# Higgs bundles and flat connections over compact Sasakian manifolds

Indranil Biswas  School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS>and  Hisashi Kasuya  Department of Mathematics, Graduate School of Science,
Osaka University, Osaka, Japan<EMAIL_ADDRESS>
###### Abstract.

Given a compact Kähler manifold $X$, there is an equivalence of categories
between the completely reducible flat vector bundles on $X$ and the polystable
Higgs bundles $(E,\,\theta)$ on $X$ with $c_{1}(E)\,=\,0\,=\,c_{2}(E)$ [Si1],
[Co], [UY], [Do2]. We extend this equivalence of categories to the context of
compact Sasakian manifolds. We prove that on a compact Sasakian manifold,
there is an equivalence between the category of semi-simple flat vector
bundles on it and the category of polystable basic Higgs bundles on it with
trivial first and second basic Chern classes. We also prove that any stable
basic Higgs bundle over a compact Sasakian manifold admits a basic Hermitian
metric that satisfies the Yang–Mills–Higgs equation.

###### Key words and phrases:

Higgs bundle, harmonic map, Yang-Mills equation, flat connection, basic
cohomology classes.

###### 2010 Mathematics Subject Classification:

53C43, 53C07, 32L05, 58E15

###### Contents

  1. 1 Introduction
  2. 2 Strongly pseudo-convex CR manifolds and Sasakian manifolds
  3. 3 Harmonic metrics on Sasakian manifolds
  4. 4 Basic vector bundles
  5. 5 Basic Higgs bundles
  6. 6 Proof of Theorem 5.2
    1. 6.1 Preliminaries
    2. 6.2 Donaldson’s functional
    3. 6.3 The heat equation on Kähler cone
    4. 6.4 Proof of Theorem 5.2
    5. 6.5 Bogomolov–Miyaoka inequality
  7. 7 Correspondence between flat bundles and Higgs bundles

##  1\. Introduction

Let $X$ be a compact Kähler manifold equipped with a Kähler form $\omega$, and
let $E$ be a holomorphic vector bundle over $X$. Uhlenbeck and Yau proved that
$E$ is polystable with respect to $\omega$ if and only if $E$ admits a
Hermitian metric that satisfies the Hermite–Einstein equation defined using
$\omega$ [UY]. This result was proved earlier by Donaldson under the extra
assumptions that $X$ is a complex projective manifold and $\omega$ represents
a rational cohomology class [Do2]. As a consequence of these theorems of
Uhlenbeck–Yau and Donaldson, a holomorphic vector bundle on $X$ admits a flat
unitary connection if and only if $E$ is polystable with
$c_{1}(E)\,=\,0\,=\,c_{2}(E)$.

Hitchin proved that a Higgs bundle $(E,\,\theta)$ of rank two and degree zero
on a compact Riemann surface $X$ admits a Hermitian metric that solves the
Yang-Mills equation if and only if $(E,\,\theta)$ is polystable [Hi]. This
implies that a polystable Higgs bundle $(E,\,\theta)$ of rank two and degree
zero produces a completely reducible flat principal
$\text{GL}(2,{\mathbb{C}})$–bundle (also called a principal
$\text{GL}(2,{\mathbb{C}})$–bundle with a semi-simple flat connection) on $X$.
Donaldson proved that a principal $\text{GL}(2,{\mathbb{C}})$–bundle on $X$
with a completely reducible flat connection admits a harmonic metric [Do3];
this produces the inverse map from the completely reducible flat
$\text{GL}(2,{\mathbb{C}})$–bundles on $X$ to the polystable Higgs bundles of
rank two and degree zero on $X$.

Simpson and Corlette extended the above bijective correspondence to the more
general context of bundles over compact Kähler manifolds [Si1], [Co]. More
precisely, Simpson and Corlette proved that for any compact Kähler manifold
$X$, there is an equivalence of categories between the completely reducible
(also called semi-simple) flat vector bundles on $X$ and the polystable Higgs
bundles $(E,\,\theta)$ on $X$ with $c_{1}(E)\,=\,0\,=\,c_{2}(E)$; this result
is explained in detail in [Si2].

We recall that the contact manifolds constitute the odd dimensional
counterpart of the symplectic manifolds. For example, while the total space of
the cotangent bundle of a smooth manifold $X$ is a typical local model of a
symplectic manifold, the local model of a contact manifold is the total space
of the projective bundle $P(T^{*}M)$. In a similar vein, compact Sasakian
manifolds can be thought of as the odd dimensional counterparts of the compact
Kähler manifolds. A compact regular Sasakian manifold is in fact the unit
circle bundle inside a holomorphic Hermitian line bundle of positive curvature
on a complex projective manifold. More generally, a compact quasi-regular
Sasakian manifold is the unit circle bundle inside a holomorphic Hermitian
line bundle of positive curvature over a complex projective orbifold. However,
the global structures of more general compact Sasakian manifolds, known as
irregular Sasakian manifolds, do not admit any such explicit description.

It may be mentioned that Sasakian manifolds were introduced by Sasaki [Sa],
[SH], which explains the terminology. During the last twenty years there has
been a very substantial increase of the interests in Sasakian manifolds,
accompanied by a flurry of research activities (see [BG] and references
therein). It is evident from these references that a large part of this recent
investigations into Sasakian manifolds were carried out by C. Boyer and K.
Galicki. Another aspect contributing to this recent activities in Sasakian
manifolds is the discovery of their relevance in the string theory in
theoretical physics. This was initiated in the works of J. Maldacena [Ma]. For
further developments in this direction, see [GMSW], [MSY], [MS] and references
therein.

Our aim here is to extend, to the context of Sasakian manifolds, the earlier
mentioned equivalence of categories between the completely reducible flat
vector bundles on a compact Kähler manifold and the polystable Higgs bundles
$(E,\,\theta)$ with $c_{1}(E)\,=\,0\,=\,c_{2}(E)$.

The main result proved here states as follows (see Theorem 7.2).

######  Theorem 1.1.

For a compact Sasakian manifold $M$, there is an equivalence between the
category of semi-simple flat bundles over $M$ and the category of polystable
basic Higgs bundles over $M$ with trivial first and second basic Chern
classes.

In [BM1], the same question was addressed for the special case of quasi-
regular Sasakian manifolds. As mentioned before, quasi-regular Sasakian
manifolds are the unit circle bundles of holomorphic Hermitian line bundles of
positive curvature over any complex projective orbifold. Using this fact
together with a result of geometric group theory [BM1, p. 3492, Lemma 2.1],
the question for quasi-regular Sasakian manifolds actually reduces to that for
complex projective manifolds. In view of the known results on Higgs bundles
over complex projective manifolds, the differential geometric and analytical
investigations needed for the proof of Theorem 1.1 could entirely be avoided
in [BM1].

To prove Theorem 1.1 we establish analogues of the theory of harmonic metrics
on flat bundles and the theory of Hermitian-Yang-Mills metrics on Higgs
bundles on compact Kähler manifolds. Just as Corlette and Simpson proved for
compact Kähler manifolds ([Co, Theorem 5.1], [Si2, Lemma 1.1]), we obtain the
characterization of harmonic flat bundles over compact Sasakian manifolds in
terms of transversally holomorphic geometry (see Theorem 4.2). It may be
mentioned that this is facilitated by a special feature of Sasakian geometry
(see Theorem 3.2) which would not hold for a general transversally Kähler
Geometry. This produces the functor that we are seeking in Theorem 1.1 from
the semi-simple flat bundles to the basic Higgs bundles.

The proof of the opposite direction in Theorem 1.1 is inspired by a recent
work of D. Baraglia and P. Hekmati [BH]. Defining the stable and polystable
basic Higgs bundles over compact Sasakian manifolds, we prove for Sasakian
manifolds the following analog of [Si1, Theorem 1] proved in the Kähler
setting (see Theorem 5.2):

######  Theorem 1.2.

For a stable basic Higgs bundle $(E,\,\theta)$ over a compact Sasakian
manifold $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$, there exists a basic
Hermitian metric $h$ on $E$ such that

$\Lambda R^{D^{h}\perp}\,=\,0\,,$

where $R^{D^{h}\perp}$ is the trace-free part of the curvature $R^{D^{h}}$ of
the canonical connection $D^{h}$ associated to $h$.

Theorem 1.2 implies the following Bogomolov–Miyaoka type inequality (see
Corollary 6.6):

######  Corollary 1.3.

Let $(E,\,\theta)$ be a polystable basic Higgs bundle of rank $r$ over a
compact Sasakian manifold $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$ with $\dim
M\,=\,2n+1$. Then

$\int_{M}\left(2c_{2,B_{\mathcal{F}_{\xi}}}(E)-\frac{r-1}{r}c_{1,B_{\mathcal{F}_{\xi}}}(E)^{2}\right)(d\eta)^{n-2}\wedge\eta\,\geq\,0\,,$

where $c_{i,B_{\mathcal{F}_{\xi}}}(E)$ is the $i$-th basic Chern class of $E$.
If the above inequality is an equality, then $R^{D^{h}\perp}\,=\,0$.

We mention that the fundamental groups of compact Sasakian manifolds were
investigated in [Ch], [Ka], [BM2].

##  2\. Strongly pseudo-convex CR manifolds and Sasakian manifolds

Let $M$ be a $(2n+1)$-dimensional real smooth manifold. A CR-structure on $M$
is an $n$-dimensional complex sub-bundle $T^{1,0}$ of the complexified tangent
bundle $TM_{\mathbb{C}}\,=\,TM\otimes_{\mathbb{R}}{\mathbb{C}}$ such that
$T^{1,0}\cap\overline{T^{1,0}}=\\{0\\}$ and $T^{1,0}$ is integrable (i.e., the
locally defined sections of $T^{1,0}$ are closed under the Lie bracket
operation). We shall denote $\overline{T^{1,0}}$ by $T^{0,1}$. For a CR-
structure $T^{1,0}$ on $M$, there is a unique sub-bundle $S$ of rank $2n$ of
the real tangent bundle $TM$ together with a vector bundle homomorphism
$I\,:\,S\,\longrightarrow\,S$ satisfying the conditions that

  1. (1)

$I^{2}\,=\,-{\rm Id}_{S}$, and

  2. (2)

$T^{1,0}$ is the $\sqrt{-1}$–eigenbundle of $I$.

A $(2n+1)$-dimensional manifold $M$ equipped with a triple $(T^{1,0},\,S,\,I)$
as above is called a CR-manifold. A contact CR-manifold is a CR-manifold $M$
with a contact $1$-form $\eta$ on $M$ such that $\ker\eta\,=\,S$. Let $\xi$
denote the Reeb vector field for the contact form $\eta$. On a contact CR-
manifold, the above homomorphism $I$ extends to entire $TM$ by setting
$I(\xi)\,=\,0$.

######  Definition 2.1.

A contact CR-manifold $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$ is a strongly
pseudo-convex CR-manifold if the Hermitian form $L_{\eta}$ on $S_{x}$ defined
by $L_{\eta}(X,Y)\,=\,d\eta(X,IY)$, $X,\,Y\,\in\,S_{x}$, is positive definite
for every point $x\,\in\,M$.

Given any strongly pseudo-convex CR-manifold
$(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$, there is a canonical Riemann metric
$g_{\eta}$ on $M$, called the Webster metric, which is defined to be

$g_{\eta}(X,Y)\,:=\,L_{\eta}(X,Y)+\eta(X)\eta(Y)\,,\ \ X,\,Y\,\in\,T_{x}M\,,\
x\,\in\,M\,.$

######  Proposition 2.2 ([Ta], [We]).

For a strongly pseudo-convex CR-manifold

$(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$

there exists a unique affine connection $\nabla^{TW}$ on $TM$ such that the
following hold:

  1. (1)

$\nabla^{TW}(C^{\infty}(S))\,\subset\,A^{1}(M,\,S)$, where $A^{k}(M,\,S)$ is
the space of differential $k$-forms on $M$ with values in the vector bundle
$S$.

  2. (2)

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

  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 $x\,\in\,M$.

The affine connection $\nabla^{TW}$ in Proposition 2.2 is called the
Tanaka–Webster connection.

######  Definition 2.3.

A Sasakian manifold is a strongly pseudo-convex CR-manifold

$(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$

satisfying any (all) of the following equivalent conditions:

  1. (1)

$[\xi,\,A^{0}(M,\,S)]\,\subset\,A^{0}(M,\,S)$.

  2. (2)

${\mathcal{L}}_{\xi}\eta\,=\,0$ and ${\mathcal{L}}_{\xi}I\,=\,0$.

  3. (3)

$T^{TW}(\xi,v)\,=\,0$ for all $v\,\in\,TM$.

For a Sasakian manifold, the curvature $R^{TW}$ of the Tanaka–Webster
connection $\nabla^{TW}$ satisfies the equation

$R^{TW}(\xi,v)\,=\,0$ (2.1)

for all $v\,\in\,TM$. See [BG] for Sasakian manifolds.

##  3\. Harmonic metrics on Sasakian manifolds

Let $M$ be a compact Riemannian manifold and $E$ a flat complex vector bundle
over $M$ equipped with a flat connection $\nabla^{E}$. For any Hermitian
metric $h$ on $E$, we have a unique decomposition

$\nabla^{E}\,=\,\nabla^{h}+\phi_{h}$ (3.1)

such that $\nabla^{h}$ is a unitary connection and $\phi_{h}$ is a $1$-form on
$M$ with values in the self-adjoint endomorphisms of $E$ with respect to $h$.

######  Theorem 3.1 ([Co]).

If a flat complex vector bundle $(E,\,\nabla^{E})$ is semi-simple (meaning,
direct sum of irreducible flat connections), then there exists a Hermitian
metric (called the harmonic metric) $h$ on $E$ such that

$(\nabla^{h})^{\ast}\phi_{h}\,=\,0\,,$

where $(\nabla^{h})^{\ast}$ is the formal adjoint operator of $\nabla^{h}$. If
the connection $\nabla^{E}$ is irreducible, then the harmonic metric is unique
up to multiplication by a constant scalar.

######  Theorem 3.2.

Let $M$ be a compact Sasakian manifold with a Reeb vector field $\xi$, and let
$(E,\,\nabla^{E})$ be a semi-simple flat complex vector bundle over $M$. Then,

$\phi_{h}(\xi)\,=\,0$

for any harmonic metric $h$ on $E$ for the flat connection $\nabla^{E}$.

###### Proof.

This theorem is proved by modifying the proof of [Pe, Theorem 4.1]. Consider
the vector bundle $\bigwedge T^{\ast}M_{\mathbb{C}}\otimes{\rm End}(E)$ over
$M$ equipped with the connection

$\widetilde{\nabla}\,=\,\nabla^{TW}\otimes{\rm Id}_{{\rm End}(E)}+{\rm
Id}_{T^{\ast}M_{\mathbb{C}}}\otimes\nabla^{h}\,.$

Denote by $Cl(M)$ the Clifford bundle of $M$ associated with the Sasakian
metric $g_{\eta}$. Then, using the canonical isomorphism
$Cl(M)\,\cong\,\bigwedge T^{\ast}M_{\mathbb{C}}$, consider $\bigwedge
T^{\ast}M_{\mathbb{C}}\otimes{\rm End}(E)$ as a Dirac bundle. Define the Dirac
operator corresponding to $\widetilde{\nabla}$ to be

${\mathcal{D}}\,=\,\sum_{i}e_{i}\cdot\widetilde{\nabla}_{e_{i}}\,,$

where $\\{e_{i}\\}$ is a local orthonormal frame for $TM$ and “$\cdot$”
denotes the Clifford multiplication. Then this ${\mathcal{D}}$ is in fact a
formal self-adjoint operator (see [Pe, Proposition 2.1]).

We have the following formula (cf. [Pe, Lemma 3.1]).

######  Lemma 3.3.

${\mathcal{D}}\phi_{h}\,=\,-d\eta\otimes\phi_{h}(\xi)$.

###### Proof.

By the flatness of the connection $\nabla^{E}$, we have
$(\nabla^{h}+\phi_{h})^{2}\,=\,0$, and this implies that

$R^{h}\,=\,-\frac{1}{2}[\phi_{h},\,\phi_{h}]$ (3.2)

and $\nabla^{h}\phi_{h}\,=\,0$, where $R^{h}$ is the curvature of the
Hermitian connection $\nabla^{h}$ on $E$ associated to $h$ (see (3.1)). By a
computation as in the proof of [Pe, Lemma 3.1], we conclude that
${\mathcal{D}}\phi_{h}-(\nabla^{h})^{\ast}\phi_{h}$ is the anti-symmetrization
of covariant derivative on $\bigwedge T^{\ast}M_{\mathbb{C}}\otimes{\rm
End}(E)$, and we have

$({\mathcal{D}}\phi_{h}-(\nabla^{h})^{\ast}\phi_{h})(X,Y)\,=\,(\nabla^{h}\phi_{h})(X,Y)-\phi_{h}(T^{TW}(X,Y))\,.$

By the harmonicity of $h$,

$(\nabla^{h})^{\ast}\phi_{h}\,=\,0\,=\,\nabla^{h}\phi_{h}$

as above. Now the lemma follows from Proposition 2.2. ∎

Consider the following formula

$\frac{1}{2}[{\mathcal{D}}^{2},\,(\xi\cdot)]\,=\,\frac{1}{2}({\mathcal{D}}^{2}(\xi\cdot)-\xi\cdot{\mathcal{D}}^{2})\,=\,-2\xi\cdot{\mathcal{R}}_{\xi}$

(see [Pe, Corollary 2.1]), where ${\mathcal{R}}_{\xi}$ is the endomorphism
defined by

${\mathcal{R}}_{\xi}\,=\,-\frac{1}{2}\sum_{i}\xi\cdot
e_{i}\cdot\widetilde{R}(\xi,e_{i})$

with $\widetilde{R}$ being the curvature of the connection
$\widetilde{\nabla}\,=\,\nabla^{TW}\otimes{\rm Id}_{{\rm End}(E)}+{\rm
Id}_{T^{\ast}M_{\mathbb{C}}}\otimes\nabla^{h}$ on
$T^{\ast}M_{\mathbb{C}}\otimes{\rm End}(E)$. For convenience, we take
$\\{e_{i}\\}$ to be a local orthonormal frame of $TM$ such that
$e_{0}\,=\,\xi$ and $e_{1},\,\cdots,\,e_{2n}$ is a local orthonormal frame of
$S$ associated with $L_{\eta}$. We have

$\frac{1}{2}\int_{M}\langle[{\mathcal{D}}^{2},(\xi\cdot)]\phi_{h},\,\xi\cdot\phi_{h}\rangle\,=\,-2\int_{M}\langle\xi\cdot{\mathcal{R}}_{\xi}\phi_{h},\,\xi\cdot\phi_{h}\rangle\,=\,-2\int_{M}\langle{\mathcal{R}}_{\xi}\phi_{h},\,\phi_{h}\rangle$
$=\,\sum_{i=0}^{2n}\int_{M}\langle\xi\cdot e_{i}\cdot
R(\xi,e_{i})\phi_{h},\,\phi_{h}\rangle\,=\,-\sum_{i=0}^{2n}\int_{M}\langle
e_{i}\cdot R(\xi,e_{i})\phi_{h},\,\xi\cdot\phi_{h}\rangle\,.$

By $R^{TW}(\xi,-)\,=\,0$ (see (2.1)), we have that

$-\sum_{i=0}^{2n}\int_{M}\langle e_{i}\cdot
R(\xi,e_{i})\phi_{h},\,\xi\cdot\phi_{h}\rangle$
$=\,-\sum_{i=1}^{2n}\int_{M}\left(\langle(R^{h}(\xi,e_{i})\phi_{h})(e_{i}),\,\phi_{h}(\xi)\rangle-\langle(R^{h}(\xi,e_{i})\phi_{h})(\xi),\,\phi_{h}(e_{i})\rangle\right)$

(see [Pe, formula (17)]). Using $R^{h}\,=\,-\frac{1}{2}[\phi_{h},\,\phi_{h}]$
(see (3.2)), this is equal to

$-\frac{1}{2}\sum_{i=1}^{2n}\int_{M}\left(\langle[[\phi_{h}(\xi),\phi_{h}(e_{i})],\phi_{h}(e_{i})],\,\phi_{h}(\xi)\rangle-\langle[[\phi_{h}(\xi),\,\phi_{h}(e_{i})],\,\phi_{h}(\xi)],\,\phi_{h}(e_{i})\rangle\right).$

Since $\phi_{h}$ is a $1$-form with values in the self-adjoint endomorphisms
of $E$, this is equal to

$-\sum_{i=1}^{2n}\int_{M}\langle[\phi_{h}(\xi),\,\phi_{h}(e_{i})],\,[\phi_{h}(\xi),\,\phi_{h}(e_{i})]\rangle.$

Thus we obtain the inequality

$\frac{1}{2}\int_{M}\langle[{\mathcal{D}}^{2},\,(\xi\cdot)]\phi_{h},\,\xi\cdot\phi_{h}\rangle\,\leq\,0\,.$
(3.3)

On the other hand, we can directly compute that

$\frac{1}{2}\int_{M}\langle[{\mathcal{D}}^{2},\,(\xi\cdot)]\phi_{h},\,\xi\cdot\phi_{h}\rangle\,=\,4\int_{M}\left(\langle\xi\cdot{\mathcal{D}}(\phi_{h}),\,\widetilde{\nabla}_{\xi}\phi_{h}\rangle+\langle\widetilde{\nabla}_{\xi}\phi_{h},\,\widetilde{\nabla}_{\xi}\phi_{h}\rangle\right)$

as done in [Pe, Page 594]. Using $(\xi\cdot)\,=\,(\eta\wedge)-i_{\xi}$ and
${\mathcal{D}}\phi_{h}\,=\,-d\eta\otimes\phi_{h}(\xi)$, where $i_{\xi}$
denotes the interior product, together with the fact that $i_{\xi}d\eta\,=\,0$
(this means that the form $d\eta$ is basic; see (4.1)), we conclude that
$\langle\xi\cdot{\mathcal{D}}(\phi_{h}),\,\widetilde{\nabla}_{\xi}\phi_{h}\rangle\,=\,0$.
Thus, we have the inequality

$\frac{1}{2}\int_{M}\langle[{\mathcal{D}}^{2},(\xi\cdot)]\phi_{h},\,\xi\cdot\phi_{h}\rangle\,=\,\int_{M}\langle\widetilde{\nabla}_{\xi}\phi_{h},\,\widetilde{\nabla}_{\xi}\phi_{h}\rangle\,\geq\,0\,.$
(3.4)

Now from (3.3) and (3.4) it follows that
$\widetilde{\nabla}_{\xi}\phi_{h}\,=\,0$.

By the same argument as in [Pe, Page 598], we have that $\phi_{h}(\xi)\,=\,0$.
∎

##  4\. Basic vector bundles

Let $M$ be a compact manifold equipped with a nonsingular foliation
$\mathcal{F}$. Then, a differential form $\omega$ on $M$ is called basic if
for every vector field $X$ on $M$ which is tangent to the leaves of
$\mathcal{F}$, the equations

$i_{X}\omega\,=\,0\,=\,{\mathcal{L}}_{X}\omega$ (4.1)

hold.

We denote by $A^{\ast}_{B_{\mathcal{F}}}(M)$ the subspace of basic forms in
the de Rham complex $A^{\ast}(M)$. It is straight-forward to check that
$A^{\ast}_{B_{\mathcal{F}}}(M)$ is a sub-complex of the de Rham complex
$A^{\ast}(M)$. Denote by

$H_{B_{\mathcal{F}_{\xi}}}^{\ast}(M)\,=\,\bigoplus_{i\geq
0}H_{B_{\mathcal{F}_{\xi}}}^{i}(M)$ (4.2)

the cohomology of the basic de Rham complex
$A^{\ast}_{B_{\mathcal{F}_{\xi}}}(M)$. Note that there is a natural
homomorphism from $H_{B_{\mathcal{F}_{\xi}}}^{i}(M)$ to the $i$-th de Rham
cohomology of $M$.

Let $E$ be a complex $C^{\infty}$ vector bundle over $M$ of rank $r$. This $E$
is said to be basic if it has local trivializations with respect to an open
covering $M\,=\,\bigcup_{\alpha}U_{\alpha}$ satisfying the condition that each
transition function $f_{\alpha\beta}\,:\,U_{\alpha}\cap
U_{\beta}\,\longrightarrow\,{\rm GL}(r,\mathbb{C})$ is basic on
$U_{\alpha}\cap U_{\beta}$, meaning it is constant on the leaves of the
foliation $\mathcal{F}$. This condition is equivalent to the condition that
$E$ has a flat partial connection in the direction of $\mathcal{F}$.

For a basic vector bundle $E$ over $M$, a differential form
$\omega\,\in\,A^{\ast}(M,\,E)$ with values in $E$ is called basic if $\omega$
is basic on every $U_{\alpha}$, meaning
$\omega_{|U_{\alpha}}\,\in\,A^{\ast}_{B_{\mathcal{F}}}(U_{\alpha})\otimes\mathbb{C}^{r}$
for every $\alpha$. Let

$A^{\ast}_{B_{\mathcal{F}}}(M,\,E)\,\subset\,A^{\ast}(M,\,E)$

denote the subspace of basic forms in the space $A^{\ast}(M,\,E)$ of
differential forms with values in $E$.

We shall consider any flat vector bundle $(E,\,\nabla^{E})$ over $M$ as a
basic vector bundle by local flat frames. Then, $A^{\ast}_{B}(M,\,E)$ is a
sub-complex of the de Rham complex $A^{\ast}(M,\,E)$ equipped with the
differential $d^{E}$ associated to the flat connection $\nabla^{E}$.

Let $(E,\,\nabla^{E})$ be a flat vector bundle over $M$. For a Hermitian
metric $h$ on $E$, consider the canonical decomposition

$\nabla^{E}\,=\,\nabla^{h}+\phi_{h}$

in (3.1).

######  Proposition 4.1.

The following two conditions are equivalent:

  * •

$\phi_{h}(X)\,=\,0$ for all $X\,\in\,T_{x}{\mathcal{F}}$ and $x\,\in\,M$.

  * •

The Hermitian structure $h$ is basic, meaning
$h\,\in\,A_{B_{\mathcal{F}}}^{0}(M,\,E^{\ast}\otimes\overline{E}^{\ast})$.

These conditions imply that $\phi_{h}\,\in\,A^{1}_{B_{\mathcal{F}}}(M,\,{\rm
End}(E))$.

###### Proof.

For local flat frames of $E$ with respect to an open covering
$M\,=\,\bigcup_{\alpha}U_{\alpha}$, we can write

$\phi_{h}\,=\,-\frac{1}{2}df_{\alpha}f_{\alpha}^{-1}$

on each $U_{\alpha}$ for certain functions $f_{\alpha}$ on $U_{\alpha}$ with
values in the positive definite Hermitian matrices $Herm^{+}_{r}$ with respect
to $h$. Then, $\phi_{h}(X)\,=\,0$ for any $X\,\in\,T_{x}{\mathcal{F}}$ if and
only if each $f_{\alpha}$ is basic. Thus the proposition follows. ∎

Let $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$ be a compact Sasakian manifold.
Then the Reeb vector field $\xi$ defines a $1$-dimensional foliation
$\mathcal{F}_{\xi}$ on $M$. It is known that the map
$I\,:\,TM\,\longrightarrow\,TM$ associated with the CR-structure $T^{1,0}$
defines a transversely complex structure on the foliated manifold
$(M,\,\mathcal{F}_{\xi})$. Furthermore, the closed basic $2$-form $d\eta$ is a
transversely Kähler structure with respect to this transversely complex
structure. Corresponding to the decomposition
$S_{\mathbb{C}}\,=\,T^{1,0}\oplus T^{0,1}$, we have the bigrading

$A^{r}_{B_{\mathcal{F}_{\xi}}}(M)_{\mathbb{C}}\,=\,\bigoplus_{p+q=r}A^{p,q}(M)$

as well as the decomposition of the exterior differential

$d_{|A^{r}_{B_{\mathcal{F}_{\xi}}}(M)_{\mathbb{C}}}\,=\,\partial_{\xi}+\overline{\partial}_{\xi}$

on $A^{r}_{B_{\mathcal{F}_{\xi}}}(M)_{\mathbb{C}}$, so that

$\partial_{\xi}\,:\,A^{p,q}_{B_{\mathcal{F}}}(M)\,\longrightarrow\,A^{p+1,q}_{B_{\mathcal{F}}}(M)\
\text{ and }\
\overline{\partial}_{\xi}\,:\,A^{p,q}_{B_{\mathcal{F}}}(M)\,\longrightarrow\,A^{p,q+1}_{B_{\mathcal{F}}}(M)\,.$

We shall now use transverse Hodge theory ([KT], [Ka]). Consider the usual
Hodge star operator

$\ast\,:\,A^{r}(M)\,\longrightarrow\,A^{2n+1-r}(M)$

associated to the Sasakian metric $g_{\eta}$ and the formal adjoint operator

$\delta\,=\,-\ast d\ast\,:\,A^{r}(M)\,\longrightarrow\,A^{r-1}(M)\,.$

We define the homomorphism

$\star_{\xi}\,:\,A^{r}_{B_{\mathcal{F}_{\xi}}}(M)\,\longrightarrow\,A^{2n-r}_{B_{\mathcal{F}_{\xi}}}(M)$

to be $\star_{\xi}\omega\,=\,\ast(\eta\wedge\omega)$ for
$\omega\,\in\,A^{r}_{B_{\mathcal{F}_{\xi}}}(M)$. Also define the operators

$\delta_{\xi}\,=\,-\star_{\xi}d\star_{\xi}\,:\,A^{r}_{B_{\mathcal{F}_{\xi}}}(M)\,\longrightarrow\,A^{r-1}_{B_{\mathcal{F}_{\xi}}}(M)\,,$
$\partial_{\xi}^{\ast}\,=\,-\star_{\xi}\overline{\partial}_{\xi}\star_{\xi}\,:\,A^{p,q}_{B_{\mathcal{F}}}(M)\,\longrightarrow\,A^{p-1,q}_{B_{\mathcal{F}}}(M)\,,$
$\overline{\partial}_{\xi}^{\ast}\,=\,-\star_{\xi}\partial_{\xi}\star_{\xi}\,:\,A^{p,q}_{B_{\mathcal{F}}}(M)\,\longrightarrow\,A^{p,q-1}_{B_{\mathcal{F}}}(M)$

and $\Lambda\,=\,-\star_{\xi}(d\eta\wedge)\star_{\xi}$. They are the formal
adjoints of $d$, $\partial_{\xi}$, $\overline{\partial}_{\xi}$ and
$(d\eta\wedge)$ respectively for the pairing

$A^{r}_{B_{\mathcal{F}_{\xi}}}(M)\times
A^{r}_{B_{\mathcal{F}_{\xi}}}(M)\,\ni\,(\alpha,\,\beta)\,\longmapsto\,\int_{M}\eta\wedge\alpha\wedge\star_{\xi}\beta\,.$

Define the Laplacian operators

$\Delta\,:\,A^{r}(M)\,\longrightarrow\,A^{r}(M)\ \ \text{ and }\ \
\Delta_{\xi}\,:\,A^{r}_{B_{\mathcal{F}_{\xi}}}(M)\,\longrightarrow\,A^{r}_{B_{\mathcal{F}_{\xi}}}(M)$

by

$\Delta\,=\,d\delta+\delta d\ \ \text{ and }\ \
\Delta_{\xi}\,=\,d\delta_{\xi}+\delta_{\xi}d$

respectively. For $\omega\,\in\,A^{r}_{B_{\mathcal{F}_{\xi}}}(M)$, since the
relation $\ast\omega\,=\,(\star_{\xi}\omega)\wedge\eta$ holds, we have the
relation

$\delta\omega\,=\,\delta_{\xi}\omega+\ast(d\eta\wedge\star_{\xi}\omega)\,.$

Thus, for $\omega\,\in\,A^{1}_{B_{\mathcal{F}_{\xi}}}(M)$, the equality
$\delta_{\xi}\omega\,=\,\delta\omega$ holds, and hence for
$f\,\in\,A^{0}_{B_{\mathcal{F}_{\xi}}}(M)$, we have that
$\Delta_{\xi}f\,=\,\Delta f$. The usual Kähler identities

$[\Lambda,\partial_{\xi}]\,=\,-\sqrt{-1}\overline{\partial}_{\xi}^{\ast}\ \
\text{ and }\ \
[\Lambda,\overline{\partial}_{\xi}]=\sqrt{-1}\partial_{\xi}^{\ast}$

hold, and these imply that

$\Delta_{\xi}\,=\,2\Delta_{\xi}^{\prime}\,=\,2\Delta_{\xi}^{\prime\prime}\,,$

where
$\Delta_{\xi}^{\prime}\,=\,\partial_{\xi}\partial_{\xi}^{\ast}+\partial_{\xi}^{\ast}\partial_{\xi}$
and
$\Delta_{\xi}^{\prime\prime}\,=\,\overline{\partial}_{\xi}\overline{\partial}_{\xi}^{\ast}+\overline{\partial}_{\xi}^{\ast}\overline{\partial}_{\xi}$.

Let $E$ be a complex $C^{\infty}$ vector bundle over $M$. We say that $E$ is
transversely holomorphic if it admits local trivializations with respect to an
open covering $M\,=\,\bigcup_{\alpha}U_{\alpha}$ such that each transition
function $f_{\alpha\beta}\,:\,U_{\alpha}\cap U_{\beta}\,\longrightarrow\,{\rm
GL}(r,\mathbb{C})$ is basic and holomorphic (i.e.,
$\overline{\partial}_{\xi}f_{\alpha\beta}\,=\,0$). For a transversely
holomorphic vector bundle $E$ over $M$, define the canonical Dolbeault
operator

$\overline{\partial}_{E}\,:\,A^{p,q}_{B_{\mathcal{F}}}(M,\,E)\,\longrightarrow\,A^{p,q+1}_{B_{\mathcal{F}}}(M,\,E)$

satisfying the following two conditions:

$\overline{\partial}_{E}(\omega\wedge
s)\,=\,(\overline{\partial}_{\xi}\omega)\wedge
s+(-1)^{p+q}\omega\wedge\overline{\partial}_{E}s$

and $\overline{\partial}_{E}\overline{\partial}_{E}\,=\,0$. Conversely, on a
basic complex vector bundle $E$, if we have an operator
$\overline{\partial}_{E}\,:\,A^{p,q}_{B_{\mathcal{F}}}(M,\,E)\,\longrightarrow\,A^{p,q+1}_{B_{\mathcal{F}}}(M,\,E)$
satisfying the above two conditions, then the differential operator
$\overline{\partial}_{E}$ defines a canonical transversely holomorphic
structure on $E$ by the Frobenius theorem [Ko, p. 9, Proposition 3.7].

Let $(E,\,\nabla^{E})$ be a flat complex vector bundle over $M$ such that $E$
is equipped with a Hermitian metric $h$. Let

$\nabla^{E}\,=\,\nabla^{h}+\phi_{h}$

be the canonical decomposition of the connection $\nabla^{E}$ (see (3.1)).
Then, using the pairing $A^{r}(M,\,E)\times
A^{2n+1-r}(M,\,E)\,\longrightarrow\,A^{2n+1}(M)$ associated with $h$, define
the Hodge star operator

$\ast_{h}\,:\,A^{r}(M,\,E)\,\longrightarrow\,A^{2n+1-r}(M,\,E)$

as well as the formal adjoint operator
$(\nabla^{h})^{\ast}\,=\,-\ast_{h}\nabla^{h}\ast_{h}$.

Assume the Hermitian structure $h$ to be basic (equivalently,
$\phi_{h}(\xi)\,=\,0$ by Proposition 4.1). Since
$\nabla^{E}_{\xi}\,=\,\nabla^{h}_{\xi}$, it follows that the unitary
connection $\nabla^{h}$ restricts to a homomorphism

$\nabla^{h}\,:\,A^{r}_{B_{\mathcal{F}}}(M,\,E)\,\longrightarrow\,A^{r+1}_{B_{\mathcal{F}}}(M,\,E)\,.$

Now, on $A^{p,q}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$, decompose

$\nabla^{h}\,=\,\partial_{h,\xi}+\overline{\partial}_{h,\xi}$

such that
$\partial_{h,\xi}\,:\,A^{p,q}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\longrightarrow\,A^{p+1,q}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$
and
$\overline{\partial}_{h,\xi}\,:\,A^{p,q}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\longrightarrow\,A^{p,q+1}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$.
Since $\nabla^{h}$ is a unitary connection, we have that

$\overline{\partial}_{\xi}h(s_{1},s_{2})\,=\,h(\overline{\partial}_{h,\xi}s_{1},s_{2})+h(s_{1},\partial_{h,\xi}s_{2})$

for $s_{1},\,s_{2}\,\in\,A^{0,0}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$. We define
the operator

$\star_{h,\xi}\,:\,A^{r}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\longrightarrow\,A^{2n-r}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$

and the formal adjoint operators

$(\nabla^{h})^{\ast}_{\xi}\,=\,-\star_{h,\xi}\nabla^{h}\star_{h,\xi}\,:\,A^{r}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\longrightarrow\,A^{r-1}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,,$

$\partial_{h,\xi}^{\ast}\,=\,-\star_{h,\xi}\overline{\partial}_{h,\xi}\star_{h,\xi}\,:\,A^{p,q}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\longrightarrow\,A^{p-1,q}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$
and

$\overline{\partial}_{h,\xi}^{\ast}\,=\,-\star_{h,\xi}\partial_{h,\xi}\star_{h,\xi}\,:\,A^{p,q}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\longrightarrow\,A^{p,q-1}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,,$

as well we $\Lambda_{h}\,:=\,-\star_{h,\xi}(d\eta\wedge)\star_{h,\xi}$ in the
same way as above. We now have the Kähler identities

$[\Lambda,\,\partial_{h,\xi}]\,=\,-\sqrt{-1}\overline{\partial}_{h,\xi}^{\ast}\
\ \text{ and }\ \
[\Lambda,\,\overline{\partial}_{h,\xi}]\,=\,\sqrt{-1}\partial_{h,\xi}^{\ast}\,.$

######  Theorem 4.2.

Let $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$ be a compact Sasakian manifold
and $(E,\,\nabla^{E})$ a flat complex vector bundle over $M$ with a Hermitian
metric $h$. Then the following two conditions are equivalent:

  * •

The Hermitian structure $h$ is harmonic, i.e.,
$(\nabla^{h})^{\ast}\phi_{h}\,=\,0$.

  * •

The Hermitian structure $h$ is basic ($\Leftrightarrow\phi_{h}(\xi)\,=\,0$ and
implying that

$\phi_{h}\,\in\,A^{1}_{B_{\mathcal{F}_{\xi}}}(M,{\rm End}(E))$

by Proposition 4.1), and for the decomposition

$\phi_{h}\,=\,\theta^{1,0}_{h,\xi}+\theta^{0,1}_{h,\xi}$

with $\theta^{1,0}_{h,\xi}\,\in\,A^{1,0}_{B_{\mathcal{F}_{\xi}}}(M,\,{\rm
End}(E))$ and $\theta^{0,1}_{h,\xi}\in
A^{0,1}_{B_{\mathcal{F}_{\xi}}}(M,\,{\rm End}(E))$,

$\overline{\partial}_{h,\xi}\overline{\partial}_{h,\xi}\,=\,0,\qquad[\theta^{1,0}_{h,\xi},\,\theta^{1,0}_{h,\xi}]\,=\,0\qquad{\rm
and}\qquad\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi}\,=\,0\,.$

###### Proof.

First suppose that $h$ is basic. Then by the relation similar to the relation
between $\delta$ and $\delta_{\xi}$, we have that

$(\nabla^{h})^{\ast}\phi_{h}\,=\,(\nabla^{h})^{\ast}_{\xi}\phi_{h}\,.$

By the same computation as in [Co, Page 376], we have that

$(\nabla^{h})^{\ast}_{\xi}\phi_{h}\,=\,\sqrt{-1}\Lambda_{h}(\partial_{h,\xi}-\overline{\partial}_{h,\xi})\phi_{h}\,=\,2\partial_{h,\xi}^{\ast}\theta^{1,0}_{h,\xi}\,.$

Thus, $h$ is harmonic if and only if
$\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi}$ is primitive, meaning
$\Lambda_{h}\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi}\,=\,0$. From this
it follows that the second condition in the theorem implies the first one.

To prove the converse, suppose that $h$ is harmonic. Then, by Theorem 3.2, $h$
is basic. By the flatness of the connection $\nabla^{E}$, we have that
$(\nabla^{h})^{2}\,=\,-\frac{1}{2}[\phi_{h},\phi_{h}]$ and
$\nabla^{h}\phi_{h}\,=\,0$. Now using
$(\nabla^{h})^{2}\,=\,-\frac{1}{2}[\phi_{h},\phi_{h}]$ it follows that

$\partial_{h,\xi}\partial_{h,\xi}\,=\,-[\theta^{1,0}_{h,\xi},\,\theta^{1,0}_{h,\xi}]\qquad{\rm
and}\qquad\overline{\partial}_{h,\xi}\overline{\partial}_{h,\xi}\,=\,-[\theta^{0,1}_{h,\xi},\,\theta^{0,1}_{h,\xi}]\,.$

Now $\nabla^{h}\phi_{h}\,=\,0$ implies that

$\partial_{h,\xi}\theta^{1,0}_{h,\xi},\qquad\overline{\partial}_{h,\xi}\theta^{0,1}_{h,\xi}\qquad{\rm
and}\qquad\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi}+\partial_{h,\xi}\theta^{0,1}_{h,\xi}=0.$

We have (cf. the proof of [Co, Theorem 5.1])

$\partial_{\xi}\overline{\partial}_{\xi}h(\theta^{1,0}_{h,\xi},\,\theta^{1,0}_{h,\xi})\,=\,h(\partial_{h,\xi}\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi},\,\theta^{1,0}_{h,\xi})+h(\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi},\,\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi})$
$=\,-h(\partial_{h,\xi}\partial_{h,\xi}\theta^{0,1}_{h,\xi},\,\theta^{1,0}_{h,\xi})+h(\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi},\,\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi})\,=\,h\left([\theta^{1,0}_{h,\xi},\,\theta^{1,0}_{h,\xi}],\,\theta^{0,1}_{h,\xi}],\,\theta^{1,0}_{h,\xi}\right)+h\left(\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi},\,\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi}\right)$
$=\,-h\left([\theta^{1,0}_{h,\xi},\,\theta^{1,0}_{h,\xi}],\,[\theta^{1,0}_{h,\xi},\,\theta^{1,0}_{h,\xi}]\right)+h\left(\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi},\,\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi}\right)\,.$

Since $\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi}$ is primitive by the
above argument, using the Lefschetz decomposition of basic forms corresponding
to the transversely Kähler form $d\eta$, and integrating the wedge product of
this equation and $(d\eta)^{n-2}\wedge\eta$, from the Stokes theorem, we
obtain that

$0\,=\,-C_{1}\int_{M}|\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi}|-C_{2}\int_{M}|[\theta^{1,0}_{h,\xi},\,\theta^{1,0}_{h,\xi}]|$

for some positive constant $C_{1}$ and $C_{2}$. Consequently, we have

$[\theta^{1,0}_{h,\xi},\,\theta^{1,0}_{h,\xi}]\,=\,0\qquad{\rm
and}\qquad\overline{\partial}_{h,\xi}\theta^{1,0}_{h,\xi}\,=\,0\,.$

Using
$\partial_{h,\xi}\partial_{h,\xi}\,=\,-[\theta^{1,0}_{h,\xi},\,\theta^{1,0}_{h,\xi}]$
it follows that $\overline{\partial}_{h,\xi}\overline{\partial}_{h,\xi}\,=\,0$
and hence the theorem is proved. ∎

##  5\. Basic Higgs bundles

Let $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$ be a compact Sasakian manifold.
As in (4.2), $H_{B_{\mathcal{F}_{\xi}}}^{\ast}(M)$ is the cohomology of the
basic de Rham complex $A^{\ast}_{B_{\mathcal{F}_{\xi}}}(M)$. By the Kähler
identities on $A^{p,q}_{B_{\mathcal{F}_{\xi}}}(M)$, as in the usual Kähler
case, we have the canonical Hodge decomposition

$H_{B_{\mathcal{F}_{\xi}}}^{r}(M)\otimes\mathbb{C}\,=\,\bigoplus_{p+q=r}H^{p,q}_{B_{\mathcal{F}_{\xi}}}(M)\,.$

Let $E$ be a complex basic vector bundle over $M$. Consider a connection
operator

$\nabla\,:\,A^{\ast}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\longrightarrow\,A^{\ast+1}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$

satisfying the equation

$\nabla(\omega s)\,=\,(d\omega)s+(-1)^{r}\omega\wedge\nabla s$

for $\omega\,\in\,A^{r}_{B_{\mathcal{F}_{\xi}}}(M)$ and
$s\,\in\,A^{0}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$. Let

$R^{\nabla}\,=\,\nabla^{2}\,\in\,A_{B_{\mathcal{F}}}^{2}({M,\,\rm End}(E))$

be the curvature of $\nabla$. For any $1\,\leq\,i\,\leq\,n$, Define
$c_{i,B_{\mathcal{F}_{\xi}}}(E,\nabla)\,\in\,A^{2i}_{B_{\mathcal{F}_{\xi}}}(M)$
by

${\rm
det}\left(I-\frac{R^{\nabla}}{2\pi\sqrt{-1}}\right)\,=\,1+\sum_{i=1}^{n}c_{i,B_{\mathcal{F}_{\xi}}}(E,\nabla)\,.$

Then, as the case of usual Chern-Weil theory, the cohomology class

$c_{i,B_{\mathcal{F}_{\xi}}}(E)\,\in\,H_{B_{\mathcal{F}_{\xi}}}^{2i}(M)$

of each $c_{i,B_{\mathcal{F}_{\xi}}}(E,\nabla)$ is independent of the choice
of the connection $\nabla$ taking $A^{\ast}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$ to
$A^{\ast+1}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$. If $E$ is a transversely
holomorphic vector bundle, just as in the case of Chern classes of holomorphic
vector bundles over compact Kähler manifolds, we have that
$c_{i,B_{\mathcal{F}_{\xi}}}(E)\,\in\,H^{i,i}_{B_{\mathcal{F}_{\xi}}}(M)$.

A basic Higgs bundle over $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$ is a pair
$(E,\,\theta)$ consisting of a transversely holomorphic vector bundle $E$ and
a section $\theta\,\in\,A^{1,0}_{B_{\mathcal{F}_{\xi}}}(M,\,{\rm End}(E))$
satisfying the following two conditions:

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

This section $\theta$ is called a Higgs field on $E$. Note that

$\theta\wedge\theta\,\in\,A^{2,0}_{B_{\mathcal{F}_{\xi}}}(M,\,{\rm
End}(E))\,.$

For a basic Higgs bundle $(E,\,\theta)$, we define the operator

$D^{\prime\prime}_{E,\theta}\,=\,\overline{\partial}_{E}+\theta\,:\,A^{\ast}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\longrightarrow\,A^{\ast+1}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,.$

Let $h$ be a Hermitian metric on $E$. Assume that $h$ is basic. Define
$\overline{\theta}_{h}\,\in\,A^{0,1}_{B_{\mathcal{F}_{\xi}}}(M,\,{\rm
End}(E))$ by

$(\theta(e_{1}),\,e_{2})\,=\,(e_{1},\,\overline{\theta}_{h}(e_{2}))$ (5.1)

for $e_{1},\,e_{2}\,\in\,E$.

Let $\nabla$ be a unitary connection on $E$ preserving $h$ associated to $h$
such that $\nabla$ actually restricts as
$\nabla\,:\,A^{\ast}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\longrightarrow\,A^{\ast+1}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$.
Define the connection

$D\,=\,\nabla+\theta+\overline{\theta}_{h}\,,$

and consider the curvature $R^{D}\,=\,D^{2}$ of $D$. Then, by the assumptions
on $\theta$ and $h$, we have that

$R^{D}\,\in\,A^{2}_{B_{\mathcal{F}_{\xi}}}(M,\,{\rm End}(E))\,.$

Define the degree of $E$ to be

${\rm deg}(E)\,=\,\frac{\sqrt{-1}}{2\pi}\int_{M}{\rm Tr}(\Lambda R^{D})\,.$

Note that we have

${\rm deg}(E)\,=\,\int_{M}c_{1,B_{\mathcal{F}_{\xi}}}(E)\wedge
d\eta\wedge\eta\,,$

and hence ${\rm deg}(E)$ depends only on $E$.

We define the canonical (Chern) connection $\nabla^{h}$ on the transversely
holomorphic Hermitian bundle $(E,\,h)$ in the following way. Take local basic
holomorphic frames $e^{\alpha}_{1},\,\cdots,\,e_{\alpha}^{\alpha}$ of $E$ with
respect to an open covering $M\,=\,\bigcup_{\alpha}U_{\alpha}$. For the
Hermitian matrices $H_{\alpha}\,=\,(h_{i\overline{j}}^{\alpha})$ with
$h_{i\overline{j}}^{\alpha}\,=\,h(e^{\alpha}_{i},e^{\alpha}_{j})$, we define

$\nabla^{h}\,=\,d+H_{\alpha}^{-1}\partial_{\xi}H_{\alpha}$ (5.2)

on each $U_{\alpha}$.

Let us consider the canonical connection
$D^{h}\,=\,\nabla^{h}+\theta+\overline{\theta}_{h}$ on $E$. We also define the
operator
$\partial_{E,h}\,:\,A^{p,q}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\longrightarrow\,A^{p+1,q}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$
such that

$\partial_{E,h}\,=\,\partial_{\xi}+H_{\alpha}^{-1}\partial_{\xi}H_{\alpha}$

on each $U_{\alpha}$; so $\partial_{E,h}$ is the $(1,\,0)$-component of
$\nabla^{h}$.

We now define the stable Higgs bundles (cf. [BH], [BS]). Denote by
${\mathcal{O}}_{B_{\mathcal{F}_{\xi}}}$ the sheaf of basic holomorphic
functions on $M$, and for a transversely holomorphic vector bundle $E$ on $M$,
denote by ${\mathcal{O}}_{B_{\mathcal{F}_{\xi}}}(E)$ the sheaf of basic
holomorphic sections of $E$. Consider
${\mathcal{O}}_{B_{\mathcal{F}_{\xi}}}(E)$ as a coherent
${\mathcal{O}}_{B_{\mathcal{F}_{\xi}}}$-sheaf.

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

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

######  Theorem 5.2.

For a stable basic Higgs bundle $(E,\,\theta)$ over a compact Sasakian
manifold $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$, there exists a basic
Hermitian metric $h$ on $E$ such that

$\Lambda R^{D^{h}\perp}\,=\,0\,,$

where $R^{D^{h}\perp}$ is the trace-free part of the curvature $R^{D^{h}}$ of
the canonical connection $D^{h}$ associated to $h$.

Theorem 5.2 will be proved in the next section.

######  Proposition 5.3.

Let $(E,\,\theta)$ be a basic Higgs bundle over a compact Sasakian manifold
$(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$. Suppose that $h$ is a basic
Hermitian metric on $E$ such that $\Lambda R^{D^{h}\perp}\,=\,0$. If
$c_{1,B_{\mathcal{F}_{\xi}}}(E)\,=\,0$ and
$c_{2,B_{\mathcal{F}_{\xi}}}(E)\,=\,0$, then the connection $D^{h}$ is flat.

###### Proof.

The arguments in the proof of [Si1, Proposition 3.4] for the usual case go
through after using the Riemann bilinear relations for basic forms on $M$. ∎

##  6\. Proof of Theorem 5.2

###  6.1. Preliminaries

The proof of Theorem 5.2 will closely follow the proof of [Si1, Theorem 1].

######  Proposition 6.1.

Let $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$ be a compact Sasakian manifold.
Let $B\,\in\,{\mathbb{R}}^{>0}$ be a positive number. There exist positive
constants $C_{1}(B)$, $C_{2}(B)$ and $C_{3}(B)$ depending on $B$ and an
increasing function $a\,:\,[0,\,+\infty)\,\longrightarrow\,[0,\,+\infty)$ with
$a(0)\,=\,0$, $a(x)\,=\,x$ for $x\,>\,1$, such that for any positive basic
function $f\,\in\,A^{0}_{B_{\mathcal{F}_{\xi}}}(M)$ on $M$ satisfying
$\Delta_{\xi}f\,\leq\,B$, the following two inequalities hold:

$\sup_{M}f\,\leq\,C_{1}(B)+C_{2}(B)\int_{M}f$

and

$\ \sup_{M}f\,\leq\,C_{3}(B)a\left(\int_{M}f\right)\,.$

###### Proof.

In view of $\Delta f\,=\,\Delta_{\xi}f$ for
$f\,\in\,A^{0}_{B_{\mathcal{F}_{\xi}}}(M)$, it suffices to prove the two
inequalities for $f\,\in\,A^{0}_{B_{\mathcal{F}_{\xi}}}(M)$ satisfying $\Delta
f\,\leq\,B$. We next note that this is already proved in [Si1, Proposition
2.1] and [Do2]. ∎

Let $(E,\,\theta)$ be a basic Higgs bundle, equipped with a basic Hermitian
metric $h$, over a compact Sasakian manifold
$(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$. We define the operator

$D^{\prime,h}_{E,\theta}\,=\,\partial_{E,h}+\overline{\theta}_{h}\,:\,A^{\ast}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\longrightarrow\,A^{\ast+1}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,,$

where $\partial_{E,h}$ and $\overline{\theta}_{h}$ are defined as in (5.1) and
(5.2) respectively. As in [Si1, Lemma 3.1], we have the following formulas:

  1. (1)

$\sqrt{-1}[\lambda,\,D^{\prime\prime}_{E,\theta}]\,=\,(D^{\prime,h}_{E,\theta})^{\ast}\qquad{\rm
and}\qquad\sqrt{-1}[\lambda,\,D^{\prime,h}_{E,\theta}]\,=\,-(D^{\prime\prime}_{E,\theta})^{\ast}\,,$

where
$(D^{\prime,h}_{E,\theta})^{\ast}\,=\,-\star_{h,\xi}\overline{\partial}_{E}\star_{h,\xi}+\star_{h,\xi}\theta\star_{h,\xi}$
and

$(D^{\prime\prime}_{E,\theta})^{\ast}\,=\,-\star_{h,\xi}\partial_{E,h}\star_{h,\xi}+\star_{h,\xi}\overline{\theta}_{\theta}\star_{h,\xi}\,;$

note that they are the formal adjoints of $D^{\prime,h}_{E,\theta}$ and
$D^{\prime\prime}_{E,\theta}$ respectively for the $L^{2}$ inner product
$A^{r}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\times
A^{r}_{B_{\mathcal{F}_{\xi}}}(M,\,E)\,\ni\,(\alpha,\beta)\,\longmapsto\,\int_{M}\langle\alpha,\,\beta\rangle$.

  2. (2)

For self-adjoint basic sections
$\sigma,\tau\,\in\,A^{0}_{B_{\mathcal{F}_{\xi}}}(M,\,{\rm end}(E))$,

$|D^{\prime\prime}_{E,\theta}(\sigma)\tau|_{h}^{2}\,=\,-\sqrt{-1}\Lambda{\rm
Tr}(D^{\prime\prime}_{E,\theta}(\sigma)\tau^{2}D^{\prime,h}_{E,\theta}(\sigma)).$

  3. (3)

If ${\mathbf{k}}\,=\,h\sigma$ for a basic positive self-adjoint section
$\sigma\,\in\,A^{0}_{B_{\mathcal{F}_{\xi}}}(M,\,{\rm end}(E))$, then

$D^{\prime,{\mathbf{k}}}_{E,\theta}\,=\,D^{\prime,h}_{E,\theta}+\sigma^{-1}D^{\prime,h}_{E,\theta}(\sigma)$

and

$\Delta_{h}^{\prime}(\sigma)\,=\,h\sqrt{-1}(\Lambda R^{D^{\mathbf{k}}}-\Lambda
R^{D^{h}})+\sqrt{-1}\Lambda
D^{\prime\prime}_{E,\theta}(\sigma)\sigma^{-1}D^{\prime,h}_{E,\theta}(\sigma)\,,$

where
$\Delta_{h}^{\prime}\,=\,(D^{\prime,h}_{E,\theta})^{\ast}D^{\prime,h}_{E,\theta}\,=\,\sqrt{-1}\Lambda
D^{\prime\prime}_{E,\theta}D^{\prime,h}_{E,\theta}$.

  4. (4)

Also,

$\Delta_{\xi}\log{\rm Tr}(\sigma)\,\leq\,2(|\Lambda
R^{D^{\mathbf{k}}}|_{\mathbf{k}}+|\Lambda R^{D^{h}}|_{h})\,.$ (6.1)

###  6.2. Donaldson’s functional

Let $(E,\,\theta)$ be a basic Higgs vector bundle over the Sasakian manifold
$(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$. Fix a basic Hermitian metric $h$ on
$E$. Denote by $S(E)$ the smooth vector bundle of self-adjoint endomorphisms
of $E$ corresponding to $h$. For smooth functions
$\phi\,:\,\mathbb{R}\,\longrightarrow\,\mathbb{R}$ and
$\Psi\,:\,\mathbb{R}\times\mathbb{R}\,\longrightarrow\,\mathbb{R}$, we define
the maps $\phi\,:\,S(E)\,\longrightarrow\,S(E)$ and
$\Psi\,:\,S(E)\,\longrightarrow\,S({\rm End}E)$ as in [Si1, Page 880].
Consider the $L^{p}$-completion $L_{B_{\mathcal{F}_{\xi}}}^{p}(S(E))$ of the
space of basic sections of $S(E)$, and also the Sobolev space
$L^{p,1}_{B_{\mathcal{F}_{\xi}}}(S(E))$ with the norm $\Arrowvert
s\Arrowvert_{L^{p,1}_{B_{\mathcal{F}_{\xi}}}(S(E))}\,=\,\Arrowvert
s\Arrowvert_{L^{p}}+\Arrowvert D^{\prime\prime}_{E,\theta}s\Arrowvert_{L^{p}}$
for any section $s$ of $S(E)$. As done in [Si1, Proposition 4.1], we can
extend $\phi\,:\,\mathbb{R}\,\longrightarrow\,\mathbb{R}$ and
$\Psi\,:\,\mathbb{R}\times\mathbb{R}\,\longrightarrow\,\mathbb{R}$ to
continuous maps

$\phi\,:\,L_{B_{\mathcal{F}_{\xi}}}^{p}(S(E))\,\longrightarrow\,L_{B_{\mathcal{F}_{\xi}}}^{p}(S(E))\,,$
$\Psi\,:\,L_{B_{\mathcal{F}_{\xi}}}^{p}(S(E))\,\longrightarrow\,{\rm
Hom}(L^{p}_{B_{\mathcal{F}_{\xi}}}({\rm
End}(E),\,L^{q}_{B_{\mathcal{F}_{\xi}}}({\rm End}(E))$

and
$\phi\,:\,L^{p,1}_{B_{\mathcal{F}_{\xi}}}(S(E))\,\longrightarrow\,L^{q,1}_{B_{\mathcal{F}_{\xi}}}(S(E))$
for $q\,<\,p$.

Denote by $\mathcal{P}$ the space of all basic Hermitian metrics on $E$. We
parametrize $\mathcal{P}$ by $A^{0}_{B_{\mathcal{F}_{\xi}}}(S(E))$ as
$A^{0}_{B_{\mathcal{F}_{\xi}}}(S(E))\,\ni\,\sigma\,\longmapsto\,h\exp({\sigma})\,\in\,\mathcal{P}$.
For $h,\,h^{\prime}\,\in\,\mathcal{P}$ with $h^{\prime}\,=\,h\exp(s)$, define

$M(h,\,h^{\prime})\,=\,\sqrt{-1}\int_{M}{\rm Tr}(s\Lambda
R^{D^{h}})+\int_{M}\left\langle\Psi_{1}(s)(D^{\prime\prime}_{E,\theta}s),\,D^{\prime\prime}_{E,\theta}s\right\rangle_{h}\,,$

where $\Psi_{1}\,:\,S(E)\,\longrightarrow\,S({\rm End}E)$ is defined by the
function

$\Psi_{1}(\lambda_{1},\lambda_{2})\,=\,\frac{e^{\lambda_{2}-\lambda_{1}}-(\lambda_{2}-\lambda_{1})-1}{(\lambda_{2}-\lambda_{1})^{2}}\,.$

By the same proof as of [Si1, Proposition 5.1], for all
$h,\,h^{\prime},\,h^{\prime\prime}\,\in\,\mathcal{P}$, the identity

$M(h,h^{\prime})+M(h^{\prime},h^{\prime\prime})\,=\,M(h,h^{\prime\prime})$

is obtained.

We have the following important estimate as in [Si1, Proposition 5.3].

######  Proposition 6.2.

Fix a positive number $B$. Suppose a basic Hermitian metric $h$ on $E$
satisfies the inequality $\sup_{M}|\Lambda R^{D^{h}}|\,\leq\,B$. If
$(E,\,\theta)$ is a stable basic Higgs bundle, then there are positive
constants $C_{1}$ and $C_{2}$ such that

$\sup_{M}|s|\,\leq\,C_{1}+C_{2}M(h,h\exp(s))$

for every $s\,\in\,A^{0}_{B}(S(E))$ with ${\rm Tr}(s)\,=\,0$ and
$\sup_{M}|\Lambda R^{D^{he^{s}}}|\,\leq\,B$.

Proposition 6.2 will be proved after Theorem 6.4.

######  Definition 6.3.

Let $E$ be a transversely holomorphic vector bundle, over a compact Sasakian
manifold $M$, equipped with a basic Hermitian metric $h$. A transversely
weakly holomorphic subbundle of $E$ is $\Pi\,\in\,L^{2,1}_{B}(S(E))$
satisfying the following two conditions

$\Pi\,=\,\Pi^{2}\qquad{\rm and}\qquad({\rm
Id}_{E}-\Pi)\overline{\partial}_{{\rm End}(E)}(\Pi)\,=\,0\,.$

In [BH, Theorem 5.7], the following result is proved.

######  Theorem 6.4 ([BH]).

Let $\Pi$ be a transversely weakly holomorphic subbundle of $E$. Then there is
a transversely coherent sheaf $\mathcal{V}$ and a transverse analytic subset
$S\,\subset\,M$ such that the following three hold:

  1. (1)

The complex codimension of $S$ in $M$ is at least two.

  2. (2)

The restriction of $\Pi$ to $M\setminus S$ is smooth and defines a
transversely holomorphic subbundle $V\,\subset\,E|_{M\setminus S}$.

  3. (3)

The restriction of $\mathcal{V}$ to $M\setminus S$ is the sheaf of basic
holomorphic sections of $V$. Moreover these properties imply that
${\mathcal{O}}_{B_{\mathcal{F}_{\xi}}}(E)/\mathcal{V}$ is torsion-free.

Now we can prove Proposition 6.2 in the same way as done for the proof of
[Si1, Proposition 5.3].

###### Proof of Proposition 6.2.

For a section $s$ as in the statement of the proposition, by the inequality

$\Delta_{\xi}\log{\rm Tr}(\sigma)\,\leq\,2(|\Lambda
R^{D^{\mathbf{k}}}|_{\mathbf{k}}+|\Lambda R^{D^{h}}|_{h})$

in (6.1) we have that $\Delta_{\xi}|s|\,\leq\,4B$, and hence using Proposition
6.1 it follows that $\sup_{M}|s|\,\leq\,C_{1}+C_{2}\Arrowvert
s\Arrowvert_{L^{1}}$.

Assume that the estimate in the statement of the proposition does not hold in
the following sense. We can find a sequence $s_{i}$ of basic sections of
$S(E)$ with ${\rm Tr}(s_{i})\,=\,0$ such that $\lim_{i\to\infty}\Arrowvert
s_{i}\Arrowvert_{L^{1}}\,=\,+\infty$ and $\Arrowvert
s_{i}\Arrowvert_{L^{1}}\,\geq\,M(h,\,h\exp(s_{i}))$.

Then, by the same arguments given between [Si1, Lemma 5.4] and [Si1, Lemma
5.7], we have a transversely weakly holomorphic subbundle $\Pi$ of $E$ such
that $({\rm Id}_{E}-\Pi)\theta(\Pi)\,=\,0$, and the inequality

$\frac{{\rm deg}(\Pi)}{{\rm Tr}(\Pi)}\,\geq\,\frac{{\rm deg}(E)}{{\rm rk}(E)}$

holds, where

${\rm deg}(\Pi)\,=\,\sqrt{-1}\int_{M}{\rm Tr}(\Pi\Lambda
R^{D^{h}})-\int_{M}|D^{\prime\prime}_{E,\theta}\Pi|^{2}\,.$

We have a sub-Higgs sheaf $\mathcal{V}$ of $(E,\,\theta)$ satisfying the
properties in Theorem 6.4. We conclude that ${\rm deg}(\Pi)\,=\,{\rm
deg}(\mathcal{V})$ as in [Si1, Lemma 3.2], and this contradicts the given
stability condition. This completes the proof of Proposition 6.2 ∎

###  6.3. The heat equation on Kähler cone

For a compact Sasakian manifold

$(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))\,,$

consider the cone $C(M)\,=\,M\times\mathbb{R}^{>0}$ and define the real
$2$-form $\omega\,\in\,A^{2}(C(M))$ to be

$\omega\,=\,2rdr\wedge\eta+r^{2}d\eta\,;$

also, define the bundle homomorphism $J\,:\,TC(M)\,\longrightarrow\,TC(M)$ by

  * •

$J(X)\,=\,I(X)$ for $X\,\in\,S$,

  * •

$J\left(r\frac{\partial}{\partial r}\right)\,=\,-\xi$, and

  * •

$J(\xi)\,=\,r\frac{\partial}{\partial r}$, where $r$ is the parameter of
$\mathbb{R}^{>0}$.

Then, the pair $(\omega,\,J)$ is a Kähler structure on the complex manifold
$C(M)$.

Consider the real $1$-dimensional foliation $\overline{\mathcal{F}}_{\xi}$ on
$C(M)$ generated by $\xi$. Denote by $A$ the $1$-parameter group of
automorphisms of the Kähler manifold $C(M)$ corresponding to
$\overline{\mathcal{F}}_{\xi}$. Then the space

$A^{\ast}(C(M))^{A}\,\subset\,A^{\ast}(C(M))$

of $A$-invariant differential forms on $C(M)$ contains the basic de Rham
complex $A^{\ast}_{B_{\overline{\mathcal{F}}_{\xi}}}(C(M))$. In particular, we
have $A^{0}(C(M))^{A}\,=\,A^{0}_{B_{\overline{\mathcal{F}}_{\xi}}}(C(M))$. For
a complex basic vector bundle $E$ over $C(M)$, we can naturally define the
$A$-action on $A^{\ast}(C(M),\,E)$ so that for $a\,\in\,A$,
$\omega\,\in\,A^{r}(C(M))$ and
$s\,\in\,A^{0}_{B_{\overline{\mathcal{F}}_{\xi}}}(C(M),\,E)$, we have
$a(\omega\otimes s)\,=\,(a^{\ast}\omega)\otimes s$.

Let $(E,\,\theta)$ be a basic Higgs bundle over
$(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$. Consider the pair
$(\widetilde{E},\,\widetilde{\theta})$ defined by the pull back of
$(E,\,\theta)$ using the projection

$\pi_{0}\,:\,C(M)\,\longrightarrow\,M$ (6.2)

defined by $(x,\,t)\,\longmapsto\,x$. Then
$(\widetilde{E},\,\widetilde{\theta})$ is a usual Higgs bundle over the Kähler
manifold $(C(M),\,\omega,\,J)$. Consider the operator

$\widetilde{D}^{\prime\prime}\,=\,\overline{\partial}_{\widetilde{E}}+\theta\,:\,A^{\ast}(C(M),\,\widetilde{E})\,\longrightarrow\,A^{\ast+1}(C(M),\,\widetilde{E})\,,$

where $\overline{\partial}_{\widetilde{E}}$ is the usual Dolbeault operator
for the holomorphic vector bundle $\widetilde{E}$. Let $\widetilde{h}$ be a
Hermitian metric on $\widetilde{E}$. Assume that $\widetilde{h}$ is basic,
i.e.,
$\widetilde{h}\,\in\,A^{0}_{B_{\overline{\mathcal{F}}_{\xi}}}(C(M),\,\widetilde{E}^{\ast}\otimes\overline{\widetilde{E}}^{\ast})$.
Then, we have the operator

$\partial_{E,\widetilde{h}}\,:\,A^{p,q}(C(M),\,\widetilde{E})\,\longrightarrow\,A^{p,q+1}(C(M),\,\widetilde{E})$

such that $\overline{\partial}_{\widetilde{E}}+\partial_{E,\widetilde{h}}$ is
the canonical Chern connection for the Hermitian holomorphic vector bundle
$(\widetilde{E},\,\widetilde{h})$. Define
$\overline{\widetilde{\theta}}_{\widetilde{h}}\,\in\,A^{0,1}(C(M),\,{\rm
End}(\widetilde{E}))$ by

$h(\widetilde{\theta}e_{1},\,e_{2})\,=\,\widetilde{h}(e_{1},\,\overline{\widetilde{\theta}}_{\widetilde{h}}e_{2})$

and also define the operator
$\widetilde{D}^{\prime}_{\widetilde{h}}\,=\,\partial_{E,\widetilde{h}}+\overline{\widetilde{\theta}}_{\widetilde{h}}\,:\,A^{\ast}(C(M),\,\widetilde{E})\,\longrightarrow\,A^{\ast+1}(C(M),\,\widetilde{E})$.
The curvature of the connection
$\widetilde{D}^{\prime\prime}+\widetilde{D}^{\prime}_{\widetilde{h}}$ will be
denoted by $\widetilde{R}^{\widetilde{h}}$.

Denote by $\widetilde{\Lambda}$ the formal adjoint of the Lefschetz operator
associated with the Kähler form $\omega$ on $C(M)$. Consider the heat equation

$\widetilde{h}^{-1}_{t}\frac{d\widetilde{h}_{t}}{dt}\,=\,-\sqrt{-1}\widetilde{\Lambda}\widetilde{R}^{\widetilde{h}_{t}\perp}\,,$
(6.3)

where $\widetilde{R}^{\widetilde{h}_{t}\perp}$ is the trace-free part of the
curvature $\widetilde{R}^{\widetilde{h}_{t}}$.

Fix a basic Hermitian metric $h_{0}$ on the basic holomorphic bundle $E$ over
the Sasakian manifold $M$. Let $\widetilde{h}_{0}\,=\,\pi^{*}_{0}h_{0}$ be the
pull-back of $h_{0}$ to a Hermitian metric on
$\widetilde{E}\,:=\,\pi^{*}_{0}E$, where $\pi_{0}$ is the projection in (6.2).

Write $\widetilde{h}_{0}^{-1}\widetilde{h}_{t}\,=\,\sigma_{t}$. Then the
equation in (6.3) becomes

$\left(\frac{d}{dt}+\Delta^{\prime}_{\widetilde{h}_{0}}\right)\sigma_{t}\,=\,-\sqrt{-1}\sigma_{t}\widetilde{\Lambda}\widetilde{R}^{\widetilde{h}_{0}\perp}+\sqrt{-1}\widetilde{\Lambda}\widetilde{D}^{\prime\prime}(\sigma_{t})\sigma^{-1}_{t}\widetilde{D}^{\prime}_{\widetilde{h}_{0}}(\sigma_{t})\,,$

where $\Delta^{\prime}_{h_{0}}$ is the Laplacian operator of
$\widetilde{D}^{\prime}_{\widetilde{h_{0}}}$. Recall that $A$ is a group of
automorphisms of the Kähler manifold $C(M)$. Since the Hermitian metric
$h_{0}$ on $E$ is basic, the action of $A$ on $A^{0}(C(M),\,{\rm
End}(\widetilde{E}))$ commutes with the operators
$\widetilde{D}^{\prime\prime}$, $\widetilde{D}^{\prime}_{h_{0}}$,
$\widetilde{\Lambda}$ and $\Delta^{\prime}_{h_{0}}$, and we have that

$\widetilde{\Lambda}\widetilde{R}^{\widetilde{h_{0}}}\,\in\,A^{0}_{B_{\overline{\mathcal{F}}_{\xi}}}(C(M),\,{\rm
End}(\widetilde{E}))\,.$

Thus, the set of solutions of the heat equation (6.3) is invariant under the
action of $A$.

The positive function $r^{2}$ on $C(M)$ is plurisubharmonic, because
$\sqrt{-1}\partial\overline{\partial}r^{2}\,=\,\omega$. Just as the results in
[Si1, Section 6] are derived using the arguments in [Ha] and [Do1], we have
the following.

######  Theorem 6.5.

Let $\epsilon$ be a positive real number. There exists a unique solution
$\widetilde{h}$, defined for all time $(0,\,+\infty)$, of the heat equation

$\widetilde{h}_{t}^{-1}\frac{d\widetilde{h}_{t}}{dt}\,=\,-\sqrt{-1}\widetilde{\Lambda}\widetilde{R}^{\widetilde{h}_{t}\perp}$

on the compact manifold $M\times[1,\,1+\epsilon]$ with boundary satisfying

${\rm det}(\widetilde{h}_{0})\,=\,{\rm det}(\widetilde{h}_{t})\,,\ \
\widetilde{h_{t}}_{|t=0}\,=\,\widetilde{h}_{0}$

together with the Neumann boundary condition $\frac{\partial h}{\partial
r}\big{|}_{r=1,1+\epsilon}\,=\,0$.

###  6.4. Proof of Theorem 5.2

Consider the solution $\widetilde{h}_{t}$ in Theorem 6.5. It was observed
above that the set of solutions of the heat equation is $A$-invariant.
Therefore, we conclude that $\widetilde{h_{t}}$ is basic by the uniqueness
property. Define the Hermitian metrics $h_{t}$ on $E$ by the pull-backs of
$\widetilde{h_{t}}$ for the embedding
$M\,\longrightarrow\,M\times[1,\,1+\epsilon]$ defined by
$x\,\longmapsto\,(x,\,1)$. In view of the Neumann boundary condition
$\frac{\partial h}{\partial r}\big{|}_{r=1,1+\epsilon}\,=\,0$ in Theorem 6.5,
we conclude that the pull-back, by this embedding, of the canonical Chern
connection
$\overline{\partial}_{\widetilde{E}}+\partial_{E,\widetilde{h}_{t}}$, on
$(\widetilde{E},\,\widetilde{h}_{t})$, is identified with the canonical Chern
connection $\nabla^{h_{t}}$ on $E$, and moreover the pull-back of
$(\widetilde{\Lambda}\widetilde{R}^{\widetilde{h}\perp})$ is identified with
$\Lambda R^{D^{h_{t}}\perp}$. Thus, $h_{t}$ satisfies the basic heat equation

$h_{t}^{-1}\frac{dh_{t}}{dt}\,=\,-\sqrt{-1}\Lambda R^{D^{h_{t}}\perp}\,.$

By the formulas in Section 6.1, for $h_{0}^{-1}h_{t}\,=\,\sigma_{t}$, this
equation can be written as

$\left(\frac{d}{dt}+\Delta^{\prime}_{{h}_{0}}\right)\sigma_{t}\,=\,-\sqrt{-1}\sigma_{t}\Lambda{R}^{{h}_{0}\perp}+\sqrt{-1}\Lambda
D^{\prime\prime}(\sigma_{t})\sigma^{-1}_{t}D^{\prime}_{h_{0}}(\sigma_{t})\,.$

By the same proof of [Si1, Lemma 7.1], we have the formula

$\frac{d}{dt}M(h_{0},h_{t})\,=\,-\int_{M}|\Lambda
R^{D^{h_{t}}\perp}|^{2}_{h_{t}}\,.$

Now we assume that the Higgs bundle $(E,\,\theta)$ is stable. Applying
Proposition 6.2, as done in [Si1, Page 895], we can take a sequence
$\\{t_{i}\\}$ of time instances, with $t_{i}\,\to\,+\infty$, such that

$\lim_{i\to+\infty}\int_{M}|\Lambda R^{D^{h_{t}}\perp}|^{2}_{h_{t_{i}}}\,=\,0$

and $h_{t_{i}}\,\to\,h_{\infty}$ weakly in $L^{2}_{1}$. By the basic Sobolev
embedding theorem [BH, Theorem 2.6], this $h_{t_{i}}$ is a Cauchy sequence in
$L^{1}$. For a positive number $B$ such that $|\Lambda
R^{D^{h_{0}}}|\,\leq\,B$, by the inequality $\Delta_{\xi}\log{\rm
Tr}(\sigma)\,\leq\,2(|\Lambda R^{D^{k}}|_{k}+|\Lambda R^{D^{h}}|_{h})$ in
(6.1), we have that

$\Delta_{\xi}\log{\rm Tr}(h_{t_{i}}^{-1}h_{t_{j}})\,\leq\,2B\,,$

and hence Proposition 6.1 implies that $\log{\rm
Tr}(h_{t_{i}}h_{t_{j}})\,\to\,0$ in $C^{0}$. Thus the convergence
$h_{t_{i}}^{-1}\,\to\,h_{\infty}$ is in $C^{0}$. By the $C^{0}$-convergence
and the uniform boundedness of $|\Lambda R^{D^{h_{t}}\perp}|_{h_{t}}$, as done
in the arguments in the proof of [Do1, Lemma 19] (also [Si1, Lemma 6.4]), we
conclude that $h_{t_{i}}$ is actually bounded in $C^{1}$.

To complete the proof, we use the basic elliptic estimate and regularity
explained in [BH, Section 2]. For the transversely elliptic operator
$\Delta_{h_{0}}^{\prime}$, as in [BH, Remark 2.2], we can non-canonically
extend $\Delta_{h_{0}}^{\prime}$ to a second order differential operator

$L\,:\,A^{0}(S(E))\,\longrightarrow\,A^{0}({\rm End}(E))\,.$

For the linear operator

$\nabla^{h}_{\xi}\,:\,A^{0}(S(E))\,\longrightarrow\,A^{0}(S(E))$

associated with the connection $\nabla^{h}$, and its formal adjoint
$(\nabla^{h}_{\xi})^{\ast}$, define the second order differential operator

$\Box\,:=\,(\nabla^{h}_{\xi})^{\ast}\nabla^{h}_{\xi}+L\,:\,A^{0}(S(E))\,\longrightarrow\,A^{0}({\rm
End}(E))\,.$

Then, by the transverse ellipticity of $\Delta_{h_{0}}^{\prime}$, the
differential operator $\Box$ is elliptic. We have
$\Box=\Delta_{h_{0}}^{\prime}$ on $A^{0}_{B_{\mathcal{F}_{\xi}}}(S(E))$.
Therefore, by the

  * •

elliptic estimate of the elliptic operator $\Box$,

  * •

the $C^{1}$-boundedness of $h_{t_{i}}$, and

  * •

the uniform boundedness of $|\Lambda R^{D^{h_{t}}\perp}|_{h_{t}}$,

we conclude that $h_{t_{i}}$ is bounded in $L^{p,2}$, and hence the
convergence $h_{t_{i}}\to h_{\infty}$ is weakly in $L^{p,2}$. Thus,
$R^{D^{h_{\infty}}}$ is defined, and $\Lambda R^{D^{h_{\infty}}\perp}\,=\,0$.

We shall prove that $h_{\infty}$ is a smooth basic Hermitian metric. For that
it is sufficient to show that $\sigma_{\infty}\,=\,h_{0}^{-1}h_{\infty}\in
A^{0}_{B_{\mathcal{F}_{\xi}}}(S(E))$. We consider the Sobolev space
$L^{p,k}(S(E))$. Then, the basic Sobolev space
$L^{p,1}_{B_{\mathcal{F}_{\xi}}}(S(E))$ is the closure of
$A^{0}_{B_{\mathcal{F}_{\xi}}}(S(E))$ in $L^{p,1}(S(E))$. By the elliptic
regularity for the elliptic operator $\Box$ (see [BH, Lemma 2.8]), we conclude
that

$\sigma_{\infty}\,\in\,L^{p,1}_{B_{\mathcal{F}_{\xi}}}(S(E))\cap
A^{0}(S(E))\,.$

We have $A^{0}_{B_{\mathcal{F}_{\xi}}}(S(E))\,=\,{\rm kernel}(\nabla^{h})$ for
the linear differential operator
$\nabla^{h}_{\xi}\,:\,A^{0}(S(E))\,\longrightarrow\,A^{0}(S(E))$. Extend
$(\nabla^{h}_{\xi})^{p,1}\,:\,L^{p,1}(S(E))\,\longrightarrow\,L^{p,0}(S(E))$
continuously. Then we have
$L^{p,1}_{B_{\mathcal{F}_{\xi}}}(S(E))\,\subset\,{\rm
kernel}((\nabla^{h})^{p,1})$. From the commutative diagram

$\textstyle{A^{0}(S(E))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla^{h}_{\xi}}$$\textstyle{L^{p,1}(S(E))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\nabla^{h}_{\xi})^{p,1}}$$\textstyle{A^{0}(S(E))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{p,0}(S(E))}$

where $A^{0}(S(E))\,\longrightarrow\,L^{p,1}(S(E))$ and
$A^{0}(S(E))\,\longrightarrow\,L^{p,0}(S(E))$ are the natural inclusions, it
follows that ${\rm kernel}((\nabla^{h}_{\xi})^{p,1})\cap
A^{0}(S(E))\,\subset\,{\rm kernel}(\nabla^{h}_{\xi})$. Thus we have
$\sigma_{\infty}\,\in\,A^{0}_{B_{\mathcal{F}_{\xi}}}(S(E))$. This completes
the proof of Theorem 5.2.

###  6.5. Bogomolov–Miyaoka inequality

The following Bogomolov–Miyaoka type inequality is derived from Theorem 5.2
just as [Si1, p. 878–879, Proposition 3.4] is derived from [Si1, p. 878,
Theorem 1].

######  Corollary 6.6.

Let $(E,\,\theta)$ be a polystable basic Higgs bundle of rank $r$ over a
compact Sasakian manifold $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$ with $\dim
M\,=\,2n+1$. Then

$\int_{M}\left(2c_{2,B_{\mathcal{F}_{\xi}}}(E)-\frac{r-1}{r}c_{1,B_{\mathcal{F}_{\xi}}}(E)^{2}\right)(d\eta)^{n-2}\wedge\eta\,\geq\,0\,,$

where $c_{i,B_{\mathcal{F}_{\xi}}}(E)$ is the $i$-th basic Chern class of $E$.
If the above inequality is an equality,, then $R^{D^{h}\perp}\,=\,0$.

##  7\. Correspondence between flat bundles and Higgs bundles

######  Proposition 7.1 (See [BH, Theorem 4.7]).

Let $(E,\,\theta)$ be a basic Higgs bundle over a compact Sasakian manifold
$(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$ with ${\rm deg}(E)=0$. Suppose that
$h$ is a basic Hermitian metric on $E$ with $\Lambda R^{D^{h}}\,=\,0$. Then
$(E,\,\theta)$ is a direct sum of stable basic Higgs bundles of degree zero.

###### Proof.

Assume that $(E,\,\theta)$ is not stable. Then there exists a sub-Higgs sheaf
${\mathcal{V}}$ of $(E,\,\theta)$ such that

  * •

${\rm rk}(\mathcal{V})\,<\,{\rm rk}(E)$ with
${\mathcal{O}}_{B_{\mathcal{F}_{\xi}}}(E)/\mathcal{V}$ is torsion-free, and

  * •

the inequality

${\rm deg}(\mathcal{V})\,\geq\,0$

holds.

Let $\pi$ be the projection to the transversely holomorphic subbundle
$V\,\subset\,E$ defined almost everywhere using $\mathcal{V}$, constructed as
in [BH, Proposition 3.21] associated with the Hermitian metric $h$. Then,
$\pi\in L^{2,1}_{B}(S(E))$, and we have the Chern-Weil formula ([Si1, Lemma
3.2])

${\rm deg}(\mathcal{V})\,=\,\sqrt{-1}\int_{M}{\rm Tr}(\pi\Lambda
R^{D^{h}})-\int_{M}|D^{\prime\prime}_{E,\theta}\pi|^{2}\,.$

By $\Lambda R^{D^{h}}\,=\,0$, we have that ${\rm deg}(\mathcal{V})\,=\,0$ and
$D^{\prime\prime}_{E,\theta}\pi\,=\,0$. By $\pi\,\in\,L^{2,1}_{B}(S(E))$, we
also have that $D^{\prime,h}_{E,\theta}\pi\,=\,0$. Using the elliptic
regularity it follows that
$\pi\,\in\,L^{p,1}_{B_{\mathcal{F}_{\xi}}}(S(E))\cap A^{0}(S(E))$, and hence
$\pi\,\in\,A^{0}_{B_{\mathcal{F}_{\xi}}}(S(E))$ by the same argument as in the
last part of the proof of Theorem 5.2.

Using $D^{\prime\prime}_{E,\theta}\pi\,=\,0$ and
$D^{\prime,h}_{E,\theta}\pi\,=\,0$, it can be seen that $\pi$ is the
projection to a globally defined basic Higgs sub-bundle $V\,\subset\,E$ of
$(E,\,\theta)$. Thus have the direct sum decomposition $E\,=\,V\oplus
V^{\perp}$ of basic Higgs bundles that satisfies the condition that ${\rm
deg}(V)\,=\,0\,=\,{\rm deg}(V^{\perp})$. Restricting $h$ to $V$ and
$V^{\perp}$, and repeating the arguments the proposition can now be proved
inductively. ∎

In view of Theorem 4.2, Theorem 5.2, Proposition 7.1 and Proposition 5.3, on a
compact Sasakian manifold $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$, we have a
bijective correspondence between

  * •

the semi-simple flat bundles $(E,\,\nabla_{E})$, and

  * •

the polystable basic Higgs bundles $(E,\,\theta)$ over $M$ with
$c_{1,B_{\mathcal{F}_{\xi}}}(E)\,=\,0\,=\,c_{2,B_{\mathcal{F}_{\xi}}}(E)$.

via harmonic metrics $h$. It should be clarified that the transversely
holomorphic structure of the vector bundle underlying $(E,\,\nabla_{E})$ is in
general different from the transversely holomorphic structure of the vector
bundle underlying the corresponding basic Higgs bundle $(E,\,\theta)$; however
the underlying $C^{\infty}$ vector bundles coincide.

In the above correspondence, considering the basic de Rham complex
$(A^{\ast}_{B_{\mathcal{F}_{\xi}}}(M,E),\,d_{E})$ with values in a semi-simple
flat bundle $(E,\,\nabla^{E})$, we have the decomposition

$d_{E}\,=\,D^{\prime\prime}_{E,\theta}+D^{\prime,h}_{E,\theta}\,.$

Denote by

$H^{\ast}_{dR,B_{\mathcal{F}_{\xi}}}(M,\,E)\ \ \text{ and }\ \
H^{\ast}_{Dol,B_{\mathcal{F}_{\xi}}}(M,\,E)$

the cohomologies of complexes
$(A^{\ast}_{B_{\mathcal{F}_{\xi}}}(M,E),\,d_{E})$ and
$(A^{\ast}_{B_{\mathcal{F}_{\xi}}}(M,\,E),\,D^{\prime\prime}_{E,\theta})$
respectively. By the Kähler identities, we have an isomorphism

$H^{\ast}_{dR,B_{\mathcal{F}_{\xi}}}(M,\,E)\,\cong\,H^{\ast}_{Dol,B_{\mathcal{F}_{\xi}}}(M,\,E)$

by the transverse Hodge theory of usual basic cohomology (see [Ka]). In
particular, a section $\varphi\,\in\,A^{\ast}_{B_{\mathcal{F}_{\xi}}}(M,\,E)$
is flat (i.e., $\nabla^{E}\varphi\,=\,0$) if and only if
$D^{\prime\prime}_{E,\theta}\varphi\,=\,0$.

As done in [Si2, Corollary 1.3], we now obtain the following result.

######  Theorem 7.2.

Let $(M,\,(T^{1,0},\,S,\,I),\,(\eta,\,\xi))$ a compact Sasakian manifold. Then
there is an equivalence between the category of semi-simple flat vector
bundles over $M$ and the category of polystable basic Higgs bundles over $M$
with trivial first and second basic Chern classes.

## Acknowledgements

We thank T. Mochizuki for pointing out Corollary 6.6. We thank D. Baraglia for
his comments. The first-named author is partially supported by a J. C. Bose
Fellowship. The second-named author is was partially supported by JSPS Grant-
in-Aid for Scientific Research Project/Area Number 19H01787.

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