# Plurisigned hermitian metrics

Daniele Angella, Vincent Guedj, Chinh H. Lu  Dipartimento di Matematica e
Informatica “Ulisse Dini”
Università di Firenze
viale Morgagni 67/A
50134 Firenze, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>https://sites.google.com/site/danieleangella/ Institut de Mathématiques de
Toulouse
Université de Toulouse
118 route de Narbonne
31400 Toulouse, France
<EMAIL_ADDRESS>https://www.math.univ-toulouse.fr/ guedj/
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, 91405,
Orsay, France<EMAIL_ADDRESS>https://www.imo.universite-paris-saclay.fr/ lu/

###### Abstract.

Let $(X,\omega)$ be a compact hermitian manifold of dimension $n$. We study
the asymptotic behavior of Monge-Ampère volumes
$\int_{X}(\omega+dd^{c}\varphi)^{n}$, when $\omega+dd^{c}\varphi$ varies in
the set of hermitian forms that are $dd^{c}$-cohomologous to $\omega$. We show
that these Monge-Ampère volumes are uniformly bounded if $\omega$ is ”strongly
pluripositive”, and that they are uniformly positive if $\omega$ is ”strongly
plurinegative”. This motivates the study of the existence of such plurisigned
hermitian metrics.

We analyze several classes of examples (complex parallelisable manifolds,
twistor spaces, Vaisman manifolds) admitting such metrics, showing that they
cannot coexist. We take a close look at $6$-dimensional nilmanifolds which
admit a left-invariant complex structure, showing that each of them admit a
plurisigned hermitian metric, while only few of them admit a pluriclosed
metric. We also study $6$-dimensional solvmanifolds with trivial canonical
bundle.

###### Key words and phrases:

Monge-Ampère volumes, hermitian metrics

###### 2010 Mathematics Subject Classification:

32W20, 32U05, 32Q15, 35A23

###### Contents

  1. 1 Preliminaries
    1. 1.1 Quasi-plurisubharmonic functions
    2. 1.2 Uniform bounds on Monge-Ampère volumes
  2. 2 Pluripositive hermitian metrics
    1. 2.1 The restriction property
    2. 2.2 Controlling $v_{+}$
    3. 2.3 Pluripositive hermitian metrics
    4. 2.4 Twistor spaces
  3. 3 Plurinegative hermitian metrics
    1. 3.1 Monge-Ampère lower bounds
    2. 3.2 Monge-Ampère bounds in dimension 3
    3. 3.3 Various obstructions
    4. 3.4 Locally conformally Kähler manifolds
  4. 4 Homogeneous examples
    1. 4.1 Complex parallelizable manifolds
    2. 4.2 Six-dimensional nilmanifolds
    3. 4.3 Higher dimensional nilmanifolds
    4. 4.4 Six-dimensional solvmanifolds with trivial canonical bundle

## Introduction

The study of complex Monge-Ampère equations on compact hermitian (non-Kähler)
manifolds has gained considerable interest in the last decade. Tosatti-
Weinkove [TW10] and then Székelyhidi-Tosatti-Weinkove [STW17] have resolved
the Gauduchon-Calabi-Yau conjecture, extending to the hermitian setting Yau’s
fundamental result [Yau78]. Associated degenerate complex Monge-Ampère
equations have been systematically studied by Dinew, Kołodziej, and Nguyen
[DK12, KN15, Din16, KN19], as well as in [LPT21, GL21a, GL21b, GL21c].

By comparison with the setting of Kähler manifolds, a key new difficulty lies
in the uniform control of Monge-Ampère volumes. Given $X$ a compact complex
manifold of complex dimension $n$ equipped with a hermitian metric $\omega$,
it is of crucial importance to decide whether

$v_{+}(\omega):=\sup\left\\{\int_{X}(\omega+dd^{c}\varphi)^{n}\;:\;\varphi\in{\mathcal{C}}^{\infty}(X)\text{
and }\omega+dd^{c}\varphi>0\right\\}$

is finite, and whether

$v_{-}(\omega):=\inf\left\\{\int_{X}(\omega+dd^{c}\varphi)^{n}\;:\;\varphi\in{\mathcal{C}}^{\infty}(X)\text{
and }\omega+dd^{c}\varphi>0\right\\}$

is bounded away from zero. Here $d=\partial+\overline{\partial}$ and
$d^{c}=\frac{1}{2i}(\partial-\overline{\partial})$.

It follows from Stokes theorem that
$v_{-}(\omega)=v_{+}(\omega)=\int_{X}\omega^{n}$ when $\omega$ is closed or,
more generally, when $dd^{c}\omega=0$ and $dd^{c}\omega^{2}=0$. The latter
conditions are however rather restrictive and it is an important open problem
to decide whether $v_{+}(\omega)$ (resp. $v_{-}(\omega)$) is always finite
(resp. positive). We refer the reader to [GL21b, Theorem C] for an
illustration of how the finiteness of $v_{+}(\omega)$ is related to a
transcendental form of Demailly’s holomorphic Morse inequalities, while
[GL21c] strongly motivates the condition $v_{-}(\omega)>0$.

It has been shown in [GL21b, Theorem A] that the condition
$v_{+}(\omega)<+\infty$ (resp. $v_{-}(\omega)>0$) is independent of the choice
of hermitian metric –it only depends on the complex structure– and is a
bimeromorphic invariant.

We further study these conditions in this article, testing them on various
classes of examples. Our first observation is the following hereditary result.

######  Theorem A.

Let $(X,\omega_{X})$ be a compact hermitian manifold and let $Y\subseteq X$ be
a closed submanifold equipped with a hermitian form $\omega_{Y}$. If
$v_{+}(X,\omega_{X})<+\infty$ then $v_{+}(Y,\omega_{Y})<+\infty$.

We then establish the finiteness of $v_{+}(X)$ (resp. positivity of
$v_{-}(X)$) when $X$ admits special pluripositive (resp. plurinegative)
hermitian metrics.

######  Theorem B.

Let $X$ be a compact complex manifold of dimension $n$.

  1. (1)

If there exists a hermitian metric $\omega$ and $\varepsilon>0$ such that
$dd^{c}\omega\geq 0$ and $dd^{c}\omega^{q}\geq\varepsilon\omega\wedge
dd^{c}\omega^{q-1},$ for $2\leq q\leq n-2,$ then $v_{+}(\omega)<+\infty$.

  2. (2)

If $n=3$ and $X$ admits a metric $\omega$ such that $dd^{c}\omega\leq 0$, then
$v_{-}(\omega)>0$.

In particular if $n=3$ and $\omega$ is pluriclosed then $0<v_{-}(\omega)\leq
v_{+}(\omega)<+\infty$.

We also provide a curvature condition to control $v_{-}(\omega)$ in higher
dimension, see Definition 3.1 and Theorem 3.2.

These conditions are always fulfilled when $\dim_{\mathbb{C}}X\leq 2$, so we
initiate a systematic study of the $3$-dimensional case. Using Hahn-Banach
theorem in the spirit of [Mic82, HL83], one can show (Theorems 2.4 and 3.5)
that there exists a pluripositive (resp. plurinegative) hermitian metric
$\omega$ on $X$ if and only if any positive current $\tau$ of bidimension
$(2,2)$ such that $dd^{c}\tau\leq 0$ (resp. $dd^{c}\tau\geq 0$) satisfies
$dd^{c}\tau=0$. Thus in dimension $3$ such plurisigned hermitian metrics
cannot coexist, i.e. the following conditions are mutually exclusive (see
Corollary 3.7):

  * •

$X$ admits a hermitian metric $\omega$ such that $dd^{c}\omega\geq 0$ and
$dd^{c}\omega\neq 0$;

  * •

$X$ admits a hermitian metric $\omega$ such that $dd^{c}\omega=0$;

  * •

$X$ admits a hermitian metric $\omega$ such that $dd^{c}\omega\leq 0$ and
$dd^{c}\omega\neq 0$;

  * •

$X$ does not admit any hermitian metric $\omega$ such that $dd^{c}\omega$ has
a sign.

Each case does occur as we show by analyzing several classes of examples:

  * •

Complex parallelisable manifolds and twistor spaces of K3 surfaces admit
pluripositive hermitian metrics (see Proposition 4.1 and Theorem 2.7).

  * •

The existence of pluriclosed hermitian metrics has been thoroughly studied in
recent years (see [FPS04, FT09, Verb14, COUV16, Ot20, ADOS22]).

  * •

Vaisman manifolds (a special type of locally conformally Kähler manifolds)
admit plurinegative hermitian metrics (Proposition 3.10).

  * •

Non-Kähler manifolds from the class ${\mathcal{C}}$ of Fujiki do not admit any
plurisgned hermitian metric (see Proposition 3.8).

This alternative is however no longer valid in higher dimension, see Example
4.5.

We take a closer look at nilmanifolds $X=\Gamma\backslash G$ of (real)
dimension $6$, where $G$ is a connected and simply connected nilpotent Lie
group, and $\Gamma$ is a discrete co-compact subgroup. There are 34
isomorphism classes of nilpotent Lie algebras in dimension 6, only 18 of which
admit a complex structure. Following [ABD11, Uga07, UV14, COUV16] we gather
them in four large families (Np), (Ni), (Nii), (Niii) and show the following.

######  Theorem C.

Consider a six-dimensional nilmanifold $X=\Gamma\backslash G$ endowed with a
left-invariant complex structure. There is always a plurisigned hermitian
metric. More precisely if $X$ is not a complex torus, then

  * •

either $X$ belongs to one of the classes (Np), (Nii), (Niii) and then any
left-invariant hermitian metric is pluripositive but not pluriclosed.

  * •

or $X$ belongs to the class (Ni) and –depending on the complex structure– it
admits a left-invariant hermitian metric wich is either pluriclosed, or
pluripositive but not pluriclosed, or else plurinegative but not pluriclosed.

This analysis largely generalizes the influential work of Fino-Parton-Salamon
[FPS04] who characterized the existence of pluriclosed metric in this context.
We refer the reader to Section 4.2 for a more precise statement.

Following [FOU15] we also analyze the case of $6$-dimensional solvmanifolds,
–i.e. compact quotients of a connected solvable Lie group by a discrete
subgroup,– endowed with left-invariant complex structures with holomorphically
trivial canonical bundle. These are gathered in ten large families (see
Section 4.4), besides the four large families of nilmanifolds that are dealt
with by the previous statement.

######  Theorem D.

Consider a six-dimensional solvmanifold endowed with a left-invariant complex
structure with holomorphically-trivial canonical bundle. Then, there is always
a plurisigned metric.

We refer the reader to Theorem 4.7 for a more precise statement, which notably
shows that in most cases the plurisigned metric is pluripositive.

######  Acknowledgements.

The authors are partially supported by the research projects HERMETIC of the
Labex CIMI, PARAPLUI of the french ANR. The first named-author is supported by
PRIN2017 (2017JZ2SW5), and by GNSAGA of INdAM.

##  1\. Preliminaries

In the whole article we let $X$ denote a compact complex manifold of complex
dimension $n\geq 1$, and we fix a hermitian form $\omega_{X}$ on $X$.

###  1.1. Quasi-plurisubharmonic functions

Let $\omega$ be a semi-positive $(1,1)$-form.

####  1.1.1. Quasi-plurisubharmonic functions

######  Definition 1.1.

A function is quasi-plurisubharmonic (quasi-psh for short) if it is locally
given as the sum of a smooth and a plurisubharmonic function.

Given an open set $U\subset X$, quasi-psh functions
$\varphi:U\rightarrow\mathbb{R}\cup\\{-\infty\\}$ satisfying
$\omega_{\varphi}:=\omega+dd^{c}\varphi\geq 0$ in the weak sense of currents
are called $\omega$-psh functions on $U$. Constant functions are $\omega$-psh
functions since $\omega$ is semi-positive.

A ${\mathcal{C}}^{2}$-smooth function $u\in\mathcal{C}^{2}(X)$ has bounded
Hessian, hence $\varepsilon u$ is $\omega$-psh on $X$ if $0<\varepsilon$ is
small enough and $\omega$ is positive (i.e. hermitian).

######  Definition 1.2.

We let ${\rm PSH}(X,\omega)$ denote the set of all $\omega$-plurisubharmonic
functions which are not identically $-\infty$.

The set ${\rm PSH}(X,\omega)$ is a closed subset of $L^{1}(X)$, for the
$L^{1}$-topology. We refer the reader to [Dem, GZ, Din16] for basic properties
of $\omega$-psh functions, and simply recall that:

  * •

$PSH(X,\omega)\subset PSH(X,\omega^{\prime})$ if $\omega\leq\omega^{\prime}$;

  * •

$PSH(X,\omega)\subset L^{r}(X)$ for $r\geq 1$; the induced $L^{r}$-topologies
are equivalent;

  * •

the subset $PSH_{A}(X,\omega):=\\{u\in PSH(X,\omega),\;-A\leq\sup_{X}u\leq
0\\}$ is compact in $L^{r}(X)$ for any $r\geq 1$ and any $A>0$.

######  Definition 1.3.

A quasi-psh function $\varphi$ has analytic singularities if it can be locally
written as

$\varphi(z)=c\log\sum_{j=1}^{s}|f_{j}(z)|^{2}+\rho(z),$

where $c>0$, the $f_{j}$’s are holomorphic functions and $\rho$ is a smooth
function.

We recall the following fundamental regularization result of Demailly [Dem92].

######  Theorem 1.4.

Any quasi-psh function is the decreasing limit of smooth quasi-psh functions.
Moreover when $\omega$ is a hermitian form, any function $\varphi\in
PSH(X,\omega)$ is the decreasing limit of functions $\varphi_{j}\in
PSH(X,\omega)$ with analytic singularities.

####  1.1.2. Monge-Ampère measure

The complex Monge-Ampère measure $(\omega+dd^{c}u)^{n}$ is well-defined for
any $\omega$-psh function $u$ which is bounded, as follows from the theory
developed by Bedford-Taylor in bounded pseudoconvex domains of
$\mathbb{C}^{n}$.

If $\beta=dd^{c}\rho$ is a Kähler form that dominates $\omega$ in a local
chart $U$, the function $u$ is $\beta$-psh in $U$ hence the positive currents
$(\beta+dd^{c}u)^{j}$ are well defined for $0\leq j\leq n$ by [BT82]. This
allows one to make the following definition.

######  Definition 1.5.

The complex Monge-Ampère measure of $u$ is

$(\omega+dd^{c}u)^{n}:=\sum_{j=0}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)(-1)^{n-j}(\beta+dd^{c}u)^{j}\wedge(\beta-\omega)^{n-j}.$

We refer to [DK12] for an adaptation of the main properties of [BT82] to this
hermitian and global context.

The mixed Monge-Ampère measures
$(\omega+dd^{c}u)^{j}\wedge(\omega+dd^{c}v)^{n-j}$ are also well defined for
any $0\leq j\leq n$, and any bounded $\omega$-psh functions $u,v$.

A basic property we shall use is the following extension of a fundamental
result of Bedford-Taylor:

######  Lemma 1.6.

Let $u,v$ be bounded $\omega$-psh functions, then $\max(u,v)\in PSH(X,\omega)$
and

$1_{\\{u<v\\}}(\omega+dd^{c}\max(u,v))^{n}=1_{\\{u<v\\}}(\omega+dd^{c}v)^{n}.$

The subtle point here is that the set $\\{u<v\\}$ is not open in the usual
sense if $v$ is not continuous, it is merely open for the plurifine topology.

###  1.2. Uniform bounds on Monge-Ampère volumes

Let $(X,\omega)$ be a compact hermitian manifold of complex dimension $n$.

######  Definition 1.7.

We consider

$v_{-}(\omega):=\inf\left\\{\int_{X}(\omega+dd^{c}u)^{n},\;u\in{\rm
PSH}(X,\omega)\cap{L}^{\infty}(X)\right\\}$

and

$v_{+}(\omega):=\sup\left\\{\int_{X}(\omega+dd^{c}u)^{n},\;u\in{\rm
PSH}(X,\omega)\cap{L}^{\infty}(X)\right\\}.$

The supremum and infimum in the definition of $v_{+}(\omega)$ and
$v_{-}(\omega)$ can be taken over ${\rm PSH}(X,\omega)\cap C^{\infty}(X)$ as
follows from Theorem 1.4 and Bedford-Taylor’s convergence theorem [BT82].
These quantities have been studied in [GL21b]. A major open problem is the
following.

######  Problem 1.8.

Understand whether $v_{+}(\omega)<+\infty$ and/or $v_{-}(\omega)>0$.

It follows from Stokes theorem that
$0<v_{-}(\omega)=v_{+}(\omega)=\int_{X}\omega^{n}<+\infty$ when $\omega$ is a
Kähler form. The same result holds true as soon as the hermitian form
satisfies

$dd^{c}\omega=0\;\;\text{ and }\;\;d\omega\wedge d^{c}\omega=0,$

a vanishing condition introduced by Guan-Li in [GL10].

This condition actually characterizes the preservation of Monge-Ampère masses,
as was observed by Chiose in [Chi16b].

######  Theorem 1.9.

[Chi16b]. The following properties are equivalent:

  1. (1)

$\int_{X}(\omega+dd^{c}\varphi)^{n}=\int_{X}\omega^{n}$ for all $\varphi\in
PSH(X,\omega)\cap L^{\infty}(X)$.

  2. (2)

$dd^{c}\omega=0\;\;\text{ and }\;\;d\omega\wedge d^{c}\omega=0$.

When $n=2$ the vanishing of $d\omega\wedge d^{c}\omega$ is automatic for
bidegree reasons, hence the preservation of Monge-Ampère volumes is equivalent
to $\omega$ being a Gauduchon metric. In higher dimension this condition is
quite restrictive and unstable.

Chiose further observed that the Guan-Li condition is moreover equivalent to

$dd^{c}\omega\geq 0\;\;\text{ and }\;\;d\omega\wedge d^{c}\omega\geq 0.$

Indeed

$dd^{c}(\omega^{n-1})=(n-1)(n-2)d\omega\wedge
d^{c}\omega\wedge\omega^{n-3}+(n-1)dd^{c}\omega\wedge\omega^{n-2}$

while Stokes theorem ensures that $\int_{X}dd^{c}(\omega^{n-1})=0$.

######  Remark 1.10.

Note that $X$ satisfying the $\partial\overline{\partial}$-Lemma is not enough
to ensure the volume preservation property
$v_{-}(\omega)=v_{+}(\omega)=\int_{X}\omega^{n}$. Some splitting-type complex
structures arising as deformations of the holomorphically-parallelizable
Nakamura solvmanifold satisfy the $\partial\overline{\partial}$-Lemma [AK17,
AOUV17], but they never admit pluriclosed metrics [AOUV17], in particular the
Guan-Li conditions are never satisfied (see also Proposition 3.8). It is a
folkore conjecture that balanced metrics always exist on compact complex
manifolds satisfying the $\partial\overline{\partial}$-Lemma [Pop17], and it
is also expected that balanced and pluriclosed metrics cannot coexist on a
compact complex manifold unless it admits Kähler metrics [FV15]. If these two
conjectures were true, the volume preservation property would never be
satisfied by compact complex non-Kähler manifolds satisfying the
$\partial\overline{\partial}$-Lemma.

In the sequel we analyze several families of compact complex manifolds
admitting a hermitian metric $\omega$ such that $dd^{c}\omega\geq 0$ (resp.
$dd^{c}\omega\leq 0$). We can further expect that $d\omega\wedge
d^{c}\omega\leq 0$ (resp. $d\omega\wedge d^{c}\omega\geq 0$), but we should
not expect, in general, that these two forms are both globally positive (resp.
negative).

We recall the following properties established in [GL21b].

######  Theorem 1.11.

[GL21b, Proposition 3.2, Theorem 3.7, Theorem 4.12]. Let $X$ be a compact
complex manifold of dimension $n$ and let $\omega$ be a hermitian metric.

  * •

The condition $v_{+}(\omega)<+\infty$ is independent of the choice of the
hermitian metric $\omega$, and it is a bimeromorphic invariant.

  * •

The condition $v_{-}(\omega)>0$ is independent of the choice of the hermitian
metric $\omega$, and it is a bimeromorphic invariant.

  * •

If $\alpha\in H^{1,1}_{BC}(X,\mathbb{R})$ is a nef class with $\alpha^{n}>0$,
then $\alpha$ contains a Kähler current (hence $X$ belongs to the Fujiki class
${\mathcal{C}}$) if and only if $v_{+}(\omega)<+\infty$.

Here $H^{1,1}_{BC}(X,\mathbb{R})$ denotes the first Bott-Chern cohomology
group of $X$. The last item is a partial answer to an important conjecture of
Demailly-Paun [DP04].

A main goal of this article is to introduce curvature conditions that allow
one to partially answer Problem 1.8, and to try and establish them on various
classes of non-Kähler manifolds.

##  2\. Pluripositive hermitian metrics

###  2.1. The restriction property

We observe in this section that the condition $v_{+}(\omega)<+\infty$ is
stable under restriction.

######  Theorem 2.1.

Let $(X,\omega_{X})$ be a compact hermitian manifold and let $Y\subset X$ be a
closed submanifold of $X$ equipped with a hermitian form $\omega_{Y}$. If
$v_{+}(X,\omega_{X})<+\infty$ then $v_{+}(Y,\omega_{Y})<+\infty$.

###### Proof.

Since the finiteness of $v_{+}$ is independent of the choice of a hermitian
metric, we work here with $\omega_{Y}=(\omega_{X})_{|Y}$.

It follows from Theorem 1.4 that one can approximate $\varphi\in
PSH(Y,\omega_{Y})$ by a decreasing sequence of smooth strictly
$\omega_{Y}$-psh functions. Since the complex Monge-Ampère operator is
continuous along decreasing sequences [BT82], it suffices to establish a
uniform bound from above on $\int_{Y}(\omega_{Y}+dd^{c}\varphi)^{k}$, where
$\varphi$ is smooth and strictly $\omega_{Y}$-psh and $k=\dim_{\mathbb{C}}Y$.

It follows from [CGZ13, Proposition 2.1] that there exists a smooth extension
$\phi\in PSH(X,\omega_{X})$ with $\phi_{|Y}=\varphi$. It is classical (see
e.g. [DP04, Lemma 2.1]) that one can find a function $\psi_{Y}\in
PSH(X,\omega_{X})$ which is smooth in $X\setminus Y$, with analytic
singularities along $Y$, and such that

$(\omega_{X}+dd^{c}\psi_{Y})^{n-k}\geq\delta_{0}[Y],$

for some $\delta_{0}>0$, where $[Y]$ denotes the current of integration along
$Y$. We infer

$\displaystyle\int_{Y}(\omega_{Y}+dd^{c}\varphi)^{k}$ $\displaystyle=$
$\displaystyle\int_{X}(\omega_{X}+dd^{c}\phi)^{k}\wedge[Y]$
$\displaystyle\leq$
$\displaystyle\delta_{0}^{-1}\int_{X}(\omega_{X}+dd^{c}\phi)^{k}\wedge(\omega_{X}+dd^{c}\psi_{Y})^{n-k}$
$\displaystyle=$
$\displaystyle\lim_{j\rightarrow+\infty}\delta_{0}^{-1}\int_{X}(\omega_{X}+dd^{c}\phi)^{k}\wedge(\omega_{X}+dd^{c}\psi_{j})^{n-k},$

where $\psi_{j}=\max(\psi_{Y},-j)\in PSH(X,\omega_{X})\cap L^{\infty}(X)$.

Observe now that $u_{j}=\frac{\phi+\psi_{j}}{2}\in PSH(X,\omega_{X})\cap
L^{\infty}(X)$ with

$(\omega_{X}+dd^{c}\phi)^{k}\wedge(\omega_{X}+dd^{c}\psi_{j})^{n-k}\leq
2^{n}(\omega_{X}+dd^{c}u_{j})^{n},$

and $\int_{X}(\omega_{X}+dd^{c}u_{j})^{n}\leq v_{+}(X,\omega_{X})$, hence

$v_{+}(Y,\omega_{Y})=\sup_{\varphi}\int_{Y}(\omega_{Y}+dd^{c}\varphi)^{k}\leq
2^{n}\delta_{0}^{-1}v_{+}(X,\omega_{X})<+\infty,$

completing the proof. ∎

###  2.2. Controlling $v_{+}$

We propose here various curvature conditions on a hermitian metric $\omega$
that ensure $v_{+}(\omega)<+\infty$.

######  Theorem 2.2.

Let $X$ be a compact complex manifold of dimension $n$. Assume there exists a
hermitian metric $\omega$ and $\varepsilon>0$ such that

  * •

$dd^{c}\omega\geq 0$ and

  * •

$dd^{c}\omega^{q}\geq\varepsilon\omega\wedge dd^{c}\omega^{q-1}$ for $2\leq
q\leq n-2$.

Then $v_{+}(\omega)<+\infty$.

In complex dimension $n=3$, the hypothesis boils down to $dd^{c}\omega\geq 0$.
We shall provide several examples of manifolds admitting such pluripositive
hermitian metrics in the sequel. Higher dimensional examples satisfying the
conditions of Theorem 2.2 will be presented in Example 4.2 and Example 4.6.

###### Proof.

When $n=3$ this observation is due to Chiose [Chi16b, Question 0.8]. We
include a proof as a warm up for the higher dimensional case. It follows from
Stokes theorem that $\int_{X}(dd^{c}u)^{3}=0$ for any $u\in{\rm
PSH}(X,\omega)\cap C^{\infty}(X)$, hence

$\int_{X}(\omega+dd^{c}u)^{3}=\int_{X}\omega^{3}+3\int_{X}\omega^{2}\wedge
dd^{c}u+3\int_{X}\omega\wedge(dd^{c}u)^{2}.$

Integrating by parts we see that

$\left|\int_{X}\omega^{2}\wedge
dd^{c}u\right|=\left|\int_{X}u{\,dd^{c}\omega^{2}}\right|\leq
B\int_{X}|u|\omega^{3}\leq C$

by compactness (see [GZ, Proposition 8.4]), as we can normalize $u$ by
$\sup_{X}u=0$. On the other hand

$\int_{X}\omega\wedge(dd^{c}u)^{2}=\int_{X}-dd^{c}\omega\wedge du\wedge
d^{c}u\leq 0$

whenever $dd^{c}\omega\geq 0$. The result follows.

We now treat the case $n\geq 4$. Observe that it suffices to deal with
$a\omega$-psh functions, where $0<a\leq 1$ is an arbitrarily small fixed
constant. Indeed if $u\in PSH(X,\omega)\cap L^{\infty}(X)$, then $au\in
PSH(X,a\omega)\cap L^{\infty}(X)$ with

$a^{n}\omega_{u}^{n}\leq((1-a)\omega+a\omega_{u})^{n},$

hence

$v_{+}(\omega)\leq a^{-n}\sup\left\\{\int_{X}(\omega+add^{c}u)^{n},\,u\in
PSH(X,\omega)\cap L^{\infty}(X)\right\\}.$

We thus fix $a>0$ and $u\in PSH(X,\omega)$ with $\sup_{X}u=0$. Stokes theorem
yields

$\displaystyle\int_{X}(\omega+add^{c}u)^{n}$ $\displaystyle=\sum_{0\leq p\leq
n}\binom{n}{p}a^{p}\int_{X}\omega^{n-p}\wedge(dd^{c}u)^{p}$
$\displaystyle=\int_{X}\omega^{n}+na\int_{X}\omega^{n-1}\wedge
dd^{c}u+\sum_{2\leq p\leq
n-1}\binom{n}{p}a^{p}\int_{X}\omega^{n-p}\wedge(dd^{c}u)^{p}$
$\displaystyle\leq C_{1}-\sum_{2\leq p\leq
n-1}\binom{n}{p}a^{p}\int_{X}du\wedge
d^{c}u\wedge(\omega_{u}-\omega)^{p-2}\wedge dd^{c}(\omega^{n-p})$
$\displaystyle=C_{1}+S.$

We decompose the sum $S$ as follows,

$\displaystyle S$ $\displaystyle=\mathop{\sum_{2\leq p\leq n-1}}_{0\leq k\leq
p-2}B_{k,p}a^{p}(-1)^{k+1}\int_{X}du\wedge
d^{c}u\wedge\omega_{u}^{p-2-k}\wedge\omega^{k}\wedge dd^{c}(\omega^{n-p})$
$\displaystyle=S_{1}+S_{2},$

where $B_{k,p}=\binom{n}{p}\binom{p-2}{k}$, $S_{1}$ is the sum for $k$ even
and $S_{2}$ is the one for $k$ odd. Using the assumption on
$dd^{c}\omega^{n-p}$ we obtain

$\displaystyle S_{1}$ $\displaystyle=-\mathop{\sum_{2\leq p\leq n-2}}_{0\leq
k=2l\leq p-2}B_{k,p}a^{p}\int_{X}du\wedge
d^{c}u\wedge\omega_{u}^{p-2-2l}\wedge\omega^{2l}\wedge dd^{c}(\omega^{n-p})$
$\displaystyle\leq-\mathop{\sum_{2\leq p\leq n-2}}_{0\leq k=2l\leq
p-2}B_{k,p}a^{p}\varepsilon\int_{X}du\wedge
d^{c}u\wedge\omega_{u}^{p-2-2l}\wedge\omega^{2l+1}\wedge
dd^{c}(\omega^{n-p-1}),$

while

$\displaystyle S_{2}$ $\displaystyle=\mathop{\sum_{3\leq p\leq n-1}}_{0\leq
k=2l+1\leq p-2}B_{k,p}a^{p}\int_{X}du\wedge
d^{c}u\wedge\omega_{u}^{p-3-2l}\wedge\omega^{2l+1}\wedge dd^{c}(\omega^{n-p})$
$\displaystyle=\mathop{\sum_{2\leq p\leq n-2}}_{{0\leq k=2l+1\leq
p-1}}B_{k,p+1}a^{p+1}\int_{X}du\wedge
d^{c}u\wedge\omega_{u}^{p-2-2l}\wedge\omega^{2l+1}\wedge
dd^{c}(\omega^{n-p-1}),$

as follows from changing $p$ in $p-1$. Now $S_{1}+S_{2}=$

$=\mathop{\sum_{2\leq p\leq n-2}}_{0\leq k=2l+1\leq
q-1}(-B_{k,p}\varepsilon+B_{k,p+1}a)a^{p}\int_{X}du\wedge
d^{c}u\wedge\omega_{u}^{p-2-2l}\wedge\omega^{2l+1}\wedge
dd^{c}(\omega^{n-p-1})\leq 0$

if $a$ is small enough, which yields $S_{1}+S_{2}\leq 0$ hence
$v_{+}(\omega)\leq a^{-n}C_{1}$. ∎

###  2.3. Pluripositive hermitian metrics

######  Definition 2.3.

A hermitian metric $\omega$ is pluripositive if $dd^{c}\omega\geq 0$,
plurinegative if $dd^{c}\omega\leq 0$, and pluriclosed if $dd^{c}\omega=0$.

Pluriclosed metrics are also often called SKT (strongly Kähler with torsion)
in the literature; a manifold $X$ is called SKT if it admits a SKT hermitian
metric.

The existence of a pluripositive hermitian metric $\omega$ is a condition that
is stable under blow-ups with smooth centers. Indeed if $\pi:Y\rightarrow X$
is the blow up of $X$ with smooth center $Z\subset X$, a hermitian metric on
$Y$ is obtained by considering

$\omega_{Y}=\pi^{*}\omega_{X}-\varepsilon\theta_{Z},$

where $\omega_{X}$ is a hermitian metric in $X\setminus Z$ with poles along
$Z$, $\theta_{Z}$ is a closed form cohomologous to the current of integration
along the exceptional divisor $\pi^{-1}(Z)$, and $0<\varepsilon$ is small.
Thus $dd^{c}\omega_{Y}=\pi^{*}dd^{c}\omega_{X}$ has the same sign as that of
$dd^{c}\omega_{X}$.

However this condition is not stable under modifications (see Proposition
3.8), and there are obstructions to the existence of pluripositive hermitian
metrics:

######  Theorem 2.4.

The following properties are equivalent.

  * •

There exists a hermitian form $\omega$ such that $dd^{c}\omega\geq 0$.

  * •

There exists no positive current $\tau$ of bidimension $(2,2)$ such that
$S=dd^{c}\tau\leq 0$ with $S\neq 0$.

When the complex dimension is $n=3$, this shows in particular that
pluripositive and plurinegative hermitian forms can not coexist (see Corollary
3.7).

###### Proof.

Observe first that the two objects cannot coexist on $X$. Indeed if $\omega$
is a hermitian form such that $dd^{c}\omega\geq 0$ and $\tau$ is a positive
current of bidimension $(2,2)$ such that $dd^{c}\tau\leq 0$, we obtain

$0\leq\int_{X}dd^{c}\omega\wedge\tau=\int_{X}\omega\wedge dd^{c}\tau\leq 0,$

which forces $dd^{c}\tau=0$ since $\omega$ is a hermitian form. Conversely
consider

${\mathcal{C}}:=\left\\{\text{negative currents }S\text{ of bidimension
}(1,1)\text{ with }\int_{X}S\wedge\omega=-1\right\\}.$

This is a compact convex set for the weak topology of currents. We set

$F:=\left\\{\text{currents }S=dd^{c}\tau,\text{ where }\tau\geq 0\text{ has
bidimension }(2,2)\right\\}.$

This is a closed set for the weak topology. There exists no positive current
$\tau$ of bidimension $(2,2)$ such that $S=dd^{c}\tau\leq 0$ with $S\neq 0$ if
and only if the sets ${\mathcal{C}}$ and $F$ are disjoints. If such is the
case, it follows from Hahn-Banach theorem that we can find a continuous
functional $\Phi$ on the set of bidimension $(1,1)$ currents that is semi-
positive on $F$ and strictly negative on ${\mathcal{C}}$. By deRham duality
the functional $\Phi$ is defined by a form $\omega$ of bidegree $(1,1)$. Now

  * •

$\Phi\geq 0$ on $F$ is equivalent to $dd^{c}\omega\geq 0$, while

  * •

$\Phi<0$ on ${\mathcal{C}}$ is equivalent to $\omega$ being hermitian,

as follows from a rescaling argument. ∎

###  2.4. Twistor spaces

Twistor spaces provide a large classe of examples of compact non-Kähler
manifolds which admit a pluripositive hermitian metric.

#### Dimension 4

Let $(M,g)$ be a compact oriented riemannian $4$-manifold. The vector bundle
$\Lambda^{2}T^{*}M$ of $2$-forms can be decomposed as a direct sum
$\Lambda^{2}T^{*}M=\Lambda_{+}\oplus\Lambda_{-},$ where $\Lambda_{\pm}$
denotes the eigenspace of the Hodge $\star$-operator corresponding to the $\pm
1$-eigenvalues of $\star$ (selfdual and anti-selfdual forms).

The riemannian curvature operator
$R:\Lambda^{2}T^{*}M\rightarrow\Lambda^{2}T^{*}M$ can be decomposed under the
action of the group of special isometries $SO(4)$ as

$R=\frac{s}{6}Id+W^{-}+W^{+}+\stackrel{{\scriptstyle\circ}}{{r}},$

where $s$ is the scalar curvature, $\stackrel{{\scriptstyle\circ}}{{r}}$ is
the trace free Ricci curvature, and $W^{\pm}$ are the trace free endomorphisms
of $\Lambda^{\pm}$. The manifold $M$ is called ASD (anti-selfdual) if
$W^{+}=0$; this definition is conformally invariant [AHS78]. A famous result
of Taubes [Taub92] provides many examples of such ASD manifolds.

######  Definition 2.5.

The twistor space $X=X(M,[g])$ of $(M,[g])$ is the total space of the sphere
bundle of self dual $2$-forms.

There is a natural almost complex structure on $X$ which is integrable if and
only if $M$ is ASD [AHS78]. In this case $X$ is a compact complex manifold of
dimension $3$ which is never Kähler unless $M=S^{4}$ is the sphere (in which
case $X=\mathbb{C}\mathbb{P}^{3}$), or $M=\mathbb{C}\mathbb{P}^{2}$ (in which
case $X$ is the flag space of $\mathbb{C}^{3}$), see [Hit81].

Despite being non-Kähler, twistor spaces have a lot of rational curves, in
particular all fibers $F\sim\mathbb{P}^{1}$ of the smooth submersion
$\pi:X\rightarrow M$. Note that there also is a holomorphic projection
$\pi:X\rightarrow\mathbb{P}^{1}$.

#### Dimension $4n$

The previous construction generalizes as follows. Let $(M,g,D)$ be a
quaternionic Kähler manifold of dimension $4n$, i.e. an oriented complete
$4n$-dimensional riemannian manifold $(M,g)$ whose holonomy group is contained
in the product $Sp(1)Sp(n)$ of quaternionic unitary groups. Such a manifold
admits a rank $3$ subbundle $D\subset End(TM)$ invariant by the Levi-Civita
connection.

######  Definition 2.6.

The twistor space $X=X(M,g,D)$ of $(M,g,D)$ is the bundle of spheres of radius
$\sqrt{2}$ of $D$.

This is a locally trivial bundle with fibre $S^{2}$ and structure group
$SO(3)$. It can be endowed with a natural metric G and an almost complex
structure $J$ that is integrable [Sal82, Theorem 4.1]. Thus $(X,J)$ is a
complex manifold of complex dimension $2n+1$.

When $(M,g)$ is hyperkähler, i.e. when the holonomy group is contained in the
quaternionic unitary group $Sp(n)$, it turns out that $(M,g)$ admits three
global $g$-orthogonal integrable Kähler structures $I,J,K$ such that
$IJ=-JI=K$. We thus obtain a pencil of complex structures

$X(M,g,D)\longrightarrow\mathbb{P}^{1}$

which is integrable. It is called the Calabi family of $(M,g,D)$.

#### Pluripositive hermitian metrics

Twistor spaces admit smooth hermitian $(1,1)$-forms $\omega$ that are balanced
(i.e. $d\omega^{n-1}=0$, see [Mic82] and [KV98, Proposition 4.5]). There is a
natural hermitian form $\omega$ whose curvature has been computed by Kaledin-
Verbitsky [KV98, Proposition 8.15] and Deschamps-LeDu-Mourougane [DLM17,
Corollary 5.6]:

######  Theorem 2.7.

Let $(M,g,D)$ be a quaternionic Kähler manifold of dimension $4n$ with
constant scalar curvature $s\leq 0$. Then the natural hermitian form $\omega$
on the twistor space $X(M,g,D)$ is pluripositive $dd^{c}\omega\geq 0$.

##  3\. Plurinegative hermitian metrics

###  3.1. Monge-Ampère lower bounds

We show that the condition $v_{-}(\omega)>0$ is satisfied if $\omega$
satisfies a special plurinegative condition.

######  Definition 3.1.

We say that a hermitian form $\omega$ satisfies the condition $Plurineg(n)$ if
it is a Gauduchon metric and either $n=2$ or $n\geq 3$ and

$\sum_{k=1}^{n-\ell-2}(-1)^{n-k-\ell}{n\choose
k}{n-k-2\choose\ell}dd^{c}\omega^{k}\wedge\omega^{n-k-2-\ell}\leq 0$

for any $\ell\in\\{0,\ldots,n-3\\}$.

The condition $Plurineg(n)$ requires the Gauduchon condition
$dd^{c}\omega^{n-1}=0$. It reduces to the latter when $n=2$, while it moreover
asks that

  * •

$dd^{c}\omega\leq 0$ when $n=3$;

  * •

$dd^{c}\omega\leq 0$ and $3dd^{c}\omega^{2}-2\omega\wedge dd^{c}\omega\leq 0$
when $n=4$;

  * •

$dd^{c}\omega\leq 0$, $dd^{c}\omega^{2}-dd^{c}\omega\wedge\omega\leq 0$, and
$2dd^{c}\omega^{3}-2\omega\wedge dd^{c}\omega^{2}+\omega^{2}\wedge
dd^{c}\omega\leq 0$ when $n=5$.

######  Theorem 3.2.

Let $X$ be a compact complex manifold of dimension $n$. If $X$ admits a
hermitian metric $\omega$ that satisfies $Plurineg(n)$ then
$v_{-}(\omega)=\int_{X}\omega^{n}>0$.

###### Proof.

Fix $u$ a smooth $\omega$-psh function. We first treat the case $n=3$ to set
the scene. The Gauduchon condition yields $\int_{X}\omega^{2}\wedge dd^{c}u=0$
while Stokes theorem ensures that $\int_{X}(dd^{c}u)^{3}=0$, thus

$\displaystyle\int_{X}(\omega+dd^{c}u)^{3}$ $\displaystyle=$
$\displaystyle\int_{X}\omega^{3}+3\int_{X}\omega^{2}\wedge
dd^{c}u+3\int_{X}\omega\wedge(dd^{c}u)^{2}+\int_{X}(dd^{c}u)^{3}$
$\displaystyle=$ $\displaystyle\int_{X}\omega^{3}+3\int_{X}-dd^{c}\omega\wedge
du\wedge d^{c}u$ $\displaystyle\geq$ $\displaystyle\int_{X}\omega^{3}$

if $dd^{c}\omega\leq 0$, so that $v_{-}(\omega)\geq\int_{X}\omega^{3}>0$.

To make the arguments clearer, we also explicitly look at the case $n=4$.
Using the binomial expansion, the Gauduchon condition and Stokes theorem, we
obtain

$\displaystyle\int_{X}\omega_{u}^{4}-\int_{X}\omega^{4}=6\int_{X}-dd^{c}\omega^{2}\wedge
du\wedge d^{c}u+4\int_{X}-dd^{c}\omega\wedge dd^{c}u\wedge du\wedge d^{c}u$
$\displaystyle=$ $\displaystyle
2\int_{X}[-3dd^{c}\omega^{2}+2dd^{c}\omega\wedge\omega]\wedge du\wedge
d^{c}u+4\int_{X}-dd^{c}\omega\wedge\omega_{u}\wedge du\wedge d^{c}u$
$\displaystyle\geq$ $\displaystyle 0$

if $Plurineg(4)$ is satisfied.

We now consider the general case.

$\displaystyle\int_{X}\omega_{u}^{n}-\int_{X}\omega^{n}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}{n\choose
k}\int_{X}\omega^{k}\wedge(dd^{c}u)^{n-k}$ $\displaystyle=$
$\displaystyle\int_{X}(dd^{c}u)^{n}+\sum_{k=1}^{n-2}{n\choose
k}\int_{X}\omega^{k}\wedge(dd^{c}u)^{n-k}+n\int_{X}u\,dd^{c}\omega^{n-1}$
$\displaystyle=$ $\displaystyle-\sum_{k=1}^{n-2}{n\choose
k}\int_{X}dd^{c}\omega^{k}\wedge(dd^{c}u)^{n-k-2}\wedge du\wedge d^{c}u,$

using the Gauduchon condition $dd^{c}\omega^{n-1}=0$. Thus

$\displaystyle\int_{X}\omega_{u}^{n}-\int_{X}\omega^{n}=-\sum_{k=1}^{n-2}{n\choose
k}\int_{X}dd^{c}\omega^{k}\wedge(\omega_{u}-\omega)^{n-k-2}\wedge du\wedge
d^{c}u$ $\displaystyle=$
$\displaystyle-\sum_{k=1}^{n-2}\sum_{\ell=0}^{n-k-2}(-1)^{n-k-\ell}{n\choose
k}{n-k-2\choose\ell}\int_{X}dd^{c}\omega^{k}\wedge\omega^{n-k-2-\ell}\wedge\omega_{u}^{\ell}\wedge
du\wedge d^{c}u$ $\displaystyle=$
$\displaystyle-\sum_{\ell=0}^{n-3}\sum_{k=1}^{n-\ell-2}(-1)^{n-k-\ell}{n\choose
k}{n-k-2\choose\ell}\int_{X}dd^{c}\omega^{k}\wedge\omega^{n-k-2-\ell}\wedge\omega_{u}^{\ell}\wedge
du\wedge d^{c}u$ $\displaystyle=$
$\displaystyle-\sum_{\ell=0}^{n-3}\int_{X}\sum_{k=1}^{n-\ell-2}(-1)^{n-k-\ell}{n\choose
k}{n-k-2\choose\ell}dd^{c}\omega^{k}\wedge\omega^{n-k-2-\ell}\wedge\omega_{u}^{\ell}\wedge
du\wedge d^{c}u.$

Therefore, if

$\sum_{k=1}^{n-\ell-2}(-1)^{n-k-\ell}{n\choose
k}{n-k-2\choose\ell}dd^{c}\omega^{k}\wedge\omega^{n-k-2-\ell}\leq 0$

for any $\ell\in\\{0,\ldots,n-3\\}$, we get that
$v_{-}(\omega)\geq\int_{X}\omega^{n}$. ∎

###  3.2. Monge-Ampère bounds in dimension 3

We focus in this section on the $3$-dimensional setting and observe that the
condition $v_{-}(X,\omega_{X})>0$ is satisfied if $X$ admits a plurinegative
hermitian metric which is not necessarily Gauduchon. Note that the restriction
of a plurinegative metric yields a plurinegative metric, but the Gauduchon
condition is in general not preserved under restriction.

######  Theorem 3.3.

Let $X$ be a compact complex manifold of dimension $3$. If $X$ admits a
hermitian metric $\omega$ such that $dd^{c}\omega\leq 0$, then
$v_{-}(\omega)>0$.

In particular if $\omega$ is pluriclosed, then $0<v_{-}(\omega)\leq
v_{+}(\omega)<+\infty$.

###### Proof.

Assume by contradiction that there exists a sequence $(u_{j})$ such that
$u_{j}\in{\rm PSH}(X,\omega)\cap C^{\infty}(X)$, $\sup_{X}u_{j}=-1$ and

$\int_{X}(\omega+dd^{c}u_{j})^{3}\to 0.$

For each $j$, let $\phi_{j}\in{\rm
PSH}(X,\omega)\cap{\mathcal{C}}^{\infty}(X)$ be the unique solution to

$(\omega+dd^{c}\phi_{j})^{3}=c_{j}u_{j}^{2}\omega^{3}$

normalized by $\sup_{X}\phi_{j}=0$. Here $c_{j}>0$ is a positive constant. The
existence of $\phi_{j}$ (and that of $c_{j}$) follows from the main result of
[TW10]. Observe that $\int_{X}|u_{j}|^{4}\omega^{3}$ is uniformly bounded away
from $0$ and infinity. Thus by [KN15, Lemma 3.13 and Theorem 5.8],

$C^{-1}\leq c_{j}\leq C,\;\phi_{j}\geq-C,$

for some uniform constant $C>0$.

Let $B$ be a positive constant such that $d\omega\wedge d^{c}\omega\leq
B\omega^{3}.$ For $u,v\in PSH(X,\omega)\cap L^{\infty}(X)$ we set
$\omega_{u}:=\omega+dd^{c}u,\omega_{v}:=\omega+dd^{c}v$. When
$dd^{c}\omega\leq 0$ one obtains

$dd^{c}(\omega_{u}\wedge\omega_{v})=dd^{c}\omega\wedge\omega_{u}+dd^{c}\omega\wedge\omega_{v}+2d\omega\wedge
d^{c}\omega\leq 2B\omega^{3},$

hence

$dd^{c}(\omega_{u_{j}}^{2}+\omega_{v_{j}}^{2}+\omega_{u_{j}}\wedge\omega_{v_{j}})\leq
6d\omega\wedge d^{c}\omega\leq 6B\omega^{3}.$

Fix $t>1$ and set $v_{j}:=\max(u_{j},\phi_{j}-t)$. Using Stokes theorem we
obtain

$\displaystyle\int_{X}(\omega_{v_{j}}^{3}-\omega_{u_{j}}^{3})$
$\displaystyle=\int_{X}(v_{j}-u_{j})dd^{c}(\omega_{u_{j}}^{2}+\omega_{v_{j}}^{2}+\omega_{u_{j}}\wedge\omega_{v_{j}})$
$\displaystyle\leq
6B\int_{X}(v_{j}-u_{j})\omega^{3}=6B\int_{\\{u_{j}<\phi_{j}-t\\}}(v_{j}-u_{j})\omega^{3}$
$\displaystyle\leq 6B\int_{\\{u_{j}<\phi_{j}-t\\}}|u_{j}|\omega^{3}$
$\displaystyle\leq\frac{6B}{tc_{j}}\int_{\\{u_{j}<\phi_{j}-t\\}}c_{j}|u_{j}|^{2}\omega^{3}$
$\displaystyle\leq\frac{6BC}{t}\int_{\\{u_{j}<\phi_{j}-t\\}}\omega_{\phi_{j}}^{3}$
$\displaystyle\leq\frac{6BC}{t}\int_{X}\omega_{v_{j}}^{3}.$

In the last line we have used the identity

${{\bf 1}_{\\{u_{j}<\phi_{j}-t\\}}\omega_{\phi_{j}}^{3}=\bf
1}_{\\{u_{j}<\phi_{j}-t\\}}\omega_{v_{j}}^{3}\leq\omega_{v_{j}}^{3}.$

Choosing $t=12BC$ we obtain $\int_{X}\omega_{v_{j}}^{3}\leq
2\int_{X}\omega_{u_{j}}^{3}$. Using [GL21b, Proposition 3.4], we arrive at a
contradiction since the functions $v_{j}$ are uniformly bounded. ∎

###  3.3. Various obstructions

####  3.3.1. Pluriclosed and Plurinegative metrics

Constructing pluriclosed hermitian metrics on compact complex manifolds is a
problem that has attracted a lot of attention in the last decades. We observe
here a rigidity property of this condition.

######  Proposition 3.4.

Let $(X,\omega)$ be a compact hermitian manifold such that $dd^{c}\omega=0$.
If $dd^{c}\omega^{2}\leq 0$ (resp. $dd^{c}\omega^{2}\geq 0$) then
$dd^{c}\omega^{j}=0$ for all $j\geq 1$.

The restrictive condition $dd^{c}\omega=0$ & $dd^{c}\omega^{2}=0$ has been
introduced by Guan-Li [GL10], and further studied by Chiose [Chi16b], it is
equivalent to the preservation of the Monge-Ampères volumes,
$v_{+}(\omega)=v_{-}(\omega)=\int_{X}\omega^{n}$ (see Theorem 1.9 and Remark
1.10).

###### Proof.

It follows from Stokes theorem that $\int_{X}dd^{c}(\omega^{n-1})=0$. Now

$\displaystyle\frac{dd^{c}(\omega^{n-1})}{n-1}$ $\displaystyle=$
$\displaystyle\left\\{dd^{c}\omega\wedge\omega+(n-2)d\omega\wedge
d^{c}\omega\right\\}\wedge\omega^{n-3}$ $\displaystyle=$
$\displaystyle\left\\{\frac{n-2}{2}dd^{c}\omega^{2}-(n-3)dd^{c}\omega\wedge\omega\right\\}\wedge\omega^{n-3},$

hence

$\frac{n-2}{2}\int_{X}dd^{c}\omega^{2}\wedge\omega^{n-3}=(n-3)\int_{X}dd^{c}\omega\wedge\omega^{n-2}.$

The conclusion follows. ∎

An adaptation of the proof of Theorem 2.4 yields the following
characterization of the existence of plurinegative metrics [Eg01, Theorem
3.3].

######  Theorem 3.5.

The following properties are equivalent.

  * •

There exists a hermitian form $\omega$ such that $dd^{c}\omega\leq 0$.

  * •

There exists no positive current $\tau$ of bidimension $(2,2)$ such that
$S=dd^{c}\tau\geq 0$ with $S\neq 0$.

A similar obstruction for the existence of pluriclosed hermitian metrics is
well-known [Eg01, Theorem 3.3]: there exists a hermitian form $\omega$ such
that $dd^{c}\omega=0$ if and only if the only $dd^{c}$-exact positive current
$S$ of bidimension $(1,1)$ is $0$.

######  Remark 3.6.

It follows from the work of Ivashkovich that one can extend meromorphic maps
with values in compact non-Kähler manifolds endowed with a plurinegative
metric (see [Iv04, Theorem 2.2]).

####  3.3.2. Mutually exclusive conditions in dimension $3$

Let $X$ be a compact complex $3$-fold. If $X$ admits a pluripositive hermitian
metric $\omega$ and another plurinegative hermitian metric $\omega^{\prime}$,
then these are actually both pluriclosed. More generally, it follows from
Theorems 2.4 and 3.5 that we have the following alternative.

######  Corollary 3.7.

Let $X$ be a compact complex $3$-fold. The following conditions are mutually
exclusive:

  * •

$X$ admits a hermitian metric $\omega$ such that $dd^{c}\omega\geq 0$ and
$dd^{c}\omega\neq 0$;

  * •

$X$ admits a hermitian metric $\omega$ such that $dd^{c}\omega=0$;

  * •

$X$ admits a hermitian metric $\omega$ such that $dd^{c}\omega\leq 0$ and
$dd^{c}\omega\neq 0$;

  * •

$X$ does not admit any hermitian metric $\omega$ such that $dd^{c}\omega$ has
a sign.

We refer the reader to the examples to follow for an illustration of each
case.

###### Proof.

Assume $\omega$ is a hermitian form such that $dd^{c}\omega\geq 0$, while
$\omega^{\prime}$ is a hermitian form such that $dd^{c}\omega^{\prime}\leq 0$.
It follows from Theorem 2.4 and Theorem 3.5 that $dd^{c}\omega=0$ and
$dd^{c}\omega^{\prime}=0$. ∎

Recall that the class ${\mathcal{C}}$ of Fujiki consists of compact complex
manifolds that are bimeromorphic to a Kähler manifold. The non-existence of
plurisigned hermitian metric occurs on non-Kähler Fujiki $3$-folds, as we now
observe.

######  Proposition 3.8.

Let $X$ be a compact complex $3$-fold in the Fujiki class ${\mathcal{C}}$.
Then $X$ does not admit any plurisigned hermitian metric, unless $X$ is
Kähler.

A celebrated example of a non-Kähler $3$-fold bimeromorphic to
$\mathbb{C}\mathbb{P}^{3}$ has been provided by Hironaka [Hir60]. This shows
that [Eg01, Theorem 6.5] is incorrect.

###### Proof.

This is [Chi14, Theorem 2.3]. We include the proof for the reader’s
convenience. Let $X$ be a compact complex $3$-fold in the Fujiki class
${\mathcal{C}}$. Let $\omega$ be a pluripositive hermitian metric. There
exists a Kähler current on $X$, i.e. a positive closed current $T$ of bidegree
$(1,1)$ which dominates a hermitian form. Up to rescaling, we can assume
$T\geq\omega$.

Then $0\leq\omega\wedge dd^{c}\omega\leq T\wedge dd^{c}\omega$, hence

$0\leq\int_{X}\omega\wedge dd^{c}\omega\leq\int_{X}T\wedge dd^{c}\omega=0,$

as follows from Stokes theorem. Thus $\omega$ is pluriclosed and it follows
from [Chi14, Theorem 2.2] that $X$ is Kähler. The proof for plurinegative
metrics is similar. ∎

###  3.4. Locally conformally Kähler manifolds

We observe in this section that a large class of non-Kähler manifolds admits a
plurinegative hermitian metric $\omega$. This ensures that $v_{-}(\omega)>0$
when $\dim_{\mathbb{C}}X=3$. We then have a closer look at the special
subfamily of diagonal Hopf $3$-folds.

####  3.4.1. Existence of plurinegative metrics

Recall that a complex manifold $(X,\omega)$ is locally conformally Kähler
(lck) if one can find local smooth conformal factors $f_{j}>0$ in an open
cover $\\{U_{j}\\}$ of $X$ such that $f_{j}\omega$ is Kähler in $U_{j}$. Thus

$d\omega=-\frac{df_{j}}{f_{j}}\wedge\omega=d(-\log f_{j})\wedge\omega$

and these glue together into a globally well defined closed $1$-form $\theta$
such that $d\omega=\theta\wedge\omega$.

The Lee form $\theta$ is unique and defines a conformal invariant, in
particular it vanishes if and only if (a conformal multiple of) the metric
$\omega$ is Kähler. We refer the reader to [DO98] for an introduction to lcK
geometry, and to [OV11, OV20, Baz18, OV] for a more recent account.

######  Definition 3.9.

One says that a compact hermitian manifold $(X,\omega)$ is locally conformally
Kähler with potential if there exists a smooth positive plurisubharmonic
function $\varphi:\tilde{X}\rightarrow\mathbb{R}_{+}^{*}$ on the universal
cover $\tilde{X}$ of $X$ such that
$\pi^{*}\omega=\frac{dd^{c}\varphi}{\varphi}$ and $\frac{\varphi\circ
f}{\varphi}=c_{f}$ is constant for all deck transformations $f$.

The manifold $(X,\omega)$ is Vaisman if $\nabla\theta=0$, where $\nabla$
denotes the Levi-Civita connection associated to $\omega$. Vaisman manifolds
are lck with potential.

The $1$-form $\theta$ decomposes as $\theta=\alpha+\overline{\alpha}$ where
$\alpha$ is a $(1,0$)-form which is $\partial$-closed and such that
$\overline{\partial}\alpha=-\partial\overline{\alpha}$. Thus

$d\omega\wedge d^{c}\omega=i\alpha\wedge\overline{\alpha}\wedge\omega^{2}\geq
0.$

When $n=\dim_{\mathbb{C}}X\geq 3$, we observe moreover that this $(3,3)$-form
is non zero if $\alpha\neq 0$, which is the case if $X$ is not Kähler.

Since $d\omega\wedge d^{c}\omega\geq 0$ does not vanish, we have observed
after Theorem 1.9 that $dd^{c}\omega$ cannot be positive. We therefore
investigate whether $\omega$ is plurinegative.

######  Proposition 3.10.

Let $(X,\omega)$ be a compact hermitian manifold which is lcK with potential.
Let $\varphi:\tilde{X}\rightarrow\mathbb{R}_{+}^{*}$ be a smooth positive
plurisubharmonic function on the universal cover of $X$ such that
$\pi^{*}\omega=\frac{dd^{c}\varphi}{\varphi}$.

If $(X,\omega)$ is Vaisman then $\psi=\log\varphi$ is plurisubharmonic and for
all $k\geq 1$

$dd^{c}\omega^{k}=-k(dd^{c}\psi)^{k+1}\leq 0.$

In particular $dd^{c}\omega\leq 0$ and

  * •

$v_{-}(X,\omega)>0$ if $\dim_{\mathbb{C}}X=3$;

  * •

there is no pluripositive hermitian metric $\tilde{\omega}$ on $X$.

Conversely if $\psi=\log\varphi$ is plurisubharmonic then $(X,\omega)$ is
Vaisman.

The last statement of this Proposition is due to Ornea-Verbitsky [OV20,
Corollary 2.4].

###### Proof.

Slightly abusing notation we identify $\omega$ with the invariant form
$\pi^{*}\omega$. We set $\psi=\log\varphi$ and observe that

$dd^{c}\psi=\frac{dd^{c}\varphi}{\varphi}-\frac{d\varphi\wedge
d^{c}\varphi}{\varphi^{2}}\;\;\text{ and }\;\;d\psi\wedge
d^{c}\psi=\frac{d\varphi\wedge d^{c}\varphi}{\varphi^{2}}$

hence $\omega=dd^{c}\psi+d\psi\wedge d^{c}\psi$. Observe that $d\psi\wedge
d^{c}\psi$ has rank $1$, while $\omega$ has rank $n$, so the rank of
$dd^{c}\psi$ is at least $n-1$.

Since $d\psi\wedge d^{c}\psi$ has rank $1$, we obtain

$\omega^{k}=(dd^{c}\psi)^{k}+k(dd^{c}\psi)^{k-1}\wedge d\psi\wedge d^{c}\psi$

and

(3.1) $dd^{c}\omega^{k}=k(dd^{c}\psi)^{k-1}\wedge dd^{c}(d\psi\wedge
d^{c}\psi)=-k(dd^{c}\psi)^{k+1}\leq 0$

if $\psi=\log\varphi$ is psh. For $k=1$ we obtain $dd^{c}\omega\leq 0$, so it
follows from Theorem 3.3 that $v_{-}(\omega)>0$ when $\dim_{\mathbb{C}}X=3$.

Assume now that $\tilde{\omega}$ is a hermitian metric on $X$ such that
$dd^{c}\tilde{\omega}\geq 0$. It follows from Stokes theorem that

$0\leq\int_{X}dd^{c}\tilde{\omega}\wedge\omega^{n-2}=\int_{X}\tilde{\omega}\wedge
dd^{c}\omega^{n-2}\leq 0,$

hence $\tilde{\omega}$ is pluriclosed and
$-dd^{c}\omega^{n-2}=(dd^{c}\psi)^{n-1}\equiv 0$. The latter equality implies
that $dd^{c}\psi$ has rank $\leq n-2$, a contradiction.

It remains to understand when $\psi=\log\varphi$ is plurisubharmonic. Recall
that $d\omega=\theta\wedge\omega$ with $\theta=-d\psi$. Set
$\theta^{c}:=\frac{1}{2i}(\alpha-\overline{\alpha})=-d^{c}\psi$ so that
$d^{c}\omega=\theta^{c}\wedge\omega$. Thus

$\beta:=dd^{c}\psi=-d\theta^{c}=\omega-\theta\wedge\theta^{c}$

is a real $(1,1)$-form whose eigenvalues with respect to $\omega$ are $1$,
with multiplicity $(n-1)$, and $\lambda=1-|\theta|_{\omega}^{2}$, since
$\beta^{n}=\omega^{n}-n\omega^{n-1}\wedge\theta\wedge\theta^{c}=\left[1-|\theta|_{\omega}^{2}\right]\omega^{n}.$
Thus $\psi$ is psh if and only if $|\theta|_{\omega}^{2}\leq 1$.

When $X$ is Vaisman then $|\theta|_{\omega}^{2}$ is constant and there exists
a unique conformal choice of $\omega$ such that $|\theta|_{\omega}^{2}\equiv
1$ (see [V82, Remark p232]). In this case $\psi$ is psh hence $\beta\geq 0$
and $\beta^{n}\equiv 0$. Conversely it follows from (3.1) that

$\beta^{n}=(dd^{c}\psi)^{n}=-\frac{1}{n-1}dd^{c}\omega^{n-1}.$

Stokes theorem ensures that $\int_{X}\beta^{n}=0$. Thus either
$|\theta|_{\omega}^{2}\equiv 1$, then $X$ is Vaisman and $\omega$ is a
Gauduchon metric [OV20, Corollary 2.4], or $\lambda$ changes signs and $\psi$
is not plurisubharmonic. ∎

######  Remark 3.11.

The existence of plurinegative metrics on Vaisman manifolds can also be
deduced from the Ornea-Verbitsky Embedding theorem for Vaisman manifolds
[OV10] and the Example 3.12 below. More generally, manifolds admitting locally
conformally Kähler metrics with potential (this class includes Vaisman
manifolds too) can be holomorphically embedded into linear Hopf manifolds
[OV10]. Therefore the problem concerning the existence of plurinegative
hermitian metrics on locally conformally Kähler manifolds with potential is
reduced to study linear Hopf manifolds (which are Vaisman if and only if the
generator of the fundamental group is diagonalizable, see e.g. [OV, Theorem
16.3]).

######  Example 3.12.

Let $X$ be a diagonal Hopf manifold $X=\mathbb{C}^{n}\setminus\\{0\\}/\sim$,
where we identify $z$ and $Az$ in $\mathbb{C}^{n}\setminus\\{0\\}$, for some
diagonal matrix $A={\rm Diag}(\lambda_{1},\ldots,\lambda_{n})$ with entries
$\lambda_{j}\in\mathbb{D}^{*}$. We choose $\alpha_{i}:=\frac{\log
2}{2(-\log|\lambda_{i}|)}$ and set

$\varphi(z)=\sum_{i=1}^{n}|z_{i}|^{2\alpha_{i}},\;\;\text{ so that
}\;\;\psi=\log\varphi\in PSH(\mathbb{C}^{n}).$

Then $\varphi\circ A(z)=\frac{1}{2}\varphi(z)$ hence
$\omega(z):=\frac{dd^{c}\varphi(z)}{\varphi(z)}=dd^{c}\psi(z)+d\psi\wedge
d^{c}\psi(z)$ defines a hermitian form on $X$ such that $dd^{c}\omega\leq 0$.

####  3.4.2. The classical Hopf $3$-fold

We consider here $X=\mathbb{C}^{3}\setminus\\{0\\}/\sim$, where we identify
$z$ and $\lambda z$ in $\mathbb{C}^{3}\setminus\\{0\\}$, for some
$\lambda\in\mathbb{D}^{*}=\\{\zeta\in\mathbb{C},0<|\zeta|<1\\}$. There is a
natural holomorphic map $f:X\rightarrow\mathbb{P}^{2}$ with elliptic fibers.

Set $\eta=\sum_{j=1}^{n}\overline{z}_{j}dz_{j}=\partial|z|^{2}$. The invariant
$(1,1)$-form $|z|^{-4}i\eta\wedge\overline{\eta}$ compensates the lack of
positivity of $f^{*}\omega_{FS}$ so that the invariant $(1,1)$ form

$\omega=f^{*}\omega_{FS}+|z|^{-2}i\eta\wedge\overline{\eta}=\frac{\sum_{j=1}^{n}idz_{j}\wedge
d\overline{z}_{j}}{|z|^{2}}=\frac{dd^{c}\varphi}{\varphi}$

induces a hermitian form on $X$ that we still denote by $\omega$. Since
$\log\varphi=\log|z|\in PSH(\mathbb{C}^{3}\setminus\\{0\\})$, it follows from
Proposition 3.10 that $dd^{c}\omega\leq 0$.

We would like to test the finiteness of $v_{+}(\omega)$. It follows from the
proof of Theorem 3.3.1 that there exists $C>0$ such that for any $u\in
PSH(X,\omega)$ normalized by $\sup_{X}u=0$, we have

$3\int_{X}(-dd^{c}\omega)\wedge du\wedge
d^{c}u-C\leq\int_{X}(\omega+dd^{c}u)^{3}\leq 3\int_{X}(-dd^{c}\omega)\wedge
du\wedge d^{c}u+C.$

It is thus tempting to think that one can reach $v_{+}(\omega)=+\infty$ by
constructing $\omega$-psh functions whose gradient does not belong to $L^{2}$.
There are several functions $v\in PSH(\mathbb{P}^{2},\omega_{FS})$ whose
gradient does not belong to $L^{2}$; they induce functions

$u=v\circ f\in PSH(X,f^{*}\omega_{FS})\subset PSH(X,\omega)$

with gradient $\nabla u\notin L^{2}(X)$. However $dv\circ f$ is proportional
to $\eta$, hence

$\int_{X}(-dd^{c}\omega)\wedge d(v\circ f)\wedge d^{c}(v\circ f)=0.$

One can construct singular $\omega$-psh functions that do not come from
$\mathbb{P}^{2}$ as follows: the function $\rho=\log dist_{\omega}(\cdot,p)$
is psh near $p\in X$ and smooth off $p$. We multiply it by a cut-off function
$\chi$ which is identically equal to $1$ near $p$. For $\varepsilon>0$ small,
the function $\varphi=\varepsilon\chi\rho$ belong to $PSH(X,\omega)$ and it
has a logarithmic singularity at $p$. However $\nabla\varphi\in L^{2}(X)$ as
the singularity of $\varphi$ is isolated. One could consider a convergent
series of such functions, this would produce examples with a discrete set of
logarithmic singularities (and possibly an uncountable polar set
$(\varphi=-\infty)$), but their global gradient would still belong to $L^{2}$.

######  Question 3.13.

Does one have $v_{+}(\omega)<+\infty$?

We tend to expect a positive answer to this problem, but a negative one would
be quite interesting as well!

##  4\. Homogeneous examples

We finally study several classes of compact (locally) homogeneous manifolds.

###  4.1. Complex parallelizable manifolds

Let $X$ be a compact complex manifolds of dimension $n$ that is complex
parallelizable, i.e. such that the holomorphic tangent bundle $T^{1,0}X$ is
holomorphically trivial. It has been shown by Wang [Wan54] that
$X=\Gamma\backslash G$ is the quotient of a connected and simply connected
complex Lie group $G$ by a discrete subgroup $\Gamma$, and that complex tori
(for which $G=\mathbb{C}^{n}$) are the only ones that are Kähler. On the other
side, they always admits balanced metrics by [AG86, Proposition 3.1] and they
cannot admit pluriclosed metrics, see [FGV19, page 7110].

######  Proposition 4.1.

A compact complex parallelizable manifold $X$ admits a hermitian metric
$\omega$ such that $dd^{c}\omega^{k}\geq 0$ for all $k\geq 1$.

In particular, $v_{+}(X,\omega)<+\infty$ when when $\dim_{\mathbb{C}}X=3$.
Moreover $X$ does not admit any plurinegative hermitian metric unless it is a
complex torus.

Complex parallelizable manifolds that arise as quotients of complex semisimple
Lie groups (e.g. compact quotients of $SL(2;\mathbb{C})$ by a lattice) have
been studied in [Yac98]. For a classification of compact complex
parallelizable solvmanifolds in complex dimension $3$ –as well as a partial
classification in dimension $4$ and $5$–, we refer the reader to [Nak75,
Theorem 1 and Section 6].

###### Proof.

By definition there exist $\varphi_{1},\ldots,\varphi_{n}\in
H^{0}(X,\Omega_{X}^{1})$ holomorphic $1$-forms that are linearly independent
at each point. We set

$\omega:=\sum_{j=1}^{n}i\varphi_{j}\wedge\overline{\varphi_{j}}.$

This is a hermitian $(1,1)$-form such that

$dd^{c}\omega=i\partial\overline{\partial}\omega=\sum_{j=1}^{n}i^{2}\partial\overline{\partial}(\varphi_{j}\wedge\overline{\varphi_{j}}).$

Since $\overline{\partial}\varphi_{j}=0$ and
$\partial\overline{\varphi_{j}}=0$ we obtain, setting
$\eta_{j}=\partial\varphi_{j}$,

$dd^{c}\omega=\sum_{j=1}^{n}-i^{2}\eta_{j}\wedge\overline{\eta_{j}}\geq 0.$

Recall indeed that if $\eta$ is a $(2,0)$-form then
$-i^{2}\eta\wedge\overline{\eta}=i^{4}\eta\wedge\overline{\eta}$ is a weakly
positive $(2,2)$-form (that is strongly positive if
$\eta=\alpha_{1}\wedge\alpha_{2}$ is decomposable).

For $k\geq 1$, we decompose

$\omega^{k}=\sum_{j_{1},\ldots,j_{k}=1}^{n}i\varphi_{j_{1}}\wedge\overline{\varphi_{j_{1}}}\wedge\cdots\wedge
i\varphi_{j_{k}}\wedge\overline{\varphi_{j_{k}}}=\sum_{J=(j_{1},\ldots,j_{k})}i^{k^{2}}\alpha_{J}\wedge\overline{\alpha_{J}},$

where $\alpha_{J}=\varphi_{j_{1}}\wedge\cdots\wedge\varphi_{j_{k}}$ is a
holomorphic $k$-form. Thus

$i\partial\overline{\partial}\omega^{k}=\sum_{J=(j_{1},\ldots,j_{k})}i^{1+k^{2}}(-1)^{k}\beta_{J}\wedge\overline{\beta_{J}},$

with $\beta_{J}=\partial\alpha_{J}$. Since
$i^{1+k^{2}}(-1)^{k}=i^{(k+1)^{2}}$, we conclude that $dd^{c}\omega^{k}\geq
0$.

Observe that $dd^{c}\omega\neq 0$ unless $\omega$ is Kähler, in which case $X$
is a complex torus. It follows therefore from Theorem 3.3 and Theorem 3.5 that
$v_{+}(X,\omega)<+\infty$ when $\dim_{\mathbb{C}}X=3$, and $X$ does not admit
a plurinegative hermitian metric unless it is a torus. ∎

In dimension higher than $3$, the conditions needed to control $v_{+}$ as in
Theorem 2.2 are not trivially satisfied. For example, take the complex
parallelizable manifold of complex dimension $n=4$ characterized by a coframe
of holomorphic $1$-forms $(\varphi^{j})_{j\in\\{1,2,3,4\\}})$ with structure
equations

$d\varphi^{1}=d\varphi^{2}=d\varphi^{3}=0,\quad
d\varphi^{4}=-\varphi^{2}\wedge\varphi^{3},$

see [Nak75, Type IV.2, page 108]. The metric
$\omega=\sum_{j=1}^{4}\sqrt{-1}\varphi^{j}\wedge\bar{\varphi}^{j}$ satisfies:

$\displaystyle dd^{c}\omega$ $\displaystyle=$
$\displaystyle\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}\geq
0,$ $\displaystyle dd^{c}\omega^{2}$ $\displaystyle=$ $\displaystyle
2\,\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}\geq
0,$

but

$\displaystyle dd^{c}\omega^{2}-\varepsilon dd^{c}\omega\wedge\omega$
$\displaystyle=$
$\displaystyle(2-\varepsilon)\,\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}$
$\displaystyle-\varepsilon\,\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}\wedge\sqrt{-1}\varphi^{4}\wedge\bar{\varphi}^{4}$

is never semi-positive for $\varepsilon>0$.

The following is an example of a complex parallelizable manifold admitting
metrics with the property of Theorem 2.2: if a lattice can be provided for the
Lie group, then we would get a compact manifold with $v_{+}<\infty$.

######  Example 4.2.

Consider the complex parallelizable solvmanifold of complex dimension $n=4$ of
type IV.5 in [Nak75, page 108], which is characterized by a coframe of
holomorphic $1$-forms with structure equations

$d\varphi_{1}=0,\quad d\varphi^{2}=\varphi^{1}\wedge\varphi^{2},$
$d\varphi^{3}=\alpha\varphi^{1}\wedge\varphi^{3},\quad
d\varphi^{4}=-(1+\alpha)\varphi^{1}\wedge\varphi^{4},$

depending on a parameter $\alpha\in\mathbb{R}\setminus\\{-1,0\\}$. It is not
known whether it admits lattices, see [Nak75, page 110]. Consider the left-
invariant metric

$\omega=a_{1}\,\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}+a_{2}\,\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}+a_{3}\,\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}+a_{4}\,\sqrt{-1}\varphi^{4}\wedge\bar{\varphi}^{4},$

where $a_{1},a_{2},a_{3},a_{4}>0$. A straightforward computation yields

$\displaystyle dd^{c}\omega$ $\displaystyle=$ $\displaystyle
a_{2}\,\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}+a_{3}|\alpha|^{2}\,\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}$
$\displaystyle+a_{4}|\alpha+1|^{2}\,\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{4}\wedge\bar{\varphi}^{4}\geq
0,$ $\displaystyle dd^{c}\omega^{2}$ $\displaystyle=$ $\displaystyle
2\,a_{2}a_{3}|\alpha+1|^{2}\,\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}$
$\displaystyle+2\,a_{2}a_{4}|\alpha|^{2}\,\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{4}\wedge\bar{\varphi}^{4}$
$\displaystyle+2\,a_{3}a_{4}\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}\wedge\sqrt{-1}\varphi^{4}\wedge\bar{\varphi}^{4}\geq
0.$

Therefore

$\displaystyle dd^{c}\omega^{2}-\varepsilon\omega\wedge dd^{c}\omega$
$\displaystyle=$
$\displaystyle\left(-|\alpha|^{2}\varepsilon+2|\alpha+1|^{2}-\varepsilon\right)a_{2}a_{3}\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}$
$\displaystyle+\left(-|\alpha+1|^{2}\varepsilon+2|\alpha|^{2}-\varepsilon\right)a_{2}a_{4}\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{4}\wedge\bar{\varphi}^{4}$
$\displaystyle+\left(-|\alpha+1|^{2}\varepsilon-|\alpha|^{2}\varepsilon+2\right)a_{3}a_{4}\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}\wedge\sqrt{-1}\varphi^{4}\wedge\bar{\varphi}^{4}.$

Therefore, if we take

$0<\varepsilon\leq\min\left\\{\frac{2|\alpha+1|^{2}}{1+|\alpha|^{2}},\frac{2|\alpha|^{2}}{1+|\alpha+1|^{2}},\frac{2}{|\alpha|^{2}+|1+\alpha|^{2}}\right\\},$

we obtain $dd^{c}\omega^{2}-\varepsilon\omega\wedge dd^{c}\omega\geq 0$.

###  4.2. Six-dimensional nilmanifolds

In this section, we consider nilmanifolds, namely, compact quotients
$\Gamma\backslash G$ of connected simply-connected nilpotent Lie groups $G$ by
co-compact discrete subgroups $\Gamma$. We recall that a Lie group $G$ is
called nilpotent if its associated Lie algebra $\mathfrak{g}$ satisfies that
the lower central series
$\\{\mathfrak{g}_{j}:=[\mathfrak{g},\mathfrak{g}_{j-1}]\\}_{j\in\mathbb{N}}$,
with $\mathfrak{g}_{0}:=\mathfrak{g}$, eventually vanishes. In dimension $6$,
according to [Mor58, Mag86], there are only $34$ isomorphism classes of
nilpotent Lie algebras over $\mathbb{R}$, and $10$ of them are reducible.

We consider left-invariant complex structures on $\Gamma\backslash G$, i.e.
complex structures that are induced by complex structures on $G$ being
invariant under left-translations. Equivalently, left-invariant complex
structures correspond to linear complex structures on the corresponding Lie
algebra satisfying an integrability condition. According to [Sal01], only $18$
of the above $34$ nilpotent Lie algebras admit left-invariant complex
structures. We notice that the existence of lattices in nilpotent Lie groups
is well-understood thanks to [Mal45], more precisely it corresponds to having
rational constant structures, a condition that is satisfied in all the above
considered cases.

Among the latter, only four may admit a hermitian metric $\omega$ which is
pluriclosed, as shown by Fino-Parton-Salamon in [FPS04]. We show in this
section that the remaining 14 classes all admit a pluripositive hermitian
metric, and we also analyze whether there exists a balanced metric, i.e. a
hermitian metric $\omega$ such that $d\omega^{2}=0$.

Left-invariant complex structures on six-dimensional nilmanifolds are gathered
in four families in [ABD11, Uga07, UV14, COUV16], some of them depending on
continuous parameters, up to linear equivalence. These families are described
by a coframe of left-invariant $(1,0)$-forms
$(\varphi^{1},\varphi^{2},\varphi^{3})$ with structure equations as follows
(we use the short-hands
$\varphi^{1\bar{2}}:=\varphi^{1}\wedge\bar{\varphi}^{2}$, etc.).

(Np):

$d\varphi^{1}=d\varphi^{2}=0$, $d\varphi^{3}=\rho\,\varphi^{12}$ where
$\rho\in\\{0,1\\}$; these are the complex parallelizable structures [Wan54,
Nak75]. The manifold is a compact quotient of a complex Lie group by a
discrete subgroup; $\rho=0$ corresponds to the complex torus, while $\rho=1$
corresponds to the Iwasawa manifold [FG86].

(Ni):

$d\varphi^{1}=d\varphi^{2}=0$,
$d\varphi^{3}=\rho\,\varphi^{12}+\varphi^{1\bar{1}}+\lambda\,\varphi^{1\bar{2}}+D\,\varphi^{2\bar{2}}$
where $\rho\in\\{0,1\\}$, $\lambda\in\mathbb{R}^{\geq 0}$, $D\in\mathbb{C}$
with $\Im D\geq 0$; this class (and the following) contains nilpotent complex
structures: the ascending series
$\\{\mathfrak{a}_{j}(J):=\\{X\in\mathfrak{g}:[X,\mathfrak{g}]+[JX,\mathfrak{g}]\subseteq\mathfrak{a}_{j-1}\\}\\}_{j\in\mathbb{N}}$,
with $\mathfrak{a}_{0}(J):=0$, eventually equals $\mathfrak{g}$. The case
$\rho=0$ corresponds to Abelian complex structures, namely, the subalgebra of
left-invariant $(1,0)$-vector fields is Abelian.

(Nii):

$d\varphi^{1}=0$, $d\varphi^{2}=\varphi^{1\bar{1}}$,
$d\varphi^{3}=\rho\varphi^{12}+B\,\varphi^{1\bar{2}}+c\,\varphi^{2\bar{1}}$
where $\rho\in\\{0,1\\}$, $B\in\mathbb{C}$, $c\in\mathbb{R}^{\geq 0}$ with
$(\rho,B,c)\neq(0,0,0)$.

(Niii):

$d\varphi^{1}=0$, $d\varphi^{2}=\varphi^{13}+\varphi^{1\bar{3}}$,
$d\varphi^{3}=\sqrt{-1}\rho\,\varphi^{1\bar{1}}\pm\sqrt{-1}(\varphi^{1\bar{2}}-\varphi^{2\bar{1}})$
where $\rho\in\\{0,1\\}$; this class contains non-nilpotent complex
structures.

We refer to [ABD11, Uga07, UV14, COUV16] and [AOUV, Table 1] for more details
on the underlying Lie algebras.

We can restrict to study left-invariant hermitian structures on
$\Gamma\backslash G$, namely, hermitian structures on $G$ being invariant by
the whole action of $G$ by left-translations. Such structures correspond to
linear hermitian structures on the associated Lie algebra. Indeed Belgun
Symmetrization Trick [Bel00, Theorem 7] (see also [FG04, Theorem 2.1])
provides the following.

######  Lemma 4.3 ([Uga07, Proposition 3.6]).

Let $\Gamma\backslash G$ be a compact quotient of a Lie group $G$ endowed with
a left-invariant complex structure. If it admits a plurisigned hermitian
metric, then it admits a left-invariant plurisigned hermitian metric.

###### Proof.

The case of pluriclosed metrics is [Uga07, Proposition 3.6]. The same argument
applies here; we include a proof for the readers’ convenience.

Since $G$ admits a compact quotient, it is unimodular [Mil76, Lemma 6.2],
namely its left-invariant Haar measure $\mu$ is also right-invariant. We
consider the symmetrization map

$\mu\colon\wedge^{\bullet}(\Gamma\backslash
G)\to\wedge^{\bullet}\mathfrak{g}^{*},$
$\mu(\alpha)(X_{1},\ldots,X_{k}):=\int_{\Gamma\backslash
G}\alpha(m)(X_{1}(m),\ldots,X_{k}(m))\mu(m),$

where $\wedge^{\bullet}\mathfrak{g}^{*}$ is identified with the subspace of
left-invariant forms. It is clear that
$\mu|_{\wedge^{\bullet}\mathfrak{g}^{*}}=\mathrm{id}$ and that
$d\circ\mu=\mu\circ d$. Since $J$ is left-invariant, we have $\mu\circ
J=J\circ\mu$. Therefore, there hold also $\partial\circ\mu=\mu\circ\partial$
and $\overline{\partial}\circ\mu=\mu\circ\overline{\partial}$.

In particular, if $\omega$ is a hermitian structure, then $\mu(\omega)$ is a
left-invariant $(1,1)$-form. It is straightforward to check that
$\mu(\omega)>0$ is still a hermitian structure. We also have:
$\partial\overline{\partial}(\mu(\omega))=\mu(\partial\overline{\partial}\omega)$.
In particular if $\omega$ is pluripositive (resp. plurinegative), then
$\mu(\omega)$ is pluripositive (resp. plurinegative). Indeed, assume that
$\omega$ is pluripositive. Then, for any $(1,0)$-form $\eta$

$dd^{c}\omega\wedge\sqrt{-1}\eta\wedge\bar{\eta}=c\sqrt{-1}\varphi^{1\bar{1}}\wedge\sqrt{-1}\varphi^{2\bar{2}}\wedge\sqrt{-1}\varphi^{3\bar{3}}$

for $c\geq 0$. We can restrict to left-invariant $(1,0)$-forms $\eta$, since
positivity is a pointwise notion. Then, thanks to [AK17, Lemma 2.5],
$\mu(\omega)\wedge\eta=\mu(\omega\wedge\eta)=\mu(c)\sqrt{-1}\varphi^{1\bar{1}}\wedge\sqrt{-1}\varphi^{2\bar{2}}\wedge\sqrt{-1}\varphi^{3\bar{3}}$
where $\mu(c)\geq 0$. ∎

We therefore only consider left-invariant hermitian structures. With respect
to a chosen coframe, it is straighforward to check that they are of the form

$\displaystyle 2\,\omega$ $\displaystyle=$
$\displaystyle\sqrt{-1}r^{2}\,\varphi^{1\bar{1}}+\sqrt{-1}s^{2}\,\varphi^{2\bar{2}}+\sqrt{-1}t^{2}\,\varphi^{3\bar{3}}$
$\displaystyle+u\,\varphi^{1\bar{2}}-\bar{u}\,\varphi^{2\bar{1}}+v\,\varphi^{2\bar{3}}-\bar{v}\,\varphi^{3\bar{2}}+z\,\varphi^{1\bar{3}}-\bar{z}\,\varphi^{3\bar{1}}$

where $r,s,t\in\mathbb{R}$, $u,v,z\in\mathbb{C}$ satisfy

$r^{2}>0,\qquad s^{2}>0,\qquad t^{2}>0,$ $r^{2}s^{2}>|u|^{2},\qquad
r^{2}t^{2}>|z|^{2},\qquad s^{2}t^{2}>|v|^{2},$
$r^{2}s^{2}t^{2}+2\Re(\sqrt{-1}\,\bar{u}z\bar{v})>t^{2}|u|^{2}+r^{2}|v|^{2}+s^{2}|z|^{2},$

in order for the metric to be positive-definite.

We compute $\sqrt{-1}\partial\overline{\partial}\omega$ for the above
families. Note that positivity and strong positivity are equivalent for
$(2,2)$-forms in dimension $3$ (see [Mic82, p. 279–280]), and that it is
enough to test positivity pairing with left-invariant forms, since it is a
pointwise notion. Recall that, by [BG88, Has89], Kähler metrics do not exist
on non-tori nilmanifolds (even with complex structures that are possibly non-
left-invariant).

(Np):

For complex paralellizable type, $\omega$ is always balanced and

$\sqrt{-1}\partial\overline{\partial}\omega=\frac{1}{2}\rho^{2}t^{2}\sqrt{-1}\varphi^{1\bar{1}}\wedge\sqrt{-1}\varphi^{2\bar{2}}$

is always strongly-positive. More precisely, any left-invariant hermitian
metric on the torus is pluriclosed (in fact, Kähler), and any left-invariant
hermitian metric on the Iwasawa manifold is pluripositive, balanced, non-
pluriclosed. The underlying Lie algebras are respectively $\mathfrak{h}_{1}$
and $\mathfrak{h}_{5}$, in the notation of [Sal01], to which we refer for more
details.

(Ni):

For nilpotent type in Family I, the balanced condition is equivalent to

$s^{2}t^{2}+{\left(r^{2}t^{2}-|z|^{2}\right)}D+{\left(-\sqrt{-1}\,t^{2}\overline{u}+v\overline{z}\right)}\lambda-|v|^{2}=0$

(it is satisfied on $\mathfrak{h}_{2}$, $\mathfrak{h}_{3}$,
$\mathfrak{h}_{4}$, $\mathfrak{h}_{5}$, $\mathfrak{h}_{6}$ for metrics such
that $r^{2}=1$, $v=z=0$, and $s^{2}+D=\sqrt{-1}\,\bar{u}\,\lambda$) and

$\sqrt{-1}\partial\overline{\partial}\omega=\frac{t^{2}}{2}\left(\lambda^{2}+\rho^{2}-2\Re
D\right)\sqrt{-1}\varphi^{1\bar{1}}\wedge\sqrt{-1}\varphi^{2\bar{2}}.$

Any hermitian metric is either

  * •:

pluripositive non-pluriclosed (when $2\Re D<\lambda^{2}+\rho^{2}$, which
happens on Lie algebras $\mathfrak{h}_{2}$, $\mathfrak{h}_{3}$,
$\mathfrak{h}_{4}$, $\mathfrak{h}_{5}$, $\mathfrak{h}_{6}$),

  * •:

or pluriclosed (when $2\Re D=\lambda^{2}+\rho^{2}$, which happens on Lie
algebras $\mathfrak{h}_{2}$, $\mathfrak{h}_{4}$, $\mathfrak{h}_{5}$,
$\mathfrak{h}_{8}$),

  * •:

or plurinegative non-pluriclosed (when $2\Re D>\lambda^{2}+\rho^{2}$, which
happens on Lie algebras $\mathfrak{h}_{2}$, $\mathfrak{h}_{3}$,
$\mathfrak{h}_{4}$, $\mathfrak{h}_{5}$).

(Nii):

for the complex structures of nilpotent type in Family II, the balanced
condition is never satisfied, and

$\sqrt{-1}\partial\overline{\partial}\omega=\frac{t^{2}}{2}\,\big{(}c^{2}+\rho^{2}+|B|^{2}\big{)}\sqrt{-1}\varphi^{1\bar{1}}\wedge\sqrt{-1}\varphi^{2\bar{2}}$

is always strongly-positive, and never zero. The underlying Lie algebras are
$\mathfrak{h}_{7}$, $\mathfrak{h}_{9}$, $\mathfrak{h}_{10}$,
$\mathfrak{h}_{11}$, $\mathfrak{h}_{12}$, $\mathfrak{h}_{13}$,
$\mathfrak{h}_{14}$, $\mathfrak{h}_{15}$, $\mathfrak{h}_{16}$.

(Niii):

For the complex structures of nilpotent type in Family II, the balanced
condition is equivalent to

$\left\\{\begin{array}[]{l}t^{2}\Re u+\Im(z\overline{v})=0\\\
\sqrt{-1}\,s^{2}\overline{z}+\overline{u}\overline{v}=0\\\
\rho=0\end{array}\right.$

(it is satisfied on $\mathfrak{h}_{19}^{-}$ for metrics such that $u=z=0$) and

$\sqrt{-1}\partial\overline{\partial}\omega=t^{2}\sqrt{-1}\varphi^{1\bar{1}}\wedge\sqrt{-1}\varphi^{2\bar{2}}+s^{2}\sqrt{-1}\varphi^{1\bar{1}}\wedge\sqrt{-1}\varphi^{3\bar{3}}$

is always strongly-positive, and never zero. The underlying Lie algebras are
$\mathfrak{h}_{19}^{-}$, $\mathfrak{h}_{26}^{+}$.

The following statement summarizes the previous discussion, it can be seen as
an extension of [FPS04] to plurisigned metrics.

######  Theorem 4.4.

Consider a six-dimensional nilmanifold $X$ endowed with a left-invariant
complex structure and assume $X$ is not a complex torus. Then, there is always
a plurisigned metric. More precisely:

  * •

either $X$ belongs to one of the families (Np), (Nii) and (Niii) then any
left-invariant hermitian metric is pluripositive but not pluriclosed.

  * •

or $X$ belongs to (Ni) and –depending on the complex structure–, every left-
invariant hermitian metric is either pluriclosed, or pluripositive but not
pluriclosed, or else plurinegative but not pluriclosed.

By Corollary 3.7 the latter three conditions are mutually exclusive.

###  4.3. Higher dimensional nilmanifolds

As noticed in [FPS04], the pluriclosed condition for left-invariant metrics on
six-dimensional nilmanifolds depends only on the complex structure, and we
noticed the same behavior for the plurisigned condition in Theorem 4.4.

This is no longer true in higher dimension for the pluriclosed condition
[EFV12, Remark 4.1]. More precisely, $8$-dimensional nilmanifolds with left-
invariant complex structures admitting pluriclosed metrics are classified into
two families in [EFV12, Section 4]. For the second family, the pluriclosed
condition only depends on the complex structure, while for the first family it
also involves the parameters of the metric. We make a similar observation for
the plurisigned condition.

######  Example 4.5.

We consider here the eight-dimensional nilmanifold with left-invariant complex
structure characterized by the structure equations

$d\varphi^{1}=0,\quad d\varphi^{2}=0,\quad
d\varphi^{3}=\varphi^{1}\wedge\bar{\varphi}^{1}+\frac{1}{2}\varphi^{2}\wedge\bar{\varphi}^{2},\quad
d\varphi^{4}=-\varphi^{1}\wedge\bar{\varphi}^{2},$

with respect to a left-invariant coframe
$\\{\varphi^{1},\varphi^{2},\varphi^{3},\varphi^{4}\\}$ of $(1,0)$-forms. It
belongs to the first family in the above mentioned classification, more
precisely, it corresponds to parameters $B_{4}=1$, $C_{4}=\frac{1}{2}$,
$F_{5}=-1$, the others zero, in the notation of [EFV12]. We consider a
Hermitian metric of the diagonal form

$\omega=\sqrt{-1}\,a_{1}\varphi^{1}\wedge\bar{\varphi}^{1}+\sqrt{-1}\,a_{2}\varphi^{2}\wedge\bar{\varphi}^{2}+\sqrt{-1}\,a_{3}\varphi^{3}\wedge\bar{\varphi}^{3}+\sqrt{-1}\,a_{4}\varphi^{4}\wedge\bar{\varphi}^{4},$

where $a_{1},a_{2},a_{3},a_{4}>0$. Observe that

$dd^{c}\omega=(-a_{3}+a_{4})\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2},$

showing that they can be either pluripositive, or pluriclosed, or
plurinegative, depending on the value of $a_{3}/a_{4}$.

Note that this example does not satisfy the condition of Theorem 2.2. Indeed,

$\displaystyle dd^{c}\omega^{2}-\varepsilon dd^{c}\omega\wedge\omega$
$\displaystyle=$ $\displaystyle\left(-\varepsilon
a_{3}(a_{4}-a_{3})+2a_{3}a_{4}\right)\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}$
$\displaystyle+\left(-\varepsilon
a_{4}(a_{4}-a_{3})-2a_{3}a_{4}\right)\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{4}\wedge\bar{\varphi}^{4}$

is not positive.

We now provide an $8$-dimensional example which satisfies the curvature
conditions of Theorem 2.2.

######  Example 4.6.

We consider again an example among the eight-dimensional nilmanifolds in the
first family of [EFV12]. More precisely, take parameters $B_{1}=1$, $G_{3}=1$,
the others zero, that is consider the structure equations

$d\varphi^{1}=0,\quad d\varphi^{2}=0,\quad
d\varphi^{3}=\varphi^{1}\wedge\varphi^{2},\quad
d\varphi^{4}=\varphi^{2}\wedge\bar{\varphi}^{1}.$

Take the diagonal metric

$\omega=\sqrt{-1}\,a_{1}\varphi^{1}\wedge\bar{\varphi}^{1}+\sqrt{-1}\,a_{2}\varphi^{2}\wedge\bar{\varphi}^{2}+\sqrt{-1}\,a_{3}\varphi^{3}\wedge\bar{\varphi}^{3}+\sqrt{-1}\,a_{4}\varphi^{4}\wedge\bar{\varphi}^{4},$

where $a_{1},a_{2},a_{3},a_{4}>0$. We compute

$\displaystyle dd^{c}\omega$ $\displaystyle=$
$\displaystyle(a_{3}+a_{4})\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\geq
0,$ $\displaystyle dd^{c}\omega^{2}$ $\displaystyle=$ $\displaystyle
2\,a_{3}a_{4}\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}$
$\displaystyle+2\,a_{3}a_{4}\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{4}\wedge\bar{\varphi}^{4},$

hence

$\displaystyle dd^{c}\omega^{2}-\varepsilon dd^{c}\omega\wedge\omega$
$\displaystyle=$ $\displaystyle a_{3}(-\varepsilon
a_{3}+(2-\varepsilon)a_{4})\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{3}\wedge\bar{\varphi}^{3}$
$\displaystyle+a_{4}((2-\varepsilon)a_{3}-\varepsilon
a_{4})\sqrt{-1}\varphi^{1}\wedge\bar{\varphi}^{1}\wedge\sqrt{-1}\varphi^{2}\wedge\bar{\varphi}^{2}\wedge\sqrt{-1}\varphi^{4}\wedge\bar{\varphi}^{4},$

which is non-negative for suitable choices of $a_{3},a_{4},\varepsilon$ (take
e.g. $a_{3}=a_{4}=1$, $\varepsilon\leq\frac{1}{2}$). In particular,
$v_{+}(\omega)<+\infty$.

###  4.4. Six-dimensional solvmanifolds with trivial canonical bundle

We now consider solvmanifolds –i.e. compact quotients of a connected solvable
Lie group by a closed subgroup– of real dimension $6$ admitting a left-
invariant complex structure with holomorphically trivial canonical bundle.

According to [FOU15] such complex structures are either nilmanifolds as above
(see [Sal01, Theorem 1.3] and [BDV09, Theorem 2.7]) or belong to one of the
classes below. We fix a coframe $\\{\varphi^{1},\varphi^{2},\varphi^{3}\\}$ of
left-invariant $(1,0)$-forms and use the same notations as in the previous
section.

(Si):

$d\varphi^{1}=A\varphi^{13}+A\varphi^{1\bar{3}}$,
$d\varphi^{2}=-A\varphi^{23}-A\varphi^{2\bar{3}}$, $d\varphi^{3}=0$, where
$A=\cos\theta+{\sqrt{-1}}\sin\theta,\theta\in[0,\pi)$. The complex structures
in this family are of splitting type in the sense of [Kas13, Assumption 1],
see [AOUV17].

(Sii):

$d\varphi^{1}=0$,
$d\varphi^{2}=-\frac{1}{2}\varphi^{13}-\big{(}\frac{1}{2}+\sqrt{-1}x\big{)}\varphi^{1\bar{3}}+\sqrt{-1}x\,\varphi^{3\bar{1}}$,
and
$d\varphi^{3}=\frac{1}{2}\varphi^{12}+\big{(}\frac{1}{2}-\frac{\sqrt{-1}}{4x}\big{)}\varphi^{1\bar{2}}+\frac{\sqrt{-1}}{4x}\varphi^{2\bar{1}}$,
where $x\in\mathbb{R}^{>0}$.

(Siii1):

$d\varphi^{1}=\sqrt{-1}(\varphi^{13}+\varphi^{1\bar{3}})$,
$d\varphi^{2}=-\sqrt{-1}(\varphi^{23}+\varphi^{2\bar{3}})$,
$d\varphi^{3}=\pm\varphi^{1\bar{1}}$.

(Siii2):

$d\varphi^{1}=\varphi^{13}+\varphi^{1\bar{3}}$,
$d\varphi^{2}=-\varphi^{23}-\varphi^{2\bar{3}}$,
$d\varphi^{3}=\varphi^{1\bar{2}}+\varphi^{2\bar{1}}$.

(Siii3):

$d\varphi^{1}=\sqrt{-1}(\varphi^{13}+\varphi^{1\bar{3}})$,
$d\varphi^{2}=-\sqrt{-1}(\varphi^{23}+\varphi^{2\bar{3}})$,
$d\varphi^{3}=\varphi^{1\bar{1}}+\varphi^{2\bar{2}}$.

(Siii4):

$d\varphi^{1}=\sqrt{-1}(\varphi^{13}+\varphi^{1\bar{3}})$,
$d\varphi^{2}=-\sqrt{-1}(\varphi^{23}+\varphi^{2\bar{3}})$,
$d\varphi^{3}=\pm(\varphi^{1\bar{1}}-\varphi^{2\bar{2}})$.

(Siv1):

$d\varphi^{1}=-\varphi^{13},\quad d\varphi^{2}=\varphi^{23},\quad
d\varphi^{3}=0$. This case corresponds to the holomorphically-parallelizable
Nakamura manifold [Nak75]. This and the next two following cases are complex
structures of splitting type in the sense of [Kas13, Assumption 1], see
[AOUV17].

(Siv2):

$d\varphi^{1}=2\sqrt{-1}\varphi^{13}+\varphi^{3\bar{3}}$,
$d\varphi^{2}=-2\sqrt{-1}\varphi^{23}+x\,\varphi^{3\bar{3}}$,
$d\varphi^{3}=0$, where $x\in\\{0,1\\}$.

(Siv3):

$d\varphi^{1}=A\,\varphi^{13}-\varphi^{1\bar{3}}$,
$d\varphi^{2}=-A\,\varphi^{23}+\varphi^{2\bar{3}}$, $d\varphi^{3}=0$, where
$A\in\mathbb{C}$ with $|A|\neq 1$.

(Sv):

$d\varphi^{1}=-\varphi^{3\bar{3}}$,
$d\varphi^{2}=\frac{\sqrt{-1}}{2}\varphi^{12}+\frac{1}{2}\varphi^{1\bar{3}}-\frac{\sqrt{-1}}{2}\varphi^{2\bar{1}}$,
$d\varphi^{3}=\frac{\sqrt{-1}}{2}(-\varphi^{13}+\varphi^{3\bar{1}})$.

We refer the reader to [Ota14, FOU15] and [AOUV, Table 2] for more details.

Thanks to Lemma 4.3, we can focus our attention to left-invariant hermitian
structures, which are of the form (4.2). We compute
$\sqrt{-1}\partial\overline{\partial}\omega$ case by case.

Si:

$\sqrt{-1}\partial\overline{\partial}\omega=2(-1)\big{(}(\Re
A)^{2}r^{2}\varphi^{1\bar{1}}+\sqrt{-1}(\Im
A)^{2}u\varphi^{1\bar{2}}-\sqrt{-1}(\Im
A)^{2}\overline{u}\varphi^{2\bar{1}}+(\Re
A)^{2}s^{2}\varphi^{2\bar{2}}\big{)}\wedge\varphi^{3\bar{3}}$. When
$(\cos\theta)^{4}\geq(\sin\theta)^{4}$, any metric is pluripositive;
otherwise, there are pluripositive metrics (taking
$|u|^{2}\leq\frac{\cos\theta}{\sin\theta}r^{2}s^{2}$). When $\cos\theta=0$,
there exist pluriclosed metrics (for $u=0$) and Kähler metrics (for $u=v=z=0$,
see [Ota14, Theorem 5.1.3]). Plurinegative metrics never exists, while
balanced metrics always exist (when $v=z=0$).

Sii:

$\sqrt{-1}\partial\overline{\partial}\omega=\left(\frac{4\,t^{2}x^{2}+t^{2}}{16\,x^{2}}\right)(-1)\varphi^{1\bar{1}}\wedge\varphi^{2\bar{2}}+\left(s^{2}x^{2}+\frac{1}{4}\,s^{2}\right)(-1)\varphi^{1\bar{1}}\wedge\varphi^{3\bar{3}}$.
Any metric is pluripositive. There is neither pluriclosed metrics, nor
plurinegative metrics, nor Kähler metrics. Balanced metrics always exist (take
$u=v=z=0$).

Siii1:

$\sqrt{-1}\partial\overline{\partial}\omega=2\big{(}\sqrt{-1}\,u\sqrt{-1}\varphi^{1\bar{2}}-\sqrt{-1}\,\overline{u}\sqrt{-1}\varphi^{2\bar{1}}\big{)}\wedge\sqrt{-1}\varphi^{3\bar{3}}$.
There exist pluriclosed metrics (take $u=0$), but neither Kähler metrics nor
balanced metrics.

Siii2:

$\sqrt{-1}\partial\overline{\partial}\omega=t^{2}(-1)\varphi^{1\bar{1}}\wedge\varphi^{2\bar{2}}+2\,r^{2}(-1)\varphi^{1\bar{1}}\wedge\varphi^{3\bar{3}}+2\,s^{2}(-1)\varphi^{2\bar{2}}\wedge\varphi^{3\bar{3}}$.
Every metric is pluripositive. There is neither pluriclosed, nor
plurinegative, nor Kähler metrics. Balanced metrics exist (take
$u\in\mathbb{R}$ and $v=z=0$).

Siii3:

$\sqrt{-1}\partial\overline{\partial}\omega=-t^{2}(-1)\varphi^{1\bar{1}}\wedge\varphi^{2\bar{2}}+2(-1)\sqrt{-1}\big{(}u\varphi^{1\bar{2}}-\overline{u}\varphi^{2\bar{1}}\big{)}\wedge\varphi^{3\bar{3}}$.
There are neither pluripositive, nor pluriclosed metrics. There exists
plurinegative metrics (take $u=0$), but neither Kähler nor balanced metrics.

Siii4:

$\sqrt{-1}\partial\overline{\partial}\omega=t^{2}(-1)\varphi^{1\bar{1}}\wedge\varphi^{2\bar{2}}+2(-1)\big{(}\sqrt{-1}\,u\varphi^{1\bar{2}}-\sqrt{-1}\,\overline{u}\varphi^{2\bar{1}}\big{)}\wedge\varphi^{3\bar{3}}$.
Pluriclosed metrics and plurinegative metrics never exist. There are
pluripositive metrics (take $u=0$), but no Kähler metrics. There are balanced
metrics (characterized by parameters $r^{2}=s^{2}$, $v=z=0$).

Siv1:

$\sqrt{-1}\partial\overline{\partial}\omega=\frac{(-1)}{2}\big{(}r^{2}\varphi^{1\bar{1}}+\sqrt{-1}\,u\varphi^{1\bar{2}}-\sqrt{-1}\,\overline{u}\varphi^{2\bar{1}}+s^{2}\varphi^{2\bar{2}}\big{)}\wedge\varphi^{3\bar{3}}$.
Every metric is pluripositive, there are neither pluriclosed nor plurinegative
metrics. There is no Kähler metric, and every metric is balanced.

Siv2:

$\sqrt{-1}\partial\overline{\partial}\omega=2(-1)\big{(}r^{2}\varphi^{1\bar{1}}+\sqrt{-1}\,u\varphi^{1\bar{2}}-\sqrt{-1}\,\overline{u}\varphi^{2\bar{1}}+s^{2}\varphi^{2\bar{2}}\big{)}\wedge\varphi^{3\bar{3}}$.
Every metric is pluripositive, there are neither pluriclosed nor plurinegative
metrics. There are no Kähler metrics and no balanced metrics.

Siv3:

$\sqrt{-1}\partial\overline{\partial}\omega=\frac{(-1)}{2}\big{(}|A-1|^{2}r^{2}\varphi^{1\bar{1}}+\sqrt{-1}\,|A+1|^{2}u\varphi^{1\bar{2}}-\sqrt{-1}\,|A+1|^{2}\overline{u}\varphi^{2\bar{1}}+|A-1|^{2}s^{2}\varphi^{2\bar{2}}\big{)}\wedge\varphi^{3\bar{3}}$.
There are pluripositive metrics: when $\Re A\leq 0$, any metric is
pluripositive; when $\Re A>0$, pluripositive metrics are characterized by
$|u|^{2}\leq\left|\frac{A-1}{A+1}\right|^{4}r^{2}s^{2}$. There are neither
pluriclosed nor plurinegative, nor Kähler metrics. Balanced metrics always
exist (take $v=z=0$).

Sv:

$\sqrt{-1}\partial\overline{\partial}\omega=\frac{(-1)}{2}\varphi^{1\bar{1}}\wedge\big{(}\sqrt{-1}\,v\varphi^{2\bar{3}}-\sqrt{-1}\,\overline{v}\varphi^{3\bar{2}}+\frac{1}{4}\,s^{2}\varphi^{3\bar{3}}\big{)}$.
There are pluripositive metrics (take $v=0$). There are neither pluriclosed,
nor plurinegative, nor Kähler, nor balanced metrics.

Notice that it is no longer true that the plurisigned property of left-
invariant metrics is completely determined by the complex structure.

We summarize the previous dicussion in the following:

######  Theorem 4.7.

Let $X$ be a non-Kähler six-dimensional solvmanifold endowed with a left-
invariant complex structure with holomorphically-trivial canonical bundle.
Then $X$ admits a plurisigned hermitian metric. More precisely:

  * •

either $X$ belongs to the class (Siii1) and there are pluriclosed metrics;

  * •

or $X$ belongs to (Siii3) and there are plurinegative non pluriclosed metrics;

  * •

or else $X$ belongs to any of the remaining classes, and it admits a left-
invariant pluripositive hermitian metric that is not pluriclosed.

Recall that these three conditions are mutually exclusive by Corollary 3.7.
The only Kähler example in the above list correspond to a special Lie algebra
${\mathcal{G}}={\mathcal{A}}_{5,17}^{0,0,1}\oplus\mathbb{R}$ of splitting type
(see [FOU15, Theorem 2.18]).

######  Remark 4.8.

If a hermitian metric is both balanced and pluriclosed, then it is Kähler
[AI01, Remark 1]. It is conjectured [FV15, Problem 3] that a compact complex
manifold admitting both pluriclosed metrics and balanced metrics also admits
Kähler metrics. The conjecture is confirmed for the above nilmanifolds and
solvmanifolds by [FV15, Theorem 6.3 and Theorem 6.4]. We notice that these
facts fail when replacing pluriclosed with plurisigned: the Iwasawa manifold
does not admit any Kähler metric [FG86, BG88, Has89], but every left-invariant
hermitian metric on it is both balanced and pluripositive.

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