Complex structures on the product of two Sasakian manifolds
Vlad Marchidanu111Partially supported by Romanian Ministry of Education and
Research, Program PN-III, Project number PN-III-P4-ID-PCE-2020-0025, Contract
30/04.02.2021
Keywords: Sasakian manifold, complex structure, Kähler manifold, LCK manifold
2020 Mathematics Subject Classification: 53C25, 53C55

Abstract. A Sasakian manifold is a Riemannian manifold whose metric cone
admits a certain Kähler structure which behaves well under homotheties. We
show that the product of two compact Sasakian manifolds admits a family of
complex structures indexed by a complex nonreal parameter, none of whose
members admits any compatible locally conformally Kähler metrics if both
Sasakian manifolds are of dimension greater than $1$. We compare this family
with another family of complex structures which has been studied in the
literature. We compute the Dolbeault cohomology groups of these products of
compact Sasakian manifolds.

###### Contents

  1. 1 Introduction
  2. 2 Sasakian manifolds
    1. 2.1 Tensorial definition of Sasakian manifolds
    2. 2.2 Sasakian manifolds via the Riemannian cone
    3. 2.3 Basic cohomology of Sasakian manifolds
    4. 2.4 The product of two Sasakian manifolds
  3. 3 The main result
  4. 4 Complex submanifolds of the product of Sasakian manifolds
  5. 5 Comparison with the  Calabi-Eckmann-Morimoto complex structures
  6. 6 The Dolbeault cohomology of the product of Sasakian manifolds

##  1 Introduction

Sasakian manifolds are the natural odd-dimensional analogue of Kähler
manifolds (see e.g. [4]). In the compact case, they are closely related to
both projective and Vaisman manifolds ([17]).

Kähler manifolds can be viewed as almost complex manifolds endowed with a
Hermitian metric such that the associated fundamental two-form is parallel
with respect to the metric connection.

Likewise, Sasakian manifolds can be thought of as almost contact manifolds
endowed with a compatible Riemannian metric satisfying certain tensorial
conditions (see [2] and Section 2.1).

Being even dimensional, a product of Sasakian manifolds is susceptible to bear
almost complex structures. Indeed, more generally, Morimoto constructed an
almost complex structure on the product of two almost contact manifolds ([15])
which proved to be integrable when the the two almost contact structures were
normal. If one starts with metric almost contact structures, then the product
metric is compatible with Morimoto’s almost complex structure. One thus
obtains an almost Hermitian structure on the product. The usual complex
structure of the Calabi-Eckmann manifold can be viewed this way. In
particular, starting with two Sasakian manifolds (whose subjacent almost
contact structures are normal), one obtains a Hermitian metric on the product.

This product Hermitian structure on a Calabi-Eckmann manifold was later
included by Tsukada in a two parameter family of Hermitian structures ([20]).
This construction was further generalized in [11] to the product of two
Sasakian manifolds. It was recently considered also in [1].

All these constructions use the tensorial definition of Sasakian manifolds and
are heavily computational. With these techniques, the authors of [1] can prove
that the considered two-parameter family of Hermitian structures is neither
Kähler nor locally conformally Kähler.

What we propose in the present paper is a shift towards the modern definition
of a Sasakian manifold as a Riemannian manifold with a Kähler structure on its
Riemannian cone (2.2). On the product of two compact Sasakian manifolds we
construct a natural family of complex structures indexed by a purely complex
parameter which we can prove that does not support neither Kähler nor locally
conformally Kähler metrics. Furthermore, we are able to characterize the
complex submanifolds of the product. Moreover, we show that our family of
complex structures does not coincide, in general, with the one in [11].
Furthermore, we compute the Dolbeault cohomology groups of these complex
manifolds.

Acknowledgements. This paper is largely a result of my stay in Rio de Janeiro.
I have learned many things from mathematicians at IMPA, but I would like to
thank in the first place Prof. Misha Verbitsky for the insightful discussions
we had and for helping me better understand mathematics. I am grateful to my
thesis advisor, Prof. Liviu Ornea, who has guided me constantly and helped me
with valuable comments and recommendations.

##  2 Sasakian manifolds

###  2.1 Tensorial definition of Sasakian manifolds

The notion of a Sasakian manifold was initially introduced by Shigeo Sasaki in
[18] as an odd-dimensional counterpart to Kähler manifolds. We recall the
tensorial definition of a Sasakian structure.

Given a smooth, odd-dimensional manifold $S$, a Sasakian structure is given by
the data $(g,\eta,\varphi,\xi)$, where $g$ is a Riemannian metric on $S$,
$\eta$ is a $1$-form, $\varphi$ is a $(1,1)$-tensor field and $\xi$ is a
vector field, satisfying the following properties for any $X,Y\in TS$:

$\begin{split}\eta\circ\varphi&=0\\\ \eta(X)&=g(X,\xi)\\\
\varphi^{2}&=-\textrm{Id}+\eta\otimes\xi\\\
g(\varphi(X),\varphi(Y))&=g(X,Y)-\eta(X)\eta(Y)\\\
(-2d\eta\otimes\xi)(X,Y)&=\varphi^{2}([X,Y])-\varphi([\varphi
X,Y])-\varphi([X,\varphi Y])-[\varphi X,\varphi Y]\\\
\textrm{Lie}_{g}\xi&=0\\\
(\nabla_{X}^{g}\varphi)Y&=g(X,Y)\xi-\eta(Y)X\end{split}$

A well-studied generalisation is that of an _almost contact structure_
$(S,\eta,\varphi,\xi)$, which occurs if we omit the presence of the metric $g$
and keep the first three conditions above, replacing $\eta(X)=g(X,\xi)$ with
$\eta(X)=1$, $\varphi(\xi)=0$. This is usually viewed as a counterpart of
almost complex geometry. See [2] for details.

In this paper we shall use the modern definition which places Sasakian
manifolds into the framework of holonomy. This approach was widely spread
following the pioneering work of C.P. Boyer and K. Galicki (see [4]).

###  2.2 Sasakian manifolds via the Riemannian cone

Let $(S,g)$ be an odd-dimensional Riemannian manifold and
$(C(S):=(S\times\mathbb{R}^{>0},g_{C(S)}=dt\otimes dt+t^{2}g)$,
$t\in\mathbb{R}^{>0}$, its Riemannian cone.

######  Definition 2.1.

A Sasakian structure is the data of a Kähler structure $(J,\omega,g_{C(S)})$
on $C(S)$ such that the homothety map $h_{\lambda}:C(S)\rightarrow C(S)$,
$h_{\lambda}(p,t):=(p,\lambda t)$ is holomorphic and satisfies
$h_{\lambda}^{*}\omega=\lambda^{2}\omega$ for each
$\lambda\in\mathbb{R}^{>0}$.

We denote by $R:=t\frac{d}{dt}$ the Euler field on $C(S)$ and by $\xi:=JR$ the
Reeb field. By definition $R$ is holomorphic, so $[R,\xi]=0$. Since $C(S)$ is
Kähler, $\xi$ is also holomorphic. When referring to $S$, we also denote by
$\xi$ the vector field $\xi\rvert_{t=1}$ on $S\times\\{1\\}\subset C(S)$.

The equivalence of the definition of Sasakian manifolds via their metric cone
with the definition formulated in Subsection 2.1 is established in [4, Section
6.5]. For our purposes, we mention that starting with a Sasakian manifold in
the above sense, one defines the tensor field $\varphi\in\textrm{End}(TS)$:

$\varphi(X):=\textrm{pr}_{TS}JX,\quad X\in TS\subset TC(S)$

where $J$ is the complex structure on $C(S)$.

We also define the $1$-form on $C(S)$, $\eta:=\frac{1}{t}Jdt$, which is
readily seen to be equal to $\frac{1}{t^{2}}i_{R}\omega$. As we did with
$\xi$, we shall also denote $\eta$ the restriction $\eta=\eta\rvert_{t=1}$ on
$S$. Then we have:

######  Proposition 2.2.

$S$ is an almost contact manifold with contact form $\eta$ and characteristic
field $\xi$. Moreover, ${\varphi^{2}=-\textrm{Id}+\eta\otimes\xi}$.

Denote by $D=\langle R,\xi\rangle^{\perp}$ the distribution
${g_{C(S)}}$-orthogonal to $\langle R,\xi\rangle$ on $C(S)$. Note that $t^{2}$
is a Kähler potential for $\omega$ and $dd^{c}(\log t)$ vanishes on $\langle
R,\xi\rangle$, the rest of its eigenvalues being positive. It follows that:

######  Proposition 2.3.

$\ker(d\eta)=\langle R,\xi\rangle$ and $(d\eta)\rvert_{D}=\omega\rvert_{D}$.
In particular $(d\eta)\rvert_{D}$ is a Kähler form.

###  2.3 Basic cohomology of Sasakian manifolds

######  Definition 2.4.

[19, Chapter 4] Let $(M,\mathcal{F})$ be a foliated manifold and consider
$F\subset TM$ to be the subbundle of vectors tangent to leaves of
$\mathcal{F}$. A form $\eta\in\Lambda^{*}M$ is called basic (with respect to
$\mathcal{F}$) if for any vector field $X\in\Gamma F$,
$\textrm{Lie}_{X}\eta=0$ and $i_{X}\eta=0$.

Denote the space of basic forms on a foliated manifold $(M,\mathcal{F},F)$ by
$\Lambda^{*}_{\textrm{bas}}M$. By Cartan’s formula, the exterior differential
$d$ maps basic forms to basic forms. Therefore, $d$ induces a cohomology on
basic forms, which we denote $H_{\textrm{bas}}^{*}M$.

We are interested in a particular type of foliations:

######  Definition 2.5.

Let $(M,\mathcal{F},F)$ be a foliated manifold.

Let $\omega_{0}\in\Lambda_{\textrm{bas}}^{*}(M)$ with $d\omega_{0}=0$ and
$g_{0}\in\textrm{Sym}_{\textrm{bas}}^{2}(T^{*}M)$ such that
$\omega_{0}|_{F}=0$ and $\omega_{0},g_{0}$ are positive definite on $TM/F$.

If the complex structure $J$ obtained from $\omega_{0}$ and $g_{0}$ is locally
integrable on any open set in $M$ where the leaf space is defined,
$(M,F,g_{0},\omega_{0})$ is called a transversally Kähler foliation.

On compact Kähler manifolds the following well-known consequence of Hodge
decomposition and Dolbeault decomposition holds.

######  Theorem 2.6.

([5, Theorem VI.8.5]) Let $M$ be a compact Kähler manifold. Denote by
$H^{p,q}_{\bar{\partial}}M$ the Dolbeault cohomology groups given by
${\bar{\partial}:\Lambda^{p,q}M\rightarrow\Lambda^{p,q+1}M}$. Then the Hodge
decomposition holds:

$H^{k}_{DR}M=\bigoplus\limits_{p+q=k}H^{p,q}_{\bar{\partial}}M$

The usefulness of transversally Kähler foliations lies in the following result
analogous to Theorem 2.6.

######  Theorem 2.7.

([17, Theorem 30.28]) Let $M$ be a compact manifold with a transversally
Kähler foliation $(M,F,g_{0},\omega_{0})$ such that $F$ is generated by
Killing vector fields and $M$ is equipped with a metric $g$ with
$g|_{TM/F}=g_{0}|_{TM/F}$. Suppose there exists $\Phi\in\Lambda^{*}(M)$ with
$d\Phi=0$, $\Phi|_{F}=0$ and $\Phi$ is a volume form on $TM/F$.

Then $H_{\textrm{bas}}(M)$ behaves just like the cohomology of a Kähler
manifold with respect to the Kähler form $\omega_{0}$. In particular,
$H_{\textrm{bas}}(M)$ admits the Hodge decomposition i.e.

$H^{k}_{\textrm{bas}}M=\bigoplus\limits_{p+q=k}H^{p,q}_{\bar{\partial}_{\textrm{bas}}}M$

where $\bar{\partial}_{\textrm{bas}}$ is the operator given locally on the
leaves of the foliation $F$ by the complex structure $J$ determined by
$\omega_{0}$ and $g_{0}$ as in Definition 2.5.

It turns out moreover that the cohomology of Sasakian manifolds is closely
related to the basic cohomology of their associated transversally Kähler
foliation. More precisely, we have:

######  Theorem 2.8.

([4, Proposition 7.4.13]) Let $S$ be a Sasakian manifold of dimension $2n+1$
with characteristic (Reeb) field $\xi$. Let $F=\langle\xi\rangle$ be the
transversally Kähler foliation generated by the Reeb field, which satisfies
the conditions of the previous theorem. Then:

$H^{k}(S)=\frac{H^{k}_{\textrm{bas}}(S)}{\rm{Im}(\omega_{0}\wedge\cdot)},\quad
k<n$

###  2.4 The product of two Sasakian manifolds

In the context of (almost) contact geometry, Morimoto was the first to
introduce an almost complex structure on the product of two almost contact
manifolds in [15]. He shows that this almost complex structure is integrable
if and only if condition (2.1) is satisfied for each factor of the product.
Building on Morimoto’s ideas, Tsukada introduced in [20] a family of complex
structures indexed by a complex nonreal parameter on the product of odd-
dimensional spheres, noting that by the same argument as in [15] these
structures are all integrable. In the same paper, Hermitian metrics associated
with each of these complex structures are introduced. Watson generalised this
family of pairs of complex structures and Hermitian metrics to products of
Sasakian manifolds ([23]). We recall the definition of this family below. In
the nomenclature of [23], we call a structure in this family a Calabi-Eckmann-
Morimoto structure, or CEM for short.

Let $S_{1}$, $S_{2}$ be Sasakian manifolds with $(1,1)$ tensors $\varphi_{1}$,
$\varphi_{2}$ and Reeb fields $\xi_{1},\xi_{2}$ respectively. Then there is a
family of complex structures $\\{J_{a,b}:a,b\in\mathbb{R},b\neq 0\\}$ on
$S_{1}\times S_{2}$:

$\displaystyle J_{a,b}(X_{1}+X_{2})$
$\displaystyle:=\varphi_{1}(X_{1})-\left(\frac{a}{b}\eta_{1}(X_{1})+\frac{a^{2}+b^{2}}{b}\eta_{2}(X_{2})\right)\xi_{1}$
$\displaystyle+\varphi_{2}(X_{2})+\left(\frac{1}{b}\eta_{1}(X_{1})+\frac{a}{b}\eta_{2}(X_{2})\right)\xi_{2}$
(1)

Let $g_{i}$ denote the Riemannian metric on the Sasakian manifold $S_{i}$,
$i=\overline{1,2}$. For each pair $(a,b),b\neq 0$ there is an associated
Hermitian ([20]) metric $g_{a,b}$ given by

$\displaystyle g_{a,b}(X_{1}+X_{2},Y_{1}+Y_{2})$
$\displaystyle:=g_{1}(X_{1},Y_{1})+a\eta_{1}(X_{1})\eta_{2}(Y_{2})+a\eta_{1}(Y_{1})\eta_{2}(X_{2})$
$\displaystyle+(a^{2}+b^{2}-1)\eta_{2}(X_{2})\eta_{2}(Y_{2})+g_{2}(X_{2},Y_{2})$
(2)

The metric data given by $g_{a,b}$ has been studied. It is shown in [11,
Theorem 1] that the metric $g_{a,b}$ is Einstein if and only if $a=0$, $S_{1}$
is Einstein, and $S_{2}$ is $\eta_{2}$-Einstein with some specific constants
(see [16], [18] for $\eta$-Einstein manifolds). The authors of [11] also
consider the property of weak $*$-Einsteiniainty for the product, which
involves the interplay of $J_{a,b}$ with $g_{a,b}$. In showing that
$(S_{1}\times S_{2},J_{a,b},g_{a,b})$ is never weakly $*$-Einstein, they also
prove that $(J_{a,b},g_{a,b})$ is never Kähler.

Further exploring this interplay, the authors of [1] study whether and when
the pairs $(J_{a,b},g_{a,b})$ satisfy a number of natural conditions which are
weaker than Kählerianity, building on previous work in [12] and [6]. The
results known about $(J_{a,b},g_{a,b})$ are summarized in the following

######  Theorem 2.9.

([12],[6],[1]) Let $S_{1}$ and $S_{2}$ be Sasakian manifolds of dimensions
$2n_{1}+1$ and $2n_{2}+1$ respectively. Consider the complex structure
$J_{a,b}$ (1) and the Hermitian metric $g_{a,b}$ (2). Then:

  1. 1.

If $n_{1}+n_{2}\geq 1$ then $(J_{a,b},g_{a,b})$ is not balanced (see [14]).

  2. 2.

$(J_{a,b},g_{a,b})$ is LCK (see [22] as well as [17, Chapter 3] for equivalent
definitions) if and only if $n_{1}=0$ and $n_{2}\geq 1$ or $n_{2}=0$ and
$n_{2}\geq 1$; if it is LCK, then it is also Vaisman.

  3. 3.

$(J_{a,b},g_{a,b})$ is SKT (see [8]) if and only if either $n_{1}=1$ and
$n_{2}=0$ or $n_{1}=0$ and $n_{2}=1$ or $a=0$ and $n_{1}=n_{2}=1$.

  4. 4.

If $n_{1}+n_{2}\geq 2$ then the condition

$n_{1}(n_{1}-1)+2an_{1}n_{2}+n_{2}(n_{2}-1)(a^{2}+b^{2})=0$

holds if and only if $(J_{a,b},g_{a,b})$ is $1$-Gauduchon (see [7] for
$k$-Gauduchon) if and only if $(J_{a,b},g_{a,b})$ is astheno-Kähler (see [9]).

  5. 5.

If $n_{1}+n_{2}\geq 3$ and $2\leq k\leq\dim_{\mathbb{C}}(S_{1}\times
S_{2})-1$, then $(J_{a,b},g_{a,b})$ is $k$-Gauduchon if and only if the
following holds:

$(n_{1}+n_{2}-k)\left(n_{1}(n_{1}-1)+2an_{1}n_{2}+n_{2}(n_{2}-1)(a^{2}+b^{2})\right)=0$

##  3 The main result

######  Theorem 3.1.

Let $S_{1}$, $S_{2}$ be compact Sasakian manifolds of respective dimensions
$2n_{i}+1$, with $n_{i}>1$. Then $S_{1}\times S_{2}$ has a family of complex
structures indexed by a complex nonreal parameter, none of whose members
admits any Kähler or LCK metrics.

###### Proof.

Step 1. Definition of the complex structure on the product.

To define the complex structure, we consider the following generalisation of
Calabi-Eckmann manifolds. Let $S$ be a Sasakian manifold and define an action
of $(\mathbb{C},+)$ on the open cone $C(S)$ by putting for
$a+bi\in\mathbb{C}$:

$(a+b\sqrt{-1})\cdot p:=\phi^{aR+bJR}_{1}(p),$ (3)

where $\phi^{X}_{t}$ denotes the flow of the vector field $X$ at time $t$.
Since $R$ and $\xi$ commute, we have

$\phi^{(cR+d\xi)+(aR+b\xi)}_{1}(p)=\phi^{cR+d\xi}_{1}(\phi^{aR+b\xi}_{1}(p))$

In other words

$((c+d\sqrt{-1})+(a+b\sqrt{-1}))\cdot
p=(c+\sqrt{-1}d)\cdot((a+b\sqrt{-1})\cdot p)$

showing that indeed (3) defines a group action.

This action is a holomorphic map $\mathbb{C}\times C(S)\rightarrow C(S)$.
Indeed, the Reeb and Euler fields act by biholomorphisms. Further, let $x\in
C(S)$ and $v\in\mathbb{C}$, $X\in T_{v}\mathbb{C}$ and $\gamma(t)$ be a curve
with tangent vector $X$ at $v$. Then
$JX=\frac{d}{dt}\rvert_{t=0}\left(\sqrt{-1}\gamma(t)\right)$. Since
$[R,\xi]=0$, one vector field is invariated by the flow of the other.
Therefore:

$\displaystyle d_{v}(w\mapsto(w\cdot x))\left(JX\right)$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\left((\sqrt{-1}\gamma(t))\cdot
x\right)$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\left(\phi_{1}^{(-\rm{Im}(\gamma(t)))R+\textrm{Re}(\gamma(t))\xi}(x)\right)$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\left(\phi_{\textrm{Re}(\gamma(t))}^{\xi}\left(\phi_{-\rm{Im}(\gamma(t))}^{R}(x)\right)\right)$
$\displaystyle=-\rm{Im}(X)R+\textrm{Re}(X)\xi=J(\textrm{Re}(X)R+\rm{Im}(X)\xi)$
$\displaystyle=Jd_{v}(w\mapsto(w\cdot x))\left(X\right)$

which shows that the map $w\mapsto(w\cdot x)$ is holomorphic for every fixed
$x\in C(S)$.

Let now $S_{i}$, $i=1,2$, be compact Sasakian manifolds with Euler fields
$R_{i}$, Reeb fields $\xi_{i}:=J_{i}R_{i}$, and consider the diagonal action
of $\mathbb{C}\times\mathbb{C}$ on the product of the cones $C(S_{1})\times
C(S_{2})$.

Fix some $\alpha\in\mathbb{C}$ with $\rm{Im}\alpha\neq 0$ and define the
subgroup ${G_{\alpha}:=\\{(t,\alpha t):t\in\mathbb{C}\\}}$ of
$(\mathbb{C}\times\mathbb{C},+)$. Clearly, $G_{\alpha}$ is isomorphic with
$\mathbb{C}$ and acts on $C(S_{1})\times C(S_{2})$. We analyze
$(C(S_{1})\times C(S_{2}))/G_{\alpha}$.

Let $r_{i}:C(S_{i})\rightarrow\mathbb{R}^{>0}$ be the projections on the
radial directions.

######  Claim 3.2.

For any $(a,b)\in{\mathbb{R}^{>0}}\times\mathbb{R}^{>0}$ and any
$x=(\tilde{p}_{1},\tilde{p}_{2})\in C(S_{1})\times C(S_{2})$ there exists a
unique $v\in\mathbb{C}\simeq G_{\alpha}$ such that $r_{1}(v\cdot x)=a$ and
$r_{2}(v\cdot x)=b$.

###### Proof.

For $\tilde{p}_{i}=(p_{i},t_{i})$, $p_{i}\in S_{i}$, $t_{i}\in\mathbb{R}^{>0}$
we have:

$\displaystyle v\cdot p_{1}$
$\displaystyle=\phi_{1}^{\textrm{Re}(v)R_{1}+\rm{Im}(v)\xi_{1}}(\tilde{p}_{1})=\phi_{1}^{\rm{Im}(v)\xi_{1}}(\phi_{1}^{\textrm{Re}(v)R_{1}}(\tilde{p}_{1}))$
$\displaystyle=\phi_{1}^{\rm{Im}(v)\xi_{1}}((p_{1},e^{\textrm{Re}(v)}t_{1}))$

Since $\rm{Im}(v)\xi_{1}$ acts only on the level sets of the cone, when we
project to the radial direction we get:

$r_{1}(v\cdot\tilde{p}_{1})=e^{\textrm{Re}(v)}t_{1}$

Similarly $r_{2}(\alpha v\cdot\tilde{p}_{2})=e^{\textrm{Re}(\alpha v)}t_{2}$.
Therefore, what we need to show is that for any strictly positive
$a,b,t_{1},t_{2}$ there exists a unique $v\in\mathbb{C}$ such that:

$\begin{cases}e^{\textrm{Re}(v)}t_{1}=a\\\
e^{\textrm{Re}(\alpha)\textrm{Re}(v)-\rm{Im}(\alpha)\rm{Im}(v)}t_{2}=b\end{cases}$

or

$\begin{cases}\textrm{Re}(v)=\log(a)-\log(t_{1})\\\
\textrm{Re}(\alpha)\textrm{Re}(v)-\rm{Im}(\alpha)\rm{Im}(v)=\log(b)-\log(t_{2})\end{cases}$

Since $\rm{Im}\alpha\neq 0$, we have

$\displaystyle\rm{Im}v=\frac{\textrm{Re}(\alpha)(\log(a)-\log(t_{1}))+\log(t_{2})-\log(b)}{\rm{Im}(\alpha)},$

and hence the solution $v$ exists and is unique.

∎

Claim 3.2 provides an identification of the quotient $(C(S_{1})\times
C(S_{2}))/G_{\alpha}$ with $S_{1}\times S_{2}$, given by an explicit formula
for $\pi:C(S_{1})\times C(S_{2})\longrightarrow S_{1}\times S_{2}$. Denote, as
in Claim 3.2,

$v(t_{1},t_{2}):=-\log(t_{1})+\frac{\sqrt{-1}}{\rm{Im}(\alpha)}(-\textrm{Re}(\alpha)\log
t_{1}+\log t_{2}).$ (4)

Now the map $\pi:C(S_{1})\times C(S_{2})\longrightarrow S_{1}\times S_{2}$ can
be described as:

$\pi((p_{1},t_{1}),(p_{2},t_{2}))=\left(\phi_{1}^{\rm{Im}(v(t_{1},t_{2}))\xi_{1}}(p_{1}),\
\phi_{1}^{\rm{Im}(\alpha v(t_{1},t_{2}))\xi_{2}}(p_{2})\right).$ (5)

Since the action of $G_{\alpha}$ defines a holomorphic map

$G_{\alpha}\times C(S_{1})\times C(S_{2})\rightarrow C(S_{1})\times C(S_{2})$

We conclude that

$(C(S_{1})\times C(S_{2}))/G_{\alpha}\simeq S_{1}\times S_{2}$

admits a complex structure compatible with the smooth product structure on
$S_{1}\times S_{2}$ and making the projection map $\pi:C(S_{1})\times
C(S_{2})\longrightarrow S_{1}\times S_{2}$ a holomorphic submersion.

Step 2. We now aim to better understand the complex structure induced by
$\pi$. More precisely, we show that on the transverse distributions of each
Sasakian, it acts like the complex structure on the cone, while it takes each
Reeb field to the span of the two Reeb fields.

To keep notation simple, we will deliberately use the same notation $\xi_{i}$
for the Reeb field(s) both on the product of the Kähler cones and on the
product of the Sasakian manifolds.

Let $X\in T_{p_{1}}S_{1}$ and $x\in\pi^{-1}(p_{1},p_{2})\subset C(S_{1})\times
C(S_{2})$ for some $p_{2}\in S_{2}$. We see $X$ as tangent in $x$ to
$C(S_{1})\times C(S_{2})$. $X$ can be extended to a vector field $\tilde{X}$
on $C(S_{1})\times C(S_{2})$, such that $\tilde{X}$ is tangent to $S_{1}$ and
moreover $\tilde{X}$ commutes with $\xi_{1}$ (and hence with all multiples of
$\xi_{1}$) in a neighborhood of $x$. We can obtain such an extension by
considering a chart on $S_{1}$ in which $\xi_{1}$ is a standard coordinate
vector field, extending the expression of $X$ in this chart to a constant
vector field and multiplying it with a bump function.

An extension $\tilde{X}$ of $X$ with $[\tilde{X},\xi_{1}]=0$ guarantees that
${d_{x}\phi_{1}^{\xi_{1}}(\tilde{X}_{x})=\tilde{X}_{\phi_{1}^{\xi_{1}}(x)}}$

Further, we have:

$\displaystyle d_{x}\pi(\tilde{X})$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\left(\pi((\phi_{t}^{\tilde{X}}(p_{1}),t_{1}),\quad(p_{2},t_{2}))\right)$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\left(\phi_{1}^{\rm{Im}(v(t_{1},t_{2}))\xi_{1}}(\phi_{t}^{\tilde{X}}(p_{1})),\quad\phi_{1}^{\rm{Im}(\alpha
v(t_{1},t_{2}))\xi_{2}}(p_{2})\right)$
$\displaystyle=\left(d_{p_{1}}\left(p\mapsto\phi_{1}^{\rm{Im}(v(t_{1},t_{2}))\xi_{1}}(p)\right)(\tilde{X}_{p_{1}}),\quad
0\right)_{\pi(x)}$ $\displaystyle=X$

Similarly, for $X\in T_{p_{2}}S_{2}$ we have:

$\displaystyle d_{x}\pi(\tilde{X})$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\left(\phi_{1}^{\rm{Im}v(t_{1},t_{2})\xi_{1}}(p_{1}),\quad\phi_{1}^{\rm{Im}(\alpha
v(t_{1},t_{2}))\xi_{2}}(\phi_{t}^{\tilde{X}}(p_{2}))\right)=X$

For the first Euler field:

$\displaystyle d_{x}\pi(R_{1})$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\pi\left((p_{1},e^{t}t_{1}),\quad(p_{2},t_{2})\right)$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\left(\phi_{1}^{\rm{Im}(v(e^{t}t_{1},t_{2}))\xi_{1}}(p_{1}),\quad\phi_{1}^{\rm{Im}(\alpha
v(e^{t}t_{1},t_{2}))\xi_{2}}(p_{2})\right)$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\left(\phi_{\rm{Im}(v(e^{t}t_{1},t_{2}))}^{\xi_{1}}(p_{1}),\quad\phi_{\rm{Im}(\alpha
v(e^{t}t_{1},t_{2}))}^{\xi_{2}}(p_{2})\right)$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\left(\rm{Im}(v(e^{t}t_{1},t_{2}))\right)\left(\xi_{1}\right)_{\pi(x)}+\frac{d}{dt}\rvert_{t=0}\left(\rm{Im}(\alpha
v(e^{t}t_{1},t_{2}))\right)\left(\xi_{2}\right)_{\pi(x)}$

Denote from now $a=\textrm{Re}\alpha,b=\rm{Im}\alpha$.

According to (4),
$v(e^{t}t_{1},t_{2})=v(t_{1},t_{2})-t\left(1+\frac{a}{b}\sqrt{-1}\right)$.
Hence

$\displaystyle d_{x}\pi(R_{1})$
$\displaystyle=-\frac{1}{b}\left(a\xi_{1}+(a^{2}+b^{2})\xi_{2}\right)_{\pi(x)}$

For the second Euler field, since, by (4),
$v(t_{1},e^{t}t_{2})=v(t_{1},t_{2})+\frac{t}{b}\sqrt{-1}$, we deduce as
before:

$\displaystyle d_{x}\pi(R_{2})$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\pi\left((p_{1},t_{1}),\quad(p_{2},e^{t}t_{2})\right)$
$\displaystyle=\frac{d}{dt}\rvert_{t=0}\left(\phi_{1}^{\rm{Im}(v(t_{1},e^{t}t_{2}))\xi_{1}}(p_{1}),\quad\phi_{1}^{\rm{Im}(\alpha
v(t_{1},e^{t}t_{2}))\xi_{2}}(p_{2})\right)$
$\displaystyle=\frac{1}{b}\left(\xi_{1}+a\xi_{2}\right)_{\pi(x)}$

In summary, we have:

$\displaystyle d_{x}\pi(X)$ $\displaystyle=X,\quad
x=((p_{1},1),(p_{2},1)),\quad X\in T_{p_{1}}S_{1}\sqcup T_{p_{2}}S_{2}$ (6)
$\displaystyle d_{x}\pi(R_{1})$
$\displaystyle=-\frac{1}{b}\left(a\xi_{1}+(a^{2}+b^{2})\xi_{2}\right)_{\pi(x)},\quad
d_{x}\pi(R_{2})=\frac{1}{b}\left(\xi_{1}+a\xi_{2}\right)_{\pi(x)}$ (7)

Step 3. The above family of complex structures does not admit any compatible
Kähler metric.

Step 3.1. Let $\eta_{1}$ be the pullback of the contact form on $S_{1}$
through $S_{1}\times S_{2}\rightarrow S_{1}$. Then $d\eta_{1}$ is a
semipositive $(1,1)$-form.

Indeed, to see that $d\eta_{1}$ is $(1,1)$, it’s enough to check that
$d\eta_{1}(z\pi_{*}X,z\pi_{*}Y)=z\bar{z}d\eta_{1}(\pi_{*}X,\pi_{*}Y)$ for
$z\in\mathbb{C}$. By (6) and holomorphicity of $\pi$:

$\displaystyle d\eta_{1}(z\pi_{*}X,z\pi_{*}Y)=$
$\displaystyle\textrm{Re}(z)^{2}d\eta_{1}(X,Y)+\rm{Im}(z)^{2}d\eta_{1}(\pi_{*}JX,\pi_{*}JY)$
$\displaystyle+\textrm{Re}(z)\rm{Im}(z)\left(d\eta_{1}(X,\pi_{*}JY)+d\eta_{1}(\pi_{*}JX,Y)\right)$

Suppose $X$ is orthogonal to $\langle R,\xi\rangle$ on its respective Sasakian
manifold. If $Y$ is also orthogonal, we are done since $\pi_{*}JY=JY$ and
$d\eta_{1}$ is transversally Kähler on the cone. Otherwise $Y$ is a multiple
of a Reeb vector $\xi_{i}$, so $\pi_{*}JY\in\langle\xi_{1},\xi_{2}\rangle$, so
$\pi_{*}JY\in\ker d\eta_{1}$ and since also $Y\in\ker d\eta_{1}$, the wanted
equality checks trivially. Finally, if $X$ is a multiple of a Reeb vector, the
wanted equality checks trivially because again $\\{X,\pi_{*}JX\\}\subset\ker
d\eta_{1}$.

Now checking semipositivity is equivalent by the holomorphicity of $\pi$ to
checking that for $X\in TC(S_{1})\times TC(S_{2})$ we have
$d\eta_{1}(\pi_{*}X,\pi_{*}JX)\geq 0$. If $X$ is tangent to either $S_{1}$ or
$S_{2}$ and is transverse to the Euler and Reeb fields, then $JX$ stays
outside the distribution generated by the Euler and Reeb fields, and so by (6)
$d\eta_{1}(\pi_{*}X,\pi_{*}JX)=d\eta_{1}(X,JX)$ and the latter is a nonegative
quantity because $d\eta_{1}$ is semipositive on the cone. If $X$ is either
$\xi_{1}$ and $\xi_{2}$ then $d\eta_{1}(\pi_{*}X,\pi_{*}JX)=0$ by (7).

Step 3.2. Suppose $S_{1}\times S_{2}$ is Kähler with Kähler form $\omega$.

$d(\eta_{1}\wedge\omega^{\dim_{\mathbb{C}}(S_{1}\times
S_{2})-1})=(d\eta_{1})\wedge\omega^{\dim_{\mathbb{C}}(S_{1}\times S_{2})-1}$

because $d\omega=0$. So by Stokes’ Theorem

$\int_{S_{1}\times
S_{2}}(d\eta_{1})\wedge\omega^{\dim_{\mathbb{C}}(S_{1}\times S_{2})-1}=0$ (8)

Because $d\eta_{1}$ is semipositive,
$d\eta_{1}\wedge\omega^{\dim_{\mathbb{C}}(S_{1}\times S_{2})-1}$ is a
semipositive volume form, which vanishes if and only if $d\eta_{1}$ vanishes.
But then $d\eta_{1}$ vanishes by (8), which contradicts the fact that
$d\eta_{1}$ is positive on the distribution transverse to $\ker d\eta_{1}$.

Step 4. Let $S_{1}$, $S_{2}$ be Sasakian manifolds of respective dimensions
$2n_{i}+1$ with $n_{i}>1$. By the Künneth formula $H^{1}(S_{1}\times
S_{2})=H^{1}(S_{1})\oplus H^{1}(S_{2})$. In view of Theorem 2.8, we can
represent forms in $H^{1}(S_{i})$ with basic forms. Hence, in view of Theorem
2.7, we can represent $[\eta]\in H^{1}(S_{1}\times S_{2})$ as
$\eta^{1,0}+\eta^{0,1}$, where $\eta^{1,0}$ is holomorphic and closed and
$\eta^{0,1}$ is antiholomorphic and closed. To see that this is the case,
suppose that $\alpha$ is a holomorphic representative of a basic class on one
of the Sasakian manifolds, say $[\alpha]\in H^{*}_{\textrm{bas}}(S_{1})$. The
fact that $\alpha$ is a basic holomorphic form implies that
$\pi_{1}^{*}\alpha$ is holomorphic, where $\pi_{1}:C(S_{1})\rightarrow S_{1}$
is the projection. We need to check that this implies that $\alpha$ is
holomorphic as a form on $S_{1}\times S_{2}$ with the complex structure
induced by the projection $\pi$ from $C(S_{1})\times C(S_{2})$. But by (6) and
(7), we obtain $\pi^{*}\alpha=\pi_{1}^{*}\alpha$ up to a constant, and hence
$\alpha$ is holomorphic.

Step 5. Assuming $S_{1}\times S_{2}$ is LCK, we represent the Lee form
$\theta$ as $\theta=\theta^{1,0}+\theta^{0,1}$ with $\theta^{1,0}$ holomorphic
and closed and $\theta^{0,1}$ antiholomorphic and closed. Thus we get
$dd^{c}\theta=0$. Then $dd^{c}(\omega^{n-1})=\omega^{n-1}\wedge\theta\wedge
J\theta$, so

$\int_{M}\omega^{n-1}\wedge\theta\wedge J\theta=0$

Combined with the fact that $\theta\wedge J\theta$ is semipositive $(1,1)$,
the above equality shows that $\omega^{n-1}\wedge\theta\wedge J\theta=0$.

So $\theta\wedge J\theta=0$. Hence $\theta=0$ since $\theta$ and $J\theta$ are
linearly independent. This shows that $S_{1}\times S_{2}$ is GCK, but then it
also admits a Kähler structure, which is a contradiction by Step 3. ∎

######  Remark 3.3.

The same proof as in Step 3 shows that $S_{1}\times S_{2}$ does not admit
balanced metrics i.e. metrics with Hermitian form $\omega$ satisfying
$d\omega^{\dim_{\mathbb{C}}(S_{1}\times S_{2})-1}=0$, since in that case we
also obtain that ${(d\eta_{1})\wedge\omega^{\dim_{\mathbb{C}}(S_{1}\times
S_{2})-1}}$ is exact.

######  Remark 3.4.

The argument developed in Steps 3 through 5 also shows that the CEM complex
structure defined by (1) does not admit any compatible locally conformally
Kähler metric.

##  4 Complex submanifolds of the product of Sasakian manifolds

Let $S_{1},S_{2}$ be compact Sasakian manifolds with
$\dim_{\mathbb{R}}S_{i}=2n_{i}+1$ and with contact forms $\eta_{1},\eta_{2}$.
Let $S_{1}\times S_{2}$ be their product with the complex structure induced by
the action of $G_{\alpha}$ on the product of their cones as in the proof of
Step 1 of Theorem 3.1.

######  Theorem 4.1.

Let $Z\subset S_{1}\times S_{2}$ be a complex submanifold of
$\dim_{\mathbb{C}}Z=k$ where the complex structure on $S_{1}\times S_{2}$ is
induced by the Calabi-Eckmann action on the product of the cones. Then $Z$ is
tangent to $\ker(d\eta_{1}+d\eta_{2})$.

###### Proof.

Let $\eta=\eta_{1}+\eta_{2}$. Then We have:

$d(\eta\wedge(d\eta)^{k-1})=d\eta\wedge(d\eta)^{k-1}=(d\eta)^{k}$

So by Stokes’ theorem we have:

$\int_{Z}(d\eta)^{k}=0$

Since outside $\ker(d\eta)$, $d\eta$ is strictly positive, and $Z$ is a
complex submanifold, we must thus have $TZ\subset\ker(d\eta)$. ∎

##  5 Comparison with the
Calabi-Eckmann-Morimoto complex structures

Consider again the principal $G_{\alpha}=\\{(v,\alpha
v):v\in\mathbb{C}\\}$-bundle

${\pi:C(S_{1})\times C(S_{2})\rightarrow S_{1}\times S_{2}}$

where $\alpha\in\mathbb{C}\setminus\mathbb{R}$.

The natural question arises whether $J_{a,b}$ defined by (1) coincides with
the complex structure induced by $\pi$.

######  Theorem 5.1.

For every fixed $\alpha\in\mathbb{C}\setminus\mathbb{R}$, the complex
structure induced by $G_{\alpha}$ does not in general coincide with the
complex structure $J_{a,b}$ for any $a,b\in\mathbb{R}$, $b\neq 0$.

###### Proof.

For general Sasakian manifolds $S_{1}$ and $S_{2}$, by uniqueness of the
complex structure making ${\pi:C(S_{1})\times C(S_{2})\rightarrow S_{1}\times
S_{2}}$ a holomorphic submersion, the complex structures on $S_{1}\times
S_{2}$ coincide if and only if for any $x=(p_{1},t_{1},p_{2},t_{2})$ and any
$X_{i}\in T_{(p_{i},t_{i})}C(S_{i})$, $i=\overline{1,2}$, we have

$J_{a,b}d_{x}\pi(X_{1}+X_{2})=d_{x}\pi(J(X_{1}+X_{2}))$

For $X_{1}=\xi_{1},X_{2}=0$ we have by (6)

$J_{a,b}\pi_{*}\xi_{1}=J_{a,b}\xi_{1}=\frac{1}{b}\left(-a\xi_{1}+\xi_{2}\right)$

while by (7)

$\pi_{*}(J\xi_{1})=-\pi_{*}(R_{1})=\frac{1}{\rm{Im}(\alpha)}\left(\textrm{Re}(\alpha)\xi_{1}+((\textrm{Re}(\alpha))^{2}+(\rm{Im}(\alpha))^{2})\xi_{2})\right)$

Furthermore,

$J_{a,b}\pi_{*}\xi_{2}=\frac{1}{b}\left(-(a^{2}+b^{2})\xi_{1}+a\xi_{2}\right)$

and

$\pi_{*}(J\xi_{2})={-\pi_{*}(R_{2})}={-\frac{1}{\rm{Im}(\alpha)}(\xi_{1}+\textrm{Re}(\alpha)\xi_{2})}.$

So if $J_{a,b}$ coincides with the structure induced by $\pi$ we obtain the
following system of equations:

$\begin{cases}\textrm{Re}(\alpha)b=-a\rm{Im}(\alpha),\quad\rm{Im}(\alpha)=b(\textrm{Re}(\alpha))^{2}+(\rm{Im}(\alpha))^{2})\\\
b=(a^{2}+b^{2})\rm{Im}(\alpha),\quad\textrm{Re}(\alpha)b=-a\rm{Im}(\alpha)\end{cases}$

This leads to the equation:

$(\rm{Im}(\alpha))^{4}+(2(\textrm{Re}(\alpha))^{2}-1)(\rm{Im}(\alpha))^{2}+(\textrm{Re}(\alpha))^{4}=0$

which implies that $|\rm{Im}(\alpha)|\leq 1$ and that
$(\textrm{Re}(\alpha))^{2}\leq\frac{1}{4}$. Hence, for $\alpha$ such that
these conditions are not met, we cannot find $(a,b)$ such that $J_{a,b}$
coincides with the complex structure induced by $\pi$.

However, on $\langle\xi_{1},\xi_{2}\rangle$ $J$ is
$-J_{\textrm{Re}\alpha,\rm{Im}\alpha}^{T}$ where the superscript is matrix
transpose. ∎

##  6 The Dolbeault cohomology of the product of Sasakian manifolds

Let $S_{1},S_{2}$ be compact Sasakian manifolds with the action of
$G_{\alpha}$ as in Step 1 of Theorem 3.1, $\alpha=a+b\sqrt{-1}$,
$a\in\mathbb{R},b\in\mathbb{R}\setminus\\{0\\}$. Denote from now
$M:=S_{1}\times S_{2}$.

Consider $\eta_{1},\eta_{2}$ the two contact forms on $M$. Let
$\eta:=\eta_{1}+\eta_{2}$, $\omega_{0}:=d\eta$ and $\eta^{0,1},\eta^{1,0}$ be
the $(0,1)$ and $(1,0)$ parts of $\eta$, respectively. Since $\omega_{0}$ is a
$(1,1)$-form and
$\omega_{0}=d\eta=\partial\eta^{0,1}+\partial\eta^{1,0}+\bar{\partial}\eta^{0,1}+\bar{\partial}\eta^{1,0}$,
we obtain that $\partial\eta^{1,0}=0$ and $\bar{\partial}\eta^{0,1}=0$.

Endow $M$ also with a Hermitian metric such that the two Reeb fields are
Killing, as follows. Consider $V:=\langle\xi_{1},\xi_{2}\rangle$ with the
frame $\\{\xi_{1},\xi_{2}\\}$ and $J_{\alpha}$ the complex structure induced
by $\pi_{\alpha}$ as in Step 1 of Theorem 3.1. Recall also that $J_{a,b}$ is
defined as in $\eqref{calabi eckmann morimoto complex structure definition}$
and the metric $g_{a,b}$ defined as in (2) is Hermitian with respect to
$J_{a,b}$. On $V$ we have $J_{\alpha}|_{V}=-(J_{a,b}|_{V})^{T}$ for
$a=\textrm{Re}(\alpha),b=\rm{Im}(\alpha)$. Hence $J_{\alpha}|_{V}$ is the
negative of the morphism induced on $V^{*}$ by $J_{a,b}|_{V}$, so
$J_{\alpha}|_{V}$ is Hermitian with respect to $(g_{a,b}|_{V})^{-1}$. Since on
$V^{\perp}$ $J_{a,b}$ coincides with $J_{\alpha}$, the metric

$\displaystyle g_{\alpha}:=$ $\displaystyle
g_{1}+g_{2}-ab^{-2}\left(\eta_{1}\otimes\eta_{2}+\eta_{2}\otimes\eta_{1}\right)$
$\displaystyle+\left(b^{-2}(a^{2}+b^{2})-1\right)\eta_{1}\otimes\eta_{1}+(b^{-2}-1)\eta_{2}\otimes\eta_{2}$

is Hermitian on $M$ with respect to $J_{\alpha}$, where $\eta_{i}$ and $g_{i}$
are the contact forms and Riemannian metrics respectively on $S_{i}$, extended
with $0$ on the Sasakian manifold they are not initially defined on. Moreover,
$\xi_{1},\xi_{2}$ are Killing with respect to $g_{\alpha}$ because each
$\xi_{i}$ is Killing with respect to $g_{i}$ and
$\textrm{Lie}_{\xi_{i}}\eta_{i}=0$ (because $S_{i}$ is contact with
characteristic field $\xi_{i}$ and by Cartan’s formula).

By a theorem of Myers and Steenrod ([13]), $\textrm{Iso}_{g_{\alpha}}(M)$ is a
Lie group, which is compact since both Sasakian manifolds are compact.
Consider $K$ to be the closure of the subgroup generated by
$\phi_{t}^{\xi_{1}}$ and $\phi_{t}^{\xi_{2}}$ inside
$\textrm{Iso}_{g_{\alpha}}(M)$. By the closed subgroup theorem, $K$ is also a
(compact) Lie group. Take $\Lambda^{*}(M)^{\textrm{inv}}$ to be all forms on
$M$ which are invariant under $K$. A standard continuity argument shows that
${(\Lambda^{*}(M))^{\textrm{inv}}=\\{\alpha\in\Lambda^{*}(M):\textrm{Lie}_{\xi_{1}}\alpha=\textrm{Lie}_{\xi_{2}}\alpha=0\\}}$.
Consider also $(\Lambda^{*}(M))_{\textrm{bas}}$ to be all the basic forms with
respect to the foliation $\langle\xi_{1},\xi_{2}\rangle$; clearly
$(\Lambda^{*}(M))_{\textrm{bas}}\subset(\Lambda^{*}(M))^{\textrm{inv}}$ (see
Definition 2.4). Locally, basic forms come from the leaf space of the
foliation.

Put
$\Lambda_{B,\eta^{0,1}}^{p,q}:=(\Lambda^{p,q})_{\textrm{bas}}\oplus\left(\eta^{0,1}\wedge\Lambda^{p,q-1}_{\textrm{bas}}\right)$.

Since $\bar{\partial}\eta^{0,1}=0$, for each $p\geq 0$ the restriction of
$\bar{\partial}$ gives a complex

$\bar{\partial}:\Lambda_{B,\eta^{0,1}}^{p,*}\rightarrow\Lambda_{B,\eta^{0,1}}^{p,*+1}.$

Now consider the operator
$L_{\omega_{0}}:\Lambda^{*}(M)\rightarrow\Lambda^{*}(M)$ to be wedge product
with $\omega_{0}$.

######  Remark 6.1.

Because $\bar{\partial}\omega_{0}=0$, for $p\geq 0$ we have that
$L_{\omega_{0}}$ is a morphism of complexes

$L_{\omega_{0}}:(\Lambda^{p,*},\bar{\partial})\rightarrow(\Lambda^{p+1,*+1},\bar{\partial})$

The restriction and corestriction of $L_{\omega_{0}}$ to invariant forms,

${L_{\omega_{0}}:\Lambda^{*}(M)^{\textrm{inv}}\rightarrow\Lambda^{*}(M)^{\textrm{inv}}}$

is well defined because
$\textrm{Lie}_{\xi_{1}}\omega_{0}=\textrm{Lie}_{\xi_{2}}\omega_{0}=0$. In
fact, $L_{\omega_{0}}$ is a well defined morphism
$L_{\omega_{0}}:\Lambda_{B,\eta^{0,1}}^{p,q}\rightarrow\Lambda_{B,\eta^{0,1}}^{p+1,q+1}$,
which follows because $\omega_{0}$ is a basic $(1,1)$-form, so whenever
$\beta\in(\Lambda^{p,q})_{\textrm{bas}}$, then
$\omega_{0}\wedge\beta\in(\Lambda^{p+1,q+1})_{\textrm{bas}}$. Together with
Remark 6.1, this shows that for each fixed $p\geq 1$, $L_{\omega_{0}}$ is a
morphism of complexes

$L_{\omega_{0}}:(\Lambda_{B,\eta^{0,1}}^{p-1,*},\bar{\partial})\longrightarrow(\Lambda_{B,\eta^{0,1}}^{p,*+1},\bar{\partial})$

Note also that $\bar{\partial}$ takes invariant form to invariant forms since
if $\beta$ is an invariant $(p,q)$-form then
$0=d\textrm{Lie}_{\xi_{i}}\beta=\textrm{Lie}_{\xi_{i}}d\beta=\textrm{Lie}_{\xi_{i}}\partial\beta+\textrm{Lie}_{\xi_{i}}\bar{\partial}\beta$
and so $\textrm{Lie}_{\xi_{i}}\bar{\partial}\beta=0$ because
$\textrm{Lie}_{\xi_{i}}\bar{\partial}\beta\in\Lambda^{p,q+1}$ and
$\textrm{Lie}_{\xi_{i}}\partial\beta\in\Lambda^{p+1,q}$.

Recall the following definition:

######  Definition 6.2.

Let $(C^{*},d_{C}),(D^{*},d_{D})$ be complexes and $f:C^{*}\rightarrow D^{*}$
be a morphism of complexes. The cone of the morphism $f$ is defined to be the
complex $(C(f),d_{f})$ with $C(f)_{i}:=C_{i+1}\oplus D_{i}$ and for $c\in
C_{i+1},d\in D_{i}$, $d_{f}(c,d):=(d_{C}(c),f(c)-d_{D}(d))$.

######  Lemma 6.3.

For each fixed $p\geq 0$, the complex
$((\Lambda^{p,*}(M))^{\textrm{inv}},\bar{\partial})$ is isomorphic to the cone
of

$L_{\omega_{0}}:(\Lambda_{B,\eta^{0,1}}^{p-1,*},\bar{\partial})\longrightarrow(\Lambda_{B,\eta^{0,1}}^{p,*+1},\bar{\partial})$

shifted by $-1$ i.e. to $C(L_{\omega_{0}})[-1]$.

###### Proof.

Forms on the tangent space of the foliation $\langle\xi_{1},\xi_{2}\rangle$
are spanned by $\eta_{1},\eta_{2}$, and hence by $\eta^{0,1},\eta^{1,0}$.
Therefore

$\displaystyle(\Lambda^{p,q})^{\textrm{inv}}$
$\displaystyle=(\Lambda^{p,q}_{\textrm{bas}})\oplus\left(\Lambda^{p-1,q}_{\textrm{bas}}\wedge\eta^{1,0}\right)\oplus\left(\Lambda^{p,q-1}_{\textrm{bas}}\wedge\eta^{0,1}\right)\oplus\left(\Lambda^{p-1,q-1}_{\textrm{bas}}\wedge\eta^{0,1}\wedge\eta^{1,0}\right)$
$\displaystyle=\Lambda^{p,q}_{B,\eta^{0,1}}\oplus\left(\Lambda^{p-1,q}_{B,\eta^{0,1}}\wedge\eta^{1,0}\right)$

The differential $\bar{\partial}$ acts on
$\left(\Lambda^{p-1,q}_{B,\eta^{0,1}}\wedge\eta^{1,0}\right)$ as
$\bar{\partial}_{\textrm{bas}}+L_{\omega_{0}}$, where

${\bar{\partial}_{\textrm{bas}}:\Lambda^{p-1,q}_{B,\eta^{0,1}}\wedge\eta^{1,0}\rightarrow\Lambda^{p-1,q+1}_{B,\eta^{0,1}}\wedge\eta^{1,0}}$

is $\bar{\partial}$ applied to the $\Lambda^{p-1,q}_{B,\eta^{0,1}}$ part,
while $L_{\omega_{0}}$ is multiplication of forms in
$\Lambda^{p-1,q}_{B,\eta^{0,1}}$ with $\bar{\partial}\eta^{1,0}=\omega_{0}$.

This suggests seeing the complex
$(\Lambda_{B,\eta^{0,1}}^{p-1,*},\bar{\partial})$ as identified with
${(\Lambda_{B,\eta^{0,1}}^{p-1,*}\wedge\eta^{1,0},\bar{\partial}_{\textrm{bas}})}$;
this identification is immediately obtained by simply dropping $\eta^{1,0}$.
Seeing $L_{\omega_{0}}$ after this identification as a morphism of complexes

$L_{\omega_{0}}:{(\Lambda_{B,\eta^{0,1}}^{p-1,*}\wedge\eta^{1,0},\bar{\partial}_{\textrm{bas}})}\longrightarrow(\Lambda_{B,\eta^{0,1}}^{p,*+1},\bar{\partial}),$

the cone of $L_{\omega_{0}}$ is in degree $q-1$:

$\displaystyle\left(C(L_{\omega_{0}})[-1]\right)_{q}=\left(C(L_{\omega_{0}})\right)_{q-1}=\left(\Lambda^{p-1,q}_{B,\eta^{0,1}}\wedge\eta^{1,0}\right)\oplus\Lambda^{p,q}_{B,\eta^{0,1}}$

Thus
$\left(C(L_{\omega_{0}})[-1]\right)_{q}=\left(\Lambda^{p,q}\right)^{\textrm{inv}}$.

At position $q-1$ of the cone, the cone differential takes an
$\alpha\wedge\eta^{1,0}\in\left(\Lambda^{p-1,q}_{B,\eta^{0,1}}\wedge\eta^{1,0}\right)$
and a $\beta\in\Lambda^{p,q}_{B,\eta^{0,1}}$ to

$\left(\bar{\partial}_{\textrm{bas}}(\alpha\wedge\eta^{1,0}),\
L_{\omega_{0}}(\alpha\wedge\eta^{1,0})-\bar{\partial}\beta\right)$

Now

$\bar{\partial}_{\textrm{bas}}(\alpha\wedge\eta^{1,0})=\left(\bar{\partial}\alpha\right)\wedge\eta^{1,0}$

and by the identification,
${L_{\omega_{0}}(\alpha\wedge\eta^{1,0})=\omega_{0}\wedge\alpha\in\Lambda^{p,q+1}_{B,\eta^{0,1}}}$.
Therefore, the action of the differential of the cone is precisely the same as
that of $\bar{\partial}$ and the complex of invariant forms is identified with
the $-1$ shift of the cone of $L_{\omega_{0}}$. ∎

Furthermore, whenever a compact group acts by holomorphic isometries on a
Hermitian manifold, its action on Dolbeault cohomology is trivial:

######  Theorem 6.4.

[10, Theorem 3.3] Let $G$ be a compact Lie group acting on a compact Hermitian
manifold $M$ by holomorphic isometries. Then the action of $G$ on Dolbeault
cohomology, given by $g\cdot[\alpha]:=[g^{*}\alpha]$ for $g\in G$ and
$[\alpha]\in H^{p,q}_{\bar{\partial}}(M)$, is trivial.

Consider the unique bi-invariant top form $\nu$ on the compact Lie group $K$
(defined above) with $\int_{K}\nu=1$. For any $\alpha\in\Lambda^{*}(M)$
consider $\overline{\alpha}:=\int_{K}(k^{*}\alpha)d\nu(k)$. Then
$\bar{\alpha}$ is an invariant ([21, Proposition 13.11]) smooth ([21,
Proposition 13.13]) form of the same degree as $\alpha$. By Theorem 6.4,
taking $\alpha$ to be $\bar{\partial}$-closed, we have for some forms
$\beta(k)$

$\displaystyle\int_{K}(k^{*}\alpha)d\nu(k)$
$\displaystyle=\int_{K}\left(\alpha+\bar{\partial}\beta(k)\right)d\nu(k)$
$\displaystyle=\alpha+\int_{K}(\bar{\partial}\beta(k))d\nu(k)=\alpha+\bar{\partial}\left(\int_{K}\beta(k)d\nu(k)\right)$

Hence, the cohomology groups $H_{\bar{\partial}}^{p,q}(M)$ are the same as the
cohomology groups of $(\Lambda^{p,*}(M)^{\textrm{inv}},\bar{\partial})$, and
hence, by Lemma 6.3,

$H_{\bar{\partial}}^{p,q}(M)=H^{q}\left(\left(C\left(L_{\omega_{0}}:\Lambda_{B,\eta^{0,1}}^{p-1,*}\rightarrow\Lambda_{B,\eta^{0,1}}^{p,*+1}\right)\right)[-1]\right).$
(9)

Now we can prove the following theorem (which has an analogue in the Vaisman
setting, [10, Theorem 4.12]).

######  Theorem 6.5.

Let $M$ be the product of two compact Sasakian manifolds with complex
structure given by (5). The Dolbeault cohomology groups of $M$ are computed
as:

$\displaystyle
H_{\bar{\partial}}^{p,q}(M)=\begin{cases}\displaystyle\frac{H_{\textrm{bas}}^{p,q}\oplus[\eta^{0,1}]\wedge
H_{\textrm{bas}}^{p,q-1}(M)}{\rm{im}(L_{\omega_{0}})},\qquad
p+q\leq\dim_{\mathbb{C}}(M)\\\\[10.00002pt]
\ker(L_{\omega_{0}})\rvert_{H_{\textrm{bas}}^{p,q}\oplus[\eta^{0,1}]\wedge
H_{\textrm{bas}}^{p,q-1}(M)},\qquad p+q>\dim_{\mathbb{C}}(M)\end{cases}$

###### Proof.

The cone of the morphism $L_{\omega_{0}}$ gives a short exact sequences of
complexes:

$0\longrightarrow\Lambda_{B,\eta^{0,1}}^{p,*+1}\longrightarrow
C(L_{\omega_{0}})\longrightarrow\left(\Lambda_{B,\eta^{0,1}}^{p-1,*}\right)[1]$

which gives rise to a long exact sequence in cohomology with connecting map
$L_{\omega_{0}}$:

$\displaystyle\cdots\longrightarrow
H^{i-1}_{\bar{\partial}}\left(\left(\Lambda_{B,\eta^{0,1}}^{p-1,*}\right)[1]\right)$
$\displaystyle\xrightarrow{L_{\omega_{0}}}H^{i}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p,*+1}\right)\longrightarrow
H^{i}(C(L_{\omega_{0}}))\longrightarrow$ $\displaystyle\longrightarrow
H^{i}_{\bar{\partial}}\left(\left(\Lambda_{B,\eta^{0,1}}^{p-1,*}\right)[1]\right)\xrightarrow{L_{\omega_{0}}}\cdots$

Taking into account shifts, degrees and (9) we thus have:

$\displaystyle\cdots\longrightarrow
H^{i}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p-1,*}\right)$
$\displaystyle\xrightarrow{L_{\omega_{0}}}H^{i}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p,*+1}\right)\longrightarrow
H^{p,i+1}_{\bar{\partial}}(M)\longrightarrow$ $\displaystyle\longrightarrow
H^{i+1}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p-1,*}\right)\xrightarrow{L_{\omega_{0}}}\cdots$

Now since $\bar{\partial}\eta^{0,1}=0$,

$H^{i}_{\bar{\partial}}(\Lambda^{p-1,*}_{B,\eta^{0,1}})=H^{p-1,i}_{\textrm{bas}}(M)\oplus[\eta^{0,1}]\wedge
H^{p-1,i-1}_{\textrm{bas}}(M)$

By Theorem 2.7, basic cohomology behaves just like the cohomology of a Kähler
manifold with Kähler form $\omega_{0}$. Hence, since $M$ is compact, by the
Hodge isomorphism theorem and the fact that the Kähler form is harmonic, the
operator

$H^{t}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{s,*}\right)\xrightarrow{L_{\omega_{0}}}H^{t}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{s+1,*+1}\right)$

is injective whenever $s+t\leq\dim_{\mathbb{C}}M-1$; by Poincaré duality, it
is surjective whenever $s+t>\dim_{\mathbb{C}}M-1$. Hence, for
$p+i\leq\dim_{\mathbb{C}}M$, we obtain the short exact sequence

$0\longrightarrow
H^{i-1}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p-1,*}\right)\xrightarrow{L_{\omega_{0}}}H^{i}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p,*}\right)\longrightarrow
H^{p,i}_{\bar{\partial}}(M)\longrightarrow 0$ (10)

while for $p+i>\dim_{\mathbb{C}}M+1$ we obtain the short exact sequence

$0\longrightarrow H^{p,i}_{\bar{\partial}}(M)\longrightarrow
H^{i}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p-1,*}\right)\xrightarrow{L_{\omega_{0}}}H^{i+1}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p,*}\right)\longrightarrow
0$ (11)

Finally, when $p+i=\dim_{\mathbb{C}}M+1$, by the Hard Lefschetz theorem
$\displaystyle{H^{i-1}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p-1,*}\right)\xrightarrow{L_{\omega_{0}}}H^{i}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p,*}\right)}$
is an isomorphism, so in particular surjective. Since, as mentioned above,
$\displaystyle{H^{i}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p-1,*}\right)\xrightarrow{L_{\omega_{0}}}H^{i+1}_{\bar{\partial}}\left(\Lambda_{B,\eta^{0,1}}^{p,*}\right)}$
is also surjective, we have the short exact sequence (11) also for the case
when $p+i=\dim_{\mathbb{C}}M+1$.

∎

## References

  * [1] A. Andrada, A. Tolcachier, _Harmonic complex structures and special hermitian metrics on products of Sasakian manifolds_ , arXiv: 2301.09706v2.
  * [2] D. E. Blair, Riemannian geometry of contact and symplectic manifolds. Second edition. Progress in Mathematics, 203. Birkhäuser Boston, Inc., Boston, MA, 2010.
  * [3] D. Blair, G. Ludden, K. Yano, _Geometry of complex manifolds similar to the Calabi-Eckmann manifolds_ , J. Diff. Geometry 9 (1974), 263-276.
  * [4] C. P. Boyer, K. Galicki, _Sasakian geometry_ , Oxford Mathematical Monographs. Oxford University Press, Oxford, 2008.
  * [5] J.-P. Demailly, Complex Analytic and Differential Geometry, Université de Grenoble I, 1997, https://www.fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf.
  * [6] A. Fino, L. Ugarte, _On generalized Gauduchon metrics_ , Proc. Edinb. Math. Soc. 56 (2013), 733–753.
  * [7] J. Fu, Z. Wang, D. Wu, _Semilinear equations, the $\gamma_{k}$ function, and generalized Gauduchon metrics_, J. Eur. Math. Soc. 15 (2013), 659–680.
  * [8] P. S. Howe, G. Papadopoulos, _Finitness and anomalies in $(4,0)$ supersymmetric sigma models_, Nucl. Phys. B381 (1992), 360-372.
  * [9] J. Jost, S.-T. Yau, _A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry_ , Acta Math. 170 (1993), 221–254.
  * [10] N. Klemyatin, _Dolbeault cohomology of compact complex manifolds with an action of a complex Lie group_ , arXiv:1909.04075
  * [11] J. C. Lee, J. H. Park, K. Sekigawa, _Hermitian structures on the product of Sasakian manifolds_ , Geom. Dedicata 161 (2012), 399-408.
  * [12] K. Matsuo, _Astheno-Kähler structures on Calabi-Eckmann manifolds_ , Colloq. Math. 115 (2009), 33–39.
  * [13] S. B. Myers, N. E. Steenrod. _The Group of Isometries of a Riemannian Manifold_ , Annals of Mathematics 40, no. 2 (1939), 400–416.
  * [14] M. L. Michelson, _On the existence of special metrics in complex geometry_ , Acta Math. 143 (1983), 261-295.
  * [15] A. Morimoto, _On normal almost contact structures_ , J. Math. Soc. Japan 15 (1963), 420-436.
  * [16] M. Okumura, _Some remarks on space with a certain contact structure_ , Tohoku Math. J. (2) 14 (1962),135–145.
  * [17] L. Ornea, M. Verbitsky, _Principles of Locally Conformally Kähler Geometry_ , arXiv: 2208.07188
  * [18] S. Sasaki, _Almost contact manifolds_ , Lecture Notes, Mathematical Institute, Tohoku University. Part 1: 1965, Part 2: 1967, Part 3: 1968.
  * [19] P. Tondeur, Geometry of foliations, Monographs in Mathematics, vol. 90, Birkhäuser Verlag, Basel, 1997.
  * [20] K. Tsukada, _Eigenvalues of the Laplacian on Calabi-Eckmann manifolds_ , J. Math. Soc. Japan 33 (1981), 673-691.
  * [21] L. Tu, Introductory Lectures on Equivariant Cohomology, Annals of Mathematics Studies 204, Princeton University Press, 2020.
  * [22] I. Vaisman, _On locally conformal almost Kähler manifolds_ , Israel J. Math. 24 (1976), 338-351.
  * [23] B. Watson, _New examples of strictly almost Kähler manifolds_ , Proc. Am. Math. Soc. 88 (1983), 541-544.

Vlad Marchidanu
University of Bucharest, Faculty of Mathematics and Informatics,
14 Academiei str., 70109 Bucharest, Romania
<EMAIL_ADDRESS>