{N}+\varepsilon\Omega_{N}\wedge\eta_{N}\text{
\ }\mathrm{on}\text{ }N\backslash S^{1},\end{array}$

and
$(\Omega_{\varepsilon}\wedge\eta_{N})|_{S^{1}}=\varepsilon(\Omega_{N}\wedge\eta_{N})|_{S^{1}}$.
We then define the basic functions $\phi_{T_{0},\varepsilon}$ to be solutions
of the transverse complex Monge-Ampére equations

(5.37)
$\begin{array}[c]{c}(\omega_{N}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\phi_{T_{0},\varepsilon})^{2}\wedge\eta_{N}=C_{\varepsilon}\Omega_{\varepsilon}\wedge\eta_{N};\sup(\phi_{T_{0},\varepsilon}-\phi_{T_{0}})=\sup(\phi_{T_{0}}-\phi_{T_{0},\varepsilon})\end{array}$

with
$C_{\varepsilon}\int_{N}\Omega_{\varepsilon}\wedge\eta_{N}=\int_{N}\omega_{N}^{2}\wedge\eta_{N}.$
Such $\phi_{T_{0},\varepsilon}$ exist and are unique due to El Kacimi-Alaoui
([EKA]) These solution functions $\phi_{T_{0},\varepsilon}$ are continuous on
$N$ and smooth on $N\backslash S^{1}.$ Furthermore, it follows from (5.36) and
[Kol2] that

$||\frac{C_{\varepsilon}\Omega_{\varepsilon}\wedge\eta_{N}}{\Omega_{N}\wedge\eta_{N}}-\frac{\omega^{\prime
n}\wedge\eta_{N}}{\Omega_{N}\wedge\eta_{N}}||_{L^{1}(N)}\rightarrow 0\text{ \
\ }\mathrm{as}\text{\quad}\varepsilon\rightarrow 0$

and thus

$|\phi_{T_{0},\varepsilon}-\phi_{T_{0}}|_{L^{\infty}(N)}\rightarrow 0\text{ \
\ }\mathrm{as}\text{\quad}\varepsilon\rightarrow 0.$

Now let $\varphi_{\varepsilon}=\varphi_{\varepsilon}(t)$ to be the basic
functions which are solutions of the transverse parabolic complex Monge-Ampére
equations

$\begin{array}[c]{c}\frac{\partial}{\partial
t}\varphi_{\varepsilon}=\log\frac{(\widehat{\omega}_{t,N}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi_{\varepsilon})^{2}\wedge\eta_{N}}{\Omega_{N}\wedge\eta_{N}},\text{
\ }\varphi_{\varepsilon}(T_{0})=\phi_{T_{0},\varepsilon},\end{array}$

for $t\in[T_{0},T^{\prime}]$. It follows from [SW2, Proposition 5.1] and
section $4$ that there exists a basic function $\varphi$ in
$C^{0}([T_{0},T^{\prime}]\times N)\cap C^{\infty}((T_{0},T^{\prime}]\times N)$
such that $\varphi_{\varepsilon}\rightarrow\varphi$ in
$L^{\infty}([T_{0},T^{\prime}]\times N)$ as $\varepsilon\rightarrow 0,$ such a
$\varphi$ is continuous on $[T_{0},T^{\prime}]\times N$ and smooth on any
compact subset of $(T_{0},T^{\prime}]\times N.$ This basic function $\varphi$
satisfies the transverse complex Monge-Ampére equation

(5.38) $\begin{array}[c]{c}\frac{\partial}{\partial
t}\varphi=\log\frac{(\widehat{\omega}_{t,N}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi)^{2}\wedge\eta_{N}}{\Omega_{N}\wedge\eta_{N}},\text{
}\varphi(T_{0})=\phi_{T_{0}},t\in(T_{0},T^{\prime}]\end{array}$

in $C^{0}([T_{0},T^{\prime}]\times N)\cap C^{\infty}((T_{0},T^{\prime}]\times
N)$. Then we define for $t\in[T_{0},T^{\prime}],$

$\begin{array}[c]{c}\omega_{T_{0},\varepsilon}(t):=T_{0}\omega_{N}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\phi_{T_{0},\varepsilon},\end{array}$

and

$\begin{array}[c]{c}\omega_{\varepsilon}(t)=\widehat{\omega}_{t,N}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi_{\varepsilon}.\end{array}$

We thus obtain estimates on $\omega_{T_{0},\varepsilon}$ and
$\omega_{\varepsilon}$ and its volume form
$\omega_{\varepsilon}^{2}\wedge\eta_{N}$ for $t\in[T_{0},T^{\prime}]$ as
Lemmas 3, 5 and 6.

######  Lemma 10.

There exist positive constants $\alpha$ and $C,$ independent of $\varepsilon$,
such that

(5.39)
$\begin{array}[c]{c}\frac{|s|_{h}^{2\alpha}}{C}\omega_{N}\leq\omega_{T,\varepsilon}\leq\frac{C}{|s|_{h}^{2\alpha}}\omega_{N}\end{array}$

and also

(5.40)
$\begin{array}[c]{c}\frac{|s|_{h}^{2\alpha}}{C}\omega_{N}\leq\omega_{\varepsilon}\leq\frac{C}{|s|_{h}^{2\alpha}}\omega_{N}\text{
\ {and} \
}\omega_{\varepsilon}^{2}\wedge\eta_{N}\leq\frac{C}{|s|_{h}^{2\alpha}}\Omega_{N}\wedge\eta_{N}\end{array}$

on $[T_{0},T^{\prime}]\times(N\backslash S^{1})$. Here we write
$(\psi|_{M\backslash V}^{-1})^{\ast}|s|_{h}^{2}$ on $N\backslash S^{1}$ as
$|s|_{h}^{2}.$ For a fix large positive integer $L$, for each integer $0\leq
m\leq L$ there exist $C_{m},$ $\alpha_{m}>0$ such that

$\begin{array}[c]{c}|(\nabla_{\mathbb{R}}^{T})^{m}g^{T}(t)|_{g_{N}^{T}}\leq\frac{C_{m}}{|s|_{h}^{2\alpha_{m}}}\text{
\ {and }
}|(\nabla_{\mathbb{R}}^{T})^{m}g_{\varepsilon}^{T}(t)|_{g_{N}^{T}}\leq\frac{C_{m}}{|s|_{h}^{2\alpha_{m}}}.\end{array}$

where $\nabla_{\mathbb{R}}^{T}$ denote the real covariant derivative with
respect to the fixed metric $g_{N}^{T}$.

We can now prove that the Sasaki-Ricci flow can be smoothly connected at time
$T_{0}$ between $[0,T_{0})\times M$ and $(T_{0},T^{\prime}]\times N$, outside
$T_{0}\times S^{1}\cong T_{0}\times V$ via the map $\psi$.

Recall that for $t\in[T_{0},T^{\prime}],$ $\varphi(t)$ is the limit of
$\varphi_{\varepsilon}$ as $\varepsilon\rightarrow 0$ and as in (5.38), the
metric

$\omega=\widehat{\omega}_{t,N}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\varphi$

is the solution of the Sasaki-Ricci flow on $N$

(5.41) $\begin{array}[c]{c}\frac{\partial}{\partial
t}\omega=-\mathrm{Ric}^{T}(\omega)\text{ }\mathrm{with}\text{
}\omega(T_{0})=\omega^{\prime},\end{array}$

for $t\in(T_{0},T^{\prime}]$. We define

$\begin{array}[c]{c}Z=([0,T_{0})\times M)\cup(T_{0}\times N\backslash
S^{1})\cup((T_{0},T^{\prime}]\times N).\end{array}$

Consider a family of metrics $\omega(t,x)$, for $(t,x)\in Z$. We define what
is $\omega$ is smooth on $Z$. If $(t,x)\in[0,T_{0})\times M$ or
$(t,x)\in(T_{0},T^{\prime}]\times N,$ then we have $\omega$ to be smooth in
the usual sense. Recall that we have the estimates for $\omega(t,x)$ on
$[0,T_{0})\times M\backslash V$ by (5.33) and on $[T_{0},T^{\prime}]\times
N\backslash S^{1}$ from (5.40). When $(t,x)=(T_{0},x)\in T_{0}\times
N\backslash S^{1}\cong T_{0}\times M\backslash V$, we take a small
neighborhood $U$ of $x$ in $M\backslash V$ and consider $\omega$ as a metric
on $(T_{0}-\delta,T_{0}+\delta)\times U$, for some $\delta>0$. We say $\omega$
is smooth at $(T_{0},x)$ if $\omega$ is smooth at $(T_{0},x)$ in the
$(T_{0}-\delta,T_{0}+\delta)\times U$. It follows from Lemma 10 that
$\omega=\omega(t)$ satisfies the Sasaki-Ricci flow (5.38) and is smooth a time
$T_{0}$ in the sense above. Hence the metric $\omega=\omega(t)$ is a smooth
solution of the Sasaki-Ricci flow in the space-time region $Z$.

This establishes the part $(3)$ in the Definition 2 of floating foliation
canonical surgical contraction under the assumption (1.2).

####  5.1.5. Gromov-Hausdorff Convergence as $t\rightarrow T_{0}^{+}$

To complete the proof of Theorem 3, it remains to show that $(N,\omega(t))$
converges in the Gromov-Hausdorff sense to $(N,d_{T})$ as $t\rightarrow
T_{0}^{+}$.

######  Lemma 11.

There exists $\delta>0$ and a uniform constant $C$ such that for
$\omega=\omega(t)$ a solution of the Sasaki-Ricci flow (5.38) satisfies

$\begin{array}[c]{c}\omega\leq C\frac{\omega_{N}}{(\psi|_{M\backslash
V}^{-1})^{\ast}|s|_{h}^{2}}\text{ \ {and} \ }\omega\leq C(\psi|_{M\backslash
V}^{-1})^{\ast}(\frac{\omega_{0}}{|s|_{h}^{2(1-\delta)}}),\end{array}$

for $t\in[T_{0},T^{\prime}].$ Here $\omega_{0}$ is the initial metric on $M$.

###### Proof.

The arguments of the proof involved is similar to the estimate as in Lemma 6,
we omit it. We refer to [SW2, Proposition 6.1] for some details as in the
Kähler-Ricci flow. ∎

Finally, it follows by the arguments of earlier sections that $(N,\omega(t))$
converges in the Gromov-Hausdorff sense to $(N,d_{T})$ as $t\rightarrow
T_{0}^{+}$. This establishes the part $(4)$ in the Definition 2 of floating
foliation canonical surgical contraction under the assumption (1.2).

This completes the proof of Theorem 3.

###  5.2. Analytic Foliation Minimal Model Program with Scaling

As a consequence of Theorem 3 and Proposition 11, we will prove our main
result in the paper on the analytic foliation minimal model program with
scaling in a compact quasi-regular Sasakian $5$-Mmanifold. We will follow the
lines of the arguments of the proof of Proposition 11 via the Sasaki-Ricci
flow.

The proof of Theorem 4 :

###### Proof.

Let $(M,\xi,\omega_{0})$ be a compact quasi-regular Sasakian $5$-manifold with
a smooth transverse Kaehler metric $\omega_{0}$. In view of the cohomological
characterization of the maximal solution of the Sasaki-Ricci flow (4.6), we
start with a pair $(M,H^{T}),$ where $M$ is a Sasakian manifold with an ample
basic divisor $H^{T}$. Let

$T_{0}=\sup\\{t>0\mathbf{\ |\ }H^{T}+tK_{M}^{T}\text{ \ is {nef}}\\}.$

Denote

$L_{0}^{T}:=H^{T}+T_{0}K_{M}^{T}$

which is a basic $Q$-line bundle and semi-ample. In fact, it follows from
Kleiman criterion that

$mL_{0}^{T}-T_{0}K_{M}^{T}$

is ample and then nef and big for some sufficiently large $m$. Then by
Kawamata criterion for base-point free, we have the semi-ample for
$L_{0}^{T}.$

Next we define a subcone

$R:=\overline{NE(M)}_{K_{M}^{T}<0}\cap(L_{0}^{T})^{\perp}$

which is a foliation extremal ray $R$ with the generic choice of $H^{T}.$ For
$V$ with $0=L_{0}^{T}\cdot V,$ we have

$K_{M}^{T}\cdot V=-\frac{1}{T_{0}}(H^{T}\cdot V)<0.$

That is the map $\Psi:M\rightarrow N$ induced from $(L_{0}^{T})^{m}$ contract
all foliation curves whose class lies in the foliation extremal ray $R$ with

$L_{0}^{T}\cdot V=0.$

and

$K_{M}^{T}\cdot V<0.$

We observe that $K_{M}^{T}\cdot V=0$ as $T_{0}\rightarrow\infty.$

Now we consider a sequence of contarctions $g(t)$ on the manifolds
$M_{0},M_{1},...,M_{k}$ on the time intervals $[0,T_{0}),$ $(T_{1},T_{2}),$
$\cdot\cdot\cdot,(T_{k-1},T_{k}).$ Denote $L_{-1}^{T}=H^{T}$ as above and

$L_{l}^{T}:=L_{l-1}^{T}+T_{i}K_{M_{l}}^{T}.$

Then the nef class $L_{l}^{T}$ is semi-ample, there exists a map
$\psi:M_{l}\rightarrow N$ where $N$ is a quasi-regular Sasakian $5$-manifold
with foliation cyclic quotient singularities. The exceptional locus of $\psi$
is a sum of irreducible foliation curves $W=\Sigma_{i}W_{i}$ with

$W_{i}\cdot L_{l}^{T}=0$

and

$K_{M}^{T}\cdot W_{i}<0.$

(I) If $L_{l}^{T}$ is big :

(i) If all foliation curves $W=\Sigma_{i}W_{i}$ do not pass the singularity on
$M_{l}$ : We show that the Sasaki-Ricci flow will perform a canonical surgical
contraction in $M_{l}$ as in Theorem 3. Since $L_{l}^{T}$ is big and all
foliation curves $W_{i}$ do not pass the singularity on $M_{l}$, thus by index
theorem we have $W_{i}^{2}<0,$ and by Adjunction Formula, all foliation curves
$W_{i}$ are foliation $(-1)$-curves and floating on $M_{l}.$ On the other
hand,

$W_{i}\cdot L_{l}^{T}=0\text{ \ \ and \ \ }W_{j}\cdot L_{l}^{T}=0,i\neq j$

and

$W_{i}^{2}=-1=W_{j}^{2}.$

Then

$(W_{i}+W_{j})\cdot L_{l}^{T}=0$

and index theorem again

$(W_{i}+W_{j})^{2}<0.$

Hence

$W_{i}\cdot W_{j}=0$

and $\\{W_{i}\\}$ is a finite number of disjoint floating foliation
$(-1)$-curves in $M_{l}.$ Thus by Sasaki analogue of Castelnuovo’s contraction
theorem (Theorem 2), Thus $\psi$ is a map blowing down the basic exceptional
curves $W_{i}$. It follows from Proposition 10 that $L_{l}^{T}$ is the pull-
back of an ample line bundle over $M_{l},$ we obtain (1.2) for some transverse
$\omega_{M_{l}}$ on $M_{l}$. It follows from Theorem 3 that the Sasaki-Ricci
flow $g(t)$ performs a foliation canonical surgical contraction with respect
to the data $W_{1},...,W_{k}$, $N$ and $\psi$.

(ii) If some of $W_{i_{0}}:=\Gamma$ pass the foliation singularity of type
$\frac{1}{r}(1,a)$ : We show that the Sasaki-Ricci flow will perform foliation
extremal contractions of foliation $K_{M}^{T}$-negative curves. Following
above notions, it follows from Theorem 6 that we can have the minimal
resolutions of foliation singularities of $M_{l}$ and $N$

$\varphi:\widetilde{M_{l}}\rightarrow M_{l}\text{ \ \ {and} \ \ \
}\widetilde{\varphi}:\widetilde{N}\rightarrow N.$

Our goal is to find
$\widetilde{\psi}:\widetilde{M_{l}}\rightarrow\widetilde{N}$ and
$\psi:M_{l}\rightarrow N$ such that the following diagram is commutative :

(5.42) $\begin{array}[c]{lcl}\widetilde{M_{l}}\supset V_{i},\text{
}\widetilde{\Gamma}&\overset{\widetilde{\psi}}{\longrightarrow}&\widetilde{N}\\\
\downarrow\varphi&\circlearrowright&\downarrow\widetilde{\varphi}\\\
M_{l}\supset\Gamma,\text{
}\mathbf{S}_{p}^{1}&\overset{\psi}{\longrightarrow}&N\ni\mathbf{S}_{q}^{1}.\end{array}$

Suppose $\widetilde{M_{l}}$ and $\widetilde{N}$ are not isomorphic; then as
they are both regular Sasakian manifolds, by Theorem 2, there must exist a
foliation $(-1)$-curve $\widetilde{\Gamma}$ such that
$\widetilde{\varphi}\circ\widetilde{\psi}(\widetilde{\Gamma})=\mathbf{S}_{q}^{1}.$
Let $\Theta$ be a set of the foliation curves in $\widetilde{M_{l}}$ such that
$\Theta\mathcal{=}$
$\widetilde{\psi}^{-1}\circ\widetilde{\varphi}^{-1}(\mathbf{S}_{q}^{1})$ and
$D^{T}=$ $\\{V_{i}\\}$ be the exceptional locus of $\varphi$ at the singular
fibre $\mathbf{S}_{p}^{1}.$ Then the Hirzebruch Jung continued fraction

$\frac{r}{a}=[b_{1},\cdot\cdot\cdot,b_{m}]$

say that

$V_{i}^{2}=-b_{i}.$

Now there is at most one such a foliation exceptional curve $V_{j}$ for some
$j$ so that $V_{j}\cdot\widetilde{\Gamma}\neq\emptyset$ with
$V_{j}^{2}=-b_{j}$. Then

(5.43) $K_{M_{l}}^{T}\cdot\Gamma<0$

and $\Gamma$ passed the foliation singularity of type $\frac{1}{r}(1,a).$ As
in section $4$, we lift the Sasaki-Ricci flow to the foliation minimal