Compact pluricanonical manifolds are Vaisman
Liviu Ornea111Partially supported by CNCS UEFISCDI, project number PN-II-ID-
PCE-2011-3-0118., and Misha Verbitsky222Partially supported by RSCF grant
14-21-00053 within AG Laboratory NRU-HSE.
Keywords: locally conformally Kähler, pluricanonical, potential, Hopf
manifold, Vaisman manifold, Monge-Ampère, Bott-Chern 2000 Mathematics Subject
Classification: 53C55.

Abstract

A locally conformally Kähler manifold is a Hermitian manifold $(M,I,\omega)$
satisfying $d\omega=\theta\wedge\omega$, where $\theta$ is a closed 1-form,
called the Lee form of $M$. It is called pluricanonical if $\nabla\theta$ is
of Hodge type $(2,0)+(0,2)$, where $\nabla$ is the Levi-Civita connection, and
Vaisman if $\nabla\theta=0$. We show that a compact LCK manifold is
pluricanonical if and only if the Lee form has constant length and the Kähler
form of its covering admits an automorphic potential. Using a degenerate
Monge-Ampère equation and the classification of surfaces of Kähler rank one,
due to Brunella, Chiose and Toma, we show that any pluricanonical metric on a
compact manifold is Vaisman. Several errata to our previous work are given in
the last Section.

###### Contents

  1. 1 Introduction
    1. 1.1 LCK manifolds
    2. 1.2 LCK manifolds with potential
    3. 1.3 Pluricanonical versus Vaisman: scheme of the proof
    4. 1.4 Some errors found
  2. 2 LCK manifolds: properness and positivity of the potential
    1. 2.1 LCK manifolds with potential: historical definition
    2. 2.2 Properness of the LCK potential
    3. 2.3 Positivity of automorphic potentials
    4. 2.4 LCK manifolds with proper and improper potential
  3. 3 Pluricanonical condition versus LCK with potential
    1. 3.1 Exterior derivative of the Lee form: preliminary computations
    2. 3.2 Pluricanonical manifolds and $(\nabla\theta)^{1,1}=0$
  4. 4 Compact pluricanonical surfaces are Vaisman
    1. 4.1 Complex surfaces of Kähler rank 1
    2. 4.2 Algebraic groups and the Jordan-Chevalley decomposition
    3. 4.3 Finding surfaces in LCK manifolds with potential
  5. 5 Pluricanonical condition versus Vaisman
    1. 5.1 Bott-Chern cohomology for Vaisman manifolds
    2. 5.2 Pluricanonical submanifolds
    3. 5.3 The main result: all compact pluricanonical manifolds are Vaisman
  6. 6 Errata
    1. 6.1 Pluricanonical condition revisited
    2. 6.2 LCK rank of Vaisman manifolds

##  1 Introduction

###  1.1 LCK manifolds

Let $(M,I)$ be a complex manifold, $\dim_{\mathbb{C}}M\geqslant 2$. It is
called locally conformally Kähler (LCK) if it admits a Hermitian metric $g$
whose fundamental 2-form $\omega(\cdot,\cdot):=g(\cdot,I\cdot)$ satisfies

$d\omega=\theta\wedge\omega,\quad d\theta=0,$ (1.1)

for a certain closed 1-form $\theta$ called the Lee form.

Definition (1.1) is equivalent to the existence of a covering $\tilde{M}$
endowed with a Kähler metric $\Omega$ which is acted on by the deck group
$\operatorname{Aut}_{M}(\tilde{M})$ by homotheties. Let

$\chi:\operatorname{Aut}_{M}(\tilde{M})\longrightarrow{\mathbb{R}}^{>0},\quad\chi(\gamma)=\frac{\tau^{*}\Omega}{\Omega},$
(1.2)

be the group homomorphism which associates to a homothety its scale factor.

For definitions and examples, see [DO] and our more recent papers.

Three subclasses of LCK manifolds will be of interest to us.

An LCK manifold $(M,\omega,\theta)$ is called Vaisman if $\nabla\theta=0$,
where $\nabla$ is the Levi-Civita connection of $g$. Note that, unlike the LCK
condition, which is conformally invariant (if $g$ is LCK, then any $e^{f}\cdot
g$ is still LCK), the Vaisman condition is not. The main example of Vaisman
manifold is the diagonal Hopf manifold ([OV8]). The Vaisman compact complex
surfaces are classified in [Be].

An LCK manifold is called with potential if it admits a Kähler covering on
which the Kähler metric has global, positive and proper potential function
which is acted on by homotheties by the deck group. Among the examples:
Vaisman manifolds, but also non-Vaisman ones, such as the non-diagonal Hopf
manifolds, [OV5].

The notion of pluricanonical LCK manifold, motivated by the theory of harmonic
maps, was introduced by G. Kokarev in [Kok]. An LCK manifold
$(M,\omega,\theta)$ is called pluricanonical if the symmetric form
$\nabla\theta$ is of type (2,0) plus (0,2), that is, $(\nabla\theta)^{1,1}=0$.
Obviously, all Vaisman manifolds are pluricanonical. In [OV6] we mentioned
(without a proof) that compact pluricanonical manifolds are the same as LCK
with potential. This claim is false: all compact pluricanonical manifolds are
LCK with potential, but not all LCK manifolds with potential admit a
pluricanonical metric.

This paper grew out from our effort to correct the above error (see Section 6)
and to clarify the definitions and relations among these three subclasses of
LCK manifolds.

In fact, as our main result here, we show that on compact complex manifolds,
pluricanonical metrics are Vaisman. To this end, we prove (1) that compact
pluricanonical manifolds admit LCK metrics with potential (generally different
from the pluricanonical one), then we prove (2) that all compact LCK manifolds
with potential do contain a finite quotient of a linear Hopf surface which, if
the ambient manifold is not Vaisman, admits no Vaisman metric, and finally, we
prove that (3) compact pluricanonical surfaces should be Vaisman, yielding a
contradiction.

So, technically, what we proved is that on a compact Hermitian manifold, the
equations $d\omega=\theta\wedge\omega$, $d\theta=0$ and
$(\nabla\theta)^{1,1}=0$ imply $\nabla\theta=0$. We don’t know whether this
could be proven directly, locally, using tensor computations. Our approach
passed through LCK manifolds with potential and, a posteriori, clarified
further their geometry.

###  1.2 LCK manifolds with potential

“LCK manifolds with potential” can be defined as LCK manifolds
$(M,\omega,\theta)$ equipped with a smooth function
$\psi\in\mathcal{C}^{\infty}(M)$,

$\omega=d_{\theta}d^{c}_{\theta}\psi,$ (1.3)

where $d_{\theta}(x)=dx-\theta\wedge x$, $d_{\theta}^{c}=Id_{\theta}I^{-1}$,
and the following properties are satisfied:

(i) $\psi>0$; (ii) the class $[\theta]\in H^{1}(M,{\mathbb{R}})$ is
proportional to a rational one. (1.4)

For more details and historical context of this definition, please see
Subsection 2.1. The differential $d_{\theta}$ is identified with the de Rham
differential with coefficients in a flat line bundle $L$ called the weight
bundle. In this context, $\psi$ should be considered as a section of $L$.
After passing to the smallest covering
$\tilde{M}\stackrel{{\scriptstyle\pi}}{{{\>\longrightarrow\>}}}M$ where
$\theta$ becomes exact, the pull-back bundle $\pi^{*}L$ can be trivialized by
a parallel section. Then the equation (1.4) becomes
$\tilde{\omega}=dd^{c}\tilde{\psi}$, where $\tilde{\omega}$ is a Kähler form
on $\tilde{M}$, and $\tilde{\psi}$ the Kähler potential.

Since $\tilde{M}\stackrel{{\scriptstyle\pi}}{{{\>\longrightarrow\>}}}M$ is the
smallest covering where $\theta$ becomes exact, its monodromy is equal to
${\mathbb{Z}}^{k}$, where $k$ is the rank of the smallest rational subspace
$V\subset H^{1}(M,{\mathbb{R}})$ such that $V\otimes_{\mathbb{Q}}{\mathbb{R}}$
contain $[\theta]$. In particular, the condition (1.4) (ii) means precisely
that $\tilde{M}\stackrel{{\scriptstyle\pi}}{{{\>\longrightarrow\>}}}M$ is a
${\mathbb{Z}}$-covering. This implies that the definition (1.3)-(1.4) is
equivalent to the historical one (2.1).

In 2.1, we prove that the condition (1.4) (i) is in fact unnecessary: it
automatically follows from (1.3).

However, the condition (1.4) (ii) is more complicated: there are examples of
LCK manifolds satisfying (1.3) and not (1.4) (ii) (Subsection 2.4). Still, any
complex manifold admitting an LCK metric $(M,\omega,\theta)$ with potential
$\psi$ satisfying (1.3), admits an LCK metric satisfying (1.3)-(1.4) in any
$\mathcal{C}^{\infty}$-neighbourhood of $(\omega,\theta)$. Therefore the
conditions (1.4) are not restrictive, and for most applications, unnecessary.

It makes sense to modify the notion of LCK manifold with potential to include
the following notion (Subsection 2.4):

Definition 1.1:  Let $(M,\omega,\theta)$ be an LCK manifold, and
$\psi\in\mathcal{C}^{\infty}(M)$ a function satisfying
$d_{\theta}d^{c}_{\theta}\psi=\omega$. Denote by $k$ the rank of the smallest
rational subspace $V\subset H^{1}(M,{\mathbb{Q}})$ such that
$V\otimes_{\mathbb{Q}}{\mathbb{R}}$ contain $[\theta]$. Then $\psi$ is called
proper potential if $k=1$ and improper potential if $k>1$.

Remark 1.2: By 2.2, the condition $k=1$ is equal to the Kähler potential
$\tilde{\psi}\in\mathcal{C}^{\infty}(\tilde{M})$ being proper in the usual
sense (that is, having compact level sets). This explains the term.

###  1.3 Pluricanonical versus Vaisman: scheme of the proof

The main result of this paper is the following theorem.

Theorem 1.3:  Let $(M,\omega,\theta)$ be a compact pluricanonical LCK
manifold. Then it is Vaisman.

Proof: 5.3.

Here we survey its proof and explain its key points.

We start from an observation which can be obtained by a straightforward
tensorial calculation. Any pluricanonical manifold $(M,\omega,\theta)$
satisfies

$d\theta^{c}=\theta\wedge\theta^{c}-|\theta|^{2}\omega,$

and, moreover, $|\theta|=\operatorname{\text{\sf const}}$, (see (3.3)). We
then rescale the metric such that $|\theta|=1$. Then the eigenvalues111The
eigenvalues of a Hermitian form $\eta$ are the eigenvalues of the symmetric
operator $L_{\eta}$ defined by the equation $\eta(x,Iy)=g(Lx,y)$. of the
(1,1)-form $d\theta^{c}$ are constant, all equal to $1$ but one which is $0$,
and hence $\omega_{0}:=-d\theta^{c}$ is a semipositive $(1,1)$-form of
constant rank $n-1$.

From now on, we assume that $M$ is a compact pluricanonical LCK manifold.

Let $\Sigma$ be the zero eigenbundle of $d\theta^{c}$. Then $\Sigma$ is
independent from the choice of the pluricanonical metric on $M$. Indeed,
suppose that two different pluricanonical metrics give exact semipositive
forms $\omega_{0}$ and $\omega_{0}^{\prime}$. Unless the corresponding zero
eigenbundles coincide, the semi-Hermitian form
$\omega_{0}+\omega_{0}^{\prime}$ would be strictly positive, which is
impossible, because it is exact.

Now consider two pluricanonical LCK metrics $(\omega,\theta)$ and
$(\omega^{\prime},\theta^{\prime})$ on $M$, with
$\theta-\theta^{\prime}=d\psi$. Then $\psi$ is a solution of the degenerate
Monge-Ampère equation

$(\omega_{0}+dd^{c}\psi)^{n}=0,\ \ \omega_{0}+dd^{c}\psi\geqslant 0.$ (1.5)

The standard argument from the theory of Monge-Ampère equations is used to
show that the function $\psi$ is constant on the leaves of $\Sigma$ (3.2).

If, in addition, $(M,\omega,\theta)$ admits a Vaisman structure, one has a
holomorphic vector field $X\in\mathcal{X}(M)$ tangent to $\Sigma$ and acting
by homotheties on its Kähler covering. Then 3.2 implies that $X$ acts on
$(M,\omega,\theta)$ by isometries lifting to non-trivial homotheties on its
Kähler covering. A theorem of Kamishima-Ornea ([KO]) then implies that
$(M,\omega,\theta)$ is Vaisman (see 5.2). This result can be stated as
follows:

Proposition 1.4:  (5.2)
Let $(M,\omega,\theta)$ be a compact pluricanonical LCK manifold admitting a
Vaisman structure. Then $(M,\omega,\theta)$ is also Vaisman.

This proposition is used to deduce our main result (1.3) as follows. First, we
prove 1.3 for surfaces. In this case, it follows from the classification
theorem due to Chiose, Toma and Brunella.

Theorem 1.5:  Let $M$ be a compact complex surface admitting an exact
semipositive form of rank 1. Then $M$ admits a Vaisman metric.

Proof: Section 4.  .

Comparing 1.3 and 1.3, we obtain 1.3 for surfaces.

To prove it in general situation, a version of a theorem of Ma. Kato is used.
In [Kat], Kato studied subvarieties in a general Hopf manifold $H$, which is
defined as a quotient of ${\mathbb{C}}^{n}\backslash 0$ by a holomorphic
contraction $A$. Note that elsewhere in this paper, we consider only linear
Hopf manifolds, for which $A$ is a linear endomorphism. The (more general)
manifolds considered by Kato are also embeddable into linear Hopf manifolds
([OV5]), hence admit an LCK metric with potential ([OV8]).

Kato proves that $H$ admits a sequence of Hopf submanifolds $H\supset
H_{1}\supset H_{2}\supset\cdots\supset H_{n}=\emptyset$ with $\dim H_{i}=\dim
H-i$.

For LCK manifolds with potential, a similar flag exists. Moreover, the
following useful result can be proven.

Theorem 1.6:  Let $M$ be an LCK manifold with potential. Then $M$ contains a
finite quotient of a linear Hopf surface $H$. Moreover, if $M$ admits no
Vaisman metric, then $H$ can be chosen to admit no Vaisman metric.

Proof: 4.3.

1.3 and 1.3 imply our main theorem easily. Indeed, any complex submanifold of
a pluricanonical manifold is again pluricanonical (5.2). Pluricanonical
surfaces are Vaisman. 1.3 implies that any non-Vaisman pluricanonical manifold
contains a surface which is not Vaisman, hence not pluricanonical:
contradiction!

###  1.4 Some errors found

This paper is much influenced by Paul Gauduchon, who discovered an error in
our result mentioned as obvious in [OV6]. In [OV6], we claimed erroneously
that an LCK metric is pluricanonical if and only if it admits an LCK
potential. This was obvious because (as we claimed) the equations for LCK with
potential and for pluricanonical metric are the same. Unfortunately, a scalar
multiplier was missing in our equation for the pluricanonical (see Subsection
3.2).

From an attempt to understand what is brought by the missing multiplier, this
paper grew, and we found an even stronger result: any compact pluricanonical
manifold is Vaisman.

However, during our work trying to plug a seemingly harmless mistake, we
discovered a much more offensive error, which has proliferated in a number of
our papers.

In [OV1], we claimed that any Vaisman manifold admits a
${\mathbb{Z}}$-covering which is Kähler. This is true for locally conformally
hyperkähler manifolds, as shown in [Ve1]. However, this result is false for
more general Vaisman manifolds, such as a Kodaira surface (6.2).

It is easiest to state this problem and its solution using the notion of “LCK
rank” (2.2), defined in [GOPP] and studied in [PV]. Briefly, LCK rank is the
smallest $r$ such that there exists a ${\mathbb{Z}}^{r}$-covering $\tilde{M}$
of $M$ such that the pullback of the LCK metric is conformally equivalent to a
Kähler metric on $\tilde{M}$.

It turns on that the Kähler rank of a Vaisman manifold can be any number
between 1 and $b_{1}(M)$ (6.2). Moreover, for each $r$, the set of all Vaisman
metrics of Kähler rank $r$ is dense in the space of all Vaisman metrics (say,
with $\mathcal{C}^{\infty}$-topology).

It is disappointing to us (and even somewhat alarming) that nobody has
discovered this important error earlier.

However, not much is lost, because the metrics which satisfy the Structure
Theorem of [OV1] are dense in the space of all LCK metrics, hence all results
of complex analytic nature remain true. To make the remaining ones correct, we
need to add “Vaisman manifold of LCK rank 1” or “Vaisman manifold with proper
potential” (Subsection 2.4) to the set of assumptions whenever [OV1] is used.

Still, we want to offer our apologies to the mathematical community for
managing to mislead our colleagues for such a long time.

For more details about our error and an explanation where the arguments of
[OV1] failed please see Subsection 6.2.

##  2 LCK manifolds: properness and positivity of the potential

###  2.1 LCK manifolds with potential: historical definition

When the notion of LCK manifold with potential was introduced in [OV5], we
assumed properness of the potential. Later, it was “proven” that the potential
is always proper ([OV7]). Unfortunately, the proof was false (see the Errata
to this paper, Section 6). In view of this error and other results in Section
6, it makes sense to generalize the notion of LCK manifold with potential to
include the manifolds with LCK rank $>1$. For the old notion of LCK with
potential we should attach “proper” to signify that the potential is a proper
function on the minimal Kähler covering.

Definition 2.1:  ([OV5]) An LCK manifold with proper potential is a manifold
which admits a Kähler covering $(\tilde{M},\tilde{\omega})$ and a smooth
function $\varphi:\,\tilde{M}\rightarrow\mathbb{R}^{>0}$ (the LCK potential)
satisfying the following conditions:

(i)

$\varphi$ is proper, _i.e._ its level sets are compact;

(ii)

The deck transform group acts on $\varphi$ by multiplication with the
character $\chi$ (see (1.2)): $\tau^{*}(\varphi)=\chi(\tau)\varphi$, where
$\tau\in\operatorname{Aut}_{M}(\tilde{M})$ is any deck transform map111In
general, differential forms $\eta\in\Lambda^{\bullet}\tilde{M}$ which satisfy
$\tau^{*}\eta=\chi(\tau)\eta$ are called automorphic. In particular, so is the
Kähler form on $\tilde{M}$..

(iii)

$\varphi$ is a Kähler potential, _i.e._ $dd^{c}\varphi=\tilde{\omega}$.

We are now able to show that an automorphic global potential is always
positive, and that once an automorphic global potential exists on a Kähler
covering, then another one, which is proper, exists too. The precise statement
is the following:

Theorem 2.2:  Let $M$ be an LCK manifold admitting a Kähler covering
$(\tilde{M},\tilde{\omega})$ and an automorphic function
$\varphi:\,\tilde{M}\rightarrow\mathbb{R}$ satisfying
$dd^{c}\varphi=\tilde{\omega}$. Then $\varphi$ is strictly positive.

Moreover, $M$ admits a covering, possibly different from $\tilde{M}$, and an
automorphic potential on it which is positive and proper, hence satisfies all
conditions of 2.1.

We prove 2.1 in the next two subsections.

###  2.2 Properness of the LCK potential

In [OV5], it was also shown that the properness condition is equivalent to the
following condition on the deck transform group of $\tilde{M}$. Recall that a
group is virtually cyclic if it contains ${\mathbb{Z}}$ as a finite index
subgroup. The following claim is clear.

Claim 2.3:  Let $M$ be a compact manifold, $\tilde{M}$ a covering, and
$\varphi:\,\tilde{M}\rightarrow{\mathbb{R}}^{>0}$ an automorphic function,
that is, a function which satisfies $\gamma^{*}\varphi=c_{\gamma}\varphi$ for
any deck transform map $\gamma$, where $c_{\gamma}$ is constant. Then
$\varphi$ is proper if and only if the deck transform group of $\tilde{M}$ is
virtually cyclic.

Definition 2.4:  Let $(M,\omega,\theta)$ be an LCK manifold. Define the LCK
rank as the dimension of the smallest rational subspace $V\subset
H^{1}(M,{\mathbb{Q}})$ such that the Lee class $[\theta]$ lies in
$V\otimes_{\mathbb{Q}}{\mathbb{R}}$.

Remark 2.5: The character
$\chi:\;\operatorname{Aut}_{M}(\tilde{M}){\>\longrightarrow\>}{\mathbb{R}}^{>0}$
can be defined on any LCK manifold, because the Kähler form $\tilde{\omega}$
is automorphic by definition:
$\tau^{*}(\tilde{\omega})=\chi(\tau)\tilde{\omega}$. Then one can see that the
LCK rank as defined above coincides with the rank of the image of
$\chi:\;\operatorname{Aut}_{M}(\tilde{M}){\>\longrightarrow\>}{\mathbb{R}}^{>0}$
which is also called the weight monodromy group of the LCK manifold. See also
[GOPP] for another interpretation of the LCK rank and see [PV] for examples on
non-Vaisman compact LCK manifolds with Kähler rank greater than 1. Clearly,
LCK rank 0 corresponds to globally conformally Kähler structures.

From 2.2 below it follows that condition (i) is equivalent to $M$ being of LCK
rank 1.

In [OV4], we managed to get rid of the need to take the covering in 2.1, by
using the Morse-Novikov (twisted) differential
$d_{\theta}:=d-\theta\wedge\cdot$, where $\theta\wedge\cdot(x)=\theta\wedge
x$, and $\theta$ is the Lee form. In [OV4] the definition of LCK manifold with
potential was restated equivalently as follows.

Definition 2.6:  Let $(M,\omega,\theta)$ be an LCK manifold of LCK rank 1.
Then $M$ is called LCK manifold with potential if there exists a positive
function $\varphi_{0}\in\mathcal{C}^{\infty}(M)$ satisfying
$d_{\theta}d^{c}_{\theta}(\varphi_{0})=\omega$, where
$d^{c}_{\theta}=Id_{\theta}I^{-1}$.

Claim 2.7:  2.1 is equivalent to 2.2.

Proof: To see that 2.1 and 2.2 are equivalent, consider the smallest covering
$\pi:\;\tilde{M}{\>\longrightarrow\>}M$ such that $\pi^{*}\theta$ is exact,
and take a function $\psi$ satisfying $d\psi=\pi^{*}\theta$. Since
$\pi^{*}\theta$ is invariant under the deck transform group $\Gamma$, for each
$\gamma\in\Gamma$ one has $\gamma^{*}\psi=\psi+c(\gamma)$, where $c(\gamma)$
is a constant. Consider the multiplicative character
$\chi:\;\Gamma{\>\longrightarrow\>}{\mathbb{R}}^{>0}$ given by
$\chi(\gamma)=e^{c(\gamma)}$. Let $\Lambda_{\chi}^{*}(M)$ denote the space of
automorphic forms on $\tilde{M}$ which satisfy
$\gamma^{*}\eta=\chi(\gamma)\eta$. The map
$\Lambda^{*}(M)\stackrel{{\scriptstyle\Psi}}{{{\>\longrightarrow\>}}}\Lambda_{\chi}^{*}(M)$
mapping $\eta$ to $e^{-\psi}\pi^{*}\eta$ makes the following diagram
commutative:

$\begin{CD}\Lambda^{*}(M)@>{\Psi}>{}>\Lambda_{\chi}^{*}(M)\\\
@V{{d_{\theta}}}V{}V@V{{d}}V{}V\\\
\Lambda^{*}(M)@>{\Psi}>{}>\Lambda_{\chi}^{*}(M)\end{CD}$

Then $\Psi$ maps a “potential” $\varphi_{0}$ in the sense of 2.2 to a
potential $\psi$ in the sense of 2.1 and vice versa. Properness of
$\Psi(\varphi_{0})$ is equivalent to $\Gamma$ being virtually cyclic, as 2.2
implies. The existence of a Kähler covering with virtually cyclic deck
transform group is clearly equivalent to $M$ having LCK rank 1.

We now show that automorphic potentials can be approximated by proper ones.
The following argument is taken from [OV4].

Claim 2.8:  Let $(M,\omega,\theta)$ be an LCK manifold, and
$\varphi\in\mathcal{C}^{\infty}(M)$ a function satisfying
$d_{\theta}d^{c}_{\theta}(\varphi)=\omega$. Then $M$ admits an LCK structure
$(\omega^{\prime},\theta^{\prime})$ of LCK rank 1, approximating
$(\omega,\theta)$ in $\mathcal{C}^{\infty}$-topology.

Proof: Replace $\theta$ by a form $\theta^{\prime}$ with rational cohomology
class $[\theta^{\prime}]$ in a sufficiently small
$\mathcal{C}^{\infty}$-neighbourhood of $\theta$, and let
$\omega^{\prime}:=d_{\theta^{\prime}}d_{\theta^{\prime}}^{c}(\varphi)$. Then
$\omega^{\prime}$ approximates $\omega$ in $\mathcal{C}^{\infty}$-topology,
hence for $\theta^{\prime}$ sufficiently close to $\theta$, the form
$\omega^{\prime}$ is positive. It is $d_{\theta^{\prime}}$-closed, because
$d_{\theta^{\prime}}^{2}=0$, hence
$0=d_{\theta^{\prime}}\omega^{\prime}=d\omega^{\prime}-\theta^{\prime}\wedge\omega^{\prime}$.
This implies that $(\omega^{\prime},\theta^{\prime})$ is an LCK structure. The
Kähler rank of an LCK manifold is the dimension of the smallest rational
subspace $W\subset H^{1}(M,{\mathbb{Q}})$ such that
$W\otimes_{\mathbb{Q}}{\mathbb{R}}$ contains the cohomology class of the Lee
form. Since $[\theta^{\prime}]$ is rational,
$(M,\omega^{\prime},\theta^{\prime})$ has LCK rank 1.

###  2.3 Positivity of automorphic potentials

Finally, we prove that automorphic potentials are necessarily positive:

Proposition 2.9:  Let $(M,\omega,\theta)$ be a compact LCK manifold,
$\dim_{\mathbb{C}}M>2$, and $\varphi\in\mathcal{C}^{\infty}(M)$ a function
satisfying $d_{\theta}d^{c}_{\theta}(\varphi)=\omega$. Then $\varphi$ is
strictly positive.

Proof: From 2.2 it is clear that it suffices to prove 2.3 assuming that
$(M,\omega,\theta)$ has LCK rank 1.

In this situation, the deck transform group of the smallest Kähler covering
$(\tilde{M},\tilde{\omega})$ is ${\mathbb{Z}}$ and, therefore, the fundamental
domains of the covering are compact. Denote by $\psi$ the automorphic Kähler
potential of $(\tilde{M},\tilde{\omega})$ (2.2).

Assume that the generator of $\Gamma$ acts on $\psi$ by multiplication with a
constant $c>1$. Then the covering $\tilde{M}$ can be written as

$\tilde{M}=\bigcup_{x>0}\psi^{-1}([x,cx])\cup\psi^{-1}(0)\cup\bigcup_{x<0}\psi^{-1}([cx,x]).$

Since $M$ has LCK rank 1, $\psi$ descends to a continuous map from $M$ to a
circle, and therefore it is proper. Therefore, $\psi^{-1}(0)$ is a compact set
on which ${\mathbb{Z}}$ cannot act freely. We conclude that
$\psi^{-1}(0)=\emptyset$. But then $\tilde{M}$ would be disconnected unless
$\bigcup_{x>0}\psi^{-1}([x,cx])$ or $\bigcup_{x<0}\psi^{-1}([cx,x])$ is empty,
and hence $\psi$ is strictly positive or strictly negative.

To show that the potential $\psi$ cannot be strictly negative, we argue by
contradiction.

Step 1. Consider the level set $S_{t}:=\psi^{-1}(t)$, $t<0$. Since $\psi$ is
plurisubharmonic, $S_{t}$ is strictly pseudoconvex. Applying the Rossi-
Andreotti-Siu theorem (cf. [R, Theorem 3, p. 245] and [AS, Proposition 3.2])
we find that $S_{t}$ is the boundary of a compact Stein domain $V_{t}$ with
boundary, uniquely determined by the CR-structure on $S_{t}$.

Step 2. Let $\tilde{M}_{[t,t^{\prime}]}$ be $V_{t}$ glued together with
$\psi^{-1}([t,t^{\prime}])$ over $S_{t}$. Since the boundary of a compact
complex manifold $\tilde{M}_{[t,t^{\prime}]}$ is strictly pseudoconvex, this
space is holomorphically convex. On the other hand, it contains no non-trivial
compact complex subvarieties without boundary. Indeed, for such a subvariety
$Z$, the restriction of $\psi$ to $Z\cap\psi^{-1}([t,t^{\prime}])$ must
achieve a maximum, unless $Z\cap\psi^{-1}([t,t^{\prime}])$ is empty. However,
a plurisubharmonic function cannot achieve a maximum on a complex subvariety.
Therefore, $Z\cap\psi^{-1}([t,t^{\prime}])=\emptyset$, and $Z$ is contained in
$V_{t}$. This is impossible, because $V_{t}$ is Stein. By Remmert’s theorem, a
holomorphically convex manifold without compact complex subvarieties is Stein.
Then the union $\tilde{M}_{t}:=\bigcup_{t^{\prime}}\tilde{M}_{[t,t^{\prime}]}$
is Stein too, as a union of an increasing family of Stein varieties.

Since $S_{t}$ is contained in $\psi^{-1}([t,0[)$ for all $t^{\prime}\in[t,0[$,
it is the boundary of a Stein subvariety within $\tilde{M}_{t}$. Therefore,
$\tilde{M}_{t}$ can be obtained as $V_{t^{\prime}}$ glued together with
$\psi^{-1}([t^{\prime},0[)$. We have identified $\tilde{M}_{t}$ and
$\tilde{M}_{t^{\prime}}$. This implies, in particular, that $\tilde{M}_{t}$
contains $\bigcup_{t}\psi^{-1}([t,0[)=\tilde{M}$, and the mapping class group
of $\tilde{M}$ acts on $\tilde{M}_{t}$ holomorphically.

We extend $\psi$ to $\tilde{M}_{t}$ by setting it to $-\infty$ on
$\tilde{M}_{t}\backslash\tilde{M}$. This gives an automorphic plurisubharmonic
function $\psi$ on $\tilde{M}_{t}$.

We have included $\tilde{M}$ into a Stein manifold $\tilde{M}_{t}$, with the
same monodromy action, and extended $\psi$ to an automorphic plurisubharmonic
function on $\tilde{M}_{t}$.

Step 3. Since $\tilde{M}_{t}$ is Stein, there exists a positive, smooth,
plurisubharmonic function $\varphi$ on $\tilde{M}_{t}$ Since $\tilde{M}_{t}$
is properly embedded to a bigger Stein domain, this function can be assumed to
be bounded. We apply the standard technique for constructing regularized
maxima of plurisubharmonic functions ([D2]; see also [OV2, Proposition 4.2]):
for a very large $C>\\!\\!>0$, define

$\xi:=\max{}_{\varepsilon}(\psi,\tilde{\varphi}-C-\sup\tilde{\varphi})$

thus obtaining a smooth plurisubharmonic function on $\tilde{M}_{t}$.
Moreover, $\xi=\psi$ on an arbitrary big neighbourhood of the boundary
including the level sets $S_{t}=\psi^{-1}(t)$ and $S_{ct}=\psi^{-1}(ct)$.

Now, $V_{t}$ can be written as $V_{t}=\psi^{-1}(]-\infty,t[)$. Then $V_{t}$ is
a Stein subset of $\tilde{M}_{t}$ with boundary $S_{t}$. By Stokes theorem, we
obtain:

$\operatorname{Vol}_{\xi}(V_{t}):=\int_{V_{t}}(dd^{c}\xi)^{n}=\int_{\partial
V_{t}}d^{c}\xi\wedge(dd^{c}\xi)^{n-1}=\int_{\partial
V_{t}}d^{c}\psi\wedge(dd^{c}\psi)^{n-1}$ (2.1)

Since the monodromy map $\tau$ maps $V_{t}$ to $V_{ct}$ multiplying $\psi$ by
$\lambda$, (2.1) gives
$\operatorname{Vol}_{\xi}(V_{t})=\lambda^{n}\operatorname{Vol}_{\xi}(V_{ct})$,
with $\lambda,c>1$. This is impossible, because $t<0$ and $V_{ct}$ is strictly
included in $V_{t}$.

###  2.4 LCK manifolds with proper and improper potential

It seems now that the equation $d_{\theta}d^{c}_{\theta}\psi=\omega$ is more
fundamental than the notion of LCK manifold with (proper) potential. For most
applications, this (more general) condition is already sufficient.

The relation between manifolds with $d_{\theta}d^{c}_{\theta}\psi=\omega$ and
LCK with potential is similar to the relation between general Vaisman
manifolds and quasiregular ones222(Quasi-)Regularity and irregularity of a
Vaisman manifold refers to the (quasi-)regularity and irregularity of the
2-dimensional canonical foliation generated by $\theta^{\sharp}$ and
$I\theta^{\sharp}$.. One could always deform an irregular Vaisman manifold to
a quasiregular one, and quasiregular Vaisman manifolds are dense in the space
of all Vaisman manifolds.

The notion of “LCK manifold with improper potential” is similar, in this
regard, to the notion of irregular Vaisman or irregular Sasakian
manifold333Here (ir)regularity refers to the 1-dimensional foliation generated
by the Reeb field., [BG].

Definition 2.10: Let $(M,\theta,\omega)$ be an LCK manifold, and $\psi$ a
function (positive by 2.3) which satisfies
$d_{\theta}d^{c}_{\theta}\psi=\omega$. Then $(M,\theta,\omega)$ is called a
manifold with improper LCK potential if its LCK rank is $\geqslant 2$, and a
manifold with proper LCK potential if it has LCK rank 1.

Remark 2.11: The expressions “proper potential” and “improper potential”, when
used for solutions of the equation $d_{\theta}d^{c}_{\theta}\psi=\omega$, as
in the above Definition, do not refer to the properness of
$\psi:M\longrightarrow{\mathbb{R}}$, which is always proper if $M$ is compact.

Note that “LCK with potential” was previously used instead of “manifold with
proper LCK potential”; now (in light of the discovery of Vaisman manifolds
having improper potential, see Section 6) it makes sense to change the
terminology by including improper potentials in the definition of LCK with
potential.

2.2 can be rephrased as follows.

Proposition 2.12: Let $(M,\omega,\theta,\psi)$ be a compact LCK manifold with
improper LCK potential. Then $(\omega,\theta,\psi)$ can be approximated in the
$\mathcal{C}^{\infty}$-topology by an LCK structure with proper LCK potential.

Remark 2.13: We have just proven that existence of an LCK metric with improper
LCK potential implies existence of a metric with proper LCK potential on the
same manifold. The converse is clearly false: when $H^{1}(M,{\mathbb{R}})$ is
1-dimensional, any Lee class is proportional to an integral cohomology class,
and any LCK structure has LCK rank 1, hence $M$ admits no metrics with
improper LCK potentials.

However, in all other situations improper potentials do exist.

Proposition 2.14:  Let $(M,\omega,\theta,\psi)$ be an LCK manifold with
potential, and suppose $b_{1}(M)>1$. Then $M$ admits an LCK metric
$(M,\omega^{\prime},\theta^{\prime},\psi)$ with improper potential and
arbitrary LCK rank between 2 and $b_{1}(M)$. Moreover,
$(\omega^{\prime},\theta^{\prime})$ can be chosen in arbitrary
$\mathcal{C}^{\infty}$-neighbourhood of $(\omega,\theta)$.

Proof: Choose a closed $\theta^{\prime}$ in a sufficiently small neighbourhood
of $\theta$, and let $V_{\theta}$ be the smallest rational subspace of
$H^{1}(M,{\mathbb{R}})$ such that $V_{\theta}\otimes_{\mathbb{Q}}{\mathbb{R}}$
contains $\theta$. Since the choice of the cohomology class
$[\theta^{\prime}]$ is arbitrary in a neighbourhood of $[\theta]$, the
dimension of $V_{\theta}$ can be chosen in arbitrary way. Choosing
$\theta^{\prime}$ sufficiently close to $\theta$, we can assume that the
(1,1)-form $\omega^{\prime}:=d_{\theta^{\prime}}d^{c}_{\theta^{\prime}}(\psi)$
is positive definite. Then $(M,\omega^{\prime},\theta^{\prime},\psi)$ is an
LCK manifold with improper potential and arbitrary LCK rank.

##  3 Pluricanonical condition versus LCK with potential

###  3.1 Exterior derivative of the Lee form: preliminary computations

For a one-form $\eta$, we set $\eta^{c}:=I\eta$ and
$I\eta(\cdot)=\eta(I\cdot)$. On an LCK manifold, we then have
$\theta^{\sharp}\hskip 2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip
2.0pt\omega=\theta^{c}$, where $\theta^{\sharp}$ is the dual vector field of
$\theta$.

Remark 3.1: Note the difference of sign with respect to other authors’
conventions who put $\omega(\cdot,\cdot)=g(I\cdot,\cdot)$ and not
$\omega(\cdot,\cdot)=g(\cdot,I\cdot)$, and define
$I\eta(\cdot)=-\eta(I\cdot)$.

As $\omega$ is nondegenerate, similarly to the Kähler case, one has:

Lemma 3.2:  On an LCK manifold $(M,I,g)$ with $\dim_{\mathbb{C}}M\geqslant 2$,
exterior multiplication with $\omega$ is injective.

It is known that the Levi-Civita connections of the local Kähler metrics glue
to a global connection, here denoted $D$, which is almost complex ($DI=0$) and
satisfies $Dg=\theta\otimes g$ – and hence it is the Weyl connection of the
couple $(g,\theta)$.

Using the well-known relation between the Levi-Civita connections of two
conformal metrics, the LCK condition is equivalent ([DO]) with:

$(\nabla_{X}I)Y=\frac{1}{2}\left(\theta(IY)X-\theta(Y)IX+g(X,Y)I\theta^{\sharp}-\omega(X,Y)\theta^{\sharp}\right)$
(3.1)

where $\sharp$ refers to the $g$-raising of indices.

From this we can derive:

Lemma 3.3: On an LCK manifold, the exterior derivative of $\theta^{c}$ is:

$d\theta^{c}(X,Y)=\left(-|\theta|^{2}\omega+\theta\wedge\theta^{c}\right)(X,Y)+\frac{1}{2}\left((\nabla_{Y})(IX)-(\nabla_{X}\theta(IY)\right)$
(3.2)

Proof: Using (3.1) we have

$\begin{split}(\nabla_{X}\theta^{c})Y&=\nabla_{X}\theta(IY)-\theta(I\nabla_{X}Y)=(\nabla_{X}\theta)(IY)+\theta(\nabla_{X}(IY))-\theta(I\nabla_{X}Y)\\\
&=(\nabla_{X}\theta)(IY)+\theta((\nabla_{X}I)Y)\\\
&=(\nabla_{X}\theta)(IY)+\frac{1}{2}\left(\theta(IY)\theta(X)-\theta(Y)\theta(IX)-\omega(X,Y)|\theta|^{2}\right)\end{split}$

Now (3.2) follows from $d\theta^{c}$ being the antisymmetrization of
$\nabla\theta^{c}$.

###  3.2 Pluricanonical manifolds and $(\nabla\theta)^{1,1}=0$

Recall that a pluricanonical LCK manifold, as defined in [Kok], [KK], is an
LCK manifold $(M,\omega,\theta)$ satisfying $(\nabla\theta)^{1,1}=0$, where
$\nabla$ is the Levi-Civita connection. This is equivalent with

$(\nabla_{IX}\theta)(IY)+(\nabla_{X}\theta)(Y)=0$

which, changing $Y$ into $IY$ and using the symmetry of $\nabla\theta$ (recall
that $d\theta=0$) gives:

$(\nabla_{Y}\theta)(IX)-(\nabla_{X}\theta)(IY)=0$

Together with (3.2) this gives (see [OV6] for a different proof) that the
pluricanonical condition is equivalent to

$d\theta^{c}=\theta\wedge\theta^{c}-|\theta|^{2}\omega$ (3.3)

We now take the exterior derivative of the above (modulo itself and using
$d\omega=\theta\wedge\omega$):

$0=-\theta\wedge
d\theta^{c}-d|\theta|^{2}\wedge\omega-|\theta|^{2}\theta\wedge\omega+=-d|\theta|^{2}\wedge\omega$

which, by 3.1, implies $|\theta|=\operatorname{\text{\sf const}}.$

We may now rescale the pluricanonical metric such that, from now on we assume
$|\theta|=1$.

Corollary 3.4:  Let $(M,\omega,\theta)$ be a LCK manifold. Then it is
pluricanonical if and only if

$d\theta^{c}=\theta\wedge\theta^{c}-\omega.$ (3.4)

and $|\theta|=1$

Remark 3.5: Condition (3.4) is equivalent to $(M,\omega,\theta)$ being “LCK
with proper or improper potential” in the sense of 1.2.

Remark 3.6: One can prove that the pluricanonical condition is also equivalent
with $\operatorname{Lie}_{\theta^{\sharp}}\omega=0$ ([G2]). See also similar
computations in the recent [AD, Section 3].

In general, on an LCK manifold with potential $\psi$, the norm of the Lee form
$d\psi$ is not constant. This, however, holds if the LCK metric is Gauduchon
(see 3.2 below).

Definition 3.7: On a complex manifold of complex dimension $n$, a Hermitian
metric whose Hermitian 2-form $\omega$ satisfies the equation
$\partial\overline{\partial}\omega^{n-1}=0$ is called Gauduchon.

Remark 3.8: On a compact Hermitian manifold, a Gauduchon metric exists in each
conformal class and it is unique up to homothety. Moreover, it is
characterized by the coclosedness of its Lee form. A Vaisman metric is a
Gauduchon metric in its conformal class, [G1].

On the other hand, it was shown in [Kok] that a pluricanonical metric has
coclosed Lee form and hence, if the manifold is compact, it is a Gauduchon
metric in the given conformal class.

Proposition 3.9:  Let $(M,\omega,\theta)$ be a compact LCK manifold with a
potential $\psi$ (proper or improper; see 1.2). Then the LCK form
$\omega=\psi^{-1}dd^{c}\psi$ is Gauduchon if and only if
$|\theta|=\operatorname{\text{\sf const}}$.

Proof: The Hermitian form $\omega$ is Gauduchon if and only if
$dd^{c}\omega^{n-1}=0$.

On the other hand, we compute $dd^{c}\omega^{n-1}$ using the equation (3.4)
which is satisfied on an LCK manifold with automorphic potential. We obtain

$dd^{c}\omega^{n-1}=(n-1)^{2}\omega^{n-1}\wedge\theta\wedge\theta^{c}+(n-1)\omega^{n-1}\wedge
d\theta^{c}.$

On the other hand,

$\omega^{n-1}\wedge\theta\wedge\theta^{c}=\frac{1}{n}|\theta|^{2}\omega^{n}$

and

$d\theta^{c}\wedge\omega^{n-1}=-\omega\wedge\omega^{n-1}+\theta\wedge\theta^{c}\wedge\omega^{n-1}=\left(\frac{1}{n}|\theta|^{2}-1\right)\omega^{n}.$

All in all we get:

$dd^{c}\omega^{n-1}=\frac{(n-1)^{2}}{n}|\theta|^{2}\omega^{n}+(n-1)\left(\frac{1}{n}|\theta|^{2}-1\right)\omega^{n}=(n-1)\big{(}|\theta|^{2}-1\big{)}\omega^{n}.$

Then $dd^{c}\omega^{n-1}=0$ if and only if $|\theta|=1$. This finishes the
proof.

We obtained the following corollary.

Corollary 3.10:  Let $M$ be a compact LCK manifold. Then the following are
equivalent.

(i)

$M$ is pluricanonical.

(ii)

$M$ is LCK manifold with potential, and its LCK metric is Gauduchon.

Claim 3.11:  Let $(M,\omega,\theta)$ be a compact LCK manifold with potential.
Consider the 2-form $\omega_{0}:=d\theta^{c}=\theta\wedge\theta^{c}-\omega$.
Then $\omega_{0}$ is semipositive if and only if $(M,\omega,\theta)$ is
pluricanonical.

Proof: On a pluricanonical manifold, the form $\omega_{0}$ has one $0$
eigenvalue and all other strictly positive. This can be seen by writing it in
a diagonal basis which includes $\theta^{\sharp}$ and $I\theta^{\sharp}$ and
taking into account that $|\theta|=1$. In particular,
$\omega_{0}^{\dim_{\mathbb{C}}M}=0$. Note that $\omega_{0}$ has the same
eigenvalues on Vaisman manifolds, see [Ve1]. Conversely, if
$\omega_{0}^{n}=0$, then $\omega_{0}$ has at least one $0$ eigenvalue.

This simple observation has strong consequences and leads to a degenerate
Monge-Ampère equation that we now discuss (compare also with [OV3], where a
similar equation is considered).

Let $\theta$ be a closed 1-form such that $\omega_{0}:=d^{c}\theta$ is a
semipositive (1,1)-form, satisfying $\omega_{0}^{n}=0$. Assume that
$\theta^{\prime}=\theta+d\psi$ is another 1-form such that
$d^{c}\theta^{\prime}$ is semipositive. Then $(d^{c}\theta-dd^{c}\psi)^{n}=0$
is a degenerate Monge-Ampère equation which can be studied in the usual way.

We write

$(d\theta^{c})^{n}-(d\theta^{\prime c})^{n}=(d\theta^{c}-d\theta^{\prime
c})\wedge P,$

where $P:=\sum_{k}d^{c}\theta^{k}\wedge(d^{c}\theta^{\prime})^{n-k-1}$ is a
semipositive $(n-1,n-1)$-form. This equation gives

$dd^{c}\psi\wedge P=0,\,\,\text{and hence}\,\,(dd^{c}\psi)\psi\wedge P=0.$

Using Stokes’ theorem and integrating by parts, we obtain

$0=\int_{M}(dd^{c}\psi)\psi\wedge P=-\int_{M}d\psi\wedge d^{c}\psi\wedge P.$
(3.5)

Now recall that the exterior product of semipositive forms is semipositive. By
assumption, $d^{c}\theta$ and $d^{c}\theta^{\prime}$ (and hence their powers)
are semipositive. Then $P$ is semipositive, as a sum of semipositive forms. As
$d\psi\wedge d^{c}\psi$ is semipositive too, $\int_{M}d\psi\wedge
d^{c}\psi\wedge P=0$ implies $d\psi\wedge d^{c}\psi\wedge P=0$.

On the other hand, $d^{c}\theta$ and $d^{c}\theta^{\prime}$ have the same
kernel $\Sigma$ of (complex) dimension $1$: otherwise, their sum would be
strictly positive. But $d^{c}\theta+d^{c}\theta^{\prime}$ is exact and hence,
by Stokes theorem, its top power must be zero, contradiction. As $P$ is a
transversal volume form on $TM/\Sigma$, it follows that $P$ has the same
kernel $\Sigma$ too. This implies:

Proposition 3.12:  Let $(M,\omega,\theta)$ and
$(M,\omega^{\prime},\theta^{\prime})$ be two pluricanonical LCK structures on
$M$, and $\omega_{0}$ and $\omega^{\prime}_{0}$ the corresponding semipositive
forms (see 3.2). Then $\ker\omega_{0}=\ker\omega_{o}^{\prime}$. Moreover, if
$\theta$ is cohomologuous to $\theta^{\prime}$, one has
$\theta-\theta^{\prime}=d\psi$, where $\psi$ is a function which is constant
on the leaves of $\Sigma=\ker\omega_{0}=\ker\omega_{0}^{\prime}$.

Proof: The forms $\omega_{0}$ and $\omega^{\prime}_{0}$ are semipositive, with
one-dimensional kernel, hence unless $\ker\omega_{0}=\ker\omega_{o}^{\prime}$,
their sum is strictly positive. In the latter case, one has
$\int_{M}(\omega_{0}+\omega^{\prime}_{0})^{n}>0$, which is impossible by
Stokes’ formula, since $\omega_{0}$ and $\omega^{\prime}_{0}$ are exact.

Finally, (3.5) implies that $d\psi$ vanishes on $\Sigma$, otherwise the
semipositive, exact form $d\psi\wedge d^{c}\psi\wedge P$ would have been
strictly positive somewhere on $M$.

##  4 Compact pluricanonical surfaces are Vaisman

###  4.1 Complex surfaces of Kähler rank 1

Recall that a compact complex surface is of Kähler rank 1 if and only if it is
not Kähler but it admits a closed semipositive (1,1)-form whose zero locus is
contained in a curve ([HL]).

Lemma 4.1: A compact pluricanonical LCK surface $M$ has Kähler rank 1.

Proof: The manifold $M$ is non-Kähler, because it admits a positive, exact
form. This form, multiplied by the Kähler one, would have given us an exact
volume form, which is impossible by Stokes’ theorem. On the other hand, $M$
admits a semipositive form by 3.2.

Recall that a Hopf surface is a finite quotient of $H$, where $H$ is a
quotient of ${\mathbb{C}}^{2}\backslash 0$ by a polynomial contraction. A Hopf
surface is diagonal if this polynomial contraction is expressed by a diagonal
matrix.

Compact surfaces of Kähler rank 1 have been classified in [CT] and [Br]. They
can be:

  1. 1.

Non-Kähler elliptic fibrations,

  2. 2.

Diagonal Hopf surfaces and their blow-ups,

  3. 3.

Inoue surfaces and their blow-ups.

Inoue surfaces have $b_{1}=1$ and hence, if they admit an automorphic
potential, this has to be proper. But all compact LCK manifolds with proper
potential can be deformed to Vaisman ones, and Inoue surfaces are not
diffeomorphic to Vaisman manifolds (this result follows from a classification
of Vaisman surfaces by F. Belgun, [Be]). Thus Inoue surfaces cannot have
automorphic potential.

A cover of a blow-up of any complex manifold cannot admit plurisubharmonic
functions because, by the lifting criterion, the projective spaces contained
in the blow-up lift to the cover. Thus blow-ups cannot have global potential.

We are left with non-Kähler elliptic fibrations and diagonal Hopf surfaces
which are known to admit Vaisman metrics, see e.g. [Be]. And hence:

Proposition 4.2:  All compact pluricanonical LCK surfaces admit Vaisman
metrics.

For further use it is convenient to list all criteria used to distinguish
Vaisman Hopf surfaces from non-Vaisman ones.

Theorem 4.3:  Let $M$ be a Hopf surface. Then the following are equivalent.

(i)

$M$ is Vaisman.

(ii)

$M$ is diagonalizable.

(iii)

$M$ has Kähler rank 1.

(iv)

$M$ contains at least two distinct elliptic curves.

Proof: Equivalence of the first three conditions is proven above. The
equivalence of (iv) and (ii) is shown by Iku Nakamura and Masahide Kato ([N,
Theorem 5.2]). Note that the cited result refers to primary Hopf surfaces, but
we can always pass to a finite covering and the number of elliptic curves will
not change because the eigenvectors for rationally independent eigenvalues
cannot be mutually exchanged, and if they were dependent, they would produce
infinitely many elliptic curves.

###  4.2 Algebraic groups and the Jordan-Chevalley decomposition

In this section we let $V:={\mathbb{C}}^{n}$.

Lemma 4.4: Let $A\in GL(V)$ be a linear operator, and $\langle A\rangle$ the
group generated by $A$. Denote by $G$ the Zariski closure of $\langle
A\rangle$ in $GL(V)$. Then, for any $v\in V$, the Zariski closure $Z_{v}$ of
the orbit $\langle A\rangle\cdot v$ is equal to the usual closure of $G\cdot
v$.

Proof: Clearly, $Z_{v}$ is $G$-invariant. Indeed, its normalizer $N(Z_{v})$ in
$GL(V)$ is an algebraic group containing $\langle A\rangle$, hence $N(Z_{v})$
contains $G$. The converse is also true: since $\langle A\rangle$ normalizes
$\langle A\rangle\cdot v$, its Zariski closure $G$ normalizes the Zariski
closure $Z_{v}$ of the orbit. Therefore, the orbit $G\cdot v$ is contained in
$Z_{v}$. Since $G\cdot v$ is a constructible set, its Zariski closure
coincides with its usual closure, [H], [Kol]. This gives $\overline{G\cdot
v}\subset Z_{v}$. As $\overline{G\cdot v}$ is algebraic and contains $\langle
A\rangle\cdot v$, the inclusion $Z_{v}\subset\overline{G\cdot v}$ is also
true.

Let $G\subset GL(V)$ be an algebraic group over ${\mathbb{C}}$. Recall that an
element $g\in G$ is called semisimple if it is diagonalizable, and unipotent
if $g=e^{n}$, where $n$ is a nilpotent element of its Lie algebra.

Theorem 4.5: (Jordan-Chevalley decomposition, [H, Section 15]) For any
algebraic group $G\subset GL(V)$, any $g\in G$ can be represented as a product
of two commuting elements $g=g_{s}g_{u}$, where $g_{s}$ is semisimple, and
$g_{u}$ unipotent. Moreover, this decomposition is unique and functorial under
homomorphisms of algebraic groups.

Corollary 4.6:  Let $M$ be a submanifold of a linear Hopf manifold
$H=(V\backslash 0)/A$, $\tilde{M}\subset V\backslash 0$ its
${\mathbb{Z}}$-covering, and $G$ the Zariski closure of $\langle A\rangle$ in
$GL(V)$. Then $\tilde{M}$ contains the $G$-orbit of each point
$v\in\tilde{M}$. Moreover, $G$ is a product of $G_{s}:=({\mathbb{C}}^{*})^{k}$
and a unipotent group $G_{u}$ commuting with $G_{s}$, and bothe of these
groups preserve $\tilde{M}$.

Proof: Let $X$ be the closure of $\tilde{M}$ in ${\mathbb{C}}^{N}$. The ideal
$I_{X}$ of $X$ is generated by polynomials, as shown in [OV5, Proof of Theorem
3.3]. As the polynomial ring is Noetherian, $I_{X}$ is finitely generated,
[AM]. Therefore, $X$ is a cone of a projective variety.

This allows us to consider the smallest algebraic group $G$ containing $A$.
Then $G$ acts naturally on $X$ and preserves it. The last assertion of 4.2 is
implied by the Jordan-Chevalley decomposition.

###  4.3 Finding surfaces in LCK manifolds with potential

Lemma 4.7:  Let $M$ be a submanifold of a linear Hopf manifold $H=(V\backslash
0)/A$, $\dim M\geqslant 3$, and $G=G_{s}G_{u}$ the Zariski closure of $\langle
A\rangle$ with its Jordan-Chevalley decomposition. Then $M$ contains a surface
$M_{0}$, with $G_{u}$ acting non-trivially on its ${\mathbb{Z}}$-covering
$\tilde{M}_{0}\subset V$.

Proof: Another form of this statement is proven by Masahide Kato ([Kat]).

We shall use induction on dimension of $M$. To prove 4.3 it would suffice to
find a subvariety $M_{1}\subset M$ of codimension 1 such that $G_{u}$ acts
non-trivially on its ${\mathbb{Z}}$-covering
$\tilde{M}_{1}\subset{\mathbb{C}}^{n}\backslash 0$. Replacing $V$ by the
smallest $A$-invariant subspace containing $\tilde{M}$, we may assume that the
intersection $\tilde{M}\cap V_{1}\neq V_{1}$ for each proper subspace
$V_{1}\subset V$. Now take a codimension 1 subspace $V_{1}\subsetneq V$ which
is $A$-invariant and such that $G_{u}$ acts on $V_{1}$ non-trivially
(equivalently, such that $A$ acts on $V_{1}$ non-diagonally). Using the Jordan
decomposition of $A$, such $V_{1}$ is easy to construct. Then
$\tilde{M}_{1}^{\prime}:=V_{1}\cap\tilde{M}$ gives a subvariety of $M$ of
codimension 1 and with non-trivial action of $G_{u}$.

Lemma 4.8:  Let $M\subset H=(V\backslash 0)/\langle A\rangle$ be a surface in
a Hopf manifold, and $G=G_{s}G_{u}$ the Zariski closure of $\langle A\rangle$
with its Jordan-Chevalley decomposition. Assume that $G_{u}$ acts on the
${\mathbb{Z}}$-covering $\tilde{M}$ non-trivially. Then $M$ is non-diagonal.

Proof: Replacing $G$ by its quotient by the subgroup acting trivially on
$\tilde{M}$ if necessary, we may assume that $G$ acts properly on a general
orbit in $\tilde{M}$. Then $G$ is at most 2-dimensional. However, it cannot be
1-dimensional because $G_{s}$ contains contractions (hence cannot be
0-dimensional) and $G_{u}$ acts non-trivially. Therefore,
$G_{s}\simeq{\mathbb{C}}^{*}$ and $G_{u}\simeq{\mathbb{C}}$.

Since $G_{s}$ acts by contractions, the quotient $S:=\tilde{M}/G_{s}$ is a
compact curve, equipped with $G_{u}$-action which has a dense orbit. The group
$G_{u}$ can act non-trivially only on a genus 0 curve, and there is a unique
open orbit $O$ of $G_{u}$, with $S\backslash O$ being one point. All complex
subvarieties of $M$ are by construction $G$-invariant, and the complement of
an open orbit is an elliptic curve, hence $M$ has only one elliptic curve, and
is non-diagonalizable by 4.1.

Theorem 4.9:  A compact LCK manifold $M$ with potential which is not Vaisman
contains an embedded non-diagonal Hopf surface.

Proof: Let $M$ be a compact LCK manifold with potential,
$\dim_{\mathbb{C}}M\geqslant 3$. Then $M$ is holomorphically embedded into a
Hopf manifold ${\mathbb{C}}^{n}\backslash 0/\langle A\rangle$, where
$A\in\mathrm{GL}(N,{\mathbb{C}})$ is a linear operator, see [OV5, Theorem
3.4]. Applying 4.3 and 4.3, we find a non-diagonal Hopf surface in $M$.

##  5 Pluricanonical condition versus Vaisman

###  5.1 Bott-Chern cohomology for Vaisman manifolds

Definition 5.1: The Bott-Chern cohomology groups $H^{*}_{BC}(M)$ of a complex
manifold $M$ are $\frac{\ker d\cap\ker d^{c}}{\operatorname{im}dd^{c}}$. Since
these groups are manifestly invariant with respect to the $U(1)$-action
inducing the Hodge decomposition, one has
$H^{*}_{BC}(M)=\bigoplus_{p,q}H^{p,q}_{BC}(M)$, where

$H^{p,q}_{BC}(M)=\frac{\ker d\cap\ker
d^{c}\cap\Lambda^{p,q}(M)}{dd^{c}(\Lambda^{p-1,q-1}(M))}.$

The Bott-Chern cohomology is relevant for complex manifolds which do not
satisfy a global $dd^{c}$ lemma, in particular for LCK manifolds (see, e.g.,
the recent [AU]).

Proposition 5.2:  Let $(M,I,\omega,\theta)$ be a compact Vaisman manifold, and
let $\psi:\;H^{1,1}_{BC}(M){\>\longrightarrow\>}H^{2}(M)$ be the tautological
map. Then $\ker\psi$ is 1-dimensional and it is generated by $d^{c}\theta$.

The proof of 5.1 will occupy the remaining part of this subsection. Let
$n=\dim_{\mathbb{C}}M$. Since $d^{*}\theta=0$ (that is $\theta$ is coclosed),
one has $d(*\theta)=d(\omega^{n-1}\wedge I(\theta))=0$ (where $*$ is the Hodge
operator). Therefore, $dd^{c}(\omega^{n-1})=0$: the Vaisman metric is
Gauduchon. This is well known.

Then for any differentiable function $f$, integration by parts and Stokes’
formula give

$\int_{M}dd^{c}f\wedge\omega^{n-1}=0.$ (5.1)

Define the degree map

$\deg:\;H^{1,1}_{BC}(M){\>\longrightarrow\>}{\mathbb{C}}\,\,\,\text{by}\,\,\,[\alpha]\mapsto\int_{M}\alpha\wedge\omega^{n-1}.$

From (5.1) it follows that the degree map is well defined.

We now define a second order elliptic operator on functions by the formula

$D(f):=\frac{dd^{c}f\wedge\omega^{n-1}}{\omega^{n}}.$

Its index is zero, because its symbol is the same as for the Laplacian on
functions, and the index of the Laplacian is zero, because it is self-dual.

Note that $\ker D$ only contains constants by the Hopf maximum principle.
Therefore, $\operatorname{coker}D$ is 1-dimensional. By (5.1),
$\operatorname{im}D$ is the space of functions $g$ such that
$\int_{M}g\omega^{n}=0$. This gives the following useful lemma:

Lemma 5.3:  Let $[\alpha]\in H^{1,1}_{BC}(M)$ be a degree 0 Bott-Chern
cohomology class. Then $[\alpha]$ can be represented by a closed (1,1)-form
$\alpha$ such that $\alpha\wedge\omega^{n-1}=0$ (such (1,1)-forms are called
primitive).

Proof: Let $\alpha_{1}$ be a (1,1)-form representing $[\alpha]$. Then
$\int_{M}\alpha_{1}\wedge\omega^{n-1}=0$, and hence
$\alpha_{1}\wedge\omega^{n-1}=dd^{c}f\wedge\omega^{n-1}$ for some
$f\in\mathcal{C}^{\infty}(M)$. Then $\alpha:=\alpha_{1}-dd^{c}f$ is primitive.

Definition 5.4: Let $M$ be a Vaisman manifold, $\theta^{\sharp}$ its Lee
field, and $\Sigma\subset TM$ the subbundle generated by $\theta^{\sharp}$ and
$I(\theta^{\sharp})$. The subbundle $\Sigma\subset TM$ is a holomorphic
foliation, called the canonical, or fundamental foliation of the Vaisman
manifold $M$ (see [Va2], [DO]).

A form $\eta$ on $M$ is called transversal or basic (with respect to $\Sigma$)
if $v\hskip 2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0pt\eta=v\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0pt(d\eta)=0$ for any vector
field $v\in\Sigma$. Locally in a neighbourhood where the leaf space of
$\Sigma$ exists, transversal forms are forms on this leaf space. See [To] and
[Va2] for the definitions of basic forms and basic cohomology with respect to
$\Sigma$.

In [Va2], transversal geometry of a Vaisman manifold is explored in depth. It
is shown that the form $\omega_{0}:=-d^{c}\theta$ is transversal and defines a
Kähler structure on the leaf space in any neighbourhood where the leaf space
of $\Sigma$ exists. Such a form is called transversally Kähler. It is
straightforward to define the transversal de Rham cohomology, the Hodge
decomposition, Lefschetz $SL(2)$-action and so forth on the space of
transversal forms. It turns out that the same properties of the Hodge
decomposition (including the Lefschetz $SL(2)$-action on cohomology) are true
for the transversal cohomology of the Vaisman manifold.

For the relation between de Rham cohomology and basic cohomology of Vaisman
manifolds, we recall the following result of T. Kashiwada ([Kas], [Va2,
Theorem 4.1]):

Theorem 5.5:  On a compact Vaisman manifold of complex dimension $n$, any
(real) harmonic $p$-form $\eta$, $0\leqslant p\leqslant n-1$, has a unique
decomposition

$\eta=\alpha+\theta\wedge\beta,$ (5.2)

with $\alpha$ and $\beta$ basic, transversally primitive, harmonic and
transversally harmonic forms.

This easily implies the following:

Lemma 5.6:  The kernel $K$ of the natural map
$H^{*}_{\Sigma}(M){\>\longrightarrow\>}H^{*}(M)$ from the basic cohomology to
the de Rham cohomology is generated (multiplicatively) by $\omega_{0}$.

Proof: Any class in $K$ can be represented by a form $\alpha$ which is
transversally harmonic and exact. Then the primitive part of $\alpha$ vanishes
by 5.1.

Lemma 5.7:  Let $\eta\in\Lambda^{1,1}(M)$ be a primitive, closed (1,1)-form on
a compact Vaisman manifold. Then $\eta$ is transversal.

Proof: The lemma is essentially [Ve2, Proposition 4.2], and its proof is
entirely similar to [Ve2, Proposition 4.2]. Choose an orthonormal basis
$z_{i}$ in $\Lambda^{1,0}(M)$ in such a way that:

$\omega=-\sqrt{-1}\>\sum_{i=0}^{n-1}z_{i}\wedge\overline{z}_{i},\quad\text{
and}\quad\omega_{0}=-\sqrt{-1}\>\sum_{i=1}^{n-1}z_{i}\wedge\overline{z}_{i}.$

Clearly $z_{0}=\theta$. It suffices to prove 5.1 for real (1,1)-forms, hence
we may write

$\sqrt{-1}\>\eta=\sum_{i}a_{i}z_{i}\wedge\overline{z}_{i}+\sum_{i\neq
j}b_{ij}z_{i}\wedge\overline{z}_{j},\quad\text{where}\,\,\,a_{i}\in{\mathbb{R}},\,\,\,b_{ij}=\overline{b}_{ji}.$

Since $\eta$ is exact, one has
$\int_{M}\eta\wedge\eta\wedge\omega_{0}^{n-2}=0$. However, at each point

$\frac{\eta\wedge\eta\wedge\omega_{0}^{n-2}}{\omega^{n}}=a_{0}\sum_{i=1}^{n-1}a_{i}-\sum
b_{0i}b_{i0}=a_{0}\sum_{i=1}^{n-1}a_{i}-\sum|b_{0i}|^{2}$ (5.3)

Since $\eta$ is primitive, $\sum_{i=0}^{n-1}a_{i}=0$, and hence
$a_{0}\sum_{i=1}^{n-1}a_{i}=-a_{0}^{2}$. Then (5.3) becomes

$\frac{\eta\wedge\eta\wedge\omega_{0}^{n-2}}{\omega^{n}}=-a_{0}^{2}-\sum|b_{0i}|^{2}\leqslant
0,$

with equality reached only when $a_{0}=0$ and $b_{0i}=0$ for all $i$. However,
the form $\eta\wedge\eta\wedge\omega_{0}^{n-2}$ is exact, hence the integral
$\eta\wedge\eta\wedge\omega_{0}^{n-2}$ cannot be positive, giving
$\eta\wedge\eta\wedge\omega_{0}^{n-2}=0$ and $a_{0}=b_{0i}=0$. The latter
equality is precisely the transversality of $\eta$.

Comparing 5.1 and 5.1, we obtain the following:

Corollary 5.8:  Let $[\alpha]\in H^{1,1}_{BC}(M)$ be a Bott-Chern class on a
compact Vaisman manifold. Then $[\alpha]$ can be represented by a transversal
form.

Proof: Since the form $\omega_{0}$ is positive, it has positive degree. Then
$[\alpha_{1}]:=[\alpha]-c[\omega_{0}]$ has degree 0, for appropriate $c$. 5.1
and 5.1 then imply that any degree 0 Bott-Chern (1,1)-class $[\alpha_{1}]$ can
be represented by a primitive, transversal form $\alpha_{1}$. As $\omega_{0}$
is closed and $\theta^{\sharp}\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip
2.0pt\omega_{0}=I(\theta^{\sharp})\hskip
2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0pt\omega_{0}=0$, the form
$\omega_{0}$ is also transversal. Then $\alpha_{1}+c\omega_{0}$ is a
transversal form representing $[\alpha]$.

Remark 5.9: For another proof of this result, see [Ts1, Theorem 2.1].

Finally we can give:

Proof of 5.1: Let $\eta\in H^{1,1}_{BC}(M)$ be a Bott-Chern cohomology class
vanishing in de Rham cohomology. By 5.1, we can represent $\eta$ by a
transversal, closed (1,1)-form. By 5.1, the kernel of the tautological map
$H^{1,1}_{\Sigma}(M){\>\longrightarrow\>}H^{1,1}_{BC}(M)$ is generated by
$\omega_{0}=d^{c}\theta$.

###  5.2 Pluricanonical submanifolds

To prove the main result, we still need several preliminary facts.

It is known ([Va1]), that if a compact complex manifold admits Kähler metrics,
then any LCK metric on it is globally conformally Kähler. The next result is
an analogue for pluricanonical versus Vaisman metrics:

Proposition 5.10:  Let $M$ be a compact complex manifold which admits a
Vaisman metric. Then any pluricanonical metric on it is Vaisman.

Proof: Let $\omega=\theta\wedge\theta^{c}-d\theta^{c}$ be a pluricanonical
form on $M$ and let $\omega^{\prime}$ be a Vaisman one, with $\theta^{\prime}$
its Lee form. Denote by $\omega_{0}$, $\omega^{\prime}_{0}$ the corresponding
exact semipositive forms, $\omega_{0}=-d^{c}\theta$,
$\omega_{0}^{\prime}=-d^{c}\theta^{\prime}$. Since both $\omega_{0}$ and
$\omega^{\prime}_{0}$ are exact, their Bott-Chern classes are proportional
(5.1). Rescaling one of these forms if necessary, we may assume that
$\omega_{0}-\omega^{\prime}_{0}=dd^{c}\varphi$, where $\varphi$ is a function
which is transversal, that is, constant on the leaves of $\Sigma$. Then
$\alpha:=\theta-\theta^{\prime}+d\varphi$ is $d$\- and $d^{c}$-closed, hence
holomorphic, and therefore transversal by [Ts2, Theorem 3.3]).

According to [KO, Theorem A], an LCK metric on $M$ is Vaisman if and only it
admits a holomorphic flow which leaves it invariant, but acts non-
isometrically on the Kähler covering. We then show that the holomorphic Lee
flow $F$ generated by
${\theta^{\prime}}^{\sharp}-\sqrt{-1}\>I({\theta^{\prime}}^{\sharp})$ (which
is tangent to the leaves of $\Sigma$) preserves the pluricanonical metric. As
$F$ is holomorphic, it is enough to show that $\omega$ is $F$-invariant.

Indeed, the form $\theta=\alpha+\theta^{\prime}-d\varphi$ is the sum of the
transversal (and hence, $F$-invariant) form $\alpha-d\varphi$ and the
$F$-invariant form $\theta^{\prime}$. Therefore, it is also $F$-invariant. As
$F$ is holomorphic, $I(\theta)$ is $F$-invariant too and also
$\omega_{0}=-d^{c}\theta$, and hence $\omega$ is $F$-invariant and Vaisman.

Remark 5.11:  The case of compact pluricanonical surfaces was treated in 4.1.
From the classification of surfaces of Kähler rank 1 it follows that any
pluricanonical surface admits a Vaisman metric.

Obviously, complex submanifolds of LCK manifolds are LCK. As we already
recalled, the Vaisman condition is inherited on compact complex submanifolds,
[Ve1]. A similar result occurs for pluricanonical manifolds:

Proposition 5.12:  A compact complex submanifold of a pluricanonical LCK
manifold is pluricanonical.

Proof: Let $M$ be a pluricanonical LCK manifold and let $i:N\hookrightarrow M$
be a compact submanifold. By 3.2 the two-form $\omega_{0}:=d\theta^{c}$ is
degenerate on $M$. Then its restriction $i^{*}\omega_{0}$ to $N$ is degenerate
too, otherwise $\int_{N}\big{(}i^{*}\omega_{0}\big{)}^{\dim_{\mathbb{C}}N}\neq
0$, contradiction with $i^{*}\omega_{0}$ being exact. This means that the
induced LCK structure on $N$ satisfies (3.3) and hence is pluricanonical.

###  5.3 The main result: all compact pluricanonical manifolds are Vaisman

Theorem 5.13:  Let $(M,I)$ be a compact complex manifold and let $g$ be an LCK
pluricanonical metric on it. Then $g$ is Vaisman.

Proof: Let $M$ be a compact pluricanonical locally conformally Kähler
manifold. As (3.4) is satisfied, the universal covering of $M$ carries an
automorphic potential which by 2.1 is strictly positive. Moreover, by the same
result, $M$ admits a locally conformally Kähler metric with potential
(possibly different from the pluricanonical metric), call it $g^{\prime}$.
This LCK metric has Lee form of length 1, by 3.2.

Now we argue by contradiction. Suppose the metric $g^{\prime}$ is not Vaisman.
Then, by 4.3, $M$ contains an embedded non-diagonal Hopf surface $H^{2}$
which, by 5.2 (applied for the initial metric on $M$), is pluricanonical and,
by 4.1, admits Vaisman metrics. But non-diagonal Hopf surfaces cannot admit
Vaisman metrics, [Be], contradiction.

As $M$ admits a Vaisman metric, $g^{\prime}$, then by 5.2, the pluricanonical
metric $g$ itself is Vaisman.

In view of this result, 3.2 now gives:

Corollary 5.14: Let $(M,\omega,\theta)$ be a compact LCK manifold with
potential. Consider the 2-form
$\omega_{0}:=d\theta^{c}=\theta\wedge\theta^{c}-\omega$. Then $\omega_{0}$ is
semipositive if and only if $(M,\omega,\theta)$ is Vaisman.

##  6 Errata

###  6.1 Pluricanonical condition revisited

In Section 3 of [OV6] the following erroneous claim was made: “We now prove
that the pluricanonical condition is equivalent with the existence of an
automorphic potential on a Kähler covering.”

Then we proceeded to make calculations purporting to show that pluricanonical
condition is equivalent to the LCK with potential condition
$d(I\theta)=\omega-\theta\wedge I\theta.$ Here, the scalar term is lost: the
correct equation (in the notation of [OV6]) is
$d(I\theta)=|\theta|^{2}\omega-\theta\wedge I\theta.$

In 3.2, we prove that this equation, indeed, implies
$d(I\theta)=\omega-\theta\wedge I\theta$.

However, the converse statement is false: as shown in 5.3, not all LCK
manifolds with potential admit a pluricanonical LCK structure, but only
Vaisman ones.

###  6.2 LCK rank of Vaisman manifolds

Recall that the LCK rank of an LCK manifold $(M,\omega,\theta)$ is the rank of
the smallest rational subspace $V$ in $H^{1}(M,{\mathbb{R}})$ such that
$V\otimes_{\mathbb{Q}}{\mathbb{R}}$ contains the cohomology class $[\theta]$.
When the LCK rank is 1, the manifold admits a ${\mathbb{Z}}$-covering which is
Kähler (Section 2).

In several papers published previously ([OV1], [OV4], [OV7]) we claimed that a
Vaisman manifold and an LCK manifold with potential always have LCK rank 1.
This is in fact false. In this section we produce a counterexample to these
claims, and explain the error.

Notice, however, that, as we prove below, any complex manifold which admits a
structure of a Vaisman manifold (or LCK manifold with potential) also admits a
structure of a Vaisman manifold (or LCK manifold with potential) with LCK rank
one, so the problems arising because of this error are all differential-
geometrical in nature; all results of complex geometry remain valid. This is
probably the reason why the error was not noticed for so many years. Moreover,
the set of Vaisman (or LCK with potential) structures with LCK rank 1 on a
given manifold is dense in the set of all Vaisman (or LCK with potential)
structures.

To construct a Vaisman (or LCK) manifold with an LCK rank bigger than 1 we use
the same construction as used in 5.2. Consider a Vaisman manifold
$(M,\omega,\theta)$, with $\omega=d_{\theta}d^{c}_{\theta}(1)$ (see 2.2 and
the following Claim) which admits a transversal holomorphic 1-form
$\alpha^{1,0}$. Examples of such Vaisman manifolds include the Kodaira
surface, which is an isotrivial elliptic fibration over an elliptic curve. In
this case, $\alpha^{1,0}$ is the pullback of a holomorphic differential of the
elliptic curve.

Now, let $\alpha:=\operatorname{Re}\alpha^{1,0}$, let
$\theta^{\prime}:=\theta+\alpha$, and consider the 2-form
$\omega^{\prime}:=d_{\theta^{\prime}}d^{c}_{\theta^{\prime}}(1)$. By
construction, this is a (1,1)-form, which is $d_{\theta^{\prime}}$-closed and
(for small values of $\alpha$ in the $\mathcal{C}^{\infty}$ norm) positive,
hence it is an LCK form. This form is in fact conformally equivalent to a
Vaisman one by Ornea-Kamishima criterion (see the proof of 1.3). By [Va2],
$H^{1}(M,{\mathbb{R}})$ is generated by the cohomology class of $[\theta]$ and
cohomology classes of the real parts of holomorphic 1-forms. This implies the
following unexpected result (already used in 5.2).

Theorem 6.1:  Let $M$ be a Vaisman manifold or LCK manifold with potential and
let $L\subset H^{1}(M,{\mathbb{R}})$ be the set of cohomology classes of all
Lee forms for the Vaisman (LCK with potential) structures on $M$. Then $L$ is
open in $H^{1}(M,{\mathbb{R}})$.

Proof: The Vaisman case was considered above. The general LCK with potential
case is elementary: given an LCK with potential structure $(M,\omega,\theta)$,
such that $\omega=d_{\theta}d^{c}_{\theta}(1)$, we can always replace $\theta$
by a closed form $\theta^{\prime}=\theta+\alpha$, with $\alpha$ sufficiently
small. Then $\omega^{\prime}=d_{\theta^{\prime}}d^{c}_{\theta^{\prime}}(1)$ is
a positive $(1,1)$-form, giving an LCK structure with potential.

Now, for a general Vaisman structure $(M,\omega,\theta)$, its LCK rank is
equal to $b_{1}(M)$, and any number between 1 and $b_{1}(M)$ can be obtained
as an LCK rank for an appropriate choice of $\theta$.

6.2 has the following consequences.

Corollary 6.2: Let $M$ be a complex manifold which admits a structure of a
Vaisman manifold (or LCK manifold with potential) $(M,\omega,\theta)$. Then
$M$ admits a structure of a Vaisman manifold (or LCK manifold with potential)
$(M,\omega^{\prime},\theta^{\prime})$ with proper potential, that is, of LCK
rank one. Moreover, such $\omega^{\prime}$ and $\theta^{\prime}$ can be chosen
in any neighbourhood of $(M,\omega,\theta)$.

Now, let us explain where the proof of [OV1] (later refined in [OV7]) failed.

Let $M$ be a compact Vaisman manifold, and $\theta^{\sharp}$ its Lee field.
Then $\theta^{\sharp}$ acts on $M$ by holomorphic isometries, and on its
smallest Kähler covering $(\tilde{M},\tilde{\omega})$ by holomorphic
homotheties. Denote by $G$ the closure of the group generated by
$e^{t\theta^{\sharp}}$. This group is a compact Lie group, because isometries
form a compact Lie group on a compact Riemannian manifold, and a closed
subgroup of a Lie group is a Lie group by Cartan’s theorem. Moreover, it is
commutative, because $\langle e^{t\theta^{\sharp}}\rangle$ is commutative, and
this gives $G=(S^{1})^{k}$.

Let $\tilde{G}$ be the group of pairs
$(\tilde{f}\in\operatorname{Aut}(\tilde{M}),\ f\in G)$, making the following
diagram commutative:

$\begin{CD}\tilde{M}@>{\tilde{f}}>{}>\tilde{M}\\\ @V{\pi}V{}V@V{}V{\pi}V\\\
M@>{f}>{}>M\end{CD}$

Then $\tilde{G}$ is a covering of $G$, and the kernel of this projection is
$\tilde{G}\cap\operatorname{Aut}_{M}(\tilde{M})$, where
$\operatorname{Aut}_{\tilde{M}}(M)$ is the deck transform group of the
covering $\tilde{M}{\>\longrightarrow\>}M$.

Consider the homomorphism
$\chi:\;\pi_{1}(M){\>\longrightarrow\>}{\mathbb{R}}^{>0}$ mapping an element
of $\pi_{1}(M)$ considered as an automorphism of $\tilde{M}$, to the Kähler
homothety constant,
$\gamma\mapsto\frac{\gamma^{*}\tilde{\omega}}{\tilde{\omega}}$. Since
$\tilde{M}$ is the smallest Kähler covering, we identify
$\operatorname{Aut}_{M}(\tilde{M})$ with
$\chi(\pi_{1}(M))\subset{\mathbb{R}}^{>0}$.

Now, let $\tilde{G}_{0}\subset\tilde{G}$ be the subgroup acting on $\tilde{M}$
by isometries. Since the group
$\tilde{G}\cap\operatorname{Aut}_{M}(\tilde{M})$ is a subgroup of
$\operatorname{Aut}_{\tilde{M}}(M)$, $\tilde{G}_{0}$ maps to its image in $G$
bijectively.

We assumed that $G_{0}$ (being the subgroup of elements of $\tilde{G}$ acting
by isometries on both $\tilde{M}$ and $M$) is closed in $G$. Then, if
$\tilde{G}_{0}\cong S^{k-1}$, this would imply that
$\tilde{G}\cong(S^{1})^{k-1}\times{\mathbb{R}}$, proving that $M$ is a
quotient of $\tilde{M}$ by ${\mathbb{Z}}$-action.

However, this is false, because $G_{0}$ is closed in $\tilde{G}$, but not
closed in $G$. This is where the argument fails.

Acknowledgments. L.O. thanks the Laboratory for Algebraic Geometry at the
Higher School of Economics in Moscow for hospitality and excellent research
environment during February and April 2014, and April 2015.

Both authors are indebted to Paul Gauduchon and Andrei Moroianu, for drawing
their attention on the insufficient motivation of their arguments in [OV6] and
for illuminating discussions, and to Victor Vuletescu for a careful reading of
a first draft of this paper.

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Liviu Ornea
University of Bucharest, Faculty of Mathematics,
14 Academiei str., 70109 Bucharest, Romania, _and_
Institute of Mathematics “Simion Stoilow” of the Romanian Academy,
21, Calea Grivitei Street 010702-Bucharest, Romania
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Misha Verbitsky
Laboratory of Algebraic Geometry, Faculty of Mathematics,
NRU HSE, 7 Vavilova Str. Moscow, Russia, _and_
Université Libre de Bruxelles, Département de Mathématique
Campus de la Plaine, C.P. 218/01, Boulevard du Triomphe
B-1050 Brussels, Belgium
<EMAIL_ADDRESS>