# Uniformizations of compact Sasakian manifolds

Hisashi Kasuya  Department of Mathematics, Graduate School of Science, Osaka
University, Osaka, Japan<EMAIL_ADDRESS>and  Natsuo Miyatake
Institute of Mathematics for Industry, Kyushu University 744 Motooka, Fukuoka
819-0395, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We give a criterion for compact Sasakian manifolds to be deformed to Sasakian
manifolds which are locally isomorphic to circle bundles of anti-canonical
bundles over Hermitian symmetric spaces as a Sasakian analogue of Simpson’s
uniformization results related to variations of Hodge structure and Higgs
bundles.

###### Key words and phrases:

Sasakian manifold, Uniformization, Variation of Hodge structures, Higgs bundle

###### 2010 Mathematics Subject Classification:

14D07, 32Q30, 32L05, 53C25,53C07

This work was partially supported by JSPS KAKENHI Grant Number 19H01787.

##  1\. Introduction

We are interested in classifications of compact Sasakian manifolds. Sasakian
geometry is an odd-dimensional counterpart of Kähler geometry. However, in
contrast to classifications of compact Kähler manifolds up to biholomorphism,
the existence of a finite dimensional local moduli space of a compact Sasakian
manifold is not given by deformation theory. Sasakian structures are usually
defined by contact CR-structures $(T^{1,0}_{M},\eta)$ on odd-dimensional
manifolds $M$ satisfying the “normality” condition. It is known that a space
of locally non-isomorphic CR structures on a compact strongly pseudo-convex CR
manifold can have arbitrary many parameters (see [BSW]). For classifying
$3$-dimensional compact Sasakian manifolds, it is a great idea to classify
Sasakian structures up to deformations of certain types. In [Bel1, Bel2],
Belgun proves that any $3$-dimensional compact Sasakian manifold is a
deformation of a standard one; locally isomorphic to one of canonical
homogeneous (left-invariant) Sasakian structures on $S^{3}$,
$\widetilde{SL}_{2}({\mathbb{R}})$, $Nil_{3}$ (see also [Ge]).

Belgun’s result is an analogue of uniformizations of compact Riemannian
surfaces. It is natural to expect that we can prove a Sasakian version of
uniformization results on higher dimensional compact Kähler manifolds in terms
of Belgun’s arguments.

The most important part of uniformizations of compact Riemannian surfaces is
to prove that every Riemannian surface of genus $\geq 2$ admits a metric of
constant negative curvature. In [Hi], Hitchin reproves this statement as a
corollary of the Kobayashi-Hitchin correspondence of Higgs bundles. In [Si1],
Simpson gives a uniformization theorem of higher dimensional compact Kähler
manifolds by generalizing Hitchin’s idea. Precisely, Simpson shows that the
existence of a uniformizing variation of Hodge structure is a criterion for
uniformizing a compact Kähler manifold by a Hermitian symmetric space. By
using this, he shows that the stability of a certain Higgs bundle is a
criterion for uniformizing a compact Kähler manifold by the unit ball.

The main result of this paper is to give a Sasakian version of Simpson’s
uniformization theorem. For a Hermitian symmetric space $D$ of non-compact
type, we can define the homogeneous Sasakian structure
$(T_{D}^{1,0},\eta_{D})$ on the circle bundle
$S^{1}(\bigwedge^{m}T^{1,0}_{D})$ of the determinant line bundle of the
holomorphic tangent bundle of $D$. We give a criterion for uniformizing
compact Sasakian manifolds by the homogeneous Sasakian structure
$(T_{D}^{1,0},\eta_{D})$.

######  Theorem 1.1.

Let $(M,T^{1,0}_{M},\eta)$ be a compact Sasakian manifold. If
$(M,T^{1,0}_{M},\eta)$ admits a uniformizing $G$-variations of Hodge structure
$(G,\rho,h)$ and $c_{1,B}(T_{M})=-C[d\eta]$ for a positive constant $C$, then
the universal covering $\widetilde{M}$ with the lifting of some
$A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost isomorphic to
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})$ with the lifting of the
homogeneous Sasakian structure $(T^{1,0}_{D},\eta_{D})$ where
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})$ is the universal covering of
$S^{1}(\bigwedge^{m}T^{1,0}_{D})$.

$c_{1,B}(T_{M})$ is the first basic Chern class of $(M,T^{1,0}_{M},\eta)$
which is also very important for studying Sasaki-Einstein geometry (see [BG]).
Note that the condition $c_{1,B}(T_{M})=C[d\eta]$ for a positive constant $C$
is a criterion for the existence of a Sasaki-Einstein metric on a compact
Sasakian manifold in contrast to our result. An $A^{1}_{B}$-deformation is a
slight modification of a second type deformation in the sense of Belgun [Bel2]
(see also a deformation of type II in [BG, Definition 7.5.9]). An almost
isomorphism is a diffeomorphism commuting with almost contact structures
associated with Sasakian manifolds equivalently Sasakian isomorphism up to
rescalings.

Combining this criterion and the Sasakian version of Simpson’s correspondence
between flat bundles and Higgs bundles ([Si1, Si]) proved in [BK, BK2], we
obtain a criterion for uniformizing compact Sasakian manifolds in terms of
stablities of Higgs bundles. For a Sasakian manifold $(M,T^{1,0}_{M},\eta)$,
we have the canonical transverse holomorphic foliation ${\mathcal{F}}_{\xi}$
and hence we can define holomorphic structures and Higgs bundle structures on
“basic” vector bundles over $(M,{\mathcal{F}}_{\xi})$. We have a canonical
Basic Higgs bundle structure $(E_{M},\,\theta_{M})$ on the vector bundle
$E_{M}=T^{1,0}_{M}\oplus{\mathbb{C}}_{M}$ where $T^{1,0}_{M}$ is the
holomorphic tangent bundle associated with a CR structure and
${\mathbb{C}}_{M}$ is the trivial line bundle.

######  Theorem 1.2.

We assume:

  1. (1)

The basic Higgs bundle $(E_{M},\,\theta_{M})$ is stable.

  2. (2)

The equality

$\int_{M}\left(2c_{2,B}(T^{1,0}_{M})-\frac{n}{n+1}c_{1,B}(T^{1,0}_{M})^{2}\right)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0$

holds.

  3. (3)

$c_{1,B}(T^{1,0}_{M})=-C[d\eta]$ for some positive constant $C$.

Then the universal covering $\widetilde{M}$ with the lifting of some
$A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost isomorphic to the
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})$ with the lifting of the
homogeneous Sasakian structure $(T^{1,0}_{D},\eta_{D})$ for
$D=SU(n,1)/S(U(n)\times U(1))$.

This result is based on the existence of Hermitian-Einstein metrics on stable
basic Higgs bundles over compact Sasakian manifolds. In [Zh], by using the
existence of Sasaki-Einstein metrics, the result of uniformization by the odd-
dimensional sphere is proved.

As a corollary of Theorem 1.2, we obtain an analogue of the existence of a
metric of constant negative curvature on a Riemannian surface of genus $\geq
2$.

######  Corollary 1.3.

Let $(M,T^{1,0}_{M},\eta)$ be a compact $3$-dimensional Sasakian manifold with
$c_{1,B}(T^{1,0})=-C[d\eta]$. Then the universal covering $\widetilde{M}$ with
the lifting of some $A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost
isomorphic to $\widetilde{PU}(1,1)$ with a left-invariant Sasakian structure.

We have $\widetilde{PU}(1,1)=\widetilde{SL}_{2}({\mathbb{R}})$. This result is
a part of Belgun’s classification in [Bel1, Bel2]. Belgun uses the non-Kähler
complex structure on $M\times S^{1}$ for a compact Sasakian manifold $M$. On
the other hand, we do not use such a complex structure.

Acknowledgement. We are grateful to Professor Takuro Mochizuki for his
valuable comments on the earlier works [BK, BK2] which led to the beginning of
this project.

##  2\. Sasakian Geometry

###  2.1. Sasakian structures on CR manifolds

Let $M$ be a $(2n+1)$-dimensional real smooth manifold. A CR-structure on $M$
is an $n$-dimensional complex involutive sub-bundle $T^{1,0}_{M}$ of the
complexified tangent bundle
$TM_{{\mathbb{C}}}\,=\,TM\otimes_{\mathbb{R}}{{\mathbb{C}}}$ such that
$T^{1,0}_{M}\cap T^{0,1}_{M}=\\{0\\}$ where
$T^{0,1}_{M}=\overline{T^{1,0}_{M}}$. Define the real $2n$-dimensional sub-
bundle $S=TM\cap(T^{1,0}_{M}\oplus T^{0,1}_{M})\subset TM$. We have the almost
complex structure $I:S\to S$ associated with the decomposition
$S_{{\mathbb{C}}}=T^{1,0}_{M}\oplus T^{0,1}_{M}$.

A strongly pseudo-convex CR structure on $M$ is a pair $(T^{1,0}_{M},\eta)$ of
a CR structure $T^{1,0}_{M}$ and a contact $1$-form $\eta$ such that
$\ker\eta=S$ and the bilinear form defined by $L_{\eta}(X,Y)=d\eta(X,IY)$ on
$S$ is a Hermitian metric on $(S,I)$. Take the Reeb vector field $\xi$
associated with the contact $1$-form $\eta$. $(T^{1,0}_{M},\eta)$ is Sasakian
if for any smooth section $X$ of $T^{1,0}_{M}$ the Lie bracket $[\xi,X]$ is
also a smooth section of $T^{1,0}_{M}$. We consider the $1$-dimensional
foliation ${\mathcal{F}}_{\xi}$ on $M$ generated by $\xi$. If
$(T^{1,0}_{M},\eta)$ is Sasakian, then $T^{1,0}_{M}$ defines a transverse
holomorphic structure on ${\mathcal{F}}_{\xi}$ and $d\eta$ is a transverse
Kähler form on ${\mathcal{F}}_{\xi}$. We say that a Sasakian structure is
quasi-regular if every leaf of ${\mathcal{F}}_{\xi}$ is closed. If $M$ is
compact and a Sasakian structure on $M$ is quasi-regular, then the flow of
$\xi$ induces an action $S^{1}\times M\to M$ (see [Wa]).

Let $(M,T^{1,0}_{M},\eta)$ be a Sasakian manifold. A differential form
$\omega$ on $M$ is called basic if the equations

$i_{\xi}\omega=0=i_{\xi}d\omega$

hold. We denote by $A^{\ast}_{B}(M)$ the subspace of basic forms in the de
Rham complex $A^{\ast}(M)$. Then $A^{\ast}_{B}(M)$ is a sub-complex of the de
Rham complex $A^{\ast}(M)$. Denote by $H_{B}^{\ast}(M)$ the cohomology of the
basic de Rham complex $A^{\ast}_{B}(M)$. We note that $d\eta\in A^{2}_{B}(M)$
and $[d\eta]\not=0\in H_{B}^{2}(M)$ if $M$ is compact. We have the bigrading
$A^{r}_{B}(M)_{{\mathbb{C}}}=\bigoplus_{p+q=r}A^{p,q}_{B}(M)$ as well as the
decomposition of the exterior differential
$d_{|A^{r}_{B}(M)_{{\mathbb{C}}}}=\partial_{B}+\overline{\partial}_{B}$ on
$A^{r}_{B}(M)_{{\mathbb{C}}}$, so that

$\partial_{B}:A^{p,q}_{B}(M)\to A^{p+1,q}_{B}(M)\ \text{ and }\
\overline{\partial}_{B}:A^{p,q}_{B}(M)\to A^{p,q+1}_{B}(M)\,.$

We note that $d\eta\in A^{1,1}_{B}(M)$. For a strongly pseudo-convex CR-
manifold $(M,T^{1,0}_{M},\eta)$ there exists a unique affine connection
$\nabla^{TW}$ on $TM$ such that the following conditions hold ([Ta, We]):

  1. (1)

$S$ is parallel with respect to $\nabla^{TW}$.

  2. (2)

$\nabla^{TW}I\,=\nabla^{TW}d\eta=\nabla^{TW}\eta\,=\nabla^{TW}\xi=\,0$.

  3. (3)

The torsion $T^{TW}$ of the affine connection $\nabla^{TW}$ satisfies the
equation

$T^{TW}(X,\,Y)\,=\,-d\eta(X,\,Y)\xi$

for all $X,\,Y\,\in\,S_{x}$ and $x\,\in\,M$.

This affine connection $\nabla^{TW}$ is called the Tanaka–Webster connection.
It is known that $(T^{1,0},\eta)$ is a Sasakian manifold if and only if
$T^{TW}(\xi,\,v)\,=\,0$ for all $v\,\in\,TM$.

Define the homomorphism $\Phi_{\xi}:TM\to TM$ by the extension of $I:S\to S$
satisfying $\Phi_{\xi}(\xi)=0$. Then, $\Phi^{2}_{\xi}=-{\rm
Id}+\xi\otimes\eta$. We call $\Phi_{\xi}$ the almost contact structure
associated with a Sasakian structure $(T^{1,0}_{M},\eta)$. We define the
Riemannian metric $g_{\eta}(X,Y)=\eta(X)\eta(Y)+d\eta(X,\Phi_{\xi}Y)$. We call
$g_{\eta}$ the Sasakian metric associated with a Sasakian structure
$(T^{1,0}_{M},\eta)$.

Let $(N,J)$ be a complex manifold. A smooth map $f:M\to N$ is a holomorphic if
$df\circ\Phi_{\xi}=J\circ df$. A holomorphic map $f:M\to N$ satisfies
$df(\xi)=0$. We note that the holomorphic condition of a smooth map $f:M\to N$
depends only on the transverse holomorphic foliation ${\mathcal{F}}_{\xi}$. A
smooth function $f:M\to{\mathbb{C}}$ is holomorphic if and only if $f\in
A_{B}^{0}(M)$ and $\bar{\partial}_{B}f=0$.

###  2.2. Morphisms, equivalences and deformations

Let $(M_{1},T^{1,0}_{M_{1}},\eta_{1})$ and $(M_{2},T^{1,0}_{M_{2}},\eta_{2})$
be two Sasakian manifolds. A smooth map $f:M_{1}\to M_{2}$ is a CR map if
$df(T^{1,0}_{M_{1}})\subset T^{1,0}_{M_{2}}$. a CR map $f:M_{1}\to M_{2}$
satisfies $f^{\ast}\eta_{2}=g(x)\eta_{1}$ for some function $g(x)$ on $M_{1}$.
By the standard arguments (see [BaK, Lemma 2.3]), we have:

######  Lemma 2.1.

Let $f:M_{1}\to M_{2}$ be a diffeomorphism. Then the following two conditions
are equivalent:

  * •

$f$ is a CR map and $f^{\ast}\eta_{2}=\eta_{1}$.

  * •

$f$ is isometry between $(M_{1},g_{\eta_{1}})$ and $(M_{2},g_{\eta_{2}})$ and
$df(\xi_{1})=\xi_{2}$.

A diffeomorphism satisfying the equivalent condition in this lemma is a
Sasakian isomorphism.

A smooth map $f:M_{1}\to M_{2}$ is a almost contact map if
$df\circ\Phi_{\xi_{1}}=\Phi_{\xi_{2}}\circ df$. A almost contact map is a CR-
map but the converse is not true. We can easily check that an almost contact
map $f:M_{1}\to M_{2}$ satisfies $f^{\ast}\eta_{2}=c\eta_{1}$ and
$df(\xi_{1x})=c\xi_{2f(x)}$ for some constant $c$. It is sufficient to check
the above function $g(x)$ is constant. By the basicness
${\mathcal{L}}_{\xi_{1}}d\eta_{1}={\mathcal{L}}_{\xi_{2}}d\eta_{2}=0$ and
$df(\xi_{1x})=c(x)\xi_{2f(x)}$, we have

${\mathcal{L}}_{\xi_{1}}(g)\eta_{1}={\mathcal{L}}_{\xi_{1}}(f^{\ast}\eta_{2})=(i_{\xi_{1}}d+di_{\xi_{1}})(f^{\ast}\eta_{2})=di_{\xi_{1}}f^{\ast}\eta_{2}=dg$

and so $dg=0$ on the sub-bundle $S={\rm ker}\eta_{1}\subset TM$. It is
sufficient to prove $dg(\xi_{1x})=0$ for any $x\in M$. Since $d\eta$ is non-
degenerate on $S$, we can take local sections $X,Y$ of $S$ such that
$d\eta_{1}(X_{x},Y_{x})\not=0$. By the property of the Tanaka-Webster
connection $\nabla^{TW}$, we have

$\nabla^{TW}_{X}Y-\nabla^{TW}_{Y}X-[X,Y]=-d\eta(X,Y)\xi.$

Since $\nabla^{TW}_{X}Y-\nabla^{TW}_{Y}X$ is a local section of $S$,

$-d\eta_{1}(X,Y)dg(\xi_{1})=dg(\nabla^{TW}_{X}Y-\nabla^{TW}_{Y}X)-X(Y(g))+Y(X(g))=0.$

This implies $dg(\xi_{1x})=0$.

Let $(M,T^{1,0}_{M},\eta)$ be a Sasakian manifold. A rescaling of
$(T^{1,0}_{M},\eta)$ is a Sasakian structure $(T^{1,0}_{M\tau},R\eta)$ for a
real number $R>0$. The Reeb vector field of $R\eta$ is $\frac{1}{R}\xi$. A
$A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is a Sasakian structure
$(T^{1,0}_{M\tau},\eta^{\tau})$ such that $\eta^{\tau}=\eta+\tau$ for $\tau\in
A^{1}_{B}$ and $T^{1,0}_{M\tau}=\\{X-\tau(X)\xi|X\in T^{1,0}_{M}\\}$. The Reeb
vector field of $\eta^{\tau}$ is $\xi$ and the transverse holomorphic
structure on ${\mathcal{F}}_{\xi}$ induced by $T^{1,0}_{M\tau}$ is same as the
one induced by $T^{1,0}_{M}$.

By the above argument on almost contact maps and the standard arguments (see
[BG, Proposition 8.1.1]), we have:

######  Lemma 2.2.

Let $f:M_{1}\to M_{2}$ be a diffeomorphism. Then the following two conditions
are equivalent:

  * •

$f$ is a Sasakian isomorphism up to rescalings.

  * •

$f$ is an almost contact map.

A diffeomorphism satisfying the equivalent condition in this lemma is an
almost isomorphism. Obviously, an almost isomorphism is a CR-diffeomorphism.

##  3\. Basic vector bundles over Sasakian manifolds

###  3.1. Basic vector bundles

We define structures on smooth vector bundles over Sasakian manifolds
$(M,T_{M}^{1,0},\eta)$ related to ${\mathcal{F}}_{\xi}$ in terms of Rawnsley’s
partial flat connections in [Ra].

A structure of basic vector bundle on a $C^{\infty}$ vector bundle $E$ is a
linear operator
$\nabla_{\xi}\,:\,{\mathcal{C}}^{\infty}(E)\,\longrightarrow\,{\mathcal{C}}^{\infty}(E)$
such that

$\nabla_{\xi}(fs)\,=\,f\nabla_{\xi}s+\xi(f)s.$

Consider the flow ${\mathbb{R}}\times M\to M$ generated by $\xi$. Then if we
have a lifting action ${\mathbb{R}}\times E\ni(t,v)\mapsto\phi_{t}(v)\in E$ of
the flow ${\mathbb{R}}\times M\to M$ on a smooth vector bundle $E$, then we
define the basic bundle structure on $E$ by

$\nabla_{\xi}(s)_{x}=\frac{d}{dt}_{|t=0}\phi_{t}(v)_{x}.$

A differential form $\omega\,\in\,A^{\ast}(M,\,E)$ with values in $E$ is
called basic if the equations

$i_{\xi}\omega=0=\nabla_{\xi}\omega$

hold where we extend $\nabla_{\xi}:A^{\ast}(M,\,E)\to A^{\ast}(M,\,E)$. Let
$A^{\ast}_{B}(M,\,E)\,\subset\,A^{\ast}(M,\,E)$ denote the subspace of basic
forms in the space $A^{\ast}(M,\,E)$ of differential forms with values in $E$.
A Hermitian metric on $E$ is called basic if it is $\nabla_{\xi}$-invariant.
Unlike the usual Hermitian metric, a basic vector bundle may not admits a
basic Hermitian metric in general. For a connection operator
$\nabla:A^{\ast}(M,\,E)\to A^{\ast+1}(M,\,E)$ such that the covariant
derivative of $\nabla$ along $\xi$ is $\nabla_{\xi}$, the curvature
$R^{\nabla}\in A^{2}(M,\,{\rm End}(E))$ is basic i.e. $R^{\nabla}\in
A^{2}_{B}(M,\,{\rm End}(E))$. For the Chern forms $c_{i}(E,\nabla)\in
A^{2i}(M)$ associated with $\nabla$, we have $c_{i}(E,\nabla)\in
A_{B}^{2i}(M)$ and we define the basic Chern classes $c_{i,B}(E)\in
H^{2i}_{B}(M)$ by the cohomology classes of $c_{i}(E,\nabla)$.

A structure of basic holomorphic bundle on a $C^{\infty}$ vector bundle $E$ is
a linear differential operator
$\nabla^{\prime\prime}\,:\,{\mathcal{C}}^{\infty}(E)\,\longrightarrow\,{\mathcal{C}}^{\infty}(E\otimes(\langle\xi\rangle\oplus
T^{0,1})^{\ast})$ such that

  * •

for any $X\in\langle\xi\rangle\oplus T^{0,1}$, and any smooth function $f$ on
$M$, the equation

$\nabla^{\prime\prime}_{X}(fs)\,=\,f\nabla^{\prime\prime}_{X}s+X(f)s$

holds for all smooth sections $s$ of $E$, and

  * •

if we extend $\nabla^{\prime\prime}$ to
$\nabla^{\prime\prime}\,:\,{\mathcal{C}}^{\infty}(E\otimes\bigwedge^{k}(\langle\xi\rangle\oplus
T^{0,1})^{\ast})\,\longrightarrow\,{\mathcal{C}}^{\infty}(E\otimes\bigwedge^{k+1}(\langle\xi\rangle\oplus
T^{0,1})^{\ast})$, then
$\nabla^{\prime\prime}\circ\nabla^{\prime\prime}\,=\,0$.

A basic holomorphic vector bundle $E$ has a canonical basic bundle structure
corresponding to the derivative $\nabla^{\prime\prime}_{\xi}$.
$\nabla^{\prime\prime}$ defines the linear operator
$\bar{\partial}_{E}:A^{p,q}_{B}(M,\,E)\to A^{p,q+1}_{B}(M,\,E)$ so that
$\bar{\partial}_{E}(f\omega)=\bar{\partial}_{B}f\wedge\omega+f\bar{\partial}_{E}(\omega)$
for $f\in A^{0}_{B}(M),\omega\in A^{p,q}_{B}(M,\,E)$. If a basic holomorphic
vector bundle $E$ admits a basic Hermitian metric $h$, as complex case, we
have a unique unitary connection $\nabla^{h}$ such that for any
$X\in\langle\xi\rangle\oplus T^{0,1}$,
$\nabla^{h}_{X}=\nabla^{\prime\prime}_{X}$.

######  Example 3.1.

Consider the ${\mathcal{C}}^{\infty}$-trivial complex line bundle
$E\,=\,M\times{\mathbb{C}}\,\longrightarrow\,M$. For any $C\in{\mathbb{R}}$,
we define the connection $\nabla^{C}\,=\,d-2\pi\sqrt{-1}C\eta$ on $E$. Then,
the curvature of $\nabla^{C}$ is $Cd\eta$. Since $d\eta\,\in\,A^{1,1}_{B}(M)$,
this $\nabla^{C}$ induces a structure of a basic holomorphic bundle.
Consequently, we have a non-trivial holomorphic vector bundle structure
$E_{C}$ on $E$ that depends on $C\,\in\,{\mathbb{C}}$. The basic cohomology
class of $Cd\eta$ is the basic first Chern class of the basic vector bundle
$E_{C}$. Thus $\\{E_{C}\\}_{C\in{\mathbb{R}}}$ is a family of basic vector
bundles such that $E_{C}\,\not\cong\,E_{C^{\prime}}$ for every
$C\,\not=\,C^{\prime}$. The standard constant Hermitian metric $h$ on the
${\mathcal{C}}^{\infty}$ trivial line bundle $E\,=\,M\times{\mathbb{C}}$ is
basic on $E_{C}$. The connection $\nabla^{C}$ is unitary for $h$, and hence
$\nabla^{C}$ is the canonical connection for the basic Hermitian metric $h$.

Consider the flow ${\mathbb{R}}\times M\to M$ generated by $\xi$. Define the
lifting action ${\mathbb{R}}\times
M\times{\mathbb{C}}\ni(t,x,v)\mapsto(x_{t},e^{-2\pi C\sqrt{-1}t}v)\in E$ of
the flow ${\mathbb{R}}\times M\to M$. Then this induces the basic bundle
structure $\nabla^{C}_{\xi}$ of $E_{C}$.

######  Example 3.2.

The Tanaka-Webster connection $\nabla^{TW}$ defines a structure of a basic
holomorphic bundle on $T^{1,0}$. Since we have $\nabla^{TW}_{\xi}(X)=[\xi,X]$,
the basic bundle structure $\nabla^{TW}_{\xi}$ is given by the action
${\mathbb{R}}\times T^{1,0}X\to T^{1,0}X$ which is the differential of the
action ${\mathbb{R}}\times M\to M$ generated by the flow of $\xi$.

###  3.2. Equivariant cohomology

Assume that $M$ is compact and a Sasakian structure on $M$ is quasi-regular.
Consider the equivariant cohomology
$H^{\ast}_{S^{1}}(M,R)=H^{\ast}(M\times_{S^{1}}ES^{1},R)$ for the action
$S^{1}\times M\to M$ with coefficients in a commutative ring $R$.
$H^{\ast}_{S^{1}}(M,{\mathbb{R}})$ is the cohomology of the Cartan model
$A^{\ast}(M)^{S^{1}}\otimes{\mathbb{R}}[u]$ with the differential $d_{S^{1}}$
so that $d_{S^{1}}u=0$, $d_{S^{1}}\omega=d\omega-i_{\xi}\omega u$ where $u$ is
of degree $2$. The natural inclusion $A^{\ast}_{B}(M)\to
A^{\ast}(M)^{S^{1}}\otimes{\mathbb{R}}[u]$ is a cochain complex homomorphism.
Since the action $S^{1}\times M\to M$ is locally free, the induced map
$H_{B}^{\ast}(M)\to H^{\ast}_{S^{1}}(M,{\mathbb{R}})$ is an isomorphism ([GS,
Section 5]).

We use the result in [MuR] which says that equivariant first Chern classes in
$H^{\ast}_{S^{1}}(M,{\mathbb{Z}})$ classify lifts of the action $S^{1}\times
M\to M$ on line bundles. For our case, basic bundle structures $\nabla_{\xi}$
on line bundles can be considered as infinitesimal lifts of the action
$S^{1}\times M\to M$. Let $L_{1}\to M,L_{2}\to M$ be line bundles admitting
lifts of the action $S^{1}\times M\to M$ and consider them as basic line
bundles. Assume that we have basic Hermitian metrics $h_{1},h_{2}$ and basic
unitary connections $\nabla_{1},\nabla_{2}$ on $L_{1}$ and $L_{2}$
respectively. We note that the equivariant first Chern classes in
$H^{\ast}_{S^{1}}(M,{\mathbb{R}})$ are images of the basic first Chern classes
via the canonical map $H_{B}^{\ast}(M)\to H^{\ast}_{S^{1}}(M,{\mathbb{R}})$.
Since $H_{B}^{\ast}(M)\to H^{\ast}_{S^{1}}(M,{\mathbb{R}})$ is an isomorphism,
we can say that if $c_{1,B}(L_{1})=c_{1,B}(L_{2})$, then $L_{1}^{l}$ and
$L_{2}^{l}$ are equivariantly isomorphic for some positive integer $l$.

######  Proposition 3.3.

If $c_{1,B}(T^{1,0}_{M})=-C[d\eta]$ for some positive constant $C$, then for
some positive integer $l$, the $l$-th power of the anti-canonical bundle
$\bigwedge^{n}T^{1,0}_{M}$ with the $S^{1}$-action given by the differential
of the action $S^{1}\times M\to M$ is equivariantly isomorphic to $E_{-lC}$.

######  Remark 3.4.

If $c_{1,B}(T^{1,0}_{M})=-C[d\eta]$ for a positive constant $C$, then the
Sasakian structure $(T^{1,0}_{M},\eta)$ is quasi-regular ([BG, Theorem
8.1.14]).

##  4\. Variations of Hodge structure and uniformizations

###  4.1. Variations of Hodge structure over Sasakian manifolds

Let $G$ be a connected semi-simple Lie group. Assume that $G$ has the finite
center and no compact factor. A Hodge structure on $G$ is a real Hodge
structure
$\mathfrak{g}_{{\mathbb{C}}}=\bigoplus_{p\in{\mathbb{Z}}}\mathfrak{g}^{p,-p}$
of weight $0$ on the Lie algebra $\mathfrak{g}$ of $G$ such that the Lie
bracket on $\mathfrak{g}$ is a morphism of real Hodge structures and
$-B_{\mathfrak{g}}$ is a polarization where $B_{\mathfrak{g}}$ is the Killing
form on $\mathfrak{g}$. Define the subalgebra
$\mathfrak{v}=\mathfrak{g}^{0,0}\cup\mathfrak{g}\subset\mathfrak{g}$ and the
subgroup $V=\exp(\mathfrak{v})\subset G$. Then $V$ is compact. The subalgebra
$\bigoplus_{p>0}\mathfrak{g}^{p.-p}\subset\mathfrak{g}_{{\mathbb{C}}}$ defines
a complex structure on $D=G/V$. The complex manifold $D$ is called the period
domain. Define the subalgebra $\mathfrak{p}=\bigoplus_{p\geq
0}\mathfrak{g}^{p.-p}$ and the subgroup $P=\exp(\mathfrak{p})\subset
G_{{\mathbb{C}}}$ in the complexification $G_{{\mathbb{C}}}$ of $G$. Then
$\mathfrak{p}$ is a parabolic subalgebra in $\mathfrak{g}_{{\mathbb{C}}}$ and
$G/V$ is a dual manifold of $G/P$ in the sense of [GrSc]. The holomorphic
tangent bundle $T^{1,0}_{D}$ is a homogeneous vector bundle given by
$(G\times\bigoplus_{p>0}\mathfrak{g}^{p.-p})/V$ and hence we have the
decomposition $T^{1,0}_{D}=\bigoplus_{p>0}E^{p,-p}$. Consider the connection
form $\lambda$ on the anti-canonical bundle $\bigwedge^{m}T^{1,0}_{D}$ given
by the Maurer-Cartan form where $m=\dim_{{\mathbb{C}}}D$. Let
$\eta_{D}=-\lambda$. Then $d\eta$ is a non-degenerate $2$-form on $D$ which is
positive on $\bigoplus_{p\,odd}E^{p,-p}$ and negative on
$\bigoplus_{p\,even}E^{p,-p}$ see [GrSc, (4.23)]. Let $(M,T^{1,0}_{M},\eta)$
be a Sasakian manifold. Denote by $\widetilde{M}$ the universal cover of $M$
at $x\in M$ and take the lifting Sasakian structure
$(T^{1,0}_{\tilde{M}},\tilde{\eta})$. A $G$-variations of Hodge structure is
$(G,\rho,h)$ such that:

  1. (1)

$G$ is a connected semi-simple Lie group equipped with a Hodge structure.

  2. (2)

$\rho:\pi_{1}(M,x)\to G$ is a representation.

  3. (3)

$h:\widetilde{M}\to D$ is a $\rho$-equivariant holomorphic map satisfying
$dh(T^{1,0}_{\widetilde{M}})\subset E^{1,-1}$.

###  4.2. Uniformizing Variations of Hodge structure

A Hodge structure on $G$ is Hermitian if
$\mathfrak{g}_{{\mathbb{C}}}=\mathfrak{g}^{1,-1}\oplus\mathfrak{g}^{0,0}\oplus\mathfrak{g}^{-1,1}$.
In this case, $D$ is a Hermitian symmetric manifold of non-compact type and
$d\eta_{D}$ is an invariant Kähler form on $D$. Regarding $\eta$ as a contact
$1$-form, $(T^{1,0}_{D},\eta_{D})$ is a $G$-invariant Sasakian structure on
the circle bundle $S^{1}(\bigwedge^{m}T^{1,0}_{D})$. We have
$S^{1}(\bigwedge^{m}T^{1,0}_{D})=(G\times S^{1})/V=G/V^{\prime}$ where
$V^{\prime}$ is the kernel of the non-trivial representation $V\to
GL(\bigwedge^{m}\mathfrak{g}^{1,-1})$. $S^{1}(\bigwedge^{m}T^{1,0}_{D})$ is
also $G$-homogeneous and the Sasakian structure $(T^{1,0}_{D},\eta_{D})$ is
homogeneous We can take the Reeb vector field $\xi_{D}$ as a fundamental
vector field of the action $S^{1}\times(G\times
S^{1})/V\ni(a,[g,t])\mapsto[g,a^{-1}t]\in(G\times S^{1})/V$. Since $D$ is
contractible, by trivializing $S^{1}(\bigwedge^{m}T^{1,0}_{D})\cong
S^{1}\times D$, we have the universal covering
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})\to
S^{1}(\bigwedge^{m}T^{1,0}_{D})$ with the infinite cyclic covering group.

Let $\Gamma\subset G$ be a discrete subgroup which acts freely and cocompactly
on $S^{1}(\bigwedge^{m}T^{1,0}_{D})=G/V^{\prime}$. We consider the compact
Sasakian manifold $(\Gamma\backslash
S^{1}(\bigwedge^{m}T^{1,0}_{D}),T^{1,0}_{D}\eta_{D})$. We observe that:

  * •

By the construction, we have $c_{1,B}(M)=-[d\eta_{D}]$.

  * •

Defining $h:\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})\to D$ by the
composition of the covering $\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})\to
S^{1}(\bigwedge^{m}T^{1,0}_{D})$ and the projection
$S^{1}(\bigwedge^{m}T^{1,0}_{D})\to D$ and $\rho:\pi_{1}(\Gamma\backslash
S^{1}(\bigwedge^{m}T^{1,0}_{D}))\to G$ by the composition the quotient
$\pi_{1}(\Gamma\backslash S^{1}(\bigwedge^{m}T^{1,0}_{D}))\to\Gamma$ by the
covering group and the inclusion $\Gamma\to G$, we have the $G$-variation of
Hodge structure $(G,\rho,h)$.

The main purpose of this section is to prove that these properties uniformize
compact Sasakian manifolds by the homogeneous Sasakian structure

Let $(M,T^{1,0}_{M},\eta)$ be a Sasakian manifold. A $G$-variation of Hodge
structure $(G,\rho,h)$ is uniformizing if a Hodge structure on $G$ is
Hermitian and the differential $dh:T^{1,0}_{\widetilde{M}}\to T^{1,0}_{D}$ is
an isomorphism at every fiber.

######  Theorem 4.1.

Let $(M,T^{1,0}_{M},\eta)$ be a compact Sasakian manifold. If
$(M,T^{1,0}_{M},\eta)$ admits a uniformizing $G$-variations of Hodge structure
$(G,\rho,h)$ and $c_{1,B}(T_{M}^{1,0})=-C[d\eta]$ for a positive constant $C$,
then the universal covering $\widetilde{M}$ with the lifting of some
$A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost isomorphic to
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})$ with the lifting of the
homogeneous Sasakian structure $(T^{1,0}_{D},\eta_{D})$.

###### Proof.

By Proposition 3.3 and Remark 3.4, for some positive integer $l$, the $l$-th
power of $\bigwedge^{n}T^{1,0}_{M}$ is equivariantly isomorphic to $E_{-lC}$.
Consider the basic holomorphic line bundle $L$ over $M$ defined by the pull-
back $h^{\ast}\bigwedge^{m}T^{1,0}_{D}$. Then, Since $f$ is $\rho$-equivariant
holomorphic and $dh:T^{1,0}_{\tilde{M}}\to T^{1,0}_{D}$ is an isomorphism at
every fiber, $L$ is identified with $\bigwedge^{m}T^{1,0}_{M}$ via the
differential $dh$. We regard the $\rho$-equivariant map $h:\widetilde{M}\to D$
as a $V$-reduction $P$ of the flat bundle $(\widetilde{M}\times
G)/\pi_{1}(M,x)$. Then $L$ is induced by $P$ associated with the
representation $V\to GL(\bigwedge\mathfrak{g}^{1,-1})$. Take an equivariant
isomorphism $L^{l}\cong E_{-lC}$. By $E_{-lC}=M\times{\mathbb{C}}$,
corresponding to $M\ni x\mapsto(x,1)\in M\times{\mathbb{C}}$, we have a
nowhere vanishing section of $L^{l}$ and it induces an $\rho$-equivariant map
$f:\widetilde{M}\to(G\times S^{1})/V=(G\times S^{1})/V=G/V^{\prime\prime}$
where $V^{\prime\prime}$ is the kernel of the representation $\Lambda^{l}$.
$f$ is a lift of $h:\widetilde{M}\to D$. $G/V^{\prime}$ is a $l$-sheeted
covering of $G/V^{\prime\prime}$. $(T^{1,0}_{D},\eta_{D})$ defines a Sasakian
structure on $G/V^{\prime\prime}$. By the equivariance of $L^{l}\cong
E_{-lC}$, we have $df(\tilde{\xi})=lC\xi_{D}$. Since $M$ is compact and $f$ is
$\rho$-equivariant, the Riemann metric $f^{\ast}g_{\eta_{D}}$ is complete on
$\widetilde{M}$. $f:\widetilde{M}\to G/V^{\prime\prime}$ is a local isometry
and hence covering map.

Let $\tau=-\tilde{\eta}+\frac{1}{C}f^{\ast}\eta_{D}$. Then, by
$\tau(\tilde{\xi})=0$ and $d\tau\in A^{2}_{B}(\widetilde{M})$, we have
$\tau\in A^{1}_{B}(\widetilde{M})$. The $A^{1}_{B}$-deformation
$(T^{1,0}_{\widetilde{M}\tau},\tilde{\eta}+\tau)$ of
$(T^{1,0}_{\widetilde{M}},\tilde{\eta})$ is almost isomorphic to
$(\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D}),T^{1,0}_{D},\eta_{D})$. Since
$f$ is $\rho$-equivariant, $(T^{1,0}_{\widetilde{M}\tau},\tilde{\eta}+\tau)$
is defined on $M$. ∎

######  Remark 4.2.

In this proof, $f:\widetilde{M}\to G/V^{\prime}$ satisfies
$df(T^{1,0}_{\widetilde{M}})=T^{1,0}_{D}$ modulo $\langle\xi\rangle$ but may
not be a CR-map. Hence, the $A_{B}^{1}$-deformation
$(T^{1,0}_{\widetilde{M}\tau},\tilde{\eta}+\tau)$ is essential.

##  5\. Higgs bundles and uniformizations

###  5.1. Higgs bundles over Sasakian manifolds

Let $(M,T^{1,0}_{M},\eta)$ be a compact Sasakian manifold. A basic Higgs
bundle over $M$ is a pair $(E,\,\theta)$ consisting of a basic holomorphic
vector bundle $E$ and $\theta\,\in\,A^{1,0}_{B}(M,\,{\rm End}(E))$ satisfying
the following two conditions:

$\overline{\partial}_{{\rm End}(E)}\theta\,=\,0\ \ \text{ and }\ \
\theta\wedge\theta\,=\,0\,.$

We define the degree of a basic holomorphic vector bundle $E$ by

${\rm deg}(E)\,:=\,\int_{M}c_{1,B}(E)\wedge(d\eta)^{n-1}\wedge\eta\,.$

Denote by ${\mathcal{O}}_{B}$ the sheaf of holomorphic functions on $M$, and
for a holomorphic vector bundle $E$ on $M$, denote by ${\mathcal{O}}_{B}(E)$
the sheaf of holomorphic sections of $E$. Consider ${\mathcal{O}}_{B}(E)$ as a
coherent ${\mathcal{O}}_{B}$-sheaf.

For a basic Higgs bundle $(E,\,\theta)$, a sub-Higgs sheaf of $(E,\,\theta)$
is a coherent ${\mathcal{O}}_{B}$-subsheaf $\mathcal{V}$ of
${\mathcal{O}}_{B}(E)$ such that
$\theta({\mathcal{V}})\,\subset\,{\mathcal{V}}\otimes\Omega_{B}$, where
$\Omega_{B}$ is the sheaf of basic holomorphic $1$-forms on $M$. By [BH,
Proposition 3.21], if ${\rm rk}(\mathcal{V})\,<\,{\rm rk}(E)$ and
${\mathcal{O}}_{B}(E)/\mathcal{V}$ is torsion-free, then there is a
transversely analytic sub-variety $S\,\subset\,M$ of complex co-dimension at
least 2 such that ${\mathcal{V}}\big{|}_{M\setminus S}$ is given by a basic
holomorphic sub-bundle $V\,\subset\,E\big{|}_{M\setminus S}$. The degree ${\rm
deg}(\mathcal{V})$ can be defined by integrating
$c_{1,B}(V)\wedge(d\eta)^{n-1}\wedge\eta$ on this complement $M\setminus S$.

######  Definition 5.1.

We say that a basic Higgs bundle $(E,\,\theta)$ is stable if $E$ admits a
basic Hermitian metric and for every sub-Higgs sheaf ${\mathcal{V}}$ of
$(E,\,\theta)$ such that ${\rm rk}(\mathcal{V})\,<\,{\rm rk}(E)$ and
${\mathcal{O}}_{B}(E)/\mathcal{V}$ is torsion-free, the inequality

$\frac{{\rm deg}(\mathcal{V})}{{\rm rk}(\mathcal{V})}\,<\,\frac{{\rm
deg}(E)}{{\rm rk}(E)}$

holds.

A basic Higgs bundle $(E,\,\theta)$ is called polystable if

$(E,\,\theta)\,=\,\bigoplus_{i=1}^{k}(E_{i},\,\theta_{i})\,,$

where each $(E_{i},\,\theta_{i})$ is a stable Higgs bundle with

$\frac{{\rm deg}(E_{i})}{{\rm rk}(E_{i})}\,=\,\frac{{\rm deg}(E)}{{\rm
rk}(E)}\,.$

Let $(E,\,\theta)$ be a basic Higgs bundle over a compact Sasakian manifold
$M$. Assume that $E$ admits a basic Hermitian metric $h$. Define
$\overline{\theta}_{h}\,\in\,A^{0,1}_{B}(M,\,{\rm End}(E))$ by
$h(\theta(e_{1}),\,e_{2})\,=\,h(e_{1},\,\overline{\theta}_{h}(e_{2}))$ for all
$e_{1},\,e_{2}\,\in\,E_{x}$ and all $x\,\in\,M$. Define the canonical
connection $D^{h}\,=\,\nabla^{h}+\theta+\overline{\theta}_{h}$ on $E$.

######  Theorem 5.2 ([BK, BK2]).

If a basic Higgs bundle $(E,\,\theta)$ is stable and satisfies

$c_{1,B}(E)\,=\,0\qquad{\rm
and}\qquad\int_{M}c_{2,B}(E)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0,$

then there exists a basic Hermitian metric $h$ so that the canonical
connection $D^{h}$ is flat and such Hermitian metric is unique up to a
positive constant. Moreover, such metric $h$ is a Harmonic metric on the flat
bundle $(E,D^{h})$ with respect to the Sasakian metric $g_{\eta}$ and hence
the flat bundle $(E,D^{h})$ is semi-simple ([Co]).

This hermitian metric is said to be a harmonic metric on $(E,\theta)$.

A polarized complex basic variation of Hodge structure of weight $w$ over $M$
is $(E=\bigoplus_{p+q=w}E^{p,q},D,H)$ so that

  1. (1)

$(E,D)$ is a complex flat vector bundle.

  2. (2)

$\bigoplus_{p+q=w}E^{p,q}$ is a direct sum of $C^{\infty}$-subbundles such
that
$D_{\xi}{\mathcal{C}}^{\infty}(E^{p,q})\subset{\mathcal{C}}^{\infty}(E^{p,q})$.

  3. (3)

(The Griffiths transversality conditions) For any $X\in T^{1,0}_{M}$ (resp.
$X\in T^{0,1}_{M}$),
$D_{X}{\mathcal{C}}^{\infty}(E^{p,q})\subset{\mathcal{C}}^{\infty}(E^{p,q})\oplus{\mathcal{C}}^{\infty}(E^{p-1,q+1})$
(resp.
$D_{X}{\mathcal{C}}^{\infty}(E^{p,q})\subset{\mathcal{C}}^{\infty}(E^{p,q})\oplus{\mathcal{C}}^{\infty}(E^{p+1,q-1})$)

  4. (4)

$H$ is a parallel Hermitian form so that decomposition
$\bigoplus_{p+q=w}E^{p,q}$ is orthogonal and $h$ is positive on $E^{p,q}$ for
even $p$ and negative for odd $p$.

Define the basic bundle structure $\nabla_{\xi}$ on $E$ and each $E^{p,q}$. By
the flatness $D^{2}=0$ and the Griffiths transversality conditions, we have:

  * •

We have $D=\nabla+\theta+\bar{\theta}$ such that $\nabla$ is a connection on
each $E^{p,q}$ and $\theta\in A^{1}(M,{\rm Hom}(E^{p,q},E^{p-1,q+1}))$,
$\bar{\theta}\in A^{1}(M,{\rm Hom}(E^{p,q},E^{p+1,q-1}))$.

  * •

$\nabla$ gives a basic holomorphic bundle structure $\nabla^{\prime\prime}$,
$\theta\in A^{1,0}(M,{\rm Hom}(E^{p,q},E^{p-1,q+1}))$, $\bar{\partial}_{{\rm
End}(E)}\theta=0$ and $\theta\wedge\theta=0$. Thus $(E,\theta)$ is a basic
Higgs bundle.

  * •

Define the basic Hermitian metric $h$ on $E$ by $(-1)^{p}H$ on each $E^{p,q}$.
Then $\nabla$ is the canonical connection associated with $h$ and
$\bar{\theta}=\overline{\theta}_{h}$ as the above sense. Thus $h$ is a
Harmonic metric on a basic Higgs bundle $(E,\theta)$.

Let $G=SU(E_{x},H_{x})$ and $V=S(\Pi_{p+q=w}U(E^{p,q}_{x},H_{x}))$. Take the
monodromy $\rho:\pi_{1}(M,x)\to GL(E_{x})$ of the flat bundle $(E,D)$. Since
$h$ is parallel, we can assume $\rho(\pi_{1}(M,x))\subset G$. Let
$\mathfrak{g}$ be the Lie algebra of $G$. The decomposition
$E=\bigoplus_{p+q=w}E^{p,q}$ gives the decomposition
$\mathfrak{g}_{{\mathbb{C}}}=\bigoplus_{r}\mathfrak{g}^{r,-r}$. This
decomposition is a Hodge structure on $G$ and the period domain is $G/V$ (cf.
[Sch]). Considering $h$ as a $V$-reduction of the flat bundle
$(\tilde{M}\times G)/\pi_{1}(M,x)$, we have a $\rho$-equivariant map
$h:\tilde{M}\to G/V$ such that $dh=\theta$ on $T^{1,0}_{\tilde{M}}$. By the
Griffiths transversality conditions, $(G,\rho,h)$ is a $G$-variation of Hodge
structure.

Let $(E,\,\theta)$ be a polystable basic Higgs bundle satisfying

$c_{1,B}(E)\,=\,0\qquad{\rm
and}\qquad\int_{M}c_{2,B}(E)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0.$

By Theorem 5.2, we have a harmonic metric $h$ on $(E,\theta)$. For any $t\in
U(1)$, we have the new polystable basic Higgs bundle $(E,\,t\theta)$ and we
can easily check that $h$ is also a harmonic metric on $(E,\,t\theta)$. Assume
that $U(1)$ acts on $E$ by holomorphic automorphisms via $\mu:U(1)\to{\rm
Aut}(E)$ so that $\mu(t)\theta\mu(t)^{-1}=t\theta$. Then, $\mu(t)$ is an
isomorphism between basic Higgs bundles $(E,\,\theta)$ and $(E,\,t\theta)$. By
the harmonicity of $h$ on both $(E,\,\theta)$ and $(E,\,t\theta)$ and the
uniqueness of harmonic metrics, we can say that the action $\mu:U(1)\to{\rm
Aut}(E)$ preserves a harmonic metric $h$. Fix $w\in{\mathbb{Z}}$, define the
sub-bundle $E^{p,w-p}$ consisting of $e\in E$ such that for any $t\in U(1)$,
$\mu(t)e=t^{p}e$. The polarized complex basic variation of Hodge structure of
weight $w$ $(E=\bigoplus_{p+q=w}E^{p,q},D^{h},H)$ is defined by $H=(-1)^{p}h$
on each $E^{p,q}$.

###  5.2. Uniformizations associated with the canonical Higgs bundles

Define the Higgs bundle $(E_{M},\,\theta_{M})$ by the following way:

  * •

$E_{M}\,=\,{\mathbb{C}}_{M}\oplus T^{1,0}_{M}$ where ${\mathbb{C}}_{M}$ is the
trivial basic holomorphic line bundle on $M$, and

  * •

$\theta_{M}\,=\,\left(\begin{array}[]{cc}0&0\\\ 1&0\end{array}\right)$ where
$1$ is the identity element in ${\rm
End}(T^{1,0}_{M})=(T^{1,0}_{M})^{\ast}\otimes T^{1,0}_{M}$ regarded as an
element in $A^{1,0}_{B}(M,{\rm Hom}({\mathbb{C}}_{M},T^{1,0}_{M}))$.

######  Theorem 5.3.

We assume:

  1. (1)

The basic Higgs bundle $(E_{M},\,\theta_{M})$ is stable.

  2. (2)

The equality

$\int_{M}\left(2c_{2,B}(T^{1,0}_{M})-\frac{n}{n+1}c_{1,B}(T^{1,0}_{M})^{2}\right)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0$

holds.

  3. (3)

$c_{1,B}(T^{1,0}_{M})=-C[d\eta]$ for some positive constant $C$.

Then the universal covering $\widetilde{M}$ with the lifting of some
$A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost isomorphic to
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})$ with the lifting of the
homogeneous Sasakian structure $(T^{1,0}_{D},\eta_{D})$ for
$D=SU(n,1)/S(U(n)\times U(1))$.

###### Proof.

By $c_{1,B}(T^{1,0}_{M})=-C[d\eta]$, $c_{1,B}(E\otimes E_{C^{\prime}})=0$ for
some $C^{\prime}\in{\mathbb{R}}$. By

$\int_{M}\left(2c_{2,B}(T^{1,0}_{M})-\frac{n}{n+1}c_{1,B}(T^{1,0}_{M})^{2}\right)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0,$

we have $\int_{M}c_{2,B}(E_{M}\otimes
E_{C^{\prime}})\wedge(d\eta)^{n-2}\wedge\eta\,=\,0.$
$\nabla^{\prime\prime}=\nabla_{|\langle\xi\rangle\oplus T^{0,1}}$ defines a
holomorphic bundle structure on $L$. Thus, $(E_{M}\otimes
E_{C^{\prime}},\,\theta\otimes{\rm Id}_{L})$ is a basic Higgs bundle. Since
$(E_{M},\,\theta_{M})$ is stable, $(E_{M}\otimes
E_{C^{\prime}}\,\theta_{M}\otimes{\rm Id}_{E_{C^{\prime}}})$ is also stable.
Let $h$ be a harmonic metric on $(E\otimes E_{C^{\prime}},\,\theta\otimes{\rm
Id}_{E_{C^{\prime}}})$. Define $E^{0,1}=T^{1,0}\otimes E_{C^{\prime}}$ and
$E^{1,0}=E_{C^{\prime}}$. We have the polarized complex basic variation of
Hodge structure of weight $1$ $(E_{M}\otimes E_{C^{\prime}}=E^{1,0}\oplus
E^{0,1},D^{h},H)$. $SU(E_{x},H_{x})\cong SU(n,1)$ and the Hodge structure on
$SU(n,1)$ is Hermitian and the period domain $D$ is the unit ball
$SU(n,1)/S(U(n)\times U(1))$. We obtain a $SU(n,1)$-variation of Hodge
structure $(SU(n,1),\rho,h)$. By the definition, for the map $h:\tilde{M}\to
SU(n,1)/S(U(n)\times U(1))$, $dh=\theta$ is an isomorphism on each fiber of
$T^{1,0}_{\widetilde{M}}$. Hence, $(SU(n,1),\rho,h)$ is uniformizing. ∎

######  Remark 5.4.

If the basic Higgs bundle $(E_{M},\theta_{M})$ is stable, then the Miyaoka-Yau
type inequality

$\int_{M}\left(2c_{2,B}(T^{1,0}_{M})-\frac{n}{n+1}c_{1,B}(T^{1,0}_{M})^{2}\right)\wedge(d\eta)^{n-2}\wedge\eta\,\geq\,0\,.$

holds. Thus the second condition means that the equality of the Miyaoka-Yau
type inequality holds.

In [Zh], Zhang proves that if $c_{1,B}(T^{1,0}_{M})=C[d\eta]$ for some
positive constant $C$ and the certain condition in terms of Sasaki-Einstein
geometry holds, then the Miyaoka-Yau type inequality holds and the equality is
a criterion for uniformizing a higher dimensional compact Sasakian manifold by
the odd-dimensional sphere.

######  Remark 5.5.

In case $D=SU(n,1)/S(U(n)\times U(1))$, the line bundle
$\bigwedge^{n}T^{1,0}_{D}$ is a homogeneous vector bundle corresponding to the
representation $S(U(n)\times U(1))\ni\left(\begin{array}[]{cc}U&0\\\ 0&{\rm
det}(U)^{-1}\end{array}\right)\mapsto{\rm det}(U)^{n+1}$. Define the subgroup
$U^{\prime}(n)=\left\\{\left(\begin{array}[]{cc}U&0\\\ 0&{\rm
det}(U)^{-1}\end{array}\right):{\rm det}(U)^{n+1}=1\right\\}$ in $SU(n,1)$. We
have $S^{1}(\bigwedge^{n}T^{1,0}_{D})=SU(n,1)/U^{\prime}(n)=PU(n,1)/SU(n)$.
Taking the universal covering $\widetilde{PU}(n,1)$ of $PU(n,1)$, we have
$\widetilde{S^{1}}(\bigwedge^{m}T^{1,0}_{D})=\widetilde{PU}(n,1)/SU(n)$. The
lifting of the Sasakian structure $(T_{D}^{1,0},\eta_{D})$ is
$\widetilde{PU}(n,1)$-invariant.

Assume $n=1$. Then $H^{2}_{B}(M)=\langle[d\eta]\rangle$. In this case, if
$c_{1,B}(T^{1,0})=-C[d\eta]$ for some positive constant $C$, then the Higgs
bundle $(E_{M},\,\theta_{M})$ is stable. By $H^{4}_{B}(M)=0$, the equality

$\int_{M}\left(2c_{2,B}(T^{1,0})-\frac{n}{n+1}c_{1,B}(T^{1,0})^{2}\right)\wedge(d\eta)^{n-2}\wedge\eta\,=\,0$

is trivial.

######  Corollary 5.6.

Let $(M,T^{1,0}_{M},\eta)$ be a compact $3$-dimensional Sasakian manifold with
$c_{1,B}(T^{1,0})=-C[d\eta]$. Then the universal covering $\widetilde{M}$ with
the lifting of some $A^{1}_{B}$-deformation of $(T^{1,0}_{M},\eta)$ is almost
isomorphic to $\widetilde{PU}(1,1)$ with a left-invariant Sasakian structure.

## References

  * [BH] D. Baraglia and P. Hekmati, A foliated Hitchin-Kobayashi correspondence, arXiv:1802.09699 (2018).
  * [Bel1] F. A. Belgun, On the metric structure of non-Kähler complex surfaces. Math. Ann. 317 (2000), no. 1, 1–40.
  * [Bel2] F. A. Belgun, Normal CR structures on compact 3-manifolds. Math. Z. 238 (2001), no. 3, 441–460.
  * [BaK] O. Baues and Y. Kamishima Locally homogeneous aspherical Sasaki manifolds. Differential Geom. Appl. 70 (2020) 41 pp.
  * [BK] I. Biswas and H. Kasuya, Higgs bundles and flat connections over compact Sasakian manifolds, Comm. Math. Phys. 385 (2021), 267–290.
  * [BK2] I. Biswas and H. Kasuya, Higgs bundles and flat connections over compact Sasakian manifolds, II: quasi-regular bundles, arXiv:2110.10644.
  * [BSW] D. Burns, S. Shnider, and R. O. Wells, Deformations of strictly pseudoconvex domains, Invent. Math. 46 (1978),237–253.
  * [BG] C. P. Boyer and K. Galicki, Sasakian geometry, Oxford Mathematical Monographs. Oxford University Press, Oxford, 2008.
  * [Co] K. Corlette, Flat G-bundles with canonical metrics,  J. Differential Geom. 28 (1988), 361–382.
  * [Hi] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59–126.
  * [Ge] H.Geiges, Normal contact structures on 3-manifolds. Tohoku Math. J.  (2) 49 (1997), no. 3, 415–422.
  * [GrSc] P. Griffiths, W. Schmid, Locally homogeneous complex manifolds, Acta Math., 123 (1969), 253–302.
  * [GS] V. Guillemin , S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer, Berlin, 1999.
  * [MuR] I. Mundet i Riera, Lifts of smooth group actions to line bundles. Bull. Lond. Math. Soc. 33(2001), no 3 351–361.
  * [Ra] J. H. Rawnsley, Flat partial connections and holomorphic structures in $\mathcal{C}^{\infty}$ vector bundles, Proc. Amer. Math. Soc. 73 (1979), 391–397.
  * [Sch] W. Schmid, Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22 (1973), 211–319.
  * [Si1] C. T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. Jour. Amer. Math. Soc. 1 (1988), no. 4, 867–918.
  * [Si] C. T. Simpson, Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. No. 75 (1992), 5–95.
  * [Ta] N. Tanaka, A Differential Geometric Study on strongly pseudoconvex CR manifolds, Lecture Notes in Math., 9, Kyoto University, 1975.
  * [Wa] A. W. Wadsley, Geodesic foliations by circles, J. Differential Geometry 10 (1975), 541–549.
  * [We] S. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25–41.
  * [Zh] X. Zhang, Energy properness and Sasakian-Einstein metrics, Comm. Math. Phys. 306 (2011), 229–260.